Abstract

Recently, significant effort has been devoted to the study of atom–photon quantum interfaces using intracavity Rydberg-blocked atomic ensembles, which may serve as the platform for many essential quantum information processing tasks. In this paper, we use a theoretical analysis of this platform where the ground-Rydberg transition is realized by a two-photon transition, and we report our recent findings regarding the Jaynes–Cummings model on optical domain and robust atom–photon quantum gates. Our implementation with typical alkali atoms, such as Rb or Cs, requires an optical cavity of moderately high finesse and the condition that the cold atomic ensemble is well within the Rydberg-blockade radius. The analysis focuses on the atomic ensemble’s collective coupling to the quantized optical field in the cavity mode, and we demonstrate its capability to serve as a controlled-PHASE gate between photonic qubit and matter qubit, where the photonic qubit is endowed with a reasonably wide frequency tuning range. The detrimental effects associated with several major decoherence factors in this system are also considered in the analysis.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Efficient quantum interfaces between atoms and light have been one of the central topics in the research frontier of quantum optics over the last three decades and are not only important for theoretical investigations of quantum electrodynamics (QED), but also are crucial for applications in quantum information processing [1] and quantum metrology [2]. In particular, two major types of experimental platforms have attracted intense attention and have turned out to be spectacularly fruitful; these are the cavity quantum electrodynamics (cQED) approach and the atomic ensemble approach. The cQED approach is primarily within the framework of one or a few isolated neutral atoms coupled to a high finesse optical cavity [3], where the second quantization of the intracavity optical field is essential. Meanwhile, the atomic ensemble approach [4] is a powerful alternative to the cQED approach, where many atoms are coerced to form a collective superposition state to enhance the atom–photon coupling. Much exciting progress has been achieved so far, including the atom–photon controlled-PHASE gate [5,6] and the photon–photon quantum gate [7], the quantum nondemolition measurement of single photon pulses [8], quantum memory for light [9,10], the quantum repeater for photonic polarization entanglement [11], and quantum networking between matter qubits [1214], to name a few. Among many potential technological breakthroughs, the phenomenon of Rydberg blockade [15] is widely perceived to be promising in enhancing the strength of a single photon’s coupling to atoms.

The rapid progress in research of Rydberg–Rydberg interactions over the past two decades [16] has already resulted in many key advances in quantum optics with neutral atoms and single-photon pulses, both theoretically and experimentally [1726]. One typical feature is that the collective coupling of many atoms to the optical field is amplified due to the Rydberg-blockade effect. Other prominent characteristics include a strong interaction strength over a long range and coherent on–off switching, which makes it advantageous to utilize the Rydberg-blockade effect with atomic ensembles [21,27,28].

Especially, the Rydberg-blockade effect is known to be a powerful tool in the study of the Jaynes–Cummings model (JCM) [29], where the field part of the JCM is instantiated either by cavity optical modes [30] or via isomorphism to collective atomic excitations [3135]. The system of an intracavity Rydberg-blocked atomic ensemble provides an interesting opportunity to combine both the cQED and atomic ensemble techniques for the atom–light quantum interface [30]. Optical nonlinearities have already been experimentally observed in a cold atomic ensemble with strong Rydberg–Rydberg interactions inside a moderate finesse cavity [20], which has been proven to be a natural consequence of Rydberg-blockade shift [16,36]. These efforts have demonstrated the unique capability of the intracavity Rydberg-blocked atomic ensemble and paved the way for further development. Moreover, thanks to the technical progress of manipulating cavity Rydberg polaritons [37], recent experimental results have confirmed JCM’s specific signatures in such systems [20,38,39].

Nevertheless, these theoretical investigations and experimental demonstrations, while being physically isomorphic to the JCM, have mainly focused, so far, on the extraction of quantum optical characteristics from the system’s response to classical light input [30,34]. Relatively less attention has been devoted to the discussion of the explicit and direct operations of such a system with a realistically quantized optical field, where the JCM’s input–output coupling is associated with single-photon or few-photon pulses. On the other hand, there exists growing interest in constructing novel quantum optical devices with single-photon level manipulation capabilities via Rydberg–Rydberg interactions [18,40,41], which strongly demands further explorations of the JCM with a Rydberg-blocked ensemble under the condition of a truly quantized optical field. Meanwhile, recent scientific advances [42] have confirmed the potential technical ability to precisely manipulate the genuine quantized intracavity optical state. All these considerations naturally lead to a push for further study of an intracavity Rydberg-blocked atomic ensemble, which would straightforwardly reveal the quantum nature of the JCM on the optical domain. The dynamics of such a hybrid system involve collective excitations of an atomic ensemble, which is inherently related to Dicke superradiance [43,44], an intriguing effect in quantum optics. Moreover, the effort along this line is capable of exploring the physical process of cQED within a parameter space not easily accessible via coupling a single atom to a high-finesse optical cavity.

Designing a robust atom–photon controlled-PHASE gate is a primary mission in the research of an atom-light quantum interface. It is widely known that the intracavity JCM on optical domain is very closely tied to such a quantum gate [5], and, therefore, the study of the JCM with an intracavity Rydberg-blocked atomic ensemble is naturally expected to be relevant as well. Indeed, recent investigations have revealed that novel controlled-PHASE gates for atom–photon, atom–atom, and photon–photon systems can be built via the help of Rydberg-mediated interactions and intracavity Rydberg electromagnetically induced transparency (EIT) [4547]. In particular, such ideas of constructing a controlled-Z (C-Z) gate have ingeniously improved the effective atom–photon coupling strength by utilizing Rydberg blockade among many atoms [4851]. These new proposals are compatible with the current mainstream experimental techniques of trapping a cold atom ensemble inside an optical cavity, and they greatly reduce the requirement on the cavity finesse as the core advantage.

In this paper, we report our latest findings for the system of an intracavity Rydberg-blocked atomic ensemble and its relationships with the JCM and the atom–photon quantum gate. We analyze in theory the two-photon interactions between the Rydberg-blocked atomic ensemble and quantized intracavity optical fields with collective coupling, where the two-photon transition is composed of the intracavity photon and one external control laser. In particular, we study the JCM with the preloaded intracavity single-photon and few-photon states as the initial condition. Thereafter, an atom–photon controlled-PHASE quantum gate is proposed, deriving from the dressed states’ property and Rydberg-blockade effect on top of the JCM dynamics embedded in this system. One prominent feature of this gate design is that it allows for a considerable amount of flexibility in the frequency of the incident single-photon pulse.

The rest of the article is organized according to the following structure. In Section 2, we offer an overall sketch of the involved physical process. In Section 3, the time evolution of the JCM is discussed, where the coupling between the intracavity field and the outside field is taken into consideration. In Section 4, the atom–photon quantum gate is discussed. Section 5 concludes the paper.

2. OVERVIEW AND FUNDAMENTALS

In this section, we sketch the basic physical system under investigation and present the rudimentary ingredients of the theoretical analysis that will follow. To begin with, the general setting of the intracavity Rydberg-blocked atomic ensemble is outlined in Fig. 1. The atomic ensemble is supposed to contain a few hundred up to a few thousand atoms, preferably in an array configuration [52,53] via far off-resonance trapping (FORT) in the optical lattice, such as the well-established experimental platforms of 2D [24] or 3D arrays [54]. It is necessary to have a geometric configuration that ensures that the entire ensemble fits within the Rydberg-blockade radius for the target Rydberg states, which can be achieved by having a trapping site spacing on the order of half a micrometer [52] for the atomic array configuration. Nevertheless, as long as the atoms’ temperature is cold enough, it is not necessary to prepare the atomic ensemble in the array configuration, since single-atom addressing is not required for the purpose here [30,49].

 

Fig. 1. Schematic of the intracavity Rydberg-blocked atomic ensemble system under investigation. The entire ensemble has a size of that is on the order of a few tens of micrometers, which is compatible with the experimentally attainable Rydberg-blockade radius. Moreover, in such a configuration, the atomic ensemble well matches the cavity mode spatially, which conveniently serves the purpose of atom–photon coupling. The control laser is of frequency ωc, while the cavity resonance frequency is ωd.

Download Full Size | PPT Slide | PDF

For this section, the cavity mirrors are assumed to have 100% perfect reflectivity and, therefore, the intracavity optical field won’t leak to the outside free space. The interaction Hamiltonian to describe such a hybrid system, including the effect of the Rydberg blockade on the form of the van der Waals interaction potential [16,49,55], is given below:

Hint=n=1N(Ω2|rnen|iGn|engn|b^)+H.c.Δen=1N|enen|Δrn=1N|rnrn|+n=1Nm>nVnm|rnrn||rmrm|,
where the one-photon detuning is Δe=ωc(ωeωg), and the two-photon detuning is Δr=ωc+ωd(ωrωg).

In Eq. (1), the Rydberg–Rydberg interaction is effectively described by the potential, Vnm, where the details of dipole–dipole interaction are hidden for the sake of simplicity. We discuss the dynamics of the system under the condition of a strong Rydberg blockade, such that a doubly occupied Rydberg state, like |rnrm, is never populated. Within the scope of our study, where the ground state is ultimately linked to the Rydberg state with adequate strength around a resonance point, the implicit assumption is that, at most, only a single excitation into the Rydberg state is practically allowed, which is a widely applicable approximation if |Vnm| is sufficiently large. Therefore, the Hamiltonian can be reduced to the form of

Hint=(Ω2|r˜e˜|iG|e˜gN|b^)+H.c.Δe|e˜e˜|Δr|r˜r˜|,
with the state of single excitation defined as
|e˜=G1Gn|en,|r˜=G1Gn|rn,
where the normalization factor is the collective atom–photon coupling strength G=(|Gn|2)1/2, and |gN stands for all N atoms in the ground state, |g.

For the situation of |Δe| much larger than G and |Ω| while the optical field contains at most a handful of photons, and upon consideration of the operator side, we may carry out the adiabatic elimination under the approximation that Ω2|r˜e˜|+iG|gNe˜|Δe|e˜e˜|0. This leads to the simple linear relation, |e˜(Δe)1(Ω2|r˜+iG|gN). The details of the adiabatic elimination procedure are a well-established subject in the literature (also see Supplement 1). Consequently, the isomorphism to the JCM is then readily obtained via combining this relation with Eq. (2), which leads to the effective two-level system Hamiltonian,

Heff=ΩG2Δe|r˜gN|+H.c.+G2Δe|gNgN|b^b^(ΔrΩ24Δe)|r˜r˜|,
which is identical to the Hamiltonian for the JCM of a single-atom coupling to optical fields with coupling strength |GΩ*2Δe|, apart from the ac Stark shift terms of G2Δe and |Ω|24Δe.

This elementary model analysis sketches the theme of the JCM for such a hybrid system, where the single emitter is replaced by many emitters interacting collectively, thanks to the Rydberg blockade. The dynamics are certainly more complicated when the cavity is coupled to the outside optical fields, especially if the focus is put on the quantum nature of the system’s response. And, that is the direction in which we are heading in the next two sections, including a revisit of the JCM with an initial condition of particularly prescribed intracavity quantum optical fields and an efficient atom–photon quantum gate.

3. JCM ON OPTICAL DOMAIN

A. Dynamics with a Single-Photon State

Here, we begin to treat the JCM dynamics with input–output coupling to the free-space modes for the intracavity Rydberg-blocked atomic ensemble system sketched in Section 2. An example of the implementation is illustrated in Fig. 2. The situation under study is that of a single-photon state for the intracavity field prepared as the initial condition; and then, the JCM dynamics mandated by Eqs. (1) and (4) are subsequently invoked.

 

Fig. 2. Schematic of the system for the study of the JCM with an intracavity Rydberg-blocked atomic ensemble, where the state initialization includes the process of feeding a nonclassical optical pulse into the cavity. The optical cavity is supposed to be single-sided, where the one end mirror is perfectly reflecting. In this particular example, the feeding and retrieving of the intracavity optical field is realized by polarization optics for the free-space optical pulse (PBS: polarizing beam splitter; QWP: quarter-wave plate). Typical parameter settings for the experimental implementation are relatively mild for the hardware nowadays. For example, the cavity finesse can be set as F5×103, the cavity decay rate can be chosen as κ2π×0.5  MHz, the cavity free spectral range can be set as FSR5  GHz, and the number of atoms in the ensemble can be chosen as 1001000. The instantiation of the Jaynes–Cummings model in such a system is relatively straightforward compared with the case of a high-finesse cavity, thanks to recent developments in single-photon pulse engineering and Rydberg atom control techniques.

