Steady state thermodynamics of two qubits strongly coupled to bosonic environments
Ketan Goyal Ryoichi Kawai
quant-ph




Current Address: Avigo Solutions, LLC 1500 District Avenue, Burlington, MA 01803, USA

Department of Physics, University of Alabama at Birmingham, Birmingham AL 35294, USA



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. 





03.65.Yz, 05.30.-d, 44.10.+i

Steady state thermodynamics of two qubits strongly coupled to bosonic environments
    Ryoichi Kawai
    December 30, 2023
==================================================================================





§ INTRODUCTION


The laws of thermodynamics and the principles of statistical mechanics tell us that every system eventually reaches a stationary state  known as the Gibbs state, 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, thermalization to the Gibbs state must involve decoherence between energy eigenstates, presumably induced by the environments surrounding the system. 

Such a decoherence process toward the Gibbs state has been investigated under the weak coupling limit<cit.>. In fact, quantum master equations based on the Born-Markovian approximation are known to converge to the Gibbs state.<cit.>.  However, it has been shown that the non-Markovian dynamics does not necessarily reach the Gibbs state.<cit.>  For a system strongly coupled to the 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.<cit.>  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 theory and quantum computing.<cit.>  In those theories, the environment does not necessarily induce decoherence in the energy eigenbasis.  Zurek<cit.> 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<cit.> can be considered as a consequence of continuous measurement of the system by the environment. A similar argument can be used for the thermalization processes, and there have been investigation of thermalization under continuous measurement.<cit.>   We investigate thermalization and heat conduction in the strong coupling regime based decoherence in the pointer basis.



§ THERMALIZATION IN THE POITER BASIS

Consider a system in the Gibbs state ρ_s^g=e^-β H_s/Z_s under the weak coupling, where H_s  β, and Z_s  are system Hamiltonian, inverse temperature and a partition function.  When the coupling energy becomes significantly larger than the system energy, the Gibbs state is continuously measured by the environments and thus projected to the pointer basis. Our main proposition is that under the strong coupling limit a system tends to relax to a stationary state  given by

    ρ_s1/Z_s∑_i  p_iρ_s^gp_i

where  |p_i⟩ is the i-th pointer state which we define below.  

Figure <ref> illustrates this proposition.  Consider the convex hull Σ_e = {ρ=∑_i Q_i e_i ;  Q_i ≥ 0 ∧∑_i Q_i = 1 } in the Liouville space.  The corners of the hull represent the pure states.  Any density operator that is diagonal in the energy eigenbasis |e_i⟩ is in Σ_u, including the Gibbs state (G in Fig <ref>).  Similarly, the convex hull  Σ_p = {ρ=∑_i P_i p_i;  P_i ≥ 0 ∧∑_i P_i = 1 } contains all possible density operators that are diagonal in the pointer basis |p_i⟩.  The density operators in the intersection of the two convex hulls are diagonal in both basis sets.  A special point I in the figure corresponds to ρ = 1/d_s I_s where I_s is an identity operator and d_s is the dimension of the system Hilbert space.  The entropy of the system reaches its maximum  value ln d_s at I.  As the coupling gets stronger, the steady state deviates from the Gibbs state (G) toward the pointer limit (P) along the projection line (GP).   The projection line is “perpendicular” to Σ_p, meaning that the diagonal elements in the pointer basis are invariant along the projection line.



§ MODEL AND NUMERICAL SIMULATION

We justify the proposition by numerically investigating the exact dynamics of a simple spin-boson model.
Following the standard open quantum system approach<cit.>, we consider an isolated system
consisting of a small subsystem ℋ_s and environments  
ℋ_b.  
The unitary evolution of the total system follows the Liouville–von Neumann equation

    i tρ_sb = 
    H_s+ H_b+V_sbρ_sb.

where H_b is the Hamiltonian of environment. For simplicity, we assume that the coupling Hamiltonian takes a bilinear form  
    V_sb = ∑_ℓ X_ℓ⊗ Y_ℓ

where X_ℓ and Y_ℓ are operators in ℋ_s and  ℋ_b, respectively.  Furthermore, we assume that X_kX_ℓ=0 so that all X_ℓ share the same eigenkets |p_j⟩ which we shall call pointer states. If there are degenerate subspaces, we choose a particular basis in the subspace such that the steady state becomes diagonal in the pointer basis.

