Approximately disentangling exponential operators

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I. INTRODUCTION
The disentangling of exponential operators is a useful tool applied, e.g., in quantum mechanics, quantum field theory, optics, or physical chemistry.Mathematically, it may be regarded as a method for the solution of linear differential equations. 1The basic idea has been developed long ago, e.g., by Feynman 2 and Glauber, 3 and developed later into different directions by many authors ͑see, e.g., Refs. 1 and 4͒.The method was reviewed recently by Popov. 5he formulation of the disentangling problem is particularly straightforward if the operators to be disentangled are members of a finite-dimensional Lie algebra with generators ͕A 1 , ... ,A m ͖.Then, under certain conditions discussed in more detail, e.g., in Ref. 1, it holds that with 1 , ... , m C given constants.Some of the i may be zero.The disentangling problem to be solved is the determination of the 1 , ... , m C for a given Lie algebra ͕A 1 , ... ,A m ͖.Here, we will assume that for the Lie algebra under consideration the relation ͑1͒ holds at least locally, and mathematical questions on the ͑global͒ existence of Eq. ͑1͒ will not be addressed. 6The aim of the present paper is to suggest a new practical way for the approximate determination of the complex constants 1 , ... , m C and to provide a suitable computer implementation.
There are various techniques that solve the disentanglement problem ͓Eq.͑1͔͒ for certain cases analytically and exactly.We briefly mention three of them: parameter differentiation, matrix representation of the Lie algebra, and a method using similarity transformations.
Parameter differentiation was first used by Glauber 3 and exposed in detail by Wilcox. 4First one introduces a parameter t into Eq.͑1͒, and then differentiates this equation with respect to t.Using the well known relation with the nested commutators ͕A , B͖ k = ͓A , ͕A , B͖ k−1 ͔ and ͕A , B͖ 0 = B as well as the assumed Lie algebra structure in order to calculate the nested commutators finally leads to a system of ordinary differential equations.If this system can be solved analytically, one may be able to obtain the solutions for 1 to m in closed form.Examples are discussed by Wilcox. 4 purely algebraic method based on a matrix representation of the Lie algebra may be useful if such a representation can be easily obtained.If the exponential of these matrices can be calculated in closed form, one obtains a system of nonlinear equations whose solutions yield the coefficients 1 to m .Examples from quantum optics are presented in Refs.7 and 8.
A method already suggested by Wilcox 4 and exposed in more detail by DasGupta 9 compares similarity transformations induced by the left hand ͑entangled͒ side of Eq. ͑1͒ and the right hand ͑disentangled͒ side of Eq. ͑1͒ on the generators of the Lie algebra.Calculations make extensive use of Eq. ͑3͒ and lead to a system of nonlinear equations in the unknown 1 to m .Note that if the identity operator I is a generator of the given Lie algebra, the similarity transformation method cannot compute the corresponding coefficient.
The approximation method proposed here uses the Baker-Campbell-Haussdorff ͑BCH͒ theorem [10][11][12] in order to rewrite the right hand side of Eq. ͑1͒.The BCH theorem asserts that the product of the exponentials of two noncommutative variables A and B may be expressed as the exponential of an infinite sum where the BCH term Z n may be expressed as a linear combination of nested commutators of the noncommuting variables A and B. If we truncate the sum in Eq. ͑4͒ at n = p, we obtain the BCH approximation of order p for the product of two exponentials.
The BCH approximation for the disentangling of exponential operators is developed in detail in Sec.II.A computer implementation is provided in Sec.III, followed by a number of examples ͑Sec.IV͒ in order to demonstrate the accuracy of the method by comparison with known exact results.Conclusions in Sec.V summarize the paper.

II. BCH APPROXIMATION
We assume that we are given a Lie algebra with generators A 1 , ... ,A m satisfying the commutation relations with the structure constants a k ij C for k =1, ... ,m and 1 Յ i Ͻ j Յ m.The component notation for a Lie element is implicitly introduced after the second equal sign in Eq. ͑5͒.It is now our goal to find an approximation for the coefficients 1 , ... , m in order to disentangle the exponential of a Lie element = ͕ 1 , ... , m ͖, The approximation is obtained as follows.In order to combine the exponentials on the right hand side of Eq. ͑6͒, we use a BCH approximation of order p repeatedly ͓see explanation after Eq. ͑4͔͒.
Then, using the structure of the Lie algebra ͓Eq.͑5͔͒, it is possible to evaluate the nested commutators in the resulting exponential and, in principle, write the right hand side of Eq. ͑6͒ in the form where f 1 p , ... , f m p : C m → C are functions that depend on the order of p of the BCH approximation.In order to obtain an approximation for i , one finally needs to solve the system of nonlinear equations
While the procedure outlined above is conceptually straightforward, a practical implementation requires a few tools.First we need a way to determine the BCH terms up to the desired order.A polynomial representation of the BCH term Z n defined in Eq. ͑4͒ is given by a weighted sum of the 2 n "words" of length n, which can be written with two letters A and B, Here, in each word X 1 s ¯Xn s , X i s stands for a factor A or B. Practical methods for the calculation of the coefficients ⌳ n s are reviewed and developed in Ref. 13.It holds that ⌳ 1 1 = ⌳ 1 2 = 1 and ⌳ n s Q for all s =1, ... ,2 n .For example, for Z 3 one obtains In order to make use of the Lie algebra structure assumed above, the polynomial representation of the BCH terms must be converted into a representation in terms of nested commutators.Such a representation is known to exist, but it is not unique.A suitable map ⌿ from polynomials to nested commutators was first developed by Dynkin, 14,15 Later a map ⌽ that yields fewer terms than the Dynkin map was conjectured by Oteo, 16 and we recently proved that this map is valid. 13The map ⌽ is defined by Here, N͑X 1 s , ... ,X n s ͒ is the number of A's in the word X 1 s ¯Xn s , e.g., N͑ABAAB͒ = 3.We proved in Ref. 13

