Abstract

The technique of shortcuts to adiabaticity (STA) has attracted broad attention due to their possible applications in quantum information processing and quantum control. However, most studies published so far have been only focused on Hermitian systems under the rotating-wave approximation (RWA). In this paper, we propose a modified shortcuts to adiabaticity technique to realize population transfer for a non-Hermitian system without RWA. We work out an exact expression for the control function and present examples consisting of two-and three-level systems with decay to show the theory. The results suggest that the shortcuts to adiabaticity technique presented here is robust for fast passages. We also find that the decay has small effect on the population transfer in the three-level system. To shed more light on the physics behind this result, we reduce the quantum three-level system to an effective two-level one with large detunings. The shortcuts to adiabaticity technique of effective two-level system is studied. Thereby the high-fidelity population transfer can be implemented in non-Hermitian systems by our method, and it works even without RWA.

© 2017 Optical Society of America

1. Introduction

Population transfer by adiabatic evolution is among the most popular methods for coherent atomic manipulation [1–10]. However, due to the requirement on adiabaticity, the manipulation is very slow. Recently, by designing nonadiabatic shortcuts to speed up quantum adiabatic process, a new technique named “shortcuts to adiabaticity” (STA) [11–22] opens a new door towards to fast and robust quantum state manipulations. The shortcut techniques include counterdiabatic control protocols, transitionless quantum driving(TQD) [23–32],“fast-forward” scaling [33], and inverse engineering based on Lewis-Riesenfeld (LR) invariants [34,35]. Among the methods of shortcuts to adiabaticity, transitionless quantum driving have been intensively studied. The key idea of transitionless quantum driving is to use a shortcut by adding extra fields to cancel the nonadiabatic couplings.

The shortcuts to adiabaticity technique has proved to be successful in designing control scheme for closed quantum systems. In practice, however, interactions between a quantum system and its surroundings is not avoidable, leading to coherence loss of the closed systems. Such a system can be described by a non-Hermitian Hamiltonian, and in the past few decades, it has drawn much attention [36], especially in the context of PT-symmetric systems [37–40]. There are many amazing phenomena predicted in PT-symmetric systems, for instance, a PT-symmetric Hamiltonian possesses a higher quantum speed efficiency than Hermitian Hamiltonian systems [41,42]. Recently, an approximation of the adiabatic condition for non-Hermitian systems were derived [43]. For non-Hermitian systems the usual approximations and criteria are not valid in general, so results that are applicable for Hermitian systems have to be reconsidered and modified. The extension of shortcuts to adiabaticity from Hermitian systems to non-Hermitian systems has been put forward for the Landau-Zener(LZ) model [44]. Therefore, we may wonder if there is a general shortcuts to adiabaticity technique for non-Hermitian PT-Hamiltonian systems? whether the adiabatic evolution proposed in Ref. [43] can be accelerated for such systems?

On the other hand, the rotating wave approximation (RWA) is in common used in atomic optics and magnetic resonance [45–47], where the counter-rotating (CR) terms are neglected. The RWA is valid in the weak coupling regime with small detuning and weak field amplitude, in this case the contribution of the counter-rotating terms to the evolution of the system is quite small. In recent experiments, efforts are made to reach a promising method for studying strong-and ultrastrong-coupling physics [48–53], where the RWA is no longer valid. Refs. [54, 55] found that the RWA may lead to faulty results. Especially, recent developments in physical implementation lead to strong coupling between qubit and cavity modes [56,57], which requires a careful consideration of the effect of counter-rotating terms. This motivate us to ask: if can we design a shortcuts to adiabaticity technique for an open system with the counter-rotating terms?

In this paper, we propose a renewed shortcuts to adiabaticity technique for non-Hermitian quantum systems beyond RWA. By using transitionless quantum driving method, we determine the exact control to speed up the adiabatic population transfer. Then, we apply it into the two-and three-level systems with decay and without RWA. We numerically calculate the population transfer dynamics with and without counter-rotating terms. The results show that the population transfer with the counter-rotating terms are more steady. On the other hand, we also find that the decay of the excited state has little effect on the population transfer in the three-level system. This is attributed to the fact that the population transfer |1〉 → |3〉 is realized by the dark state. In the case of large detuning, we reduce the three-level system to an effective two-level system. The shortcuts to adiabaticity technique is then applied to the effective two-level system, which might shed more light on the present scheme.

This paper is organized as follows. In Sec. 2, we study the shortcuts to adiabaticity applied to a decaying [58–60] two-level system without RWA. We numerically compare the population transfer with and without counter-rotating terms. In Sec. 3, we study the shortcuts to adiabaticity applied to a decaying [61, 62] three-level system without RWA. We numerically calculate the population transfer with and without counter-rotating terms. The effect of the decay on the population transfer in the three-level system is also discussed in this section. Sec. 4 is devoted to discussion and conclusion.

2. Transitionless driving applied to a decaying two-level atom beyond the RWA

In this section, we present a shortcuts to adiabaticity technique for an open two-level quantum system beyond RWA. Assume the two-level system interacts with a laser electric field with linear polarization in x direction E⃗(t) = E0(t) cos[ωLt]x⃗, the Hamiltonian of the two-level system reads,

HS(t)=2[ΩR(t)(|21|+|12|)(eiωLt+eiωLt)+ω0(t)(|22||11|)],
where |1〉 (|2〉) is the ground (excited) state of the two-level atom, ω0 is the transition frequency, ΩR(t) and ωL are the strength and the frequency of the classical field. Defining
HL(t)=iU˙(t)U(t)=ωL2(|22||11|),
with unitary operator
U(t)=ei0tHL(t)dt=eiωLt/2|22|eiωLt/2|11|,
we can transform the Hamiltonian (1) into the interaction picture, i.e.,
H(t)=U(t)[HS(t)HL(t)]U(t)=2[Δ(t)Ω(t)Ω(t)Δ(t)].
Here the bases are |1〉 = [1, 0]T and |2〉 = [0, 1]T with T denoting transposition.

With

Ω(t)=ΩR(t)(1+e2iωLt),
and
Δ(t)=ω0(t)ωL,
in which e±2Lt are the counter-rotating terms. To describe the decay of the upper level, we add an imaginary energy iΓ to the energy of the excited state. The Hamiltonian then becomes non-Hermitian,
H˜0(t)=2[Δ(t)Ω(t)Ω(t)Δ(t)iΓ].
The dynamics of the two-level quantum system is described by the Schrödinger equation,
it[C1(t)C2(t)]=H˜0(t)[C1(t)C2(t)],
where C1(t) and C2(t) are probability amplitudes of the two bare states |1〉 and |2〉, respectively.

To design a shortcut to adiabaticity for this non-Hermitian system without RWA, we have to explore the left and right eigenstates of the non-Hermitian Hamiltonian (7). Straightforward calculations lead to left eigenstates of 0(t) as

|φ+(t)=sinβ|1+cosβeiωLt|2,|φ(t)=cosβeiωLt|1sinβ|2,
where the mixing angle β = β(t) is complex and defined as
β=12arctan2ΩRcosωLtΔiΓ/2,
and the right eigenstates can also be determined,
|φ^+(t)=sinβ*|1+cosβ*eiωLt|2,|φ^(t)=cosβ*eiωLt|1sinβ*|2,
where the asterisk means “complex conjugate”.

By the spirit of the shortcut to adiabaticity [23–30,32], the counterdiabatic term can be given by [63]

H˜CD(t)=i[|tφ+(t)φ^+(t)|φ^+(t)|tφ+(t)|φ+(t)φ^+(t)|+|tφ(t)φ^(t)|φ^(t)|tφ(t)|φ(t)φ^(t)|],
where
φ^±(t)|tφ±(t)=±iωLcos2β2,φ^(t)|tφ±(t)=(±β˙2iωL2sinβ)e±iωLt,
which is equivalent to
H˜CD(t)=[ωL2sin2β(iβ˙2+ωL4sin2β)eiωLt(iβ˙2+ωL4sin2β)eiωLtωL2sin2β].
Here, we should stress that the counterdiabatic terms presented in Eq. (14) is beyond RWA. However, we can easily obtain the counterdiabatic term for the RWA case by ignoring the counter-rotating terms in H0(t). Applying the rotating wave approximation to get rid of the counter-rotating terms, we end up with
H0(t)=2[Δ(t)ΩR(t)ΩR(t)Δ(t)iΓ].

By the same procedure as above, HCD reads

HCD(t)=[0Ωa(t)Ωa(t)0],
where Ωa(t) = iθ̇/2 is the auxiliary pulse with
θ˙=Ω˙R[Δ(t)iΓ/2]ΩR(t)(Δ˙iΓ˙/2)ΩR2(t)+[Δ(t)iΓ/2]2,
which plays the role of the Rabi frequency for a fast-driving field. In principle, HCD(t) can drives the dynamics along the instantaneous eigenstates of H0(t) in arbitrarily short time, but due to the practical limitations, e.g., the laser power, the performance would depend on those limitations.

The Hamiltonian (16) has off-diagonal terms with real and imaginary parts, see Fig. 1. Practically, realizing the Hamiltonian (16) is not straightforward. From the Hamiltonian (16), we can see that the off-diagonal terms are not complex conjugate to each other except that the real part of Ωa is zero. The imaginary parts, shown by the dip in Fig. 1(a), can be realized by a complementary laser with orthogonal polarization [12], the real parts constitute a non-Hermitian contribution to the Hamiltonian. They are, however, contribute very small to the dynamics. Ignoring the real part of Ωa, i.e., setting ΩaiIm[Ωa], we plot the Figs. 1(b) and 1(c). We can see that the population transfer is almost the same. So the Hamiltonian (16) can be realized approximately. When the real parts can not be ignored, the realization mentioned above is not valide anymore.

 

Fig. 1 Real (red solid line) and imaginary (blue dashed line) parts of Ωa (a). The population evolution is driven by Ωa (b), and [ImΩa] (c). The parameters chosen are Γ = 2π × 0.5Mhz, Ω0 = 2π × 5Mhz, δ = 2π × 300Mhz, tf = 2 × 10−10s, t0 = 10−10s.

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In the RWA regime, i.e., drop the terms with e ±2Lt= 0. It can be verified that Eq. (14) is the same as Eq. (16). Figure 2 shows the population of the ground state |1〉 (red dash lines) and the excited state |2〉 (blue solid lines) with and without RWA. In our case, we use the shortcuts to adiabaticity technology to speed up the passage, in which pulses do not need satisfy adiabaticity condition. It requires an additional laser field. Therefore we provide an example in which the adiabaticity condition fails and then apply the Hamiltonian in Eq. (16) to remedy this problem and achieve a fast full decay. Here, the coherent controls in the Hamiltonians 0(t) and H0(t) are chosen to be the Allen-Eberly (AE) drivings [64], i.e.,

ΩR(t)=Ω0sech[π(ttf/2)2t0],Δ(t)=2δ2t0πtanh[π(ttf/2)2t0].
Here, we select the appropriate parameters [43,63], in the RWA regime, we use Hamiltonian H0(t) and the counterdiabatic Hamiltonian HCD(t) to describe the two-level system (see Fig. 2(a)). We find that the population is transferred from the ground state into the excited state completely. However, when the RWA fails, counter-rotating terms must be taken into account, HCD(t) can not be used as the counterdiabatic Hamiltonian to accelerate the adiabatic dynamics governed by the reference Hamiltonian 0(t), see Fig. 2(b). Therefore, in the regime beyond RWA, we have to redesign the counterdiabatic terms of the shortcuts to adiabaticity according to Eq. (14). As shown in Fig. 2(c), if we apply Hamiltonian (14) into the dynamics governed by Hamiltonian 0(t), the population transfer efficiency can be enhanced remarkably. Our results suggest that shortcuts to adiabaticity technique well developed for Hermitian systems can be extended to non-Hermitian one. It is envisaged that our study could stimulate further studies of coherent population transfer techniques of multilevel systems for non-Hermitian systems. In the next section, we mainly discuss three-level non-Hermitian system.

