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 non-interacting bosonic modes, while the fermionic chain is represented by a chain of interacting fermions with nearest-neighbor interactions. We compare 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 Lorentz-Drude type bath, which is the same as the conventional approximations. On the other hand, the decay rate is $1/t^{3}$ for Ohmic type and $1/t$ for white noise, which doesn't 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 Ohmic spectrum, when the fermionic chain is uniformly-coupled, we have $J_{T}\propto1/t^{6}$ and $J_{TI}\propto1/t^{3}$. Remarkably, for perfect state transfer (PST) couplings, there always exists an oscillating quantum energy current $J_{TI}$. Moreover, it is interesting that $J_{T}$ is proportional to $(N-1)^{1/2}$ at certain times for PST couplings under Lorentz-Drude or Ohmic bath.
We study the asymptotic entanglement of two quantum harmonic oscillators nonlinearly coupled to an environment. Coupling to independent baths and a common bath are investigated. Numerical results obtained using the Wangsness-Bloch-Redfield method are supplemented by analytical results in the rotating wave approximation. The asymptotic negativity as function of temperature, initial squeezing and coupling strength, is compared to results for systems with linear system-reservoir coupling. We find that due to the parity conserving nature of the coupling, the asymptotic entanglement is considerably…
Fluctuation and dissipation are by-products of coupling to the `environment.' The Caldeira-Leggett model, a successful paradigm of quantum Brownian motion, views the environment as a collection of harmonic oscillators linearly coupled to the system. However, symmetry considerations may forbid a linear coupling, e.g. for a neutral particle in quantum electrodynamics. We argue that nonlinear couplings can lead to a fundamentally different behavior. Specifically, we consider a heavy particle quadratically coupled to quantum fluctuations of the bath. In one dimension the particle undergoes anomalo…
We calculate the entanglement-assisted and unassisted channel capacities of an exactly solvable spin star system, which models the quantum dephasing channel. The capacities for this non-Markovian model exhibit a strong dependence on the coupling strengths of the bath spins with the system, the bath temperature, and the number of bath spins. For equal couplings and bath frequencies, the channel becomes periodically noiseless.
We investigate the effects of the initial correlation on the dynamics of open system in the exactly solvable pure dephasing model. We show that the role of the initial correlation come into play through a phase function and a weight factor, which would perform oscillations during time evolution, and find that the decoherence of a qubit coupled to a boson bath is more enhanced with respect to a spin bath in the short time. We also demonstrate that the trace distance between two states of a qubit can increase above its initial value, and that the initial correlation can provide another resource …
We study a model of dephasing (decoherence) in a two-state quantum system (qubit) coupled to a bath of harmonic oscillators. An exact analytic solution for the reduced dynamics of a two-state system in this model has been obtained previously for factorizing initial states of the combined system. We show that the model admits exact solutions for a large class of correlated initial states which are typical in the theory of quantum measurements. We derive exact expressions for the off-diagonal elements of the qubit density matrix, which hold for an arbitrary strength of coupling between the qubit…
We study the bath dynamics in the dephasing model of a two-state quantum system (qubit) coupled to an environment of harmonic oscillators. This model was shown [Morozov et al., Phys. Rev. A, 2012, 85, 022101] to admit the analytic solution for the qubit and environment dynamics. Using this solution, we derive the exact expression for the bath reduced density matrix in the presence of initial qubit-environment correlations. We obtain the non-equilibrium phonon distribution function and discuss in detail the time behavior of the bath energy. It is shown that only the inclusion of dynamic correla…
Exact analytical results for the dynamics of two interacting qubits each of which is embedded in its own spin star bath are presented. The time evolution of the concurrence and the purity of the two-qubit system is investigated for finite and infinite numbers of environmental spins. The effect of qubit-qubit interactions on the steady state of the central system is investigated
We study the entropy dynamics of a dephasing model, where a two-level system (TLS) is coupled with a squeezed thermal bath via non-demolition interaction. This model is exactly solvable, and the time dependent states of both the TLS and its bath can be obtained exactly. Based on these states, we calculate the entropy dynamics of both the TLS and the bath, and find that the dephasing rate of the system relies on the squeezing phase of the bath. In zero temperature and high temperature limits, we prove that both the system and bath entropy increases monotonically. Moreover, we find that the deph…
A quantum many-body model is presented with features similar to those of certain particle detectors. The energy spectrum contains a single metastable 'ready'-state and macroscopically-distinct 'pointer' states. Measurements do not pose paradoxes or require interventions outside the field theory formalism. Transitions into classical-like states can be triggered by a single particle with help of the thermal bath. Schroedinger cat states are associated with superpositions of inequivalent vacua, thus relating wavefunction collapse to the dynamics of symmetry breaking in phase transformations.
