PHYSICAL REVIEW A 90, 052104 (2014) Exact non-Markovian master equations for multiple qubit systems: Quantum-trajectory approach Yusui Chen,1,* J. Q. You,2,† and Ting Yu1,‡ 1Center for Controlled Quantum Systems and the Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA 2Laboratory for Quantum Optics and Quantum Information, Beijing Computational Science Research Center, Beijing 100084, China (Received 27 March 2014; published 5 November 2014) A wide class of exact master equations for a multiple qubit system can be explicitly constructed by using the corresponding exact non-Markovian quantum-state diffusion equations. These exact master equations arise naturally from the quantum decoherence dynamics of qubit system as a quantum memory coupled to a collective colored noisy source. The exact master equations are also important in optimal quantum control, quantum dissipation, and quantum thermodynamics. In this paper, we show that the exact non-Markovian master equation for a dissipative N-qubit system can be derived explicitly from the statistical average of the corresponding non-Markovian quantum trajectories. We illustrated our general formulation by an explicit construction of a three-qubit system coupled to a non-Markovian bosonic environment. This multiple qubit master equation offers an accurate time evolution of quantum systems in various domains, and paves the way to investigate the memory effect of an open system in a non-Markovian regime without any approximation. DOI: 10.1103/PhysRevA.90.052104 PACS number(s): 03.65.Yz, 03.67.Bg, 03.65.Ud, 32.90.+a I. INTRODUCTION a non-Markovian environment has been studied extensively by employing an exact or an approximate non-Markovian A quantum open system, its temporal evolution governed master equation, which has many experimental applications by a master equation or a stochastic Schrodinger¨ equation, has in quantum device, quantum information, and quantum op- attracted widespread interest due to its applications in various tics [21–25]. research fields such as nonequilibrium quantum dynamics, The exact master equations provide a fundamental de- quantum control, quantum cooling, quantum decoherence, scription to non-Markovian quantum open systems. Even and quantum dissipation [1–12]. The quantum dynamics of if the exact stochastic Schrodinger¨ equation is known, it an open system is commonly formulated in the system plus is still highly desirable to derive the corresponding master environment framework where the state of the open system is equation due to its conceptual importance in understanding described by a reduced density operator. Typically, deriving quantum decoherence and quantum-classical transition as well the master equation governing the reduced density operator as its wide applications in quantum optics, condensed matter involves several important elements regarding fine details of physics, and quantum information processing [1,12]. the environment and the coupling between the system and In the case of a non-Markovian open system, deriving environment. In the conventional quantum optics where the a non-Markovian master equation is a notoriously difficult quantized radiation field is treated as an environment, the problem due to the lack of a systematic tool that is applicable master equation for an atomic system weakly coupled to to a generic open quantum system irrespective of the system- the radiation field is systematically derived, which applies the environment coupling strength and the environment frequency Markov approximation and takes the standard Lindblad form distribution [26,27]. For a quantum system coupled to a (setting = 1) [13] bosonic or fermionic bath, a systematic method is formulated =− + † − † − † called non-Markovian quantum-state-diffusion method (QSD) ρ˙ i[Hs,ρ] (2Li ρLi Li Li ρ ρLi Li ). (1) i or stochastic Schrodinger¨ equation approach [28–30]. In the non-Markovian QSD method, the quantum dynamics Here, Hs is the Hamiltonian of the system of interest, and Li represented by a stochastic differential equation is driven by are a set of system operators called Lindblad operators which ∗ a Gaussian type of process zt . By construction, applying couple the system to the environment. the ensemble average on all possible stochastic processes, An environment can bring about various physical phenom- one can get the reduced density matrix of the interested ena to the open quantum system [12,14]. For example, in system. For many models, such as multilevel atom and mul- the case of two-qubit system coupled to two local bosonic tiple qubit system, the exact non-Markovian dynamics have baths, a Markov environment typically induces both irre- been numerically studied by using the non-Markovian QSD versible decoherence and disentanglement [15,16]. However, approach [31–36]. the non-Markovian