Shortcut to adiabatic passage in a three-level system via a chosen path and its application in a complicated system

Jin-Lei Wu; Xin Ji; Shou Zhang

doi:10.1364/OE.25.021084

1. Introduction

For various quantum technologies such as coherence manipulation of atoms and molecules [1], highly-precise measurement [2], and quantum information processing [3, 4], the ideal method of coherent control of quantum state should be as fast and robust as possible. For manipulating quantum state with interacting fields, the method of resonant pulses is fast but usually sensitive to parameter fluctuations. On the contrary, adiabatic passage (AP) as well as its variants is immune to parameter fluctuations and decoherence but of quite long evolutive time because of the limitation of the adiabatic criterion. In recent several years, many researchers focus on speeding up the evolution of AP but keeping the advantage of robustness against parameter fluctuations and decoherence. Methods [5–9] for constructing “shortcuts to adiabatic passage (STAP)” to speed up AP have been proposed, such as invariant-based inverse engineering [10–12], transitionless quantum driving [13–16], iterative interaction pictures [17–19], and other optimized or extended methods [20–24]. Besides, some remarkable achievements by means of STAP have been implemented in experiment [25–28].

Two basic ideas are widely used to construct STAP for speeding up AP. One is to add counterdiabatic new terms into the original Hamiltonian to cancel non-adiabatic transitions (e.g., counterdiabatic driving [8] and transitionless quantum driving [14]), and the other is to change the time dependence of the original Hamiltonian without adding new terms (e.g., invariant-based shortcuts [10]). Also, the combination of the two ideas has been applied for constructing STAP. For example, multiple Schrödinger dynamics [6] and the dressed-state method [9] change the time dependence of the original Hamiltonian by adding new terms. The combination of the two ideas, in many cases, has showed more advantages for experimental implementations of STAP. For transitionless quantum driving or counterdiabatic driving, the addition of counterdiabatic terms is not easy to achieve in experiment because non-existent or unavailable couplings are involved. For invariant-based shortcuts, the experimental realizability of driving pulses may be a challenge in some cases [29, 30]. For multiple Schrödinger dynamics and the dressed-state method, however, the difficulties mentioned above can be usually avoided, that is to say, both of the driving pulses and Hamiltonian can be designed with highly experimental feasibility [31,32].

In this paper, with the combination of the two basic ideas, we construct a STAP in a three-level system by adding two auxiliary driving pulses and choosing a dressed state acting as evolutive path. The Hamiltonian has the identical form as the original one, and the difficulty of the feasibility of driving pulses does not increase yet. Besides, the population of the intermediate state, generally with dissipation, can be restrained by controlling the introduced parameter. Furthermore, as an example, we apply this method to speed up the AP scheme [33] proposed by Chen et al. for generating the three-dimensional (3D) entanglement between a single atom and a Bose-Einstein condensate (BEC). In such a complicated quantum electrodynamics system including a single ⁸⁷Rb atom and a ⁸⁷Rb BEC trapped, respectively, in two bimodal optical cavities connected by an optical fiber, we greatly simplify the original Hamiltonian, with the assistance of quantum Zeno dynamics, into a relatively simple three-level Hamiltonian. Then we use the proposed STAP method and choose appropriate parameters to successfully accelerate the generation of the 3D entanglement between a single atom and a BEC. In addition, adequate numerical simulations prove that the STAP method has strong robustness against parameter fluctuations and decoherence.

This paper is organized as follows. In Sec. 2, we show our proposal for constructing STAP in a three-level system. In Sec. 3, we use the proposed STAP method in a complicated system to accelerate the AP scheme for generating the 3D entanglement. In Sec. 4, we discuss the effectiveness, feasibility, and robustness of the proposed STAP method in the complicated system by numerical simulations. The conclusion appears in Sec. 5.

2. STAP in three-level system

For a three-level system, e.g., a Λ-type atomic system, there are two lower energy levels |A〉 and |C〉 interacting resonantly with the upper level |B〉 via the time-dependent pump pulse Ω_p (t) and Stokes pulse Ω_s (t), respectively. The Hamiltonian is written as (ħ = 1)

H (t) = Ω_{p} (t) | A 〉 〈 B | + Ω_{s} (t) | B 〉 〈 C | + H . c .,

whose instantaneous eigenstates with corresponding eigenenergies ±Ω(t) and 0, respectively, are

| ψ_{\pm} (t) 〉 = \frac{1}{\sqrt{2}} [\sin θ_{1} (t) | A 〉 + | B 〉 + \cos θ_{1} (t) | C 〉], | ψ_{0} (t) 〉 = \cos θ_{1} (t) | A 〉 - \sin θ_{1} (t) | C 〉,