Download Full Size | PPT Slide | PDF

The initial condition can be realized by efficiently and deterministically loading a prescribed single-photon optical pulse into a cavity [42]. This loading process is supposed to be carried out with the control laser shut off such that the intracavity optical field is off-resonant with the atomic transition |g|e. That is to say, the cavity seems to be empty for the incoming optical field when ΔeG. As soon as the deterministic single-photon loading stage completes, the control laser flashes onto the ensemble to enable the JCM process.

Now, the state vector for the entire system needs to include the free-space optical field component, and it reads as

|Ψ(t)=dωϕs(ω,t)a^s(ω)|gN,Øb,Øa+Cb(t)b^|gN,Øb,Øa+m=1NCem(t)|gN1em,Øb,Øa+m=1NCrm(t)|gN1rm,Øb,Øa,a,
where |Øa denotes the vacuum of free-space optical modes outside the cavity, |Øb denotes the vacuum of intracavity optical modes, and subscript s stands for the components of free space.

Under such circumstances, the excitation of the atomic ensemble caused by the intracavity single-photon optical state is, in essence, a collective excitation, which can be readily observed from Eq. (5). Therefore, in order to examine the process in a little more detailed manner, we’d like to define the coefficients with respect to collective state basis:

Ce=G1Gn*Cen,Cr=G1Gn*Crn,
where the subscript n denotes the numbering of atoms.

The coupling through the cavity mirror between the intracavity mode and the free-space mode can be described by the interaction Hamiltonian:

Hio=idωgs(ω)(a^s(ω)b^b^a^s(ω)).
According to the quantum input–output formalism (see Supplement 1 for more details), the equations describing the dynamics can be obtained from Eqs. (1) and (7). Under the collective state basis specified in Eq. (6), the equations of motion are formulated as
ddtCb(t)=GCeκ2Cb(t),
ddtCe(t)=GCb(t)+iΩ*2Cr(t)+iΔeCe(t)Γe2Ce(t),
ddtCr(t)=iΩ2Ce(t)+iΔrCr(t)Γr2Cr(t),
where κ is the effective cavity optical-field decay rate, Γe is the spontaneous emission rate of state |e, and Γr is the spontaneous emission rate of state |r.

A numerical example is presented in Fig. 3, according to the dynamics governed by Eq. (8). The signature of quantum Rabi oscillation is clear in Fig. 3(a), as it belongs to the strong-coupling regime, according to the cQED terminology. The decay of the population within the system is dominated by the photon leaking out of the cavity since the Rydberg state a has relatively long lifetime. In the effective two-level atom picture, this amounts to the case where the coupling to the excited state is strong while the spontaneous emission rate of the excited state is relatively much smaller. This type of parameter setting and the initial state preparation are not easily accessible in the typical platform of a single atom coupled to high-finesse cavity [56].

 

Fig. 3. Numerical simulation of the quantum Rabi oscillations of the intracavity atom–photon interaction, single-photon–state case. Parameters are set as Δe=2π×200  MHz, Δr=0, Γe=2π×1  MHz, Γr=2π×0.01  MHz, κ=2π×0.5  MHz. For (a), Ω=2π×20  MHz, G=2π×10  MHz; for (b), Ω=2π×5  MHz, G=2π×2.5  MHz.

Download Full Size | PPT Slide | PDF

Under the condition of a relatively large Δe, the adiabatic elimination is again applicable, which can lead to a straightforward effective two-level atom description. Analogous to Eq. (4), for the intermediate state, we have Ce(t)=(iΔeΓe2)1(GCb(t)iΩ*2Cr(t)), which leads to the following equations from Eq. (8):

ddtCb(t)=iG2Δe+iΓe2Cb(t)GΩ*2(Δe+iΓe2)Cr(t)κ2Cb(t),ddtCr(t)=GΩ2(Δe+iΓe2)Cb(t)i|Ω|24(Δe+iΓe2)Cr(t)+iΔrCr(t).
Apart from the ac Stark shift terms, this can be identified as the dynamics for a typical cQED system of a single atom coupled to an optical cavity. Moreover, we may effectively recognize the collective cooperativity as C=|GΩ2Δe|212κΓr. Since Rydberg states usually have a relatively long lifetime, the situation generally satisfies |GΩ2Δe|Γr. Therefore, when |GΩ2Δe|κ and C1, the dynamics belong to the strong-coupling regime; on the other hand, when κ|GΩ2Δe| and C1, the dynamics belong to the Purcell regime. Compared to the case of a single atom, the enhancement factor in the effective atom–photon coupling strength for N atoms is N for uniform coupling. With other settings kept the same way, the cavity finesse can be reduced by a factor of N in order to reach the same level of cooperativity strength.

B. Dynamics with an Optical Coherent State

In Subsection 3.A, the discussion is devoted to the case where the initial condition is prepared as a deterministic single-photon state for the intracavity optical field. On the other hand, the situation of an optical coherent state with a small mean photon number is also frequently encountered in the study of quantum optics. It is worthwhile to investigate such a situation, with merits from the experimental side as well as the theoretical side. Practically, such an initial state is commonly prepared via a weak coherent optical pulse incident upon a cavity in a configuration like Fig. 2.

Throughout the rest of this section, the assumption of a strong Rydberg blockade is kept so that, at most, a single excitation is allowed into the Rydberg state. For the contents of Subsection 3.A, this assumption stays pragmatically redundant since the optical field does not contain a multiphoton component and, therefore, naturally the Rydberg excitation number can not exceed 1.

We use the quantum input–output theory together with the quantum jump approach [57,58] to treat the time evolution of this system, which is also known as the approach of the Monte–Carlo wave function (MCWF). To begin with, we extend the definition of the system’s wave-function coefficients, (Ce,Cr), specified in Eq. (6) to include the multiphoton case so that we can set Cb,n, Ce,n, Cr,n to denote the coefficients with respect to the Fock-state basis, |n, on the photonic side. Then, the time evolution without considering any decay is given by the equations below for n1:

ddtCb,n(t)=nGCe,n1(t),
ddtCe,n1(t)=nGCb,n(t)+iΩ*2Cr,n1(t)+iΔeCe,n1(t),
ddtCr,n1(t)=iΩ2Ce,n1(t)+iΔrCr,n1(t),
where Cb,n, Ce,n1, and Cr,n1 form a closed loop. In the numerical instantiation, we manually set a cutoff in n depending on |α| from the initial condition’s optical coherent state, |α. In particular, on top of the dynamics governed by Eqs. (4) and (10), the MCWF method also takes the decays into consideration in order to numerically resolve the time evolution, including the atomic spontaneous emissions, (Γr,Γe), and the decay of intracavity optical field, κ (see Supplement 1 for more details).

If the initial intracavity optical field is prepared as a coherent state with a mean photon number that is not too large, the phenomenon of collapse and revival of the quantum Rabi oscillation is anticipated when the cavity decay is very small. An example of the numerical simulation for such case is shown in Fig. 4, where the photon loss is negligible over the time duration towards revival. However, from a practical point of view, such a small κ is hard to achieve experimentally. For a reasonably large value of κ, the dynamics are different. Typically, after some short period of time, the intracavity photon number decreases by a lot due to the loss from cavity emission. This changes the amount of ac Stark shift induced by the intracavity optical field and ultimately causes the two-photon transition to the Rydberg state to be off-resonant. See the corresponding numerical simulation result in Fig. 5 for an example. The common feature of the relatively fast oscillations in both Figs. 4 and 5 can be recognized as the semiclassical Rabi oscillations of the effective Rabi frequency, ΩG2Δe|α|. Roughly speaking, the re-emission from the ensemble’s collective excitation into the cavity mode may be recognized as a form of superradiance [44]. In other words, the Rabi oscillation can alternatively be viewed as the absorption and re-emission of the photonic state in the cavity mode by the many intracavity emitters.

 

Fig. 4. Numerical simulation of the quantum Rabi oscillations of the intracavity atom–photon interaction, coherent-state version. The artificial condition of almost no cavity decay is imposed to show the quantum revival, while the atomic decays are retained in the calculations. Parameters are set as G=2π×2.5  MHz, Ω=2π×20  MHz, Δe=2π×100  MHz, Δr=0, Γe=2π×1  MHz, Γr=2π×0.01  MHz, κ=2π×104  MHz. The initial condition is |α|=4 for the optical coherent state, |α, while the cutoff is set at 50. It is averaged over 10,000 MCWF traces.

Download Full Size | PPT Slide | PDF

 

Fig. 5. Numerical simulation of the quantum Rabi oscillations of the intracavity atom–photon interaction, coherent-state version. A moderate cavity decay into the free-space environment is considered here. Parameters are set as G=2π×2.5  MHz, Ω=2π×20  MHz, Δe=2π×100  MHz, Δr=0, Γe=2π×1  MHz, Γr=2π×0.01  MHz, κ=2π×0.1  MHz. The initial condition is |α|=4 for the optical coherent state, |α, while the cutoff is set at 50. It is averaged over 10,000 MCWF traces.

Download Full Size | PPT Slide | PDF

The discussions so far have hinted that such a system allows not only the study of the JCM in an interesting parameter sector but also potential applications in spin squeezing and superradiant lasing. The role of the Rydberg blockade is essential since it guarantees that the entire atomic ensemble behaves a lot more like a single emitter. This ensures that the quantum Rabi oscillation is taking place even when it is driven by an optical coherent state, which can be regarded as the classical field. On the contrary, a classical optical pulse driving a medium of uncorrelated emitters is hardly capable of yielding this demonstrated behavior. With the contents discussed in this section, we observe that the stimulated Raman approach with a two-photon transition has unique characteristics, when compared with the approaches of intracavity Rydberg EIT [4850] or a single atom coupled to a high-finesse cavity [5,6].

4. ATOM–PHOTON QUANTUM GATE

A. Controlled-PHASE Gate via Rydberg Blockade

The isomorphism to the JCM discussed in Section 2 already hints that the system of the intracavity Rydberg-blocked atomic ensemble may have potential applications for atom–photon entanglement in quantum optics, where the cavity resonance frequency is detuned by a significant amount from the atomic resonance frequency. Inspired by the strong-coupling regime of the atom–photon interaction discussed in Section 3, the analogy with the typical cQED scenario leads to the straightforward observation that a controlled-PHASE gate can be constructed accordingly on the same platform.

The purpose is to establish a controlled-Z quantum gate between a photonic qubit and a matter qubit; the basic idea of the mechanism is illustrated in Fig. 6. The instantiation of the matter qubit may be in the form of a distinguished single atom in the ensemble [24,26,28] or a spin wave embedded in the entire ensemble [10,21,27,59]. In both cases, the qubit register states can be chosen as the hyperfine states of the ground level for typical alkali atoms. The gate protocol can be chosen to include the process of exciting the matter qubit from the ground state to the Rydberg state, just like the very original Rydberg-blockade gate proposal [15,16]. As shown in Fig. 6, the single-photon pulse incidence is ultimately reflected from the cavity, where it gains a phase shift depending on the matter qubit’s state during the process. The Rabi frequency and detuning of the control laser are supposed to ensure that the two-photon transition, together with the intracavity optical field, is on resonance for the atoms, whose details will be discussed quantitatively later. For simplicity, let’s assume the two register states of the matter qubit are |g and |r. When the matter qubit is sitting at the state |g, it does not exert any substantial impact on the other atoms, and the two-photon transition from |g to |r exactly holds. Therefore, in this case, the system’s resonance frequency is effectively shifted and henceforth the incident single-photon pulse cannot enter the cavity [5]. When the matter qubit is sitting at the state |r, it influences the rest of atoms via Rydberg blockade such that the two-photon transition for the cavity field and the control laser is out of resonance. Therefore, the incident single-photon pulse enters the cavity freely before it gets reflected eventually. Briefly speaking, upon reflection, the single-photon pulse gains a conditional phase shift, 0 or π, depending on whether the matter qubit state is |g or |r.

 

Fig. 6. Outline for the basic principles of the atom–photon gate with an intracavity Rydberg-blocked atomic ensemble, where the qubit state on the atom side is abstracted into the internal states of a single atom within the ensemble. The optical cavity is supposed to be single-sided, where the one end mirror is perfectly reflecting. The frequency of the incident single-photon pulse is resonant with the optical cavity, but not necessarily so with the atomic transition |g|e. Here, the matter qubit is instantiated in the form of a single atom within the atomic ensemble. In such a configuration, a strong Rydberg blockade is presumed to take place between state |r of the qubit atom and state |r for the rest atoms of the ensemble, as a consequence of the Förster resonance structure in Eq. (11). Nevertheless, the single qubit atom does not have to be the same species as the other atoms in the ensemble [26,50]. In principle, this gate protocol does not induce mechanical forces between atoms since the underlying mechanism belongs to the category of a Rydberg-blockade gate [28].

