Shortcuts to adiabaticity in non-Hermitian quantum systems without rotating-wave approximation

Hong Li; H. Z. Shen; S. L. Wu; X. X. Yi

doi:10.1364/OE.25.030135

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[ω_Lt]x⃗, the Hamiltonian of the two-level system reads,

\begin{array}{l} H_{S} (t) & = & \frac{ℏ}{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 |)], \end{array}

where |1〉 (|2〉) is the ground (excited) state of the two-level atom, ω₀ is the transition frequency, Ω_R(t) and ω_L are the strength and the frequency of the classical field. Defining

\begin{array}{l} H_{L} (t) & = & i ℏ \dot{U} (t) U^{†} (t) \\ = & \frac{ℏ ω_{L}}{2} (| 2 〉 〈 2 | - | 1 〉 〈 1 |), \end{array}

with unitary operator

\begin{array}{l} U (t) & = & e^{- \frac{i}{ℏ} \int_{0}^{t} H_{L} (t^{'}) d t^{'}} \\ = & e^{- i ω_{L} t / 2} | 2 〉 〈 2 | - e^{i ω_{L} t / 2} | 1 〉 〈 1 |, \end{array}

we can transform the Hamiltonian (1) into the interaction picture, i.e.,

\begin{array}{l} H (t) & = & U^{†} (t) [H_{S} (t) - H_{L} (t)] U (t) \\ = & \frac{ℏ}{2} [\begin{matrix} - Δ (t) & Ω (t) \\ Ω (t) & Δ (t) \end{matrix}] . \end{array}

Here the bases are |1〉 = [1, 0]^T and |2〉 = [0, 1]^T with T denoting transposition.

With

Ω (t) = Ω_{R} (t) (1 + e^{- 2 i ω_{L} t}),

and

Δ (t) = ω_{0} (t) - ω_{L},

in which e^±2iω_Lt 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,

{\tilde{H}}_{0} (t) = \frac{ℏ}{2} [\begin{matrix} - Δ (t) & Ω (t) \\ Ω (t) & Δ (t) - i Γ \end{matrix}] .

The dynamics of the two-level quantum system is described by the Schrödinger equation,

i ℏ \frac{\partial}{\partial t} [\begin{matrix} C_{1} (t) \\ C_{2} (t) \end{matrix}] = {\tilde{H}}_{0} (t) [\begin{matrix} C_{1} (t) \\ C_{2} (t) \end{matrix}],

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 H̃₀(t) as

\begin{array}{l} | 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 〉, \end{array}

where the mixing angle β = β(t) is complex and defined as

β = \frac{1}{2} arctan \frac{2 Ω_{R} cos ω_{L} t}{Δ - i Γ / 2},

and the right eigenstates can also be determined,

\begin{array}{l} | {\hat{φ}}_{+} (t) 〉 = sin β^{*} | 1 〉 + cos β^{*} e^{i ω_{L} t} | 2 〉, \\ | {\hat{φ}}_{-} (t) 〉 = cos β^{*} e^{- i ω_{L} t} | 1 〉 - sin β^{*} | 2 〉, \end{array}

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]

\begin{array}{l} {\tilde{H}}_{CD} (t) = i ℏ [| \partial_{t} φ_{+} (t) 〉 〈 {\hat{φ}}_{+} (t) | - 〈 {\hat{φ}}_{+} (t) | \partial_{t} φ_{+} (t) 〉 | φ_{+} (t) 〉 〈 {\hat{φ}}_{+} (t) | \\ + | \partial_{t} φ_{-} (t) 〉 〈 {\hat{φ}}_{-} (t) | - 〈 {\hat{φ}}_{-} (t) | \partial_{t} φ_{-} (t) 〉 | φ_{-} (t) 〉 〈 {\hat{φ}}_{-} (t) |], \end{array}

where

\begin{array}{l} 〈 {\hat{φ}}_{\pm} (t) | \partial_{t} φ_{\pm} (t) 〉 = \pm i ω_{L} {cos}^{2} \frac{β}{2}, \\ 〈 {\hat{φ}}_{\mp} (t) | \partial_{t} φ_{\pm} (t) 〉 = (\pm \frac{\dot{β}}{2} - i \frac{ω_{L}}{2} sin β) e^{\pm i ω_{L} t}, \end{array}

which is equivalent to

{\tilde{H}}_{CD} (t) = ℏ [\begin{matrix} \frac{ω_{L}}{2} {sin}^{2} β & (i \frac{\dot{β}}{2} + \frac{ω_{L}}{4} sin 2 β) e^{- i ω_{L} t} \\ (- i \frac{\dot{β}}{2} + \frac{ω_{L}}{4} sin 2 β) e^{i ω_{L} t} & - \frac{ω_{L}}{2} {sin}^{2} β \end{matrix}] .