We study the analytically solvable Ising model of a single qubit system coupled to a spin bath. The purpose of this study is to analyze and elucidate the performance of Markovian and non-Markovian master equations describing the dynamics of the system qubit, in comparison to the exact solution. We find that the time-convolutionless master equation performs particularly well up to fourth order in the system-bath coupling constant, in comparison to the Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the infinite bath correlation time in this model. A recently proposed p…
We demonstrate through an exactly solvable model that collective coupling to any thermal bath induces effectively nonlinear couplings in a quantum many-body (multi-spin) system. The resulting evolution can drive an uncorrelated large-spin system with high probability into a macroscopic quantum-superposition state. We discuss possible experimental realizations.
The interrelationship between the non-Markovian stochastic Schrödinger equations and the corresponding non-Markovian master equations is investigated in the finite temperature regimes. We show that the general finite temperature non-Markovian trajectories can be used to derive the corresponding non-Markovian master equations. A simple, yet important solvable example is the well-known damped harmonic oscillator model in which a harmonic oscillator is coupled to a finite temperature reservoir in the rotating wave approximation. The exact convolutionless master equation for the damped harmonic os…
An exactly solvable model for a quantum measurement is discussed which is governed by hamiltonian quantum dynamics. The $z$-component $\hat s_z$ of a spin-1/2 is measured with an apparatus, which itself consists of magnet coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state according to the sign of $s_z$ of the tested spin. The quantum measurement goes in two steps. On a timescale $1/\sqrt{N}$ the off-diagonal elem…
An exactly solvable model for a quantum measurement is discussed, that integrates quantum measurements with classical measurements. The z-component of a spin-1/2 test spin is measured with an apparatus, that itself consists of magnet of N spin-1/2 particles, coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state, according to the sign of s_z of the test spin. The quantum measurement goes in two steps. On a timescale …
A key quantity characterizing a time-periodically forced quantum system coupled to a heat bath is the energy flowing in the steady state through the system into the bath, where it is dissipated. We derive a general expression which allows one to compute this energy dissipation rate for a heat bath consisting of a large number of harmonic oscillators, and work out two analytically solvable model examples. In particular, we distinguish between genuine transitions effectuating a change of the systems's Floquet state, and pseudo-transitions preserving that state; the latter are shown to yield an i…
We study the full counting statistics for interacting quantum many-body spin systems weakly coupled to the environment. In the leading order in the system-bath coupling we derive exact spin current statistics for a large class of parity symmetric spin-1/2 systems driven by a pair of Markovian baths with local coupling operators. Interestingly, in this class of systems the leading order current statistics are universal and do not depend on details of the Hamiltonian. Furthermore, in the specific case of symmetrically boundary driven anisotropic Heisenberg ($XXZ$) spin 1/2 chain we derive explic…
We study the relationship between one-dimensional fermion gas-impurity models and quantum dissipative systems, via the method of constructive bosonisation and unitary transformation. Starting from an anisotropic Coqblin-Schrieffer model, a new, exactly solvable, three-level quantum dissipative system is derived as a generalisation of the standard spin-half spin-boson model. The new system has two environmental oscillator baths with ohmic coupling, and admits arbitrary detuning between the three levels. All tunnelling matrix elements are equal, up to one complex phase which is itself a function…
A new post-Markovian quantum master equation is derived, that includes bath memory effects via a phenomenologically introduced memory kernel k(t). The derivation uses as a formal tool a probabilistic single-shot bath-measurement process performed during the coupled system-bath evolution. The resulting analytically solvable master equation interpolates between the exact Nakajima-Zwanzig equation and the Markovian Lindblad equation. A necessary and sufficient condition for complete positivity in terms of properties of k(t) is presented, in addition to a prescription for the experimental determin…
The measurement of a spin-$\half$ is modeled by coupling it to an apparatus, that consists of an Ising magnetic dot coupled to a phonon bath. Features of quantum measurements are derived from the dynamical solution of the measurement, regarded as a process of quantum statistical mechanics. Schrödinger cat terms involving both the system and the apparatus, die out very quickly, while the registration is a process taking the apparatus from its initially metastable state to one of its stable final states. The occurrence of Born probabilities can be inferred at the macroscopic level, by looking at…