 

Fig. 2 The evolution of the population for a ndecaying two-level system with decay rate Γ = 2π × 0.5MHz. The population transfer is implemented by the shortcuts to adiabaticity technique under RWA [Figs. 2(a), and 2(b)], and implemented by the shortcuts to adiabaticity technique beyond RWA [Fig. 2(c)]. Note that all the counter-rotating terms are neglected in Fig. 2(a), but are considered in Fig. 2(b). The population of level |1〉 (dashed red), and |2〉 (solid blue) are shown. The other parameters: Ω0 = 2π × 5MHz, δ = 2π × 300MHz, tf = 2 × 10−10s, t0 = 10−10s, ωL = 2π × 10GHz.

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3. Transitionless driving scheme applied to a decaying three-level atom beyond RWA

3.1. HCD(t) applied to a decaying three-level atom

In this section, we consider a three-level system of Λ-type with decay in the excited state. The two lower states are long-lived. We consider the state |1〉 is initially populated, and the other state |3〉 as the target state, and |2〉 as the excited state. Levels |1〉 and |2〉 are coupled by the Pump laser Ω̄p(t), while Levels |2〉 and |3〉 are coupled by the Stokes laser Ω̄s(t). We assume that there is a decay in the excited state |2〉 with rate Γ. Such a system can be described by the following time-dependent Schrödinger equation,

iddtc(t)=H¯0(t)c(t),
where the vector c(t) = [c1(t), c2(t), c3(t)]T is constructed by the three probability amplitudes of states |1〉, |2〉, and |3〉. Without the rotating wave approximation, the reference Hamiltonian of the Λ-type three-level system can be depicted as [61,62,65]
H¯0(t)=2[0Ω¯p(t)0Ω¯p*(t)2ΔpiΓΩ¯s(t)0Ω¯s*(t)2(ΔpΔs)],
where the detunings are defined by Δp = ωp − (E2E1)/ħ, Δs = ωs − (E2E3)/ħ for the Λ configuration. Ω̄p(t) = Ωp(t)(1+e−2pt) and Ω̄s(t) = Ωs(t)(1+e−2st) are the pump and Stokes pulses (beyond RWA). The imaginary term Γ describes the decay from |2〉. Conventional stimulated Raman adiabatic passage (STIRAP) relies on the two-photon resonance, i.e., Δ = Δp = Δs.

The mechanism of the population transfer in STIRAP and the limitation of adiabaticity can be easily understood by introducing the so-called adiabatic states. They are connected to the bare states |1〉, |2〉 and |3〉 by the eigenstates of 0(t),

|E0(t)=1Ξ0(t)[Ω¯s(t)|1Ω¯p*(t)|3],|E+(t)=1Ξ1(t)[Ω¯p(t)|1+ε+(t)|2+Ω¯s*(t)|3],|E(t)=1Ξ2(t)[Ω¯p(t)|1+ε(t)|2+Ω¯s*(t)|3],
where Ξ0(t)=|Ω¯p(t)|2+|Ω¯s(t)|2, Ξ1(t)=ε+2(t)+Ξ02(t), and Ξ2(t)=ε2(t)+Ξ02(t). The eigenvalues of 0(t) corresponding to the eigenstates are,
E0(t)=0,E±(t)=2ε±(t),
with
ε±(t)=(ΔiΓ/2)±(ΔiΓ/2)2+Ξ02(t).
In the STIRAP, the |E0(t)〉 eigenstate is a dark state without population on state |2〉. The protocol allows us to produce an effcient transfer from |1〉 to |3〉 following the evolution of the dark state |E0(t)〉. Since 0(t) is non-Hermitian, the instantaneous eigenstates of the Hamiltonian H¯0(t) have to be considered, which reads
|E^0(t)=1Ξ0(t)[Ω¯s(t)|1Ω¯p*(t)|3],|E^+(t)=1Ξ1(t)[Ω¯p(t)|1+ε^+(t)|2+Ω¯s*(t)|3],|E^(t)=1Ξ2(t)[Ω¯p(t)|1+ε^(t)|2+Ω¯s*(t)|3],
with ε^±(t)=(Δ+iΓ/2)±(Δ+iΓ/2)2+Ξ02(t).

In order to realize the shortcuts to adiabaticity beyond RWA, we first write out three orthogonal projection operators corresponding to the relevant instantaneous eigenstates:

Π0(t)=|E0(t)E^0(t)|=1Ξ02(t)[|Ω¯s(t)|20Ω¯p(t)Ω¯s(t)000Ω¯p*(t)Ω¯s*(t)0|Ω¯p(t)|2],Π1(t)=|E1(t)E^1(t)|=1Ξ12(t)[|Ω¯p(t)|2Ω¯p(t)ε+(t)Ω¯p(t)Ω¯s(t)Ω¯p*(t)ε+(t)ε+2(t)Ω¯s(t)ε+(t)Ω¯p*(t)Ω¯s*(t)Ω¯s(t)ε+(t)|Ω¯s(t)|2],Π2(t)=|E2(t)E^2(t)|=1Ξ22(t)[|Ω¯p(t)|2Ω¯p(t)ε(t)Ω¯p(t)Ω¯s(t)Ω¯p*(t)ε(t)ε2(t)Ω¯s(t)ε(t)Ω¯p*(t)Ω¯s*(t)Ω¯s(t)ε(t)|Ω¯s(t)|2].
By the same procedure, the Hermitian CD Hamiltonian becomes
H¯CD=ijkΠj(t)tH¯0(t)Πk(t)Ek(t)Ej(t)=iΞ12(t)Ξ22(t)[|Ω¯p(t)|2B+CDΩ¯p(t)A2iΓΩ¯s(t)GΩ¯p(t)Ω¯s(t)B+CFΩ¯p*(t)A*2iΓΩ¯s*(t)G*Ξ02(t)BΩ¯s(t)A*2iΓΩ¯p(t)G*Ω¯p*(t)Ω¯s*(t)B*CF*Ω¯s*(t)A+2iΓΩ¯p*(t)G|Ω¯s|2BCD],
with
A=(2ΔiΓ)[Ω¯p*(t)Ω¯˙p(t)+Ω¯s(t)Ω¯˙s*(t)],B=Ω¯p*(t)Ω¯˙p*(t)+Ω¯s(t)Ω¯˙s*(t)H.C.,C=[|ε+(t)|2+|ε(t)|2+2Ξ02(t)]/Ξ02(t),D=|Ω¯p(t)|2[Ω¯s*(t)Ω¯˙s*(t)H.C.]+|Ω¯s(t)|2[(Ω¯p*(t)Ω¯˙p(t)H.C.)],F=Ω¯p2(t)[Ω¯s(t)Ω¯p*(t)Ω¯p*(t)Ω¯˙s*(t)]+Ω¯s2(t)[Ω¯s*(t)Ω¯˙p(t)Ω¯p(t)Ω¯˙s*(t)],G=Ω¯s*(t)Ω¯˙p(t)Ω¯p(t)Ω¯˙s*(t).

Correspondently, we give the auxiliary driving for the shortcuts to adiabaticity under RWA

HCD(t)=i[0A(t)C(t)A(t)0B(t)C(t)B(t)0],
where
A(t)=sinθ(t)ϕ˙(t),B(t)=cosθ(t)ϕ˙(t),C(t)=θ˙(t),
with ϕ˙(t)=Ω˙(t)(ΔiΓ/2)Ω(t)[Δ˙iΓ˙/2]2[Ω2(t)+(ΔpiΓ/2)2], θ˙(t)=Ω˙p(t)Ωs(t)Ω˙s(t)Ωp(t)Ω2(t), and Ω(t)=Ωp2+Ωs2, respectively.

We would like to address that when the transitions |1〉 ↔ |2〉 and |2〉 ↔ |3〉 are allowed, the transition |1〉 ↔ |3〉 is in general forbidden. But for artificial atom [66–70], the forbiddenness of transition is lifted.

The Pump pulse and Stokes pulse in our numerical analysis can be described by Ωp(t)=Ω0exp[(tτtf/2T)2] and Ωs(t)=Ω0exp[(t+τtf/2T)2]. The Gaussion pulses have the same shapes and strengths but separated by a delay of 2τ. The entire interaction time of the system evolution is tf. The constant loss rate Γ is a constant. The initial conditions for the populations are P1(0) = 1, P2(0) = 0 and P3(0) = 0. In Fig. 3(a) and 3(b), we can find that the shortcuts to adiabaticity technique under the RWA does not works well if considering the counter-rotating effects, but beyond the RWA is not.

 

Fig. 3 Transition probability P1→3 of the fast stimulated Raman by the shortcuts to adiabaticity under and beyond RWA. Fig. 3(a) is plotted by using the shortcuts to adiabaticity technique under RWA, but consider the CR terms. Fig. 3(b) is plotted by using the shortcuts to adiabaticity technique beyond RWA. We set the parameters: tf = 30ns, T = tf/6, τ = tf/10, Ω0 = 2π×200MHz, Δ = 2π×200MHz, Γ = 2π×100MHz, ωp = 100MHz, ωs = 80MHz. The dashed red, dotted green, and solid blue lines describe the time-dependent population of the state |1〉, |2〉, and |3〉, respectively.

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We use the fidelity between the finial state |Ψ(tf)〉 and the target state |3〉 to characterize the population transfer efficiency (see Fig. 4), i.e., F = |〈3 |Ψ(tf)〉|2. F = |〈3| Ψ(tf)〉|2. Figs. 4(a) and 4(c) show the fidelity beyond RWA, where we use HCD(t) (Eq. (28)) as the countdiabatic Hamiltonian. Figs. 4(b) and 4(d) are also for the results beyond RWA, but CD (Eq. (26)) is chosen as the counterdiabatic Hamiltonian. The parameters used in the numerical simulation are chosen as follows: In Figs. 4(a) and 4(b), we choose Ω0 = 2π × [100, 800]MHz and Γ = 2π × [100, 600]MHz, respectively. In Figs. 4(c) and 4(d), the parameters are Δ = 2π × [100, 800]MHz and Γ = 2π × [100, 600]MHz, respectively. The final evolution time is 30ns. The numerical results show that the shortcuts to adiabaticity technique with the counterdiabatic Hamiltonian HCD(t) does not works well if the counter-rotating terms are taken into account (see Figs. 4(a) and 4(c)). The fidelity is sensitive to the decay rate and pulse parameters. The required fidelities can be reached only in narrow range of parameters. This is due to the matching-lost between the adiabatic reference Hamiltonian 0(t) and the counterdiabatic Hamiltonian HCD(t). To better the performance, we take CD(t) (Eq. (26)) as the counterdiabatic Hamiltonian to replace HCD(t), which are illustrated in Figs. 4(b) and 4(d). When the reference Hamiltonian and the counterdiabatic Hamiltonian are matching, the fidelity is not affected by any parameters in the coherent control field, and it is robust against the decay of the excited state.