environment with a finite memory time can It is known that the Markov master equation for the open assist in regenerating quantum coherence and entanglement system may be derived from the corresponding stochastic in the system [17–20]. Some interesting physics induced by unrevealing [14,37–39]. There are also some examples in the non-Markovian case where the exact master equation can be recovered from the non-Markovian QSD equation, *[email protected] but we need to point out that these works are derived in †[email protected] special conditions, such as the single-spin system [31,33,36] ‡[email protected] or quantum Brownian motion [40,41]. For a multiple qubit 1050-2947/2014/90(5)/052104(6) 052104-1 ©2014 American Physical Society YUSUI CHEN, J. Q. YOU, AND TING YU PHYSICAL REVIEW A 90, 052104 (2014) system, deriving the exact master equations from the stochastic A. Non-Markovian QSD equation Schrodinger¨ equation is still an open problem. Unlike creating The non-Markovian diffusive stochastic Schrodinger¨ equa- thousands of trajectories to recover the reduced density matrix tion is given by [29] in the QSD approach, the exact master equation can be solved ∗ ∗ ∗ ∂t ψt (z ) = (−iHs + Lz )ψt (z ) deterministically, so that it can significantly improve the nu- t merical efficiency. More importantly, the exact master equation t − † δ ∗ may allow an analytical solution of non-Markovian dynamics. L dsα(t,s) ∗ ψt (z ), (3) 0 δzs Such kinds of analytical evaluations give rise to useful infor- ∗ where ψt (z ) is the pure stochastic wave function of the mation about quantum dissipation and decoherence in the non- ∗ ∗ =− iωk t three-qubit system, and zt i k gkzk e is the complex Markovian regime. In this paper, we present a generic exact M ∗ = Gaussian stochastic process with zero mean [zt ] 0, master equation for a multiple qubit dissipative system coupled ∗ ∗ ∗ and correlations M[z z ] = 0 and M[z z ] = α(t,s). Note to a non-Markovian bosonic bath. The methodology used in t s t s that α(t,s) is the correlation function of the bath, which this paper can also be extended to a multilevel atomic system determines environment memory time and dictates the tran- coupled to a quantized radiation field [42]. Our exact master sition from non-Markovian to Markov regimes. The symbol equation provides a systematic tool in dealing with quantum 2 2 M[...] = d z e−|z| ... means ensemble average operation coherence and optimal quantum control in a non-Markovian π ∗ regime [43,44]. on all stochastic trajectories zt . Our paper is organized as follows. In Sec. II, we introduce The stochastic Schrodinger¨ equation (3) can be transformed into a time-local form when the functional derivative of noise a three-qubit system and show the principle idea and the detail ∗ ∗ δ ∗ is replaced by an operator O(t,s,z )ψt (z ) = ∗ ψt (z ) acting of analytical derivation of exact master equation for the three- δzs qubit system. In Sec. III, we show some numerical simulation on the system’s current state. In the Markov limit O operator must be the same as Lindblad operator L, therefore, the results by applying the new master equation approach. In ∗ Sec. IV, we start a general discussion on the derivation of consistent initial condition for O operator is O(t,t,z ) = L.By ∂ δ δ ∂ the consistency condition ∗ ψt = ∗ ψt ,theO operator the master equation for the N-qubit system. ∂t δzs δzs ∂t satisfies the following time-evolution equation ∗ = − + ∗ − † ¯ ∗ ∗ ∂t O(t,s,z ) [ iHs Lzt L O(t,z ),O(t,s,z )] II. EXACT NON-MARKOVIAN MASTER EQUATION δO¯ (t,z∗) − L† , (4) An N-qubit system representing a carrier of quantum δz∗ s information or memory is assumed to be coupled to one ¯ ∗ = t ∗ where O(t,z ) 0 dsα(t,s)O(t,s,z ). or more dissipative environments described by a set of For the three-qubit system with dissipative coupling, the harmonic oscillators. To be specific, now we consider a functional expansion of the O operator contains at most the three-qubit model to illustrate our method of deriving the exact two-fold noises [36] master equation from the non-Markovian QSD equation for a t multiple qubit system. A more generic N-qubit model can O(t,s,z∗) = O (t,s) + ds z∗ O (t,s,s ) 0 1 s1 1 1 be treated in a similar way. The total Hamiltonian for our 0 three-qubit system coupled to a bosonic bath may be written t + ds ds z∗ z∗ O (t,s,s ,s ), (5) as [8] 1 2 s1 s2 2 1 2 0 where O0(t,s),O1(t,s,s1),O2(t,s,s1,s2) are three 8 × 8ma- Htot = Hs + Hint + Hb, trices not containing any noise. One can get the evolution equations for O by plugging the solution (5) into Eq. (4) 3 2 i ωj j x x y y ∗ Hs = σ + Jxy σ σ + + σ σ + , δO¯ (t,z ) 2 z j j 1 j j 1 [−iH + Lz∗ − L†O¯ (t,z∗),O(t,s,z∗)] − L† j=1 j=1 s t δz∗ (2) s = † + † t Hint L gkbk L gkbk, = + ∗ ∂t O0(t,s) ∂t ds1zs O1(t,s,s1) k k 1 0 † t H = ω b b , ∗ ∗ b k k k + ∂ ds ds z z O (t,s,s ,s ).