PHYSICAL REVIEW RESEARCH 1, 033018 (2019) Steady state thermodynamics of two qubits strongly coupled to bosonic environments Ketan Goyal* and Ryoichi Kawai Department of Physics, University of Alabama at Birmingham, Birmingham, Alabama 35294, USA (Received 3 June 2019; published 11 October 2019) When a quantum system is placed in thermal environments, we often assume that the system relaxes to the Gibbs state in which decoherence takes place in the system energy eigenbasis. However, when the coupling between the system and the environments is strong, the stationary state is not necessarily the Gibbs state due to environment-induced decoherence, which can be interpreted as a continuous measurement by the environment. Based on the einselection proposed by Zurek, we postulate that the Gibbs state is projected onto the pointer basis due to the continuous measurement. We justify the proposition by exact numerical simulation of a pair of coupled qubits interacting with boson gases. Furthermore, we demonstrate that heat conduction in nonequilibrium steady states can be suppressed in the strong coupling limit also by the environment-induced decoherence. DOI: 10.1103/PhysRevResearch.1.033018 I. INTRODUCTION the system density operator becomes diagonal in the pointer basis under the strong coupling limit. This einselection [18] The laws of thermodynamics and the principles of statis- can be considered as a consequence of continuous measure- tical mechanics tell us that every system eventually reaches ment of the system by the environment. A similar argu- a stationary state known as the Gibbs state, which is the ment can be used for the thermalization processes, and there hallmark of thermal equilibrium. The density operator of have been investigation of thermalization under continuous the Gibbs state is notably a function of only the system measurement [20,21]. We investigate thermalization and heat Hamiltonian and is thus diagonal in the energy eigenbasis. conduction in the strong coupling regime based decoherence The coherence between energy eigenstates is completely de- in the pointer basis. stroyed. Therefore, thermalization to the Gibbs state must involve decoherence between energy eigenstates, presumably induced by the environments surrounding the system. II. THERMALIZATION IN THE POITER BASIS −β Such a decoherence process toward the Gibbs state has G HS Consider a system in the Gibbs state ρ = e /ZS under been investigated under the weak coupling limit [1]. In fact, S the weak coupling, where HS, β, and ZS are system Hamil- quantum master equations based on the Born-Markovian ap- tonian, inverse temperature, and a partition function. When proximation are known to converge to the Gibbs state [2]. the coupling energy becomes significantly larger than the However, it has been shown that the non-Markovian dynamics system energy, the Gibbs state is continuously measured by does not necessarily reach the Gibbs state [3–8]. For a system the environments and thus projected to the pointer basis. Our strongly coupled to the environments, its equilibrium state main proposition is that under the strong coupling limit a cannot be expressed with the system Hamiltonian alone, and system tends to relax to a stationary state given by an effective Hamiltonian based on the potential of mean force ρ t→∞ | |ρG| |, has been developed [9–16]. The resulting stationary state is no S −−−→ pi pi S pi pi (1) longer diagonal in the system energy eigenstates. i Environment-induced decoherence has been intensively in- where |p is the ith pointer state, which we define below. vestigated in the context of quantum measurement theory and i Figure 1 illustrates this proposition. Consider the con- quantum computing [17]. In those theories, the environment vex hull ={ρ = Q |e e |; Q 0 ∧ Q = 1} in does not necessarily induce decoherence in the energy eigen- E i i i i i i i the Liouville space. The corners of the hull represent basis. Zurek [18,19] showed that the decoherence takes place the pure states. Any density operator that is diagonal in among so-called pointer states determined by the coupling | the energy eigenbasis ei is in E, including the Gibbs Hamiltonian between a system and environments. In general, = state (G in Fig. 1). Similarly, the convex hull P {ρ = | | ∧ = } i Pi pi pi ; Pi 0 i Pi 1 contains all possible density operators that are diagonal in the pointer basis |pi. *Present address: Avigo Solutions, LLC, 1500 District Avenue, The density operators in the intersection of the two convex Burlington, MA 01803, USA. hulls are diagonal in both basis sets. A special point I in 1 the figure corresponds