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arXiv:1908.07746 · Physical Review A

Quantum energy transfer between a nonlinearly coupled bosonic bath and a fermionic chain: An exactly solvable model

Zhao-Ming Wang, Da-Wei Luo, Baowen Li, Lian-Ao Wu

Published Apr 30, 2020• 2 citations
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Abstract

The evolution of a quantum system towards thermal equilibrium is usually studied by approximate methods, which have their limits of validity and should be checked against analytically solvable models. In this paper, we propose an analytically solvable model to investigate the energy transfer between a bosonic bath and a fermionic chain which are nonlinearly coupled to each other. The bosonic bath consists of an infinite collection of noninteracting bosonic modes, while the fermionic chain is represented by a chain of interacting fermions with nearest-neighbor interactions. We compare the behaviors of the temperature-dependent energy current ${J}_{T}$ and temperature-independent energy current ${J}_{TI}$ for different bath configurations. With respect to the bath spectrum, ${J}_{T}$ decays exponentially for a Lorentz-Drude-type bath, which is the same as the conventional approximations. On the other hand, the decay rate is $1/{t}^{3}$ for the Ohmic type and $1/t$ for white noise, which does not have conventional counterparts. For the temperature-independent current ${J}_{TI}$, the decay rate is divergent for the Lorentz-Drude-type bath, $1/{t}^{4}$ for the Ohmic bath, and $1/t$ for the white noise. When further considering the dynamics of the fermionic chain, the current will be modulated based on the envelope from the bath. As an example, for a bosonic bath with an Ohmic spectrum, when the fermionic chain is uniformly coupled, we have ${J}_{T}\ensuremath{\propto}1/{t}^{6}$ and ${J}_{TI}\ensuremath{\propto}1/{t}^{3}$. Moreover, it is interesting that ${J}_{T}$ is proportional to ${(N\ensuremath{-}1)}^{1/2}$ at certain times for perfect state transfer couplings under a Lorentz-Drude or Ohmic bath.

