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arXiv:1107.5010

On the Fermi Function of Squeezed Coherent States

Maurice A. de Gosson

Published Jul 25, 2011• 0 citations
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Abstract

Fermi observed in 1930 that the state of a quantum system may be defined in two different (but equivalent) ways, namely by its wavefunction $Ψ$ or by a certain function $g_F$ on phase space canonically associated with $Ψ$. In this Note we study we study Fermi's function when $Ψ$ is a squeezed coherent state. We relate it with the Wigner transform of $Ψ$, thus generalising a previous observation of Benenti and Strini. We show that the symplectic capacity of the phase space ellipsoid $g_{F}(x,p)\leq0$ is bounded by $h/2$ and $nh/2$ (n the number of degrees of freedom).

DOI
10.48550/arxiv.1107.5010

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Fermi observed in 1930 that the state of a quantum system may be defined in two different (but equivalent) ways, namely by its wavefunction ΨΨΨ or by a certain function gFg_FgF​ on phase space canonically associated with ΨΨΨ. In this Note we study we study Fermi's function when ΨΨΨ is a squeezed coherent state. We relate it with the Wigner transform of ΨΨΨ, thus generalising a previous observation of Benenti and Strini. We show that the symplectic capacity of the phase space ellipsoid gF(x,p)≤0g_{F}(x,p)\leq0gF​(x,p)≤0 is bounded by h/2h/2h/2 and nh/2nh/2nh/2 (n the number of degrees of freedom).
Fermi observed in 1930 that the state of a quantum system may be defined in two different (but equivalent) ways, namely by its wavefunction Ψ\PsiΨ or by a certain function gFg_FgF​ on phase space canonically associated with Ψ\PsiΨ. In this Note we study we study Fermi's function when Ψ\PsiΨ is a squeezed coherent state. We relate it with the Wigner transform of Ψ\PsiΨ, thus generalising a previous observation of Benenti and Strini. We show that the symplectic capacity of the phase space ellipsoid gF(x,p)≤0g_{F}(x,p)\leq0gF​(x,p)≤0 is bounded by h/2h/2h/2 and nh/2nh/2nh/2 (n the number of degrees of freedom).
This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.
Abstract I nspired by our work with Basil Hiley, we continue the exploration of Fermi’s Ansatz. Back in 1930, Fermi showed that a wavefunction could be associated with a hypersurface in phase space. Fermi’s approach therefore leads to a geometric picture of quantum states that is based on Fermi ellipsoids. Our investigations with Basil Hiley revealed that the internal energy provided by Bohm’s quantum potential sustains these Fermi blobs whenever a quantum process is unfolding. Moreover, this internal energy is associated with a noncommutative quantum phase space (coarse-grained by quantum blobs - minimum uncertainty phase-space cells that are in a bijective correspondence with the squeezed coherent states familiar from quantum optics). In this paper, we use Fermi’s method in order to gain further insights into the invariance of quantum blobs under canonical transformations. Starting with a real Gaussian, we multiply it by a complex exponential (“chirp”) and thereby obtain a generalized coherent state. The associated change of gauge enables us to make contact with Fermi’s approach. As in de Broglie-Bohm theory, the phase and magnitude of the wavefunction feature explicitly here and are revealed to play a key role. Subsequently, we find the Fermi Hamiltonian operator and associated eigenvalue equation which the generalized coherent state satisfies. Considering Hamilton’s equations written in compact form, we show that the Hamiltonian flow determined by the Fermi Hamiltonian consists of symplectic matrices that belong to a one-parameter subgroup of Sp( n ) - we call this subgroup the canonical group of the generalized coherent state. The lift of this canonical group to the metaplectic group is found to leave the generalized Gaussian invariant; whereas the canonical group itself leaves the associated phase-space quantum blob invariant. We conclude that Fermi’s Ansatz provides an instructive method for proving these invariance results. It sheds light on subtle aspects of quantum processes that other approaches might miss.
A new operator representation, called squeezed coherent state representation, is introduced since Husimi operator has the form of squeezed coherent state. We fisrt introduce its specific integral expression. When κ = 1, this representation is reduced to the usual P function. As an example, we calculate the squeezed coherent state representation for thermal field to illustrate a difference between P function and the squeezed coherent state representation. Especially, in order to better apply this representation to quantum optics, we reveal the integral transformations between the squeezed coherent state representation, respectively, and the following three functions: Wigner function, Q function, and Husimi function.
We compute the photon-number distribution, the Q (α) distribution function and the wavefunctions in the momentum and position representation for a single mode squeezed number state using generating functions which allow one to obtain any matrix element in the squeezed number state representation from the matrix elements in the squeezed coherent state representation. For highly squeezed number states we discuss the previously unnoted oscillations which appear in the Q (α) function. We also note that these oscillations can be related to the photon-number distribution oscillations and to the momentum representation of the wavefunction.