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arXiv:1803.03334 · Physical Review A

Connecting dissipation and noncommutativity: A Bateman system case study

Sayan Kumar Pal, Partha Nandi, Biswajit Chakraborty

Published Jun 11, 2018• 16 citations
arXivHTMLPDF

Abstract

We present an approach to the problem of quantization of the damped harmonic oscillator. To start with, we adopt the standard method of doubling the degrees of freedom of the system (Bateman form) and then, by introducing some new parameters, we get a generalized coupled set of equations from the Bateman form. Using the corresponding time-independent Lagrangian, quantum effects on a pair of Bateman oscillators embedded in an ambient noncommutative space (Moyal plane) are analyzed by using both path integral and canonical quantization schemes within the framework of the Hilbert--Schmidt operator formulation. Our method is distinct from those existing in the literature and where the ambient space was taken to be commutative. Our quantization shows that we end up again with a Bateman system except that the damping factor undergoes renormalization. Strikingly, the corresponding expression shows that the renormalized damping factor can be nonzero even if ``bare'' one is zero to begin with. In other words, noncommutativity can act as a source of dissipation. Conversely, the noncommutative parameter $\ensuremath{\theta}$, taken to be a free one now, can be fine tuned to get a vanishing renormalized damping factor. This indicates in some sense a ``duality'' between dissipation and noncommutativity. Our results match the existing results in the commutative limit.

