Diagonal Representation of Density Matrix Using q-coherent States

Transcription

1 Proceedings of Institute of Mathematics of NAS of Ukraine 24, Vol. 5, Part 2, Diagonal Representation of Density Matrix Using -Coherent States R. PARTHASARATHY and R. SRIDHAR The Institute of Mathematical Sciences, Chennai 6 3, India sarathy@imsc.res.in, sridhar@imsc.res.in A -analogue of the diagonal representation of the density matrix is derived using -boson coherent states. The -generalization of the self-reproducing property of ρ(z,z) and a self consistent self-reproducing kernel K(z,z) are obtained. Some applications to non-linear issues in optics are indicated Introduction A diagonal representation of the uantum mechanical density matrix by means of standard bosonic oscillator coherent states was obtained by Sudarshan [] and Glauber [2]. A remarkable feature of this representation is that the average expectation value of normal ordered operators becomes the same as that of a classical function for a probability distribution over complex plane, thereby bringing one-to-one correspondence between classical complex representation and uantum mechanical density matrices. The study of Quantum Groups has led to a nonlinear realization of Lie algebras and this resulted in the construction of SU (2) algebra using -bosonic oscillators by Macfarlane [3] and Biedenharn [4]. A corresponding -fermion oscillator and SU (2) algebra had been proposed by Parthasarathy and Viswanathan [5]. Such -oscillators effectively deal with non-ideal systems which have interactions and provide a method to study non-linear excitations of EM fields. The use of -bosonic oscillator to describe density matrix has been made by Nelson and Fields [6]. In this contribution, we reconsider this issue in more detail and present some new results regarding the self-reproducing property and relation between the expansion coefficients of the density matrix in the Fock space description and -coherent state description. 2 -boson coherent states The -boson oscillator algebra for -annihilation, -creation and -number operators {a, a,n} is [3, 4] aa aa =, [N,a] = a, [N,a ]=a, () and the deformation parameter is real and <. Further N a a. The Fock space F is built upon a vacuum state with a =. The n-boson normalized Fock space state is given by n = [n]! (a ) n, (2) where { n n N {}} and [n] is the -number [n] = n. (3)

2 9 R. Parthasarathy and R. Sridhar Further it follows, a n = [n] n, a n = [n +] n +, N n = n n, a a n =[n] n. (4) And [n]! = [n][n ][n 2] [2][]. A -boson coherent state is defined as { z = e z 2 } 2 n= z n [n]! n, (5) where z is a complex variable and e x is the -exponential function. This is a meromorphic function [7] with no zeroes and simple poles at x k = k /( ). The first pole occurs at x =/( ). Further it can be seen that z is an eigenstate of a with eigenvalue z. The most important property of z is that if z is another such -coherent state, then z z = { } e z 2 e z 2 2 e zz, (6) which means that such states are not orthogonal. But z z =. Further, by writing z = re iθ and treating the integration over the bounded variable θ as ordinary integration while that over r 2 as -integration, it follows from (5) π d 2 z z z = π where R 2 =/( ). 2π R 2 dθ 2 dr2 z z =, (7) 3 Diagonal representation of density matrix The density matrix has the expansion ρ = n,m= ρ(n, m) n m, (8) in terms of Fock space states. From the definition of the -coherent state (5), it can be shown [8] that { ( ) [n]![m]! d (n+m) } 2π dθ n m = [n + m]! dr 2π er2 e i(m n)θ re iθ re iθ, (9) where ( d dr ) stands for -differentiation. Substituting this in (8), we obtain the diagonal representation of the density matrix in terms of -boson coherent states as ρ = n,m= ρ(n, m) [n]![m]! [n + m]! { ( d dr ) (n+m) 2π a -analogue of the result of [, 2]. Expanding ρ as ρ = d 2 zφ(z) z z, r= } dθ 2π er2 e i(m n)θ re iθ re iθ r=, ()

