Feng-Yang Zhang,Zhi-Qing Feng and Chong Li

1 School of Physics and Materials Engineering,Dalian Minzu University,Dalian 116600,China

2 Department of Physics,Tsinghua University,Beijing,100084,China

3 School of Physics,Dalian University of Technology,Dalian 116024,China

Abstract A shortcut to the adiabatic process is an effective method for quantum information processing.The fast and robust quantum information transfer can be implemented by this method.The energetic cost is an important measurement for the shortcut.In this paper,we investigate how to inhibit the energetic cost in stimulated Raman shortcut-to-adiabatic passage in a three-level system.The energetic cost can be manipulated by adjusting detuning of the system and the energetic cost takes the minimum with one-photon resonance condition.

Keywords: Quantum information,shortcut-to-adiabatic passage,energetic cost

1.Introduction

Coherent manipulation of the quantum state is a crucial process for quantum information processing and quantum computing.How to implement high fidelity quantum state transfer is still an open question .The technique of the stimulated Raman adiabatic passage (STIRAP),which is robust against fluctuations of experimental parameters,has been widely applied in different physical systems [1-4].This technique uses partially overlapping pulses to perfectly implement the population transfer of quantum states.However,this approach requires slow evolution of the quantum system.The decoherence may happen before the quantum information processing has been achieved.Therefore,the fast and highfidelity quantum state transfer is an interesting topic.

On the other hand,several approaches [5] have been proposed to speed up adiabatic quantum state transfer,such as,transitionless quantum driving [6,7],Lewis-Riesenfeld invariant-based inverse engineering[8-11],and dressed states[12,13].The transitionless quantum driving shows that by adding a supplementary Hamiltonian to the original Hamiltonian,so that the system evolves exactly following the adiabatic reference and diabatic transition can be suppressed.This shortcut approach has been demonstrated in recent experiments with optical lattice [14],nitrogen-vacancy (NV)center in diamonds[15,16],and cold atoms[17].Also,many theoretical schemes have been proposed to realize the transitionless process [10,18-23].For example,the scheme of a stimulated Raman shortcut-to-adiabatic passage (STIRSAP)in a quantum three-level system by the transitionless quantum driving has been proposed [18].One considers a three-level system,which is described by a time-dependent Hamiltonian H0(t).The technique of transitionless quantum driving needs to construct an additional counterdiabatic field Hcd(t).So,the evolution of the system is guided by a joint Hamiltonian H0(t) + Hcd(t).This provides a fast and robust approach to quantum state transfer for a three-level system due to the additional counterdiabatic field Hcd(t).However,‘there is no such thing as a free lunch.’ Hence,it is very interesting to study the energetic cost of a counterdiabatic driving field.Recently,a family of energetic cost functionals was introduced in [24].The cost of the implementation of the fast rotation gates was analyzed [25].The relationship between quantum speed limit and energetic cost in shortcuts to adiabaticity was studied [26].It is necessary to investigate the energetic cost of implementing STIRSAP in a quantum threelevel system by the transitionless quantum driving.

In this paper,we discuss how to inhibit the energetic cost of a STIRSAP in a quantum three-level system by the transitionless quantum driving.The expression of energetic cost for the three-level system is obtained.We find that the energetic cost can be controlled by adjusting detuning of the system.Under one-photon resonance condition (Δ=0),the energetic cost takes the minimum.

Figure 1.Λ-type three-level system driven by a pump and a Stokes field.Ωp(Ωs)presents the Rabi frequency of the pump(Stokes)field.Δ is the detuning.

2.Stimulated Raman Shortcut-to-adiabatic passage

We briefly review the STIRSAP in the three-level systems.The configuration of the quantum system is shown in figure 1.The excited state is labelled as |2〉 and two ground states are|1〉 and |3〉.For the two-photon resonance case,the Hamiltonian within the rotating wave approximation is [1]

Here,Ωs(t) and Ωp(t) are Rabi frequencies of Stokes and pump laser fields,and the detuning Δ=(E2- E1)/? - ωs=(E2- E3)/? - ωp,ωsand ωpare the laser frequencies of Stokes and pump laser,and Ej=1,2,3are barebasis state energies.The instantaneous eigenstates are given by

and their corresponding eigenvalues are given by

where parameters θ(t),φ(t),and Ω0(t) are defnied by tanθ(t)=tanφ(t) =Ω0(t) [Δ (t) +andrespectively.When adiabatic conditionsand Ω0? 1/T(T is the operation time)are satisfied,the perfect population transfer from|1〉to|3〉is realized by the dark state |λ0(t)〉.If Stokes and pump pulses are not adiabatic at all,the population is not completely transferred between quantum states |1〉 and |3〉 [27].