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

H_{0} (t) = \frac{ℏ}{2} [\begin{matrix} - Δ (t) & Ω_{R} (t) \\ Ω_{R} (t) & Δ (t) - i Γ \end{matrix}] .

By the same procedure as above, H_CD reads

H_{CD} (t) = ℏ [\begin{matrix} 0 & Ω_{a} (t) \\ - Ω_{a} (t) & 0 \end{matrix}],

where Ω_a(t) = iθ̇/2 is the auxiliary pulse with

\dot{θ} = \frac{{\dot{Ω}}_{R} [Δ (t) - i Γ / 2] - Ω_{R} (t) (\dot{Δ} - i \dot{Γ} / 2)}{Ω_{R}^{2} (t) + {[Δ (t) - i Γ / 2]}^{2}},

which plays the role of the Rabi frequency for a fast-driving field. In principle, H_CD(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 Ω_a 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 Ω_a, i.e., setting Ω_a ≃ iIm[Ω_a], 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 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, t_f = 2 × 10⁻¹⁰s, t₀ = 10⁻¹⁰s.

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In the RWA regime, i.e., drop the terms with e ^±2iω_Lt= 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 H̃₀(t) and H₀(t) are chosen to be the Allen-Eberly (AE) drivings [64], i.e.,

\begin{array}{l} Ω_{R} (t) = Ω_{0} sech [\frac{π (t - t_{f} / 2)}{2 t_{0}}], \\ Δ (t) = \frac{2 δ^{2} t_{0}}{π} tanh [\frac{π (t - t_{f} / 2)}{2 t_{0}}] . \end{array}

Here, we select the appropriate parameters [43,63], in the RWA regime, we use Hamiltonian H₀(t) and the counterdiabatic Hamiltonian H_CD(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, H_CD(t) can not be used as the counterdiabatic Hamiltonian to accelerate the adiabatic dynamics governed by the reference Hamiltonian H̃₀(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 H̃₀(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, t_f = 2 × 10⁻¹⁰s, t₀ = 10⁻¹⁰s, ω_L = 2π × 10GHz.

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

3.1. H_CD(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 Ω̄_p(t), while Levels |2〉 and |3〉 are coupled by the Stokes laser Ω̄_s(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,

i ℏ \frac{d}{d t} c (t) = {\bar{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]

{\bar{H}}_{0} (t) = \frac{ℏ}{2} [\begin{matrix} 0 & {\bar{Ω}}_{p} (t) & 0 \\ {\bar{Ω}}_{p}^{*} (t) & 2 Δ_{p} - i Γ & {\bar{Ω}}_{s} (t) \\ 0 & {\bar{Ω}}_{s}^{*} (t) & 2 (Δ_{p} - Δ_{s}) \end{matrix}],

where the detunings are defined by Δ_p = ω_p − (E₂ − E₁)/ħ, Δ_s = ω_s − (E₂ − E₃)/ħ for the Λ configuration. Ω̄_p(t) = Ω_p(t)(1+e^−2iω_pt) and Ω̄_s(t) = Ω_s(t)(1+e^−2iω_st) 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 = Δ_p = Δ_s.

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 H̄₀(t),

\begin{array}{l} | E_{0} (t) 〉 = \frac{1}{Ξ_{0} (t)} [{\bar{Ω}}_{s} (t) | 1 〉 - {\bar{Ω}}_{p}^{*} (t) | 3 〉], \\ | E_{+} (t) 〉 = \frac{1}{Ξ_{1} (t)} [{\bar{Ω}}_{p} (t) | 1 〉 + ε_{+} (t) | 2 〉 + {\bar{Ω}}_{s}^{*} (t) | 3 〉], \\ | E_{-} (t) 〉 = \frac{1}{Ξ_{2} (t)} [{\bar{Ω}}_{p} (t) | 1 〉 + ε_{-} (t) | 2 〉 + {\bar{Ω}}_{s}^{*} (t) | 3 〉], \end{array}

