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Journal of Physics A Mathematical and General

Dynamic equivalence of a two-dimensional quantum electron gas and a classical harmonic oscillator chain with an impurity mass

M. Howard Lee, J. Florencio, J. M. Hong

Published Apr 21, 1989• 53 citations
Publisher

Abstract

There is an exact equivalence in the time-dependent behaviour of a zero-temperature two-dimensional interacting electron gas at long wavelengths and a classical harmonic oscillator chain with one impurity mass. The mass difference m-m 0 , where m 0 is the impurity mass, acts as the electron-electron interaction. Time evolution is asymmetric in m-m 0 about m=m 0 .

DOI
10.1088/0305-4470/22/8/005

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There is an exact equivalence in the time-dependent behaviour of a zero-temperature two-dimensional interacting electron gas at long wavelengths and a classical harmonic oscillator chain with one impurity mass. The mass difference m-m 0 , where m 0 is the impurity mass, acts as the electron-electron interaction. Time evolution is asymmetric in m-m 0 about m=m 0 .
Relaxation phenomena in a class of nondissipative systems with two highly disparate time scales (i.e., Brownian systems) have unique commonalities quantifiable via two scalars. It is shown that the exactly solvable problem of the dynamics of a weakly linked impurity spin in a s=1/2 XY chain belongs to this dynamical universality class and so does that of a heavy mass in an infinite harmonic oscillator chain and a spinless quasi two-dimensional attractive Fermi gas in the long-wavelength limit. The case of the strongly linked impurity in the XY chain is also discussed along with the corresponding limits in the harmonic oscillator chain and the electron-gas problems. \textcopyright{} 1996 The American Physical Society.
In the following, we study the dissipative time-evolution of a quantum chain consisting of three coupled harmonic oscillators, the first and third of which weakly interact quadratically with two independent thermal baths in equilibrium at different temperatures. Due to the quadratic form of the total Hamiltonian, the unitary dynamics of the compound system is formally analytically solvable and defines a one-parameter group of Gaussian maps which enables us to solve the exact dynamics of the chain numerically. Following the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) approach to open quantum systems, one can perform the rotating wave approximation with respect to the interacting, or non-interacting chain Hamiltonian and respectively derive the so-called global and local master equations. The solutions of the ensuing different master equations can then be compared with the exact one, possibly sorting out the two approaches in correspondence to different time-scales of the system. We derive the steady states of the open chain quantum dynamics in the two approaches and show that the behaviour of fidelity between them versus inter-oscillator coupling depends on the two bath temperatures, revealing the existence of a temperature-dependent critical inter-oscillator coupling strength that determines the domain of validity of each approach. When the newly found coupling is less than this critical value, the local approach outperforms the global approach, whereas for larger inter-oscillator coupling, the global approach is a better approximation of the exact evolution. This critical value of inter-oscillator coupling depends on the two bath temperatures, which then play a crucial role in deciding the best possible approximating open dynamics.
We investigate a temporal evolution of an impurity atom in a one-dimensional trapped Bose gas following a sudden change of the boson-impurity interaction strength. Our focus is on the effects of inhomogeneity due to the harmonic confinement. These effects can be described by an effective one-body model where both the mass and the spring constant are renormalized. This is in contrast to the classic renormalization, which addresses only the mass. We propose an effective single-particle Hamiltonian and apply the multilayer multiconfiguration time-dependent Hartree method for bosons to explore its validity. Numerical results suggest that the effective mass is smaller than the impurity mass, which means that it cannot straightforwardly be extracted from translationally invariant models.