Institut für Numerische und Angewandte Mathematik

Institut für Numerische und Angewandte Mathematik
Disentangling exponential operators
D. Scholz, V.G. Voronov, M. Weyrauch
Nr. 2009-19
Preprint-Serie des
Instituts für Numerische und Angewandte Mathematik
Lotzestr. 16-18
D - 37083 Göttingen

Disentangling exponential operators
Daniel Scholz
Institute for Numerical and Applied Mathematics, Georg-August-University Göttingen, Germany
dscholz@math.uni-goettingen.de
Volodymyr G. Voronov
Faculty of Physics, Taras Shevchenko National University of Kyiv, Ukraine
v_voronov@univ.kiev.ua
Michael Weyrauch
Physikalisch-Technische Bundesanstalt, Braunschweig, Germany
michael.weyrauch@ptb.de
Manuscript (Version of October 21, 2009)
Abstract
A new method for the approximate disentangling of exponential operators based on
the Baker-Campbell-Haussdorff theorem is suggested and implemented in a computer
program. The operators to be disentangled must form a finite-dimensional Lie algebra.
The accuracy of the method is tested and demonstrated in several explicitly calculated
examples, where exact analytic solutions are available.
Keywords: Disentangling, Lie groups, Lie algebras, Baker-Campbell-Hausdorff series.
PACS: 02.10.-v, 02.60.Pn
1

1 Introduction
The disentangling of exponential operators is a useful tool applied e.g. in quantum mechan-
ics, quantum field theory, optics, or physical chemistry. Mathematically it may be regarded
as a method for the solution of linear differential equations [1]. The 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, 4]. The method was reviewed recently by
Popov [5].
The formulation of the disentangling problem is particularly straightforward, if the oper-
ators to be disentangled are members of a finite dimensional Lie algebra with generators
{A1, . . . , Am}. Then, under certain conditions discussed in more detail e.g. in Ref. [1], it
holds that
exp(ξ1A1 + . . . + ξmAm) = exp(σ1A1) · . . . · exp(σmAm), (1)
with ξ1, . . . , ξm ∈ C given constants. Some of the ξi may be zero. The disentangling problem
to be solved is the determination of the σ1, . . . , σm ∈ C for a given Lie algebra {A1, . . . , An}.
Here, we will assume that for the Lie algebra under consideration the relation (1) holds at
least locally, and mathematical questions on the (global) existence of Eq. (1) will not be
addressed [6]. The aim of the present paper is to suggest a new practical way for the
approximate determination of the complex constants σ1, . . . , σm ∈ C and to provide a
suitable computer implementation.
There are various techniques which solve the disentanglement problem Eq. (1) 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 [4]:
First one introduces a parameter t into Eq. (1)
exp(t(ξ1A1 + . . . + ξmAm)) = exp(σ1(t)A1) · . . . · exp(σm(t)Am)
and then differentiates this equation with respect to t. Using the well known relation
exp(A) · B · exp(−A) =
∞�
k=0
1
· {A, B}k
(2)
k!
with the nested commutators {A, B}k = [A, {A, B}k−1] and {A, B}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
2

may be able to obtain the solutions for σ1 to σm in closed form. Examples are discussed
by Wilcox [4].
A 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 non-linear equations whose solutions
yield the coefficients σ1 to σm. Examples from quantum optics are presented in Refs. [7, 8].
A method already suggested by Wilcox [4] and exposed in more detail by DasGupta [9]
compares similarity transformations induced by the left hand (entangled) side of Eq. (1)
and the right hand (disentangled) side of Eq. (1) on the generators of the Lie algebra.
Calculations make extensive use of Eq. (2) and lead to a system of non-linear equations
in the unknown σ1 to σm. 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 (BCH) the-
orem [10, 11, 12] in order to rewrite the right hand side of Eq. (1). The BCH theorem asserts
that the product of the exponentials of two non-commutative variables A and B may be
expressed as the exponential of an infinite sum
�
exp(A) · exp(B) = exp
A + B +
∞�
n=2
Zn
�
, (3)
where the BCH terms Zn may be expressed as linear combinations of nested commutators
of the non-commuting variables A and B. If we truncate the sum in Eq. (3) at n = p we
obtain an approximation for the product of two exponentials of order p.
The BCH approximation for the disentangling of exponential operators is developed in
detail in Section 2. A computer implementation is provided in Section 3 followed by a
number of examples (Section 4) in order to demonstrate the accuracy of the method by
comparison with known exact results. Conclusions in Section 5 summarize the paper.
2 BCH approximation
We assume that we are given a Lie algebra with generators A1, . . . , Am satisfying the com-
mutation relations
[Ai, Aj] = a ij
1 · A1 + . . . + a ij m · Am = {a ij
1 , . . . , aij m} (4)
3