where

Ξ_{0} (t) = \sqrt{{| {\bar{Ω}}_{p} (t) |}^{2} + {| {\bar{Ω}}_{s} (t) |}^{2}}

,

Ξ_{1} (t) = \sqrt{ε_{+}^{2} (t) + Ξ_{0}^{2} (t)}

, and

Ξ_{2} (t) = \sqrt{ε_{-}^{2} (t) + Ξ_{0}^{2} (t)}

. The eigenvalues of H̄₀(t) corresponding to the eigenstates are,

E_{0} (t) = 0, E_{\pm} (t) = \frac{ℏ}{2} ε_{\pm} (t),

with

ε_{\pm} (t) = (Δ - i Γ / 2) \pm \sqrt{{(Δ - i Γ / 2)}^{2} + Ξ_{0}^{2} (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 H̄₀(t) is non-Hermitian, the instantaneous eigenstates of the Hamiltonian

{\bar{H}}_{0}^{†} (t)

have to be considered, which reads

\begin{array}{l} | {\hat{E}}_{0} (t) 〉 = \frac{1}{Ξ_{0} (t)} [{\bar{Ω}}_{s} (t) | 1 〉 - {\bar{Ω}}_{p}^{*} (t) | 3 〉], \\ | {\hat{E}}_{+} (t) 〉 = \frac{1}{Ξ_{1} (t)} [{\bar{Ω}}_{p} (t) | 1 〉 + {\hat{ε}}_{+} (t) | 2 〉 + {\bar{Ω}}_{s}^{*} (t) | 3 〉], \\ | {\hat{E}}_{-} (t) 〉 = \frac{1}{Ξ_{2} (t)} [{\bar{Ω}}_{p} (t) | 1 〉 + {\hat{ε}}_{-} (t) | 2 〉 + {\bar{Ω}}_{s}^{*} (t) | 3 〉], \end{array}

with

{\hat{ε}}_{\pm} (t) = (Δ + i Γ / 2) \pm \sqrt{{(Δ + i Γ / 2)}^{2} + Ξ_{0}^{2} (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:

\begin{array}{l} Π_{0} (t) = | E_{0} (t) 〉 〈 {\hat{E}}_{0} (t) | = \frac{1}{Ξ_{0}^{2} (t)} [\begin{matrix} {| {\bar{Ω}}_{s} (t) |}^{2} & 0 & - {\bar{Ω}}_{p} (t) {\bar{Ω}}_{s} (t) \\ 0 & 0 & 0 \\ - {\bar{Ω}}_{p}^{*} (t) {\bar{Ω}}_{s}^{*} (t) & 0 & {| {\bar{Ω}}_{p} (t) |}^{2} \end{matrix}], \\ Π_{1} (t) = | E_{1} (t) 〉 〈 {\hat{E}}_{1} (t) | = \frac{1}{Ξ_{1}^{2} (t)} [\begin{matrix} {| {\bar{Ω}}_{p} (t) |}^{2} & {\bar{Ω}}_{p} (t) ε_{+} (t) & {\bar{Ω}}_{p} (t) {\bar{Ω}}_{s} (t) \\ {\bar{Ω}}_{p}^{*} (t) ε_{+} (t) & ε_{+}^{2} (t) & {\bar{Ω}}_{s} (t) ε_{+} (t) \\ {\bar{Ω}}_{p}^{*} (t) {\bar{Ω}}_{s}^{*} (t) & {\bar{Ω}}_{s} (t) ε_{+} (t) & {| {\bar{Ω}}_{s} (t) |}^{2} \end{matrix}], \\ Π_{2} (t) = | E_{2} (t) 〉 〈 {\hat{E}}_{2} (t) | = \frac{1}{Ξ_{2}^{2} (t)} [\begin{matrix} {| {\bar{Ω}}_{p} (t) |}^{2} & {\bar{Ω}}_{p} (t) ε_{-} (t) & {\bar{Ω}}_{p} (t) {\bar{Ω}}_{s} (t) \\ {\bar{Ω}}_{p}^{*} (t) ε_{-} (t) & ε_{-}^{2} (t) & {\bar{Ω}}_{s} (t) ε_{-} (t) \\ {\bar{Ω}}_{p}^{*} (t) {\bar{Ω}}_{s}^{*} (t) & {\bar{Ω}}_{s} (t) ε_{-} (t) & {| {\bar{Ω}}_{s} (t) |}^{2} \end{matrix}] . \end{array}