DOI
10.1103/physreva.97.062110

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Journal of Mathematical Physics•Jul 1, 2018•5 citations
We present an approach to the problem of quantization of the damped harmonic oscillator. To start with, we adopt the standard method of doubling the degrees of freedom of the system (Bateman form) and then, by introducing some new parameters, we get a generalized coupled set of equations from the Bateman form. Using the corresponding time-independent Lagrangian, quantum effects on a pair of Bateman oscillators embedded in an ambient noncommutative space (Moyal plane) are analyzed by using both path integral and canonical quantization schemes within the framework of the Hilbert--Schmidt operator formulation. Our method is distinct from those existing in the literature and where the ambient space was taken to be commutative. Our quantization shows that we end up again with a Bateman system except that the damping factor undergoes renormalization. Strikingly, the corresponding expression shows that the renormalized damping factor can be nonzero even if bare'' one is zero to begin with. In other words, noncommutativity can act as a source of dissipation. Conversely, the noncommutative parameter $\ensuremath{\theta}$, taken to be a free one now, can be fine tuned to get a vanishing renormalized damping factor. This indicates in some sense a duality'' between dissipation and noncommutativity. Our results match the existing results in the commutative limit.
Abstract In this work, we present a quantization scheme for the damped harmonic oscillator (QDHO) using a framework known as momentous quantum mechanics. Our method relies on a semiclassical dynamical system derived from an extended classical Hamiltonian, where the phase-space variables are given by expectation values of observables and quantum dispersions. The significance of our study lies in its potential to serve as a foundational basis for the effective description of open quantum systems (OQS), and the description of dissipation in quantum mechanics. By employing the Bateman’s dual model as the initial classical framework, and undergoing quantization, we demonstrate that our description aligns exceptionally well with the well-established Lindblad master equation for the QDHO. Furthermore, the robust formulation of our approach suggests a broad applicability in the context of OQS, extending implementations into more general systems, making it a versatile and powerful tool to study diverse phenomena.
The so-called Bateman system for the damped harmonic oscillator is reduced to a genuine dual dissipation system (DDS) by setting the mass to zero. We explore herein the condition under which the canonical quantization of the DDS is consistently performed. The roles of the observable and auxiliary coordinates are discriminated. The results show that the complete and orthogonal Fock space of states can be constructed on the stable vacuum if an anti-Hermite representation of the canonical Hamiltonian is adopted. The amplitude of the one-particle wavefunction is consistent with the classical solution. The fields can be quantized as bosonic or fermionic. For bosonic systems, the quantum fluctuation of the field is directly associated with the dissipation rate.
In a recent paper a slightly modified version of the Bateman system, originally proposed to describe a damped harmonic oscillator, was proposed. This system is really different from the Bateman’s one, in the sense that this latter cannot be recovered for any choice of its parameters. In this paper we consider this system and we show that, at a quantum level, it is not necessarily dissipative. In particular we show that the Hamiltonian of the system, when quantized, produces different behaviors, depending on some relations between its parameters. In fact, it gives rise to either a two dimensional (standard) harmonic oscillator, or to two independent oscillators, one of which is again standard , and a second one which is an inverted oscillator. The two cases are analyzed in terms of bosonic or pseudo-bosonic ladder operators, and the appearance of distributions for the inverted oscillator is commented.
Bateman's dual oscillator embeds a damped harmonic oscillator and an independent anti-damped adjoint coordinate in a conservative doubled system. We develop a split-quaternionic formulation that remains consistent from the classical equations through ordinary complex canonical quantization. The doubled dynamics is written as a two-sided hypercomplex evolution whose conserved indefinite quadratic form is the split norm, while the sign of the generator square unifies the underdamped, overdamped, and critical regimes. The Bateman Lagrangian is recovered as a real scalar action on the split-complex configuration subalgebra. After quantization, operator-valued split quaternions belong to the complexified algebra C⊗RHs≃M2(C)\mathbb C\otimes_{\mathbb R}\mathbb H_s\simeq M_2(\mathbb C)C⊗R​Hs​≃M2​(C). Noncommuting mechanical components require a symmetrized quantum split norm, which exactly reproduces the symmetrically ordered canonical Hamiltonian. Null idempotents resolve the crossed canonical structure of the doubled phase space. In particular, damping yields [my^˙,mx^˙]=iℏmγ[m\dot{\widehat y},m\dot{\widehat x}]=\mathrm{i}\hbar mγ[my​˙​,mx˙]=iℏmγ and a Robertson--Schrödinger uncertainty relation. The limit γ→0γ\to0γ→0 retains the doubled positive/negative geometry; the ordinary single-oscillator Heisenberg relation is recovered only after selecting the positive normal mode, or by undoubling before quantization. The scalar split action gives the exact quadratic propagator and Gaussian evolution. A formal trace over the auxiliary coordinate is finally compared with high-temperature Caldeira--Leggett dynamics, clarifying that the doubled model is a coherent zero-noise embedding rather than a microscopic reservoir theory.
Quantization of a damped harmonic oscillator leads to so called Bateman's dual system. The corresponding Bateman's Hamiltonian, being a self-adjoint operator, displays the discrete family of complex eigenvalues. We show that they correspond to the poles of energy eigenvectors and the corresponding resolvent operator when continued to the complex energy plane. Therefore, the corresponding generalized eigenvectors may be interpreted as resonant states which are responsible for the irreversible quantum dynamics of a damped harmonic oscillator.
Bateman’s dual system (BDS) is a linear time-reversal invariant system that consists of damping and amplifying coordinate variables subjected to a harmonic oscillator potential. It is known that canonical quantization of the BDS in the underdamping region suffers from pathologies such as the non-stationary vacuum and the breakdown of Heisenberg’s uncertainty principle. In this paper, following a previous study that targeted the pathology-free massless BDS, the massive BDS is quantized by decomposing it into two independent effectively massless subsystems with reduced degrees of freedom. In each subsystem, the amplifying variable is the conjugate momentum of a damping variable that obeys a non-canonical quantization condition. By virtue of general scaling invariance including U(1), the variables in BDS are treated as non-self-adjoint without changing the total degrees of freedom. The physical states are constructed with time-reversal normalization and are shown to satisfy the time-independent uncertainty relation. The original massive BDS that fulfills the canonical quantization condition is reconstructed by superposing the two subsystems. The same method is also applied to the underdamped BDS to obtain the time-independent uncertainty relations. The expectation values of the coordinate operators represent the complex representation of the solution of the classical equation of motion. The scaling invariance may be broken by coupling with the environment.