Steady State Thermodynamics of Two Qubits Strongly Coupled to Bosonic Environments

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. spaces could be formed, which are protected from decoherence due In order to demonstrate the proposition, we consider a to symmetry. We avoid the decoherence-free subspace by using two simple model consisting of a pair of identical qubits S1 and independent environments.

033018-2 STEADY STATE THERMODYNAMICS OF TWO QUBITS … PHYSICAL REVIEW RESEARCH 1, 033018 (2019)

+ − S± = , · with anti- ( ) and regular ( ) commutators [X ]±. (a) (c) (d) (n) The dissipation kernel K (t ) and noise kernel K (t )are 0.5 ρ 22 ρ ρ respectively the real and imaginary part of the correlation 22, 33 = 0.4 ρ function C(t ) YB (t )YB (t0 ) 0, where the expectation value 11 ρ is taken with the initial environment state B (t0 ). The time 0.3 ρ ←− 44 ordering operator T in Eq. (10) chronologically orders the ρ , ρ ± 0.2 11 44 superoperators S (t ).

Diagonal elements 0.1 ρ Kato and Tanimura [23] showed that Eq. (10) can be 33 numerically evaluated if the spectral density of environments 0 is of the Drude-Lorentz type: 0.2 (b) (d) 2λγω g(ω) = , (12) 0.1 ρ 2 2 ω + γ 14 0 where γ and λ are the response rate of environment and the overall coupling strength between qubit S and environment ρ ρ , ρ , ρ , ρ -0.1 14 ρ 12 13 24 34 B, respectively. Then, the environmental correlation can be 23 expressed with reasonable accuracy as [24] Off-diagonal elements -0.2

−γ C(t ) ≈ λ[c e + 2 δ(t )], (13) 0123401234 λ λ where c = 2/β − γ − iγ and = γβ/6. B B Following Kato and Tanimura [23], we introduce a set of auxiliary operators FIG. 2. Stationary state density matrix, diagonal (top) and off- diagonal elements (bottom), are plotted as a function of coupling ←− n t −γ (t−s) strength λ . In the left panels, (a) and (b), the matrix is evaluated ζn ,n (t ) = T −i dse G(s) B 1 2 t0 in the eigenbasis of the system Hamiltonian H , and in the right S t t1 − −γ(t −t ) −λ dt1dt2S (t1 )e 1 2 G (t2 ) panels, (c) and (d), the pointer basis is used. The parameter values ×e t0 t0 ω = λ = . T = . γ = . 0 1, S 1 55, 1 5, and B 0 15 are used. The Gibbs t − − −λ dt1S (t1 )S (t1 ) density matrix in the energy eigenbasis is shown as dotted lines at the ×e t0 ρ (t ), (14) S 0 left end, and the strong coupling limit (pointer state limit) predicted by the present proposition is shown as dashed lines at the right end. where G = /β − γ S− − γ S+ . (t ) (2 ) (t ) i j (t ) (15) Index n associated with environment B runs using the energy eigenbasis and the pointer basis. The density from 0 through infinity. Only the first three lowest matrix in the energy eigenbasis shows that the Gibbs state is ρ = order auxiliary operators are needed for S (t ) realized only at the weak coupling limit. The diagonal ele- ζ η = λ ζ − S− ζ η = 0,0(t ), 1 1[ 1,0(t ) i 1 1 (t ) 0,0(t )], and 2 ments deviate from the Gibbs state as the coupling increases. λ ζ − S− ζ 2[ 0,1(t ) i 2 2 (t ) 0,0(t )]. However, the dynamics of The off-diagonal elements indicate that the superposition of auxiliary operators is determined by an infinite set of coupled eigenstates |e1 and |e4 grows rapidly and thus decoherence ODEs or so-called hierarchical equations of motion [23] does not fully take place in the energy eigenbasis. Both the

