Canonical equivalence of quantum systems, multimode squeezed states and Robertson relation

2012

We describe a six-parameter family of the minimum-uncertainty squeezed states for the harmonic oscillator in nonrelativistic quantum mechanics. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. We show that the product of the variances attains the required minimum value 1/4 only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied. The overlap coefficients between the squeezed, or generalized harmonic, and the Fock states are explicitly evaluated in terms of hypergeometric functions. The corresponding oscillating photons statistics are discussed and an application to quantum optics and cavity quantum electrodynamics is mentioned.

Pramana, 1997

A definition of coherent states is proposed as the minimum uncertainty states with equal variance in two hermitian non-commuting generators of the Lie algebra of the hamiltonian. That approach classifies the coherent states into distinct classes. The coherent states of a harmonic oscillator, according to the proposed approach, are shown to fall in two classes. One is the familiar class of Glauber states whereas the other is a new class. The coherent states of spin constitute only one class. The squeezed states are similarly defined on the physical basis as the states that give better precision than the coherent states in a process of measurement of a force coupled to the given system. The condition of squeezing based on that criterion is derived for a system of spins.

Journal of Physics B: Atomic, Molecular and Optical Physics, 2002

We introduce nonlinear canonical transformations that yield effective, solvable Hamiltonians of multiphoton down-conversion processes, and we define the associated non-Gaussian multiphoton squeezed states as the coherent states of the multiphoton Hamiltonians. We study in detail the four-photon processes and the associated non-Gaussian four-photon squeezed states. The realization of squeezing, the behaviour of the field statistics, and the structure of the phase space distributions show that these states realize a natural four-photon generalization of the two-photon squeezed states.

International Journal of Applied Mathematics and Theoretical Physics, 2015

We present a study on linear canonical transformation in the framework of a phase space representation of quantum mechanics that we have introduced in our previous work . We begin with a brief recall about the so called phase space representation. We give the definition of linear canonical transformation with the transformation law of coordinate and momentum operators. We establish successively the transformation laws of mean values, dispersions, basis state and wave functions. Then we introduce the concept of isodispersion linear canonical transformation.

Physical Review A, 1994

The two quantum invariant operators are found from the time-dependent Hamiltonian of the harmon- ic oscillator with an auxiliary condition. The solution of the Schrodinger equation for the system, such as the eigenfunctions, eigenvalues, and minimum uncertainty, is derived by utilizing these invariant operators. The coherent states of this system are not the squeezed states, and the eigenfunction of the in- variant operator is not the eigenfunction of the Hamiltonian of the system unless it is in the invariant representation. The squeezing function, which is an eigenfunction of the Hamiltonian of the system in the invariant representation and which also gives the minimum uncertainty, is obtained by a set of uni- tary transformed operators, i.e. , squeezing operators.

The dynamics of the coherent and squeezed states is described and differentiated by means of the probability flux vector. The mean and variance values of the photon number operator used in quantum optics are calculated, by employing squeezed states defined in the coherent-state quantum phase-space representation.

Physical Review A, 1998

General Hamiltonian systems related by linear canonical transformations, which are combinations of the scale and gauge transformations, are considered. Using path integrals, it is proven that each quantum Hamiltonian can be expressed by the classical Hamiltonian whose canonical variables are replaced by their corresponding quantum operators. The relation between the Schro ¨dinger solution and the propagator for the transformed ͑new͒ and for the original ͑old͒ systems are evaluated by applying the unitary operator which describes the linear relationship between their quantum operators as the corresponding relations between their classical forms. It is shown that the uncertainty relations between the canonical position and momentum operator depend on the gauge function chosen, and satisfy Heisenberg's uncertainty principle. The uncertainty relations between the canonical position and kinetic momentum operator ͑distinguished from the momentum operator͒ do not depend on the gauge function chosen, and may not satisfy Heisenberg's uncertainty principle. By the gauge transformation, a single system has innumerable Schro ¨dinger equations, but the quantum averages of the function of the position and kinetic momentum operators are invariant for all solutions, as for classical cases.

We show that the parametrized Wave-Packet Phase Space representation, which has been studied earlier by one of the authors. is equivalent to a Squeezed States Phase Space Representation of quantum mechanics.

A preferred form for the path integral discretization is suggested that allows the implementation of canonical transformations in quantum theory.

Journal of Physics A: Mathematical and Theoretical, 2013

We provide a coarse but intuitive classification of squeezing in quantum systems with SU (n) symmetries. This classification is based on the non-equivalent paths (classical trajectories) in the corresponding phase-space. The example of SU (3) is studied in details.

Eprint Arxiv Quant Ph 9510001, 1995

We propose a displacement-operator approach to some aspects of squeezed states for general multiphoton systems. The explicit displacement-operators of the squeezed vacuum and the coherent states are achieved and expresses as the ordinary exponential form. As a byproduct the coherent states of the q-oscillator are obtained by the usual exponential displacement-operator.

Physical Review A, 2002

We introduce a linear, canonical transformation of the fundamental single-mode field operators a and a † that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states. This generalization is obtained by adding to the linear transformation a nonlinear function of any of the fundamental quadrature operators X1 and X2, making the original Bogoliubov transformation quadrature-dependent. Remarkably, the conditions of canonicity do not impose any constraint on the form of the nonlinear function, and lead to a set of nontrivial algebraic relations between the c-number coefficients of the transformation. We examine in detail the structure and the properties of the new quantum states defined as eigenvectors of the transformed annihilation operator b. These eigenvectors define a class of multiphoton squeezed states. The structure of the uncertainty products and of the quasiprobability distributions in phase space shows that besides coherence properties, these states exhibit a squeezing and a deformation (cooling) of the phase-space trajectories, both of which strongly depend on the form of the nonlinear function. The presence of the extra nonlinear term in the phase of the wave functions has also relevant consequences on photon statistics and correlation properties. The non quadratic structure of the associated Hamiltonians suggests that these states be generated in connection with multiphoton processes in media with higher nonlinearities.

Canadian Journal of Physics, 1999

Canonical transformations using the idea of quantum generating functions are applied to construct a quantum Hamilton-Jacobi theory, based on the analogy with the classical case. An operator and a c-number form of the time-dependent quantum Hamilton-Jacobi equation are derived and used to find dynamical solutions of quantum problems. The phase-space picture of quantum mechanics is discussed in connection with the present theory.PACS Nos.: 03.65-w, 03.65Ca, 03.65Ge

Lecture Notes in Mathematics, 1978

We discuss through the examples of the oscillator with centrifugal potential and the Coulomb problem in dilated form, the general procedure for finding the representation in quantum mechanics-of non-linear and non-bijective canonical transformations. The ambiguity group associated with the canonical transformations and the irreducible representations of this group, which lead to the concept of ambiguity spin, suggest also that some features of the spectra of quantum operators are already present in the classical picture.

Physica Scripta, 2015

The effect of phase space general noncommutativity on producing deformed coherent squeezed states is examined. A two-dimensional noncommutative quantum system supported by a deformed mathematical structure similar to that of Hadamard billiards is obtained and their components behavior are monitored in time. It is assumed that the independent degrees of freedom are two free 1D harmonic oscillators (HO's), so the system Hamiltonian does not contain interaction terms.