with the structure constants a ij m ∈ C for 1 ≤ i < j ≤ m. The component notation for a Lie
element is implicitly introduced after the second equal sign in Eq. (4). It is now our goal to
find an approximation for the coefficients σ1, . . . , σm in order to disentangle the exponential
of a Lie element ξ = {ξ1, . . . , ξm},
exp(ξ1A1 + . . . + ξmAm) = exp(σ1A1) · . . . · exp(σmAm). (5)
The approximation is obtained as follows: We first use the Baker-Campbell-Hausdorff the-
orem Eq. (3) up to order p repeatedly in order to combine all exponentials on the right
hand side of Eq. (5). Then, using the structure of the Lie algebra Eq. (4), it is possible to
evaluate the nested commutators in the resulting exponential, and, in principle, write the
right hand side of Eq. (5) in the form
�
exp(σ1A1) · . . . · exp(σmAm) ≈ exp f p
1 (σ1, . . . , σm) · A1 + . . . + f p �
m(σ1, . . . , σm) · Am
where f p
1 , . . . , f p m : C m → C are functions which depend on the order p of the BCH approx-
imation.
In order to obtain an approximation for the σi one finally needs to solve the system of
non-linear equations
⎛
f p
1 (τ1, . . . , τm)
F p (τ1, . . . , τm) =
⎜
⎝
.
f p m(τ1, . . . , τm)
⎟
⎠ =
⎞
⎛
⎜
⎝
ξ1
.
ξm
⎞
⎟ . (6)
⎠
Let (τ1, . . . , τm) be a solution of this system. We then obtain the approximation σk ≈ τk
for k = 1, . . . , m. Note that the functions f p
k (τ1, . . . , τm) for k = 1, . . . , m are polynomials
of degree p − 1.
While the procedure outlined above is conceptually straightforward, a practical implemen-
tation requires a few tools, which were partly developed recently [13]. First we need a way
to determine the BCH terms to the desired order. A polynomial representation of the BCH
terms Zn defined in Eq. (3) is given by
Zn = �
t1,u1,...,tn,un
Λn(A t1 B u1 . . . A tn B un ) · A t1 B u1 . . . A tn B un
where the sum is over all t1, u1, . . . , tn, un ∈ {0, 1} with tk + uk = 1 for k = 1, . . . , n. It
holds that Λ1(A) = Λ1(B) = 1 and Λn(A t1 B u1 . . . A tn B un ) ∈ Q. E.g., for Z3 one obtains
Z3 = 1
12
1 1 1
AAB − ABA + ABB +
6 12 12
4
1 1
BAA − BAB +
6 12 BBA,

and, therefore, e.g. Λ3(ABB) = 1/12.
In order to make use of the Lie algebra structure assumed above the polynomial repre-
sentation 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 Φ which yields fewer terms than the Dynkin map was conjectured by Oteo [16], and
we recently proved that this map is valid [13].
The map Φ is defined as follows: For general polynomials
where Λ s n ∈ Q and X s i
Φ(P (A, B)) =
P (A, B) =
m�
s=1
Λ s n · X s 1 . . . X s n
∈ {A, B} for i = 1, . . . , n and s = 1, . . . , m, define
m�
s=1
X s 1 =A, Xs 2 =B
Λ s n
N(X s 1 , . . . , Xs n) · [[· · · [[[A, B], Xs 3], X s 4], . . .], X s n].
Here, N(X s 1 , . . . , Xs n) is the number of A generators in the set {X s 1 , . . . , Xs n}, e.g. N(ABAAB) = 3.
We proved in Ref. [13] the following theorem:
For all n ≥ 2 it holds that Φ(Zn) = Zn.
E.g., for Z3 one obtains
Z3 = 1
12
1 1 1
AAB − ABA + ABB +
6 12 12
= − 1
1
[[A, B], A] + [[A, B], B] = Φ(Z3).
12 12
1 1
BAA − BAB +
6 12 BBA
With the tools collected above it is possible to explicitly calculate the function F p in Eq. (6)
to the desired order p with the help of a computer.
Finally, we need a suitable method to solve the system of non-linear equations given in
Eq. (6). In general, there are several solutions of such a polynomial system, and we need
to pick out that solution which for fixed i and ξ → ei = (0, . . . , 0, 1, 0, . . . , 0) continuously
evolves into (σ1, . . . , σm) = ei, which must hold according to Eq. (5).
Numerically, we semi-continuously evolve the desired solution from the trivial solution for
ξ = e1 by solving several auxiliary problems
F p (τ) = t · ξ + (1 − t) · e1 for t = 1 2
,
M M
5
M − 1
, . . . , , 1,
M

