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Modern Physics Letters A

DYNAMICAL HAMILTONIAN FOR PREASSIGNED TIME-EVOLUTION OF FERMIONIC SQUEEZING

Hong-Yi Fan, Lin Jing-Xian

Published Apr 30, 1999• 1 citation
Publisher

Abstract

We derive the general form of dynamical Hamiltonian generating the preassigned time-evolution of multimode fermionic squeezing. The fermionic squeezing is a quantum-mechanical image in fermionic Hilbert space corresponding to an orthogonal transformation in pseudo-classical Grassmann number space. The derivation is mainly based on the orthogonal properties of the transformation matrices. It is shown that some new fermionic operator realization of SO (2n) group and some new Hamiltonians can be obtained via this approach.

DOI
10.1142/s0217732399000900

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We derive the general form of dynamical Hamiltonian generating the preassigned time-evolution of multimode fermionic squeezing. The fermionic squeezing is a quantum-mechanical image in fermionic Hilbert space corresponding to an orthogonal transformation in pseudo-classical Grassmann number space. The derivation is mainly based on the orthogonal properties of the transformation matrices. It is shown that some new fermionic operator realization of SO (2n) group and some new Hamiltonians can be obtained via this approach.
For the two-mode fermionic squeezing operators we find that their coherent state projection operator representation make up a loyal representation, which is homomorphic to an SO(4) group, though the fermionic coherent states are not mutual orthogonal. Thus the result of successively operating with many fermionic squeezing operators on a state can be equivalent to a single operation. The fermionic squeezing operators are mappings of orthogonal transformations in Grassmann number pseudo-classical space in the fermionic coherent state representation.
The design of squeezing by a time-dependent harmonic oscillator is considered. The predetermined general time-dependent squeezing transformations are taken as the starting point from which the time evolution operator U(t,0) and then the corresponding time-dependent Hamiltonian are derived. The technique of integration within an ordered product of operators and the coherent-state representation are used in the intermediate derivations, where U(t,0) is obtained as an operator image of the classical symplectic transformation in phase space by a phase-space integral over coherent states.
In this Paper we present an approach to Quantum Mechanical Canonical Transformations. Our main result is that Time Dependent Quantum Canonical Transformations can always be cast in the form of Squeezing Operators. We revise the main properties of these operators in regard to its Lie group properties, how two of them can be combined to yield another operator of the same class and how can also be decomposed and fragmented. In the second part of the paper we show how this procedure works extremely well for the Time Dependent Quantum Harmonic Oscillator. The issue of the systematic construction of Quantum Canonical Transformations is also discussed along the lines of Dirac, Wigner and Schwinger ideas and to the more recent work by Lee. The main conclusion is that the Classical Phase Space Transformation can be maintained in the operator formalism but the construction of the Quantum Canonical Transformation is not clearly related to the Classical Generating Function of a Classical Canonical Transformation. We propose the road of Squeezing Operators rather than the old one attached to Quantum Operators constructed under the guideline of the exponential of the Classical Generating Function.
In this paper we generalize our previous results of the generalized one-mode harmonic oscillator to the generalized two-mode case. Systematic use is made of the SO(3,2) dynamical group and we are able to write a general form for the exact time evolution operator in terms of squeezing operators of one and two modes. A complete classification of the exact solutions is made and we derive them explicitly whenever possible. The relevant results on algebraic decomposition, coherent state generators and classification of the solutions are shown in tables. A plethora of soluble Hamiltonians already treated in the literature, which appear to be particular cases of the general formalism presented herein, are analysed as well as new cases, which to the authors' knowledge, have not yet been considered.
We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a + , which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.
We provide a Hilbert space approach to quantum mechanics where space and time are treated on an equal footing. Our approach replaces the standard dependence on an external classical time parameter with a spacetime-symmetric algebraic structure, thereby unifying the axioms that traditionally distinguish the treatment of spacelike and timelike separations. Standard quantum evolution can be recovered from timelike correlators, defined by means of a quantum action operator, a quantum version of the action of classical mechanics. The corresponding map also provides an alternative perspective on the path integral formulation that, in the case of fermions, does not require the use of Grassmann variables. In addition, the formalism can be interpreted in terms of generalized quantum states, codifying both the conventional information of a quantum system at a given time and its evolution. We show that these states are solutions to a quantum principle of stationary action grounded in timelike correlations and pseudo-entropies.