3 Diagonal Representation of Density Matrix Using -Coherent States 9 with d 2 zφ(z) = (to ensure that Tr ρ = ), the expansion coefficients ρ(n, m) inf and φ(z) in -coherent states, are related by ρ(n, m) = d 2 z n z m { } zφ(z) e z 2 [n]![m]! R 2 = π d ( r 2) φ(r 2 ) r2n { [n]! where in the last step the θ integration is performed. It follows ρ(n, n) =. n 4 Some applications e r2 } δn,m, () As a first application, we consider the expansion of normal ordered operators, F = p,s C p,s (a ) p a s. Then using a z = z z, and taking matrix elements of F in between -coherent states, we find C p,s = min(p,s) k= ( ) k k(k )/2 [k]! p k F s k, (2) [s k]![p k]! relating C p,s to the Fock space matrix elements of F. As a second application, we consider the average expectation value of F. Using ρ = d 2 zφ(z) z z, we find, Tr (ρf )= C p,s d 2 zφ(z) z p z s. (3) p,s The following results are to be noted. First, Tr (ρf ) is now the expectation value of complex classical function z p z s for a distribution φ(z). Second for φ(z) =φ ( r 2), this becomes, π p,s R 2 C p,s d ( r 2) φ ( r 2) r 2p. For φ(r 2 )= π e r2, -Gaussian, tr (ρf )= p C p,p [p]!. As a third application, we consider an operator O = n to express B n (a a) n. First of all, it is possible (a a) n = s C n s (a ) s a s. (4) The coefficients Cs n satisfy a recursion relation Cs n+ = Cs n s +[s]cs n. Then we find, Tr (ρo) =π n R 2 B n Cs n d ( r 2) φ ( r 2) r 2s. (5) n s= For φ(r 2 )a-gaussian, (5) simplifies to Tr (ρo) = n B n Cs n [s]!. n s=

4 92 R. Parthasarathy and R. Sridhar 5 Self reproducing property Now we take up the important feature of ρ, namely, the self-reproducing property. The matrix elements of ρ in between two -coherent states is calculated to be z ρ z = { } ρ(n, m) e z 2 e z 2 2 z m z n. (6) n,m [m]![n]! Introduce ρ(z,z)= z m z n { ρ(n, m) n,m [m]![n]! e z 2 } e z 2. (7) Then, ρ(z,z)= z ρ z. Now we use the completeness relation (7) for the -coherent states, to write ρ(z,z)= z ρ z = d 2 ξ z ρ ξ ξ z π = d 2 ξk(ξ,z)ρ(z,ξ), (8) where the reproducing kernel K(ξ,z) is defined as K(ξ,z) = π ξ z. (9) (8) gives the self-reproducing property of ρ(z,z)withk(ξ,z) as the self-reproducing kernel. The theory of self-reproducing kernel has been studied by Aronszajn [9]. The above kernel (9) satisfies the reuired properties of a self-reproducing kernel. Namely, first, it satisfies the matrix multiplication property, d 2 ξk(z,ξ)k(ξ,z )=K(z,z ). (2) Second, since ρ is Hermitian, the kernel satisfies the Hermiticity property, K(z,ξ) = K(ξ,z). (2) The explicit expression for the kernel is, K(ξ,z) = { } e ξ 2 e z 2 2 e ξz, (22) π from which it follows K(z,z) >. (23) 6 -coherent states in physical situations We now give an attempt to use -coherent states to some physical situations. The fluctuations of photon number in the standard Fock space description is zero. However, optic field described by the standard coherent state has large uncertainty in photon number.