where the parameter t controls the ‘distance’ of the auxiliary problem to the problem we
want to solve (t = 1). The parameter M defines the number of auxiliary problems to be
considered.
For each step in this procedure, we solve the non-linear system of equations (6) using
Newton’s iteration method
τ k+1 = τ k �
− DF (τ k �−1 ) · (F (τ k ) − ξ)
with the Jacobian DF p (τ1, . . . , τm) of F p (τ1, . . . , τm) given by
DF p (τ1, . . . , τm) =
⎛
⎜
⎝
∂f p
1
∂τ1 (τ1, . . . , τm) . . .
.
∂f p m
∂τ1 (τ1, . . . , τm) . . .
∂f p
1
∂τm (τ1, . . . , τm)
.
∂f p m
∂τm (τ1, . . . , τm)
As the starting vector τ 0 for each Newton iteration we use the solution of the previous step
starting with τ 0 = e1 for t = 1
M . Summarizing, we suggest the following solution approach:
Input: Absolute accuracy ε > 0 and number of steps M ∈ N.
( 1 ) Set τ := e1, ˆτ := e1, k := 0, and t := 0.
( 2 ) If k = M, stop. Else set k = k + 1 and t = k/M.
( 3 ) Set τ = ˆτ and
� �−1 �
�
ˆτ = τ − DF (τ) · F (τ) − (t · ξ + (1 − t) · e1) .
( 4 ) If �τ − ˆτ�2 < ε, go to Step ( 2 ). Else go to Step ( 3 ).
Output: Approximation ˆτ.
In all examples in Section 4 we applied this technique using ε = 10 −10 and M = 10.
3 Computer implementation
In this section we provide a Mathematica 7 [17] implementation of the BCH approximation
method outlined in the previous section. Of course, similar implementations are easily
possible in other languages.
6
⎞
⎟
⎠ .

The implementation assumes that the commutation relations of the Lie algebra are coded
as K[i,j]={a_1^{ij},...,a_m^{ij}} for all 1 ≤ i < j ≤ m. For example, for the SU(1,1)
algebra with generators {K+, K0, K−}, which is defined by the commutation relations
[K+, K0] = −K+ [K0, K−] = −2K0 [K0, K−] = −K−,
one needs to enter the following Mathematica statements:
K[1,2] = {-1,0,0};
K[1,3] = {0,-2,0};
K[2,3] = {0,0,-1};
The commutator of two Lie elements a and b is then calculated by the following function:
cm[a_List,b_List]:=Module[{range,vor,com},
range = Range[Length[a]];
vor = Times @@@ Flatten[Outer[List,a,b], 1];
com = K @@@ Flatten[Outer[List,range,range], 1];
vor.com];
Using again the SU(1,1) algebra as an example, the Lie element 2K0 + K− is coded by
{0,2,1} and the Lie element K+ +K− is coded by {1,0,1}. Hence, we obtain for example:
cm[{0,2,1},{1,0,1}] = 2*K[2,1]+2*K[2,3]+K[3,1]+K[3,3] = {2,2,-2}
Nested commutators are computed by the code:
nestcm[a_List,b_List]:=Module[{oplist},
oplist = Flatten[Table @@@ Partition[Riffle[Partition[a,1], b, {1,-2,2}],2], 1];
Fold[cm, First[oplist], Rest[oplist]] ];
We compute the BCH terms up to approximation order p using Goldberg’s method [18]. A
slightly different variant of the code below is discussed and documented in more detail in
Ref. [13].
G[1] = 1;
G[s_]:= G[s] =Expand[1/s*D[t*(t-1)*G[s-1],t]];
GC[W_List] := GC[W] =Module[{m,m1,m2,k},
m = Length[W];
m1 = Floor[m/2];
m2 = Floor[(m-1)/2];
Integrate[t^m1*(t-1)^m2*Product[G[W[[k]]],{k, m}], {t,0,1}]];
Goldberg[n_,l_List] := Module[{L},
L = Select[Flatten[Permutations /@ IntegerPartitions[n], 1], First[#] == 1 &];
Plus @@ ((GC[Sort[#]]/(Plus @@ First /@ Partition[#,2,2,1,{}])*nestcm @@ {#,l}) & /@ L)];
BCH[n_Integer] := BCH[n] = Plus @@ {f,g} + Sum[Goldberg[k, {f,g}], {k,2,n}];
7

