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arXiv:1312.3016 · International Journal of Geometric Methods in Modern Physics

Basic properties of coherent-squeezed states revisited

Kazuyuki Fujii, Hiroshi Ôike

Published Mar 4, 2014• 2 citations
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Abstract

In this paper, we treat coherent-squeezed states of Fock space once more and study some basic properties of them from a geometrical point of view. Since the set of coherent-squeezed states {|α, β〉|α, β ∈ C} makes a real four-dimensional surface in the Fock space ℱ (which is of course not flat), we can calculate its metric. On the other hand, we know that coherent-squeezed states satisfy the minimal uncertainty of Heisenberg under some condition imposed on the parameter space {α, β}, so that we can study the metric from the viewpoint of uncertainty principle. Then, we obtain a surprising simple form (at least to us). We also make a brief review on Holonomic Quantum Computation by use of a simple model based on nonlinear Kerr effect and coherent-squeezed operators.

DOI
10.1142/s0219887814500510

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In this paper, we treat coherent-squeezed states of Fock space once more and study some basic properties of them from a geometrical point of view. Since the set of coherent-squeezed states {|α, β〉|α, β ∈ C} makes a real four-dimensional surface in the Fock space ℱ (which is of course not flat), we can calculate its metric. On the other hand, we know that coherent-squeezed states satisfy the minimal uncertainty of Heisenberg under some condition imposed on the parameter space {α, β}, so that we can study the metric from the viewpoint of uncertainty principle. Then, we obtain a surprising simple form (at least to us). We also make a brief review on Holonomic Quantum Computation by use of a simple model based on nonlinear Kerr effect and coherent-squeezed operators.
Aiming towards a geometric description of quantum theory, we study the coherent states-induced metric on the phase space, which provides a geometric formulation of the Heisenberg uncertainty relations (both the position-momentum and the time-energy ones). The metric also distinguishes the original uncertainty relations of Heisenberg from the ones that are obtained from non-commutativity of operators. Conversely, the uncertainty relations can be written in terms of this metric only, hence they can be formulated for any physical system, including ones with non-trivial phase space. Moreover, the metric is a key ingredient of the probability structure of continuous-time histories on phase space. This fact allows a simple new proof the impossibility of the physical manifestation of the quantum Zeno and anti-Zeno paradoxes. Finally, we construct the coherent states for a spinless relativistic particle, as a non-trivial example by which we demonstrate our results.
The Pegg-Barnett phase-operator formalism utilizes a finite basis set to represent operators of the harmonic oscillator; this enables the phase to be represented by a Hermitian operator, but rests on taking the dimensionality of the basis set to infinity for observable quantities. Simultaneously, in their approach Pegg and Barnett consider quantum states of a harmonic oscillator which are normalized in the Fock space, i.e., the dimensionality of the basis set in which the states of the harmonic oscillator are defined is supposed to be infinite, while the phase operator is defined in the finite-dimensional basis set. In this paper we address the problem of a consistent definition of a coherent state within a finite state basis. We employ displacement operators to define such coherent states and numerically evaluate observables as a function of the size of the basis set. We investigate phase properties of these coherent states. We find that if the dimensionality of the state space is much larger than the mean occupation number of the coherent states, then the results obtained in the finite-dimensional basis are applicable in the case of a ordinary quantum-mechanical harmonic oscillator. These coherent states are minimum uncertainty states with respect to quadrature operators (i.e., the position and momentum operators) and do not exhibit quadrature squeezing. A weakly excited (compared with the dimensionality of the state space) coherent state in finite-dimensional basis is not strictly speaking a minimum uncertainty state with respect to the number and phase operators. We give definitions of amplitude and phase squeezing and show that weakly excited coherent states can be amplitude squeezed. In the high-intensity limit (again compared with the dimensionality of the state space) these states exhibit phase squeezing.
As an aid to understanding the displacement operator definition of squeezed states for arbitrary systems, we investigate the properties of systems where there is a Holstein-Primakoff or Bogoliubov transformation. In these cases the ladder-operator or minimum-uncertainty definitions of squeezed states are equivalent to an extent displacement-operator definition. We exemplify this in a setting where there are operators satisfying [A, Aå] = 1, but the A's are not necessarily the Fock space a's; the multiboson system. It has been previously observed that the ground state of a system often can be shown to to be a coherent state. We demonstrate why this must be so. We close with a discussion of an alternative, effective definition of displacement-operator squeezed states.
An introductory survey on the Schroedinger uncertainty relation and its minimization states is presented with minimal number of formulas and some historical points. The case of the two canonical observables, position and momentum, is discussed in greater detail: basic properties of the two subsets of minimization states (canonical squeezed and coherent states) are reviewed and compared. The case of two non-canonical observables is breafly outlined. Stanfard SU(1,1) and SU(2) group-related coherent states can be defined as states that minimize Schroedinger inequality for the three pairs of generators simultaneously. The symmetry of the Heisenberg and Schroedinger relations is also discussed, and two natural generalizations to the cases of several observables and several states are noted.
Recently a fff-deformed Fock space which is spanned by ∣n>λ|n>_λ∣n>λ​ has been introduced. These bases are indeed the eigen-states of a deformed non-Hermitian Hamiltonian. In this contribution, we will use a rather new non-orthogonal basis vectors for the construction of coherent and squeezed states, which in special case lead to the earlier known states. For this purpose, we first generalize the previously introduced Fock space spanned by ∣n>λ|n>_λ∣n>λ​ bases, to a new one, spanned by an extended two-parameters bases ∣n>λ1,λ2|n>_{λ_{1},λ_{2}}∣n>λ1​,λ2​​. These bases are now the eigen-states of a non-Hermitian Hamiltonian Hλ1,λ2=aλ1,λ2†a+1/2H_{λ_{1},λ_{2}}=a^{\dagger}_{λ_{1},λ_{2}}a+1/2Hλ1​,λ2​​=aλ1​,λ2​†​a+1/2, where aλ1,λ2†=a†+λ1a+λ2a^{\dagger}_{λ_{1},λ_{2}}=a^{\dagger}+λ_{1}a + λ_{2}aλ1​,λ2​†​=a†+λ1​a+λ2​ and aaa are respectively, the deformed creation and ordinary bosonic annihilation operators. The bases ∣n>λ1,λ2|n>_{λ_{1},λ_{2}}∣n>λ1​,λ2​​ are non-orthogonal (squeezed states), but normalizable. Then, we deduce the new representations of coherent and squeezed states, in our two-parameters Fock space. Finally, we discuss the quantum statistical properties, as well as the non-classical properties of the obtained states, numerically.