Download Full Size | PPT Slide | PDF

Here, we begin a quantitative analysis for this gate protocol. The interaction Hamiltonian for the atom–photon interaction of this intracavity atomic ensemble system, including the Rydberg-blockade effects, is

Hp1=n=1N(Ω2|rnen|iGn|engn|b^)+H.cΔen=1N|enen|Δrn=1N|rnrn|+n=1NVn|rnpn||rp|+H.c.+δpn=1N|pnpn||pp|,
where |r, |p are the Rydberg states of the qubit atom; |r, |p are the Rydberg states of the other atoms; and, those two pairs of states form the Förster resonance |rr|pp. The term δp is the small Förster energy penalty term from the Rydberg product states, |pp.

From a realistic point of view, the values of Δe and Gn are fixed to begin with, as the cavity hardware is already chosen and the incoming single-photon pulse’s frequency needs to be resonant with the cavity. On the other hand, flexibility is granted for the control laser in terms of Ω and Δr. For example, under the specific condition of the typical Autler–Townes effect, it is fair to set Δe=Δr=Δ.

The state vector for the complete system is similar to the form described by Eq. (5). Note that the difference with respect to Eq. (5) is that now the state may definitely contain one intracavity Rydberg excitation in the qubit atom:

|Ψ(t)=dωφs(ω,t)a^s(ω)|gN,r,Øb,Øa+Cb(t)b^|gN,r,Øb,Øa+m=1NCem(t)|gN1,em,r,Øb,Øa+m=1NCrm(t)|gN1rm,r,Øb,Øa+m=1NCpm(t)|gN1pm,p,Øb,Øa.
Based upon Eqs. (11) and (12), we proceed to compute the conditional phase shifts according to the quantum input–output theory, together with the quantum jump approach. The control laser parameters are assumed to be time-independent, and then the formal calculation can be carried out on the Fourier domain. From this point on, δ is used to label the frequency components of the incident single-photon pulse, where δ=0 denotes a component that’s completely resonant with the cavity.

We make a few definitions here: the ac Stark shift induced by the atom–photon interaction on the ground level, Δac; the dressing factor to the atom–photon coupling strength, η; the effective relative frequency shift for coupling to the dressed states, Δdr; and the effective Rydberg-blockade shift, Bm, where the subscript m is the index for the atom

Δac=G2Δe+δ+iΓe2,η=14|Ω|2(iΔeΓe2+iδ)2,Δdr=Δri4|Ω|2Γe2iΔeiδ,Bm=|Vm|2Γp2+iδpiδ.
With the above preparations, the conditional phase shifts can be readily expressed (see Supplement 1 for more details). When the qubit atom is sitting at the state |r in the cavity, the reflection coefficient is
R(δ)=1κ{κ2iδiΔacηm=1N|Gm|2Γr2iδ+iΔdr+Bm}1.
On the other hand, when there is no stored excitation in the qubit atom, the reflection coefficient reads
R(δ)=1κ{κ2iδiΔacηG2Γr2iδ+iΔdr}1.
In the limit that G2/(δκ), (Bκ)/G2, (Δeκ)/G2, where B conceptually denotes the effective Rydberg-blockade shift, the conditional phase shifts obey R1, R1, and the performance of this controlled-Z gate is ideal. Recent advances in utilizing the dark state to enhance the fidelity of the atom–atom controlled-PHASE gate [60] are also expected to offer future improvements in the atom–photon gate protocol under discussion here (see also a recent advance in utilizing dark-state technology to improve an atom–photon gate in Ref. [51]).

The above analysis, together with Eqs. (14) and (15), is also essentially applicable to the case where the matter qubit is realized via a spin wave embedded in the entire atomic ensemble. The encoding (write) and retrieval (read) of a single excitation in the form of a spin wave within the atom ensemble inside the optical cavity are robust and efficient; recent experimental progress has confirmed that a single excitation of a spin wave in an intracavity atomic ensemble can be efficiently prepared and retrieved [59,61], allowing a single-photon pulse to be transferred out from the atom–cavity system with near unity efficiency. As long as state |r differs from state |r, which is common in the state choices when implementing a Rydberg blockade, the operational laser frequency for controlling the spin-wave states and the laser frequency for controlling the two-photon transition of the gate protocol will usually be more than a few hundred gigahertz apart, and therefore the cross talk can be made minimal. Meanwhile, this gate protocol can work as an atom–photon quantum gate for single-photon pulses endowed with polarization encodings, as well as time-frequency encodings, such as frequency-bin encoding [62].

The physics interpretation of this gate mechanism is naturally related to the original proposal of atom–photon controlled-Z gate [3,5]. Roughly speaking, if the Rydberg-blockade strength is strong enough and the Rydberg-state lifetime is long enough, the scaling relation between the fidelity and the collective cooperativity of the system behave in the same way as those of a single two-level atom coupled to a high-finesse cavity. In the meantime, it also contains more practical flexibility compared with the previously known schemes; in particular, there exists a relatively wide tuning range for the cavity resonance frequency, namely the one-photon detuning, Δe. And therefore, this gate protocol can even be used as a photon–photon gate to entangle single-photon pulses with different frequencies up to a few hundred megahertz. The choice of sizable Ω, Δe is also a distinguishable feature as we focus on the two-photon transition and Autler–Townes effect, when compared with the recent advances reported in Ref. [51], which focuses on the Rydberg dressing and the dark states of the intracavity Rydberg EIT system.

B. Estimation of Gate Performance

In this subsection, the estimated performance and related numerical simulations of the atom–photon controlled-Z gate are provided. Since we have already checked the conditional phase shifts of the reflection coefficient, we may then calculate the Choi–Jamiolkowski fidelity [4850,6365] of the proposed gate operation. According to the arguments made in Refs. [49,50], the average fidelity for the atom–photon quantum gate is

Fz=116|2+RR|2.
Next, we are going to compute Fz via numerical simulations with typical parameter settings. The intracavity ensemble is set as a 3D atomic array of 10×10×10 with the site spacing being 0.37 μm. The single-atom qubit is placed at 1.5 sites away from the top layer center of the array. The Rydberg–Rydberg interaction parameters are taken from Rb87 atoms, where |r is regarded as 81S and |r is regarded as 84S. For a larger principal quantum number, it is possible to get an even stronger Förster resonance. The single-atom single-photon coupling is assumed to be the same as G0 for every atom in the ensemble. The angular dependence of the Rydberg–Rydberg interaction is also taken into consideration according to Ref. [55]. If the qubit state is realized via a spin wave embedded in the entire atomic ensemble, the gate fidelity results will be similar, whose details are omitted here for simplicity.

Sample results of the numerical simulations are presented in Figs. 7 and 8. As has already been revealed by previous theoretical deductions, we observe that the implementation of this gate protocol does not require ultra-high finesse from the optical cavity to get a strong effective atom–photon coupling. Meanwhile, it clearly offers a rather versatile parameter range. More specifically, it is able to serve as an interface where the photon frequency is endowed with a reasonable frequency dynamic range. Generally speaking, major sources limiting the best attainable fidelity of the system include the number of atoms in the ensemble, the Rydberg-blockade strength, and the lifetime of the Rydberg state.

 

Fig. 7. Numerical simulation of the gate’s fidelities with respect to different single-atom single-photon coupling strengths, G0. Particular parameters for this simulation include Ω=2π×100  MHz, Δe=2π×1000  MHz, Γe=2π×1  MHz, Γr=Γp=2π×0.01  MHz, κ=2π×1  MHz. While all other parameters are not changed, continuing to increase the strength of G won’t unlimitedly enhance the fidelity. This is because, as G increases beyond certain point, it will cause a power broadening effect, which harms the desired conditional phase shift. This can also be observed quantitatively from Eqs. (14) and (15).

Download Full Size | PPT Slide | PDF

 

Fig. 8. Numerical simulation of the gate’s fidelities to demonstrate its tuning range in Δe, with respect to different Rabi frequency settings of the control laser. Particular parameters for this simulation includes G0=2π×0.5  MHz, Γr=Γp=2π×0.01  MHz, κ=2π×1  MHz. The practical strategy of choosing the value of Δe will involve considerations to ensure that |Δe|Γe and |Δe|G.

Download Full Size | PPT Slide | PDF

A relatively straightforward way to comprehend the working principle of this gate protocol is the Autler–Townes effect. In the strict sense, this is referring to the situation that Δe is the same as Δr and Ω=2Δe, while discounting the ac Stark shifts. The two dressed states of the ensemble atoms are (|e+|r)/2 and (|e|r)/2, where only one of them is hitting the resonance to prevent the single-photon pulse from entering the cavity. Otherwise, if Rydberg blockade is turned on, this resonance with one of the two dressed states is broken, and the single-photon pulse may enter the cavity. If the system is liberated from the very strict definition of the Autler–Townes effect, it can be extended from this special case to the general concept of a two-photon stimulated Raman transition where the magnitude of Ω is a sizable fraction of |Δe|, not necessarily 2 or 1. This is tied to the relative strength of the state |e in the dressed state, which is actually resonantly coupled to the ground state.

For such a configuration to implement the atom–photon controlled-PHASE gate, several potential upgrades can be considered toward reaching an even higher fidelity. They include the reduction of the effective intermediate state decay, Γe, choosing Rydberg states with longer lifetimes, Stark tuning to obtain a stronger Förster resonance, and increasing the number of atoms in the ensemble. Another point calling for caution is the sign of the Rydberg-blockade shift. If the magnitude of the Rydberg-blockade shift is so large that the intermediate detuning, Δe, no longer dominates, its sign will be chosen so that the system will be pulled toward a direction that is away from both of the two dressed states’ resonances.

5. CONCLUSION AND OUTLOOK

In summary, we have offered an analysis for the interaction between an intracavity Rydberg-blocked atomic ensemble and cavity optical fields, with the emphasis put on the quantum optical properties of the dynamics. The fundamental relation with the JCM is explored, where the atoms participate in the atom–photon interaction collectively with the cavity field due to the Rydberg blockade. Moreover, an atom–photon controlled-PHASE quantum gate is constructed using the insights gained through the study of the JCM for this hybrid system, and its performance is also investigated. The exact number of atoms inside the ensemble does not play an essential role in the dynamics of the system, as long as the total number is adequate to enhance the collective atom–photon coupling, while the functionality of the system is robust against atom loss [66]. The behavior of this hybrid system clearly shows signatures from the well-known Autler–Townes effect and the two-photon stimulated Raman transition. More specifically, the cavity optical field couples the atom ground state to a dressed state made from an intermediate state |e and a Rydberg state |r. This feature enables the exploration of parameter regions that were not easy to access previously in the study of cQED.

One essential problem for the atom–photon gate in the real world is its fidelity under operational conditions, where technical noises and systematic imperfections are always present. Therefore, we want to emphasize that the long-standing goal of efficiently realizing high-fidelity atom–photon and photon–photon quantum gates remains a difficult challenge; enormous theoretical and experimental efforts are devoted to the research of this topic. We hope that our work offers help with this pursuit.

Funding

National Natural Science Foundation of China (NSFC) (11174370, 61632021); National Key R&D Program of China (2016YFA0301504, 2016YFA0301903); Basic Research Program Fund at Interdisciplinary Center for Quantum Information (ICQI), National University of Defense Technology.

Acknowledgment

The authors also acknowledge the hospitality of the Key Laboratory of Quantum Optics and Center of Cold Atom Physics, Shanghai Institute of Optics and Fine Mechanics. The authors gratefully thank Professor Liang Liu and Professor Mark Saffman whose help offers enormous momentum to this work. The authors also thank Professor Peng Xu and Professor Xiaodong He for enlightening discussions.

 

See Supplement 1 for supporting content.