BCH[p] returns all BCH terms up to the order of p in a commutator representation for
non-commutating variables f and g. Here, unlike in Ref. [13], we directly applied the map
Φ as defined in the previous section.
The set of non-linear equations (6) is set up and solved using the command sigmas. The
non-commuting variables f and g in the BCH terms are replaced by the appropriate Lie
elements:
sigmas[n_Integer,xi_List,eps_Rational,M_Integer] :=
Module[{sm,m,s,F,DF,k,tau,tau1,tau2,t},
sm = DiagonalMatrix[Table[s[i], {i,m=Length[xi]}]];
tau = tau1 = UnitVector[m,1];
K[x_Integer, y_Integer] := -K[y, x] /; x > y;
K[x_Integer, y_Integer] := Table[0, {m}] /; (x == y);
F = Expand[Fold[BCH[n] /. {f -> #1, g -> #2} &, sm[[1]], Rest[sm]]];
DF = Table[D[F[[i]],s[j]], {i,m}, {j,m}];
For[k = 1, k #[[j]]].((F /. s[j_] :> #[[j]])
- (t*xi+(1-t)*tau1)) &, tau, Norm[#1-#2] >= eps &, 2]];
tau2
];
The variable F holds the function F p (σ1, . . . , σm) and the rest of the code implements
Newton’s iteration method as described in the previous section.
Thus, the function sigmas[p,xi,epsilon,M] returns the desired approximation for the
coefficients σ1 to σm. Here, p is the BCH approximation order, xi is the vector ξ =
(ξ1, . . . , ξm) as given in Eq. (1), and epsilon and M are the parameters ε and M, which
control Newton’s iteration method as described in the previous section.
4 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.
4.1 Two-dimensional Lie algebra
Consider the Lie algebra {A, B} with [A, B] = λB and λ ∈ C. We look for σ1, σ2 ∈ C
such that
exp(αA + βB) ≈ exp(σ1A) · exp(σ2B).
8