The state of the system is represented by reduced density ρ_s = _bρ_sb which obeys the equation of motion

    i tρ_s = H_sρ_s + ∑_ℓX_ℓη_ℓ

where we introduced a new operator,

    η_ℓ≡_b{ρ_sb Y_ℓ}  ∈ℋ_s .

Note that the time evolution of the system needs only limited information on the state of the environments through η_ℓ.

In order to demonstrate the proposition, we consider a simple model consisting of a 
pair of identical qubits S_1 and S_2 whose Hamiltonian is given by 

    H_s = ω_0/2σ^z_1	+ 
    ω_0/2σ^z_2 + λ_s(σ^+_1σ^-_2 + σ^-_1σ^+_2 )
 
where σ_ℓ^z,±,(ℓ=1,2) are usual Pauli matrices for the ℓ-th qubit, and ω_0 and λ_s 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.

Each qubit S_ℓ is coupled to its own environment  B_ℓ.[If two qubits share the same environment, decoherence-free subspaces could be formed, which is protected from decoherence due to symmetry.  We avoid the decoherence free subspace by using two independent environments.]
The environments are assumed to be ideal Bose gases whose Hamiltonians are given by H_b_ℓ 
= ∑_k 
ω_ℓ(k)  a^†_ℓ(k) a_ℓ(k), where a^†_ℓ(k) 
and 
a_ℓ(k) are creation and annihilation operators for the k-th mode in B_ℓ. The interaction Hamiltonian between S_ℓ and B_ℓ assumes a simple bilinear form X_ℓ⊗ Y_ℓ
where X_ℓ=σ^x_ℓ and Y_ℓ = 
∑_k ϵ_ℓ(k) [a^†_ℓ(k) + a_ℓ(k) ].  The coupling strength between the system and the k-th mode in B_ℓ is denoted as ϵ_ℓ(k).   



The pointer states in this model are the simultaneous eigenkets of X_1 and X_2 and denoted as |p_1⟩=|0 0⟩, |p_2⟩=|0 1⟩, |p_3⟩=|1 0⟩, and |p_4⟩=|1 1⟩, where |0⟩ and |1⟩ are the eigenkets of σ^x.


When the coupling is weak, the stationary state is the Gibbs state

    ρ_s∑_j ρ^e_jje_j

where ρ^e_jj = e^-β e_j /∑_i e^-β e_i.  Under the strong coupling limit,
Proposition (<ref>) claims that the stationary state density is given by

    ρ_s∑_j ρ_jj^pp_j

where ρ^p_jj = p_jρ_s^gp_j/Z_s can be explicitly expressed as 


    ρ_11^p=ρ_44^p   =   1/4 ( 1- 
        sinhβλ_s/coshβω_0 + coshβλ_s )
    ρ_22^p=ρ_33^p   =   1/4 ( 1+ 
        sinhβλ_s/coshβω_0 + coshβλ_s )
    



Now we show the transition from the Gibbs limit (<ref>) to the pointer limit (<ref>) by numerically solving Eq. (<ref>). 
Assuming that the total system is initially in a product state ρ(t_0) = ρ_s (t_0) ⊗ρ_b (t_0) with the environment in a thermal state ρ_b(t_0) = ∏_ℓexp(-β_ℓ H_b_ℓ)/Z_b_ℓ, we obtain a formally exact expression of the system density in the interaction picture<cit.>
   
