The single mode q-Bose field for -1<or=q<1 is shown to be a bounded operator with a continuous spectrum lying in the interval between +or-(2/(1-q)) 1 2/, thus becoming unbounded in the Bose limit (q to 1). The generalized eigenstates are determined in terms of q-Hermite polynomials whose properties are exploited to obtain orthonormality relations. It is observed that these are connected to important results in combinatorics. It is shown that the eigenstates are products of two q-exponentials of the q-creation operator acting on the Fock vacuum, in contrast to ordinary coherent states which involve only a single q-exponential of the creation operator. A method of representing operators in normal-ordered form based on this representation is described. The spectral resolution is employed to compute the asymptotic temporal behaviour of the particle number expectation value in states produced by coupling the q-Bose field to a source. It is shown that the limit does not commute with the limit q to 1. Specifically it is shown that for O<or=q<1 the asymptotic growth is linear in time in contrast to the quadratic growth for q=1.
The single mode q-Bose field for -1<or=q<1 is shown to be a bounded operator with a continuous spectrum lying in the interval between +or-(2/(1-q)) 1 2/, thus becoming unbounded in the Bose limit (q to 1). The generalized eigenstates are determined in terms of q-Hermite polynomials whose properties are exploited to obtain orthonormality relations. It is observed that these are connected to important results in combinatorics. It is shown that the eigenstates are products of two q-exponentials of the q-creation operator acting on the Fock vacuum, in contrast to ordinary coherent states which involve only a single q-exponential of the creation operator. A method of representing operators in normal-ordered form based on this representation is described. The spectral resolution is employed to compute the asymptotic temporal behaviour of the particle number expectation value in states produced by coupling the q-Bose field to a source. It is shown that the limit does not commute with the limit q to 1. Specifically it is shown that for O<or=q<1 the asymptotic growth is linear in time in contrast to the quadratic growth for q=1.
The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper, a natural approach has been applied using exclusively the operational properties of the Hermite and Hermite-like polynomials and showing their fundamental role in this field. Closed-form results in terms of polynomials, exponentials, and simple algebraic functions are the major contribution of the paper.
At the end of the preceding chapter, we showed that the photon number states | n 〉 have a uniform phase distribution over the range 0 to 2π. Essentially, then, there is no well-defined phase for these states and, as we have already shown, the expectation value of the field operator for a number state vanishes. It is frequently suggested (see, for example, Sakurai) that the classical limit of the quantized field is the limit in which the number of photons becomes very large such that the number operator becomes a continuous variable. However, this cannot be the whole story since the mean field 〈 n | Ê x | n 〉 = 0 no matter how large the value of n . We know that at a fixed point in space a classical field oscillates sinusoidally in time. Clearly this does not happen for the expectation value of the field operator for a number state. In this chapter we present a set of states, the coherent states, which do give rise to a sensible classical limit; and, in fact, these states are the “most classical” quantum states of a harmonic oscillator, as we shall see. Eigenstates of the annihilation operator and minimum uncertainty states In order to have a non-zero expectation value of the electric field operator or, equivalently, of the annihilation and creation operators, we are required to have a superposition of number states differing only by ±1.
Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.
Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.
In place of the usual commutation relation [â,â†]=1 we consider the generalized commutation relation characteristic of the para-Bose oscillators, viz, [â, Ĥ]=Ĥ where Ĥ is the Hamiltonian (1/2)(ââ†+â†â). The number states and the representation of various operators in the basis formed by these states are obtained. We then introduce the para-Bose coherent states defined as the eigenstates of â for this generalized case. We consider some of the properties of these coherent states and also show that the uncertainty product 〈 (Δq̂)2〉 〈 (Δp̂)2〉 in this case could be made arbitrarily small.
The energy, position, and momentum eigenstates of a para-Bose oscillator system were considered in paper I. Here we consider the Bargmann or the analytic function description of the para-Bose system. This brings in, in a natural way, the coherent states ‖z;α〉 defined as the eigenstates of the annihilation operator ?. The transformation functions relating this description to the energy, position, and momentum eigenstates are explicitly obtained. Possible resolution of the identity operator using coherent states is examined. A particular resolution contains two integrals, one containing the diagonal basis ‖z;α〉〈z;α‖ and the other containing the pseudodiagonal basis ‖z;α〉〈−z;α‖. We briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations. The problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible α dependence of this minimum value is considered.