to ρ = IS, where IS is an identity dS Published by the American Physical Society under the terms of the operator and dS is the dimension of the system Hilbert space. Creative Commons Attribution 4.0 International license. Further The entropy of the system reaches its maximum value ln dS distribution of this work must maintain attribution to the author(s) at I. As the coupling gets stronger, the steady state deviates and the published article’s title, journal citation, and DOI. from the Gibbs state (G) toward the pointer limit (P) along the 2643-1564/2019/1(3)/033018(5) 033018-1 Published by the American Physical Society KETAN GOYAL AND RYOICHI KAWAI PHYSICAL REVIEW RESEARCH 1, 033018 (2019) S2 whose Hamiltonian is given by ω ω H = 0 σ z + 0 σ z + λ (σ +σ − + σ −σ +), (6) S 2 1 2 2 S 1 2 1 2 z,± where σ , ( = 1, 2) are usual Pauli matrices for the th ω λ qubit and 0 and S are the qubit excitation energy and the internal coupling strength, respectively. We write the energy eigenstates as |e j, ( j = 1,...,4) with eigenvalue e j starting from the ground state. 1 Each qubit S is coupled to its own environment B. The environments are assumed to be ideal Bose gases whose FIG. 1. Schematic representation of Proposition 1. The Gibbs = ω † Hamiltonians are given by HB k (k) a (k)a(k), where state on the convex hull E is projected onto another convex hull † a (k) and a(k) are creation and annihilation operators for P. As the coupling strength increases, the steady state deviates from the kth mode in B. The interaction Hamiltonian between S the Gibbs state (G) along the projection line toward the pointer limit B X ⊗ Y X = (P). The maximal entropy state (I) is located on the intersection of and assumes a simple bilinear form , where σ x = † + and Y k (k)[a (k) a(k)]. The coupling strength the two convex hulls. We note that P is closer to I than G, and the entropy increases as the steady state moves toward the pointer limit. between the system and the kth mode in B is denoted as (k). The pointer states in this model are the simultaneous eigenkets of X1 and X2 and denoted as |p1=|00, |p2= projection line (GP). The projection line is “perpendicular” to |01, |p3=|10, and |p4=|11, where |0 and |1 are the σ x P, meaning that the diagonal elements in the pointer basis eigenkets of . are invariant along the projection line. When the coupling is weak, the stationary state is the Gibbs state, III. MODEL AND NUMERICAL SIMULATION ρ t→∞ ρe | |, S −−−→ jj e j e j (7) j We justify the proposition by numerically investigating the −β −β exact dynamics of a simple spin-boson model. Following the ρe = e j / ei where jj e i e . Under the strong coupling limit, standard open quantum system approach [2], we consider Proposition 1 claims that the stationary state density is given H an isolated system consisting of a small subsystem S and by environments HB. The unitary evolution of the total system ρ t→∞ ρp | |, follows the Liouville–von Neumann equation S −−−→ jj p j p j (8) j ∂ ρ = + + ,ρ , p G i SB [HS HB VSB SB] (2) where ρ =p |ρ |p /Z can be explicitly expressed as ∂t jj j S j S where HB is the Hamiltonian of environment. For simplicity, 1 sinh βλ ρp = ρp = 1 − S , (9a) we assume that the coupling Hamiltonian takes a bilinear 11 44 βω + βλ 4 cosh 0 cosh S form, βλ p p 1 sinh S = ⊗ , ρ = ρ = 1 + . (9b) VSB X Y (3) 22 33 βω + βλ 4 cosh 0 cosh S Now we show the transition from the Gibbs limit (7)to H H where X and Y are operators in S and B, respectively. the pointer limit (8) by numerically solving Eq. (4). As- Furthermore, we assume that [Xk, X] = 0 so that all X share suming that the total system is initially in a product state |p ρ = ρ ⊗ ρ the same eigenkets j , which we shall call pointer states. If (t0 ) S (t0 ) B (t0 ) with the environment in a thermal ρ = −β / there are degenerate subspaces, we choose a particular basis state B (t0 ) exp( HB ) ZB , we obtain a formally ex- in the subspace such that the steady state becomes diagonal in act expression of the system density in the interaction picture the pointer basis. [22] The state of the system is represented by reduced density ←− t t1 − dt1dt2K (t1,t2 ) ρ = T t0 t0 ρ , ρS = TrBρSB, which obeys the equation of motion S (t ) e S (t0 ) (10) d K i ρS = [HS,ρS] + [X,η], (4) where the super operator j is defined by dt − (n) − K(t1, t2 ) = S (t1) K (t1 − t2 ) S (t2 ) where we introduced a new operator, − (d) + +iS (t1) K (t1 − t2 ), S (t2 ) (11) { } η ≡ TrB ρSBY ∈HS. (5) Note that the time evolution of the system needs only limited 1If two qubits share the same environment, decoherence-free sub- information on the state of the environments through η.