Responses of small quantum systems subjected to finite baths

TL;DR

This study investigates how small quantum harmonic oscillator systems respond to external forces when coupled with finite baths, highlighting the importance of finite size effects in quantum open systems.

Contribution

It provides a detailed analysis of the response behavior of finite quantum systems under external forces, using canonical transformations and Husimi's method, emphasizing finite size effects.

Findings

Response to uniform force is not proportional to system size when coupled to a bath.

Spatial correlations depend on system size and temperature.

Finite system size significantly affects quantum open system behavior.

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 $H_f= -f(t) \sum_{k=1}^{N_S} Q_k \: e^{i 2 \pi k u/N_S}$, where the wavevector $u$ is $u=0$ and $u=-N_S/2$ for uniform and staggered forces, respectively. The spatial correlation $\Gamma_m$ 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.