Solving quantum dynamics with a Lie algebra decoupling method Soa Qvarfort12and Igor Pikovski23 1Nordita KTH Royal Institute of Technology and Stockholm University Hannes Alfv ens v ag 12 SE-106

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Solving quantum dynamics with a Lie algebra decoupling method

Soﬁa Qvarfort1,2and Igor Pikovski2,3

1Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes Alfv´ens v¨ag 12, SE-106

91 Stockholm, Sweden

2Department of Physics, Stockholm University, AlbaNova University Center, SE-106 91 Stockholm,

Sweden

3Department of Physics, Stevens Institute of Technology, Castle Point on the Hudson, Hoboken, New

Jersey 07030, USA

E-mail: sofia.qvarfort@fysik.su.se, igor.pikovski@fysik.su.se

Abstract. At the heart of quantum technology development is the control of quantum systems at the

level of individual quanta. Mathematically, this is realised through the study of Hamiltonians and the use

of methods to solve the dynamics of quantum systems in various regimes. Here, we present a pedagogical

introduction to solving the dynamics of quantum systems by the use of a Lie algebra decoupling theorem.

As background, we include an overview of Lie groups and Lie algebras aimed at a general physicist audience.

We then prove the theorem and apply it to three well-known examples of linear and quadratic Hamiltonian

that frequently appear in quantum optics and related ﬁelds. The result is a set of diﬀerential equations that

describe the most Gaussian dynamics for all linear and quadratic single-mode Hamiltonian with generic

time-dependent interaction terms. We also discuss the use of the decoupling theorem beyond quadratic

Hamiltonians and for solving open-system dynamics.

arXiv:2210.11894v1 [quant-ph] 21 Oct 2022

Solving quantum dynamics with a Lie algebra decoupling method 2

Contents

1 Introduction 2

2 Introduction to Lie algebras and Lie groups 3

2.1 Liegroups.................................................. 3

2.2 Liealgebras................................................. 4

2.3 Link between Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 The Lie algebra decoupling theorem 5

3.1 Proof of the decoupling theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 A recipe for decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Hamiltonian with linear terms 10

5 Hamiltonian with quadratic terms 12

6 Most general Gaussian Hamiltonian 13

7 Discussion 15

7.1 Casimir elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7.2 Extension to open dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 Conclusions 17

Appendix A Lie algebra decoupling theorem in phase space 17

Appendix B Calculations for Section 4: Hamiltonian with linear terms 19

Appendix C Calculation for Section 5: Hamiltonian with quadratic terms 21

Appendix D Calculations for Section 6: Most general Gaussian Hamiltonian 22

1. Introduction

Quantum physics results in unique phenomena, such as quantum superpositions and entanglement, that can

be harnessed for applications ranging from quantum technologies to tests of fundamental physics. However,

developing such applications requires exact knowledge and control of individual quanta. As new experimental

platforms and hybrid systems emerge, novel dynamics and interaction regimes become increasingly accessible.

In order to fully unlock the potential of these systems, it is crucial to be able to model their quantum

dynamics. To mention just a few examples, quantum dynamics plays a crucial role for quantum control [1],

quantum information processing [2], and quantum sensing [3].

Beyond quantum technologies, searches for new eﬀects in fundamental physics often result in prediction

for changes in quantum dynamics. To detect these often extremely weak eﬀects, it is crucial to be able to

model the system dynamics exactly. As an example, the search for a quantum theory of gravity has resulted

in the study of modiﬁcations to the usual algebras used in quantum theory [4,5]. Such a modiﬁed algebra

necessarily leads to changes in the dynamics of quantum systems. In order to predict observable eﬀects, we

typically require methods to construct the resulting decoupled unitary operations, at least perturbatively [6].

