Hydrodynamic interpretation of generic squeezed coherent states: A kinetic theory

Abstract

The hydrodynamic interpretation of quantum mechanics treats a system of particles in an effective manner. In this work, we investigate squeezed coherent states within the hydrodynamic interpretation. The Hamiltonian operator in question is time dependent, n-dimensional and in quadratic order. We start by deriving a phase space Wigner probability distribution and an associated equilibrium entropy for the squeezed coherent states. Then, we decompose the joint phase space distribution into two portions: a marginal position distribution and a momentum distribution that is conditioned on the post-selection of positions. Our conditionally averaged momenta are shown to be equal to the Bohm's momenta whose connection to the weak measurements is already known. We also keep track of the corresponding classical system evolution by identifying shear, magnification and rotation components of the symplectic phase space dynamics. This allows us to pinpoint which portion of the underlying classical motion appears in which quantum statistical concept. We show that our probability distributions satisfy the Fokker-Planck equations exactly. They can be used to decompose the equilibrium entropy into the missing information in positions and in momenta as in the Sackur-Tetrode entropy of the classical kinetic theory. Eventually, we define a quantum pressure, a quantum temperature and a quantum internal energy which are related to each other in the same fashion as in the classical kinetic theory. We show that the quantum potential incorporates the kinetic part of the internal energy and the fluctuations around it. This allows us to suggest a quantum conditional virial relation. In the end, we show that the kinetic internal energy is linked to the fractional Fourier transformer part of the underlying classical dynamics similar to the case where the energy of a quantum oscillator is linked to its Maslov index.

arXiv comment27 pages, 1 figure

References in corpus (33)

• The Real Symplectic Groups in Quantum Mechanics and Optics
• Inflation and squeezed quantum states
• Measuring Gaussian quantum information and correlations using the Renyi entropy of order 2
• Classical-Like Description of Quantum Dynamics by Means of Symplectic Tomography
• Grounding Bohmian Mechanics in Weak Values and Bayesianism
• Bohmian Mechanics Revisited
• The significance of the imaginary part of the weak value
• Cosmological Inflation and the Quantum Measurement Problem
• Entropic Uncertainty Relations in Quantum Physics
• Quantum Signatures of the Optomechanical Instability
• First demonstration of 6 dB quantum noise reduction in a kilometer scale gravitational wave observatory
• Contextual Values of Observables in Quantum Measurements
• The Wigner Entropy Production Rate
• Laser Theory for Optomechanics: Limit Cycles in the Quantum Regime
• Experimental determination of irreversible entropy production in out-of-equilibrium mesoscopic quantum systems
• Theory of an optomechanical quantum heat engine
• On the Thermodynamic Origin of the Quantum Potential
• Contextual-value approach to the generalized measurement of observables
• Squeezed Light Induced Two-photon Absorption Fluorescence of Fluorescein Biomarkers
• Bohm's Quantum Potential as an Internal Energy
• Reversible work extraction in a hybrid opto-mechanical system
• Bohmian quantum trajectories from coherent states
• Entropy Production in Continuously Measured Gaussian Quantum Systems
• Quantum Blobs
• Comparing Wigner, Husimi and Bohmian distributions: Which one is a true probability distribution in phase space?
• Short Time Quantum Propagator and Bohmian Trajectories
• An optomechanical heat engine with feedback-controlled in-loop light
• Wigner entropy production and heat transport in linear quantum lattices
• A Quantum Heat Machine from Fast Optomechanics
• Classical and quantum-mechanical phase space distributions
• On the classical limit of Bohmian mechanics for Hagedorn wave packets
• How the Weak Variance of Momentum Can Turn Out to be Negative
• Mean Value of the Quantum Potential and Uncertainty Relations