 

Fig. 4 Fidelity as a function of Γ, Ω0 and Δ. The fidelity is defined as the population on the state |3〉 with initial state |1〉. The population transfer is realized by using the STA technique under RWA but with counter-rotating terms [Figs. 4(a) and 4(c)], and without RWA [Figs. 4(b) and 4(d)]. In Figs. 4(a) and 4(b), the parameters chosen are, tf = 30ns, Ω0 = 2π × [100, 800]MHz, Γ = 2π × [100, 600]MHz, ωp = 100MHz, ωs = 80MHz. In Figs. 4(c) and 4(d), the parameters are set as: tf = 30ns, Δ = 2π × [100, 800]MHz, Γ = 2π × [100, 600]MHz, ωp = 100MHz, ωs = 80MHz. We can clearly find that the fidelity without RWA is robust against the parameter fluctuations.

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It turned out that the previous shortcuts to adiabaticity technique under RWA does not work well if the counter-rotating terms are included. Therefore, the study of shortcuts to adiabaticity technique beyond RWA is necessary. The robustness to the population loss of the excited state was investigated and it was shown that high fidelity can be achieved even with large decay rates. The decay of the excited state has little effect on the population transfer. This is due to the fact that in the STIRAP the dark states are used, the excited state is almost not population during the evolution of the system. We hope that this work may open a new door towards to the experimental realization of population transfer with decaying quantum system in the near future.

3.2. Effective two-level system for large single-photon detuning

When the detuning Δ(t) ≫ Ωp,s(t), the middle state |2〉 can be eliminated adiabatically by setting ċ2 ≃ 0 in Eq. (19). The three-level system can be then reduced to an effective two-level system (with levels |1〉 and |3〉) [71–73]:

Heff(t)=2[Δeff(t)Ωeff(t)Ωeff*(t)Δeff(t)],
where Δeff(t) and Ωeff(t) are effective detuning and Rabi frequency, respectively,
Δeff(t)=Ω¯p(t)Ω¯p(t)2ΔiΓ,Ωeff(t)=|Ω¯p(t)|2|Ω¯s(t)|24Δ2iΓ.

In the large detuning limit, the three-level system reduces to an effective two-level system described by the Hamiltonian (30). Once the effective two-level Hamiltonian is obtained, we can calculate the counter-adiabatic driving according to [74]

HCD(t)=i2M(t)[0P(t)P*(t)Q(t)],
with
P(t)=Ω˙eff(t)Δeff(t)Δ˙eff(t)Ωeff(t),Q(t)=Ω˙eff*(t)Ωeff(t)Ω˙eff(t)Ωeff*,M(t)=|Ωeff|2+Δeff2.
Thus the total Hamiltonian reads H(t) = Heff(t) + HCD(t). We can directly apply the effective two-level model in the analysis of our three-level system with large single-photon detuning. The Pump pulse and Stokes pulse in our numerical analysis still can be chosen as Ωp(t)=Ω0exp[(tτtf/2T)2] and Ωs(t)=Ω0exp[(t+τtf/2T)2]. We now compare the performance of the above effective two-level system shortcuts to adiabaticity and three-level system shortcuts to adiabaticity protocols. We set the detuning Δ = 2π × 2.5GHz. With Ω0 = 2π × 0.16GHz, tf = 30ns, T = tf/6, and τ = tf/10, the dynamics described by the original Hamiltonian (20) is not adiabatic at all, and population can not be completely transferred from |1〉 to |3〉. Figs. 5(a) and 5(b) show the population transfer by shortcuts to adiabaticity of the effective two-level system and that of the three-level system, respectively. We can find that they are almost the same.

 

Fig. 5 The population evolution of the effective two-level system by shortcuts to adiabaticity (a), and the three-level system by shortcuts to adiabaticity (b). Here the population of level |1〉 (dashed red), and |3〉 (solid blue) is presented. We set the parameters as: tf = 30ns, T = tf /6, τ = tf/10, Ω0 = 2π × 0.16GHz, Δ = 2π × 2.5GHz, Γ = 2π × 0.16GHz, ωp = 0.1GHz, ωs = 0.08GHz. We can find that the population evolution given by the effective two-level model and by the three-level model is almost the same.

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4. Conclusion

Employing transitionless quantum driving, we have presented a method to design a shortcut to adiabaticity for a non-Hermitian system [58, 59] without RWA. We find that the previous shortcuts to adiabaticity technique with RWA does not work well if the counter-rotating terms can not be neglected, instead the present scheme leads to a high performance to finish the shortcut. Meanwhile we found that the decay of the excited state has little effect on the adiabatic population transfer in the three-level system. Our method is robust against the fluctuation of parameters, as the fidelity is still unity for a wide range of Γ, Ω0 and Δ. In the case of large detuning, we reduce the three-level system to an effective two-level system by using adiabatic elimination. The shortcuts to adiabaticity technique of effective two-level system is also studied and discussed.

Funding

National Natural Science Foundation of China (NSFC) Grants Nos. 11534002, 61475033, 11775048, and 11705025; China Postdoctoral Science Foundation Grant Nos. 2016M600223 and 2017T100192; Fundamental Research Funds for the Central Universities No. 2412017QD005.

References and links

1. N. V. Vitanov, K. A. Suominen, and B. W. Shore, “Creation of coherent atomic superpositions by fractional stimulated Raman adiabatic passage,” J. Phys. B 32, 4535 (1999). [CrossRef]  

2. Y. X. Du, Z. T. Liang, W. Huang, H. Yan, and S. L. Zhu, “Experimental observation of double coherent stimulated Raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014). [CrossRef]  

3. L. F. Wei, J. R. Johansson, L. X. Cen, S. Ashhab, and F. Nori, “Controllable coherent population transfers in superconducting qubits for quantum computing,” Phys. Rev. Lett. 100, 113601 (2008). [CrossRef]   [PubMed]  

4. T. Halfmann and K. Bergmann, “Coherent Population Transfer and Dark Resonances in SO2,” J. Chem. Phys. 104, 7068 (1996). [CrossRef]  

5. K. Bergmann, H. Theuer, and B. W. Shore, “Coherent Population Transfer Among Quantum States of Atoms and Molecules,” Rev. Mod. Phys. 70, 1003 (1998). [CrossRef]  

6. L. Giannelli and E. Arimondo, “Three-level superadiabatic quantum driving,” Phys. Rev. A 89, 033419 (2014). [CrossRef]  

7. Y. Sun and H. Metcalf, “Nonadiabaticity in stimulated Raman adiabatic passage,” Phys. Rev. A 90, 033408 (2014). [CrossRef]  

8. N. Wiebe and N. S. Babcock, “Improved error-scaling for adiabatic quantum evolutions,” New J. Phys. 14, 013024 (2012). [CrossRef]  

9. K. Takahashi, “How fast and robust is the quantum adiabatic passage?” J. Phys. A: Math. Theor. 46, 315304 (2013). [CrossRef]  

10. B. T. Torosov and N. V. Vitanov, “Composite Stimulated Raman Adiabatic Passage,” Phys. Rev. A 87, 043418 (2013). [CrossRef]  

11. X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A 86, 033405 (2012). [CrossRef]  

12. X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to Adiabatic Passage in Two-and Three-Level Atoms,” Phys. Rev. Lett. 105, 123003 (2010). [CrossRef]  

13. M. Demirplak and S. A. Rice, “Adiabatic Population Transfer with Control Fields,” J. Phys. Chem. A 107, 9937 (2003). [CrossRef]  

14. X. B. Huang, Y. H. Chen, and Z. Wang, “Fast generation of three-qubit Greenberger-Horne-Zeilinger state based on the Lewis-Riesenfeld invariants in coupled cavities,” Sci. Rep. 6, 25707 (2016). [CrossRef]   [PubMed]  

15. J. Sun, S. F. Lu, and F. Liu, “Speedup in adiabatic evolution based quantum algorithms,” Sci. China-Phys. Mech. Astron. 55, 1630 (2012). [CrossRef]  

16. S. Ibáñez, X. Chen, and J. G. Muga, “Improving shortcuts to adiabaticity by iterative interaction pictures,” Phys. Rev. A 87, 043402 (2013). [CrossRef]  

17. K. Paul and A. K. Sarma, “High-fidelity entangled Bell states via shortcuts to adiabaticity,” Phys. Rev. A 94, 052303 (2016). [CrossRef]  

18. Y. H. Chen, Y. Xia, Q. C. Wu, B. H. Huang, and J. Song, “Method for constructing shortcuts to adiabaticity by a substitute of counterdiabatic driving terms,” Phys. Rev. A 93, 052109 (2016). [CrossRef]  

19. S. Campbell, G. De Chiara, M. Paternostro, G. M. Palma, and R. Fazio, “Shortcut to Adiabaticity in the Lipkin-Meshkov-Glick Model,” Phys. Rev. Lett. 114, 177206 (2015). [CrossRef]   [PubMed]  

20. M. Okuyama and K. Takahashi, “From Classical Nonlinear Integrable Systems to Quantum Shortcuts to Adiabaticity,” Phys. Rev. Lett. 117, 070401 (2016). [CrossRef]   [PubMed]  

21. S. He, S. L. Su, D. Y. Wang, W. M. Sun, C. H. Bai, A. D. Zhu, H. F. Wang, and S. Zhang, “Efcient shortcuts to adiabatic passage for three-dimensional entanglement generation via transitionless quantum driving,” Sci. Rep. 6, 30929 (2016). [CrossRef]  

22. K. S. Kumar, A. Vepsäläinen, S. Danilin, and G. S. Paraoanu, “Stimulated Raman adiabatic passage in a three-level superconducting circuit,” Nat. Commun. 7, 10628 (2016). [CrossRef]   [PubMed]  

23. S. Martnez-Garaot, E. Torrontegui, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity in three-level systems using Lie transforms,” Phys. Rev. A 89, 053408 (2014). [CrossRef]  

24. T. Opatrný and K. Mømer, “Partial suppression of nonadiabatic transitions,” New J. Phys. 16, 015025 (2014). [CrossRef]  

25. S. Ibáñez, X. Chen, E. Torrontegui, J. G. Muga, and A. Ruschhaupt, “Multiple Schrödinger Pictures and Dynamics in Shortcuts to Adiabaticity,” Phys. Rev. Lett. 109, 100403 (2012). [CrossRef]  

26. Y. H. Chen, Q. C. Wu, B. H. Huang, Y. Xia, and J. Song, “Method for constructing shortcuts to adiabaticity by a substitute of counterdiabatic driving terms,” Phys. Rev. A 93, 052109 (2016). [CrossRef]  

27. K. Takahashi, “Transitionless quantum driving for spin systems,” Phys. Rev. E 87, 062117 (2013). [CrossRef]  

28. X. Shi and L. F. Wei, “High-efficiency single-photon Fock state production by transitionless quantum driving, Laser,” Phys. Lett. 12, 015204 (2015).

