Steady State Thermodynamics of Two Qubits Strongly Coupled to Bosonic Environments

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 AL 35294, USA (Dated: August 13, 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 continuous measurement by the environments. 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 non-equilibrium steady states can be suppressed in the strong coupling limit also by the environment-induced decoherence.

PACS numbers: 03.65.Yz, 05.30.-d, 44.10.+i

I. INTRODUCTION can be used for the thermalization processes, and there have been investigation of thermalization under continu- The laws of thermodynamics and the principles of sta- ous measurement.[20, 21] We investigate thermalization tistical mechanics tell us that every system eventually and heat conduction in the strong coupling regime based reaches a stationary state known as the Gibbs state, decoherence in the pointer basis. which is the hallmark of thermal equilibrium. The density operator of the Gibbs state is notably a function of only the system Hamiltonian and is thus diagonal in the energy eigenbasis. The coherence between energy eigenstates is completely destroyed. Therefore, thermaliza- II. THERMALIZATION IN THE POITER BASIS tion to the Gibbs state must involve decoherence between energy eigenstates, presumably induced by the environ- g −βHs ments surrounding the system. Consider a system in the Gibbs state ρs = e /Zs Such a decoherence process toward the Gibbs state under the weak coupling, where Hs β, and Zs are system has been investigated under the weak coupling limit[1]. Hamiltonian, inverse temperature and a partition func- In fact, quantum master equations based on the Born- tion. When the coupling energy becomes signiﬁcantly Markovian approximation are known to converge to the larger than the system energy, the Gibbs state is contin- Gibbs state.[2]. However, it has been shown that the uously measured by the environments and thus projected non-Markovian dynamics does not necessarily reach the to the pointer basis. Our main proposition is that under Gibbs state.[3–8] For a system strongly coupled to the the strong coupling limit a system tends to relax to a environments, its equilibrium state cannot be expressed with the system Hamiltonian alone, and an eﬀective Hamiltonian based on the potential of mean force has been developed.[9–16] The resulting stationary state is no longer diagonal in the system energy eigenstates. Environment-induced decoherence has been intensively investigated in the context of quantum measurement the-

arXiv:1712.09129v3 [quant-ph] 9 Aug 2019 ory and quantum computing.[17] In those theories, the environment does not necessarily induce decoherence in the energy eigenbasis. Zurek[18, 19] showed that the decoherence takes place among so-called “pointer states” determined by the coupling Hamiltonian between a system and environments. In general, the system density operator becomes diagonal in the pointer basis under the strong coupling limit. This einselection[18] can be FIG. 1. Schematic representation of Proposition (1). The Gibbs state on the convex hull Σ is projected onto another considered as a consequence of continuous measurement e convex hull Σp. As the coupling strength increases, the steady of the system by the environment. A similar argument state deviates from the Gibbs state (G) along the projection line toward the pointer limit (P ). The maximal entropy state (I) is located on the intersection of the two convex hulls. ∗ Current Address: Avigo Solutions, LLC 1500 District Avenue, Noting that P is closer to I than G, the entropy increases Burlington, MA 01803, USA as the steady state moves toward the pointer limit. 2 stationary state given by Energy eigenbasis Pointer basis

t→∞ 1 X g ρs −−−→ |piihpi| ρs |piihpi| (1) Zs 0.5 ρ i 22 ρ ρ 22, 33 0.4 ρ where |pii is the i-th pointer state which we deﬁne below. 11 Figure 1 illustrates this proposition. Consider the con- 0.3 ρ 44 P P ρ , ρ vex hull Σe = {ρ = i Qi |eiihei| ; Qi ≥ 0 ∧ i Qi = 1} 0.2 11 44 in the Liouville space. The corners of the hull repre-