By the same procedure, the Hermitian H̄_CD Hamiltonian becomes

\begin{array}{l} {\bar{H}}_{CD} = i ℏ \sum_{j \neq k} \sum \frac{Π_{j} (t) \partial_{t} {\bar{H}}_{0} (t) Π_{k} (t)}{E_{k} (t) - E_{j} (t)} \\ = \frac{i ℏ}{Ξ_{1}^{2} (t) Ξ_{2}^{2} (t)} [\begin{matrix} {| {\bar{Ω}}_{p} (t) |}^{2} B + C D & {\bar{Ω}}_{p} (t) A - 2 i Γ {\bar{Ω}}_{s} (t) G & {\bar{Ω}}_{p} (t) {\bar{Ω}}_{s} (t) B + C F \\ - {\bar{Ω}}_{p}^{*} (t) A^{*} - 2 i Γ {\bar{Ω}}_{s}^{*} (t) G^{*} & - Ξ_{0}^{2} (t) B & - {\bar{Ω}}_{s} (t) A^{*} - 2 i Γ {\bar{Ω}}_{p} (t) G^{*} \\ - {\bar{Ω}}_{p}^{*} (t) {\bar{Ω}}_{s}^{*} (t) B^{*} - C F^{*} & {\bar{Ω}}_{s}^{*} (t) A + 2 i Γ {\bar{Ω}}_{p}^{*} (t) G & {| {\bar{Ω}}_{s} |}^{2} B - C D \end{matrix}], \end{array}

with

\begin{array}{l} A = (2 Δ - i Γ) [{\bar{Ω}}_{p}^{*} (t) {\dot{\bar{Ω}}}_{p} (t) + {\bar{Ω}}_{s} (t) {\dot{\bar{Ω}}}_{s}^{*} (t)], \\ B = {\bar{Ω}}_{p}^{*} (t) {\dot{\bar{Ω}}}_{p}^{*} (t) + {\bar{Ω}}_{s} (t) {\dot{\bar{Ω}}}_{s}^{*} (t) - H . C ., \\ C = [{| ε_{+} (t) |}^{2} + {| ε_{-} (t) |}^{2} + 2 Ξ_{0}^{2} (t)] / Ξ_{0}^{2} (t), \\ D = {| {\bar{Ω}}_{p} (t) |}^{2} [{\bar{Ω}}_{s}^{*} (t) {\dot{\bar{Ω}}}_{s}^{*} (t) - H . C .] + {| {\bar{Ω}}_{s} (t) |}^{2} [({\bar{Ω}}_{p}^{*} (t) {\dot{\bar{Ω}}}_{p} (t) - H . C .)], \\ F = {\bar{Ω}}_{p}^{2} (t) [{\bar{Ω}}_{s} (t) {\bar{Ω}}_{p}^{*} (t) - {\bar{Ω}}_{p}^{*} (t) {\dot{\bar{Ω}}}_{s}^{*} (t)] + {\bar{Ω}}_{s}^{2} (t) [{\bar{Ω}}_{s}^{*} (t) {\dot{\bar{Ω}}}_{p} (t) - {\bar{Ω}}_{p} (t) {\dot{\bar{Ω}}}_{s}^{*} (t)], \\ G = {\bar{Ω}}_{s}^{*} (t) {\dot{\bar{Ω}}}_{p} (t) - {\bar{Ω}}_{p} (t) {\dot{\bar{Ω}}}_{s}^{*} (t) . \end{array}

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

H_{CD} (t) = i ℏ [\begin{matrix} 0 & A (t) & C (t) \\ - A (t) & 0 & - B (t) \\ - C (t) & B (t) & 0 \end{matrix}],

where

\begin{array}{l} A (t) = sin θ (t) \dot{ϕ} (t), \\ B (t) = cos θ (t) \dot{ϕ} (t), \\ C (t) = \dot{θ} (t), \end{array}

with

\dot{ϕ} (t) = \frac{{\dot{Ω}}^{'} (t) (Δ - i Γ / 2) - Ω^{'} (t) [\dot{Δ} - i \dot{Γ} / 2]}{2 [Ω^{' 2} (t) + {(Δ_{p} - i Γ / 2)}^{2}]}