29. X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011). [CrossRef]  

30. X. K. Song, Q. Ai, J. Qiu, and F. G. Deng, “Physically feasible three-level transitionless quantum driving with multiple Schrödinger dynamics,” Phys. Rev. A 93, 052324 (2016). [CrossRef]  

31. X. K. Song, H. Zhang, Q. Ai, J. Qiu, and F. G. Deng, “Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm,” New J. Phys. 18, 023001 (2016). [CrossRef]  

32. M. V. Berry, “Transitionless quantum driving,” J. Rhys. A: Math. Theor 42, 365303 (2009).

33. S. Masuda and K. Nakamura, “Acceleration of adiabatic quantum dynamics in electromagnetic fields,” Phys. Rev. A 84, 043434 (2011). [CrossRef]  

34. A. Kiely and A. Ruschhaupt, “Inhibiting unwanted transitions in population transfer in two-and three-level quantum systems,” J. Phys. B 47, 115501 (2014). [CrossRef]  

35. S. Masuda and S. A. Rice, “Rapid Coherent Control of Population Transfer in Lattice Systems,” Phys. Rev. A 89, 033621 (2014). [CrossRef]  

36. N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University, Cambridge, 2011).

37. C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having P-T Symmetry,” Phys. Rev. Lett. 80, 5243(1998). [CrossRef]  

38. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 957 (2007). [CrossRef]  

39. A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 205 (2002). [CrossRef]  

40. A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A 36, 7081 (2003). [CrossRef]  

41. R. Uzdin, U. Günther, S. Rahav, and N. Moiseyev, “Time-dependent Hamiltonians with 100 0.000000e+000volution speed efficiency,” J. Phys. 45, 415304 (2012).

42. C. Hang, G. Huang, and V. V. Konotop, “PT-Symmetry with a System of Three-Level Atoms,” Phys. Rev. Lett. 110, 083604 (2013). [CrossRef]  

43. S. Ibáñez and J. G. Muga, “Adiabaticity condition for non-Hermitian Hamiltonians,” Phys. Rev. A 89, 033403 (2014). [CrossRef]  

44. S. A. Reyes, F. A. Olivares, and L. Morales-Molina, “Landau-Zener-Stückelberg interferometry in PT-symmetric optical waveguides,” J. Phys. A: Math. Theor. 45, 444027 (2012). [CrossRef]  

45. S. Chan, M. D. Reid, and Z. Ficek, “Entanglement evolution of two remote and non-identical Jaynes-Cummings atoms,” J. Phys. B 42, 065507(2009). [CrossRef]  

46. E. K. Irish, “Generalized rotating-wave approximation for arbitrarily large coupling,” Phys. Rev. Lett. 99, 173601 (2007). [CrossRef]   [PubMed]  

47. A. B. Klimov, I. Sainz, and S. M. Chumakov, “Resonance expansion versus the rotating-wave approximation,” Phys. Rev. A 68, 063811 (2003). [CrossRef]  

48. T. Liu, K. L. Wang, and M. Feng, “The generalized analytical approximation to the solution of the single-mode spin-boson model without rotating-wave approximation,” Europhys. Lett. 86, 54003 (2009). [CrossRef]  

49. S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A 81, 042311 (2010). [CrossRef]  

50. J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep Strong Coupling Regime of the Jaynes-Cummings Model,” Phys. Rev. Lett. 105, 263603 (2010). [CrossRef]  

51. A. Fedorov, A. K. Feofanov, P. Macha, P. Forn-Díaz, C. J. P. M. Harmans, and J. E. Mooij, “Strong Coupling of a Quantum Oscillator to a Flux Qubit at Its Symmetry Point,” Phys. Rev. Lett. 105, 060503 (2010). [CrossRef]   [PubMed]  

52. E. K. Irish, “Generalized Rotating-Wave Approximation for Arbitrarily Large Coupling,” Phys. Rev. Lett. 99, 173601 (2007). [CrossRef]   [PubMed]  

53. J. Bourassa, J. M. Gambetta, A. A. Abdumalikov Jr., O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009). [CrossRef]  

54. J. Larson, “Absence of Vacuum Induced Berry Phases without the Rotating Wave Approximation in Cavity QED,” Phys. Rev. Lett. 108, 033601 (2012). [CrossRef]   [PubMed]  

55. Z. Sun, J. Ma, X. Wang, and F. Nori, “Photon-assisted Landau-Zener transition: Role of coherent superposition states,” Phys. Rev. A 86, 012107 (2012). [CrossRef]  

56. J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature 474, 589 (2011). [CrossRef]   [PubMed]  

57. T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nature Phys. 6, 772 (2010). [CrossRef]  

58. K. Paul and A. K. Sarma, “Shortcut to adiabatic passage in a waveguide coupler with a complex-hyperbolic-secant scheme,” Phys. Rev. A 91, 053406 (2015). [CrossRef]  

59. M. B. Kenmoe, S. E. M. Tchouobiap, C. K. Sadem, A. B. Tchapda, and L. C. Fai, “Non-adiabatic and adiabatic transitions at level crossing with decay: two-and three-level systems,” J. Phys. A 48, 095303 (2015). [CrossRef]  

60. R. Schilling, M. Vogelsberger, and D. A. Garanin, “Nonadiabatic transitions for a decaying two-level system: geometrical and dynamical contributions,” J. Phys. A: Math. Gen. 39, 13727 (2006). [CrossRef]  

61. R. Garcia Fernandez, A. Ekers, L. P. Yatsenko, N. V. Vitanov, and K. Bergmann, “Control of Population Flow in Coherently Driven Quantum Ladders,” Phys. Rev. Lett. 95, 043001 (2005). [CrossRef]   [PubMed]  

62. I. I. Boradjiev and N. V. Vitanov, “Stimulated Raman adiabatic passage with unequal couplings: Beyond two-photon resonance,” Phys. Rev. A 81, 053415 (2010). [CrossRef]  

63. S. Ibánez, S. Martinez-Garaot, X. Chen, E. Torrontegui, and J. G. Muga, “Shortcuts to adiabaticity for non-Hermitian systems,” Phys. Rev. A 84, 023415 (2011). [CrossRef]  

64. S. Ibáñez, Y. C. Li, X. Chen, and J. G. Muga, “Pulse design without the rotating-wave approximation,” Phys. Rev. A 92, 062136 (2015). [CrossRef]  

65. N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, “Stimulated Raman adiabatic passage in physics, chemistry, and beyond,” Rev. Mod. Phys. 89, 015006 (2017). [CrossRef]  

66. Y. X. Liu, J. Q. You, L. F. Wei, C. P. Sun, and F. Nori, “Optical Selection Rules and Phase-Dependent Adiabatic State Control in a Superconducting Quantum Circuit,” Phys. Rev. Lett. 95, 087001 (2005). [CrossRef]   [PubMed]  

67. L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, “Quantum Routing of Single Photons with a Cyclic Three-Level System,” Phys. Rev. Lett. 111, 103604 (2013). [CrossRef]  

68. O. Astafiev, K. Inomata, A. O. Niskanen, T. Yamamoto, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Single artificial-atom lasing,” Nature 449, 588(2007). [CrossRef]   [PubMed]  

69. A. A. Abdumalikov Jr., O. Astafiev, A. M. Zagoskin, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Electromagnetically Induced Transparency on a Single Artificial Atom,” Phys. Rev. Lett. 104, 193601(2010). [CrossRef]   [PubMed]  

70. O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov Jr., Yu. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance Fluorescence of a Single Artificial Atom,” Science 327, 840 (2010). [CrossRef]   [PubMed]  

71. N. V. Vitanov and S. Stenholm, “Analytic properties and effective two-level problems in stimulated Raman adiabatic passage,” Phys. Rev. A 55, 648 (1997). [CrossRef]  

72. G. S. Vasilev, A. Kuhn, and N. V. Vitanov, “Optimum pulse shapes for stimulated Raman adiabatic passage,” Phys. Rev. A 80, 013417 (2009). [CrossRef]  

73. N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, “Laser-induced population transfer by adiabatic passage techniques,” Annu. Rev. Phys. Chem. 52, 763 (2001). [CrossRef]   [PubMed]  

74. J. W. Chen and L. F. Wei, “Implementation speed of deterministic population passages compared to that of Rabi pulses,” Phys. Rev. A 91, 023405 (2015). [CrossRef]  

