Non-Markovian quantum interference in multilevel quantum systems: Exact master equation approach Yusui Chen1,2,∗ J. Q. You1,† and Ting Yu1,2‡ 1Laboratory for Quantum Optics and Quantum Information, Beijing Computational Science Research Center, Beijing 100084, China 2Center for Quantum Science and Engineering and Department of Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA (Dated: December 20, 2017) We study the non-Markovian quantum interference phenomenon of a multi-state atomic system coupled to a bosonic dissipative environment by using the exact master equations derived in this paper. Two examples involving four-level systems with or without external control fields are inves- tigated. Our results show that non-Markovian master equations are capable of exhibiting quantum interference phenomena in a temporal domain that has not fully explored before. In particular, we show that the environmental memory is instrumental in the onset of non-Markovian quantum interference pattern in different time scales. PACS numbers: 03.65.Yz, 03.67.Bg, 03.65.Ud, 32.90.+a I. INTRODUCTION In recent years, the non-Markovian quantum-state dif- fusion equation has provided a powerful tool to study the Open quantum system approach has been used ex- non-Markovian dynamics of open quantum systems cou- tensively recently to study important quantum phenom- pled to a bosonic environment [23–27]. However, general ena involving the system-environment interaction, such non-Markovian master equations are still of fundamental as quantum decoherence, atom emission and absorption, importance for decoherence analysis and quantum dy- quantum dissipation, quantum transport, quantum inter- namics simulations [28–31]. Recently, we have success- ference and quantum control, etc. [1–13]. In the case of fully derived a set of exact non-Markovian master equa- quantum interference, many important observations have tions from the non-Markovian quantum-state diffusion been made for the steady states of quantum multilevel equation [22]. Here, we show that the temporal quantum systems embedded in a continuum medium based on a interference for multilevel systems can also be studied rig- Markov master equation [4]. With the recent advance in orously in the framework of the non-Markovian master non-Markovian open quantum system theory, we will be equations. able to study the temporal behaviour of quantum coher- In this paper, our primary purpose is to study the ence dynamics for different time scales interpolating the quantum inference phenomena in multilevel open systems short-time limit and the long-time limit. However, stud- arising from many important physical processes ranging ies on the dynamics of quantum multilevel systems are from quantum information processing, quantum control, technically difficult because the non-Markovian master quantum coherence transfer, and quantum decoherence equation for the system of interest lacks when the quan- [3–13]. First, we systematically establish a theoretical tum system is coupled to an environment that cannot be approach to deriving the exact non-Markovian master approximated by a Markov reservoir. equations to be used for our investigation on the tempo- The non-Markovian evolution depends on the history ral quantum inference. The idea is that the time-local of the system, known as the memory effect [14–21]. The non-Markovian master equation for a multilevel system non-Markovian master equations of multilevel open sys- may be obtained from the corresponding non-Markovian arXiv:1712.06795v1 [quant-ph] 19 Dec 2017 tems contain all the information about quantum infer- quantum-state diffusion equation. For this purpose, we ence patterns due to couplings to a structured medium. use a simple four-level atom as an example to illustrate Typically the open system and its environment will be- our theoretical methods. Our major results are presented come entangled due to their interactions and the system’s in the second example where we study the quantum in- state will be described by a density matrix [1]. If the in- terference phenomena for a four-level atomic system con- formation of environmental memory recorded in the den- trolled by the external optical pulses. In each case, we sity matrix is accessible, then the interference patterns of show that the exact non-Markovian master equation can the quantum system will be time-dependent, and will be be derived, and can be used to investigate the temporal modified by the memory before the system settles down behaviors of quantum inference. to its steady state. The