,

\dot{θ} (t) = \frac{{\dot{Ω}}_{p} (t) Ω_{s} (t) - {\dot{Ω}}_{s} (t) Ω_{p} (t)}{Ω^{2} (t)}

, and

Ω^{'} (t) = \sqrt{Ω_{p}^{2} + Ω_{s}^{2}}

, 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 $Ω_{p} (t) = Ω_{0} exp [- {(\frac{t - τ - t_{f} / 2}{T})}^{2}]$ and $Ω_{s} (t) = Ω_{0} exp [- {(\frac{t + τ - t_{f} / 2}{T})}^{2}]$ . 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 t_f. 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 P_1→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: t_f = 30ns, T = t_f/6, τ = t_f/10, Ω₀ = 2π×200MHz, Δ = 2π×200MHz, Γ = 2π×100MHz, ω_p = 100MHz, ω_s = 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(t_f)〉 and the target state |3〉 to characterize the population transfer efficiency (see Fig. 4), i.e., F = |〈3 |Ψ(t_f)〉|². F = |〈3| Ψ(t_f)〉|². Figs. 4(a) and 4(c) show the fidelity beyond RWA, where we use H_CD(t) (Eq. (28)) as the countdiabatic Hamiltonian. Figs. 4(b) and 4(d) are also for the results beyond RWA, but H̄_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 H_CD(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 H̄₀(t) and the counterdiabatic Hamiltonian H_CD(t). To better the performance, we take H̄_CD(t) (Eq. (26)) as the counterdiabatic Hamiltonian to replace H_CD(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, t_f = 30ns, Ω₀ = 2π × [100, 800]MHz, Γ = 2π × [100, 600]MHz, ω_p = 100MHz, ω_s = 80MHz. In Figs. 4(c) and 4(d), the parameters are set as: t_f = 30ns, Δ = 2π × [100, 800]MHz, Γ = 2π × [100, 600]MHz, ω_p = 100MHz, ω_s = 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) ≫ Ω_p,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]:

H_{eff} (t) = \frac{ℏ}{2} [\begin{matrix} - Δ_{eff} (t) & Ω_{ef f} (t) \\ Ω_{eff}^{*} (t) & Δ_{eff} (t) \end{matrix}],

where Δ_eff(t) and Ω_eff(t) are effective detuning and Rabi frequency, respectively,

\begin{array}{l} Δ_{eff} (t) = - \frac{{\bar{Ω}}_{p} (t) {\bar{Ω}}_{p} (t)}{2 Δ - i Γ}, \\ Ω_{eff} (t) = \frac{{| {\bar{Ω}}_{p} (t) |}^{2} - {| {\bar{Ω}}_{s} (t) |}^{2}}{4 Δ - 2 i Γ} . \end{array}

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]

H_{CD} (t) = \frac{i ℏ}{2 M (t)} [\begin{matrix} 0 & P (t) \\ - P^{*} (t) & Q (t) \end{matrix}],

with

\begin{array}{l} P (t) = {\dot{Ω}}_{eff} (t) Δ_{eff} (t) - {\dot{Δ}}_{eff} (t) Ω_{eff} (t), \\ Q (t) = {\dot{Ω}}_{eff}^{*} (t) Ω_{eff} (t) - {\dot{Ω}}_{eff} (t) Ω_{eff}^{*}, \\ M (t) = {| Ω_{eff} |}^{2} + Δ_{eff}^{2} . \end{array}

Thus the total Hamiltonian reads H(t) = H_eff(t) + H_CD(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

Ω_{p} (t) = Ω_{0} exp [- {(\frac{t - τ - t_{f} / 2}{T})}^{2}]

and

Ω_{s} (t) = Ω_{0} exp [- {(\frac{t + τ - t_{f} / 2}{T})}^{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, t_f = 30ns, T = t_f/6, and τ = t_f/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: t_f = 30ns, T = t_f /6, τ = t_f/10, Ω₀ = 2π × 0.16GHz, Δ = 2π × 2.5GHz, Γ = 2π × 0.16GHz, ω_p = 0.1GHz, ω_s = 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.

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Shortcuts to adiabaticity in non-Hermitian quantum systems without rotating-wave approximation

Abstract

1. Introduction

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

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

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

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

4. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (33)

Optics Express

Abstract

1. Introduction

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

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

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

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

4. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (33)

Optics Express

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