with

Ω (t) = \sqrt{Ω_{p} {(t)}^{2} + Ω_{s} {(t)}^{2}}

and θ₁(t) = arctan[Ω_p (t)/Ω_s (t)]. We transform time-dependent |ψ_±,₀(t)〉 into the time-independent adiabatic frame by the unitary transformation U₀(t) = ∑_n_=±,0 |ψ_n〉〈ψ_n(t)| with time-independent |ψ_±,0〉 being the basis. Then in the adiabatic frame, Hamiltonian (1) reads

\begin{array}{l} H_{1} (t) = U_{0} (t) H (t) U_{0}^{†} (t) - i U_{0} (t) {\dot{U}}_{0}^{†} (t) \\ = Ω (t) (| ψ_{+} 〉 〈 ψ_{+} | - | ψ_{-} 〉 〈 ψ_{-} |) + \frac{{\dot{θ}}_{1} (t)}{\sqrt{2}} (i | ψ_{+} 〉 〈 ψ_{0} | + i | ψ_{-} 〉 〈 ψ_{0} | + H . c .), \end{array}

in which the second term represents the non-adiabatic coupling. Based on the technique of AP [34], if the adiabatic criterion

| {\dot{θ}}_{1} (t) | ≪ \sqrt{2} Ω (t)

is satisfied well, the non-adiabatic coupling can be neglected, and if the system is initially in the dark state |ψ₀(t)〉 the system will approximatively evolve along |ψ₀(t)〉. Therefore, an arbitrary superposition of |A〉 and |C〉 can be obtained by setting θ₁(t). For example, supposing the evolution starts from t = 0 and ends at t = t_f, one can achieve the quantum state transfer from |A〉 to |C〉 by setting θ₁(0) = 0 and θ₁(t_f) = π/2, but a very long evolutive time is needed because of the limitation of the adiabatic criterion.

Now we combine the two basic ideas to speed up AP. In order to ensure that the Hamiltonian is of the same form as Hamiltonian (1), we choose its form as

H^{'} (t) = [Ω_{p} (t) + Ω_{p}^{'} (t)] | A 〉 〈 B | + [Ω_{s} (t) + Ω_{s}^{'} (t)] | B 〉 〈 C | + H . c .,

where

Ω_{p}^{'} (t)

and

Ω_{s}^{'} (t)

are two auxiliary pulses added into the pump and Stokes pulses, respectively. Then we need to find a dressed state as the evolutive path which can guarantee the same final state as |ψ₀(t_f)〉. This state can be chosen as

| Ψ_{0} (t) 〉 = \cos θ_{2} (t) [\cos θ_{1} (t) | A 〉 + \sin θ_{1} (t) | C 〉] + e^{i ϕ} \sin θ_{2} (t) | B 〉 .

As long as the introduced parameter θ₂(t) satisfies the boundary condition θ₂(0) = θ₂(t_f) = 0 and |Ψ₀(t)〉 acts as the evolutive path, the same final state as |ψ₀(t_f)〉 will be obtained, obviously. In order to form a complete set of orthogonal states, the other two dressed states |Ψ_± (t)〉, which satisfy 〈Ψ_i (t)|Ψ_j (t)〉 = δ_ij (i, j = ±, 0; δ_ij = 1 when i = j or δ_ij = 0 when i ≠ j) and ∑_n_=±,0 |Ψ_n(t)〉〈Ψ_n(t)| = 1, can be chosen as

| Ψ_{\pm} (t) 〉 = \frac{1}{\sqrt{2}} {[\sin θ_{1} (t) \mp i \sin θ_{2} (t) \cos θ_{1} (t)] | A 〉 \mp \cos θ_{2} (t) | B 〉 - [\cos θ_{1} (t) \pm i \sin θ_{2} (t) \sin θ_{1} (t)] | C 〉},

for which we have set ϕ = π/2 for simplicity.

In order to guarantee |Ψ₀(t)〉 can become the evolutive path, $Ω_{p}^{'} (t)$ and $Ω_{s}^{'} (t)$ must be skill-fully designed to eliminate the interactions of |Ψ₀(t)〉 with |Ψ_±(t)〉. Similarly to the transformation from Hamiltonian (1) to (3), now we transform Hamiltonian (4) into a new frame by the unitary transformation U₁(t) = ∑_n_=±,0 |Ψ_n〉〈Ψ_n(t)|. In this new frame, Hamiltonian (4) becomes

\begin{array}{l} H_{1}^{'} (t) = U_{1} (t) H^{'} (t) U_{1}^{†} (t) - i U_{1} (t) {\dot{U}}_{1}^{†} (t) \\ = λ (t) (| Ψ_{+} 〉 〈 Ψ_{+} | - | Ψ_{-} 〉 〈 Ψ_{-} |) + [η_{+} (t) | Ψ_{+} 〉 〈 Ψ_{0} | + η_{-} (t) | Ψ_{-} 〉 〈 Ψ_{0} | + H . c .], \end{array}