As shown e.g. in Ref. [9] by the similarity transformation method mentioned in the intro-
duction, it holds that
σ1 = α and σ2 = β
λα · (1 − e−λα ).
Using the BCH approximation up to order p = 4 one finds
exp(σ1A) · exp(σ2B) ≈
�
exp σ1A + σ2B + 1
2 σ1σ2λB + 1
12 σ2 1σ2λ 2 =
�
B
�
�
exp σ1 · A + σ2 1 + 1
2 σ1λ + 1
12 σ2 1λ 2
� �
· B .
Hence, we obtain
F 4 (τ1, τ2) =
⎛
⎝
and the solution of this system is
τ1
�
1
τ2 1 + 2τ1λ + 1
12τ 2 1 λ2�
τ1 = α and τ2 =
⎞
⎠ =
β
⎛
⎝ α
β
1 + 1 1
2αλ + 12α2 . (7)
λ2 Table 1 presents numerical results for different approximation orders p and selected values
for α, β, and λ.
p {α, β, λ} {�τ1 − σ1�, �τ2 − σ2�}
4 {1, 1, 1} {0, 5.41 · 10 −4 }
8 {1, 1, 1} {0, 3.22 · 10 −7 }
12 {1, 1, 1} {0, 2.06 · 10 −10 }
4 {1 + i, 1 − i, 1} {0, 2.90 · 10 −3 }
8 {1 + i, 1 − i, 1} {0, 6.89 · 10 −6 }
12 {1 + i, 1 − i, 1} {0, 1.76 · 10 −8 }
4 { 1
2 , 1 − i, 2} {0, 7.65 · 10−4 }
8 { 1
2 , 1 − i, 2} {0, 4.56 · 10−7 }
12 { 1
2 , 1 − i, 2} {0, 2.91 · 10−10 }
Table 1: Disentangling coefficients for the Lie algebra {A, B} with [A, B] = λB.
For the present example it is furthermore possible to address questions of convergence:
Obviously, the denominator in Eq. (7) results from the expansion,
x
1 − exp(−x) =
∞�
(−)
n=0
n Bnx n
n!
⎞
⎠
1 1
= 1 + x +
2 12 x2 + 1
180 x3 + . . .
which is a relation well known from one possible definition of the Bernoulli numbers Bn. It
is known that this series converges for 0 < |x| < 2π, which in this case defines the radius
9

of convergence for our method to work. Specifically, it must hold that 0 < |αλ| < 2π. In
Figure 1, we show numerical results for α = β = 1 and 0 ≤ λ ≤ 10. The exact solution
for σ2 = σ2(λ) 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 λ ≥ 2π.
Figure 1: (Color online) Disentangling coefficient σ2 for the two-dimensional Lie algebra
{A, B} for 0 ≤ λ ≤ 10. Exact solution (thin black curve), approximation order p = 4
(dotted green curve), approximation order p = 9 (dashed blue curve), and approximation
order p = 12 (solid red curve).
4.2 Four-dimensional Lie algebra
In our second example we consider the Lie algebra {A, B, D, I} with the commutation
relations
[A, B] = −λD [A, D] = −µB [A, I] = 0
[B, D] = −αI [B, I] = 0 [D, I] = 0
for λ, µ, α ∈ C. Our goal is to approximate the coefficients σ1, . . . , σ4 ∈ C such that
exp(A + B) ≈ exp(σ1A) · exp(σ2B) · exp(σ3D) · exp(σ4I).
Exact solutions are e.g. given in Ref. [19]
σ1 = 1, σ2 = sinh(√ λµ)
√ λµ
, σ3 = cosh(√λµ) − 1
, σ4 =
µ
α
4µ ·
10
� sinh(2 √ λµ)
√ λµ
�
− 2 .

The BCH approximation up to order p = 4 yields the system of equations
F 4 ⎛
(τ1, τ2, τ3, τ4) =
⎜
⎝
τ1
τ2 − 1
2τ1τ3µ + 1
12τ 2 1 τ2λµ
τ3 − 1
2τ1τ2λ + 1
12τ1τ3µλ ⎞ ⎛ ⎞
⎟
⎠
=
1
⎜ ⎟
⎜ ⎟
⎜ 1 ⎟
⎜ ⎟
⎜ ⎟ .
⎜ 0 ⎟
⎝ ⎠
0
τ4 − 1
2τ2τ3α − 1
12τ1τ 2 1
2 λα + 12τ1τ 2 3 µα
Numerical comparison between the exact solution and the BCH approximation can be found
in Table 2.
p {λ, µ, α} {�τ1 − σ1�, . . . , �τ4 − σ4�}
4 {1, 1, 1} {0, 2.27 · 10 −3 , 1.73 · 10 −3 , 9.96 · 10 −4 }
8 {1, 1, 1} {0, 1.35 · 10 −6 , 1.03 · 10 −6 , 2.34 · 10 −6 }
12 {1, 1, 1} {0, 8.64 · 10 −10 , 6.58 · 10 −10 , 2.62 · 10 −9 }
4 {1 − i 1
2 , 2 + i, 2} {0, 3.64 · 10−3 , 3.03 · 10−3 , 6.60 · 10−3 }
8 {1 − i 1
2 , 2 + i, 2} {0, 3.38 · 10−6 , 2.82 · 10−6 , 1.23 · 10−5 }
12 {1 − i 1
2 , 2 + i, 2} {0, 3.37 · 10−9 , 2.81 · 10−9 , 1.95 · 10−8 }
4 {2, i
2 , −1} {0, 1.50 · 10−3 , 2.80 · 10 −3 , 6.96 · 10 −3 }
8 {2, i
2 , −1} {0, 8.93 · 10−7 , 1.66 · 10 −6 , 7.36 · 10 −6 }
12 {2, i
2 , −1} {0, 5.71 · 10−10 , 1.06 · 10 −9 , 6.76 · 10 −9 }
Table 2: Disentangling coefficients for the four-dimensional Lie algebra.
4.3 SU(1,1) algebra
Here and in the following examples we will use the same notations as given in Refs. [7, 9].
The SU(1,1) Lie algebra {K+, K0, K−} satisfies the commutation relations
[K+, K0] = −K+ [K+, K−] = −2K0 [K0, K−] = −K−.
For complex parameters α denote by α ∗ the conjugate of α. Our aim is to approximate the
disentanglement coefficients σ1 to σ3 in
exp(αK+ − α ∗ K−) = exp(σ1K+) · exp(σ2K0) · exp(σ3K−).
Rewriting α = λ · e iθ with λ ≥ 0 and 0 ≤ θ < 2π, the exact solutions [7] are given by
σ1 = e iθ · tanh(λ), σ2 = − 2 log(cosh(λ)), σ3 = − e −iθ · tanh(λ).
Table 3 presents numerical results of the disentanglement coefficients using the BCH ap-
proximation.
11

