Initial value formulation of a quantum damped harmonic oscillator
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arXiv
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| Format: | Preprint |
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2023
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| author | Agarwal, Nishant Chu, Yi-Zen |
| author_facet | Agarwal, Nishant Chu, Yi-Zen |
| contents | The in-in formalism and its influence functional generalization are widely used to describe the out-of-equilibrium dynamics of unitary and open quantum systems, respectively. In this paper, we build on these techniques to develop an effective theory of a quantum damped harmonic oscillator and use it to study initial state-dependence, decoherence, and thermalization. We first consider a Gaussian initial state and quadratic influence functional and obtain general equations for the Green's functions of the oscillator. We solve the equations in the specific case of time-local dissipation and use the resulting Green's functions to obtain the purity and unequal-time two-point correlations of the oscillator. We find that the dynamics must include a non-vanishing noise term to yield physical results for the purity and that the oscillator decoheres in time such that the late-time density operator is thermal. We show that the frequency spectrum or unequal-time correlations can, however, distinguish between the damped oscillator and an isolated oscillator in thermal equilibrium, and obtain a generalized fluctuation-dissipation relation for the damped oscillator. We briefly consider time-nonlocal dissipation as well, to show that the fluctuation-dissipation relation is satisfied for a specific choice of dissipation kernels. Lastly, we develop a double in-out path integral approach to go beyond Gaussian initial states and show that our equal-time results for time-local dissipation are in fact non-perturbative in the initial state. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_04829 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Initial value formulation of a quantum damped harmonic oscillator Agarwal, Nishant Chu, Yi-Zen High Energy Physics - Theory Statistical Mechanics Quantum Physics The in-in formalism and its influence functional generalization are widely used to describe the out-of-equilibrium dynamics of unitary and open quantum systems, respectively. In this paper, we build on these techniques to develop an effective theory of a quantum damped harmonic oscillator and use it to study initial state-dependence, decoherence, and thermalization. We first consider a Gaussian initial state and quadratic influence functional and obtain general equations for the Green's functions of the oscillator. We solve the equations in the specific case of time-local dissipation and use the resulting Green's functions to obtain the purity and unequal-time two-point correlations of the oscillator. We find that the dynamics must include a non-vanishing noise term to yield physical results for the purity and that the oscillator decoheres in time such that the late-time density operator is thermal. We show that the frequency spectrum or unequal-time correlations can, however, distinguish between the damped oscillator and an isolated oscillator in thermal equilibrium, and obtain a generalized fluctuation-dissipation relation for the damped oscillator. We briefly consider time-nonlocal dissipation as well, to show that the fluctuation-dissipation relation is satisfied for a specific choice of dissipation kernels. Lastly, we develop a double in-out path integral approach to go beyond Gaussian initial states and show that our equal-time results for time-local dissipation are in fact non-perturbative in the initial state. |
| title | Initial value formulation of a quantum damped harmonic oscillator |
| topic | High Energy Physics - Theory Statistical Mechanics Quantum Physics |
| url | https://arxiv.org/abs/2303.04829 |