REFERENCES

1. A. Galindo and M. A. Martín-Delgado, “Information and computation: classical and quantum aspects,” Rev. Mod. Phys. 74, 347–423 (2002). [CrossRef]  

2. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011). [CrossRef]  

3. A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015). [CrossRef]  

4. K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010). [CrossRef]  

5. L.-M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004). [CrossRef]  

6. A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014). [CrossRef]  

7. B. Hacker, S. Welte, G. Rempe, and S. Ritter, “A photon-photon quantum gate based on a single atom in an optical resonator,” Nature 536, 193–196 (2016). [CrossRef]  

8. A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349–1351 (2013). [CrossRef]  

9. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurášek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432, 482–486 (2004). [CrossRef]  

10. K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008). [CrossRef]  

11. N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011). [CrossRef]  

12. H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008). [CrossRef]  

13. S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012). [CrossRef]  

14. C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A 89, 022317 (2014). [CrossRef]  

15. M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001). [CrossRef]  

16. M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010). [CrossRef]  

17. M. Saffman and T. G. Walker, “Creating single-atom and single-photon sources from entangled atomic ensembles,” Phys. Rev. A 66, 065403 (2002). [CrossRef]  

18. J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011). [CrossRef]  

19. A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011). [CrossRef]  

20. V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012). [CrossRef]  

21. Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336, 887–889 (2012). [CrossRef]  

22. D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013). [CrossRef]  

23. H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014). [CrossRef]  

24. K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015). [CrossRef]  

25. C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, “Single-photon absorber based on strongly interacting Rydberg atoms,” Phys. Rev. Lett. 117, 223001 (2016). [CrossRef]  

26. Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017). [CrossRef]  

27. M. Ebert, M. Kwon, T. G. Walker, and M. Saffman, “Coherence and Rydberg blockade of atomic ensemble qubits,” Phys. Rev. Lett. 115, 093601 (2015). [CrossRef]  

28. M. Saffman, “Quantum computing with atomic qubits and Rydberg interactions: progress and challenges,” J. Phys. B 49, 202001 (2016). [CrossRef]  

29. E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963). [CrossRef]  

30. C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010). [CrossRef]  

31. R. G. Unanyan and M. Fleischhauer, “Efficient and robust entanglement generation in a many-particle system with resonant dipole-dipole interactions,” Phys. Rev. A 66, 032109 (2002). [CrossRef]  

32. D. Møller, L. B. Madsen, and K. Mølmer, “Quantum gates and multiparticle entanglement by Rydberg excitation blockade and adiabatic passage,” Phys. Rev. Lett. 100, 170504 (2008). [CrossRef]  

33. T. C. V. Opatrný and K. Mølmer, “Spin squeezing and Schrödinger-cat-state generation in atomic samples with Rydberg blockade,” Phys. Rev. A 86, 023845 (2012). [CrossRef]  

34. I. I. Beterov, T. Andrijauskas, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, and S. Bergamini, “Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms,” Phys. Rev. A 90, 043413 (2014). [CrossRef]  

35. T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, “Arbitrary Dicke-state control of symmetric Rydberg ensembles,” Phys. Rev. Lett. 117, 213601 (2016). [CrossRef]  

36. A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: a perturbative approach,” Phys. Rev. A 92, 043841 (2015). [CrossRef]  

37. J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine, A. Sommer, and J. Simon, “Observation and characterization of cavity Rydberg polaritons,” Phys. Rev. A 93, 041802 (2016). [CrossRef]  

38. J. Lee, M. J. Martin, Y.-Y. Jau, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms,” Phys. Rev. A 95, 041801 (2017). [CrossRef]  

39. A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014). [CrossRef]  

40. C. R. Murray and T. Pohl, “Coherent photon manipulation in interacting atomic ensembles,” Phys. Rev. X 7, 031007 (2017). [CrossRef]  

41. C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018). [CrossRef]  

42. C. Liu, Y. Sun, L. Zhao, S. Zhang, M. M. T. Loy, and S. Du, “Efficiently loading a single photon into a single-sided Fabry–Perot cavity,” Phys. Rev. Lett. 113, 133601 (2014). [CrossRef]  

43. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954). [CrossRef]  

44. M. O. Scully, “Collective Lamb shift in single photon Dicke superradiance,” Phys. Rev. Lett. 102, 143601 (2009). [CrossRef]  

45. D. Paredes-Barato and C. S. Adams, “All-optical quantum information processing using Rydberg gates,” Phys. Rev. Lett. 112, 040501 (2014). [CrossRef]  

46. M. Khazali, K. Heshami, and C. Simon, “Photon-photon gate via the interaction between two collective Rydberg excitations,” Phys. Rev. A 91, 030301 (2015). [CrossRef]  

47. O. Lahad and O. Firstenberg, “Induced cavities for photonic quantum gates,” Phys. Rev. Lett. 119, 113601 (2017). [CrossRef]  

48. Y. M. Hao, G. W. Lin, K. Xia, X. M. Lin, Y. P. Niu, and S. Q. Gong, “Quantum controlled-phase-flip gate between a flying optical photon and a Rydberg atomic ensemble,” Sci. Rep. 5, 10005 (2015). [CrossRef]  

49. S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016). [CrossRef]  

50. A. C. J. Wade, M. Mattioli, and K. Mølmer, “Single-atom single-photon coupling facilitated by atomic-ensemble dark-state mechanisms,” Phys. Rev. A 94, 053830 (2016). [CrossRef]  

51. F. Motzoi and K. Mølmer, “Precise single-qubit control of the reflection phase of a photon mediated by a strongly-coupled ancilla-cavity system,” New J. Phys. 20, 053029 (2018). [CrossRef]  

52. L. H. Pedersen and K. Mølmer, “Few qubit atom-light interfaces with collective encoding,” Phys. Rev. A 79, 012320 (2009). [CrossRef]  

53. A. E. B. Nielsen and K. Mølmer, “Deterministic multimode photonic device for quantum-information processing,” Phys. Rev. A 81, 043822 (2010). [CrossRef]  

54. Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, “Single-qubit gates based on targeted phase shifts in a 3D neutral atom array,” Science 352, 1562–1565 (2016). [CrossRef]  

55. I. I. Beterov and M. Saffman, “Rydberg blockade, Förster resonances, and quantum state measurements with different atomic species,” Phys. Rev. A 92, 042710 (2015). [CrossRef]  

56. J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008). [CrossRef]  

57. J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992). [CrossRef]  

58. M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998). [CrossRef]  

59. A. T. Black, J. K. Thompson, and V. Vuletić, “On-demand superradiant conversion of atomic spin gratings into single photons with high efficiency,” Phys. Rev. Lett. 95, 133601 (2005). [CrossRef]  

60. D. Petrosyan, F. Motzoi, M. Saffman, and K. Mølmer, “High-fidelity Rydberg quantum gate via a two-atom dark state,” Phys. Rev. A 96, 042306 (2017). [CrossRef]  

61. E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014). [CrossRef]  

62. B.-Q. Ou, C. Liu, Y. Sun, and P.-X. Chen, “Deterministically swapping frequency-bin entanglement from photon-photon to atom-photon hybrid systems,” Phys. Rev. A 97, 023839 (2018). [CrossRef]  

63. A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,” Rep. Math. Phys. 3, 275–278 (1972). [CrossRef]  

64. M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra Appl. 10, 285–290 (1975). [CrossRef]  

65. A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2005). [CrossRef]  

66. W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. A. Galindo and M. A. Martín-Delgado, “Information and computation: classical and quantum aspects,” Rev. Mod. Phys. 74, 347–423 (2002).
    [Crossref]
  2. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
    [Crossref]
  3. A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015).
    [Crossref]
  4. K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010).
    [Crossref]
  5. L.-M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004).
    [Crossref]
  6. A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
    [Crossref]
  7. B. Hacker, S. Welte, G. Rempe, and S. Ritter, “A photon-photon quantum gate based on a single atom in an optical resonator,” Nature 536, 193–196 (2016).
    [Crossref]
  8. A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349–1351 (2013).
    [Crossref]
  9. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurášek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432, 482–486 (2004).
    [Crossref]
  10. K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008).
    [Crossref]
  11. N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011).
    [Crossref]
  12. H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
    [Crossref]
  13. S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
    [Crossref]
  14. C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A 89, 022317 (2014).
    [Crossref]
  15. M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
    [Crossref]
  16. M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
    [Crossref]
  17. M. Saffman and T. G. Walker, “Creating single-atom and single-photon sources from entangled atomic ensembles,” Phys. Rev. A 66, 065403 (2002).
    [Crossref]
  18. J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
    [Crossref]
  19. A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
    [Crossref]
  20. V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
    [Crossref]
  21. Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336, 887–889 (2012).
    [Crossref]
  22. D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
    [Crossref]
  23. H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
    [Crossref]
  24. K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
    [Crossref]
  25. C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, “Single-photon absorber based on strongly interacting Rydberg atoms,” Phys. Rev. Lett. 117, 223001 (2016).
    [Crossref]
  26. Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
    [Crossref]
  27. M. Ebert, M. Kwon, T. G. Walker, and M. Saffman, “Coherence and Rydberg blockade of atomic ensemble qubits,” Phys. Rev. Lett. 115, 093601 (2015).
    [Crossref]
  28. M. Saffman, “Quantum computing with atomic qubits and Rydberg interactions: progress and challenges,” J. Phys. B 49, 202001 (2016).
    [Crossref]
  29. E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
    [Crossref]
  30. C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010).
    [Crossref]
  31. R. G. Unanyan and M. Fleischhauer, “Efficient and robust entanglement generation in a many-particle system with resonant dipole-dipole interactions,” Phys. Rev. A 66, 032109 (2002).
    [Crossref]
  32. D. Møller, L. B. Madsen, and K. Mølmer, “Quantum gates and multiparticle entanglement by Rydberg excitation blockade and adiabatic passage,” Phys. Rev. Lett. 100, 170504 (2008).
    [Crossref]
  33. T. C. V. Opatrný and K. Mølmer, “Spin squeezing and Schrödinger-cat-state generation in atomic samples with Rydberg blockade,” Phys. Rev. A 86, 023845 (2012).
    [Crossref]
  34. I. I. Beterov, T. Andrijauskas, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, and S. Bergamini, “Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms,” Phys. Rev. A 90, 043413 (2014).
    [Crossref]
  35. T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, “Arbitrary Dicke-state control of symmetric Rydberg ensembles,” Phys. Rev. Lett. 117, 213601 (2016).
    [Crossref]
  36. A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: a perturbative approach,” Phys. Rev. A 92, 043841 (2015).
    [Crossref]
  37. J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine, A. Sommer, and J. Simon, “Observation and characterization of cavity Rydberg polaritons,” Phys. Rev. A 93, 041802 (2016).
    [Crossref]
  38. J. Lee, M. J. Martin, Y.-Y. Jau, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms,” Phys. Rev. A 95, 041801 (2017).
    [Crossref]
  39. A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
    [Crossref]
  40. C. R. Murray and T. Pohl, “Coherent photon manipulation in interacting atomic ensembles,” Phys. Rev. X 7, 031007 (2017).
    [Crossref]
  41. C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018).
    [Crossref]
  42. C. Liu, Y. Sun, L. Zhao, S. Zhang, M. M. T. Loy, and S. Du, “Efficiently loading a single photon into a single-sided Fabry–Perot cavity,” Phys. Rev. Lett. 113, 133601 (2014).
    [Crossref]
  43. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
    [Crossref]
  44. M. O. Scully, “Collective Lamb shift in single photon Dicke superradiance,” Phys. Rev. Lett. 102, 143601 (2009).
    [Crossref]
  45. D. Paredes-Barato and C. S. Adams, “All-optical quantum information processing using Rydberg gates,” Phys. Rev. Lett. 112, 040501 (2014).
    [Crossref]
  46. M. Khazali, K. Heshami, and C. Simon, “Photon-photon gate via the interaction between two collective Rydberg excitations,” Phys. Rev. A 91, 030301 (2015).
    [Crossref]
  47. O. Lahad and O. Firstenberg, “Induced cavities for photonic quantum gates,” Phys. Rev. Lett. 119, 113601 (2017).
    [Crossref]
  48. Y. M. Hao, G. W. Lin, K. Xia, X. M. Lin, Y. P. Niu, and S. Q. Gong, “Quantum controlled-phase-flip gate between a flying optical photon and a Rydberg atomic ensemble,” Sci. Rep. 5, 10005 (2015).
    [Crossref]
  49. S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
    [Crossref]
  50. A. C. J. Wade, M. Mattioli, and K. Mølmer, “Single-atom single-photon coupling facilitated by atomic-ensemble dark-state mechanisms,” Phys. Rev. A 94, 053830 (2016).
    [Crossref]
  51. F. Motzoi and K. Mølmer, “Precise single-qubit control of the reflection phase of a photon mediated by a strongly-coupled ancilla-cavity system,” New J. Phys. 20, 053029 (2018).
    [Crossref]
  52. L. H. Pedersen and K. Mølmer, “Few qubit atom-light interfaces with collective encoding,” Phys. Rev. A 79, 012320 (2009).
    [Crossref]
  53. A. E. B. Nielsen and K. Mølmer, “Deterministic multimode photonic device for quantum-information processing,” Phys. Rev. A 81, 043822 (2010).
    [Crossref]
  54. Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, “Single-qubit gates based on targeted phase shifts in a 3D neutral atom array,” Science 352, 1562–1565 (2016).
    [Crossref]
  55. I. I. Beterov and M. Saffman, “Rydberg blockade, Förster resonances, and quantum state measurements with different atomic species,” Phys. Rev. A 92, 042710 (2015).
    [Crossref]
  56. J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008).
    [Crossref]
  57. J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
    [Crossref]
  58. M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
    [Crossref]
  59. A. T. Black, J. K. Thompson, and V. Vuletić, “On-demand superradiant conversion of atomic spin gratings into single photons with high efficiency,” Phys. Rev. Lett. 95, 133601 (2005).
    [Crossref]
  60. D. Petrosyan, F. Motzoi, M. Saffman, and K. Mølmer, “High-fidelity Rydberg quantum gate via a two-atom dark state,” Phys. Rev. A 96, 042306 (2017).
    [Crossref]
  61. E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014).
    [Crossref]
  62. B.-Q. Ou, C. Liu, Y. Sun, and P.-X. Chen, “Deterministically swapping frequency-bin entanglement from photon-photon to atom-photon hybrid systems,” Phys. Rev. A 97, 023839 (2018).
    [Crossref]
  63. A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,” Rep. Math. Phys. 3, 275–278 (1972).
    [Crossref]
  64. M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra Appl. 10, 285–290 (1975).
    [Crossref]
  65. A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2005).
    [Crossref]
  66. W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
    [Crossref]

