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College Physics

Hyperbolic sine and cosine operators together with odd and even coherent states

Tian Li-jie

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

The hyperbolic sine and cosine operators are introduced by the agency of translational operator in Fock space, thier action on the vocuum state give the new expressions of odd and even coherent states. And the quasi-probability distribution function and non-classical properties of odd and even coherent states are studied.

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The quantization of phase is still an open problem. In the approach of Susskind and Glogower so called cosine and sine operators play a fundamental role. Their eigenstates in the Fock representation are related with the Chebyshev polynomials of the second kind. Here we introduce more general cosine and sine operators whose eigenfunctions in the Fock basis are related in a similar way with arbitrary orthogonal polynomial sets on the intervall [-1,1]. To each polynomial set defined in terms of a weight function there corresponds a pair of cosine and sine operators. Depending on the symmetry of the weight function we distinguish generalized or extended operators. Their eigenstates are used to define cosine and sine representations and probability distributions. We consider also the inverse arccosine and arcsine operators and use their eigenstates to define cosine-phase and sine-phase distributions, respectively. Specific, numerical and graphical results are given for the classical orthogonal polynomials and for particular Fock and coherent states.
New states of electromagnetic field, i.e. even and odd negative binomial states are introduced here. These states interpolate between the even (odd) coherent states and the even (odd) quasi-thermal states. Various statistical properties of these states such as the mean photon number, second-order coherence function, normal squeezing are calculated. The Wigner function for these states are also discussed. These results may be useful for carrying out systematic study of the non-classical properties of `Schrödinger cat states' as one moves from even (odd) coherent state to even (odd) quasi-thermal state.
Even/odd coherent states can be generated from a vacuum in the following way \\a±)=D(a±)\\0), (9) where even/odd displacement operators read [1,7] D(a+) = cosh( Both even and odd coherent states are normalized eigenstates of operator a2 a2\\a±)=a2\\a±), (10) where a is an arbitrary complex number. It is worth mentioning that by applying \noperator a to an even coherent state |a+), one obtains an odd coherent state with \nthe same label a, but with a different normalization constant: a\\a+) = Q\\/tanh\\a\\ 2 |a_). (11) Similarly, \na\\a-) — axjcoth\\a\\2 |a+). (12) The decomposition of the even and odd coherent states in terms of number \nstates can be obtained by using equation (3) | a + ) = T V V ^ l V s ^ 1 + ( - ! ) " Q > ) , ( 1 3 ) |a_> - 7 V _ e - l a l 2 / 2 ^ 1 ^ 1 z l ^ Q n | n ) . ( 1 4 ) The even coherent state can only be expressed in terms of even number states \nand the odd coherent state can only be expressed in terms of odd number states. \nConsequently, the probability of finding an odd number of photons in the even \ncoherent states equals zero, as does the probability of finding an even number of \nphotons in the odd coherent states. Thus the probability distribution functions for \nthese states strongly oscillate: \\a\\4k \nfor n=2k P(+)(n) = \\ coshH 2 (2 fc ) ! \n0 for n=2k+l \n0 for n=2k P ( _)(n) = ( | a | W L sinh|a:| 2(2£; + l ) ! \nfor n=2k+l Expectation values of the first-order moments for the annihilation and creation \noperators in the even and odd coherent states are equal to zero ( a ± | a | a ± ) = 0 , (15) MULTIMODE EVEN AND ODD COHERENT STATES 223 due to Eqs. (11), (12), and the orthogonality property ( a ± | a T ) = 0. The expectation values of the second-order moments are (a±\\a2\\a±) = a2, (a+\\a*a\\a+) = \\a\\2 tanh | a | 2 , (a_|a + a|a_) = \\a\\2 coth | a | 2 . (16) Defining the quadratures of the electromagnetic field mode as Xx = (a + a f ) /2 , X2 = (a-a f ) /(2i), one obtains the following expressions for their variances in the even coherent state: 4 A X 2 = 2\\a\\2 tanh \\a\\2 + 2 |a | 2 cos 20 + 1, (17) 4 A X 2 = 2 |a | 2 tanh \\a\\2 - 2\\a\\2 cos 261 + 1. (18) We have introduced the modulus and phase of the complex coherent state amplitude as a=|a|e^. Eqs. (17) and (18) show that for 9=TT/2 the first quadrature \nhas some amount of squeezing for small values of |a | , while for 0=0 the second \nquadrature is squeezed. There is a possibility of squeezing alternatively in both the \nquadratures depending upon the phase of the complex amplitude a. For odd coherent states we get the same expressions as in (17) and (18), but tanh \\ a\\2 should \nbe replaced by coth \\ a\\2. Also, odd coherent states do not exhibit the property of \nsecond-order squeezing.
In this paper a new class of finite-dimensional even and odd nonlinear pair coherent states (EONLPCSs), which can be realized via operating the superposed evolution operators D±(τ) on the state |q, 0), is constructed, then their orthonormalized property, completeness relations and some nonclassical properties are discussed. It is shown that the finite-dimensional EONLPCSs possess normalization and completeness relations. Moreover, the finite-dimensional EONLPCSs exhibit remarkably different sub-Poissonian distributions and phase probability distributions for different values of parameters q,η and ζ.