d − − diagonal and off-diagonal elements approach the pointer limit ζ , t =−γ n + γ n ζ , t − λ S t S t n1 n2 ( ) ( 1 1 2 2 ) n1 n2 ( ) 1 1 1 ( ) 1 ( ) predicted by Eq. (8). dt − − When the matrix elements of the same density operator are + λ2 2S (t )S (t ) ζn ,n (t ) 2 2 1 2 evaluated in the pointer basis, all of the off-diagonal elements − − iλ1 S ζn +1,n (t ) − n1G1(t )ζn −1,n (t ) tend to vanish as the coupling strength increases, suggesting 1 1 2 1 2 − that full decoherence takes place in the pointer basis. The − λ S ζ , + − G ζ , − i 2 2 n1 n2 1(t ) n2 2(t ) n1 n2 1(t ) (16) diagonal elements are remarkably insensitive to the coupling ζ = ζ = strength and in good agreement with Eq. (9b) regardless of with the initial condition n1,n2 (t0 ) 0 except for 0,0(t0 ) ρ = the coupling strength. The invariance of the diagonal elements S (t0 ). The infinite hierarchy is truncated at depth d 50 such that higher depth auxiliary operators do not significantly confirms that the projection is perpendicular to the convex hull contribute to the first two depths. P. (See Fig. 1.) In Fig. 3, the deviation of the steady state from the Gibbs state and its approach√ to the√ pointer limit are measured by fidelity F (ρ,ρ ) = (tr{ ρρ ρ})2.Atλ = 4, IV. RESULTS AND DISCUSSION B the distance between the steady state and the pointer limit First, we investigate the equilibrium situation where the nearly vanishes. initial states of the two environments are identical (λ1 = λ2 ≡ Through the continuous measurement, the environments λ = ≡ γ = γ ≡ γ B, T1 T2 T , 1 2 ). We tried more than ten differ- gain information of the system and the system loses informa- ent initial densities, and all converged to the same stationary tion. Accordingly, the entropy of the system increases [18,25]. state. In Fig. 2, the matrix elements of the stationary state As the coupling gets stronger, more information is expected to density are plotted as a function of the coupling strength λB be lost and thus the entropy goes up monotonically. Figure 3

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0.03 (a) (a) 1.00 λ S=1.55 λ =1.05 0.96 0.02 S λ S=0.55

Fidelity λ 0.92 S=0.10 0.01 0.88 heat current

1.3 (b) 0 (b) 0.1 ρ 1.2 14 Entropy 0.0 ρ ρ 1.1 13, 24

01234 -0.1 ρ ρ , ρ λ 23 12 34

B Off-diagonal elements 01234 ρ FIG. 3. (a) The ﬁdelity between the steady state S and the Gibbs λ B state ρG (dashed line) and between ρS and the pointer limit ρP (solid line). They show that the steady state deviates from the Gibbs state FIG. 4. Vanishing heat due to environment-induced decoherence. and approaches the pointer limit. (b) The entropy of the steady state The upper panel (a) shows the steady state heat current with T = 2 also deviates from the Gibbs limit (dotted line) and approaches the B1 and T = 1. Notably, the heat current vanishes at the strong coupling pointer limit (dashed line). See Fig. 2 for the parameter values. B2 limit. The lower panel (b) shows the decoherence in the pointer basis

for λS = 1.55. The dotted lines are the equilibrium density matrix = + / = . confirms the increase of the von Neumann entropy which at the effective temperature T (T1 T2 ) 2 1 5. The deviation λ = converges to the pointer limit (8) at the strong coupling limit. from the equilibrium density is seen only around B 1, where the As further evidence of continuous measurement by envi- heat current reaches its maximum. ronments, we also investigated a nonequilibrium steady state. When different temperatures are used, heat flows through the asymmetry in temperature, the decoherence power of the two system. Heat from the environment B to the system can be environments becomes nearly identical and eventually the computed as asymmetry in the off-diagonal element responsible for the heat conduction vanishes. =− { ˆ ,η }. J iTrS [X ]− HS (17) In conclusion, we claim that the “thermal equilibrium” of a small quantum system is not the Gibbs state when Figure 4 shows the steady state heat current as a function the coupling to the environments is strong. Because of the of the coupling strength. In the weak coupling regime, the continuous measurement by the environment, the stationary current increases linearly as expected from the linear response state loses the coherency between the pointer states and thus theory. However, the heat current reaches its maximum and the density is diagonal in the pointer basis rather than in the dies off rather quickly as the coupling becomes stronger. This energy eigenbasis. We further claim that the the stationary suppression of heat is predicted earlier as a consequence of state density at the strong coupling limit is the Gibbs state the quantum Zeno effect [26] based on a heuristic argument projected onto the pointer basis. The diagonal elements in the and is observed by Kato-Tanimura [23]. pointer basis appear to be insensitive to the coupling strength. The present results show that indeed the decoherence due We have demonstrated this proposition by exact numerical to environments is responsible for the suppression of heat. The calculation using the hierarchical equations of motion. This off-diagonal elements of the steady state density look almost strong coupling limit can be used as a benchmark test for identical to those in the stationary state at a single effective analytic models such as the Hamiltonian of mean force. temperature T = (T1 + T2 )/2 However, there is small but significant difference where the heat current is strong. The ACKNOWLEDGMENTS elements ρ13 and ρ24 deviate from ρ12 and ρ34 due to the difference in decoherence power between the two environments. We would like to thank Janet Anders, Ala-Nissila, Sa- In general, a higher temperature environment causes stronger har Alipour, and Erik Aurell for helpful discussion during decoherence [27]. However, it also depends on the coupling NORDITA programs. We also thank James Cresser for inter- strength as well. When the coupling strength overcomes the esting discussion.

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