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arXiv:1201.0565

Responses of small quantum systems subjected to finite baths

Hideo Hasegawa

Published Jan 3, 2012• 0 citations
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Abstract

We have studied responses to applied external forces of the quantum $(N_S+N_B)$ model for $N_S$-body interacting harmonic oscillator (HO) system subjected to $N_B$-body HO bath, by using canonical transformations combined with Husimi's method for a driven quantum HO [K. Husimi, Prog. Theor. Phys. {\bf 9}, 381 (1953)]. It has been shown that the response to a uniform force expressed by the Hamiltonian: $H_f= -f(t) \sum_{k=1}^{N_S} Q_k$ is generally not proportional to $N_S$ except for no system-bath couplings, where $f(t)$ expresses its time dependence and $Q_k$ denotes a position operator of $k$th particle of the system. We have calculated also the response to a space- and time-dependent force expressed by $H_f= -f(t) \sum_{k=1}^{N_S} Q_k \: e^{i 2 πk u/N_S}$, where the wavevector $u$ is $u=0$ and $u=-N_S/2$ for uniform and staggered forces, respectively. The spatial correlation $Γ_m$ for a pair of positions of $Q_k$ and $Q_{k+m}$ has been studied as functions of $N_S$ and the temperature. Our calculations have indicated an importance of taking account of finite $N_S$ in studying quantum open systems which generally include arbitrary numbers of particles.