We study an analytically solvable model for decoherence of a two spin system embedded in a large spin environment. As a measure of entanglement, we evaluate the concurrence for the Bell states (Einstein-Podolsky-Rosen pairs). We find that while for two separate spin baths all four Bell states lose their coherence with the same time dependence, for a common spin bath, two of the states decay faster than the others. We explain this difference by the relative orientation of the individual spins in the pair. We also examine how the Bell inequality is violated in the coherent regime. Both for one b…
We introduce and study the adiabatic dynamics of free-fermion models subject to a local Lindblad bath and in the presence of a time-dependent Hamiltonian. The merit of these models is that they can be solved exactly, and will help us to study the interplay between non-adiabatic transitions and dissipation in many-body quantum systems. After the adiabatic evolution, we evaluate the excess energy (average value of the Hamiltonian) as a measure of the deviation from reaching the target final ground state. We compute the excess energy in a variety of different situations, where the nature of the b…
We present analytical results for the time-dependent information entropy in exactly solvable two-state (qubit) models. The first model describes dephasing (decoherence) in a qubit coupled to a bath of harmonic oscillators. The entropy production for this model in the regimes of "complete" and "incomplete" decoherence is discussed. As another example, we consider the damped Jaynes-Cummings model describing a spontaneous decay of a two-level system into the field vacuum. It is shown that, for all strengths of coupling, the open system passes through the mixed state with the maximum information e…
The spin-boson model is analytically studied using displaced Fock states (DFS) without discretization of the continuum bath. In the orthogonal displaced Fock basis, the ground-state wavefunction can be systematically improved in a controllable way. Interestingly, the zeroth-order DFS reproduces exactly the well known Silbey-Harris results. In the framework of the second-order DFS, the magnetization and the entanglement entropy are exactly calculated. It is found that the magnetic critical exponent $β$ is converged to $0.5$ in the whole sub-Ohmic bath regime $0<s<1$, compared with that by the e…
Based on the system-reservoir description we propose a simple solvable microscopic model for a nonequilibrium bath. This captures the essential features of a nonstationary quantum Markov process. We establish an appropriate generalization of the fluctuation-dissipation relation pertaining to this process and explore the essential modifications of the Bloch equations to reveal the nonexponential decay of the Bloch vector components and transient spectral broadening in resonance fluorescence. We discuss a simple experimental scheme to verify the theoretical results.
Solving realistic quantum systems coupled to an environment is a challenging task. Here we develop a hierarchical functional derivative (HFD) approach for efficiently solving the non-Markovian quantum trajectories of an open quantum system embedded in a bosonic bath. An explicit expression for arbitrary order HFD equation is derived systematically. Moreover, it is found that for an analytically solvable model, this hierarchical equation naturally terminates at a given order and thus becomes exactly solvable. This HFD approach provides a systematic method to study the non-Markovian quantum dyna…
We obtain an explicit solution for the stationary state populations of a dissipative Fano model, where a discrete excited state is coupled to a continumm set of states; both excited set of states are reachable by photo-excitation from the ground state. The dissipative dynamic is described by a Liouville equation in Lindblad form and the field intensity can take arbitrary values within the model. We show that the continuum states population as a function of laser frequency can always be expressed as a Fano profile plus a Lorentzian function with effective parameters whose explicit expressions a…
Relying on an exact time evolution scheme, we identify a novel transient energy transfer phe- nomenon in an exactly-solvable quantum microscopic model consisting of a three-level system coupled to two non-Markovian zero-temperature bosonic baths through two separable quantum channels. The dynamics of this model can be solved exactly using the quantum-state-diffusion equation formalism, demonstrating finite intervals of unidirectional energy flow across the system, typically, from the non-Markovian environment towards the more Markovian bath. Furthermore, when introducing a spatial asymmetry in…
Small quantum systems non-weakly coupled to a bath become in the quantum regime surrounded by a cloud of photons or phonons, which modifies their thermodynamic behavior. Exactly solvable examples are the Brownian motion of a quantum particle in a harmonic confining potential and coupled to a harmonic quantum thermal bath, e.g. an ion in a Penning trap, and a spin immersed in a bosonic bath, as occurs in NMR physics. It appears that the Clausius inequality $\dbarrm Q\le T\d S$ can be violated. For non-adiabatic changes of system parameters the rate of energy dissipation can be negative, and, ou…