    ρ_s(t) = 𝒯∏_ℓ  
        e^-∫_t_0^t∫_t_0^t_1t_1t_2𝒦_ℓ(t_1,t_2)ρ_s(t_0)

where the super operator 𝒦_j is defined by

    𝒦_ℓ(t_1,t_2)    =   𝒮^-_ℓ(t_1)   
        K^(n)_ℓ(t_1-t_2) 𝒮^-_ℓ(t_2) 
          + i 𝒮^-_ℓ(t_1)  K^(d)_ℓ(t_1-t_2), 
        𝒮^+_ℓ(t_2)

with anti(+) and regular(-) commutators 𝒮^±_ℓ=X_ℓ·_±.   The 
dissipation kernel K^(d)_ℓ(t) and noise kernel K^(n)_ℓ(t) 
are respectively the real and imaginary part of the correlation function 
C_ℓ(t)=Y_b_ℓ(t) Y_b_ℓ(t_0)_0 where the expectation value is taken 
with the initial environment state ρ_b_ℓ(t_0). The time 
ordering operator 𝒯 in Eq. (<ref>) chronologically orders the 
super operators 𝒮^±_ℓ(t).



Kato and Tanimura<cit.> showed that Eq (<ref>) can be numerically evaluated if  the spectral density of environments is of the Drude–Lorentz type:

    g_ℓ(ω)= 
    2 λ_ℓγ_ℓω/ω^2+γ_ℓ^2

where γ_ℓ and λ_ℓ are the response rate of environment and 
the overall coupling strength between qubit S_ℓ and environment B_ℓ, respectively.
Then, the environmental correlation can be expressed with reasonable accuracy as<cit.>

    C_ℓ(t)    ≈   λ_ℓ [ c_ℓ  e^-γ_ℓ + 2 Δ_ℓ δ(t)  ]

where c_ℓ = 2/β_ℓ -γ_ℓΔ_ℓ - i γ_ℓ and Δ_ℓ = γ_ℓβ_ℓ / 6. 



Following Kato and Tanimura<cit.>,  we introduce a set of auxiliary operators

    ζ_n_1,n_2(t) =  𝒯∏_ℓ{[ -i∫_t_0^ts e^-γ_ℓ (t-s)𝒢_ℓ (s) ]^n_ℓ . 
     . ×  e^- λ_ℓ∫_t_0^t∫_t_0^t_1t_1t_2𝒮^-_ℓ(t_1) e^-γ_ℓ(t_1-t_2)𝒢_ℓ(t_2)  . 
     . ×  e^- λ_ℓΔ_ℓ∫_t_0^tt_1𝒮^-_ℓ(t_1) 𝒮^-_ℓ(t_1) }ρ_s(t_0)

where 

    𝒢_ℓ(t) = ( 2/β_ℓ - γ_ℓΔ_ℓ ) 𝒮^-_ℓ(t) - i γ_ℓ𝒮^+_j(t).

Index n_ℓ associated with environment B_ℓ runs from 0 through infinity. Only the first three lowest order auxiliary operators are needed for   ρ_s(t) = ζ_0,0(t), η_1 = λ_1 [ζ_1,0(t) - i Δ_1𝒮^-_1(t) ζ_0,0(t)  ] and η_2 = λ_2 [ζ_0,1(t) - i Δ_2𝒮^-_2(t) ζ_0,0(t)  ].  However, 
the dynamics of auxiliary operators is determined by an infinite set of coupled ODEs or so-called  hierarchical equations of motion<cit.>

    tζ_n_1,n_2 (t) =  -(γ_1 n_1 + γ_2 n_2) ζ_n_1,n_2(t) 
    
    - [λ_1Δ_1𝒮^-_1(t)𝒮^-_1(t)+λ_2Δ_2𝒮^-_2(t)𝒮^-_2(t)  ] ζ_n_1,n_2(t) 
    
    -iλ_1 [𝒮^-_1ζ_n_1+1,n_2(t) - n_1𝒢_1(t) ζ_n_1-1,n_2(t) ] 
    
    -iλ_2 [𝒮^-_2ζ_n_1,n_2+1(t) - n_2𝒢_2(t) ζ_n_1,n_2-1(t)]

with the initial condition ζ_n_1, n_2(t_0) = 0 except for ζ_0, 0(t_0) = ρ_s(t_0).   The infinite hierarchy is truncated at depth d=50 such that higher depth auxiliary operators do not significantly  contribute to the first two depths.   