Diagonal elements 0.1 ρ sent the pure states. Any density operator that is di- 33 agonal in the energy eigenbasis |eii is in Σu, including 0 the Gibbs state (G in Fig 1). Similarly, the convex hull 0.2 P P Σp = {ρ = Pi |piihpi| ; Pi ≥ 0 ∧ Pi = 1} contains i i 0.1 ρ all possible density operators that are diagonal in the 14 pointer basis |pii. The density operators in the inter- 0 section of the two convex hulls are diagonal in both ba- -0.1 ρ ρ , ρ , ρ , ρ sis sets. A special point I in the ﬁgure corresponds to 14 ρ 12 13 24 34 1 23 ρ = Is where Is is an identity operator and ds is the ds Off-diagonal elements -0.2 dimension of the system Hilbert space. The entropy of 0 1 2 3 4 0 1 2 3 4 the system reaches its maximum value ln ds at I. As the λ λ coupling gets stronger, the steady state deviates from the B B Gibbs state (G) toward the pointer limit (P ) along the projection line (GP ). The projection line is “perpendic- FIG. 2. Stationary state density matrix, diagonal (top) and ular” to Σp, meaning that the diagonal elements in the oﬀ-diagonal elements (bottom), are plotted as a function of pointer basis are invariant along the projection line. coupling strength λb. In the left panel the matrix is evaluated in the eigenbasis of the system Hamiltonian Hs and in the right panel the pointer basis is used. The parameter val- III. MODEL AND NUMERICAL SIMULATION ues ω0 = 1, λs = 1.55, T = 1.5, γb = 0.15 are used. The Gibbs density matrix in the energy eigenbasis is shown as red dashed lines, and the strong coupling limit (pointer state We justify the proposition by numerically investigating limit) predicted by the present proposition is shown as blue the exact dynamics of a simple spin-boson model. Fol- dashed lines. lowing the standard open quantum system approach[2], we consider an isolated system consisting of a small sub- system Hs and environments Hb. The unitary evolution Note that the time evolution of the system needs only of the total system follows the Liouville–von Neumann limited information on the state of the environments equation through η`. ∂ In order to demonstrate the proposition, we consider i ρ = [H + H + V , ρ ]. (2) a simple model consisting of a pair of identical qubits S ∂t sb s b sb sb 1 and S2 whose Hamiltonian is given by where Hb is the Hamiltonian of environment. For sim- ω0 z ω0 z + − − + plicity, we assume that the coupling Hamiltonian takes a Hs = σ1 + σ2 + λs σ1 σ2 + σ1 σ2 (6) bilinear form 2 2 X where σz,±, (` = 1, 2) are usual Pauli matrices for the `- Vsb = X` ⊗ Y` (3) ` ` th qubit, and ω0 and λs the qubit excitation energy and the internal coupling strength, respectively. We write the where X` and Y` are operators in H and H , respectively. s b energy eigenstates as |eji , (j = 1, ··· , 4) with eigenvalue Furthermore, we assume that [Xk,X`] = 0 so that all X` ej starting from the ground state. share the same eigenkets |pji which we shall call pointer Each qubit S` is coupled to its own environment states. If there are degenerate subspaces, we choose a B`.[22] The environments are assumed to be ideal particular basis in the subspace such that the steady state Bose gases whose Hamiltonians are given by H = becomes diagonal in the pointer basis. b` P ω (k) a†(k)a (k), where a†(k) and a (k) are creation The state of the system is represented by reduced den- k ` ` ` ` ` and annihilation operators for the k-th mode in B`. sity ρs = Trb ρsb which obeys the equation of motion The interaction Hamiltonian between S` and B` assumes x d X a simple bilinear form X` ⊗ Y` where X` = σ` and i ρs = [Hs, ρs] + [X`, η`] (4) h i dt P † ` Y` = k `(k) a`(k) + a`(k) . The coupling strength where we introduced a new operator, between the system and the k-th mode in B` is denoted as `(k). η` ≡ Trb {ρsbY`} ∈ Hs. (5) The pointer states in this model are the simultane- 3 ous eigenkets of X1 and X2 and denoted as |p1i = |0 0i, |p i = |0 1i, |p i = |1 0i, and |p i = |1 1i, where |0i and F(ρ , ρ ) 2 3 4 1.00 S P |1i are the eigenkets of σx. When the coupling is weak, the stationary state is the 0.96 F(ρ , ρ ) Gibbs state S G Fidelity 0.92 t→∞ X e F(ρ , ρ ) ρs −−−→ ρjj |ejihej| (7) 0.88 P G j 1.3 pointer limit e −βej P −βei where ρjj = e / i e . Under the strong coupling limit, Proposition (1) claims that the stationary state 1.2