It can however be challenging to treat quantum dynamics analytically. The core of the diﬃculty lies

in the non-commutativity of operators that enter into the Hamiltonian. While it is always possible to write

down the formal solution to the dynamics in terms of an exponential operator, this operator cannot always

be tractably applied to the initial quantum state. The notion of solving quantum dynamics can therefore be

understood as ﬁnding a closed-form expression of the time-evolution operator that facilitates the computation

of any quantity of the system. Ultimately, this is equivalent to solving Schr¨odinger’s diﬀerential equation.

In many ﬁelds of quantum science, it is often suﬃcient to use approximations such as considering only

small quantum perturbations around classical solutions, or to average over many systems. As long as the

Solving quantum dynamics with a Lie algebra decoupling method 3

eﬀects of interest are weak, such perturbative solutions can be used to accurately model the behaviour of

experiments in the laboratory. Many mathematical methods have been developed to express, manipulate

and truncate exponential operators [7–12]. In order to control the full quantum behaviour of individual

quantum systems, however, such as in quantum optics, quantum information and for quantum technologies,

it can be necessary to go beyond these approximations.

One mathematical technique that can be used to solve quantum dynamics exactly makes use of the

Lie algebra induced by the Hamiltonian. The method is based on a Lie algebra decoupling theorem, which

was put forwards by Wei and Norman in 1963 [13]. At its core is the observation that a ﬁnite Lie algebra

generated by a Hamiltonian can be used as a basis that allows for a set of scalar diﬀerential equations to

be derived. Should the resulting diﬀerential equations have analytic solutions, the system dynamics can be

solved exactly.

In this work, we provide a pedagogical introduction to the Lie algebra decoupling theorem and its

application. Speciﬁcally, we apply the theorem towards solving the dynamics of three Hamiltonians that

frequently appear in quantum optics and adjacent ﬁelds. The Hamiltonians we consider have both linear

and quadratic interaction terms, where each interaction term depends on time in an arbitrary fashion. As

such, the theorem allows us to provide a treatment of the most general case of Gaussian dynamics.

The work is structured as follows. Section 2provides a mathematical introduction to Lie groups and Lie

algebras. Section 3contains a proof of the decoupling theorem, as well as a straight-forward recipe for how

it can be applied. Then, in Sections 4and 5, we apply the theorem to solve the dynamics of a Hamiltonian

with time-dependent linear and quadratic interaction terms, respectively. There are also ways in which

these solutions can be combined to represent the most general Gaussian dynamics, which we demonstrate in

Section 6. The work is concluded with a discussion of symmetries and extensions to open-system dynamics in

Section 7, as well as some ﬁnal remarks in Section 8. In the appendices that follow, we provide an extension

of the decoupling theorem to phase space, as well as detailed calculations for each section in the main text.

2. Introduction to Lie algebras and Lie groups

The Lie algebra decoupling theorem can be understood and applied without deep knowledge of Lie algebras,

but for the interested reader, we here provide a basic mathematical introduction to Lie groups, Lie algebras,

and the connection between them. In each case, we start the discussion with a formal deﬁnition, then provide

a few examples that commonly appear in physics.

2.1. Lie groups

To understand Lie algebras, it helps to ﬁrst introduce the concept of a Lie group. We start by recounting

the formal deﬁnition of a group.

Deﬁnition (group) A group (G, ∗)is a set Gwith group elements gand a binary operation ∗such that

G×G→G, which satisﬁes three conditions:

(i) Associativity: For any three elements x, y, z ∈G, we have (x∗y)∗z=x∗(y∗z).

(ii) Identity: The group must include an identity element ∈G. It must hold that ∀g∈G, multiplying gby

the identity element leaves ginvariant, such that ∗g=g∗=g.

(iii) Inverse: For each element in the group, there must be an inverse element. That is, for each element

g∈Gthere is some inverse element ¯g∈Gsuch that g∗¯g=¯g∗g=.