with three time-dependent parameters

\begin{array}{l} λ (t) = \cos θ_{2} (t) [Ω_{s}^{'} (t) \cos θ_{1} (t) - Ω_{p}^{'} (t) \sin θ_{1} (t) + Ω (t) \cos 2 θ_{1} (t)] + {\dot{θ}}_{1} (t) \sin θ_{2} (t), \\ η_{\pm} (t) = i {{\dot{θ}}_{1} (t) \cos θ_{2} (t) - \sin θ_{2} (t) [Ω_{s}^{'} (t) \cos θ_{1} (t) - Ω_{p}^{'} (t) \sin θ_{1} (t) + Ω (t) \cos 2 θ_{1} (t)]} \mp [Ω_{p}^{'} (t) \cos θ_{1} (t) + Ω_{s}^{'} (t) \sin θ_{1} (t) + Ω (t) \sin 2 θ_{1} (t) + {\dot{θ}}_{2} (t)] . \end{array}

\begin{matrix} Ω_{p}^{'} (t) = - \sin θ_{1} (t) [{\dot{θ}}_{1} (t) \cot θ_{2} (t) + Ω (t)] - {\dot{θ}}_{2} (t) \cos θ_{1} (t), \\ Ω_{s}^{'} (t) = \cos θ_{1} (t) [{\dot{θ}}_{1} (t) \cot θ_{2} (t) - Ω (t)] - {\dot{θ}}_{2} (t) \sin θ_{1} (t) . \end{matrix}

Because adding

Ω_{p, s}^{'} (t)

cancels the interactions among {|Ψ_0,± (t)〉}, the evolution of the system will exactly follow |Ψ₀(t)〉 with |Ψ₀(0)〉 being the initial state. Then, by controlling θ_1,2(t), one can obtain the same final state as |ψ₀(t_f)〉 without the limitation of the adiabatic criterion, which implies that the evolutive time can be infinitely short in theory. Therefore, a STAP in three-level system will be constructed by changing driving pulses

\begin{matrix} Ω_{p} (t) \to Ω_{p}^{″} (t) = Ω_{p} (t) + Ω_{p}^{'} (t) = - {\dot{θ}}_{1} (t) \sin θ_{1} (t) \cot θ_{2} (t) - {\dot{θ}}_{2} (t) \cos θ_{1} (t), \\ Ω_{s} (t) \to Ω_{s}^{″} (t) = Ω_{s} (t) + Ω_{s}^{'} (t) = - {\dot{θ}}_{1} (t) \cos θ_{1} (t) \cot θ_{2} (t) - {\dot{θ}}_{2} (t) \sin θ_{1} (t) . \end{matrix}

and controlling the boundary values of θ_1,2(t). Besides, it is worth mentioning that in |Ψ₀(t)〉, the population of |B〉 can be controlled by controlling θ₂(t), which will be very useful for depressing the effects of dissipation on the desired evolution of the system.

As an illustration, in the following, we achieve the quantum state transfer from |A〉 to |C〉 by using the proposed STAP method. In order to satisfy the conditions θ₂(0) = θ₂(t_f) = 0, θ₁(0) = 0, and θ₁(t_f) = π/2, we adopt the parameters [35]

θ_{1} (t) = \frac{π t}{2 t_{f}} - \frac{1}{3} \sin (\frac{2 π t}{t_{f}}) + \frac{1}{24} \sin (\frac{4 π t}{t_{f}}), θ_{2} (t) = \frac{Θ}{2} [1 - \cos (\frac{2 π t}{t_{f}})] .

In Figs. 1(a)–1(d), we plot the time-dependence of the populations of the three states |A〉, |B〉 and |C〉 with different values of Θ, for which the population of the state |k〉 (k = A, B, C) is defined by P_k = |〈k| Ψ(t)〉|² with |Ψ (t)〉 being the state at time t of the system governed by Hamiltonian (4). All of Figs. 1(a)–1(d) give the perfect quantum state transfer from |A〉 to |C〉 with an arbitrarily t_f, which indicates that the STAP method we propose is effective. Meanwhile, through comparing Figs. 1(a)–1(d), one can easily find that the value of Θ has great effects on the population of |B〉 during the evolution, and Θ should be as small as possible for constraining the population of |B〉. In the next section, we will apply the proposed STAP method in a complicated system to further show the effectiveness of the STAP method.

Fig. 1 Time-dependence of populations of three states |A〉, |B〉 and |C〉 with different values of Θ and an arbitrarily t_f.

