# Geometric phase for an adiabatically evolving open quantum system

**ABSTRACT** We derive an elegant solution for a two-level system evolving adiabatically under the influence of a driving field with a time-dependent phase, which includes open system effects such as dephasing and spontaneous emission. This solution, which is obtained by working in the representation corresponding to the eigenstates of the time-dependent Hermitian Hamiltonian, enables the dynamic and geometric phases of the evolving density matrix to be separated and relatively easily calculated.

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Page 1

arXiv:quant-ph/0406018v1 2 Jun 2004

Geometric phase for an adiabatically evolving open quantum system

Ingo Kamleitner,1James D. Cresser,2and Barry C. Sanders1,3

1Australian Centre for Quantum Computer Technology,

Macquarie University, Sydney, New South Wales 2109, Australia

2Department of Physics, Macquarie University, Sydney, New South Wales 2109, Australia.

3Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, Canada

(Dated: February 1, 2008)

We derive an elegant solution for a two-level system evolving adiabatically under the influence of

a driving field with a time-dependent phase, which includes open system effects such as dephasing

and spontaneous emission. This solution, which is obtained by working in the representation cor-

responding to the eigenstates of the time-dependent Hermitian Hamiltonian, enables the dynamic

and geometric phases of the evolving density matrix to be separated and relatively easily calculated.

PACS numbers:

I. INTRODUCTION

The discovery by Berry [1, 2] that a (non-degenerate)

state of a quantum system can acquire a phase of purely

geometric origin when the Hamiltonian of the system un-

dergoes a cyclic, adiabatic change has lead to an explo-

sion of interest in this and related geometric phases in

quantum mechanics, both from a theoretical perspective,

and from the point of view of possible applications, the

latter including applications to optics (where the geo-

metric phase was first discovered [3]), NMR and molec-

ular physics, and to quantum computing [4, 5]. Since

Berry’s work, and the demonstration that Berry’s phase

can be understood as a holonomy associated with the

parallel transport of the quantum state [2], there have

been numerous proposals for generalizations. The first

of these was due to Wilczek and Zee [6] who, by consid-

ering a Hamiltonian with non-degenerate eigenstates, es-

tablished the existence of an intimate connection between

Berry’s phase and non-Abelian gauge theories. The re-

striction to changes occurring adiabatically was relaxed

in the work of Aharonov and Anandan [7] while Anandan

[8] generalized the geometric phase to the non-adiabatic

non-Abelian case. The restriction of cyclicity was re-

moved by Samuel and Bhandari [9] and by Pati [11]. All

of this work is concerned with geometric phases of pure

states of closed systems and is now standard, though [9]

indicated extensions to taking account of quantum mea-

surements and consequent non-unitary evolution. A nice

overview, theoretical as well as experimental, is given in

[10].

More recently attention has turned to studying geo-

metric phases for mixed states, though there is not yet

a standard description for geometric phases associated

with mixed states.

As realistic systems always interact with their environ-

ment, and as an open system is almost always to be found

in a mixed state, open systems are a natural source of

problems involving the geometric phases of mixed states.

Garrison and Wright [12] were the first to touch on this

issue in a phenomenological way, by describing open sys-

tem evolution in terms of a non-Hermitean Hamiltonian.

This was, in fact, a pure state analysis, so it did not,

strictly speaking, directly address the problem of geomet-

ric phases for a mixed state, but this work raised issues

which could potentially have a bearing on the analysis of

the mixed state problem. In fact, they did point out that

a proper treatment of an open system would require mak-

ing use of the density operator approach. Nevertheless,

they arrived at an interesting result, a complex geomet-

ric phase for dissipative evolution. This is a result that

has been recently put into doubt by a master equation

treatment by Fonseca Romero et al and Aguiar Pinto and

Thomaz [13, 14].

The first complete open systems analyses of geometric

phase for a mixed state, from two different perspectives,

is to be found in the papers of Ellinas et al [15] and Gam-

liel and Freed [16]. The former worked with the standard

master equation for the density operator of a multilevel

atom subject to radiative damping and driven by a laser

field with a time-dependent phase. What is of interest

in their approach is that it entailed introducing eigenma-

trices of the Liouville superoperator of the master equa-

tion for the damped system. The system Hamiltonian

was allowed to vary adiabatically, with the result that a

non-degenerate eigenmatrix acquires a geometric phase

as well as a dynamic phase. In [16], the effect of the envi-

ronment was modelled as an external classical stochastic

influence which, when averaged, gives rise to the relax-

ation terms of the master equation for the system. In

both cases the effects of any geometric phase was then

shown to be present in measurable quantities such as the

inversion of a two state system.