DOI
10.48550/arxiv.1201.0565

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We have studied responses to applied external forces of the quantum (NS+NB)(N_S+N_B)(NS​+NB​) model for NSN_SNS​-body interacting harmonic oscillator (HO) system subjected to NBN_BNB​-body HO bath, by using canonical transformations combined with Husimi's method for a driven quantum HO [K. Husimi, Prog. Theor. Phys. {\bf 9}, 381 (1953)]. It has been shown that the response to a uniform force expressed by the Hamiltonian: Hf=−f(t)∑k=1NSQkH_f= -f(t) \sum_{k=1}^{N_S} Q_kHf​=−f(t)∑k=1NS​​Qk​ is generally not proportional to NSN_SNS​ except for no system-bath couplings, where f(t)f(t)f(t) expresses its time dependence and QkQ_kQk​ denotes a position operator of kkkth particle of the system. We have calculated also the response to a space- and time-dependent force expressed by Hf=−f(t)∑k=1NSQk ei2πku/NSH_f= -f(t) \sum_{k=1}^{N_S} Q_k \: e^{i 2 πk u/N_S}Hf​=−f(t)∑k=1NS​​Qk​ei2πku/NS​, where the wavevector uuu is u=0u=0u=0 and u=−NS/2u=-N_S/2u=−NS​/2 for uniform and staggered forces, respectively. The spatial correlation ΓmΓ_mΓm​ for a pair of positions of QkQ_kQk​ and Qk+mQ_{k+m}Qk+m​ has been studied as functions of NSN_SNS​ and the temperature. Our calculations have indicated an importance of taking account of finite NSN_SNS​ in studying quantum open systems which generally include arbitrary numbers of particles.
We have studied the specific heat of the (NS + NB) model for an NS-body harmonic oscillator (HO) system which is strongly coupled to an NB-body HO bath without dissipation. The system specific heat of CS(T) becomes NSkB at T → ∞ and vanishes at T = 0 in accordance with the third law of thermodynamics. The calculated CS(T) at low temperatures is not proportional to NS and shows an anomalous temperature dependence, strongly depending on NS, NB, and the system-bath coupling. In particular at very low (but finite) temperatures, it may become negative for a strong system-bath coupling, which is in contrast with non-negative specific heat of a HO system with NS = 1 reported by Ingold, Hänggi, and Talkner [Phys. Rev. E 79, 061105 (2005)]. Our calculation indicates an importance of taking account of finite NS in studying open quantum systems which may include an arbitrary number of particles in general.
We develop an exact analytical framework for studying the nonequilibrium dynamics of interacting quantum harmonic oscillators coupled to a modelled thermal reservoir and driven by external classical fields. The total Hamiltonian is diagonalized via successive Bogoliubov transformations, enabling a nonperturbative treatment of dissipation and driving. For two coupled oscillators with an external source applied to the first, we derive explicit energy expressions and analyze their dependence on coupling and bath parameters. The phase-space structure is characterized through Husimi QQQ-functions for initial separable coherent and number states. We obtain the reduced density matrix elements in the number-state basis and demonstrate that, in the absence of driving and at zero temperature, the reduced density matrix of the main system satisfies the Lindblad master equation. A generating function for the Husimi function is introduced to facilitate computation of higher-order correlations. The analysis is then generalized to nnn interacting oscillators, with analytical energy expressions provided and the case n=3n=3n=3 examined in detail. Our results establish a versatile and exact framework for driven-dissipative quantum systems, with potential applications in quantum thermodynamics, open quantum systems, and many-body physics.
When an open quantum system is driven by an external time-dependent force, the coupling of the driving to the central system is usually included, whereas the impact of the driving field on the bath is neglected. We investigate the effect of a quantum bath of linearly driven harmonic oscillators on the relaxation dynamics of a quantum two-level system which is not directly driven. In particular, we calculate the frequency-dependent response of the system when the bath is subject to Dirac and Gaussian driving pulses. We show that a time-retarded effective force on the system is induced by the driven bath which depends on the full history of the perturbation and the spectral characteristics of the underlying bath. In particular, when a structured Ohmic bath with a pronounced Lorentzian peak is considered, the dynamical response of the system to a driven bath is qualitatively different than that of the undriven bath. Specifically, additional resonances appear which can be directly associated with a Jaynes-Cummings-like effective energy spectrum.
The prevailing method for addressing strong-coupling thermodynamics typically involves open quantum systems coupled to infinite baths in equilibrium through the Hamiltonian of mean force (HMF). However, its applicability to finite baths remains limited. In this work, we transcend this limit by considering the impacts of system-bath coupling which is not only on the system but also on the finite bath feedback. When the bath and system sizes are comparable, they act as each other's effective 'bath'. We introduce a parameter α within [0,1], where α and 1-α act as weighting factors to distribute the system-bath coupling between the system and the bath. Our approach innovatively incorporates the effect of coupling on the respective effective 'bath' into both the system and bath in the statistical factor form. We generalize the HMF and propose quantum Hamiltonians of mean forces to handle the distributed coupling of the system to finite baths. The additional parameter α is determined by minimizing the joint free energy density and is a new thermodynamic quantity representing the finite bath's influence. Examining the damped quantum harmonic oscillator with a finite bath, we find that α affects both states, leading to a complex phase diagram with unique phenomena at critical temperatures T_{L} and T_{R}, including valleys, peaks, negative values, and discontinuities in entropy and specific heat. Notably, these anomalies disappear when α→1, both at high temperatures and in the thermodynamic limit, indicating the negligible influence of coupling on the bath states and reverting to the HMF. Thus, studying finite-sized baths holds substantial significance in small quantum systems.
We determine the frequency-dependent response characteristics of a quantum system to a driven Caldeira-Leggett bath. The bath degrees of freedom are explicitly driven by an external time-dependent force, in addition to the direct time-dependent forcing of the system itself. After general considerations of driven Caldeira-Leggett baths, we consider the Rubin model of a chain of quantum particles coupled by linear springs as an important model of a quantum dissipative system. We show that in the presence of time-dependent driving of the chain, this model can be mapped to a quantum system which couples to a driven Caldeira-Leggett bath. The effect of the bath driving is captured by a time-dependent force on the central system, which is, in principle, non-Markovian in nature. We study two specific examples, the exactly solvable case of a harmonic potential and a quantum two-state system for which we assume a weak system-bath coupling. We evaluate the dynamical response to a periodic driving of the system and the bath. The dynamic susceptibility is shown to be altered qualitatively by the bath drive: The dispersive part is enhanced at low frequencies and acquires a maximum at zero frequency. The absorptive part develops a shoulder-like behavior in this frequency regime. These features seem to be generic for quantum systems in a driven Caldeira-Leggett bath.
The motion of a particle subject to an external force and coupled to a large number of harmonic oscillators is reducible to a nonlinear integro-differential equation for the position operator. This is achieved by eliminating the degrees of freedom corresponding to the motion of the oscillators. Upon this elimination the commutator of the position and the mechanical momentum of the particle becomes time dependent and a c number for a quadratic potential, but in general a q number for other fields of force. For a harmonically bound particle interacting with the heat bath the time evolution of the position and momentum operators are given in terms of the solutions of an integro-differential equation. Certain forms of the coupling of the motion of the particle to the heat bath allow for further reduction of these equations to quartic differential equations with constant coefficients. For these cases the time dependence of the position-momentum commutator, the energy of the particle as well as the heat bath are studied.