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Communications in Theoretical Physics

Quasiprobability Distribution of Squeezed Thermal Spin States in Ferromagnet

QINFENG XU, Ze Cheng

Published Mar 1, 2008• 2 citations
Publisher

Abstract

We study squeezed properties of magnon squeezed thermal spin states by using the distribution of Q function in the ferromagnet. It is found that the distribution of Q function strongly depends on the temperature T and coupling parameter γ. Below the transition temperature T c , the distribution Q function in the squeezed thermal spin state presents a richer structure than in the normal state. Non-classical effects have been observed. In the transition from the normal to the squeezed thermal spin state, the phase symmetry of the magnon system is spontaneously broken.

DOI
10.1088/0253-6102/49/3/52

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We study squeezed properties of magnon squeezed thermal spin states by using the distribution of Q function in the ferromagnet. It is found that the distribution of Q function strongly depends on the temperature T and coupling parameter γ. Below the transition temperature T c , the distribution Q function in the squeezed thermal spin state presents a richer structure than in the normal state. Non-classical effects have been observed. In the transition from the normal to the squeezed thermal spin state, the phase symmetry of the magnon system is spontaneously broken.
In this paper, we introduce the self-consistent field approximation to treat with the nonlinear interaction among spin waves. Then temperature-dependent Bogoliubov transformation is introduced to generate a new representation which engenders the transition from the zero temperature to the finite temperature. At last, temperature-dependent quantum fluctuation properties of magnons are discussed in the thermal field. At lower temperature, we find that the fluctuation of spin-component at some given time regions can be below the zero-point fluctuation level of the vacuum state and exhibit a periodical squeezing behavior. In particular, these squeezed effects vanish with the increasing of temperature. These squeezing effects differ from the previous studies.
Phase properties of squeezed number states and squeezed thermal states are studied. Exact analytical formulae for phase distributions based on different phase approaches are derived and illustrated graphically. It is shown that the phase quasiprobability distribution P (W) ( theta ) associated with the Wigner function does not depend on the photon number of the initial number state and has the same form for both kinds of squeezed states under consideration while the Pegg-Barnett phase distribution and the phase quasiprobability distributions associated with the Q function and the Glauber-Sudarshan P function, approach P (W) ( theta ) in the limit of highly excited states.
Much attention has been given in the past to two classes of squeezed states: the squeezed vacuum and coherent states. Here we study the effects of squeezing on number states and on thermal field states. The statistical properties of these various squeezed states are discussed using the second-order correlation functions. The quasiprobabilities of the Wigner, Q, and positive P representations are calculated and compared for the squeezed states. The Glauber P representation for the squeezed thermal state explicitly shows the limit of its applicability. The photon number distributions of the squeezed number and squeezed thermal states are extensively discussed and new interference effects in phase space are shown to lead to highly structured number distributions.
We investigate nonclassical properties of the field states generated by subtracting any number of photons from the squeezed thermal state (STS). It is found that the normalization factor of photon-subtracted STS (PSSTS) is a Legendre polynomial of squeezing parameter rrr and average photon number $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ of the thermal state. Expressions of several quasiprobability distributions of PSSTS are derived analytically. Furthermore, the nonclassicality is discussed in terms of the negativity of the Wigner function (WF). It is shown that the WF of single PSSTS always has negative values if $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}<\mathrm{sinh}{}^{2}r$ at the phase space center. The decoherence effect on PSSTS is then included by analytically deriving the time evolution of WF. The results show that the WF of single PSSTS has negative value if $2\ensuremath{\kappa}t<\mathrm{ln}{1\ensuremath{-}(2\mathrm{n\ifmmode \bar{}\else \={}\fi{}}+1)(\mathrm{n\ifmmode \bar{}\else \={}\fi{}}\ensuremath{-}\mathrm{sinh}{}^{2}r)/[(2\mathfrak{N}+1)(\mathrm{n\ifmmode \bar{}\else \={}\fi{}}\mathrm{cosh}2r+\mathrm{sinh}{}^{2}r)]}$, which is dependent not only on average number N\mathfrak{N}N of thermal photons in the environment, but also on $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ and rrr.
We calculate exactly the quantum mechanical, temporal Wigner quasiprobability density for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state. The Wigner function allows us to calculate the fluctuations in photon number and the quadrature variance. We contrast the difference between the non-classicality criteria, which is independent of the displacement parameter α, based on the Glauber–Sudarshan quasiprobability distribution P(β) and the classical/non-classical behaviour of the Mandel QM(τ) parameter, which depends strongly on α. We find a phase transition as a function of α such that at the critical point αc, QM(τ), as a function of τ, goes from strictly classical, for |α|<|αc|, to a mixed classical/non-classical behaviour, for |α|>|αc|.