Since then, research has been increasing rapidly into

the problem of defining a geometric phase for mixed

quantum states for both unitary and non-unitary evo-

lution, motivated to a very large extent by the need to

understand the effects of decoherence in quantum com-

putational processes that exploit geometric phases as a

means of constructing intrinsically fault-tolerant quan-

tum logic gates. This issue has been addressed from two

points of view, the first holistic in nature wherein the

aim is to identify a geometric phase to be associated with

the mixed state itself, and the second, essentially the ap-

Page 2

2

proach of [15], which works with the pure state geometric

phases of an appropriate set of parallel transported basis

states, which then gives rise to geometric phase factors

in the off-diagonal elements of the density operator. No

geometric phase is explicitly associated with the mixed

state itself; instead observable quantities that will exhibit

the effects of the geometric phases of the underlying basis

states are determined.

The former, holistic approach was first introduced in a

formal way by Uhlmann [17], and in a different way, based

on phase-sensitive measurements via interferometry, by

Sj¨ oqvist et al [18] for unitary evolution of a mixed state,

and later for non-unitary evolution [19, 20]. The phase

defined in this way is not the same in all respects to that

proposed by Uhlmann [21].

The latter kind of approach has been used only for

open systems, and involves working with the (Markovian)

master equation of the open system. The approaches

used involve either solving the master equation of the sys-

tem [13, 14, 15, 16], or employing a quantum trajectory

analysis [22, 23, 24] to unravel the dynamics into pure

state trajectories, and calculating the geometric phases

associated with individual pure state trajectories. Noise

of a classical origin, such as stochastic fluctuations of the

parameters of the Hamiltonian have also been studied by

Chiara and Palma [25]. In essence, the common feature

is not so much to propose a new definition of geometric

phase for a mixed state as to show how the underlying

existence of a geometric phase will nevertheless show up

in the observed behaviour of an open quantum system.

It is this perspective that is adopted in the work to be

presented in this paper.

Here we introduce an elegant approach for solving the

central master equation which is based on introducing a

unitary transformation due to Kato [26] described in the

classic text by Messiah [27].

The new picture is defined via a time-dependent

unitary transformation A†(t), usually referred to as a

rotating axis transformation, which is such that the

transformed system Hamiltonian has time independent

eigenspaces. This method is extended by showing that

under the conditions of adiabatic evolution, all the infor-

mation on geometric phase for a closed loop is contained

within A(t), and is regained by transforming back to the

original picture. The goals of this approach is its sim-

plicity, since one needs only to calculate the geometric

phases for the eigenstates of the Hamiltonian H(t), and

the fact that dynamic and geometric phase are separated

in a clear way. In fact, perfect separation of geomet-

ric and dynamical contributions is obtained provided the

Hamiltonian evolution is adiabatic and the coupling to

the environment is weak. This approach bears some sim-

ilarity to that used by Fonseca RomeroFonseca[13] who

make use of several unitary transformations to separate

the geometric phase from the dynamic phase. However,

the rationale for their transformations, and the origin of

the geometric phase, is somewhat elusive in their anal-

ysis. In contrast to [13], with the transformation intro-

duced here, the parallel transport condition is essential

and explains the appearance of the geometric phase.

Within the approach used here, it is possible to show

explictly how to achieve, under certain circumstances,

operational cancellation of the dynamic phase, thereby

making the geometric phase accessible in experiments.

An example of where this is possible is given in Sec-

tion III.

This paper is organized as follows. In Section II we

present the main ideas. In Section III we look at sev-

eral examples. In Section V we summarize our results

while in Section IV we indicate possible new directions,

including generalisations to non-cyclic evolution and non-

Abelian holonomies. An analog to the adiabatic theorem

is proved for a general Lindblad equation [28] in the Ap-

pendix.

