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Journal of Mathematical Physics

Poincaré disk as a model of squeezed states of a harmonic oscillator

Ian Chi, Martin Fraas, Tina Tan

Published Jun 1, 2025• 0 citations
Publisher

Abstract

Single-mode squeezed states exhibit a direct correspondence with points on the Poincaré disk. In this study, we delve into this correspondence and describe the motions of the disk generated by a quadratic Hamiltonian. This provides a geometric representation of squeezed states and their evolution. We discuss applications in bang-bang and adiabatic control problems involving squeezed states.

DOI
10.1063/5.0252902

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Single-mode squeezed states exhibit a direct correspondence with points on the Poincaré disk. In this study, we delve into this correspondence and describe the motions of the disk generated by a quadratic Hamiltonian. This provides a geometric representation of squeezed states and their evolution. We discuss applications in bang-bang and adiabatic control problems involving squeezed states.
The phase states of a harmonic oscillator are considered and used to define an analytic representation in the unit disk. The Weyl semigroup in the number-angle phase space is studied. More general states called negative binomial states are introduced, and the corresponding analytic representations in the unit disk are defined. M\"obius conformal mappings are used for the study of SU(1,1) transformations. They define alternative types of one-mode squeezing based on the discrete series of SU(1,1) representations.
In a squeezed state, the variance in one canonical variable may be suppressed below that normally associated with either the ground state or a coherent state, at the expense of an expansion in the variance of the conjugate variable. Squeezed states are usually discussed in the context of quantized light fields. The principal properties of squeezed states are demonstrated using the motion of quantum mechanical simple harmonic oscillators. The motion of a single oscillator is discussed to introduce the key concepts of squeezing, including quadrature operators, and the error contours of the Wigner function describing the quantum quasiprobability distributions in phase space of the oscillator motion. The main topic of two-mode squeezing is then addressed, in which the fluctuations in a system of two oscillators are tightly correlated. The existence of squeezing is demonstrated in normal mode coordinates representing motion of superpositions of the motion of the two oscillators. The fluctuations within a single oscillator are shown to increase when squeezing increases in the normal modes, generating thermal noise in individual mode subsystems.
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.
A simple, and elegant geometrical representation is developed to describe the concept of coherence and squeezing for angular momentum operators. Angular momentum squeezed states were obtained by applying Bogoliubov transformation on the angular momentum coherent states in Schwinger representation [Phys. Rev. {\bf{A 51}} (1995)]. We present the geometrical phase space description of angular momentum coherent and squeezed states and relate with the harmonic oscillator. The unique feature of our geometric representation is the portraying of the expectation values of the angular momentum components accompanied by their uncertainties. The bosonic representation of the angular momentum coherent and squeezed states is compared with the conventional one mentioning the advantages of this representation of angular momentum in context of coherence and squeezing. Extension of our work on single mode squeezing to double mode squeezing is presented and compared with the single mode one. We also point out the possible applications of the geometrical representation in analyzing the accuracy of interferometers and in studying the behavior, dynamics of an ensemble of quantum-mechanical two-level systems and its interaction with radiation.
Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.