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Journal of Mathematical Physics

Generation of excited coherent states for a charged particle in a uniform magnetic field

B. Mojaveri, Alireza Dehghani

Published Apr 1, 2015• 6 citations
Publisher

Abstract

We introduce excited coherent states, β,α;n≔a†nβ,α, where n is an integer and states β,α denote the coherent states of a charged particle in a uniform magnetic field. States β,α minimize the Schrödinger-Robertson uncertainty relation while having the nonclassical properties. It has been shown that the resolution of identity condition is realized with respect to an appropriate measure on the complex plane. Some of the nonclassical features such as sub-Poissonian statistics and quadrature squeezing of these states are investigated. Our results are compared with similar Agarwal’s type photon added coherent states (PACSs) and it is shown that, while photon-counting statistics of β,α,n are the same as PACSs, their squeezing properties are different. It is also shown that for large values of β, while they are squeezed, they minimize the uncertainty condition. Additionally, it has been demonstrated that by changing the magnitude of the external magnetic field, Bext, the squeezing effect is transferred from one component to another. Finally, a new scheme is proposed to generate states β,α;n in cavities.

DOI
10.1063/1.4917545

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We introduce excited coherent states, β,α;n≔a†nβ,α, where n is an integer and states β,α denote the coherent states of a charged particle in a uniform magnetic field. States β,α minimize the Schrödinger-Robertson uncertainty relation while having the nonclassical properties. It has been shown that the resolution of identity condition is realized with respect to an appropriate measure on the complex plane. Some of the nonclassical features such as sub-Poissonian statistics and quadrature squeezing of these states are investigated. Our results are compared with similar Agarwal’s type photon added coherent states (PACSs) and it is shown that, while photon-counting statistics of β,α,n are the same as PACSs, their squeezing properties are different. It is also shown that for large values of β, while they are squeezed, they minimize the uncertainty condition. Additionally, it has been demonstrated that by changing the magnitude of the external magnetic field, Bext, the squeezing effect is transferred from one component to another. Finally, a new scheme is proposed to generate states β,α;n in cavities.
A new scheme is proposed to design excited coherent states. where the states βββ,ααα denote the Glauber two variable minimum uncertainty coherent states, which minimize minimum uncertainty conditions while carrier nonclassical properties too and n is an integer. They are converted into the Agarwal's type of the photon added coherent states, arbitrary Fock states and the Glauber two variable coherent states, respectively depending on which of the parameters $\b{eta}$,ααα and nnn equal to zero. It has been shown that the resolution of identity condition is realized with respect to an appropriate measure on the complex plane, too. We have compared our results with the similar quantum states of Agarwal's type and seen that in our case amount of quantum fluctuations are much controllable. Moreover, we have also more flexibility to establish and set-out of their features. Also, there is a discussion on the statistical properties, can unveil (non-)classical properties of these states. For instance their Poissonian statistics are significant similar to what we saw earlier in the Glauber two variable coherent states. Interestingly, depending on the particular choice of the parameters of the above scenarios, we are able to determine the status of compliance with squeezing properties in field quadratures. The last stage is devoted to some theoretical framework to generate them in cavities.
We introduce states defined by \ensuremath{\Vert}\ensuremath{\alpha},m〉=${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathit{m}}$\ensuremath{\Vert}\ensuremath{\alpha}〉 up to a normalization constant, where \ensuremath{\Vert}\ensuremath{\alpha}〉 is a coherent state and m an integer. We study the mathematical and physical properties of such states. We demonstrate phase squeezing and the sub-Poissonian character of the fields in such states. We study in detail the quasiprobability distributions and the distribution of the field quadrature. We also show how such states can be produced in nonlinear processes in cavities.
The two-mode photon-added state defined by |?, q; m>?=?a+mb+m|?, q> up to a normalization constant is introduced, where |?, q> is a pair coherent state and m an integer. It is shown that the fields in such a state have remarkable quantum features such as sub-Poissonian statistics, violations of Cauch-Schwarz inequalities, correlations in the number fluctuations and squeezing. We examine the influence of photon-adding on the properties of pair coherent state, and propose a method of generating two-mode photon-added states in the context of cavity quantum electrodynamics.
Quantum statistical properties of general superpositions of coherent states of the type Nα,βα+eiξβ have been studied. Formulas for fluctuations (variances) of the coordinate and momentum quadratures and fluctuations of the number of photons in these states are obtained. The amplitude Nα,βnαeiφ+eiξnβeiφ,φ=φα=φβ and phase Nα,βneiφα+eiξneiφβ,n=nα=nβ superpositions of coherent states are considered separately. In the case of amplitude superpositions, a field under certain conditions occurs in a quadrature-squeezed state. At the same time, the photon distribution in the case of amplitude superpositions is always super-Poissonian. In the case of phase superpositions, a field under certain conditions occurs in quadrature-squeezed states for small average numbers of photons. For certain parameters of coherent states, the photon statistics in phase superpositions is sub-Poissonian. Using amplitude superpositions, it is possible to generate a microscopic electromagnetic field with the average photon number below or on the order of unity, as well as mesoscopic and even macroscopic fields with arbitrary large average photon numbers, characterized by significant quadrature squeezing. Using phase superpositions, it is possible to create under certain conditions the electromagnetic field states with sub-Poissonian statistics (squeezed distribution of photon numbers) for an arbitrary large average photon number. Expressions for a ‘strict’ coordinate–momentum uncertainty relation (Cauchy inequality) are obtained and analysed in cases of general, amplitude and phase superpositions of coherent states.
The most general displaced number 'coherent' states, based on the Heisenberg, su (2) and su (1, 1) Lie algebras symmetries, are constructed. They depend on two parameters, and can be converted into the well-known photon-added, two variable Glauber coherent states and displaced number states respectively, depending on which of the parameters is equal to zero. The relations of the Weyl–Heisenberg algebra guarantee a corresponding resolution of the identity conditions. A discussion of the statistical properties of these states is included. Significant are their squeezing properties, which can be raised by increasing the energy and angular momentum quantum numbers n and m . The maximum squeezing is obtained for B ext = 0. Depending on the particular choice of parameters in the above scenarios, we are able to determine the status of compliance with Poissonian statistics. In the limiting case, we obtain a major result about the non-classical properties of the Glauber minimum uncertainty coherent states. In other words, in addition to the requirement to minimize uncertainty conditions, they carry non-classical features too. Finally, a theoretical framework is proposed to generate them.
Two new classes of nonclassical states are defined by exciting Hillery's even and odd coherent states. Their physical properties are investigated numerically. It is found that they do not exhibit the squeezing effect but exhibit remarkable characteristics of sub-Poissonian distribution and antibunching. In sharp contrast to the case of excited Glauber coherent states, it is observed that the uncertainty of the field quadrature in excited even and odd coherent states increases with the degree of excitation. It is also seen that the Wigner functions associated with them take negative values for wider regions of parameters compared to those of Hillery's even and odd coherent states.