Download Full Size | PDF

3. Application of STAP in complicated system

In this section, we apply the proposed STAP method in the AP scheme [33] proposed by Chen et al. for generating 3D entanglement between a single atom and a BEC. In such a complicated quantum electrodynamics system, a single ⁸⁷Rb atom and a ⁸⁷Rb BEC are trapped, respectively, in two bimodal optical cavities connected by an optical fiber. The interaction Hamiltonian of the whole system is written as

\begin{array}{l} H_{I} (t) = H_{al} (t) + H_{acf}, \\ H_{al} (t) = Ω_{A} (t) {| e_{0} 〉}_{A} 〈 g_{a} | + \sum_{k = L, R} \sqrt{N} Ω_{B} (t) {| E_{k} 〉}_{B} 〈 G_{k} | + H . c ., \\ H_{acf} = \sqrt{N} g_{B} a_{B}_{, L} {| E_{R} 〉}_{B} 〈 G_{0} | + \sqrt{N} g_{B} a_{B}_{, R} {| E_{L} 〉}_{B} 〈 G_{0} | + \sum_{k = L, R} [g_{A} a_{A, k} {| e_{0} 〉}_{A} 〈 g_{k} | +_{v} b_{k} (a_{A, k}^{†} + a_{B, k}^{†})] + H . c ., \end{array}

for which H_al(t) (H_acf) is the interaction between the atom or BEC and the classical laser fields (the cavity-fiber system). If the atom and BEC are prepared in the states |g_a〉_A and |G₀〉_B respectively, and the cavities and fiber modes are in the vacuum states, the system will evolve in a single excitation subspace spanned by

\begin{array}{l} | ϕ_{1} 〉 = {| g_{a} 〉}_{A} {| G_{0} 〉}_{B} {| 0 〉}_{c A} {| 0 〉}_{f} {| 0 〉}_{c B}, | ϕ_{2} 〉 = {| e_{0} 〉}_{A} {| G_{0} 〉}_{B} {| 0 〉}_{c A} {| 0 〉}_{f} {| 0 〉}_{c B}, \\ | ϕ_{3} 〉 = {| g_{L} 〉}_{A} {| G_{0} 〉}_{B} {| L 〉}_{c A} {| 0 〉}_{f} {| 0 〉}_{c B}, | ϕ_{4} 〉 = {| g_{R} 〉}_{A} {| G_{0} 〉}_{B} {| R 〉}_{c A} {| 0 〉}_{f} {| 0 〉}_{c B}, \\ | ϕ_{5} 〉 = {| g_{L} 〉}_{A} {| G_{0} 〉}_{B} {| 0 〉}_{c A} {| L 〉}_{f} {| 0 〉}_{c B}, | ϕ_{6} 〉 = {| g_{R} 〉}_{A} {| G_{0} 〉}_{B} {| 0 〉}_{c A} {| R 〉}_{f} {| 0 〉}_{c B}, \\ | ϕ_{7} 〉 = {| g_{L} 〉}_{A} {| G_{0} 〉}_{B} {| 0 〉}_{c A} {| 0 〉}_{f} {| L 〉}_{c B}, | ϕ_{8} 〉 = {| g_{R} 〉}_{A} {| G_{0} 〉}_{B} {| 0 〉}_{c A} {| 0 〉}_{f} {| 0 〉}_{c B}, \\ | ϕ_{9} 〉 = {| g_{L} 〉}_{A} {| E_{R} 〉}_{B} {| 0 〉}_{c A} {| 0 〉}_{f} {| 0 〉}_{c B}, | ϕ_{10} 〉 = {| g_{R} 〉}_{A} {| E_{L} 〉}_{B} {| 0 〉}_{c A} {| 0 〉}_{f} {| 0 〉}_{c B}, \\ | ϕ_{11} 〉 = {| g_{L} 〉}_{A} {| G_{R} 〉}_{B} {| 0 〉}_{c A} {| 0 〉}_{f} {| 0 〉}_{c B}, | ϕ_{12} 〉 = {| g_{R} 〉}_{A} {| G_{L} 〉}_{B} {| 0 〉}_{c A} {| 0 〉}_{f} {| 0 〉}_{c B} . \end{array}

where the kets |L(R)〉_{c A}₍_cB₎ and |L(R)〉_f denote a single phonon in the L(R)-polarized modes of the cavity A(B) and the fiber, respectively.

According to the theory of quantum Zeno dynamics [36, 37], the eigenstates of the atomcavity-fiber interaction Hamiltonian H_acf can be split into several eigenspaces (i.e., quantum Zeno subspaces), and the eigenstates in the same Zeno subspace are with the same eigenvalue. By choosing the quantum Zeno limit condition Ω_A₍_B₎(t) ≪ g_A₍_B₎, v, the couplings between different subspaces can be ignored, which means that the evolution of the system will be limited in the subspace where the initial state exists in. Because the initial state |ϕ₁〉 is the dark state of H_acf (i.e., H_acf|ϕ₁〉 = 0), the whole system will approximatively evolve in the Zeno subspace consisting of dark states of H_acf,

H_{P} = {| ϕ_{1} 〉, | ϕ_{d} 〉, | ϕ_{11} 〉, | ϕ_{12} 〉},

corresponding to the projections

P^{α} = | α 〉 〈 α |, (| α 〉 \in H_{P}) .