References

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  1. N. V. Vitanov, K. A. Suominen, and B. W. Shore, “Creation of coherent atomic superpositions by fractional stimulated Raman adiabatic passage,” J. Phys. B 32, 4535 (1999).
    [Crossref]
  2. Y. X. Du, Z. T. Liang, W. Huang, H. Yan, and S. L. Zhu, “Experimental observation of double coherent stimulated Raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
    [Crossref]
  3. L. F. Wei, J. R. Johansson, L. X. Cen, S. Ashhab, and F. Nori, “Controllable coherent population transfers in superconducting qubits for quantum computing,” Phys. Rev. Lett. 100, 113601 (2008).
    [Crossref] [PubMed]
  4. T. Halfmann and K. Bergmann, “Coherent Population Transfer and Dark Resonances in SO2,” J. Chem. Phys. 104, 7068 (1996).
    [Crossref]
  5. K. Bergmann, H. Theuer, and B. W. Shore, “Coherent Population Transfer Among Quantum States of Atoms and Molecules,” Rev. Mod. Phys. 70, 1003 (1998).
    [Crossref]
  6. L. Giannelli and E. Arimondo, “Three-level superadiabatic quantum driving,” Phys. Rev. A 89, 033419 (2014).
    [Crossref]
  7. Y. Sun and H. Metcalf, “Nonadiabaticity in stimulated Raman adiabatic passage,” Phys. Rev. A 90, 033408 (2014).
    [Crossref]
  8. N. Wiebe and N. S. Babcock, “Improved error-scaling for adiabatic quantum evolutions,” New J. Phys. 14, 013024 (2012).
    [Crossref]
  9. K. Takahashi, “How fast and robust is the quantum adiabatic passage?” J. Phys. A: Math. Theor. 46, 315304 (2013).
    [Crossref]
  10. B. T. Torosov and N. V. Vitanov, “Composite Stimulated Raman Adiabatic Passage,” Phys. Rev. A 87, 043418 (2013).
    [Crossref]
  11. X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A 86, 033405 (2012).
    [Crossref]
  12. X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to Adiabatic Passage in Two-and Three-Level Atoms,” Phys. Rev. Lett. 105, 123003 (2010).
    [Crossref]
  13. M. Demirplak and S. A. Rice, “Adiabatic Population Transfer with Control Fields,” J. Phys. Chem. A 107, 9937 (2003).
    [Crossref]
  14. X. B. Huang, Y. H. Chen, and Z. Wang, “Fast generation of three-qubit Greenberger-Horne-Zeilinger state based on the Lewis-Riesenfeld invariants in coupled cavities,” Sci. Rep. 6, 25707 (2016).
    [Crossref] [PubMed]
  15. J. Sun, S. F. Lu, and F. Liu, “Speedup in adiabatic evolution based quantum algorithms,” Sci. China-Phys. Mech. Astron. 55, 1630 (2012).
    [Crossref]
  16. S. Ibáñez, X. Chen, and J. G. Muga, “Improving shortcuts to adiabaticity by iterative interaction pictures,” Phys. Rev. A 87, 043402 (2013).
    [Crossref]
  17. K. Paul and A. K. Sarma, “High-fidelity entangled Bell states via shortcuts to adiabaticity,” Phys. Rev. A 94, 052303 (2016).
    [Crossref]
  18. Y. H. Chen, Y. Xia, Q. C. Wu, B. H. Huang, and J. Song, “Method for constructing shortcuts to adiabaticity by a substitute of counterdiabatic driving terms,” Phys. Rev. A 93, 052109 (2016).
    [Crossref]
  19. S. Campbell, G. De Chiara, M. Paternostro, G. M. Palma, and R. Fazio, “Shortcut to Adiabaticity in the Lipkin-Meshkov-Glick Model,” Phys. Rev. Lett. 114, 177206 (2015).
    [Crossref] [PubMed]
  20. M. Okuyama and K. Takahashi, “From Classical Nonlinear Integrable Systems to Quantum Shortcuts to Adiabaticity,” Phys. Rev. Lett. 117, 070401 (2016).
    [Crossref] [PubMed]
  21. S. He, S. L. Su, D. Y. Wang, W. M. Sun, C. H. Bai, A. D. Zhu, H. F. Wang, and S. Zhang, “Efcient shortcuts to adiabatic passage for three-dimensional entanglement generation via transitionless quantum driving,” Sci. Rep. 6, 30929 (2016).
    [Crossref]
  22. K. S. Kumar, A. Vepsäläinen, S. Danilin, and G. S. Paraoanu, “Stimulated Raman adiabatic passage in a three-level superconducting circuit,” Nat. Commun. 7, 10628 (2016).
    [Crossref] [PubMed]
  23. S. Martnez-Garaot, E. Torrontegui, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity in three-level systems using Lie transforms,” Phys. Rev. A 89, 053408 (2014).
    [Crossref]
  24. T. Opatrný and K. Mømer, “Partial suppression of nonadiabatic transitions,” New J. Phys. 16, 015025 (2014).
    [Crossref]
  25. S. Ibáñez, X. Chen, E. Torrontegui, J. G. Muga, and A. Ruschhaupt, “Multiple Schrödinger Pictures and Dynamics in Shortcuts to Adiabaticity,” Phys. Rev. Lett. 109, 100403 (2012).
    [Crossref]
  26. Y. H. Chen, Q. C. Wu, B. H. Huang, Y. Xia, and J. Song, “Method for constructing shortcuts to adiabaticity by a substitute of counterdiabatic driving terms,” Phys. Rev. A 93, 052109 (2016).
    [Crossref]
  27. K. Takahashi, “Transitionless quantum driving for spin systems,” Phys. Rev. E 87, 062117 (2013).
    [Crossref]
  28. X. Shi and L. F. Wei, “High-efficiency single-photon Fock state production by transitionless quantum driving, Laser,” Phys. Lett. 12, 015204 (2015).
  29. X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
    [Crossref]
  30. X. K. Song, Q. Ai, J. Qiu, and F. G. Deng, “Physically feasible three-level transitionless quantum driving with multiple Schrödinger dynamics,” Phys. Rev. A 93, 052324 (2016).
    [Crossref]
  31. X. K. Song, H. Zhang, Q. Ai, J. Qiu, and F. G. Deng, “Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm,” New J. Phys. 18, 023001 (2016).
    [Crossref]
  32. M. V. Berry, “Transitionless quantum driving,” J. Rhys. A: Math. Theor 42, 365303 (2009).
  33. S. Masuda and K. Nakamura, “Acceleration of adiabatic quantum dynamics in electromagnetic fields,” Phys. Rev. A 84, 043434 (2011).
    [Crossref]
  34. A. Kiely and A. Ruschhaupt, “Inhibiting unwanted transitions in population transfer in two-and three-level quantum systems,” J. Phys. B 47, 115501 (2014).
    [Crossref]
  35. S. Masuda and S. A. Rice, “Rapid Coherent Control of Population Transfer in Lattice Systems,” Phys. Rev. A 89, 033621 (2014).
    [Crossref]
  36. N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University, Cambridge, 2011).
  37. C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having P-T Symmetry,” Phys. Rev. Lett. 80, 5243(1998).
    [Crossref]
  38. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 957 (2007).
    [Crossref]
  39. A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 205 (2002).
    [Crossref]
  40. A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A 36, 7081 (2003).
    [Crossref]
  41. R. Uzdin, U. Günther, S. Rahav, and N. Moiseyev, “Time-dependent Hamiltonians with 100 0.000000e+000volution speed efficiency,” J. Phys. 45, 415304 (2012).
  42. C. Hang, G. Huang, and V. V. Konotop, “PT-Symmetry with a System of Three-Level Atoms,” Phys. Rev. Lett. 110, 083604 (2013).
    [Crossref]
  43. S. Ibáñez and J. G. Muga, “Adiabaticity condition for non-Hermitian Hamiltonians,” Phys. Rev. A 89, 033403 (2014).
    [Crossref]
  44. S. A. Reyes, F. A. Olivares, and L. Morales-Molina, “Landau-Zener-Stückelberg interferometry in PT-symmetric optical waveguides,” J. Phys. A: Math. Theor. 45, 444027 (2012).
    [Crossref]
  45. S. Chan, M. D. Reid, and Z. Ficek, “Entanglement evolution of two remote and non-identical Jaynes-Cummings atoms,” J. Phys. B 42, 065507(2009).
    [Crossref]
  46. E. K. Irish, “Generalized rotating-wave approximation for arbitrarily large coupling,” Phys. Rev. Lett. 99, 173601 (2007).
    [Crossref] [PubMed]
  47. A. B. Klimov, I. Sainz, and S. M. Chumakov, “Resonance expansion versus the rotating-wave approximation,” Phys. Rev. A 68, 063811 (2003).
    [Crossref]
  48. T. Liu, K. L. Wang, and M. Feng, “The generalized analytical approximation to the solution of the single-mode spin-boson model without rotating-wave approximation,” Europhys. Lett. 86, 54003 (2009).
    [Crossref]
  49. S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A 81, 042311 (2010).
    [Crossref]
  50. J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep Strong Coupling Regime of the Jaynes-Cummings Model,” Phys. Rev. Lett. 105, 263603 (2010).
    [Crossref]
  51. A. Fedorov, A. K. Feofanov, P. Macha, P. Forn-Díaz, C. J. P. M. Harmans, and J. E. Mooij, “Strong Coupling of a Quantum Oscillator to a Flux Qubit at Its Symmetry Point,” Phys. Rev. Lett. 105, 060503 (2010).
    [Crossref] [PubMed]
  52. E. K. Irish, “Generalized Rotating-Wave Approximation for Arbitrarily Large Coupling,” Phys. Rev. Lett. 99, 173601 (2007).
    [Crossref] [PubMed]
  53. J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009).
    [Crossref]
  54. J. Larson, “Absence of Vacuum Induced Berry Phases without the Rotating Wave Approximation in Cavity QED,” Phys. Rev. Lett. 108, 033601 (2012).
    [Crossref] [PubMed]
  55. Z. Sun, J. Ma, X. Wang, and F. Nori, “Photon-assisted Landau-Zener transition: Role of coherent superposition states,” Phys. Rev. A 86, 012107 (2012).
    [Crossref]
  56. J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature 474, 589 (2011).
    [Crossref] [PubMed]
  57. T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nature Phys. 6, 772 (2010).
    [Crossref]
  58. K. Paul and A. K. Sarma, “Shortcut to adiabatic passage in a waveguide coupler with a complex-hyperbolic-secant scheme,” Phys. Rev. A 91, 053406 (2015).
    [Crossref]
  59. M. B. Kenmoe, S. E. M. Tchouobiap, C. K. Sadem, A. B. Tchapda, and L. C. Fai, “Non-adiabatic and adiabatic transitions at level crossing with decay: two-and three-level systems,” J. Phys. A 48, 095303 (2015).
    [Crossref]
  60. R. Schilling, M. Vogelsberger, and D. A. Garanin, “Nonadiabatic transitions for a decaying two-level system: geometrical and dynamical contributions,” J. Phys. A: Math. Gen. 39, 13727 (2006).
    [Crossref]
  61. R. Garcia Fernandez, A. Ekers, L. P. Yatsenko, N. V. Vitanov, and K. Bergmann, “Control of Population Flow in Coherently Driven Quantum Ladders,” Phys. Rev. Lett. 95, 043001 (2005).
    [Crossref] [PubMed]
  62. I. I. Boradjiev and N. V. Vitanov, “Stimulated Raman adiabatic passage with unequal couplings: Beyond two-photon resonance,” Phys. Rev. A 81, 053415 (2010).
    [Crossref]
  63. S. Ibánez, S. Martinez-Garaot, X. Chen, E. Torrontegui, and J. G. Muga, “Shortcuts to adiabaticity for non-Hermitian systems,” Phys. Rev. A 84, 023415 (2011).
    [Crossref]
  64. S. Ibáñez, Y. C. Li, X. Chen, and J. G. Muga, “Pulse design without the rotating-wave approximation,” Phys. Rev. A 92, 062136 (2015).
    [Crossref]
  65. N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, “Stimulated Raman adiabatic passage in physics, chemistry, and beyond,” Rev. Mod. Phys. 89, 015006 (2017).
    [Crossref]
  66. Y. X. Liu, J. Q. You, L. F. Wei, C. P. Sun, and F. Nori, “Optical Selection Rules and Phase-Dependent Adiabatic State Control in a Superconducting Quantum Circuit,” Phys. Rev. Lett. 95, 087001 (2005).
    [Crossref] [PubMed]
  67. L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, “Quantum Routing of Single Photons with a Cyclic Three-Level System,” Phys. Rev. Lett. 111, 103604 (2013).
    [Crossref]
  68. O. Astafiev, K. Inomata, A. O. Niskanen, T. Yamamoto, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Single artificial-atom lasing,” Nature 449, 588(2007).
    [Crossref] [PubMed]
  69. A. A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Electromagnetically Induced Transparency on a Single Artificial Atom,” Phys. Rev. Lett. 104, 193601(2010).
    [Crossref] [PubMed]
  70. O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Yu. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance Fluorescence of a Single Artificial Atom,” Science 327, 840 (2010).
    [Crossref] [PubMed]
  71. N. V. Vitanov and S. Stenholm, “Analytic properties and effective two-level problems in stimulated Raman adiabatic passage,” Phys. Rev. A 55, 648 (1997).
    [Crossref]
  72. G. S. Vasilev, A. Kuhn, and N. V. Vitanov, “Optimum pulse shapes for stimulated Raman adiabatic passage,” Phys. Rev. A 80, 013417 (2009).
    [Crossref]
  73. N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, “Laser-induced population transfer by adiabatic passage techniques,” Annu. Rev. Phys. Chem. 52, 763 (2001).
    [Crossref] [PubMed]
  74. J. W. Chen and L. F. Wei, “Implementation speed of deterministic population passages compared to that of Rabi pulses,” Phys. Rev. A 91, 023405 (2015).
    [Crossref]