2018 (3)

C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018).
[Crossref]

F. Motzoi and K. Mølmer, “Precise single-qubit control of the reflection phase of a photon mediated by a strongly-coupled ancilla-cavity system,” New J. Phys. 20, 053029 (2018).
[Crossref]

B.-Q. Ou, C. Liu, Y. Sun, and P.-X. Chen, “Deterministically swapping frequency-bin entanglement from photon-photon to atom-photon hybrid systems,” Phys. Rev. A 97, 023839 (2018).
[Crossref]

2017 (5)

D. Petrosyan, F. Motzoi, M. Saffman, and K. Mølmer, “High-fidelity Rydberg quantum gate via a two-atom dark state,” Phys. Rev. A 96, 042306 (2017).
[Crossref]

O. Lahad and O. Firstenberg, “Induced cavities for photonic quantum gates,” Phys. Rev. Lett. 119, 113601 (2017).
[Crossref]

C. R. Murray and T. Pohl, “Coherent photon manipulation in interacting atomic ensembles,” Phys. Rev. X 7, 031007 (2017).
[Crossref]

J. Lee, M. J. Martin, Y.-Y. Jau, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms,” Phys. Rev. A 95, 041801 (2017).
[Crossref]

Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
[Crossref]

2016 (8)

M. Saffman, “Quantum computing with atomic qubits and Rydberg interactions: progress and challenges,” J. Phys. B 49, 202001 (2016).
[Crossref]

C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, “Single-photon absorber based on strongly interacting Rydberg atoms,” Phys. Rev. Lett. 117, 223001 (2016).
[Crossref]

B. Hacker, S. Welte, G. Rempe, and S. Ritter, “A photon-photon quantum gate based on a single atom in an optical resonator,” Nature 536, 193–196 (2016).
[Crossref]

T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, “Arbitrary Dicke-state control of symmetric Rydberg ensembles,” Phys. Rev. Lett. 117, 213601 (2016).
[Crossref]

J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine, A. Sommer, and J. Simon, “Observation and characterization of cavity Rydberg polaritons,” Phys. Rev. A 93, 041802 (2016).
[Crossref]

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

A. C. J. Wade, M. Mattioli, and K. Mølmer, “Single-atom single-photon coupling facilitated by atomic-ensemble dark-state mechanisms,” Phys. Rev. A 94, 053830 (2016).
[Crossref]

Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, “Single-qubit gates based on targeted phase shifts in a 3D neutral atom array,” Science 352, 1562–1565 (2016).
[Crossref]

2015 (7)

I. I. Beterov and M. Saffman, “Rydberg blockade, Förster resonances, and quantum state measurements with different atomic species,” Phys. Rev. A 92, 042710 (2015).
[Crossref]

M. Khazali, K. Heshami, and C. Simon, “Photon-photon gate via the interaction between two collective Rydberg excitations,” Phys. Rev. A 91, 030301 (2015).
[Crossref]

Y. M. Hao, G. W. Lin, K. Xia, X. M. Lin, Y. P. Niu, and S. Q. Gong, “Quantum controlled-phase-flip gate between a flying optical photon and a Rydberg atomic ensemble,” Sci. Rep. 5, 10005 (2015).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: a perturbative approach,” Phys. Rev. A 92, 043841 (2015).
[Crossref]

A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015).
[Crossref]

M. Ebert, M. Kwon, T. G. Walker, and M. Saffman, “Coherence and Rydberg blockade of atomic ensemble qubits,” Phys. Rev. Lett. 115, 093601 (2015).
[Crossref]

K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
[Crossref]

2014 (8)

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

I. I. Beterov, T. Andrijauskas, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, and S. Bergamini, “Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms,” Phys. Rev. A 90, 043413 (2014).
[Crossref]

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref]

C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A 89, 022317 (2014).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

C. Liu, Y. Sun, L. Zhao, S. Zhang, M. M. T. Loy, and S. Du, “Efficiently loading a single photon into a single-sided Fabry–Perot cavity,” Phys. Rev. Lett. 113, 133601 (2014).
[Crossref]

D. Paredes-Barato and C. S. Adams, “All-optical quantum information processing using Rydberg gates,” Phys. Rev. Lett. 112, 040501 (2014).
[Crossref]

E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014).
[Crossref]

2013 (2)

A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349–1351 (2013).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

2012 (4)

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336, 887–889 (2012).
[Crossref]

T. C. V. Opatrný and K. Mølmer, “Spin squeezing and Schrödinger-cat-state generation in atomic samples with Rydberg blockade,” Phys. Rev. A 86, 023845 (2012).
[Crossref]

2011 (4)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011).
[Crossref]

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

2010 (4)

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
[Crossref]

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010).
[Crossref]

C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010).
[Crossref]

A. E. B. Nielsen and K. Mølmer, “Deterministic multimode photonic device for quantum-information processing,” Phys. Rev. A 81, 043822 (2010).
[Crossref]

2009 (2)

M. O. Scully, “Collective Lamb shift in single photon Dicke superradiance,” Phys. Rev. Lett. 102, 143601 (2009).
[Crossref]

L. H. Pedersen and K. Mølmer, “Few qubit atom-light interfaces with collective encoding,” Phys. Rev. A 79, 012320 (2009).
[Crossref]

2008 (4)

J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008).
[Crossref]

D. Møller, L. B. Madsen, and K. Mølmer, “Quantum gates and multiparticle entanglement by Rydberg excitation blockade and adiabatic passage,” Phys. Rev. Lett. 100, 170504 (2008).
[Crossref]

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008).
[Crossref]

H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
[Crossref]

2005 (2)

A. T. Black, J. K. Thompson, and V. Vuletić, “On-demand superradiant conversion of atomic spin gratings into single photons with high efficiency,” Phys. Rev. Lett. 95, 133601 (2005).
[Crossref]

A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2005).
[Crossref]

2004 (2)

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurášek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432, 482–486 (2004).
[Crossref]

L.-M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004).
[Crossref]

2002 (3)

A. Galindo and M. A. Martín-Delgado, “Information and computation: classical and quantum aspects,” Rev. Mod. Phys. 74, 347–423 (2002).
[Crossref]

M. Saffman and T. G. Walker, “Creating single-atom and single-photon sources from entangled atomic ensembles,” Phys. Rev. A 66, 065403 (2002).
[Crossref]

R. G. Unanyan and M. Fleischhauer, “Efficient and robust entanglement generation in a many-particle system with resonant dipole-dipole interactions,” Phys. Rev. A 66, 032109 (2002).
[Crossref]

2001 (1)

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[Crossref]

2000 (1)

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

1998 (1)

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
[Crossref]

1992 (1)

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[Crossref]

1975 (1)

M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra Appl. 10, 285–290 (1975).
[Crossref]

1972 (1)

A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,” Rep. Math. Phys. 3, 275–278 (1972).
[Crossref]

1963 (1)

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
[Crossref]

1954 (1)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[Crossref]

Adams, C. S.

D. Paredes-Barato and C. S. Adams, “All-optical quantum information processing using Rydberg gates,” Phys. Rev. Lett. 112, 040501 (2014).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Andrijauskas, T.

I. I. Beterov, T. Andrijauskas, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, and S. Bergamini, “Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms,” Phys. Rev. A 90, 043413 (2014).
[Crossref]

Baldwin, C. H.

T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, “Arbitrary Dicke-state control of symmetric Rydberg ensembles,” Phys. Rev. Lett. 117, 213601 (2016).
[Crossref]

Bergamini, S.

I. I. Beterov, T. Andrijauskas, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, and S. Bergamini, “Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms,” Phys. Rev. A 90, 043413 (2014).
[Crossref]

Beterov, I. I.

I. I. Beterov and M. Saffman, “Rydberg blockade, Förster resonances, and quantum state measurements with different atomic species,” Phys. Rev. A 92, 042710 (2015).
[Crossref]

I. I. Beterov, T. Andrijauskas, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, and S. Bergamini, “Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms,” Phys. Rev. A 90, 043413 (2014).
[Crossref]

Biedermann, G. W.

J. Lee, M. J. Martin, Y.-Y. Jau, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms,” Phys. Rev. A 95, 041801 (2017).
[Crossref]

T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, “Arbitrary Dicke-state control of symmetric Rydberg ensembles,” Phys. Rev. Lett. 117, 213601 (2016).
[Crossref]

Bimbard, E.

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: a perturbative approach,” Phys. Rev. A 92, 043841 (2015).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Black, A. T.

A. T. Black, J. K. Thompson, and V. Vuletić, “On-demand superradiant conversion of atomic spin gratings into single photons with high efficiency,” Phys. Rev. Lett. 95, 133601 (2005).
[Crossref]

Bochmann, J.

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008).
[Crossref]

Boddeda, R.

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: a perturbative approach,” Phys. Rev. A 92, 043841 (2015).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014).
[Crossref]

Borregaard, J.

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

Braun, C.

C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018).
[Crossref]

Brion, E.

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: a perturbative approach,” Phys. Rev. A 92, 043841 (2015).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010).
[Crossref]

Brown, K. R.

C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A 89, 022317 (2014).
[Crossref]

Büchler, H. P.

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

Busche, H.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Carr, A. W.

K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
[Crossref]

Castin, Y.

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[Crossref]

Chen, P.-X.

B.-Q. Ou, C. Liu, Y. Sun, and P.-X. Chen, “Deterministically swapping frequency-bin entanglement from photon-photon to atom-photon hybrid systems,” Phys. Rev. A 97, 023839 (2018).
[Crossref]

Choi, K. S.

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008).
[Crossref]

Choi, M.-D.

M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra Appl. 10, 285–290 (1975).
[Crossref]

Cirac, J. I.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurášek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432, 482–486 (2004).
[Crossref]

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[Crossref]

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

Cote, R.

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[Crossref]

Cummings, F. W.

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
[Crossref]

Dalibard, J.

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[Crossref]

Das, S.

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

de Riedmatten, H.

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011).
[Crossref]

Deng, H.

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008).
[Crossref]

Deutsch, I. H.

J. Lee, M. J. Martin, Y.-Y. Jau, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms,” Phys. Rev. A 95, 041801 (2017).
[Crossref]

T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, “Arbitrary Dicke-state control of symmetric Rydberg ensembles,” Phys. Rev. Lett. 117, 213601 (2016).
[Crossref]

Dicke, R. H.

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[Crossref]

Du, S.

C. Liu, Y. Sun, L. Zhao, S. Zhang, M. M. T. Loy, and S. Du, “Efficiently loading a single photon into a single-sided Fabry–Perot cavity,” Phys. Rev. Lett. 113, 133601 (2014).
[Crossref]

Duan, L. M.

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[Crossref]

Duan, L.-M.

C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A 89, 022317 (2014).
[Crossref]

L.-M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004).
[Crossref]

Dudin, Y. O.

Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336, 887–889 (2012).
[Crossref]

Dür, W.

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

Ebert, M.

M. Ebert, M. Kwon, T. G. Walker, and M. Saffman, “Coherence and Rydberg blockade of atomic ensemble qubits,” Phys. Rev. Lett. 115, 093601 (2015).
[Crossref]

Entin, V. M.

I. I. Beterov, T. Andrijauskas, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, and S. Bergamini, “Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms,” Phys. Rev. A 90, 043413 (2014).
[Crossref]

Erbel, C.

J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008).
[Crossref]

Esslinger, T.

C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010).
[Crossref]

Fedder, H.

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

Figueroa, E.

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

Firstenberg, O.

O. Lahad and O. Firstenberg, “Induced cavities for photonic quantum gates,” Phys. Rev. Lett. 119, 113601 (2017).
[Crossref]

Fiurášek, J.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurášek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432, 482–486 (2004).
[Crossref]

Fleischhauer, M.

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

R. G. Unanyan and M. Fleischhauer, “Efficient and robust entanglement generation in a many-particle system with resonant dipole-dipole interactions,” Phys. Rev. A 66, 032109 (2002).
[Crossref]

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[Crossref]

Galindo, A.