5 Diagonal Representation of Density Matrix Using -Coherent States 93 This is seen by considering z a a z = z 2 = N for an ordinary coherent state. Also, N 2 = z a aa a z = z 2 + z 4. Then the uncertainty in the photon number is n = N 2 N 2 = z. (24) For large z, the variance of the photon number is large. If we choose to describe the optic field by a -coherent state, the photon number is taken as photon -number. Then using -coherent states, we have z a a z = N = z 2. On the other hand, z a aa a z = N 2 = z 2 + z 4. Then, n = z +( ) z 2. (25) By comparing this with (24), we see that the uncertainity in photon -number depends nonlinearly on z. For <, the photon -number has less uncertainity. Further, the probability of finding the optic field in number state is n z 2, which for ordinary coherent state is P n = e, n! which is just a Poisson distribution. If we use -coherent states, then z 2 z 2n P n = {e z 2 z 2n } [n]!, which is -Poisson distribution. We now consider the propagation of light in a non-linear medium like Kerr medium. This is usually described by the Hamiltonian [] as H = ωa a + 2 χ(a ) 2 a 2, (26) where a, a are the usual annihilation and creation operators with aa a a = and χ represents the interaction strength of the light with the non-linear medium. Using photon number states, { n H n = ω n + χ } 2ω n(n ). (27) Instead of describing the propagation by the Hamiltonian (26), we desire to use a Hamiltonian H = ωa a, (28) where now a, a are -annihilation and creation operators (). Then it follows n H n = ω[n]. (29) The important observation is that (29) will describe the content of (27). By setting = ɛ, we have from (3), n H n = ω {n + ɛ } 2 n(n ), (3) keeping terms up to ɛ. Thus (3) has the same content of (27), for ɛ = χ ω. However H is simple, like a free Hamiltonian and the interaction is shifted to -number. As we have kept terms linear in ɛ, such description will be valid for high freuency propagation.

6 94 R. Parthasarathy and R. Sridhar 7 Summary We have obtained an explicit diagonal representation of the uantum mechanical density matrix using -coherent states. Certain classes of operators are studied for their statistical averages. The expansion coefficients of ρ in Fock space and -coherent state representation are related. The Cs n coefficients introduced in (4) with their recursion relation, are -Stirling numbers of the second kind and they enter in -coherent state description. The self-reproducing property of ρ(z,z) is established. In this, the completeness relation of -coherent states (7) plays a crucial role. The self-reproducing kernel is found and it satisfies the reuired Hermiticity property and matrix multiplication rule. Applications representing the use of -coherent states in the description of light propagating in non-linear medium are illustrated. Acknowledgements The author (RP) wishes to thank the Institute of Mathematical Sciences for providing the travel support to participate in the V International Conference on Symmetry in Nonlinear Mathematical Physics, held at the Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv and Professor Nikitin and the organisers for the excellent academic atmosphere provided during the conference. [] Sudarshan E.C.G., Euivalence of semiclassical and uantum mechanical descriptions of statistical light beams, Phys. Rev. Lett., 963, V., N 7, [2] Glauber R.J., Coherent and incoherent states of radiation field, Phys. Rev., 963, V.34, N 6, [3] Macfarlane A.J., On -analogue of the uantum harmonic oscillator and the uantum group SU (2), J. Phys. A, 989, V.22, N 23, [4] Biedenharn L.C., The uantum group SU (2) and -analogue of the boson operators, J. Phys. A, 989, V.22, N 8, L873 L878. [5] Parthasarathy R. and Viswanathan K.S., A -analogue of the supersymmetric oscillator and its -superalgebra, J. Phys. A, 99, V.24, N 9, [6] Nelson C.A. and Fields M.H., Number and phase uncertainities of the -analogue uantized field, Phys. Rev. A, 995, V.5, N 3, [7] Perelomov A.M., On the completeness of some subsystems of -deformed coherent states, Helv. Phys. Acta, 966, V.68, [8] Parthasarathy R. and Sridhar R., A diagonal representation of the uantum density matrix using -boson coherent states, Phys. Lett. A, 22, V.35, N 3 4, 5. [9] Aronszajn N., Theory of reproducing kernels, Trans. Am. Math. Soc., 95, V.68, [] Peng J.S and Li G.X., Introduction to modern uantum optics, Singapore, World Scientific, 998.