2017 (1)

N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, “Stimulated Raman adiabatic passage in physics, chemistry, and beyond,” Rev. Mod. Phys. 89, 015006 (2017).
[Crossref]

2016 (9)

X. B. Huang, Y. H. Chen, and Z. Wang, “Fast generation of three-qubit Greenberger-Horne-Zeilinger state based on the Lewis-Riesenfeld invariants in coupled cavities,” Sci. Rep. 6, 25707 (2016).
[Crossref] [PubMed]

M. Okuyama and K. Takahashi, “From Classical Nonlinear Integrable Systems to Quantum Shortcuts to Adiabaticity,” Phys. Rev. Lett. 117, 070401 (2016).
[Crossref] [PubMed]

S. He, S. L. Su, D. Y. Wang, W. M. Sun, C. H. Bai, A. D. Zhu, H. F. Wang, and S. Zhang, “Efcient shortcuts to adiabatic passage for three-dimensional entanglement generation via transitionless quantum driving,” Sci. Rep. 6, 30929 (2016).
[Crossref]

K. S. Kumar, A. Vepsäläinen, S. Danilin, and G. S. Paraoanu, “Stimulated Raman adiabatic passage in a three-level superconducting circuit,” Nat. Commun. 7, 10628 (2016).
[Crossref] [PubMed]

K. Paul and A. K. Sarma, “High-fidelity entangled Bell states via shortcuts to adiabaticity,” Phys. Rev. A 94, 052303 (2016).
[Crossref]

Y. H. Chen, Y. Xia, Q. C. Wu, B. H. Huang, and J. Song, “Method for constructing shortcuts to adiabaticity by a substitute of counterdiabatic driving terms,” Phys. Rev. A 93, 052109 (2016).
[Crossref]

Y. H. Chen, Q. C. Wu, B. H. Huang, Y. Xia, and J. Song, “Method for constructing shortcuts to adiabaticity by a substitute of counterdiabatic driving terms,” Phys. Rev. A 93, 052109 (2016).
[Crossref]

X. K. Song, Q. Ai, J. Qiu, and F. G. Deng, “Physically feasible three-level transitionless quantum driving with multiple Schrödinger dynamics,” Phys. Rev. A 93, 052324 (2016).
[Crossref]

X. K. Song, H. Zhang, Q. Ai, J. Qiu, and F. G. Deng, “Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm,” New J. Phys. 18, 023001 (2016).
[Crossref]

2015 (6)

X. Shi and L. F. Wei, “High-efficiency single-photon Fock state production by transitionless quantum driving, Laser,” Phys. Lett. 12, 015204 (2015).

S. Campbell, G. De Chiara, M. Paternostro, G. M. Palma, and R. Fazio, “Shortcut to Adiabaticity in the Lipkin-Meshkov-Glick Model,” Phys. Rev. Lett. 114, 177206 (2015).
[Crossref] [PubMed]

S. Ibáñez, Y. C. Li, X. Chen, and J. G. Muga, “Pulse design without the rotating-wave approximation,” Phys. Rev. A 92, 062136 (2015).
[Crossref]

J. W. Chen and L. F. Wei, “Implementation speed of deterministic population passages compared to that of Rabi pulses,” Phys. Rev. A 91, 023405 (2015).
[Crossref]

K. Paul and A. K. Sarma, “Shortcut to adiabatic passage in a waveguide coupler with a complex-hyperbolic-secant scheme,” Phys. Rev. A 91, 053406 (2015).
[Crossref]

M. B. Kenmoe, S. E. M. Tchouobiap, C. K. Sadem, A. B. Tchapda, and L. C. Fai, “Non-adiabatic and adiabatic transitions at level crossing with decay: two-and three-level systems,” J. Phys. A 48, 095303 (2015).
[Crossref]

2014 (8)

S. Ibáñez and J. G. Muga, “Adiabaticity condition for non-Hermitian Hamiltonians,” Phys. Rev. A 89, 033403 (2014).
[Crossref]

A. Kiely and A. Ruschhaupt, “Inhibiting unwanted transitions in population transfer in two-and three-level quantum systems,” J. Phys. B 47, 115501 (2014).
[Crossref]

S. Masuda and S. A. Rice, “Rapid Coherent Control of Population Transfer in Lattice Systems,” Phys. Rev. A 89, 033621 (2014).
[Crossref]

S. Martnez-Garaot, E. Torrontegui, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity in three-level systems using Lie transforms,” Phys. Rev. A 89, 053408 (2014).
[Crossref]

T. Opatrný and K. Mømer, “Partial suppression of nonadiabatic transitions,” New J. Phys. 16, 015025 (2014).
[Crossref]

L. Giannelli and E. Arimondo, “Three-level superadiabatic quantum driving,” Phys. Rev. A 89, 033419 (2014).
[Crossref]

Y. Sun and H. Metcalf, “Nonadiabaticity in stimulated Raman adiabatic passage,” Phys. Rev. A 90, 033408 (2014).
[Crossref]

Y. X. Du, Z. T. Liang, W. Huang, H. Yan, and S. L. Zhu, “Experimental observation of double coherent stimulated Raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
[Crossref]

2013 (6)

K. Takahashi, “How fast and robust is the quantum adiabatic passage?” J. Phys. A: Math. Theor. 46, 315304 (2013).
[Crossref]

B. T. Torosov and N. V. Vitanov, “Composite Stimulated Raman Adiabatic Passage,” Phys. Rev. A 87, 043418 (2013).
[Crossref]

S. Ibáñez, X. Chen, and J. G. Muga, “Improving shortcuts to adiabaticity by iterative interaction pictures,” Phys. Rev. A 87, 043402 (2013).
[Crossref]

K. Takahashi, “Transitionless quantum driving for spin systems,” Phys. Rev. E 87, 062117 (2013).
[Crossref]

L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, “Quantum Routing of Single Photons with a Cyclic Three-Level System,” Phys. Rev. Lett. 111, 103604 (2013).
[Crossref]

C. Hang, G. Huang, and V. V. Konotop, “PT-Symmetry with a System of Three-Level Atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

2012 (8)

S. A. Reyes, F. A. Olivares, and L. Morales-Molina, “Landau-Zener-Stückelberg interferometry in PT-symmetric optical waveguides,” J. Phys. A: Math. Theor. 45, 444027 (2012).
[Crossref]

J. Larson, “Absence of Vacuum Induced Berry Phases without the Rotating Wave Approximation in Cavity QED,” Phys. Rev. Lett. 108, 033601 (2012).
[Crossref] [PubMed]

Z. Sun, J. Ma, X. Wang, and F. Nori, “Photon-assisted Landau-Zener transition: Role of coherent superposition states,” Phys. Rev. A 86, 012107 (2012).
[Crossref]

R. Uzdin, U. Günther, S. Rahav, and N. Moiseyev, “Time-dependent Hamiltonians with 100 0.000000e+000volution speed efficiency,” J. Phys. 45, 415304 (2012).

X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A 86, 033405 (2012).
[Crossref]

N. Wiebe and N. S. Babcock, “Improved error-scaling for adiabatic quantum evolutions,” New J. Phys. 14, 013024 (2012).
[Crossref]

J. Sun, S. F. Lu, and F. Liu, “Speedup in adiabatic evolution based quantum algorithms,” Sci. China-Phys. Mech. Astron. 55, 1630 (2012).
[Crossref]

S. Ibáñez, X. Chen, E. Torrontegui, J. G. Muga, and A. Ruschhaupt, “Multiple Schrödinger Pictures and Dynamics in Shortcuts to Adiabaticity,” Phys. Rev. Lett. 109, 100403 (2012).
[Crossref]

2011 (4)

S. Masuda and K. Nakamura, “Acceleration of adiabatic quantum dynamics in electromagnetic fields,” Phys. Rev. A 84, 043434 (2011).
[Crossref]

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[Crossref]

J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature 474, 589 (2011).
[Crossref] [PubMed]

S. Ibánez, S. Martinez-Garaot, X. Chen, E. Torrontegui, and J. G. Muga, “Shortcuts to adiabaticity for non-Hermitian systems,” Phys. Rev. A 84, 023415 (2011).
[Crossref]

2010 (8)

I. I. Boradjiev and N. V. Vitanov, “Stimulated Raman adiabatic passage with unequal couplings: Beyond two-photon resonance,” Phys. Rev. A 81, 053415 (2010).
[Crossref]

A. A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Electromagnetically Induced Transparency on a Single Artificial Atom,” Phys. Rev. Lett. 104, 193601(2010).
[Crossref] [PubMed]

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Yu. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance Fluorescence of a Single Artificial Atom,” Science 327, 840 (2010).
[Crossref] [PubMed]

T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nature Phys. 6, 772 (2010).
[Crossref]

S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A 81, 042311 (2010).
[Crossref]

J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep Strong Coupling Regime of the Jaynes-Cummings Model,” Phys. Rev. Lett. 105, 263603 (2010).
[Crossref]

A. Fedorov, A. K. Feofanov, P. Macha, P. Forn-Díaz, C. J. P. M. Harmans, and J. E. Mooij, “Strong Coupling of a Quantum Oscillator to a Flux Qubit at Its Symmetry Point,” Phys. Rev. Lett. 105, 060503 (2010).
[Crossref] [PubMed]

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to Adiabatic Passage in Two-and Three-Level Atoms,” Phys. Rev. Lett. 105, 123003 (2010).
[Crossref]

2009 (5)

M. V. Berry, “Transitionless quantum driving,” J. Rhys. A: Math. Theor 42, 365303 (2009).

S. Chan, M. D. Reid, and Z. Ficek, “Entanglement evolution of two remote and non-identical Jaynes-Cummings atoms,” J. Phys. B 42, 065507(2009).
[Crossref]

G. S. Vasilev, A. Kuhn, and N. V. Vitanov, “Optimum pulse shapes for stimulated Raman adiabatic passage,” Phys. Rev. A 80, 013417 (2009).
[Crossref]

J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009).
[Crossref]

T. Liu, K. L. Wang, and M. Feng, “The generalized analytical approximation to the solution of the single-mode spin-boson model without rotating-wave approximation,” Europhys. Lett. 86, 54003 (2009).
[Crossref]

2008 (1)