A. Galindo and M. A. Martín-Delgado, “Information and computation: classical and quantum aspects,” Rev. Mod. Phys. 74, 347–423 (2002).
[Crossref]

Gauguet, A.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Georgakopoulos, A.

J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine, A. Sommer, and J. Simon, “Observation and characterization of cavity Rydberg polaritons,” Phys. Rev. A 93, 041802 (2016).
[Crossref]

Gilchrist, A.

A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2005).
[Crossref]

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Gisin, N.

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011).
[Crossref]

Gong, S. Q.

Y. M. Hao, G. W. Lin, K. Xia, X. M. Lin, Y. P. Niu, and S. Q. Gong, “Quantum controlled-phase-flip gate between a flying optical photon and a Rydberg atomic ensemble,” Sci. Rep. 5, 10005 (2015).
[Crossref]

Gorniaczyk, H.

C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, “Single-photon absorber based on strongly interacting Rydberg atoms,” Phys. Rev. Lett. 117, 223001 (2016).
[Crossref]

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

Gorshkov, A. V.

C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

Grangier, P.

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: a perturbative approach,” Phys. Rev. A 92, 043841 (2015).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Grankin, A.

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: a perturbative approach,” Phys. Rev. A 92, 043841 (2015).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014).
[Crossref]

Guerlin, C.

C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010).
[Crossref]

Hacker, B.

B. Hacker, S. Welte, G. Rempe, and S. Ritter, “A photon-photon quantum gate based on a single atom in an optical resonator,” Nature 536, 193–196 (2016).
[Crossref]

Hahn, C.

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

Hammerer, K.

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010).
[Crossref]

Hao, Y. M.

Y. M. Hao, G. W. Lin, K. Xia, X. M. Lin, Y. P. Niu, and S. Q. Gong, “Quantum controlled-phase-flip gate between a flying optical photon and a Rydberg atomic ensemble,” Sci. Rep. 5, 10005 (2015).
[Crossref]

He, X.

Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
[Crossref]

Heshami, K.

M. Khazali, K. Heshami, and C. Simon, “Photon-photon gate via the interaction between two collective Rydberg excitations,” Phys. Rev. A 91, 030301 (2015).
[Crossref]

Hilliard, A. J.

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Hofferberth, S.

C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018).
[Crossref]

C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, “Single-photon absorber based on strongly interacting Rydberg atoms,” Phys. Rev. Lett. 117, 223001 (2016).
[Crossref]

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

Honer, J.

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

Iakoupov, I.

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

Isenhower, L.

K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
[Crossref]

Jaksch, D.

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[Crossref]

Jamiolkowski, A.

A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,” Rep. Math. Phys. 3, 275–278 (1972).
[Crossref]

Jau, Y.-Y.

J. Lee, M. J. Martin, Y.-Y. Jau, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms,” Phys. Rev. A 95, 041801 (2017).
[Crossref]

T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, “Arbitrary Dicke-state control of symmetric Rydberg ensembles,” Phys. Rev. Lett. 117, 213601 (2016).
[Crossref]

Jaynes, E. T.

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
[Crossref]

Jones, M. P. A.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Julsgaard, B.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurášek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432, 482–486 (2004).
[Crossref]

Kalb, N.

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref]

Keating, T.

J. Lee, M. J. Martin, Y.-Y. Jau, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms,” Phys. Rev. A 95, 041801 (2017).
[Crossref]

T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, “Arbitrary Dicke-state control of symmetric Rydberg ensembles,” Phys. Rev. Lett. 117, 213601 (2016).
[Crossref]

Khazali, M.

M. Khazali, K. Heshami, and C. Simon, “Photon-photon gate via the interaction between two collective Rydberg excitations,” Phys. Rev. A 91, 030301 (2015).
[Crossref]

Kim, J.

C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A 89, 022317 (2014).
[Crossref]

Kimble, H. J.

H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
[Crossref]

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008).
[Crossref]

L.-M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004).
[Crossref]

Knight, P. L.

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
[Crossref]

Kumar, A.

Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, “Single-qubit gates based on targeted phase shifts in a 3D neutral atom array,” Science 352, 1562–1565 (2016).
[Crossref]

Kuzmich, A.

Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336, 887–889 (2012).
[Crossref]

Kwon, M.

M. Ebert, M. Kwon, T. G. Walker, and M. Saffman, “Coherence and Rydberg blockade of atomic ensemble qubits,” Phys. Rev. Lett. 115, 093601 (2015).
[Crossref]

Lahad, O.

O. Lahad and O. Firstenberg, “Induced cavities for photonic quantum gates,” Phys. Rev. Lett. 119, 113601 (2017).
[Crossref]

Langfahl-Klabes, G.

J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008).
[Crossref]

Langford, N. K.

A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2005).
[Crossref]

Laurat, J.

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008).
[Crossref]

Lee, J.

J. Lee, M. J. Martin, Y.-Y. Jau, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms,” Phys. Rev. A 95, 041801 (2017).
[Crossref]

T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, “Arbitrary Dicke-state control of symmetric Rydberg ensembles,” Phys. Rev. Lett. 117, 213601 (2016).
[Crossref]

Lichtman, M. T.

K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
[Crossref]

Lin, G. W.

Y. M. Hao, G. W. Lin, K. Xia, X. M. Lin, Y. P. Niu, and S. Q. Gong, “Quantum controlled-phase-flip gate between a flying optical photon and a Rydberg atomic ensemble,” Sci. Rep. 5, 10005 (2015).
[Crossref]

Lin, X. M.

Y. M. Hao, G. W. Lin, K. Xia, X. M. Lin, Y. P. Niu, and S. Q. Gong, “Quantum controlled-phase-flip gate between a flying optical photon and a Rydberg atomic ensemble,” Sci. Rep. 5, 10005 (2015).
[Crossref]

Liu, C.

B.-Q. Ou, C. Liu, Y. Sun, and P.-X. Chen, “Deterministically swapping frequency-bin entanglement from photon-photon to atom-photon hybrid systems,” Phys. Rev. A 97, 023839 (2018).
[Crossref]

C. Liu, Y. Sun, L. Zhao, S. Zhang, M. M. T. Loy, and S. Du, “Efficiently loading a single photon into a single-sided Fabry–Perot cavity,” Phys. Rev. Lett. 113, 133601 (2014).
[Crossref]

Liu, M.

Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
[Crossref]

Liu, Y.

Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
[Crossref]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Löw, R.

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

Loy, M. M. T.

C. Liu, Y. Sun, L. Zhao, S. Zhang, M. M. T. Loy, and S. Du, “Efficiently loading a single photon into a single-sided Fabry–Perot cavity,” Phys. Rev. Lett. 113, 133601 (2014).
[Crossref]

Lukin, M. D.

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[Crossref]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Madsen, L. B.

D. Møller, L. B. Madsen, and K. Mølmer, “Quantum gates and multiparticle entanglement by Rydberg excitation blockade and adiabatic passage,” Phys. Rev. Lett. 100, 170504 (2008).
[Crossref]

Maller, K. M.

K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
[Crossref]

Martin, M. J.

J. Lee, M. J. Martin, Y.-Y. Jau, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms,” Phys. Rev. A 95, 041801 (2017).
[Crossref]

Martín-Delgado, M. A.

A. Galindo and M. A. Martín-Delgado, “Information and computation: classical and quantum aspects,” Rev. Mod. Phys. 74, 347–423 (2002).
[Crossref]

Mattioli, M.

A. C. J. Wade, M. Mattioli, and K. Mølmer, “Single-atom single-photon coupling facilitated by atomic-ensemble dark-state mechanisms,” Phys. Rev. A 94, 053830 (2016).
[Crossref]

Maunz, P.

C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A 89, 022317 (2014).
[Crossref]

Maxwell, D.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Mirgorodskiy, I.

C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018).
[Crossref]

C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, “Single-photon absorber based on strongly interacting Rydberg atoms,” Phys. Rev. Lett. 117, 223001 (2016).
[Crossref]

Moehring, D. L.

J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008).
[Crossref]

Møller, D.

D. Møller, L. B. Madsen, and K. Mølmer, “Quantum gates and multiparticle entanglement by Rydberg excitation blockade and adiabatic passage,” Phys. Rev. Lett. 100, 170504 (2008).
[Crossref]

Mølmer, K.

F. Motzoi and K. Mølmer, “Precise single-qubit control of the reflection phase of a photon mediated by a strongly-coupled ancilla-cavity system,” New J. Phys. 20, 053029 (2018).
[Crossref]

D. Petrosyan, F. Motzoi, M. Saffman, and K. Mølmer, “High-fidelity Rydberg quantum gate via a two-atom dark state,” Phys. Rev. A 96, 042306 (2017).
[Crossref]

A. C. J. Wade, M. Mattioli, and K. Mølmer, “Single-atom single-photon coupling facilitated by atomic-ensemble dark-state mechanisms,” Phys. Rev. A 94, 053830 (2016).
[Crossref]

T. C. V. Opatrný and K. Mølmer, “Spin squeezing and Schrödinger-cat-state generation in atomic samples with Rydberg blockade,” Phys. Rev. A 86, 023845 (2012).
[Crossref]

C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010).
[Crossref]

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
[Crossref]

A. E. B. Nielsen and K. Mølmer, “Deterministic multimode photonic device for quantum-information processing,” Phys. Rev. A 81, 043822 (2010).
[Crossref]

L. H. Pedersen and K. Mølmer, “Few qubit atom-light interfaces with collective encoding,” Phys. Rev. A 79, 012320 (2009).
[Crossref]

D. Møller, L. B. Madsen, and K. Mølmer, “Quantum gates and multiparticle entanglement by Rydberg excitation blockade and adiabatic passage,” Phys. Rev. Lett. 100, 170504 (2008).
[Crossref]

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[Crossref]

Monroe, C.

C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A 89, 022317 (2014).
[Crossref]

Motzoi, F.

F. Motzoi and K. Mølmer, “Precise single-qubit control of the reflection phase of a photon mediated by a strongly-coupled ancilla-cavity system,” New J. Phys. 20, 053029 (2018).
[Crossref]

D. Petrosyan, F. Motzoi, M. Saffman, and K. Mølmer, “High-fidelity Rydberg quantum gate via a two-atom dark state,” Phys. Rev. A 96, 042306 (2017).
[Crossref]

Mucke, M.

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

Mücke, M.

J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008).
[Crossref]

Murray, C. R.

C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018).
[Crossref]

C. R. Murray and T. Pohl, “Coherent photon manipulation in interacting atomic ensembles,” Phys. Rev. X 7, 031007 (2017).
[Crossref]

Neuzner, A.

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

Nielsen, A. E. B.

A. E. B. Nielsen and K. Mølmer, “Deterministic multimode photonic device for quantum-information processing,” Phys. Rev. A 81, 043822 (2010).
[Crossref]

Nielsen, M. A.

A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2005).
[Crossref]

Ningyuan, J.

J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine, A. Sommer, and J. Simon, “Observation and characterization of cavity Rydberg polaritons,” Phys. Rev. A 93, 041802 (2016).
[Crossref]

Niu, Y. P.

Y. M. Hao, G. W. Lin, K. Xia, X. M. Lin, Y. P. Niu, and S. Q. Gong, “Quantum controlled-phase-flip gate between a flying optical photon and a Rydberg atomic ensemble,” Sci. Rep. 5, 10005 (2015).
[Crossref]

Nogrette, F.

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Nolleke, C.

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

Opatrný, T. C. V.

T. C. V. Opatrný and K. Mølmer, “Spin squeezing and Schrödinger-cat-state generation in atomic samples with Rydberg blockade,” Phys. Rev. A 86, 023845 (2012).
[Crossref]

Otterbach, J.

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

Ou, B.-Q.

B.-Q. Ou, C. Liu, Y. Sun, and P.-X. Chen, “Deterministically swapping frequency-bin entanglement from photon-photon to atom-photon hybrid systems,” Phys. Rev. A 97, 023839 (2018).
[Crossref]

Ourjoumtsev, A.

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: a perturbative approach,” Phys. Rev. A 92, 043841 (2015).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Papoular, D. J.

Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
[Crossref]

Paredes-Barato, D.

D. Paredes-Barato and C. S. Adams, “All-optical quantum information processing using Rydberg gates,” Phys. Rev. Lett. 112, 040501 (2014).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Parigi, V.

E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Paris-Mandoki, A.

C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018).
[Crossref]

C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, “Single-photon absorber based on strongly interacting Rydberg atoms,” Phys. Rev. Lett. 117, 223001 (2016).
[Crossref]

Pedersen, L. H.