4.4 SU(2) algebra
p α {�τ1 − σ1�, . . . , �τ3 − σ3�}
4 1 {1.28 · 10 −2 , 3.72 · 10 −2 , 6.70 · 10 −3 }
8 1 {4.89 · 10 −4 , 1.40 · 10 −3 , 2.81 · 10 −4 }
12 1 {2.19 · 10 −5 , 6.31 · 10 −5 , 1.29 · 10 −5 }
4 1 + i
2
8 1 + i
2
12 1 + i
2
{1.91 · 10 −2 , 6.63 · 10 −2 , 7.39 · 10 −3 }
{1.07 · 10 −3 , 3.64 · 10 −3 , 4.96 · 10 −4 }
{7.23 · 10 −5 , 2.48 · 10 −4 , 3.48 · 10 −5 }
4 1
2 + i {1.91 · 10−2 , 6.63 · 10 −2 , 7.39 · 10 −3 }
8 1
2 + i {1.07 · 10−3 , 3.64 · 10 −3 , 4.96 · 10 −4 }
12 1
2 + i {7.23 · 10−5 , 2.48 · 10 −4 , 3.48 · 10 −5 }
Table 3: Disentangling coefficients for the SU(1,1) algebra.
The SU(2) Lie algebra has three generators {J+, J0, J−} which satisfy the commutation
relations
[J+, J0] = −J0 [J+, J−] = 2J0 [J0, J−] = −J−.
Here, we want to calculate the disentanglement
exp(αJ+ − α ∗ J−) = exp(σ1J+) · exp(σ2J0) · exp(σ3J−).
Rewriting again α = λ · e iθ , the exact solutions as derived in Refs. [7, 9] are
σ1 = e iθ · tan(λ), σ2 = 2 log(sec(λ)), σ3 = − e −iθ · tan(λ).
Numerical results can be found in Table 4.
Moreover, Figure 2 presents some results for α ∈ R and 0 ≤ α ≤ 1.5. The exact solutions
σ1 = σ1(α) to σ3 = σ3(α) 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.
4.5 The six-dimensional double photon algebra
In our last example we turn to the double photon algebra {K+, A + , K0, I, K−, A}, defined
by the following commutation relations:
12