Here,

| ϕ_{d} 〉 = K [\sqrt{N} v g_{B} | ϕ_{2} 〉 - \sqrt{N} g_{A} g_{B} (| ϕ_{5} 〉 + | ϕ_{6} 〉) + v g_{A} (| ϕ_{9} 〉 + | ϕ_{10} 〉)],

with the normalization factor

K = 1 / \sqrt{N v^{2} g_{B}^{2} + 2 N g_{A}^{2} g_{B}^{2} + 2 v^{2} g_{A}^{2}}

. Therefore, we can obtain an effective Hamiltonian of the system as [38, 39]

\begin{array}{l} H_{e f f} (t) = \sum_{α} P^{α} H_{al} (t) P^{α} \\ = K \sqrt{N} v [g_{B} Ω_{A} (t) | ϕ_{1} 〉 〈 ϕ_{d} | + g_{A} Ω_{B} (t) (| ϕ_{11} 〉 + 〈 ϕ_{12} |) 〈 ϕ_{d} |] + H . c .. \end{array}

Here we adopt the same parameters as in Ref. [33], i.e., g_A = g_B = g, v = 100g and N = 10⁴, and hence the effective Hamiltonian (17) becomes

H_{e f f} (t) = Ω_{A} (t) | ϕ_{1} 〉 〈 ϕ_{d} | + \sqrt{2} Ω_{B} (t) | Φ 〉 〈 ϕ_{d} |] + H . c .,

with

| Φ 〉 = (| ϕ_{11} 〉 + | ϕ_{12} 〉) / \sqrt{2}

.

Hamiltonian (18) has the same form as Hamiltonian (1). The target 3D entanglement $| Ψ_{3 D} 〉 = (| ϕ_{1} 〉 - \sqrt{2} | Φ 〉) / \sqrt{3}$ , therefore, can be rapidly achieved based on the STAP method we give in the last section. The necessary conditions are: (i) the pulses $Ω_{A}^{″} (t) = Ω_{p}^{″} (t)$ and $Ω_{B}^{″} (t) = Ω_{s}^{″} (t) / \sqrt{2}$ replace Ω_A(t) and Ω_B(t); (ii) $Ω_{A (B)}^{″} (t) ≪ g$ ; (iii) θ₂(0) = θ₂(t_f) = 0; (iv) θ₁(0) = 0, and $θ_{1} (t_{f}) = - \arctan \sqrt{2}$ . Based on the four necessary conditions above, in the next section we will discuss the effectiveness, feasibility, and robustness of the proposed STAP method in such a complicated system by numerical simulations.

4. Numerical simulations

Firstly we use the pulses $Ω_{A}^{″} (t)$ and $Ω_{B}^{″} (t)$ to replace Ω_A(t) and Ω_B(t) in Hamiltonian (12) to satisfy the condition (i). Then in order to meet the conditions (iii) and (iv), we set θ₁(t) and θ₂(t) as

\begin{array}{l} θ_{1} (t) = - \frac{2 \arctan \sqrt{2}}{π} [\frac{π t}{2 t_{f}} - \frac{1}{3} \sin (\frac{2 π t}{t_{f}}) + \frac{1}{24} \sin (\frac{4 π t}{t_{f}})], \\ θ_{2} (t) = \frac{Θ}{2} [1 - \cos (\frac{2 π t}{t_{f}})] . \end{array}

From Eqs. (10) and (19), we learn that

Ω_{A (B)}^{″} (t)

is strongly dependent on t_f and Θ, and the condition (ii) does not allow a too small t_f as well as Θ. For choosing a pair of appropriate t_f and Θ, in Fig. 2(a) we plot a contour image for the final fidelity F(t_f) = |〈Ψ₃_D| Ψ(t_f)〉² with |Ψ(t_f)〉 being the final state at time t = t_f governed by Hamiltonian (12). Figure 2(a) gives a wide range of the values of t_f and Θ for a relatively high F(t_f). For guaranteeing a relatively small value of Θ but with a high F(t_f), we choose Θ = 0.25. Then, with Θ = 0.25, in Fig. 2(b) we plot F(t_f) versus t_f by using the proposed STAP method. Meanwhile, we also plot F(t_f) versus tf in the AP scheme [33] proposed by Chen et al. with different values of Ω₀. Here, we adopt the same Ω_A₍_B₎(t) as in Ref. [33] but starting at t = 0,

\begin{array}{l} Ω_{A} (t) = Ω_{0} \exp [- {(t - 3 t_{f} / 8 - t_{0})}^{2} / 200 τ^{2}], \\ Ω_{B} (t) = \frac{1}{2} Ω_{0} \exp [- {(t - 3 t_{f} / 8 - t_{0})}^{2} / 200 τ^{2}] + Ω_{0} \exp [- {(t - 3 t_{f} / 8)}^{2} / 200 τ^{2}], \end{array}

with τ = t_f/80 and t₀ = 20τ. Since the adiabatic criterion

| {\dot{θ}}_{1} (t) | ≪ \sqrt{2} Ω (t)

requires either a very long evolutive time or very large eigenenergies, in Fig. 2(b), for the AP scheme a larger Ω₀ will shorten the operation time, which will however consume more resources. Besides, the limitation of the condition (ii) requires that Ω₀ can not be too large, so when Ω₀ = 20g the final fidelity never increases up to near unity even with an infinitely large evolutive time, which indicates that shortening the evolutive time by increasing the system energy is limited. For the STAP scheme, when the final evolutive time is t_f = 25/g the final fidelity is near unity. In following discussions, we pick t_f = 30/g as the final evolutive time for the sake of robustness.

