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arXiv:quant-ph/0207035 · Journal of Physics A Mathematical and General

Creating quanta with an annihilation operator

Salomon Sylvain Mizrahi, V. V. Dodonov

Published Oct 2, 2002• 47 citations
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Abstract

The asymmetric nature of the boson 'destruction' operator â and its 'creation' partner ↠is made apparent by applying them to a quantum state |ψ⟩ different from the Fock state | n ⟩. We show that it is possible to increase (by many times or by any quantity) the mean number of quanta in the new 'photon-subtracted' state â|ψ⟩. Moreover, for certain 'hyper-Poissonian' states |ψ⟩ the mean number of quanta in the (normalized) state â|ψ⟩ can be much greater than in the 'photon-added' state â†|ψ⟩. The explanation of this 'paradox' is given and some examples elucidating the meaning of Mandel's q -parameter and the exponential phase operators are considered.

DOI
10.1088/0305-4470/35/41/315

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The asymmetric nature of the boson 'destruction' operator â and its 'creation' partner ↠is made apparent by applying them to a quantum state |ψ⟩ different from the Fock state | n ⟩. We show that it is possible to increase (by many times or by any quantity) the mean number of quanta in the new 'photon-subtracted' state â|ψ⟩. Moreover, for certain 'hyper-Poissonian' states |ψ⟩ the mean number of quanta in the (normalized) state â|ψ⟩ can be much greater than in the 'photon-added' state â†|ψ⟩. The explanation of this 'paradox' is given and some examples elucidating the meaning of Mandel's q -parameter and the exponential phase operators are considered.
In atomic, nuclear and particle physics we come across several situations where particles or radiation quanta (photons) are created or destroyed. For example in the beta decay of the neutron, a neutron is annihilated and a proton, electron and antineutrino are created. In some situations the interaction of radiation with matter can result in creation or annihilation of a pair of electron and positron. Such processes cannot be understood on the basis of (non-relativistic or relativistic) single particle wave equations because in such theories the probability ψ∗ψdτ (of finding the particle in the region of space dτ) when integrated over the whole space, results in unity at all times and so the existence of the particle is always guaranteed. Hence in the framework of single particle wave equation it is not possible to understand the creation or annihilation of particles. Even a simple process in which an atom undergoes a transition from an excited to the ground state by spontaneous emission of a photon in the absence of a perturbing electromagnetic field cannot be explained within this framework.
Inverses of the harmonic oscillator creation and annihilation operators by their actions on the number states are introduced. Three of the two photon annihilation operators, viz., a(sup +/-1)a, aa(sup +/-1), and a(sup 2), have normalizable right eigenstates with nonvanishing eigenvalues. The eigenvalue equation of these operators are discussed and their normalized eigenstates are obtained. The Fock state representation in each case separates into two sets of states, one involving only the even number states while the other involving only the odd number states. It is shown that the even set of eigenstates of the operator a(sup +/-1)a is the customary squeezed vacuum S(sigma) O greater than.
The photon-added nonlinear coherent states, defined as adaggerm∣alpha,f>{a}^{\\dagger m}|\\alpha, f>adaggerm∣alpha,f> up to a normalization constant, are still nonlinear coherent states with the nonlinear function f(N−m)[1−m/(N+1)]f(N-m)[1-m/(N+1)]f(N−m)[1−m/(N+1)]. Here % |\\alpha, f> is the nonlinear coherent state with the nonlinear function f(N)f(N)f(N), NNN is the number operator, adaggera^{\\dagger}adagger is the boson creation operator and mmm is a non-negative integer. We introduce a new type of nonlinear coherent states with the nonlinear function f(N+m)[1+m/(N+1)]f(N+m)[1+m/(N+1)]f(N+m)[1+m/(N+1)] obtained from the photon-added nonlinear coherent states with negative values of m.m.m. The nonlinear coherent states corresponding to the positive and negative values of mmm are shown to be the result of nonunitarily deforming the number states ∣m>|m>∣m> and ∣0>|0>∣0>, respectively. As an example, we study the sub-Poissonian statistics and squeezing effects of the photon-added geometric states with negative values of mmm in detail.
We introduce states a † m | ξ ⟩ and a m | ξ ⟩, where| ξ ⟩ is a two-mode squeezed vacuum state. We examine the properties of these states by analytic derivation. Because of the entanglement between the modes of the field, photon adding (or substracting) for one of the modes results in interesting outcomes. For a two-mode squeezed vacuum state, the effect of annihilation of m a-photons is just the same as creation of m b-photons. Annihilation of photons of mode a (or mode b ) actually increases the mean number of photons of mode a and mode b . We also calculate Mandel Q parameter, cross-correlation function and squeezing of the field.