density is given by Entropy 1.1 t→∞ X p Gibbs limit ρs −−−→ ρjj |pjihpj| (8) 0 1 2 3 4 j λ B p g where ρjj = hpj|ρs |pji /Zs can be explicitly expressed as FIG. 3. The ﬁdelity (upper panel) between the steady state 1 sinh βλ ρ , the Gibbs state ρ , and the pointer limit ρ shows that the ρp = ρp = 1 − s (9a) s g p 11 44 steady state deviates from the Gibbs state and approaches the 4 cosh βω0 + cosh βλs pointer limit. The entropy of the steady (lower panel) state p p 1 sinh βλs also deviates from the Gibbs limit and approaches the pointer ρ22 = ρ33 = 1 + (9b) 4 cosh βω0 + cosh βλs limit. See Fig. 2 for the parameter values. Now we show the transition from the Gibbs limit (7) to the pointer limit (8) by numerically solving Eq. (4). where c` = 2/β` − γ`∆` − iγ` and ∆` = γ`β`/6. Assuming that the total system is initially in a product Following Kato and Tanimura[24], we introduce a set state ρ(t0) = ρs(t0) ⊗ ρb(t0) with the environment in a Q of auxiliary operators thermal state ρb(t0) = ` exp(−β`Hb` )/Zb` , we obtain a formally exact expression of the system density in the ( n ←− Z t ` interaction picture[23] Y −γ`(t−s) ζn1,n2 (t) = T −i ds e G`(s) t0 R t R t1 ` ←− Y − dt1dt2K (t1,t2) t t0 ` ρ (t) = T e 0 ρ (t ) (10) R t R t1 − −γ (t1−t2) s s 0 −λ` dt1dt2S (t1)e ` G`(t2) t0 t0 ` ` ×e R t − − −λ`∆` dt1S (t1)S (t1)o t0 ` ` where the super operator Kj is deﬁned by ×e ρs(t0) (14)

− (n) − K`(t1, t2) = S` (t1) K` (t1 − t2) S` (t2) where − (d) + +iS (t1) K (t1 − t2), S (t2) (11) − + ` ` ` G`(t) = (2/β` − γ`∆`) S` (t) − iγ`Sj (t). (15) ± with anti(+) and regular(-) commutators S = [X`, ·] . ` ± Index n` associated with environment B` runs from (d) (n) The dissipation kernel K` (t) and noise kernel K` (t) 0 through infinity. Only the first three lowest are respectively the real and imaginary part of the corre- order auxiliary operators are needed for ρs(t) = − lation function C`(t) = hYb` (t)Yb` (t0)i0 where the expec- ζ0,0(t), η1 = λ1 ζ1,0(t) − i∆1S1 (t)ζ0,0(t) and η2 = tation value is taken with the initial environment state − ←− λ2 ζ0,1(t) − i∆2S2 (t)ζ0,0(t) . However, the dynamics ρb` (t0). The time ordering operator T in Eq. (10) of auxiliary operators is determined by an infinite set ± chronologically orders the super operators S` (t). of coupled ODEs or so-called hierarchical equations of Kato and Tanimura[24] showed that Eq (10) can be motion[24] numerically evaluated if the spectral density of environments is of the Drude–Lorentz type: d ζ (t) = −(γ n + γ n ) ζ (t) dt n1,n2 1 1 2 2 n1,n2 2λ`γ`ω g`(ω) = (12) − − − − 2 2 − λ1∆1S1 (t)S1 (t) + λ2∆2S2 (t)S2 (t) ζn1,n2 (t) ω + γ` − iλ S−ζ (t) − n G (t)ζ (t) where γ and λ are the response rate of environment 1 1 n1+1,n2 1 1 n1−1,n2 ` ` − and the overall coupling strength between qubit S` and − iλ2 S2 ζn1,n2+1(t) − n2G2(t)ζn1,n2−1(t) (16) environment B , respectively. Then, the environmental ` with the initial condition ζ (t ) = 0 except for correlation can be expressed with reasonable accuracy n1,n2 0 as[25] ζ0,0(t0) = ρs(t0). The infinite hierarchy is truncated at depth d = 50 such that higher depth auxiliary operators −γ` C`(t) ≈ λ` c` e + 2∆` δ(t) (13) do not significantly contribute to the first two depths. ρ ρ 33, 44

