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Mathematical Notes

Squeezed states and their applications to quantum evolution

Alexander Mikhailovich Chebotarev, Timur Tlyachev, A. A. Radionov

Published Apr 1, 2011• 8 citations
Publisher

Abstract

In this paper, we consider quantum multidimensional problems solvable by using the second quantization method. A multidimensional generalization of the Bogolyubov factorization formula, which is an important particular case of the Campbell-Baker-Hausdorff formula, is established. The inner product of multidimensional squeezed states is calculated explicitly; this relationship justifies a general construction of orthonormal systems generated by linear combinations of squeezed states. A correctly defined path integral representation is derived for solutions of the Cauchy problem for the Schrödinger equation describing the dynamics of a charged particle in the superposition of orthogonal constant ( E,H )-fields and a periodic electric field. We show that the evolution of squeezed states runs over compact one-dimensional matrix-valued orbits of squeezed components of the solution, and the evolution of coherent shifts is a random Markov jump process which depends on the periodic component of the potential.

DOI
10.1134/s0001434611030308

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In this paper, we consider quantum multidimensional problems solvable by using the second quantization method. A multidimensional generalization of the Bogolyubov factorization formula, which is an important particular case of the Campbell-Baker-Hausdorff formula, is established. The inner product of multidimensional squeezed states is calculated explicitly; this relationship justifies a general construction of orthonormal systems generated by linear combinations of squeezed states. A correctly defined path integral representation is derived for solutions of the Cauchy problem for the Schrödinger equation describing the dynamics of a charged particle in the superposition of orthogonal constant ( E,H )-fields and a periodic electric field. We show that the evolution of squeezed states runs over compact one-dimensional matrix-valued orbits of squeezed components of the solution, and the evolution of coherent shifts is a random Markov jump process which depends on the periodic component of the potential.
A symplectically covariant quantum-mechanical uncertainty relation more accurate than previously known ones is derived for multidimensional systems. It is shown that the quantum-mechanical description of a linear Hamiltonian system in terms of squeezed states is completely equivalent to its description in terms of a phase-space distribution function. A new approach to the semiclassical limit is proposed, based on the use of squeezed states. By analyzing explicit formulas for squeezed states, a semiclassical asymptotic form of the solution to the Cauchy problem for a multidimensional Schrödinger equation is found in the limit of ħ → 0. The behavior of the semiclassical solution in the neighborhood of a caustic is analyzed in the one-dimensional case, and the phase shift across the caustic is determined. General properties and examples of squeezed states are discussed that point to the fundamental importance of squeezed states for developing a nonrelativistic quantum-mechanical description of a system of charged particles in an electromagnetic field in the dipole approximation.
We consider the multimode generalization of the normally ordered factorization formula of squeezings. This formula allows us to establish relationships between various representations of squeezed states, to calculate partial traces, mean values, and variations. The main results are expressed in terms of the matrix representation of canonical transformations, which is a convenient and numerically stable mathematical tool. Explicit representations are given for the inner product and the composition of generalized multimode squeezings. Explicitly solvable evolution problems are considered.
In this paper, we present a purely algebraic construction of the normal factorization of multimode squeezed states and calculate their inner products. This procedure allows one to orthonormalize bases generated by squeezed states. We calculate several correct representations of the normalizing constant for the normal factorization, discuss an analog of the Maslov index for squeezed states, and show that the Jordan decomposition is a useful mathematical tool for problems with degenerate Hamiltonians. As an application of this theory, we consider a nontrivial class of squeezing problems which are solvable in any dimension.
The decay of coherence when a quantum system interacts with a much larger environment is usually described by a master equation for the system reduced density matrix and emphasizes the evolution of an entire ensemble. We consider two methods that have been developed recently to simulate the evolution of single realizations. Quantum-state diffusion involves both diffusion, where the individual quantum trajectory fluctuates through a Wiener process deriving from the environment, and localization to a coherent state, an eigenstate of the relevant Lindblad operator describing the coupling of the system to the environment. We demonstrate the localization process for different initial states and utilize the Wigner function to depict this localization in phase space. We concentrate on quantum states that can be expressed as a superposition of appropriate coherent states. For an initial superposition of two coherent states (a Schr\"odinger ``cat''), one of the two components will dominate the evolution. For initial Fock states, which can be described as a continuous superposition of coherent states on a ring, localization takes place when one coherent state is selected from that ring where each component has nearly the same energy as the original Fock state. We also consider the localization from a nonclassical squeezed ground state, which can be expressed as a superposition of coherent states along a line in phase space. The second simulation method considered is the state vector Monte Carlo, or ``quantum jump,'' approach, which relates to the direct counting of decay quanta. In the case of an initial Schr\"odinger ``cat,'' we find that when no quantum is detected the ``cat'' shrinks, but when a quantum is detected, the Schr\"odinger ``cat'' ``jumps'' from one type of ``cat'' to another with different internal phase. For an initial squeezed state we show how quantum jumps lead to individual realizations which are superpositions of two squeezed states.
Coherent states are special types of wavefunctions that minimize a generalized uncertainty principle for a suitable pair of operators. Equivalently, they are eigenstates of an appropriate annihilation operator. Their applications are extensive throughout physics including in quantum optics, nuclear physics, quantum field theory, path integral formulations, and quantum information through the study of entanglement and quantum measurement. This thesis explores two main topics. First, we consider the Schrödinger evolution of a Gaussian coherent state under a non-Hermitian Hamiltonian. We develop a symbol calculus and use it to construct an approximate solution to the time-dependent Schrödinger equation. We find the evolution equations of the center and the Gaussian matrix of the coherent state, which form a system. This result generalizes the previously-known case where the classical Hamiltonian is quadratic. In the second part of the thesis, we apply a quantum version of dimensional reduction to construct Gaussian coherent states in the Bargmann space of complex projective space. The semiclassical properties of these reduced states are controlled by a suitable notion of symbol. Making use of these properties, we provide norm estimates and a propagation result for Hermitian Hamiltonians. As a special case of these reduced states, we define and examine spin-squeezed states that live naturally in the Bargmann space of the Riemann sphere.
Functional integrals provide a powerful and useful formalism for deriving both perturbative and nonperturbative results in quantum theory. Here we consider their application to the problem of formulating the dynamics of coherent and squeezed states. We derive the functional integral representation of the coherent state density matrix by combining the closed time path representation of the density matrix in terms of functional integrals in configuration space with the projection of coherent states onto configuration space states. In this formalism the atomic variables can, without loss of generality, be integrated out exactly yielding a single functional integral which describes the combined quantum dynamics of both the atomic and the coherent states.