Fig. 2 (a) Contour image for the final fidelity F(t_f) versus Θ and t_f. (b) Contrast between the STAP scheme and AP scheme. The parameters used here are chosen as g_A = g_B = g, v = 100g, N = 10⁴, τ = t_f/80 and t₀ = 20τ.

Download Full Size | PDF

Next, with t_f = 30/g and Θ = 0.25, we plot the time dependence of the pulses $Ω_{A, B}^{″} (t)$ with the amplitude about 0.2g roughly meeting the condition (ii). Now back to Fig. 2(b), with the same value of Ω₀ = 0.2g, the AP scheme needs about t_f = 150/g to reach a same high F(t_f) but the STAP scheme just needs t_f = 30/g, which indicates that the proposed STAP method greatly shortens the evolutive time. The feasibility of the pulses is a crucial element for the implementation of the scheme in practice. From Eqs. (10) and (19), we know that the pulses have very complex time dependence, so for higher experimental feasibility we seek two superpositions of Gaussian functions by curve fitting to replace $Ω_{A, B}^{″} (t)$ , respectively,

{\tilde{Ω}}_{A} (t) = \sum_{m = 1}^{2} Ω_{1 m} \exp [- {(t - τ_{1 m})}^{2} / χ_{1 m}^{2}], {\tilde{Ω}}_{B} (t) = \sum_{m = 1}^{2} Ω_{2 m} \exp [- {(t - τ_{2 m})}^{2} / χ_{2 m}^{2}],

with related parameters {Ω₁₁ = 0.0247g, Ω₁₂ = 0.1885g, τ₁₁ = 5.3194/g, τ₁₂ = 18.09/g, χ₁₁ = 4.9455/g, χ₁₂ = 6.013/g} for

{\tilde{Ω}}_{A} (t)

and {Ω₂₁ = 0.209g, Ω₂₂ = 0.0759g, τ₂₁ = 12.878/g, τ₂₂ = 21.2/g, χ₂₁ = 6.7059/g, χ₂₂ = 5.3892/g for

{\tilde{Ω}}_{B} (t)

. From Fig. 3(a), we see that the curve of

{\tilde{Ω}}_{A, B} (t)

coincides very well with that of

Ω_{A, B}^{″} (t)

. In order to further see the effectiveness of

{\tilde{Ω}}_{A, B} (t)

, in Fig. 3(b) we plot time dependence of the fidelity of adopting

Ω_{A, B}^{″} (t)

or

{\tilde{Ω}}_{A, B} (t)

. The two curves are very close to each other and both reach up to near unity. However, for generating an entangled state, the symbol of success may not only be a high fidelity but also ideal population transfer of quantum states. Therefore, with

{\tilde{Ω}}_{A, B} (t)

, in Fig. 4 we plot the time-dependent populations of the states |ϕ_1,11,12〉 and the sum population of excited states including atom(BEC)-excited states |ϕ_2,9,10〉, cavity-excited states |ϕ_3,4,7,8〉 and fiber-excited states |ϕ_5,6〉. Figure 4 shows that the desired population transfer for the target state |Ψ₃_D〉 is obtained at the final time, and even the excited states are seldom populated during the whole evolution. In a word, Figs. 3(b) and 4 fully demonstrate the effectiveness of

{\tilde{Ω}}_{A, B} (t)

used in the proposed STAP method.

Fig. 3 (a) Time dependence of $Ω_{A, B}^{″} (t)$ and ${\tilde{Ω}}_{A, B} (t)$ . (b) Comparison between the fidelities of adopting $Ω_{A, B}^{″} (t)$ and ${\tilde{Ω}}_{A, B} (t)$ . t_f = 30/g, Θ = 0.25, and other parameters are the same as in Fig. 2.

Download Full Size | PDF

Fig. 4 Time-dependent populations of the states |ϕ_1,11,12〉 and the sum population of excited states with ${\tilde{Ω}}_{A, B} (t)$ . The parameters used here are the same as in Fig. 3.