Groups arise in many separate context in physics. Perhaps one of the simplest examples of a group is the

set of discrete rotations that leave a square invariant. That is, the group elements correspond to rotation

actions (acting on a square) and the group operation corresponds to the composition of actions. This group

is known as Z4and contains rotations of zero, 90○, 180○, and 270○degrees. This set of rotations satisﬁes the

group axioms, since any combination of the group elements satisﬁes the same symmetry. That is, we can

rotate the square by ﬁrst 90○, then 180○, and it remains invariant. The associativity criterion is satisﬁed by

a proper representation of the group elements, such as a matrix. Lastly, the identity is equivalent to not

rotating the square, and the inverse can easily be constructed by rotating the square in the reverse direction

by the same amount of degrees.

Solving quantum dynamics with a Lie algebra decoupling method 4

We now proceed to discuss Lie groups. They are continuous groups and play an ubiquitous role in

physics and mathematics. For example, in quantum mechanics, the set of unitary time-evolution operators

form a Lie-group, as we will see below. We proceed with a formal deﬁnition of a Lie group.

Deﬁnition (Lie group): A Lie group is a set Gwith two structures:

(i) Gis a group with the structure discussed in the deﬁnition of a group shown above.

(ii) Gis a smooth and real manifold. Smoothness means that the group operation and inverse map are

diﬀerentiable. The group is described by a set of real parameters that describe the group elements.

Examples of Lie groups include, for example, the translation group, the special unitary group SU(n),

the group of all invertible linear maps, and the special orthogonal group in three dimensions SO(3). For all

of these groups, the inverse and the zero elements can be constructed. The real line has continuous elements,

where the inverse can be constructed by subtraction, and the identity element is zero. The special unitary

group SU(n) describes n×nunitary matrices with determinant 1. Furthermore, the inverse element can

be obtained through complex conjugation, and the identity is the n×nidentity matrix. Finally, the SO(3)

group describes the rotation of vectors in three dimensions. The inverse element can be constructed through

orthogonality, and matrix multiplication automatically satisﬁes associativity. In fact, most continuous

rotation groups are Lie groups. A list of Lie groups can be found in Ref [14].

2.2. Lie algebras

We are now ﬁnally ready to properly deﬁne a Lie algebra. Lie algebras are the key object of interest in

this work, and the link between Lie groups and Lie algebras underpins the Lie algebra decoupling theorem.

Given a Lie group, it is always possible to construct a Lie algebra from the group, and sometimes there are

advantages to studying the algebra rather than the group itself. We begin with the basic deﬁnition of a Lie

algebra [15].

Deﬁnition (Lie algebra): A Lie algebra is a vector space 

gover some ﬁeld F‡, together with a binary

operation [⋅,⋅]∶g×g→g(the Lie bracket) which must satisfy the following axioms:

(i) Bilinearity, such that [ax +by, z]=a[x, z]+b[y, z]and [z, ax +by]=a[z, x]+b[z, y]for all scalars a, b in

the ﬁeld Fand all elements x, y, z in g.

(ii) Alternativity, which means that the Lie bracket is zero for the same element: [x, x]=0.

(iii) The Jacoby identity, which states that

[x, ]y, x]]+[z, [x, y]]+[y, [z, x]]=0 (1)

We ﬁnd that the commutator bracket [A, B]=AB−BA, which is commonly used in quantum physics, satisﬁes

these criteria. In fact, the commutator bracket is often used and is a measure of how non-commutative an

algebra is.

2.3. Link between Lie groups and Lie algebras

The next question is how Lie groups connect with Lie algebras. While this can be discussed to great

mathematical detail, we here provide an example that is hopefully more intuitive to the quantum physicist.

To begin with, we consider a Lie group Lwith elements G(α)∈L, where αis some real parameter. To

determine the action of the element near identity, we can slightly perturb G(α)for a small αjto ﬁnd

G(α)≈+iδαjXj,(2)

where we have deﬁned the generator ˆ

Xj. Then, performing this small perturbation many times in addition

to the identity operation, we ﬁnd

∞

∑

k1+iαjXj

kk≡eiαjXj=D(α).(3)

‡A ﬁeld is a fundamental algebraic structure in the form of a set, where where addition, subtraction, multiplication, and

division are deﬁned.

