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Communications in Theoretical Physics

New Convenient Way for Deriving Exponential Operators' Disentangling Formulas and Their Applications (II)

Fan Hong‐Yi, Liyun Hu

Published Mar 1, 2009• 12 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/51/3/25

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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.
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.
在前面的纸(Commun。Theor。Phys。51 (2009 ) 321 ) 我们为解开指数的操作员推荐了一个方便方法。在这个工作，我们使用这个方法解开由尖动量操作员组成的指数的操作员。我们主要解开 exp 的形式[ 2hJz +gJ++ kJ ]作为订的 exp (...J+) exp (... Jz ) exp (... J )，我们采用 Schwinger Bose 实现 J = ba ，J+= ab ， Jz =( aa bb )完成这项任务的 /2 ，没有呼吁谎言代数学方法。注意这个操作员的解开与它在正常的订的分解不同。
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.