the following theorem:
For all n Ն 2, it holds that ⌽͑Z n ͒ = Z n .For example, for Z 3 one obtains Alternatively, with the Dynkin map ⌿, one would obtain
With the tools collected above, it is possible to explicitly calculate the function F p in Eq. ͑8͒ to the desired order p with the help of a computer.Finally, we need a suitable method to solve the system of nonlinear equations given in Eq. ͑8͒.In general, there are several solutions of such a polynomial system, and we need to pick out the correct one.Our procedure to do this is based on the following observation: each solution = ͕ 1 , 2 , ... , m ͖ of the set of equations, of course, depends on the parameters = ͕ 1 , 2 , ... , m ͖.However, for = e 1 = ͕1,0, ... ,0͖ the solution must be 1 = 1 , 2 =0, ... , m = 0, according to Eq. ͑6͒.Therefore, as a function of a solution ͑͒ must fulfill the boundary condition ͑e 1 ͒ = ͕ 1 ,0, ... ,0͖.Our procedure constructs the function ͑͒ starting from this boundary, i.e., we start from the trivial solution for = e 1 and then solve several auxiliary problems where the parameter t controls the "distance" of the auxiliary problem to the problem we want to solve ͑t =1͒.The parameter M defines the number of auxiliary problems to be considered.For each step in this procedure, we solve the nonlinear system of Eq. ͑8͒ using Newton's iteration method with DF p ͑ 1 , ... , m ͒ denoting the Jacobian of F p ͑ 1 , ... , m ͒.As the starting vector 0 for each Newton iteration, we use the solution of the previous step starting with 0 = e 1 for t =1/ M. In summary, we suggest the following solution approach.
Input: Absolute accuracy Ͼ0 and number of auxiliary problems M N 1.Set ª e 1 , ˆª e 1 , k ª 0, and In all examples in Sec.IV we applied this technique using =10 −10 and M = 10.

III. COMPUTER IMPLEMENTATION
In this section we provide a MATHEMATICA7 ͑Ref.17͒ implementation of the BCH approximation method outlined in Sec.II.Of course, similar implementations are easily possible in other languages.
The set of nonlinear equations ͑8͒ is set up and solved using the command sigmas.The noncommuting variables f and g in the BCH terms are replaced by the appropriate Lie elements: The variable F holds the function F p ͑ 1 , ... , m ͒ and the rest of the code implements Newton's iteration method, as described in Sec.II.
Thus, the function sigmas[p,xi,epsilon,M] returns the desired approximation for the coefficients 1 to m .Here, p is the BCH approximation order, xi is the vector = ͑ 1 , ... , m ͒ as given in Eq. ͑1͒, and epsilon and M are the parameters and M, which control Newton's iteration method, as described in Sec.II.

IV. EXAMPLES
In order to study the applicability of our implementation of the BCH approximation for the disentangling of exponential operators, we here present a few numerical examples.For all examples, analytical results are available for comparison.Our program was run on a standard personal computer with 1.8 GHz and 1024 Mbytes of memory.The CPU times in Tables I-V correspond to this machine.

A. Two-dimensional Lie algebra
Consider the Lie algebra ͕A , B͖ with ͓A , B͔ = B and C. We look for 1 , 2 C such that exp͑␣A + ␤B͒ Ϸ exp͑ 1 A͒ • exp͑ 2 B͒.͑19͒ As shown, e.g., in Ref. 9 by the similarity transformation method mentioned in Sec.I, it holds that Using the BCH approximation up to order p = 4, one finds

͑22͒
Hence, we obtain and the solution of this system is

͑24͒
Table I presents numerical results for different approximation orders p and selected values for ␣, ␤, and .
For the present example, it is furthermore possible to address questions of convergence.Comparing the exact result for 2 ͓Eq.͑20͔͒ with the BCH approximation 2 ͓Eq.͑24͔͒, one asserts that the denominator in Eq. ͑24͒ results from the expansion which is a relation well known from one possible definition of the Bernoulli numbers B n .It is known that this series converges for 0 Ͻ ͉x͉ Ͻ 2, which in this case defines the radius of convergence for our method to work.Specifically, it must hold that 0 Ͻ ͉␣͉ Ͻ 2. In Fig. 1, we show numerical results for ␣ = ␤ = 1 and 0 ՅՅ10.The exact solution for 2 = 2 ͑͒ corresponds to the thin black curve, and the BCH approximation for orders p =4, p = 9, and p = 12 are illustrated by the dotted, dashed, and solid curves, respectively.It can be seen that one cannot expect convergence for values Ն2.

