Even and odd coherent states and tomographic representation of quantum mechanics and quantum optics VI. Man'ko

Abstract

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.