ρ ρ 11, 22

IV. RESULTS AND DISCUSSION 0.03 λ =1.55 First, we investigate the equilibrium situation where S λ =1.05 the initial states of the two environments are identical 0.02 S λ (λ1 = λ2 ≡ λb, T1 = T2 ≡ T , γ1 = γ2 ≡ γ). We tried S=0.55 λ more than ten different initial densities, and all converged S=0.10 to the same stationary state. In Fig. 2, the matrix el- 0.01 ements of the stationary state density are plotted as a heat current function of the coupling strength λb using the energy eigenbasis and the pointer basis. The density matrix in 0 the energy eigenbasis shows that the Gibbs state is re- alized only at the weak coupling limit. The diagonal 0.1 ρ elements deviate from the Gibbs state as the coupling 14 increases. The off-diagonal elements indicate that the su- perposition of eigenstates |e i and |e i grows rapidly and 0.0 ρ ρ 1 4 13, 24 thus decoherence does not fully take place in the energy eigenbasis. Both the diagonal and off-diagonal elements -0.1 ρ ρ , ρ approach the pointer limit predicted by Eq. (8). 23 12 34

When the matrix elements of the same density oper- Off-diagonal elements ator are evaluated in the pointer basis, all of the off- 0 1 2 3 4 λ diagonal elements tend to vanish as the coupling strength B increases, suggesting that full decoherence takes place in the pointer basis. The diagonal elements are remarkably insensitive to the coupling strength and in good agree- FIG. 4. Vanishing heat due to environment-induced decoherence. The upper panel shows the steady-state heat current ment with Eq. (9) regardless of the coupling strength. with Tb1 = 2 and Tb2 = 1. Notably, the heat current van- The invariance of the diagonal elements confirms that ishes at the strong coupling limit. The lower panel shows the projection is perpendicular to the convex hull Σ . p the decoherence in the pointer basis for λs = 1.55. The dot- (See Fig. 1.) In Fig. 3 the deviation of the steady state ted lines are the equilibrium density matrix at the effective from the Gibbs state and its approach to the pointer limit temperature T = (T1 + T2)/2 = 1.5. The deviation from the 0 √ 0√ 2 are measured by fidelity F (ρ, ρ ) = tr{ ρ ρ ρ} . At equilibrium density is seen only around λb = 1 where the heat current reaches its maximum. λb = 4, the distance between the steady state and the pointer limit nearly vanishes. Through the continuous measurement, the environments gain information of the system and the system The present results show that indeed the decoherence loses information. Accordingly, the entropy of the sys- due to environments is responsible for the suppression tem increases.[18, 26] As the coupling gets stronger, more of heat. The off-diagonal elements of the steady state information is expected to be lost and thus the entropy density look almost identical to those in the stationary goes up monotonically. Figure 3 confirms the increase of state at a single effective temperature T = (T1 + T2)/2 the von Neumann entropy which converges to the pointer However, there is small but significant difference where limit (8) at the strong coupling limit. the heat current is strong. The elements ρ13 and ρ24 As further evidence of continuous measurement by deviate from ρ12 and ρ34 due to the difference in deco- environments, we also investigated a non-equilibrium herence power between the two environments. In gen- steady state. When different temperatures are used, heat eral a higher temperature environment causes stronger decoherence.[28] However, it also depends on the cou- flows through the system. Heat from the environment B` to the system can be computed as pling strength as well. When the coupling strength over- comes the asymmetry in temperature, the decoherence power of the two environments becomes nearly identical h ˆ i J` = −i Trs X`, η` Hs . (17) − and eventually the asymmetry in the off-diagonal element responsible for the heat conduction vanishes. Figure 4 shows the steady state heat current as a function In conclusion, we claim that the “thermal equilibrium” of the coupling strength. In the weak coupling regime, of a small quantum system is not the Gibbs state when the current increases linearly as expected from the lin- the coupling to the environments is strong. Due to con- ear response theory. However, the heat current reaches tinuous measurement by the environment, the stationary its maximum and dies off rather quickly as the coupling state loses the coherency between the pointer states and becomes stronger. This suppression of heat is predicted thus the density is diagonal in the pointer basis rather earlier as a consequence of the quantum zeno effect[27] than in the energy eigenbasis. We further claim that the based on a heuristic argument and is observed by Kato- the stationary state density at the strong coupling limit Tanimura[24]. is the Gibbs state projected onto the pointer basis. The Elements of density matrix 5 diagonal elements in the pointer basis appear to be insen- NORDITA programs. We also thank James Cresser for sitive to the coupling strength. We have demonstrated interesting discussion. this proposition by exact numerical calculation using the hierarchical equations of motion. This strong coupling limit can be used as a bench mark test for analytic mod- els such as the Hamiltonian of mean force.

ACKNOWLEDGMENTS

We would like to thank Janet Anders, Ala-Nissila, Sa- har Alipour, and Erik Aurell for helpful discussion during

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