p α {�τ1 − σ1�, . . . , �τ3 − σ3�}
4 1 {5.14 · 10 −2 , 9.13 · 10 −2 , 9.55 · 10 −2 }
8 1 {1.19 · 10 −2 , 1.23 · 10 −2 , 1.51 · 10 −2 }
12 1 {7.38 · 10 −4 , 7.54 · 10 −4 , 9.22 · 10 −4 }
4 1 + i
2
8 1 + i
2
12 1 + i
2
{8.52 · 10 −1 , 8.25 · 10 −1 , 1.17 · 10 −0 }
{7.99 · 10 −2 , 7.12 · 10 −2 , 1.03 · 10 −1 }
{7.14 · 10 −3 , 6.36 · 10 −3 , 9.04 · 10 −3 }
4 1
2 + i {8.53 · 10−1 , 8.25 · 10 −1 , 1.17 · 10 −0 }
8 1
2 + i {7.99 · 10−2 , 7.12 · 10 −2 , 1.03 · 10 −1 }
12 1
2 + i {7.14 · 10−3 , 6.36 · 10 −3 , 9.04 · 10 −3 }
Table 4: Disentangling coefficients for the SU(2) algebra.
Figure 2: (Color online) Numerical results for the SU(2) algebra for 0 ≤ α ≤ 1.5. Exact so-
lutions (thin black curves), approximation order p = 3 (dotted green curves), approximation
order p = 7 (dashed blue curves), and approximation order p = 11 (solid red curves).
13

[K+, A + ] = 0 [K+, K0] = −K+ [K+, I] = 0
[K+, K−] = −2K0 [K+, A] = −A + [A + , K0] = − 1
2 A+
[A + , I] = 0 [A + , K−] = −A [A + , A] = −I
[K0, I] = 0 [K0, K−] = −K− [K0, A] = − 1
2 A
[I, K−] = 0 [I, A] = 0 [K−, A] = 0.
With α = λ · e iθ and ζ = µ · e iφ , our goal is to approximate the disentanglement form
exp(αK+ + ζA + − α ∗ K− − ζ ∗ A)
= exp(σ1K+) · exp(σ2A + ) · exp(σ3K0) · exp(σ4I) · exp(σ5K−) · exp(σ6A).
The exact coefficients as given in Ref. [9] are
σ1 = e iθ · tanh(λ),
µ
σ2 = −
λ cosh(λ) ·
σ3 = −2 log(cosh(λ)),
µ
σ4 = −
2
λ2 cosh(λ) ·
σ5 = −e −iθ · tanh(λ),
µ
σ6 = −
λ cosh(λ) ·
�
(cosh(λ) − 1) · e i(θ−φ) − sinh(λ) · e iφ�
,
�
�
(cosh(λ) − 1) + i sin(θ − 2φ) · (sinh(λ) − λ cosh(λ)) ,
�
(cosh(λ) − 1) · e −i(θ−φ) + sinh(λ) · e −iφ�
.
We believe that the result for σ6 given in Ref. [9] has a misprint for the sign before the sinh
term. Finally, results using the BCH approximation are presented in Table 5.
p {α, ζ} {�τ1 − σ1�, . . . , �τ6 − σ6�}
4 {1, 1} {1.28 · 10 −2 , 6.86 · 10 −3 , 3.72 · 10 −2 , 1.86 · 10 −3 , 6.71 · 10 −3 , 7.00 · 10 −3 }
8 {1, 1} {4.89 · 10 −4 , 2.44 · 10 −4 , 1.40 · 10 −3 , 6.04 · 10 −5 , 2.81 · 10 −4 , 2.84 · 10 −4 }
12 {1, 1} {2.19 · 10 −5 , 1.11 · 10 −5 , 6.31 · 10 −5 , 2.92 · 10 −6 , 1.29 · 10 −5 , 1.25 · 10 −5 }
4 {1 + i 1
2 , 2 + i} {1.91 · 10−2 , 2.03 · 10−2 , 6.63 · 10−2 , 1.43 · 10−2 , 7.39 · 10−3 , 3.04 · 10−2 }
8 {1 + i 1
2 , 2 + i} {1.07 · 10−3 , 1.16 · 10−3 , 3.64 · 10−3 , 6.81 · 10−4 , 4.96 · 10−4 , 1.54 · 10−3 }
12 {1 + i 1
2 , 2 + i} {7.23 · 10−5 , 7.93 · 10−5 , 2.46 · 10−4 , 4.62 · 10−5 , 3.48 · 10−5 , 1.02 · 10−4 }
4 { 1
2
8 { 1
2
12 { 1
2
3i 1 2i
+ 7 , − 2 + 3 } {2.54 · 10−3 , 3.94 · 10−3 , 4.00 · 10−3 , 3.15 · 10−3 , 2.05 · 10−3 , 3.22 · 10−3 }
+ 3i
7
+ 3i
7
, − 1
2
, − 1
2
+ 2i
3 } {2.07 · 10−5 , 3.28 · 10 −5 , 3.28 · 10 −5 , 2.64 · 10 −5 , 1.71 · 10 −5 , 2.40 · 10 −5 }
+ 2i
3 } {1.91 · 10−7 , 3.03 · 10 −7 , 3.02 · 10 −7 , 2.44 · 10 −7 , 1.58 · 10 −7 , 2.16 · 10 −7 }
Table 5: Disentangling coefficients for the double photon algebra.
14