DOI
10.1103/physreva.101.042130

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The evolution of a quantum system towards thermal equilibrium is usually studied by approximate methods, which have their limits of validity and should be checked against analytically solvable models. In this paper, we propose an analytically solvable model to investigate the energy transfer between a bosonic bath and a fermionic chain which are nonlinearly coupled to each other. The bosonic bath consists of an infinite collection of noninteracting bosonic modes, while the fermionic chain is represented by a chain of interacting fermions with nearest-neighbor interactions. We compare the behaviors of the temperature-dependent energy current JT{J}_{T}JT​ and temperature-independent energy current JTI{J}_{TI}JTI​ for different bath configurations. With respect to the bath spectrum, JT{J}_{T}JT​ decays exponentially for a Lorentz-Drude-type bath, which is the same as the conventional approximations. On the other hand, the decay rate is 1/t31/{t}^{3}1/t3 for the Ohmic type and 1/t1/t1/t for white noise, which does not have conventional counterparts. For the temperature-independent current JTI{J}_{TI}JTI​, the decay rate is divergent for the Lorentz-Drude-type bath, 1/t41/{t}^{4}1/t4 for the Ohmic bath, and 1/t1/t1/t for the white noise. When further considering the dynamics of the fermionic chain, the current will be modulated based on the envelope from the bath. As an example, for a bosonic bath with an Ohmic spectrum, when the fermionic chain is uniformly coupled, we have JT\ensuremath∝1/t6{J}_{T}\ensuremath{\propto}1/{t}^{6}JT​\ensuremath∝1/t6 and JTI\ensuremath∝1/t3{J}_{TI}\ensuremath{\propto}1/{t}^{3}JTI​\ensuremath∝1/t3. Moreover, it is interesting that JT{J}_{T}JT​ is proportional to (N\ensuremath−1)1/2{(N\ensuremath{-}1)}^{1/2}(N\ensuremath−1)1/2 at certain times for perfect state transfer couplings under a Lorentz-Drude or Ohmic bath.
The motion of a free quantum particle in a thermal environment is usually described by the quantum Langevin equation, where the effect of the bath is encoded through a dissipative and a noise term, related to each other via the fluctuation dissipation theorem. The quantum Langevin equation can be derived starting from a microscopic model of the thermal bath as an infinite collection of harmonic oscillators prepared in an initial equilibrium state. The spectral properties of the bath oscillators and their coupling to the particle determine the specific form of the dissipation and noise. Here we investigate in detail the well-known Rubin bath model, which consists of a one-dimensional harmonic chain with the boundary bath particle coupled to the Brownian particle. We show how in the limit of infinite bath bandwidth, we get the Drude model, and a second limit of infinite system-bath coupling gives the Ohmic model. A detailed analysis of relevant equilibrium correlation functions, such as the mean squared displacement, velocity autocorrelation functions, and response function are presented, with the aim of understanding the various temporal regimes. In particular, we discuss the quantum-to-classical crossover time scales where the mean square displacement changes from a ∼lnt to a ∼t dependence. We relate our study to recent work using linear response theory to understand quantum Brownian motion.
We study the transfer of heat in the nonequilibrium spin-boson model with an Ohmic dissipation. In the nonadiabatic limit we derive a formula for the thermal conductance based on a rate equation formalism at the level of the noninteracting blip approximation, valid for temperatures T>T(K), with T(K) as the Kondo temperature. We evaluate this expression analytically assuming either weak or strong couplings, and demonstrate that our results agree with exact relations. Far-from-equilibrium situations are further examined, showing a close correspondence to the linear response limit.
We study the statistics of thermal energy transfer in the nonequilibrium (two-bath) spin-boson model. This quantum many-body impurity system serves as a canonical model for quantum energy transport. Our method makes use of the Majorana fermion representation for the spin operators, in combination with the Keldysh nonequilibrium Green's function approach. We derive an analytical expression for the cumulant generating function of the model in the steady state limit, and show that it satisfies the Gallavotti–Cohen fluctuation symmetry. We obtain analytical expressions for the heat current and its noise, valid beyond the sequential and the co-tunneling regimes. Our results satisfy the quantum mechanical bound for heat current in interacting nanojunctions. Results are compared with other approximate theories, as well as with a non-interacting model, a fully harmonic thermal junction.
We study the energy transport in a system of two half-infinite XXZ chains initially kept separated at different temperatures, and later connected and let free to evolve unitarily. By changing independently the parameters of the two halves, we highlight, through bosonization and time-dependent matrix-product-state simulations, the different contributions of low-lying bosonic modes and of fermionic quasiparticles to the energy transport. In the simulations we also observe that the energy current reaches a finite value which only slowly decays to zero. The general picture that emerges is the following. Since integrability is only locally broken in this model, a preequilibration behavior may appear. In particular, when the sound velocities of the bosonic modes of the two halves match, the low-temperature energy current is almost stationary and described by a formula with a nonuniversal prefactor interpreted as a transmission coefficient. Thermalization, characterized by the absence of any energy flow, occurs only on longer time scales which are not accessible with our numerics.
We propose an alternative concept for the dynamics of a quantum bath, the Chebyshev space, and a method based on this concept, the Chebyshev space method. The Chebyshev space is an abstract vector space that exactly represents the fermionic or bosonic bath degrees of freedom, without a discretization of the bath density of states. Relying on Chebyshev expansions, the Chebyshev space representation of a bath has very favorable properties with respect to extremely precise and efficient calculations of ground state properties, static and dynamical correlations, and time evolution for a great variety of quantum systems. The aim of the present work is to introduce the Chebyshev space in detail and to demonstrate the capabilities of the Chebyshev space method. Although the central idea is derived in full generality, the focus is on model systems coupled to fermionic baths. In particular, we address quantum impurity problems, such as an impurity in a host or a bosonic impurity with a static barrier, and the motion of a wave packet on a chain coupled to leads. For the bosonic impurity, the phase transition from a delocalized electron to a localized polaron in arbitrary dimension is detected. For the wave packet on a chain, we show how the Chebyshev space method implements different boundary conditions, including transparent boundary conditions replacing infinite leads. Furthermore, the self-consistent solution of the Holstein model in infinite dimension is calculated. With the examples, we demonstrate how highly accurate results for system energies, correlation and spectral functions, and time dependence of observables are obtained with modest computational effort.
Continuing our work on the nature and existence of fluctuation-dissipation relations (FDR) in linear and nonlinear open quantum systems [J.-T. Hsiang, B. L. Hu, and S.-Y. Lin, Phys. Rev. D 100, 025019 (2019); J.-T. Hsiang, B. L. Hu, S.-Y. Lin, and K. Yamamoto, Phys. Lett. B 795, 694 (2019); J.-T. Hsiang and B. L. Hu, Physics (Utrecht) 1, 430 (2019); J.-T. Hsiang and B. L. Hu, Phys. Rev. D 101, 125003 (2020)], here we consider such relations when a linear system is in a nonequilibrium steady state (NESS). With the model of two-oscillators (considered as a short harmonic chain with the two ends) each connected to a thermal bath of different temperatures we find that when the chain is fully relaxed due to interaction with the baths, the relation that connects the noise kernel and the imaginary part of the dissipation kernel of the chain in one bath does not assume the conventional form for the FDR in equilibrium cases. There exists an additional term we call the ``bias current'' that depends on the difference of the bath's initial temperatures and the interoscillator coupling strength. We further show that this term is related to the steady heat flow between the two baths when the system is in an NESS. The ability to know the real-time development of the interheat exchange (between the baths and the end-oscillators) and the intraheat transfer (within the chain) and their dependence on the parameters in the system offers possibilities for quantifiable control, and in the design of quantum heat engines, or thermal devices.