B. Four-dimensional Lie algebra
In We remark that, in general, every ordering of the exponentials would yield different disentanglement coefficients i .Exact solutions are, e.g., given in Ref. 19, The BCH approximation up to order p = 4 yields the system of equations Numerical comparison between the exact solution and the BCH approximation can be found in Table II.

C. SU"1,1… algebra
Here and in the following examples we will use the same notations as given in Refs.7 and 9.The SU͑1,1͒ Lie algebra ͕K + , K 0 , K − ͖ satisfies the commutation relations For complex parameters ␣, denote by ␣ ‫ء‬ the conjugate of ␣.Our aim is to approximate the disentanglement coefficients 1 to 3 in Rewriting ␣ = •e i with Ն0 and 0 Յ Ͻ 2, the exact solutions 7 are given by 1 = e i tanh͑͒, 2 = − 2 log͑cosh͑͒͒, 3 = − e −i tanh͑͒.͑32͒ Table III presents numerical results of the disentanglement coefficients using the BCH approximation.

D. SU"2… algebra
The SU͑2͒ Lie algebra has three generators ͕J + , J 0 , J − ͖, which satisfy the commutation relations Here, we want to calculate the disentanglement Rewriting again ␣ = •e i , the exact solutions as derived in Refs.7 and 9 are

͑35͒
Numerical results can be found in Table IV.Moreover, Fig. 2 presents some results for ␣ R and 0 Յ ␣ Յ 1.5.The exact solutions 1 = 1 ͑␣͒ to 3 = 3 ͑␣͒ correspond to the thin black curves and the BCH approximation for orders p =3, p = 7, and p = 11 are illustrated by dotted, dashed, and solid curves, respectively.

E. The six-dimensional double photon algebra
In our last example we turn to the double photon algebra ͕K + , A + , K 0 , I , K − , A͖, defined by the following commutation relations: ͓I,K − ͔ = 0, ͓I,A͔ = 0, ͓K − ,A͔ = 0. ͑40͒ With ␣ = •e i and = •e i , our goal is to approximate the disentanglement form The exact coefficients as given in Ref. 9 are We believe that the result for 6 given in Ref. 9 has a misprint for the sign before the sinh term.Finally, results using the BCH approximation are presented in Table V.

V. CONCLUSION
In this paper, we suggested a general method for the approximate disentangling of exponential operators that satisfy a finite-dimensional Lie algebra.We provide a computer implementation that determines the disentangling coefficients as defined in Eq. ͑1͒ for, in principle, arbitrary Lie algebras.The method uses the BCH theorem in an essential way, and therefore, we expect that it converges only in a finite convergence radius centered at zero for all parameters of a given problem.The limits of convergence have not been investigated in detail.
The computer implementation provided is a basic demonstration of the method, but does not control accuracy by, e.g., comparing different approximation orders.Accuracy control could easily be added in a more sophisticated version of the implementation.Furthermore, the code is optimized for simplicity, but not for running time.
Five numerical examples with known exact analytic solutions demonstrate that the method yields very good approximations for various parameter sets.For example, we obtained a quite good approximate disentangling for the four-dimensional Lie algebra as defined in Sec.IV B already for an approximation order p = 8.For the three-dimensional SU͑2͒ algebra, the same accuracy was achieved for an order of p Ն 14.As can be seen from our numerical results presented in Sec.IV, the CPU times and the accessible orders of approximation strongly depend on the dimension of the given Lie algebra.For four-dimensional algebras, the approximation order 12 can be obtained in a few seconds.On the other hand, in higher-dimensional algebras the required CPU times increase rapidly as can be seen, e.g., in Table V.However, for specific applications, much more efficient implementations of the proposed method could be possible.
As explained in the text, we need to pick out from the set of solutions of a system of nonlinear equations the one that fulfills a certain boundary condition.This is done in several steps starting from a trivial initial auxiliary problem.The number of auxiliary problems to be considered is controlled by the parameter M to be given to our program.Although we used M =10 in all numerical examples, there might be problems for which our method picks the correct solution only for some larger values of M. This would need to be carefully controlled by suitable tests.

TABLE I .
Comparison of exact and approximate disentangling coefficients for the Lie algebra ͕A , B͖ with ͓A , B͔ = B defined in Sec.IV A.

TABLE II .
Comparison of exact and approximate disentangling coefficients for the four-dimensional Lie algebra defined in Sec.IV B.

TABLE III .
Comparison of exact and approximate disentangling coefficients for the SU͑1,1͒ algebra defined in Sec.IV C.