We describe the dynamics of a qubit interacting with a bosonic mode coupled to a zero-temperature bath in the deep strong coupling (DSC) regime. We provide an analytical solution for this open system dynamics in the off-resonance case of the qubit-mode interaction. Collapses and revivals of parity chain populations and the oscillatory behavior of the mean photon number are predicted. At the same time, photon number wave packets, propagating back and forth along parity chains, become incoherently mixed. Finally, we investigate numerically the effect of detuning on the validity of the analytical…
Decoherence of a center spin or qubit in a spin bath is essentially determined by the many-body bath evolution. We develop a cluster-correlation expansion (CCE) theory for the spin bath dynamics relevant to the qubit decoherence problem. A cluster correlation term is recursively defined as the evolution of a group of bath spins divided by the cluster correlations of all the subgroups. The so-defined correlation accounts for the authentic collective excitations within a given group. The bath propagator is the product of all possible cluster correlation terms. For a finite-time evolution as in t…
We study the non-Markovian effects on the dynamics of entanglement in an exactly-solvable model that involves two independent oscillators each coupled to its own stochastic noise source. First, we develop Lie algebraic and functional integral methods to find an exact solution to the single-oscillator problem which includes an analytic expression for the density matrix and the complete statistics, i.e., the probability distribution functions for observables. For long bath time-correlations, we see non-monotonic evolution of the uncertainties in observables. Further, we extend this exact solutio…
We perform a study on quantum entropy production, different kinds of correlations, and their interplay in the driven Caldeira-Leggett model of quantum Brownian motion. The model, taken with a large but finite number of bath modes, is exactly solvable, and the assumption of a Gaussian initial state leads to an efficient numerical simulation of all desired observables in a wide range of model parameters. Our study is composed of three main parts. We first compare two popular definitions of entropy production, namely the standard weak-coupling formulation originally proposed by Spohn and later on…
We study the emergence over time of a universal, uniform distribution of quantum states supported on a finite subsystem, induced by projectively measuring the rest of the system. Dubbed deep thermalization, this phenomenon represents a form of equilibration in quantum many-body systems stronger than regular thermalization, which only constrains the ensemble-averaged values of observables. While there exist quantum circuit models of dynamics in one dimension where this phenomenon can be shown to arise exactly, these are special in that deep thermalization occurs at precisely the same time as re…
We present a self contained formalism modelled after the Brownian motion of a quantum harmonic oscillator for describing the performance of microscopic Brownian heat engines like Carnot, Stirling and Otto engines. Our theory, besides reproducing the standard thermodynamics results in the steady state enables permits us to study the role dissipation plays in determining the efficiency of Brownian heat engines under actual laboratory conditions. In particular, we analyse in detail the dynamics associated with decoupling a system in equilibrium with one bath and recoupling it to another bath and …
Traditional approach on quantum Zeno effect (QZE) and quantum anti-Zeno effect (QAZE) in open quantum systems (implicitly) assumes the bath (environment) state returning to its original state after each instantaneous projective measurement on the system and thus ignores the cross-correlations of the bath operators between different Zeno intervals. However, this assumption is not generally true, especially for a bath with a considerably non-negligible memory effect and for a system repeatedly projected into an initial general superposition state. We find that in stark contrast to the result of …
We show that work can be extracted from a two-level system (spin) coupled to a bosonic thermal bath. This is possible due to different initial temperatures of the spin and the bath, both positive (no spin population inversion) and is realized by means of a suitable sequence of sharp pulses applied to the spin. The extracted work can be of the order of the response energy of the bath, therefore much larger than the energy of the spin. Moreover, the efficiency of extraction can be very close to its maximum, given by the Carnot bound, at the same time the overall amount of the extracted work is m…
We study the role of the system-bath coupling for the generalized canonical thermalization [S. Popescu, et al., Nature Physics 2,754(2006) and S. Goldstein et al., Phys. Rev. Lett. 96, 050403(2006)] that reduces almost all the pure states of the "universe" [formed by a system S plus its surrounding heat bath $B$] to a canonical equilibrium state of S. We present an exactly solvable, but universal model for this kinematic thermalization with an explicit consideration about the energy shell deformation due to the interaction between S and B. By calculating the state numbers of the "universe" and…