§ RESULTS AND DISCUSSION


First, we investigate the equilibrium situation where the initial states of the two environments are identical (λ_1=λ_2≡λ_b, T_1=T_2≡ T, γ_1=γ_2≡γ). We tried more than ten different initial densities, and all converged to the same stationary state.
In Fig. <ref>, the matrix elements of the stationary state density are plotted as a function of the coupling strength λ_b using the energy eigenbasis and the pointer basis.
The density matrix in the energy eigenbasis shows that the Gibbs state is realized only at the weak coupling limit. The diagonal elements deviate from the Gibbs state as the coupling increases. The off-diagonal elements indicate that the superposition of eigenstates |e_1⟩ and |e_4⟩ grows rapidly and thus decoherence does not fully take place in the energy eigenbasis.  Both the diagonal and off-diagonal elements approach the pointer limit predicted by Eq. (<ref>).

When the matrix elements of the same density operator are evaluated in the pointer basis, all of the off-diagonal elements tend to vanish as the coupling strength increases, suggesting that full decoherence takes place in the pointer basis.  The diagonal elements are remarkably insensitive to the coupling strength and in good agreement with Eq. (<ref>) regardless of the coupling strength.  The invariance of the diagonal elements confirms that the projection is perpendicular to the convex hull Σ_p. (See Fig. <ref>.) In  Fig. <ref> the deviation of the steady state from the Gibbs state and its approach to the pointer limit are measured by fidelity F(ρ,ρ') =  ( {√(ρ) ρ' √(ρ)} )^2.  At λ_b=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 loses information.  Accordingly, the entropy of the system increases.<cit.>  As the coupling gets stronger, more information is expected to be lost and thus the entropy goes up monotonically.  Figure <ref> confirms the increase of the von Neumann entropy which converges to the pointer limit (<ref>)  at the strong coupling limit.

As further evidence of continuous measurement by environments, we also investigated a non-equilibrium steady state.  When different temperatures are used, heat flows through the system.
Heat from the environment B_ℓ to the system can be computed as

    J_ℓ = -i _s{X̂_ℓη_ℓ_-  H_s}.

Figure <ref> shows the steady state heat current as a function of the coupling strength. In the weak coupling regime, the current increases linearly as expected from the linear response theory.  However, the heat current reaches its maximum and dies off rather quickly as the coupling becomes stronger.  This suppression of heat is predicted earlier as a consequence of the quantum zeno effect<cit.> based on a heuristic argument and is observed by Kato-Tanimura<cit.>.  




The present results show that indeed the decoherence due to environments is responsible for the suppression of heat. The off-diagonal elements of the steady state density look almost identical to those in the stationary state at a single effective temperature T=(T_1+T_2)/2  However, there is small but significant difference where the heat current is strong.  The elements ρ_13 and ρ_24 deviate from ρ_12 and ρ_34 due to the difference in decoherence power between the two environments.  In general a higher temperature environment causes stronger decoherence.<cit.> However, it also depends on the coupling strength as well.  When the coupling strength overcomes  the asymmetry in temperature, the decoherence power of the two environments becomes nearly identical and eventually the asymmetry in the off-diagonal element responsible for the heat conduction vanishes.     

In conclusion, we claim that the “thermal equilibrium” of a small quantum system is not the Gibbs state when the coupling to the environments is strong.  Due to continuous measurement by the environment, the stationary state loses the coherency between the pointer states and thus the density is diagonal in the pointer basis rather than in the energy eigenbasis. We further claim that the the stationary state density at the strong coupling limit is the Gibbs state projected onto the pointer basis. The diagonal elements in the pointer basis appear to be insensitive to the coupling strength.  We have demonstrated 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 models such as the Hamiltonian of mean force. 


We would like to thank Janet Anders, Ala-Nissila, Sahar Alipour, and Erik Aurell for helpful discussion during NORDITA programs.  We also thank James Cresser for interesting discussion.












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