II. THE ROTATING AXIS TRANSFORMATION

FOR NONDEGENERATE MULTILEVEL

SYSTEMS

A.Geometric Phases for a Closed System

For the present we consider the case of a closed system

so as to introduce the basic method employed here. Sup-

pose we have a system with Hamiltonian H(t), a function

of time due to the dependence of H on parameters whose

values can be changed in time. This Hamiltonian will

have instantaneous eigenvectors |n(t)? with eigenvalues

En(t):

H(t)|n(t)? = En(t)|n(t)?.(1)

For simplicity we assume Ei(t) ?= Ej(t) for i ?= j. This

restriction will be removed in section IV to obtain non-

Abelian holonomies. For an adiabatically slow change

in the system parameters, these eigenvectors will also

change in such as way as to satisfy the parallel trans-

port condition [2]:

?n(t)|d

dt|n(t)? = 0. (2)

At this point we introduce a unitary operator A(t) via

the equation

A(t)|n(0)? = |n(t)?. (3)

This completely defines A(t). Note that because of the

path dependence of the parallel transported eigenstates

|n(t)?, the operator A(t) has a non-integrable nature,

and, as we see later, will contain the information on the

geometric phase.

This unitary operator can now be used to remove the

time dependence of the eigenstates of the Hamiltonian.

Thus, if we define

HA≡ A†HA(4)

Page 3

3

we note that the eigenvectors of HAare now just |n(0)?

and hence are time independent.

Schr¨ odinger equation is then

?

where |ψA? = A†|ψ?. If |n(t)? is parallel transported,

then, to the lowest-order adiabatic approximation, one

can neglect the terms containing A†(t)˙A(t) so that the

transformed Schr¨ odinger equation becomes [27]

The transformed

HA|ψA? = i¯ hA†˙A|ψA? +d

dt|ψA?

?

(5)

HA|ψA? = i¯ hd

dt|ψA?. (6)

The solution of Eq. (6) contains no geometric contribu-

tion — it gives the dynamic contribution to the phase of

any adiabatically evolving state. So we have extracted

the dynamics from the geometric influence of the time-

varying Hamiltonian. The geometric contribution is en-

tirely contained within the operator A. To obtain this

information we have to transform back to the original

picture in terms of the states |ψ?:

|ψ? = A|ψA?.

If the Hamiltonian undergoes a closed loop in time T,

i.e. H(T) = H(0), then the parallel transported eigen-

states |n(T)? return to the initial eigenstates |n(0)?

up to the geometric phase.

diag(eiϕ1,...,eiϕN) where ϕnis the geometric phase as-

sociated with the eigenstate |n(T)?. This result can be

readily generalized if the system is in a mixed state ρ.

Introducing the notation

(7)

Hence we have A(T) =

ρA(t) = A†(t)ρ(t)A(t) (8)

we obtain

ρ(T) =A(T)ρA(T)A†(T)

=diag(eiϕ1,...,eiϕN)ρA(T)

× diag(e−iϕ1,...,e−iϕN).

holonomic transformation

diagonal elements of the density operator ρA

ei(ϕj−ϕi), which is the difference of the geometric phases

of the eigenstates of the Hamiltonian H(T).

(9)

This multipliestheoff-

ijby a phase

B. Open System and Master Equation

Systems that are coupled to a reservoir (or environ-

ment) can usually be described by a reduced density

operator that evolves according to a master equation,

which, in many cases, can be written in the Lindblad

form [28]

˙ ρ(t) =

1

i¯ h[H(t),ρ(t)] +1

2

k

?

α=1

LΓα[ρ(t)](10)

for

LΓ[ρ] ≡ 2ΓρΓ†− Γ†Γρ − ρΓ†Γ ,

and where ρ(t) is the density operator for the system of

interest, ˙ ρ(t) is its derivative with respect to time, H(t)

is the system Hamiltonian and the dissipation operators

Γαarise due to the presence of the reservoir. To obtain a

geometric phase we once again consider an adiabatically

changing Hamiltonian H(t), with H(T) = H(0), as in the

preceding section. Upon introducing the operator A(t)

defined in Eq. (3) we can transform the master equation

with the unitary operator A†(t), which leads us to the

new master equation

(11)

˙ ρA(t) =1

i¯ h[HA(t),ρA(t)] + ρA(t)A†(t)˙A(t)

− A†(t)˙A(t)ρA(t) +1

2

k

?

α=1

LΓA

α[ρA(t)].(12)

Note that ΓA

when Γα is not. As shown in the Appendix, the terms

containing A†(t)˙A(t) can be neglected, as in the case of

a unitary evolution but with the additional requirement

that the damping is weak. This establishes the parallel

transport condition Eq. (2) for weakly damped systems

in an adiabatic evolution. We then get

α≡ A†ΓαA is in general time-dependent even

˙ ρA(t) =

1

i¯ h[HA(t),ρA(t)] +1

2

k

?

α=1

LΓA

α[ρA(t)]. (13)

The solution of Eq. (13) contains no geometric contri-

butions. To regain ρ(T) we have to transform back to

the original picture as in the unitary case, Eq. (9).