5 Conclusion
In this paper we suggested a general method for the approximate disentangling of expo-
nential operators which satisfy a finite dimensional Lie algebra. We provide a computer
implementation which determines the disentangling coefficients σ as defined in Eq. (1) for, in
principle, arbitrary Lie algebras. The method uses the Baker-Campbell-Haussdorff 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 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 approximation using the four-dimensional Lie algebra as defined in Subsec-
tion 4.2 already for an approximation order p = 8. For the three-dimensional SU(2) algebra,
the same accuracy was achieved for an order of p ≥ 14.
As explained in the text we need to smoothly evolve the solution for a given problem from a
trivial initial auxiliary problem in a number of steps. The number of auxiliary problems to
be considered is controlled by the parameter M for Newton’s iteration procedure. Although
we used M = 10 in all numerical examples, there might be problems for which Newton’s
method only picks the correct solution for some larger values of M. This would need to be
carefully controlled by suitable tests.
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15

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17

Institut für Numerische und Angewandte Mathematik
Universität Göttingen
Lotzestr. 16-18
D - 37083 Göttingen
Telefon: 0551/394512
Telefax: 0551/393944
Email: trapp@math.uni-goettingen.de URL: http://www.num.math.uni-goettingen.de
Verzeichnis der erschienenen Preprints 2009:
2009-01 H. Harbrecht, T. Hohage A Newton method for reconstructing non
star-shaped domains in electrical impedance
tomography
2009-02 A. Schöbel, A. Kratz A bicriteria approach for robust timetabling
2009-03 S. Cicerone, G. D’Angelo,
G. Di Stefano, D. Frigioni,
A. Navarra, M. Schachtebeck,
A. Schöbel
2009-04 M. Ehrgott, L. Shao,
A. Schöbel
Recoverable robustness in shunting and timetabling
An Approximation Algorithm for Convex Multiobjective
Programming Problems
2009-05 L. Nannen, A. Schädle Transparent boundary conditions for
Helmholtz-type problems using Hardy space
infinite elements
2009-06 S. Soussi, T. Hohage Riesz bases and Jordan form of the translation
operator in semi-infinite periodic waveguides
2009-07 M. Schachtebeck Algorithmic Approaches to the Capacitated Delay
Management Problem
2009-08 M. Schachtebeck, A. Schöbel To wait or not to wait and who goes first? Delay
Management with Priority Decisions
2009-09 M. Körner, J. Brimberg, H.
Juel, A. Schöbel
2009-10 J. Brimberg, H. Juel, M.
Körner, A. Schöbel
General minisum circle location - extended abstract
-
Locating a minisum circle on the plane with arbitrary
norm
2009-11 M. Körner Minimizing the door-to-door distance
2009-12 A. Schöbel, D. Scholz The theoretical and empirical rate of convergence
for geometric branch-and-bound methods
2009-13 M. Schmidt, A. Schöbel Location of speed-up subnetworks

2009-14 T. Dollevoet, D. Huisman, M.
Schmidt, A. Schöbel
Delay management with re-routing of
passengers
2009-15 F. Cakoni, R. Kress, C. Schuft Integral equations for shape and impedance reconstruction
in corrosion detection
2009-16 D. Colton, R. Kress Inverse scattering
2009-17 F. Delbary, R. Kress Electrical impedance tomography with point
sources
2009-18 O. Ivanyshyn, R. Kress Identification of sound-soft 3D obstacles from
phaseless data
2009-19 D. Scholz, V.G. Voronov, M.
Weyrauch
Disentangling exponential operators