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American Journal of Physics

Disentanglement formulas: An alternative derivation and some applications to squeezed coherent states

Ananda Dasgupta

Published Nov 1, 1996• 24 citations
Publisher

Abstract

An alternative derivation of the disentanglement formulas for exponential operators, also known as the Baker–Campbell–Housdorff formulas is given for several Lie algebras. The method is especially suited to Lie algebras having no nontrivial center. We also look at some results that can be obtained by the use of these formulas in the case of squeezed coherent states.

DOI
10.1119/1.18183

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An alternative derivation of the disentanglement formulas for exponential operators, also known as the Baker–Campbell–Housdorff formulas is given for several Lie algebras. The method is especially suited to Lie algebras having no nontrivial center. We also look at some results that can be obtained by the use of these formulas in the case of squeezed coherent states.
In the preceding paper [Commun. Theor. Phys. 51 (2009) 321] we have recommended a convenient method for disentangling exponential operators in the form of exp{B + C}, trying to find an operator A that satisfies [A, B] = C, and [A,[A,B]] = 0, then from the Baker–Hausdorff formula we have exp{B + C} = exp{B+[A, B]} = e A e B e −A . After arranging e A e B = e B e A e W , the disentangling exp{B + C} = e B e W is obtained. In this work we use this method to two-mode case, especially, derive the normal ordering form of exp[h(a † a + b † b) + ga † b † + kab] without appealing to Lie algebra method.
Abstract The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. For non-commutative operators X and Y, according to the Campbell-Baker-Hausdorff-Dynkin theorem, e X+Y is not equivalent to e X e Y , but is instead given by the well-known infinite series formula. For a Lie algebra of a basis of three operators {X, Y, Z}, such that [X, Y] = κ Z for scalar κ and cyclic permutations, here it is proven that e a X+b Y is equivalent to e p Z e q X e−p Z for scalar p and q. Extensions for e a X+b Y+c Z are also provided. This method is useful for the dynamics of atomic and molecular nuclear and electronic spins in constant and oscillatory transverse magnetic and electric fields.
In the preceding paper ( Commun. Theor. Phys. 51 (2009) 321 ) we have recommended a convenient method for disentangling exponential operators. In this work we use this method for disentangling exponential operators composed of angular momentum operators. We mainly disentangle the form of exp[2hJ z + gJ + + kJ − ] as the ordering exp(...J + )exp(...J z )exp(...J − ), we employ the Schwinger Bose realization J − = b † a, J + = a † b, J z = (a † a − b † b)/2 to fulfil this task, without appealing to Lie algebra method. Note that this operator's disentangling is different from its decomposition in normal ordering.
Ordered operator expansions for operators forming physically important low-dimensional Lie algebras are derived in a simple unified way. Starting with the Zassenhaus formula for the disentangling of exponential operators, series expansions of both undisentangled and disentangled exponentials and comparison of the operator coefficients of equal powers of an ordering parameter alpha leads to ordered operator expansions. This 'comparison method' gives an alternative simple derivation of some already known formulae and a number of new formulae in the physical and chemical applications of the harmonic oscillator and for master equation problems with nearest-neighbour transition probabilities. The 'comparison method' cannot be applied to the angular momentum algebra directly. By a slight modification it can be used to derive from one matrix element or trace of J x k J y l J z n all possible combinations k, l, n by simply comparing powers of ordering parameters.