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International Journal of Modern Physics B

ENTROPY OF A HARMONIC OSCILLATOR IN CONTACT WITH A SQUEEZED RESERVOIR

Robert R. Tucci

Published Feb 10, 1991• 6 citations
Publisher

Abstract

We consider a harmonic oscillator (h.o.) in contact with a non-minimum uncertainty squeezed reservoir (but isolated from contact with other non-squeezed reservoirs). We calculate the h.o.’s density matrix and thermodynamic entropy. We interpret the derivatives of the entropy in terms of two temperatures, one for each quadrature of the reservoir. A change in the total (random) energy of the h.o. is shown to equal the sum of changes in the energies of each h.o. quadrature separately (a version of the First Law of thermodynamics). A change in the total entropy of the h.o. system is likewise shown to equal the sum of changes in the entropies of each h.o. quadrature separately (Second Law of thermodynamics). We also present equations that correspond to the so called “Fundamemental equation” and “Gibbs-Duhem equation” for the h.o system under consideration.

DOI
10.1142/s021797929100033x

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We consider a harmonic oscillator (h.o.) in contact with a non-minimum uncertainty squeezed reservoir (but isolated from contact with other non-squeezed reservoirs). We calculate the h.o.’s density matrix and thermodynamic entropy. We interpret the derivatives of the entropy in terms of two temperatures, one for each quadrature of the reservoir. A change in the total (random) energy of the h.o. is shown to equal the sum of changes in the energies of each h.o. quadrature separately (a version of the First Law of thermodynamics). A change in the total entropy of the h.o. system is likewise shown to equal the sum of changes in the entropies of each h.o. quadrature separately (Second Law of thermodynamics). We also present equations that correspond to the so called “Fundamemental equation” and “Gibbs-Duhem equation” for the h.o system under consideration.
We study the entropy of a two level atom in contact with a non-minimum uncertainty squeezed reservoir and a coherent drive. Let δ be the phase of the coherent drive, and ΩR/γ the Rabi frequency divided by the atomic decay rate. For large, near minimum uncertainty squeezing, we find that the atomic quantum state is approximately a pure state, or, equivalently, a minimum entropy state, if ΩR/γ and δ satisfy (ΩR/γ)| cos δ| = 1/2. (Previous workers have only noted the occurrence of a pure state at δ = 0 or 180°.) We find that the atom saturates separately in each quadrature of the coherent drive. We also find that the thermodynamic order of the atom (given by the magnitude of the Bloch vector or by the negative of the atomic entropy) may increase as the background fluctuations or disorder increase. This behavior does not occur with non-squeezed reservoirs; it may be interpreted as a negative temperature effect, if atomic temperatures are defined in terms of the derivatives of the atomic entropy with respect to the variances of the two reservoir quadratures. We give an extensive collection of plots of the atomic entropy and temperatures. We find that a harmonic oscillator always cools down to lower temperatures than a 2 level atom if both bodies are in contact with the same squeezed reservoir.
CONFINED HARMONIC OSCILLATOR.The aim of this study was a didactic presentation of the Shannon entropy in the quantum theory context, followed by application to the case of a one-dimensional harmonic oscillator in its ground state, both in the free case and confined case.The study of these systems allows us to highlight notions such as location or delocation of a particle, a possible interpretation that the Shannon entropy can adopt.The Shannon entropy in position (S x ) and momentum (S p ) spaces were calculated for both systems, beyond the entropic sum (S t = S x + S p ).With this procedure it was possible to identify trends in the behavior of the Shannon entropy (S x and S p ) and test the compliance of the entropic uncertainty relation (S t = S x + S p ≥ n(1 + ln(p))).
We explore the perspective of considering the squeezed thermal reservoir as an equilibrium reservoir in a generalized Gibbs ensemble with two noncommuting conserved quantities. We outline the main properties of such a reservoir in terms of the exchange of energy, both heat and work, and entropy, giving some key examples to clarify its physical interpretation. This allows for a correct and insightful interpretation of all thermodynamical features of the squeezed thermal reservoir, as well as other similar nonthermal reservoirs, including the characterization of reversibility and the first and second laws of thermodynamics.
We consider the global thermal state of classical and quantum harmonic oscillators that interact with a reservoir. Ohmic damping of the oscillator can be exactly treated with a 1D scalar field reservoir, whereas general non-Ohmic damping is conveniently treated with a continuum reservoir of harmonic oscillators. Using the diagonalized Hamiltonian of the total system, we calculate a number of thermodynamic quantities for the damped oscillator: the mean force internal energy, mean force free energy, and another internal energy based on the free-oscillator Hamiltonian. The classical mean force energy is equal to that of a free oscillator, for both Ohmic and non-Ohmic damping no matter how strong the coupling to the reservoir. In contrast, the quantum mean force energy depends on the details of the damping and diverges for strictly Ohmic damping. These results give additional insight into the steady-state thermodynamics of open systems with arbitrarily strong coupling to a reservoir, complementing results for energies derived within dynamical approaches (e.g. master equations) in the weak-coupling regime.
Boltzmann’s statistical interpretation of the second law of thermodynamics indicates that systems evolve from less probable to more probable states. In an isolated system, entropy reaches its maximum at equilibrium, corresponding to the greatest number of accessible microstates. The quantum uncertainty principle, a fundamental feature of quantum mechanics, reflects the intrinsic probabilistic nature of physical systems. This study proposes that the equilibrium state defined by maximal entropy and microstate accessibility also represents a condition of maximal intrinsic uncertainty. We examine the connection between Boltzmann’s entropy framework and the quantum mechanical uncertainty relation between position and momentum, focusing on localized harmonic oscillator systems. Properties of the uncertainty product are analyzed in detail. Both the uncertainty principle and the second law of thermodynamics are rooted in probabilistic foundations. Our findings suggest that the uncertainty principle provides a conceptual perspective on thermodynamic behavior.
The time-dependent behaviour of harmonic oscillator squeezed states is examined from the viewpoint of position and momentum space Shannon entropies and their sum. Analytical expressions for these entropies are obtained from a symplectic group approach. The entropy sum returns to the entropic uncertainty bound at different time intervals, and is characterized by the presence of either a maximum (minimum) in the position space Shannon entropy and a minimum (maximum) in the momentum space one. The maximal value of the entropy sum is characterized by equality of position and momentum Shannon entropies. The transition of the entropy sum from the bound to its maximal value is studied, and relations between position and momentum space Shannon entropies are probed. The relation between the position and momentum space entropies is deformed as the value of the squeezing parameter increases. An analysis of the dependence of the Shannon entropies on the squeezing parameter shows different behaviours at different times. Both position and momentum space Shannon entropies exhibit a linear dependence on the parameter when their sum lies on the uncertainty bound. Here, the entropy sum is independent of the parameter. At other times, the relation between the entropies and squeezing parameter is non-linear, and extremal points are present. The entropy sum here is seen to be an increasing function of the parameter. Kullback–Leibler distances between squeezed and unsqueezed densities illustrate that the position or momentum space densities of the squeezed system can revert back to the unsqueezed ones during the time evolution. The phase-space Wigner function, entropy and mutual information are also examined and discussed. The statistical correlation as measured by mutual information is the entropy sum minus its value at the uncertainty bound.