L. F. Wei, J. R. Johansson, L. X. Cen, S. Ashhab, and F. Nori, “Controllable coherent population transfers in superconducting qubits for quantum computing,” Phys. Rev. Lett. 100, 113601 (2008).
[Crossref] [PubMed]

2007 (4)

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 957 (2007).
[Crossref]

O. Astafiev, K. Inomata, A. O. Niskanen, T. Yamamoto, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Single artificial-atom lasing,” Nature 449, 588(2007).
[Crossref] [PubMed]

E. K. Irish, “Generalized rotating-wave approximation for arbitrarily large coupling,” Phys. Rev. Lett. 99, 173601 (2007).
[Crossref] [PubMed]

E. K. Irish, “Generalized Rotating-Wave Approximation for Arbitrarily Large Coupling,” Phys. Rev. Lett. 99, 173601 (2007).
[Crossref] [PubMed]

2006 (1)

R. Schilling, M. Vogelsberger, and D. A. Garanin, “Nonadiabatic transitions for a decaying two-level system: geometrical and dynamical contributions,” J. Phys. A: Math. Gen. 39, 13727 (2006).
[Crossref]

2005 (2)

R. Garcia Fernandez, A. Ekers, L. P. Yatsenko, N. V. Vitanov, and K. Bergmann, “Control of Population Flow in Coherently Driven Quantum Ladders,” Phys. Rev. Lett. 95, 043001 (2005).
[Crossref] [PubMed]

Y. X. Liu, J. Q. You, L. F. Wei, C. P. Sun, and F. Nori, “Optical Selection Rules and Phase-Dependent Adiabatic State Control in a Superconducting Quantum Circuit,” Phys. Rev. Lett. 95, 087001 (2005).
[Crossref] [PubMed]

2003 (3)

A. B. Klimov, I. Sainz, and S. M. Chumakov, “Resonance expansion versus the rotating-wave approximation,” Phys. Rev. A 68, 063811 (2003).
[Crossref]

A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A 36, 7081 (2003).
[Crossref]

M. Demirplak and S. A. Rice, “Adiabatic Population Transfer with Control Fields,” J. Phys. Chem. A 107, 9937 (2003).
[Crossref]

2002 (1)

A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 205 (2002).
[Crossref]

2001 (1)

N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, “Laser-induced population transfer by adiabatic passage techniques,” Annu. Rev. Phys. Chem. 52, 763 (2001).
[Crossref] [PubMed]

1999 (1)

N. V. Vitanov, K. A. Suominen, and B. W. Shore, “Creation of coherent atomic superpositions by fractional stimulated Raman adiabatic passage,” J. Phys. B 32, 4535 (1999).
[Crossref]

1998 (2)

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent Population Transfer Among Quantum States of Atoms and Molecules,” Rev. Mod. Phys. 70, 1003 (1998).
[Crossref]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having P-T Symmetry,” Phys. Rev. Lett. 80, 5243(1998).
[Crossref]

1997 (1)

N. V. Vitanov and S. Stenholm, “Analytic properties and effective two-level problems in stimulated Raman adiabatic passage,” Phys. Rev. A 55, 648 (1997).
[Crossref]

1996 (1)

T. Halfmann and K. Bergmann, “Coherent Population Transfer and Dark Resonances in SO2,” J. Chem. Phys. 104, 7068 (1996).
[Crossref]

Abdumalikov, A. A.

A. A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Electromagnetically Induced Transparency on a Single Artificial Atom,” Phys. Rev. Lett. 104, 193601(2010).
[Crossref] [PubMed]

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Yu. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance Fluorescence of a Single Artificial Atom,” Science 327, 840 (2010).
[Crossref] [PubMed]

J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009).
[Crossref]

Ai, Q.

X. K. Song, Q. Ai, J. Qiu, and F. G. Deng, “Physically feasible three-level transitionless quantum driving with multiple Schrödinger dynamics,” Phys. Rev. A 93, 052324 (2016).
[Crossref]

X. K. Song, H. Zhang, Q. Ai, J. Qiu, and F. G. Deng, “Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm,” New J. Phys. 18, 023001 (2016).
[Crossref]

Arimondo, E.

L. Giannelli and E. Arimondo, “Three-level superadiabatic quantum driving,” Phys. Rev. A 89, 033419 (2014).
[Crossref]

Ashhab, S.

S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A 81, 042311 (2010).
[Crossref]

L. F. Wei, J. R. Johansson, L. X. Cen, S. Ashhab, and F. Nori, “Controllable coherent population transfers in superconducting qubits for quantum computing,” Phys. Rev. Lett. 100, 113601 (2008).
[Crossref] [PubMed]

Astafiev, O.

A. A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Electromagnetically Induced Transparency on a Single Artificial Atom,” Phys. Rev. Lett. 104, 193601(2010).
[Crossref] [PubMed]

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Yu. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance Fluorescence of a Single Artificial Atom,” Science 327, 840 (2010).
[Crossref] [PubMed]

J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009).
[Crossref]

O. Astafiev, K. Inomata, A. O. Niskanen, T. Yamamoto, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Single artificial-atom lasing,” Nature 449, 588(2007).
[Crossref] [PubMed]

Babcock, N. S.

N. Wiebe and N. S. Babcock, “Improved error-scaling for adiabatic quantum evolutions,” New J. Phys. 14, 013024 (2012).
[Crossref]

Bai, C. H.

S. He, S. L. Su, D. Y. Wang, W. M. Sun, C. H. Bai, A. D. Zhu, H. F. Wang, and S. Zhang, “Efcient shortcuts to adiabatic passage for three-dimensional entanglement generation via transitionless quantum driving,” Sci. Rep. 6, 30929 (2016).
[Crossref]

Bender, C. M.

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 957 (2007).
[Crossref]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having P-T Symmetry,” Phys. Rev. Lett. 80, 5243(1998).
[Crossref]

Bergmann, K.

N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, “Stimulated Raman adiabatic passage in physics, chemistry, and beyond,” Rev. Mod. Phys. 89, 015006 (2017).
[Crossref]

R. Garcia Fernandez, A. Ekers, L. P. Yatsenko, N. V. Vitanov, and K. Bergmann, “Control of Population Flow in Coherently Driven Quantum Ladders,” Phys. Rev. Lett. 95, 043001 (2005).
[Crossref] [PubMed]

N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, “Laser-induced population transfer by adiabatic passage techniques,” Annu. Rev. Phys. Chem. 52, 763 (2001).
[Crossref] [PubMed]

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent Population Transfer Among Quantum States of Atoms and Molecules,” Rev. Mod. Phys. 70, 1003 (1998).
[Crossref]

T. Halfmann and K. Bergmann, “Coherent Population Transfer and Dark Resonances in SO2,” J. Chem. Phys. 104, 7068 (1996).
[Crossref]

Berry, M. V.

M. V. Berry, “Transitionless quantum driving,” J. Rhys. A: Math. Theor 42, 365303 (2009).

Blais, A.

J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having P-T Symmetry,” Phys. Rev. Lett. 80, 5243(1998).
[Crossref]

Boradjiev, I. I.

I. I. Boradjiev and N. V. Vitanov, “Stimulated Raman adiabatic passage with unequal couplings: Beyond two-photon resonance,” Phys. Rev. A 81, 053415 (2010).
[Crossref]

Bourassa, J.

J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009).
[Crossref]

Campbell, S.

S. Campbell, G. De Chiara, M. Paternostro, G. M. Palma, and R. Fazio, “Shortcut to Adiabaticity in the Lipkin-Meshkov-Glick Model,” Phys. Rev. Lett. 114, 177206 (2015).
[Crossref] [PubMed]

Casanova, J.

J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep Strong Coupling Regime of the Jaynes-Cummings Model,” Phys. Rev. Lett. 105, 263603 (2010).
[Crossref]

Cen, L. X.

L. F. Wei, J. R. Johansson, L. X. Cen, S. Ashhab, and F. Nori, “Controllable coherent population transfers in superconducting qubits for quantum computing,” Phys. Rev. Lett. 100, 113601 (2008).
[Crossref] [PubMed]

Chan, S.

S. Chan, M. D. Reid, and Z. Ficek, “Entanglement evolution of two remote and non-identical Jaynes-Cummings atoms,” J. Phys. B 42, 065507(2009).
[Crossref]

Chen, J. W.

J. W. Chen and L. F. Wei, “Implementation speed of deterministic population passages compared to that of Rabi pulses,” Phys. Rev. A 91, 023405 (2015).
[Crossref]

Chen, X.

S. Ibáñez, Y. C. Li, X. Chen, and J. G. Muga, “Pulse design without the rotating-wave approximation,” Phys. Rev. A 92, 062136 (2015).
[Crossref]

S. Martnez-Garaot, E. Torrontegui, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity in three-level systems using Lie transforms,” Phys. Rev. A 89, 053408 (2014).
[Crossref]

S. Ibáñez, X. Chen, and J. G. Muga, “Improving shortcuts to adiabaticity by iterative interaction pictures,” Phys. Rev. A 87, 043402 (2013).
[Crossref]

X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A 86, 033405 (2012).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Real (red solid line) and imaginary (blue dashed line) parts of Ω a (a). The population evolution is driven by Ω a (b), and [ImΩ a ] (c). The parameters chosen are Γ = 2π × 0.5Mhz, Ω0 = 2π × 5Mhz, δ = 2π × 300Mhz, tf = 2 × 10−10s, t0 = 10−10s.
Fig. 2
Fig. 2 The evolution of the population for a ndecaying two-level system with decay rate Γ = 2π × 0.5MHz. The population transfer is implemented by the shortcuts to adiabaticity technique under RWA [Figs. 2(a), and 2(b)], and implemented by the shortcuts to adiabaticity technique beyond RWA [Fig. 2(c)]. Note that all the counter-rotating terms are neglected in Fig. 2(a), but are considered in Fig. 2(b). The population of level |1〉 (dashed red), and |2〉 (solid blue) are shown. The other parameters: Ω0 = 2π × 5MHz, δ = 2π × 300MHz, tf = 2 × 10−10s, t0 = 10−10s, ωL = 2π × 10GHz.
Fig. 3
Fig. 3 Transition probability P1→3 of the fast stimulated Raman by the shortcuts to adiabaticity under and beyond RWA. Fig. 3(a) is plotted by using the shortcuts to adiabaticity technique under RWA, but consider the CR terms. Fig. 3(b) is plotted by using the shortcuts to adiabaticity technique beyond RWA. We set the parameters: tf = 30ns, T = tf/6, τ = tf/10, Ω0 = 2π×200MHz, Δ = 2π×200MHz, Γ = 2π×100MHz, ωp = 100MHz, ωs = 80MHz. The dashed red, dotted green, and solid blue lines describe the time-dependent population of the state |1〉, |2〉, and |3〉, respectively.
Fig. 4
Fig. 4 Fidelity as a function of Γ, Ω0 and Δ. The fidelity is defined as the population on the state |3〉 with initial state |1〉. The population transfer is realized by using the STA technique under RWA but with counter-rotating terms [Figs. 4(a) and 4(c)], and without RWA [Figs. 4(b) and 4(d)]. In Figs. 4(a) and 4(b), the parameters chosen are, tf = 30ns, Ω0 = 2π × [100, 800]MHz, Γ = 2π × [100, 600]MHz, ωp = 100MHz, ωs = 80MHz. In Figs. 4(c) and 4(d), the parameters are set as: tf = 30ns, Δ = 2π × [100, 800]MHz, Γ = 2π × [100, 600]MHz, ωp = 100MHz, ωs = 80MHz. We can clearly find that the fidelity without RWA is robust against the parameter fluctuations.
Fig. 5
Fig. 5 The population evolution of the effective two-level system by shortcuts to adiabaticity (a), and the three-level system by shortcuts to adiabaticity (b). Here the population of level |1〉 (dashed red), and |3〉 (solid blue) is presented. We set the parameters as: tf = 30ns, T = tf /6, τ = tf/10, Ω0 = 2π × 0.16GHz, Δ = 2π × 2.5GHz, Γ = 2π × 0.16GHz, ωp = 0.1GHz, ωs = 0.08GHz. We can find that the population evolution given by the effective two-level model and by the three-level model is almost the same.