Download Full Size | PDF

In the following, we investigate the robustness of the STAP scheme against parameter fluctuations. Here we define δx = x′ − x as the deviation of parameter x, for which x denotes the ideal value and x′ denotes the actual value. In Fig. 5(a), we consider the effects of the variations in the parameters on F(t_f) for the target state |Ψ₃_D〉, in which the variations in $\bar{Ω}$ donates the collective variations in Ω₁₁, Ω₁₂, Ω₂₁ and Ω₂₂. As shown in Fig. 5(a) the effects of the variations in $\bar{Ω}$ and t_f on F(t_f) are quite slight, and even when $| δ t_{f} / t_{f} | = | δ \bar{Ω} / \bar{Ω} | = 0.1$ , F(t_f) is still over 0.992, which thus prove that the STAP scheme is robust against parameter fluctuations. Then we consider the effects of decoherence induced by the atom spontaneous radiation and the photon leakage on F(t_f) in Fig. 5(b) by numerically solving the Markov master equation just similar to Ref. [33], in which γ and κ are the spontaneous radiation rate and photon leakage rate, respectively. Though F(t_f) in the STAP scheme is slightly more sensitive to decoherence than that in the AP scheme of Ref. [33], we can still say that the STAP scheme is robust against de-coherence, because in Fig. 5(b), F(t_f) is near 0.98 even when γ = 0.02g and κ = 0.05g but the practical values of γ and κ could be far below these two values under the current experimental conditions [40].

Fig. 5 (a) Effects of variations in $\bar{Ω}$ and t_f on F(t_f). (b) Effects of the decoherence on F(t_f). The parameters used here are the same as in Fig. 3.

Download Full Size | PDF

5. Conclusion

In this paper, we construct a STAP in a three-level system by choosing a dressed state acting as an evolutive path. With the addition of two auxiliary pulses, the couplings between different dressed states are eliminated. Besides, the population of the lossy intermediate state can be depressed by controlling the introduced parameter. Furthermore, the proposed STAP method can be successfully applied in a complicated quantum system to speed up the AP scheme. With adequate numerical stimulations and discussions, we prove the feasibility, effectiveness and robustness of the STAP method in a complicated system. The proposed STAP method could be further applied to the fields such as constructions of quantum gates [41, 42], transports of atoms [43, 44], or other tasks in more complicated quantum systems [45–47].

Funding

National Natural Science Foundation of China (11464046 and 61465013).

References and links

1. P. Král, I. Thanopulos, and M. Shapiro, “Colloquium: Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53 (2007). [CrossRef]

2. T. W. Hänsch, “Nobel lecture: Passion for precision,” Rev. Mod. Phys. 78, 1297 (2006). [CrossRef]

3. J. Stolze and D. Suter, Quantum Computing: A Short Course from Theory to Experiment, 2nd ed. (Wiley-VCH, Berlin, 2008).

4. S. B. Zheng, “Nongeometric conditional phase shift via adiabatic evolution of dark eigenstates: A new approach to quantum computation,” Phys. Rev. Lett. 95, 080502 (2005). [CrossRef] [PubMed]

5. 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] [PubMed]

6. 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]

7. E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117 (2013). [CrossRef]

8. A. del Campo, “Shortcuts to adiabaticity by counterdiabatic driving,” Phys. Rev. Lett. 111, 100502 (2013). [CrossRef]

9. A. Baksic, H. Ribeiro, and A. A. Clerk, “Speeding up adiabatic quantum state tansfer by using dressed states,” Phys. Rev. Lett. 116, 230503 (2016). [CrossRef]

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

11. S. Martínez-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]

12. E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, “Hamiltonian engineering via invariants and dynamical algebra,” Phys. Rev. A 89, 043408 (2014). [CrossRef]

13. M. Demirplak and S. A. Rice, “On the consistency, extremal, and global properties of counterdiabatic fields,” J. Chem. Phys. 129, 154111 (2008). [CrossRef] [PubMed]

14. M. V. Berry, “Transitionless quantum driving,” J. Phys. A: Math. Theor. 42, 365303 (2009). [CrossRef]

15. J. F. Schaff, X. L. Song, P. Capuzzi, P. Vignolo, and G. Labeyrie, “Shortcut to adiabaticity for an interacting Bose-Einstein condensate,” Europhys. Lett. 93, 23001 (2011). [CrossRef]

16. A. del Campo and K. Sengupta, “Controlling quantum critical dynamics of isolated systems,” Eur. Phys. J. Spec. Top. 224, 189 (2015). [CrossRef]

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

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

19. B. H. Huang, Y. H. Chen, Q. C. Wu, J. Song, and Y. Xia, “Fast generating Greenberger–Horne–Zeilinger state via iterative interaction pictures,” Laser Phys. Lett. 13, 105202 (2016). [CrossRef]

20. B. T. Torosov, G. Della Valle, and S. Longhi, “Non-Hermitian shortcut to adiabaticity,” Phys. Rev. A 87, 052502 (2013). [CrossRef]

21. 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]

22. J. Song, Z. J. Zhang, Y. Xia, X. D. Sun, and Y. Y. Jiang, “Fast coherent manipulation of quantum states in open systems,” Opt. Express 24, 21674–21683 (2016). [CrossRef] [PubMed]