L. H. Pedersen and K. Mølmer, “Few qubit atom-light interfaces with collective encoding,” Phys. Rev. A 79, 012320 (2009).
[Crossref]

Petrosyan, D.

D. Petrosyan, F. Motzoi, M. Saffman, and K. Mølmer, “High-fidelity Rydberg quantum gate via a two-atom dark state,” Phys. Rev. A 96, 042306 (2017).
[Crossref]

Pfau, T.

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

Piotrowicz, M. J.

K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
[Crossref]

Plenio, M. B.

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
[Crossref]

Pohl, T.

C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018).
[Crossref]

C. R. Murray and T. Pohl, “Coherent photon manipulation in interacting atomic ensembles,” Phys. Rev. X 7, 031007 (2017).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

Polzik, E. S.

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010).
[Crossref]

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurášek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432, 482–486 (2004).
[Crossref]

Pritchard, J. D.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Raussendorf, R.

C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A 89, 022317 (2014).
[Crossref]

Reiserer, A.

A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015).
[Crossref]

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref]

A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349–1351 (2013).
[Crossref]

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

Rempe, G.

B. Hacker, S. Welte, G. Rempe, and S. Ritter, “A photon-photon quantum gate based on a single atom in an optical resonator,” Nature 536, 193–196 (2016).
[Crossref]

A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015).
[Crossref]

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref]

A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349–1351 (2013).
[Crossref]

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008).
[Crossref]

Ritter, S.

B. Hacker, S. Welte, G. Rempe, and S. Ritter, “A photon-photon quantum gate based on a single atom in an optical resonator,” Nature 536, 193–196 (2016).
[Crossref]

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref]

A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349–1351 (2013).
[Crossref]

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

Ruthven, A.

C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A 89, 022317 (2014).
[Crossref]

Ryabtsev, I. I.

I. I. Beterov, T. Andrijauskas, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, and S. Bergamini, “Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms,” Phys. Rev. A 90, 043413 (2014).
[Crossref]

Ryou, A.

J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine, A. Sommer, and J. Simon, “Observation and characterization of cavity Rydberg polaritons,” Phys. Rev. A 93, 041802 (2016).
[Crossref]

Saffman, M.

D. Petrosyan, F. Motzoi, M. Saffman, and K. Mølmer, “High-fidelity Rydberg quantum gate via a two-atom dark state,” Phys. Rev. A 96, 042306 (2017).
[Crossref]

M. Saffman, “Quantum computing with atomic qubits and Rydberg interactions: progress and challenges,” J. Phys. B 49, 202001 (2016).
[Crossref]

M. Ebert, M. Kwon, T. G. Walker, and M. Saffman, “Coherence and Rydberg blockade of atomic ensemble qubits,” Phys. Rev. Lett. 115, 093601 (2015).
[Crossref]

K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
[Crossref]

I. I. Beterov and M. Saffman, “Rydberg blockade, Förster resonances, and quantum state measurements with different atomic species,” Phys. Rev. A 92, 042710 (2015).
[Crossref]

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
[Crossref]

M. Saffman and T. G. Walker, “Creating single-atom and single-photon sources from entangled atomic ensembles,” Phys. Rev. A 66, 065403 (2002).
[Crossref]

Sangouard, N.

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011).
[Crossref]

Schine, N.

J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine, A. Sommer, and J. Simon, “Observation and characterization of cavity Rydberg polaritons,” Phys. Rev. A 93, 041802 (2016).
[Crossref]

Schmidt, J.

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

Scully, M. O.

M. O. Scully, “Collective Lamb shift in single photon Dicke superradiance,” Phys. Rev. Lett. 102, 143601 (2009).
[Crossref]

Sherson, J.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurášek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432, 482–486 (2004).
[Crossref]

Shlyapnikov, G. V.

Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
[Crossref]

Simon, C.

M. Khazali, K. Heshami, and C. Simon, “Photon-photon gate via the interaction between two collective Rydberg excitations,” Phys. Rev. A 91, 030301 (2015).
[Crossref]

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011).
[Crossref]

Simon, J.

J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine, A. Sommer, and J. Simon, “Observation and characterization of cavity Rydberg polaritons,” Phys. Rev. A 93, 041802 (2016).
[Crossref]

Sommer, A.

J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine, A. Sommer, and J. Simon, “Observation and characterization of cavity Rydberg polaritons,” Phys. Rev. A 93, 041802 (2016).
[Crossref]

Sørensen, A. S.

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010).
[Crossref]

Specht, H. P.

J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008).
[Crossref]

Stanojevic, J.

E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Sun, Y.

B.-Q. Ou, C. Liu, Y. Sun, and P.-X. Chen, “Deterministically swapping frequency-bin entanglement from photon-photon to atom-photon hybrid systems,” Phys. Rev. A 97, 023839 (2018).
[Crossref]

K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
[Crossref]

C. Liu, Y. Sun, L. Zhao, S. Zhang, M. M. T. Loy, and S. Du, “Efficiently loading a single photon into a single-sided Fabry–Perot cavity,” Phys. Rev. Lett. 113, 133601 (2014).
[Crossref]

Szwer, D. J.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Thompson, J. K.

A. T. Black, J. K. Thompson, and V. Vuletić, “On-demand superradiant conversion of atomic spin gratings into single photons with high efficiency,” Phys. Rev. Lett. 95, 133601 (2005).
[Crossref]

Tresp, C.

C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018).
[Crossref]

C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, “Single-photon absorber based on strongly interacting Rydberg atoms,” Phys. Rev. Lett. 117, 223001 (2016).
[Crossref]

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

Tretyakov, D. B.

I. I. Beterov, T. Andrijauskas, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, and S. Bergamini, “Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms,” Phys. Rev. A 90, 043413 (2014).
[Crossref]

Tualle-Brouri, R.

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Unanyan, R. G.

R. G. Unanyan and M. Fleischhauer, “Efficient and robust entanglement generation in a many-particle system with resonant dipole-dipole interactions,” Phys. Rev. A 66, 032109 (2002).
[Crossref]

Uphoff, M.

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

Usmani, I.

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: a perturbative approach,” Phys. Rev. A 92, 043841 (2015).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

Vidal, G.

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

Vitrant, N.

E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014).
[Crossref]

Vuletic, V.

A. T. Black, J. K. Thompson, and V. Vuletić, “On-demand superradiant conversion of atomic spin gratings into single photons with high efficiency,” Phys. Rev. Lett. 95, 133601 (2005).
[Crossref]

Wade, A. C. J.

A. C. J. Wade, M. Mattioli, and K. Mølmer, “Single-atom single-photon coupling facilitated by atomic-ensemble dark-state mechanisms,” Phys. Rev. A 94, 053830 (2016).
[Crossref]

Walker, T. G.

M. Ebert, M. Kwon, T. G. Walker, and M. Saffman, “Coherence and Rydberg blockade of atomic ensemble qubits,” Phys. Rev. Lett. 115, 093601 (2015).
[Crossref]

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
[Crossref]

M. Saffman and T. G. Walker, “Creating single-atom and single-photon sources from entangled atomic ensembles,” Phys. Rev. A 66, 065403 (2002).
[Crossref]

Wang, J.

Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
[Crossref]

Wang, Y.

Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, “Single-qubit gates based on targeted phase shifts in a 3D neutral atom array,” Science 352, 1562–1565 (2016).
[Crossref]

Weatherill, K. J.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Weber, B.

J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008).
[Crossref]

Weimer, H.

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

Weiss, D. S.

Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, “Single-qubit gates based on targeted phase shifts in a 3D neutral atom array,” Science 352, 1562–1565 (2016).
[Crossref]

Welte, S.

B. Hacker, S. Welte, G. Rempe, and S. Ritter, “A photon-photon quantum gate based on a single atom in an optical resonator,” Nature 536, 193–196 (2016).
[Crossref]

Wu, T.-Y.

Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, “Single-qubit gates based on targeted phase shifts in a 3D neutral atom array,” Science 352, 1562–1565 (2016).
[Crossref]

Xia, K.

Y. M. Hao, G. W. Lin, K. Xia, X. M. Lin, Y. P. Niu, and S. Q. Gong, “Quantum controlled-phase-flip gate between a flying optical photon and a Rydberg atomic ensemble,” Sci. Rep. 5, 10005 (2015).
[Crossref]

Xia, T.

K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
[Crossref]

Xu, P.

Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
[Crossref]

Yakshina, E. A.

I. I. Beterov, T. Andrijauskas, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, and S. Bergamini, “Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms,” Phys. Rev. A 90, 043413 (2014).
[Crossref]

Zeng, Y.

Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
[Crossref]

Zhan, M.

Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
[Crossref]

Zhang, S.

C. Liu, Y. Sun, L. Zhao, S. Zhang, M. M. T. Loy, and S. Du, “Efficiently loading a single photon into a single-sided Fabry–Perot cavity,” Phys. Rev. Lett. 113, 133601 (2014).
[Crossref]

Zhao, L.

C. Liu, Y. Sun, L. Zhao, S. Zhang, M. M. T. Loy, and S. Du, “Efficiently loading a single photon into a single-sided Fabry–Perot cavity,” Phys. Rev. Lett. 113, 133601 (2014).
[Crossref]

Zimmer, C.

C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, “Single-photon absorber based on strongly interacting Rydberg atoms,” Phys. Rev. Lett. 117, 223001 (2016).
[Crossref]

Zoller, P.

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[Crossref]

J. Phys. B (1)

M. Saffman, “Quantum computing with atomic qubits and Rydberg interactions: progress and challenges,” J. Phys. B 49, 202001 (2016).
[Crossref]

Linear Algebra Appl. (1)

M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra Appl. 10, 285–290 (1975).
[Crossref]

Nat. Photonics (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Nature (6)

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref]

B. Hacker, S. Welte, G. Rempe, and S. Ritter, “A photon-photon quantum gate based on a single atom in an optical resonator,” Nature 536, 193–196 (2016).
[Crossref]

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurášek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432, 482–486 (2004).
[Crossref]

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008).
[Crossref]

H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
[Crossref]

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

New J. Phys. (2)

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

F. Motzoi and K. Mølmer, “Precise single-qubit control of the reflection phase of a photon mediated by a strongly-coupled ancilla-cavity system,” New J. Phys. 20, 053029 (2018).
[Crossref]

Phys. Rev. (1)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[Crossref]

Phys. Rev. A (20)

M. Khazali, K. Heshami, and C. Simon, “Photon-photon gate via the interaction between two collective Rydberg excitations,” Phys. Rev. A 91, 030301 (2015).
[Crossref]

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

A. C. J. Wade, M. Mattioli, and K. Mølmer, “Single-atom single-photon coupling facilitated by atomic-ensemble dark-state mechanisms,” Phys. Rev. A 94, 053830 (2016).
[Crossref]

L. H. Pedersen and K. Mølmer, “Few qubit atom-light interfaces with collective encoding,” Phys. Rev. A 79, 012320 (2009).
[Crossref]

A. E. B. Nielsen and K. Mølmer, “Deterministic multimode photonic device for quantum-information processing,” Phys. Rev. A 81, 043822 (2010).
[Crossref]

I. I. Beterov and M. Saffman, “Rydberg blockade, Förster resonances, and quantum state measurements with different atomic species,” Phys. Rev. A 92, 042710 (2015).
[Crossref]

A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2005).
[Crossref]

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

D. Petrosyan, F. Motzoi, M. Saffman, and K. Mølmer, “High-fidelity Rydberg quantum gate via a two-atom dark state,” Phys. Rev. A 96, 042306 (2017).
[Crossref]

B.-Q. Ou, C. Liu, Y. Sun, and P.-X. Chen, “Deterministically swapping frequency-bin entanglement from photon-photon to atom-photon hybrid systems,” Phys. Rev. A 97, 023839 (2018).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: a perturbative approach,” Phys. Rev. A 92, 043841 (2015).
[Crossref]

J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine, A. Sommer, and J. Simon, “Observation and characterization of cavity Rydberg polaritons,” Phys. Rev. A 93, 041802 (2016).
[Crossref]

J. Lee, M. J. Martin, Y.-Y. Jau, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms,” Phys. Rev. A 95, 041801 (2017).
[Crossref]

C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010).
[Crossref]

R. G. Unanyan and M. Fleischhauer, “Efficient and robust entanglement generation in a many-particle system with resonant dipole-dipole interactions,” Phys. Rev. A 66, 032109 (2002).
[Crossref]

T. C. V. Opatrný and K. Mølmer, “Spin squeezing and Schrödinger-cat-state generation in atomic samples with Rydberg blockade,” Phys. Rev. A 86, 023845 (2012).
[Crossref]

I. I. Beterov, T. Andrijauskas, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, and S. Bergamini, “Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms,” Phys. Rev. A 90, 043413 (2014).
[Crossref]

K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
[Crossref]