Although Stokes and pump pulses are not adiabatic at all,the transitionless quantum driving shows that the fast and perfect population transfer from quantum states|1〉to|3〉can be achieved by adding a supplementary Hamiltonian Hcdto the original Hamiltonian H0.The transitionless quantum driving equates to counter-diabatic driving.Also,a shortcut quantum state transfer is implemented by modifying the pump and Stokes pulses.The total Hamiltonian is [18]

where H0=∑|λn(t)〉En〈λn(t)|and Hcd=i∑|?tλn(t)〉〈λn(t)|,respectively,and Enare eigenvalues of the original Hamiltonian H0and its corresponding instantaneous eigenstates|λn(t)〉.The dynamics of the system will behave ideally adiabatically along the eigenvectors of the original Hamiltonian.For the STIRSAP case,the supplementary Hamiltonian is given by [18]

3.Energetic cost

3.1.Three-level system

In this section,we discuss the energetic cost of implementing STIRSAP in the three-level system by the transitionless quantum driving.The expense of energy in transitionless quantum driving can be written in general as [24]where νt,nis a setup dependent constant,the index of the norm n depends on the nature of the applied fields,and the norm is defined by the Frobenius normwith an operator A.The norm plays the most crucial role in defining the cost of driving.So,the expression of energetic cost can be reduced as [26]

In order to evaluate C(T),we adopt the basis of eigenstates of the original Hamiltonian.So,the energetic cost becomes

For the STIRSAP in the three-level system,according to equations (2),(3) and (7) we obtain the expression of the energetic cost as follows

where the first and second terms are the energetic cost of the adiabatic Hamiltonian H0and the supplementary Hamiltonian Hcd.This explicitly shows that a transitionless evolution has an energetic cost larger than its corresponding adiabatic evolution.C″(T)is independent with parameter φ.Comparing equations (9) with (10),we obtain the inequalityC′(T) ≥C″(T),which means that the energetic cost is inhibited further.In order to avoid the expense of energy in transitionless quantum driving,we need to control system work in a resonance interaction regime.

3.2.Effective two-level system

On one-photon resonance (Δ=0),the Hamiltonian (1) of a three-level system is reduced to an equivalent two-level model with an effective Rabi frequency Ωp(t)/2 and an effective detuning Ωs(t)/2 [28].The effective Hamiltonian can be written as [27,28]

whose instantaneous eigenstates are

with eigenvalues

The supplementary Hamiltonian of transitionless quantum driving is [18]

From equations(10)and(15)we obtain the Σ(T)=C″(T)/2.Therefore,the quantum three-level system is reduced as an effective two-level model,the energetic cost of shortcut population transfer is half only than former.

4.Discussion and conclusion

In recent experiments,the STIRSAP has been realized with a NV center in diamond [16] and cold atoms [17],respectively.An NV center has a relatively long coherence time and the possibility of coherent manipulation at room temperature [29].And it has a rich energy level structure,which includes ground state,metastable,and excited state.The ground state is a spin triplet with a zero-field splitting,Dgs=2.87 GHz,between|ms=0〉and the nearly degenerate sublevels|ms=± 1〉in the absence of external magnetic field[30].The degeneracy of spin sublevels|ms=± 1〉can be removed by an external magnetic field.The three-level system is composed of |ms=0〉,|ms=- 1〉,and |ms=1〉 (or |ms=- 1〉,|ms=1〉,and excited state).Also,this three-level system exists in the cold87Rb atom ensemble,two ground states |F=1,mF=0〉,|F=2,mF=0〉 and one excited state 52P3/2.

In summary,the energetic cost of a STIRSAP in the three-level system was studied.Through calculating energetic cost function,we found that the energetic cost can be inhibited by adjusting detuning Δ.When the detuning Δ=0,the three-level system can be reduced as an effective two-level system.The energetic cost of shortcut population transfer is inhibited further.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No.11505024 and the Natural Science Foundation of Liaoning Province under Grant No.20180550944.