Solving quantum dynamics with a Lie algebra decoupling method 5

We can now also deﬁne the generator Xjas the rate of change with respect to the parameter αj:

Xj≡−i∂

∂αj

D(α)j

,(4)

We note that Eq. (3) is, in fact, the deﬁnition of the exponential map. The Xjare generators, which form a

Lie algebra. The Lie algebra then generates the group together with the real parameters αj. In other words,

given a Lie algebra with a set of nelements one can always use the exponential map to generate a Lie group.

3. The Lie algebra decoupling theorem

Equipped with some knowledge of Lie groups and Lie algebras, we are now ready to study the Lie algebra

decoupling theorem, ﬁrst outlined by Wei and Norman in 1963 [13]. This section closely follows the proof ﬁrst

developed in Ref [13], but with slightly diﬀerent notation in order to be consistent with modern conventions

in quantum information and quantum optics. See also Ref [16] for a presentation of these methods in the

context of optomechanical systems. For convenience, we set 

h=1 in this section.

Intuitively, the Lie algebra decoupling methods separates a dynamical problem into the notion of

directions of evolution (where the directions are deﬁned by the algebra), from the speed of the evolution

(the amount with which each algebra element is applied to the quantum state). For example, in continuous

variables quantum information, rotation, displacement and squeezing operators are often used to describe

the trajectory of a quantum state in phase space.

The Lie algebra decoupling method eﬀectively transforms the problem of solving an operator-valued

linear diﬀerential equation into that of solving a coupled system of diﬀerential equations of real coeﬃcients.

One advantage of using this method is that problems which would have required the use of large numerical

Hilbert spaces can instead be treated by solving a set of potentially coupled scalar diﬀerential equations.

While these equations do not always have analytic solutions and might similarly have to be solved using

numerical methods, errors due to the limited size of numerical Hilbert spaces can be avoided.

In short, the Lie algebra decoupling method is concerned with solving the Schr¨odinger equation.

Consider the ﬁrst-order diﬀerential equation

dˆ

U(t)

dt =−iˆ

H(t)ˆ

U(t),(5)

where ˆ

H(t)is the Hamiltonian and ˆ

U(t)is a time-evolution operator. The formal solution to ˆ

U(t)is given

U(t)=Texp −i



h∫t

0dt′ˆ

H(t),(6)

where Tindicates time-ordering.

We then assume that the Hamiltonian ˆ

H(t)can be written as a ﬁnite sum with mterms§of constant

operators ˆ

Hjand general time-dependent coeﬃcients Gj(t):

H(t)=m

∑

j=1

Gj(t)ˆ

Hj.(7)

The set {ˆ

Hj}with j=1,2,...,m reproduces the Hamiltonian ˆ

H(t). It can be extended to a larger set

with n≥melements by taking the commutator of the elements in {ˆ

Hj}and adding the result to the set of

Hamiltonian terms. One could then write the original Hamiltonian as:

H(t)=n

∑

j=1

Gj(t)ˆ

Hj,(8)

§A Hamiltonian with an inﬁnite number of unique terms would by extension also generate an inﬁnite Lie algebra. The Lie

algebra decoupling theorem holds ﬁnite Lie algebras, which is why we also assume that the initial Hamiltonian can be written

as a sum over ﬁnite m.

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SolvingquantumdynamicswithaLiealgebradecouplingmethodSoaQvarfort1;2andIgorPikovski2;31Nordita,KTHRoyalInstituteofTechnologyandStockholmUniversity,HannesAlfvensvag12,SE-10691Stockholm,Sweden2DepartmentofPhysics,StockholmUniversity,AlbaNovaUniversityCenter,SE-10691Stockholm,Sweden3DepartmentofPhysi...

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