23. Q. C. Wu, Y. H. Chen, B. H. Huang, J. Song, Y. Xia, and S. B. Zheng, “Improving the stimulated Raman adiabatic passage via dissipative quantum dynamics,” Opt. Express 24, 22847–22864 (2016). [CrossRef] [PubMed]

24. Y. H. Chen, Z. C. Shi, J. Song, Y. Xia, and S. B. Zheng, “Optimal shortcut approach based on an easily obtained intermediate Hamiltonian,” Phys. Rev. A 95, 062319 (2017). [CrossRef]

25. M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, and O. Morsch, “High-fidelity quantum driving,” Nat. Phys. 8, 147–152 (2012). [CrossRef]

26. Y. X. Du, Z. T. Liang, Y. C. Li, X. X. Yue, Q. X. Lv, W. Huang, X. Chen, H. Yan, and S. L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016). [CrossRef] [PubMed]

27. S. An, D. Lv, A. del Campo, and K. Kim, “Shortcuts to adiabaticity by counterdiabatic driving for trapped-ion displacement in phase space,” Nat. Commun. 7, 12999 (2016). [CrossRef] [PubMed]

28. B. B. Zhou, A. Baksic, H. Ribeiro, C. G. Yale, F. J. Heremans, P. C. Jerger, A. Auer, G. Burkard, A. A. Clerk, and D. D. Awschalom, “Accelerated quantum control using superadiabatic dynamics in a solid-state lambda system,” Nat. Phys. 13, 330–334 (2017). [CrossRef]

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

30. 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]

31. Y. H. Kang, Y. H. Chen, Q. C. Wu, B. H. Huang, J. Song, and Y. Xia, “Fast generation of W states of superconducting qubits with multiple Schrödinger dynamics,” Sci. Rep. 6, 36737 (2016). [CrossRef]

32. J. L. Wu, X. Ji, and S. Zhang, “Fast adiabatic quantum state transfer and entanglement generation between two atoms via dressed states,” Sci. Rep. 7, 46255 (2017). [CrossRef] [PubMed]

33. L. B. Chen, P. Shi, C. H. Zheng, and Y. J. Gu, “Generation of three-dimensional entangled state between a single atom and a Bose-Einstein condensate via adiabatic passage,” Opt. Express 20, 14547–14555 (2012). [CrossRef] [PubMed]

34. 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]

35. Y. H. Kang, Y. H. Chen, Z. C. Shi, J. Song, and Y. Xia, “Fast preparation of W states with superconducting quantum interference devices by using dressed states,” Phys. Rev. A 94, 052311 (2016). [CrossRef]

36. P. Facchi and S. Pascazio, “Quantum Zeno subspaces,” Phys. Rev. Lett. 89, 080401 (2002). [CrossRef] [PubMed]

37. P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. 196, 012017 (2009).

38. X. Q. Shao, L. Chen, S. Zhang, Y. F. Zhao, and K. H. Yeon, “Deterministic generation of arbitrary multi-atom symmetric Dicke states by a combination of quantum Zeno dynamics and adiabatic passage,” Europhys. Lett. 90, 50003 (2010). [CrossRef]

39. J. L. Wu, C. Song, X. Ji, and S. Zhang, “Fast generation of three-dimensional entanglement between two spatially separated atoms via invariant-based shortcut,” J. Opt. Soc. Am. B 33, 2026–2032 (2016). [CrossRef]

40. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005). [CrossRef]

41. 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]

42. M. Palmero, S. Martínez-Garaot, D. Leibfried, D. J. Wineland, and J. G. Muga, “Fast phase gates with trapped ions,” Phys. Rev. A 95, 022328 (2017). [CrossRef]

43. A. Benseny, J. Gillet, and T. Busch, “Spatial adiabatic passage via interaction-induced band separation,” Phys. Rev. A 93, 033629 (2016). [CrossRef]

44. A. Tobalina, M. Palmero, S. Martínez-Garaot, and J. G. Muga, “Fast atom transport and launching in a nonrigid trap,” Sci. Rep. 7, 5753 (2017). [CrossRef] [PubMed]

45. S. Deffner, “Shortcuts to adiabaticity: suppression of pair production in driven Dirac dynamics,” New J. Phys. 18, 012001 (2016). [CrossRef]

46. J. G. Muga, M. A. Simón, and A. Tobalina, “How to drive a Dirac system fast and safe,” New J. Phys. 18, 021005 (2016). [CrossRef]

47. X. K. Song, F. G. Deng, L. Lamata, and J. G. Muga, “Robust state preparation in quantum simulations of Dirac dynamics,” Phys. Rev. A 95, 022332 (2017). [CrossRef]

Shortcut to adiabatic passage in a three-level system via a chosen path and its application in a complicated system

Abstract

1. Introduction

2. STAP in three-level system

3. Application of STAP in complicated system

4. Numerical simulations

5. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (21)

Optics Express