Elsevier

Physics Letters A

Volume 380, Issue 43, 23 October 2016, Pages 3595-3600
Physics Letters A

Flexible and experimentally feasible shortcut to quantum Zeno dynamic passage

https://doi.org/10.1016/j.physleta.2016.08.051Get rights and content

Highlights

  • We propose a general scheme to speed up quantum Zeno dynamics based on Lyapunov control theory.

  • The acceleration Hamiltonian in our scheme is flexible and experimentally feasible.

  • The acceleration of an entanglement preparation scheme is analyzed to explain our idea in detail.

Abstract

We propose and discuss a theoretical scheme to speed up Zeno dynamic passage by an external acceleration Hamiltonian. This scheme is a flexible and experimentally feasible acceleration because the acceleration Hamiltonian does not adhere rigidly to an invariant relationship, whereas it can be a more general form uj(t)Hcj. Here Hcj can be arbitrarily selected without any limitation, and therefore one can always construct an acceleration Hamiltonian by only using realizable Hcj. Applying our scheme, we finally design an experimentally feasible Hamiltonian as an example to speed up an entanglement preparation passage.

Introduction

In order to achieve some appropriate approximate conditions and to simplify physics system, the evolution speed has to be sacrificed sometimes, which was a common practice in quantum information processing (QIP). For a successful QIP, however, one necessary prerequisite is that the evolution time should be short enough to avoid the influence of decoherence [1], [2]. An ideal solution for reconciling this contradiction is to append an external Hamiltonian in the system in order to ensure that the evolutions are similar to the results adopted approximation conditions. This basic principle of approximation acceleration had been applied to speed up adiabatic passages successfully in recent years [3], [4], [5], [6], [7], [8], [9], [10], but the discussions about the accelerations on other approximations are still rare in both theoretical and experimental areas. In addition, another common defect of existing acceleration schemes is that almost all acceleration Hamiltonians are designed based on a fixed expression (H1(t)=iħn|tλnλn|). There remain some difficulties in achieving those schemes in experiments because the corresponding acceleration Hamiltonian may consist of some non-physical interactions. For example, Chen's scheme [3] needed a transition between two ground states of a Λ-type atom and Lu's scheme [6] required a swap-gate like term |gf12fg|+H.c. in his acceleration Hamiltonian. Detuning driving fields may realize some of those non-physical interactions to some extent [6], [11], [12]. But obtaining an effective interaction is bounded to introduce other approximate conditions. Therefore, it is still an open question to design an acceleration Hamiltonian by only using reasonable interactions.

In this letter, we try to improve above two defects, i.e., (a): The acceleration scheme is extended into other common approximations; (b): A general scheme is proposed (we call it “flexible scheme”) so that the acceleration Hamiltonian, for a certain system, can be always divided to allowed interactions. We believe that such a scheme is universal and feasible in experiments.

In recent years, quantum Zeno dynamic [13], [14] was also an approximation used widely to simplify Hamiltonian in entanglement preparation or quantum gate realization in a long evolution time (gt102) [15], [16], [17], [18], [19], [20], [21]. Unlike adiabatic approximation, Zeno approximation acceleration only requires the system to evolve into a specific subspace instead of a specific state. In other words, Zeno approximation acceleration corresponds to a more relaxed restriction and it is more suitable for flexible designs. Thus, in this letter, we discuss how to speed up a Zeno dynamic process in detail, and present an example of the entanglement preparation to explain the fixed scheme and flexible scheme more intuitively. We demonstrate that the evolution time takes on an obvious reduction after the acceleration, and boundaries of decay rates are also relaxed. And above all, the generators of the flexible acceleration Hamiltonian in our example are exactly the ones of system Hamiltonian, which provides a promising platform for advancing the maneuverability of QIP.

Before the in-depth discussion, we firstly give a brief introduction about the Zeno dynamic and quantum Lyapunov control. Suppose a dynamical evolution of the whole system is governed by the Hamiltonian H=H0+HI=H0+KHm, where H0 is the subsystem Hamiltonian to be investigated and HI=KHm is an additional interaction Hamiltonian to perform the continuous coupling with the constant K. Under the strong coupling condition K, the subsystem investigated is dominated by the time-evolution operator (ħ=1) [13], [14]:U0(t)=limKexp(iKHmt)U(t). On the other hand, the time-evolution operator of this subsystem can also be expressed as: U0(t)=exp(itnPnH0Pn), where Pn is the eigenprojection of the Hm with the eigenvalue λn. The time-evolution operator of the whole system can then be simplified as:U(t)exp(iKHmt)U0(t)=exp[itn(KλnPn+PnH0Pn)], and we can obtain an effective Hamiltonian in the following form: Heff=n(KλnPn+PnH0Pn). Because the Zeno condition requires a weaker H0 compared with Hm, the evolution time will be quite long in this case.

For a quantum system, the aim of quantum control is to make the system evolve to a specified target quantum state (or a target subspace) by designing appropriate time-varying control fields. The core idea of quantum Lyapunov control is to design an auxiliary function V involving both quantum state and control field. V can be regarded as a Lyapunov function if V0 and the system converges to the target state given by its saddle point V=0 [22], [23], [24]. The Lyapunov control theory has demonstrated that the system can be controlled into the target state (subspace) only if the control fields are designed to meet V˙0, i.e., the Lyapunov function is a monotonically nonincreasing function corresponding to whole evolution process in the time domain, and it tends to its minimum finally with the help of control fields [25], [26].

Section snippets

“Rough” acceleration

Similar to adiabatic approximation acceleration, a simple idea of Zeno acceleration is to compensate the terms neglected in the approximate processes, in other words, those terms reappear in system Hamiltonian, that is,HR=UHeffUH0HI, consequently, the total Hamiltonian Ht=H0+HI+HR will be precisely equal to Heff after a diagonalization but without any approximation. Therefore some restrictions of key parameters are no longer necessary, which provides a prerequisite for the process of

Example in entanglement preparation

Entanglement preparation is one of the most important issues in the field of QIP [27], [28]. In this section, we introduce and analyze a general entanglement preparation scheme based on Zeno dynamic to show the necessity of the shortcut and to explain our acceleration scheme in more detail. In the frame of cavity quantum electrodynamic (QED) system, the sketch of this entanglement preparation scheme is shown in Fig. 1. In the interaction picture, the Hamiltonian of the whole system can be

Discussion and outlook

In this letter, we have proposed a flexible and realizable acceleration scheme to speed up Zeno dynamic passage that has already been used widely in QIP. Unlike the efforts to eliminate the effects of neglected terms, the basic idea of our acceleration is to drive the system for evolving within an appointed subspace. The acceleration Hamiltonian is discussed with a general form HF=uj(t)Hcj instead of the traditional fixed form HR=UHeffUH. Thus, our acceleration Hamiltonian can be designed

Acknowledgements

All authors thank Jiong Cheng, Wenzhao Zhang and Yang Zhang for the useful discussion. This research was supported by the National Natural Science Foundation of China (Grant No. 11175033, No. 11574041, No. 11505024 and No. 11447135.) and the Fundamental Research Funds for the Central Universities (DUT13LK05).

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