We present an approach that allows quantifying decoherence processes in an open quantum system subject to external time-dependent control. Interactions with the environment are modeled by a standard bosonic heat bath. We develop two unitarity-preserving approximation schemes to calculate the reduced density matrix. One of the approximations relies on a short-time factorization of the evolution operator, while the other utilizes expansion in terms of the system-bath coupling strength. Applications are reported for two illustrative systems: an exactly solvable adiabatic model, and a model of a r…
We study the entanglement entropy of a one-parameter family of exactly solvable gravities in the 2-dimensional asymptotically-flat space. The islands and Page curves of eternal, evaporating and bath-removed black holes are investigated. The different theories in this parameter class are identified through field redefinitions which leave the island invariant. The Page transition is found to occur at the first a third of the black hole life time in the evaporating case for this family of solutions. In addition, we consider gluing the equilibrium black hole and the evaporating one along a null tr…
We propose a spin-star network, where a central spin-$1/2$ is coupled with XXZ interaction to $N$ outer spin-$1/2$ particles, as a quantum fuel. If the network is in thermal equilibrium with a cold bath, the central spin can have an effective temperature larger than the bath one and scaling nonlinearly with $N$. The nonlinearity can be tuned to $N^2, N^3$ or $N^4$ with the anisotropy parameter of the coupling. Using a stream of central-spin particles to pump a micromaser cavity, we calculate the dynamics of the cavity field using a coarse-grained master equation. Our study reveals that the cen…
The Brownian motion of a quantum particle in a harmonic confining potential and coupled to a harmonic quantum thermal bath is exactly solvable. It is shown that at low enough temperatures the stationary state is non-Gibbsian due to an entanglement with the bath. In physical terms, this happens when the cloud of bath modes around the particle starts to play a non-trivial role. Indeed, equilibrium thermodynamics of the total system, particle plus bath, does not imply standard equilibrium thermodynamics for the particle itself at low $T$. Various formulations of the second law are found to be inv…
Describing matter at near absolute zero temperature requires understanding a system's quantum ground state and the low energy excitations around it, the quasiparticles, which are thermally populated by the system's contact to a heat bath. However, this paradigm breaks down if thermal equilibration is obstructed. This paper presents solvable examples of quantum many-body Hamiltonians of systems that are unable to reach their ground states as the environment temperature is lowered to absolute zero. These examples, three dimensional generalizations of quantum Hamiltonians proposed for topological…
This work addresses quantum adiabatic decoherence of many-body spin systems coupled with a boson field in the framework of open quantum systems theory. We generalize the traditional spin-boson model by considering a system-environment interaction Hamiltonian that represents a partition of non-interacting subsystems and highlights the collective correlation that appears exclusively due to the coupling with a common environment. Remarkably, this simple, exactly solvable model encompasses relevant aspects of a many-body open quantum system and features the subtle quantum effects that arise when t…
We discuss the influence of a zero-temperature environment on a coherent quantum system. First, we calculate the reduced density operator of the system in the framework of the well-known, exactly solvable model of an oscillator coupled to a bath of harmonic oscillators. Then, we propose the sketch of an Aharonov--Bohm-like interferometer showing, through interference measurements, the decrease of the coherence length of the system due to the interaction with the environment, even in the zero temperature limit.
An open harmonic chain: Exact vs global and local reduced dynamics
Sharif University of Technology（谢里夫理工大学）；University of Trieste（的里雅斯特大学）；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste（意大利国家核物理研究所（INFN）的里雅斯特分部）
We define the entropy S and uncertainty function of a squeezed system interacting with a thermal bath, and study how they change in time by following the evolution of the reduced density matrix in the influence functional formalism. As examples, we calculate the entropy of two exactly solvable squeezed systems: an inverted harmonic oscillator and a scalar field mode evolving in an inflationary universe. For the inverted oscillator with weak coupling to the bath, at both high and low temperatures, $S\to r $, where r is the squeeze parameter. In the de Sitter case, at high temperatures, $S\to (1…
When deriving a master equation for a multipartite weakly-interacting open quantum systems, dissipation is often addressed \textit{locally} on each component, i.e. ignoring the coherent couplings, which are later added `by hand'. Although simple, the resulting local master equation (LME) is known to be thermodynamically inconsistent. Otherwise, one may always obtain a consistent \textit{global} master equation (GME) by working on the energy basis of the full interacting Hamiltonian. Here, we consider a two-node `quantum wire' connected to two heat baths. The stationary solution of the LME and …
