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
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  and quantum metrology . 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 , where the second quantization of the intracavity optical field is essential. Meanwhile, the atomic ensemble approach  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 , the quantum nondemolition measurement of single photon pulses , quantum memory for light [9,10], the quantum repeater for photonic polarization entanglement , and quantum networking between matter qubits [12–14], to name a few. Among many potential technological breakthroughs, the phenomenon of Rydberg blockade  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  has already resulted in many key advances in quantum optics with neutral atoms and single-photon pulses, both theoretically and experimentally [17–26]. 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) , where the field part of the JCM is instantiated either by cavity optical modes  or via isomorphism to collective atomic excitations [31–35]. 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 . Optical nonlinearities have already been experimentally observed in a cold atomic ensemble with strong Rydberg–Rydberg interactions inside a moderate finesse cavity , 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 , 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  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 , 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) [45–47]. 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 [48–51]. 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  or 3D arrays . 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  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].
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:
In Eq. (1), the Rydberg–Rydberg interaction is effectively described by the potential, , 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 , 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 is sufficiently large. Therefore, the Hamiltonian can be reduced to the form of
For the situation of much larger than and 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 . This leads to the simple linear relation, . 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,
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.
The initial condition can be realized by efficiently and deterministically loading a prescribed single-photon optical pulse into a cavity . 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 . That is to say, the cavity seems to be empty for the incoming optical field when . 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
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:
The coupling through the cavity mirror between the intracavity mode and the free-space mode can be described by the interaction Hamiltonian: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
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 .
Under the condition of a relatively large , 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 , which leads to the following equations from Eq. (8):
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, , specified in Eq. (6) to include the multiphoton case so that we can set , , to denote the coefficients with respect to the Fock-state basis, , on the photonic side. Then, the time evolution without considering any decay is given by the equations below for :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, , and the decay of intracavity optical field, (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 is hard to achieve experimentally. For a reasonably large value of , 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, . Roughly speaking, the re-emission from the ensemble’s collective excitation into the cavity mode may be recognized as a form of superradiance . 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.
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 [48–50] 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 and . When the matter qubit is sitting at the state , it does not exert any substantial impact on the other atoms, and the two-photon transition from to 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 . When the matter qubit is sitting at the state , 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 , depending on whether the matter qubit state is or .
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
From a realistic point of view, the values of and 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 and . For example, under the specific condition of the typical Autler–Townes effect, it is fair to set .
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: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, is used to label the frequency components of the incident single-photon pulse, where 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, ; the dressing factor to the atom–photon coupling strength, ; the effective relative frequency shift for coupling to the dressed states, ; and the effective Rydberg-blockade shift, , where the subscript is the index for the atomSupplement 1 for more details). When the qubit atom is sitting at the state in the cavity, the reflection coefficient is 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. ).
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 differs from state , 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 .
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, . 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 , 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. , 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 [48–50,63–65] of the proposed gate operation. According to the arguments made in Refs. [49,50], the average fidelity for the atom–photon quantum gate is55]. 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.
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 is the same as and , while discounting the ac Stark shifts. The two dressed states of the ensemble atoms are and , 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 is a sizable fraction of , not necessarily 2 or 1. This is tied to the relative strength of the state 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, , 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, , 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 . 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 and a Rydberg state . 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.
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.
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.
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