Equations (33)

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H S ( t ) = 2 [ Ω R ( t ) ( | 2 1 | + | 1 2 | ) ( e i ω L t + e i ω L t ) + ω 0 ( t ) ( | 2 2 | | 1 1 | ) ] ,
H L ( t ) = i U ˙ ( t ) U ( t ) = ω L 2 ( | 2 2 | | 1 1 | ) ,
U ( t ) = e i 0 t H L ( t ) d t = e i ω L t / 2 | 2 2 | e i ω L t / 2 | 1 1 | ,
H ( t ) = U ( t ) [ H S ( t ) H L ( t ) ] U ( t ) = 2 [ Δ ( t ) Ω ( t ) Ω ( t ) Δ ( t ) ] .
Ω ( t ) = Ω R ( t ) ( 1 + e 2 i ω L t ) ,
Δ ( t ) = ω 0 ( t ) ω L ,
H ˜ 0 ( t ) = 2 [ Δ ( t ) Ω ( t ) Ω ( t ) Δ ( t ) i Γ ] .
i t [ C 1 ( t ) C 2 ( t ) ] = H ˜ 0 ( t ) [ C 1 ( t ) C 2 ( t ) ] ,
| φ + ( t ) = sin β | 1 + cos β e i ω L t | 2 , | φ ( t ) = cos β e i ω L t | 1 sin β | 2 ,
β = 1 2 arctan 2 Ω R cos ω L t Δ i Γ / 2 ,
| φ ^ + ( t ) = sin β * | 1 + cos β * e i ω L t | 2 , | φ ^ ( t ) = cos β * e i ω L t | 1 sin β * | 2 ,
H ˜ CD ( t ) = i [ | t φ + ( t ) φ ^ + ( t ) | φ ^ + ( t ) | t φ + ( t ) | φ + ( t ) φ ^ + ( t ) | + | t φ ( t ) φ ^ ( t ) | φ ^ ( t ) | t φ ( t ) | φ ( t ) φ ^ ( t ) | ] ,
φ ^ ± ( t ) | t φ ± ( t ) = ± i ω L cos 2 β 2 , φ ^ ( t ) | t φ ± ( t ) = ( ± β ˙ 2 i ω L 2 sin β ) e ± i ω L t ,
H ˜ CD ( t ) = [ ω L 2 sin 2 β ( i β ˙ 2 + ω L 4 sin 2 β ) e i ω L t ( i β ˙ 2 + ω L 4 sin 2 β ) e i ω L t ω L 2 sin 2 β ] .
H 0 ( t ) = 2 [ Δ ( t ) Ω R ( t ) Ω R ( t ) Δ ( t ) i Γ ] .
H CD ( t ) = [ 0 Ω a ( t ) Ω a ( t ) 0 ] ,
θ ˙ = Ω ˙ R [ Δ ( t ) i Γ / 2 ] Ω R ( t ) ( Δ ˙ i Γ ˙ / 2 ) Ω R 2 ( t ) + [ Δ ( t ) i Γ / 2 ] 2 ,
Ω R ( t ) = Ω 0 sech [ π ( t t f / 2 ) 2 t 0 ] , Δ ( t ) = 2 δ 2 t 0 π tanh [ π ( t t f / 2 ) 2 t 0 ] .
i d d t c ( t ) = H ¯ 0 ( t ) c ( t ) ,
H ¯ 0 ( t ) = 2 [ 0 Ω ¯ p ( t ) 0 Ω ¯ p * ( t ) 2 Δ p i Γ Ω ¯ s ( t ) 0 Ω ¯ s * ( t ) 2 ( Δ p Δ s ) ] ,
| E 0 ( t ) = 1 Ξ 0 ( t ) [ Ω ¯ s ( t ) | 1 Ω ¯ p * ( t ) | 3 ] , | E + ( t ) = 1 Ξ 1 ( t ) [ Ω ¯ p ( t ) | 1 + ε + ( t ) | 2 + Ω ¯ s * ( t ) | 3 ] , | E ( t ) = 1 Ξ 2 ( t ) [ Ω ¯ p ( t ) | 1 + ε ( t ) | 2 + Ω ¯ s * ( t ) | 3 ] ,
E 0 ( t ) = 0 , E ± ( t ) = 2 ε ± ( t ) ,
ε ± ( t ) = ( Δ i Γ / 2 ) ± ( Δ i Γ / 2 ) 2 + Ξ 0 2 ( t ) .
| E ^ 0 ( t ) = 1 Ξ 0 ( t ) [ Ω ¯ s ( t ) | 1 Ω ¯ p * ( t ) | 3 ] , | E ^ + ( t ) = 1 Ξ 1 ( t ) [ Ω ¯ p ( t ) | 1 + ε ^ + ( t ) | 2 + Ω ¯ s * ( t ) | 3 ] , | E ^ ( t ) = 1 Ξ 2 ( t ) [ Ω ¯ p ( t ) | 1 + ε ^ ( t ) | 2 + Ω ¯ s * ( t ) | 3 ] ,
Π 0 ( t ) = | E 0 ( t ) E ^ 0 ( t ) | = 1 Ξ 0 2 ( t ) [ | Ω ¯ s ( t ) | 2 0 Ω ¯ p ( t ) Ω ¯ s ( t ) 0 0 0 Ω ¯ p * ( t ) Ω ¯ s * ( t ) 0 | Ω ¯ p ( t ) | 2 ] , Π 1 ( t ) = | E 1 ( t ) E ^ 1 ( t ) | = 1 Ξ 1 2 ( t ) [ | Ω ¯ p ( t ) | 2 Ω ¯ p ( t ) ε + ( t ) Ω ¯ p ( t ) Ω ¯ s ( t ) Ω ¯ p * ( t ) ε + ( t ) ε + 2 ( t ) Ω ¯ s ( t ) ε + ( t ) Ω ¯ p * ( t ) Ω ¯ s * ( t ) Ω ¯ s ( t ) ε + ( t ) | Ω ¯ s ( t ) | 2 ] , Π 2 ( t ) = | E 2 ( t ) E ^ 2 ( t ) | = 1 Ξ 2 2 ( t ) [ | Ω ¯ p ( t ) | 2 Ω ¯ p ( t ) ε ( t ) Ω ¯ p ( t ) Ω ¯ s ( t ) Ω ¯ p * ( t ) ε ( t ) ε 2 ( t ) Ω ¯ s ( t ) ε ( t ) Ω ¯ p * ( t ) Ω ¯ s * ( t ) Ω ¯ s ( t ) ε ( t ) | Ω ¯ s ( t ) | 2 ] .
H ¯ CD = i j k Π j ( t ) t H ¯ 0 ( t ) Π k ( t ) E k ( t ) E j ( t ) = i Ξ 1 2 ( t ) Ξ 2 2 ( t ) [ | Ω ¯ p ( t ) | 2 B + C D Ω ¯ p ( t ) A 2 i Γ Ω ¯ s ( t ) G Ω ¯ p ( t ) Ω ¯ s ( t ) B + C F Ω ¯ p * ( t ) A * 2 i Γ Ω ¯ s * ( t ) G * Ξ 0 2 ( t ) B Ω ¯ s ( t ) A * 2 i Γ Ω ¯ p ( t ) G * Ω ¯ p * ( t ) Ω ¯ s * ( t ) B * C F * Ω ¯ s * ( t ) A + 2 i Γ Ω ¯ p * ( t ) G | Ω ¯ s | 2 B C D ] ,
A = ( 2 Δ i Γ ) [ Ω ¯ p * ( t ) Ω ¯ ˙ p ( t ) + Ω ¯ s ( t ) Ω ¯ ˙ s * ( t ) ] , B = Ω ¯ p * ( t ) Ω ¯ ˙ p * ( t ) + Ω ¯ s ( t ) Ω ¯ ˙ s * ( t ) H . C . , C = [ | ε + ( t ) | 2 + | ε ( t ) | 2 + 2 Ξ 0 2 ( t ) ] / Ξ 0 2 ( t ) , D = | Ω ¯ p ( t ) | 2 [ Ω ¯ s * ( t ) Ω ¯ ˙ s * ( t ) H . C . ] + | Ω ¯ s ( t ) | 2 [ ( Ω ¯ p * ( t ) Ω ¯ ˙ p ( t ) H . C . ) ] , F = Ω ¯ p 2 ( t ) [ Ω ¯ s ( t ) Ω ¯ p * ( t ) Ω ¯ p * ( t ) Ω ¯ ˙ s * ( t ) ] + Ω ¯ s 2 ( t ) [ Ω ¯ s * ( t ) Ω ¯ ˙ p ( t ) Ω ¯ p ( t ) Ω ¯ ˙ s * ( t ) ] , G = Ω ¯ s * ( t ) Ω ¯ ˙ p ( t ) Ω ¯ p ( t ) Ω ¯ ˙ s * ( t ) .
H CD ( t ) = i [ 0 A ( t ) C ( t ) A ( t ) 0 B ( t ) C ( t ) B ( t ) 0 ] ,
A ( t ) = sin θ ( t ) ϕ ˙ ( t ) , B ( t ) = cos θ ( t ) ϕ ˙ ( t ) , C ( t ) = θ ˙ ( t ) ,
H eff ( t ) = 2 [ Δ eff ( t ) Ω ef f ( t ) Ω eff * ( t ) Δ eff ( t ) ] ,
Δ eff ( t ) = Ω ¯ p ( t ) Ω ¯ p ( t ) 2 Δ i Γ , Ω eff ( t ) = | Ω ¯ p ( t ) | 2 | Ω ¯ s ( t ) | 2 4 Δ 2 i Γ .
H CD ( t ) = i 2 M ( t ) [ 0 P ( t ) P * ( t ) Q ( t ) ] ,
P ( t ) = Ω ˙ eff ( t ) Δ eff ( t ) Δ ˙ eff ( t ) Ω eff ( t ) , Q ( t ) = Ω ˙ eff * ( t ) Ω eff ( t ) Ω ˙ eff ( t ) Ω eff * , M ( t ) = | Ω eff | 2 + Δ eff 2 .

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