C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A 89, 022317 (2014).
[Crossref]

M. Saffman and T. G. Walker, “Creating single-atom and single-photon sources from entangled atomic ensembles,” Phys. Rev. A 66, 065403 (2002).
[Crossref]

Phys. Rev. Lett. (21)

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[Crossref]

L.-M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004).
[Crossref]

C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, “Single-photon absorber based on strongly interacting Rydberg atoms,” Phys. Rev. Lett. 117, 223001 (2016).
[Crossref]

Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
[Crossref]

M. Ebert, M. Kwon, T. G. Walker, and M. Saffman, “Coherence and Rydberg blockade of atomic ensemble qubits,” Phys. Rev. Lett. 115, 093601 (2015).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, “Arbitrary Dicke-state control of symmetric Rydberg ensembles,” Phys. Rev. Lett. 117, 213601 (2016).
[Crossref]

D. Møller, L. B. Madsen, and K. Mølmer, “Quantum gates and multiparticle entanglement by Rydberg excitation blockade and adiabatic passage,” Phys. Rev. Lett. 100, 170504 (2008).
[Crossref]

E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014).
[Crossref]

A. T. Black, J. K. Thompson, and V. Vuletić, “On-demand superradiant conversion of atomic spin gratings into single photons with high efficiency,” Phys. Rev. Lett. 95, 133601 (2005).
[Crossref]

J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008).
[Crossref]

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[Crossref]

O. Lahad and O. Firstenberg, “Induced cavities for photonic quantum gates,” Phys. Rev. Lett. 119, 113601 (2017).
[Crossref]

M. O. Scully, “Collective Lamb shift in single photon Dicke superradiance,” Phys. Rev. Lett. 102, 143601 (2009).
[Crossref]

D. Paredes-Barato and C. S. Adams, “All-optical quantum information processing using Rydberg gates,” Phys. Rev. Lett. 112, 040501 (2014).
[Crossref]

C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018).
[Crossref]

C. Liu, Y. Sun, L. Zhao, S. Zhang, M. M. T. Loy, and S. Du, “Efficiently loading a single photon into a single-sided Fabry–Perot cavity,” Phys. Rev. Lett. 113, 133601 (2014).
[Crossref]

Phys. Rev. X (1)

C. R. Murray and T. Pohl, “Coherent photon manipulation in interacting atomic ensembles,” Phys. Rev. X 7, 031007 (2017).
[Crossref]

Proc. IEEE (1)

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
[Crossref]

Rep. Math. Phys. (1)

A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,” Rep. Math. Phys. 3, 275–278 (1972).
[Crossref]

Rev. Mod. Phys. (6)

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
[Crossref]

A. Galindo and M. A. Martín-Delgado, “Information and computation: classical and quantum aspects,” Rev. Mod. Phys. 74, 347–423 (2002).
[Crossref]

A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015).
[Crossref]

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010).
[Crossref]

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011).
[Crossref]

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
[Crossref]

Sci. Rep. (1)

Y. M. Hao, G. W. Lin, K. Xia, X. M. Lin, Y. P. Niu, and S. Q. Gong, “Quantum controlled-phase-flip gate between a flying optical photon and a Rydberg atomic ensemble,” Sci. Rep. 5, 10005 (2015).
[Crossref]

Science (3)

Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, “Single-qubit gates based on targeted phase shifts in a 3D neutral atom array,” Science 352, 1562–1565 (2016).
[Crossref]

Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336, 887–889 (2012).
[Crossref]

A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349–1351 (2013).
[Crossref]

Supplementary Material (1)

NameDescription
» Supplement 1       supplemental material

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1. Schematic of the intracavity Rydberg-blocked atomic ensemble system under investigation. The entire ensemble has a size of that is on the order of a few tens of micrometers, which is compatible with the experimentally attainable Rydberg-blockade radius. Moreover, in such a configuration, the atomic ensemble well matches the cavity mode spatially, which conveniently serves the purpose of atom–photon coupling. The control laser is of frequency ω c , while the cavity resonance frequency is ω d .
Fig. 2.
Fig. 2. Schematic of the system for the study of the JCM with an intracavity Rydberg-blocked atomic ensemble, where the state initialization includes the process of feeding a nonclassical optical pulse into the cavity. The optical cavity is supposed to be single-sided, where the one end mirror is perfectly reflecting. In this particular example, the feeding and retrieving of the intracavity optical field is realized by polarization optics for the free-space optical pulse (PBS: polarizing beam splitter; QWP: quarter-wave plate). Typical parameter settings for the experimental implementation are relatively mild for the hardware nowadays. For example, the cavity finesse can be set as F 5 × 10 3 , the cavity decay rate can be chosen as κ 2 π × 0.5    MHz , the cavity free spectral range can be set as FSR 5    GHz , and the number of atoms in the ensemble can be chosen as 100 1000 . The instantiation of the Jaynes–Cummings model in such a system is relatively straightforward compared with the case of a high-finesse cavity, thanks to recent developments in single-photon pulse engineering and Rydberg atom control techniques.
Fig. 3.
Fig. 3. Numerical simulation of the quantum Rabi oscillations of the intracavity atom–photon interaction, single-photon–state case. Parameters are set as Δ e = 2 π × 200    MHz , Δ r = 0 , Γ e = 2 π × 1    MHz , Γ r = 2 π × 0.01    MHz , κ = 2 π × 0.5    MHz . For (a), Ω = 2 π × 20    MHz , G = 2 π × 10    MHz ; for (b), Ω = 2 π × 5    MHz , G = 2 π × 2.5    MHz .
Fig. 4.
Fig. 4. Numerical simulation of the quantum Rabi oscillations of the intracavity atom–photon interaction, coherent-state version. The artificial condition of almost no cavity decay is imposed to show the quantum revival, while the atomic decays are retained in the calculations. Parameters are set as G = 2 π × 2.5    MHz , Ω = 2 π × 20    MHz , Δ e = 2 π × 100    MHz , Δ r = 0 , Γ e = 2 π × 1    MHz , Γ r = 2 π × 0.01    MHz , κ = 2 π × 10 4    MHz . The initial condition is | α | = 4 for the optical coherent state, | α , while the cutoff is set at 50. It is averaged over 10,000 MCWF traces.
Fig. 5.
Fig. 5. Numerical simulation of the quantum Rabi oscillations of the intracavity atom–photon interaction, coherent-state version. A moderate cavity decay into the free-space environment is considered here. Parameters are set as G = 2 π × 2.5    MHz , Ω = 2 π × 20    MHz , Δ e = 2 π × 100    MHz , Δ r = 0 , Γ e = 2 π × 1    MHz , Γ r = 2 π × 0.01    MHz , κ = 2 π × 0.1    MHz . The initial condition is | α | = 4 for the optical coherent state, | α , while the cutoff is set at 50. It is averaged over 10,000 MCWF traces.
Fig. 6.
Fig. 6. Outline for the basic principles of the atom–photon gate with an intracavity Rydberg-blocked atomic ensemble, where the qubit state on the atom side is abstracted into the internal states of a single atom within the ensemble. The optical cavity is supposed to be single-sided, where the one end mirror is perfectly reflecting. The frequency of the incident single-photon pulse is resonant with the optical cavity, but not necessarily so with the atomic transition | g | e . Here, the matter qubit is instantiated in the form of a single atom within the atomic ensemble. In such a configuration, a strong Rydberg blockade is presumed to take place between state | r of the qubit atom and state | r for the rest atoms of the ensemble, as a consequence of the Förster resonance structure in Eq. (11). Nevertheless, the single qubit atom does not have to be the same species as the other atoms in the ensemble [26,50]. In principle, this gate protocol does not induce mechanical forces between atoms since the underlying mechanism belongs to the category of a Rydberg-blockade gate [28].
Fig. 7.
Fig. 7. Numerical simulation of the gate’s fidelities with respect to different single-atom single-photon coupling strengths, G 0 . Particular parameters for this simulation include Ω = 2 π × 100    MHz , Δ e = 2 π × 1000    MHz , Γ e = 2 π × 1    MHz , Γ r = Γ p = 2 π × 0.01    MHz , κ = 2 π × 1    MHz . While all other parameters are not changed, continuing to increase the strength of G won’t unlimitedly enhance the fidelity. This is because, as G increases beyond certain point, it will cause a power broadening effect, which harms the desired conditional phase shift. This can also be observed quantitatively from Eqs. (14) and (15).
Fig. 8.
Fig. 8. Numerical simulation of the gate’s fidelities to demonstrate its tuning range in Δ e , with respect to different Rabi frequency settings of the control laser. Particular parameters for this simulation includes G 0 = 2 π × 0.5    MHz , Γ r = Γ p = 2 π × 0.01    MHz , κ = 2 π × 1    MHz . The practical strategy of choosing the value of Δ e will involve considerations to ensure that | Δ e | Γ e and | Δ e | G .

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

H int = n = 1 N ( Ω 2 | r n e n | i G n | e n g n | b ^ ) + H.c. Δ e n = 1 N | e n e n | Δ r n = 1 N | r n r n | + n = 1 N m > n V n m | r n r n | | r m r m | ,
H int = ( Ω 2 | r ˜ e ˜ | i G | e ˜ g N | b ^ ) + H.c. Δ e | e ˜ e ˜ | Δ r | r ˜ r ˜ | ,
| e ˜ = G 1 G n | e n , | r ˜ = G 1 G n | r n ,
H eff = Ω G 2 Δ e | r ˜ g N | + H.c. + G 2 Δ e | g N g N | b ^ b ^ ( Δ r Ω 2 4 Δ e ) | r ˜ r ˜ | ,
| Ψ ( t ) = d ω ϕ s ( ω , t ) a ^ s ( ω ) | g N , Ø b , Ø a + C b ( t ) b ^ | g N , Ø b , Ø a + m = 1 N C e m ( t ) | g N 1 e m , Ø b , Ø a + m = 1 N C r m ( t ) | g N 1 r m , Ø b , Ø a , a ,
C e = G 1 G n * C e n , C r = G 1 G n * C r n ,
H io = i d ω g s ( ω ) ( a ^ s ( ω ) b ^ b ^ a ^ s ( ω ) ) .
d d t C b ( t ) = G C e κ 2 C b ( t ) ,
d d t C e ( t ) = G C b ( t ) + i Ω * 2 C r ( t ) + i Δ e C e ( t ) Γ e 2 C e ( t ) ,
d d t C r ( t ) = i Ω 2 C e ( t ) + i Δ r C r ( t ) Γ r 2 C r ( t ) ,
d d t C b ( t ) = i G 2 Δ e + i Γ e 2 C b ( t ) G Ω * 2 ( Δ e + i Γ e 2 ) C r ( t ) κ 2 C b ( t ) , d d t C r ( t ) = G Ω 2 ( Δ e + i Γ e 2 ) C b ( t ) i | Ω | 2 4 ( Δ e + i Γ e 2 ) C r ( t ) + i Δ r C r ( t ) .
d d t C b , n ( t ) = n G C e , n 1 ( t ) ,
d d t C e , n 1 ( t ) = n G C b , n ( t ) + i Ω * 2 C r , n 1 ( t ) + i Δ e C e , n 1 ( t ) ,
d d t C r , n 1 ( t ) = i Ω 2 C e , n 1 ( t ) + i Δ r C r , n 1 ( t ) ,
H p 1 = n = 1 N ( Ω 2 | r n e n | i G n | e n g n | b ^ ) + H.c Δ e n = 1 N | e n e n | Δ r n = 1 N | r n r n | + n = 1 N V n | r n p n | | r p | + H.c. + δ p n = 1 N | p n p n | | p p | ,
| Ψ ( t ) = d ω φ s ( ω , t ) a ^ s ( ω ) | g N , r , Ø b , Ø a + C b ( t ) b ^ | g N , r , Ø b , Ø a + m = 1 N C e m ( t ) | g N 1 , e m , r , Ø b , Ø a + m = 1 N C r m ( t ) | g N 1 r m , r , Ø b , Ø a + m = 1 N C p m ( t ) | g N 1 p m , p , Ø b , Ø a .
Δ a c = G 2 Δ e + δ + i Γ e 2 , η = 1 4 | Ω | 2 ( i Δ e Γ e 2 + i δ ) 2 , Δ d r = Δ r i 4 | Ω | 2 Γ e 2 i Δ e i δ , B m = | V m | 2 Γ p 2 + i δ p i δ .
R ( δ ) = 1 κ { κ 2 i δ i Δ a c η m = 1 N | G m | 2 Γ r 2 i δ + i Δ d r + B m } 1 .
R ( δ ) = 1 κ { κ 2 i δ i Δ a c η G 2 Γ r 2 i δ + i Δ d r } 1 .
F z = 1 16 | 2 + R R | 2 .

Metrics