paper is organized as follows. In Sec. II, we will briefly review the principle ideas of quantum-state diffu- sion equations (QSD) approach, and show how to derive ∗ [email protected] the general master equations for the multilevel systems. † [email protected] In Sec. III, we apply the non-Markovian master equa- ‡ [email protected] tions to investigate two examples with some brief discus- 2 sions on non-Markovian features. We conclude in Sec. operator in the basis of noise, we expand the O operator IV. and rewrite it in the form of t ∗ O(t,s)= O0 + ds1zs1 O1(t,s,s1) II. EXACT NON-MARKOVIAN MASTER ˆ0 EQUATIONS t ∗ ∗ + ds1ds2zs1 zs2 O2(t,s,s1,s2)+ ..., (5) ¨0 For a generic multilevel quantum system coupled to a zero-temperature bosonic environment, the total Hamil- where the terms Oj (j = 0, 1, 2, 3...) are the operators tonian is given by [11] (we set ~ = 1) according to the jth order of the noise. By consistency ∂ δ δ ∂ condition ∗ ψt = ∗ ψt, the O operator can be ∂t δzs δzs ∂t Htot = Hs + Hint + Hb, (1) formally determined by its evolution equation, † † ∗ Hint = L gkbk + L gkbk, ¯ ∗ † † δO Xk Xk ∂ O = −iH + Lz − L O,¯ O − L , (6) t s t δz∗ † s Hb = ωkbkbk, Xk ¯ ∗ t ∗ where O(t,z ) = 0 dsα(t,s)O(t,s,z ). Based on the exact QSD equation´ (4), the reduced density matrix ρ where L is the Lindblad operator of the system, describ- t is recovered from ensemble average over all trajectories. ing the coupling between the system and its environment. The formal master equation can be written as For a dissipative coupling, the Lindblad operator L con- tained in the usual form of interaction term is generally a ∗ † ∂tρt = −i [Hs, ρt]+ LM[z Pt] − L M[OP¯ t] ladder operator in the form of L = |nihn +1|, which t n † ¯† typically satisfies the following condition:P + M[ztPt]L −M[PtO ]L, ∗ LN =0, (2) where ρt = M[|ψt(z )ihψt(z)]. We denote the stochastic ∗ reduced density matrix as Pt = |ψt(z )ihψt(z)|. By the ? where the integer N is the number of energy levels. Note Novikov theorem [ ], the ensemble average can be calcu- † ∗ t ∗ δ lated by M[z Pt]= dsM[z zs]M[ Pt]. Since |ψti is that bk(bk) is the creation (annihilation) operator for the t 0 t δzs δ † kth mode in the bosonic environment and gk is the com- independent of noise´ z , we have M[ P ] = M[P O ]. t δzs t t plex coupling constant between the system and the kth Then the above formal master equation can be further environment mode, described by the correlation function written as of the bath at zero temperature, † † ∂tρt = −i [Hs, ρt]+ L, M[PtO¯ ] − L , M[OP¯ t] . (7) 2 −iωk (t−s) α(t,s)= |gk| e . (3) Xk When the environment is Markov and the correspond- ∗ ing correlation function α(t,s)= δ(t,s), then M[zt Pt]= In the QSD approach, the dynamics of quantum sys- M[PtL] = ρtL, and Eq. (7) is reduced to the master tems is described by a stochastic Schr¨odinger equation. equation of the standard Lindblad form [2]. However, The total state Ψtot for system and environment can O operator generally contains noise in non-Markovian † be expanded in the basis of Bargmann coherent states regime, so M[PtO¯ ] cannot reduce to a simply form. By 1 2 ∗ Ψtot(t) = π d z||zi|ψt(z )i, where ψt is the stochastic repeatedly using Novikov theorem, we can write the nth- wave function´ for the multilevel system. The formal QSD order noise term as equation at zero temperature [23, 24] is M[zs1 zs3 ...zs2n−1 Pt] ∗ ∗ ∗ t ∂tψt(z ) = (−iHs + Lzt ) ψt(z ) δPt = ds α , M zs3 ...zs2 −1 t δ ˆ 2 1 2 n δz∗ † ∗ 0 s2 − L dsα(t,s) ∗ ψt(z ), (4) ˆ δz n n 0 s t δ = ds2, .., ds2n αj,j+1 M Pt , ∗ ∗ iωk t ˆ δz∗ where zt = −i k gkzke is the complex Gaussian 0 j=1 j=1 s2j ∗ Y Y stochastic processP with correlation function M[zt zs] = 2 2 (8) d z −|z| α(t,s). The symbol M[..] = π e .. means ensem- ble average operation over all´ possible stochastic trajec- where αi,j = α(si,sj ). Next, by applying this inference ∗ † tories zt . The QSD equation may be numerically solved of Novikov theorem on M[PtO¯ ], we can have an infinite ∗ by introducing the so-called O operator: O(t,s)ψt(z )= series of increasing power order of O operator and O† δ ∗ ∗ ψt(z ), whose Markov limit is the Lindblad opera- δzs operator. Now let us recall the identity function of O tor L. In most cases, the O operators contain complex operator in Eq. (6), the right side contains up to (N−2)th components. In order to clearly show the structure of O order of noise, while for the left side, the commutator 3 † th term [L O¯j , Ok] generates up to (j + k) order of noise. Next, the ensemble average term R(t) in the formal mas- Naturally, the two sides of Eq.