ρ(T) =A(T)ρA(T)A†(T)

=diag(eiϕ1,...,eiϕN)ρA(T)

× diag(e−iϕ1,...,e−iϕN)

with ϕnbeing the geometric phase for the state |n?.

(14)

III. EXAMPLES FOR TWO-LEVEL SYSTEMS

A.Optical Resonance with Spontaneous Emission

We consider a two level atom in a classical reso-

nant laser field. In the rotating-wave approximation the

Hamiltonian for this system is

?

The detuning ∆, the coupling strength Ω and the phase

φ are properties of the laser. To induce a geometric phase

we change the phase φ(t) of the laser field slowly in com-

parison to E/¯ h = (Ω2+1

H = ¯ h

∆

2

Ωe−iφ

−∆

Ωeiφ

2

?

(15)

4∆2)

1

2, which is the absolute

Page 4

4

value of the eigenenergies of the Hamiltonian divided by

¯ h. The eigenvalue equation is

H(t)| + (t)? = E| + (t)?

H(t)| − (t)? = −E| − (t)?

(16)

(17)

with

| + (t)? = e−iφ(t)sin2 θ

2cosθ

2|e? + eiφ(t)cos2 θ

2sinθ

2|g?

(18)

| − (t)? = −e−iφ(t)cos2 θ

2sinθ

2|e? + eiφ(t)sin2 θ

2cosθ

2|g?

(19)

and

sinθ

2=

?

?

E −1

2¯ h∆

2E

(20)

cosθ

2=

E +1

2¯ h∆

2E

(21)

|e?and |g?denotes the excited state and the ground state

of the two level atom, respectively. Note that |+?and

|−?satisfy the parallel transport condition as required in

Section II.

Furthermore we want to include spontaneous emission

as a source of dissipation. In the weak coupling limit the

master equation is known to be

˙ ρ(t) =

1

i¯ h[H(t),ρ(t)] +1

2LΓ[ρ(t)]. (22)

for

Γ =

√

λ σ−=

√

λ

?0 0

1 0

?

. (23)

Here λ denotes the spontaneous emission rate. The task

here is to solve Eq. (22) in the adiabatic and weak damp-

ing limit. As in Section II we define the operator A(t)

by A(t)| ± (0)? = | ± (t)?. After the transformation of

Eq. (22) with A†(t), the Hamiltonian is not diagonal.

Hence we carry out another transformation with an op-

erator B†that is defined by

B†| + (0)? =|e?

B†| − (0)? =|g?. (24)

As B†is time-independent, we obtain no term B† ˙B in the

master equation. We can carry out both transformations

together with the operator C†(t) = B†A†(t), which turns

out to be

?e−iφ(t) sin2 θ

2

The master equation for ρC(t) = C†(t)ρ(t)C(t) is, from

Eq. (13) and Eq. (22),

C(t) =

2cosθ

2sinθ

2−e−iφ(t)cos2 θ

eiφ(t)sin2 θ

2sinθ

2cosθ

2

eiφ(t) cos2 θ

2

?

. (25)

˙ ρC(t) =

1

i¯ h[HC,ρC(t)] +1

2LΓC(t)[ρC(t)] (26)

for

HC= C†(t)H(t)C(t) =

?E0

0 −E

?

(27)

and

ΓC(t) =C†(t)ΓC(t)

√

=

λ

?

cosθ

2eiφ(t)cos θ

2sinθ

2

sin2 θ

−cosθ

2e−iφ(t) cosθ

2sinθ

cos2 θ

2

?

. (28)

We show in the Appendix that only the absolute value

of the off-diagonal elements of the dissipation operator

ΓC(t) contributes in the solution of Eq. (26). Using only

the absolute values of these elements does not simplify

the calculation, but multiplying them with a time inde-

pendent factor eiβand averaging over all 0 < β < 2π

does. Doing so, we first rewrite Eq. (26)

˙ ρC(t) =1

i¯ h[HC,ρC(t)] +

1

4π

?2π

0

LΓC[ρC(t)]dβ. (29)

In Eq. (29) we substitute the Lindblad operator ΓCwith

the new Lindblad operators ΓC

?cosθ

βwith

ΓC

β=

√

λ

2sinθ

2eiβ

2

sin2 θ

−cosθ

2e−iβ

2sinθ

cos2 θ

2

?

,0 ≤ β < 2π

(30)

as explained previously. Finally we get a new master

equation, which is equivalent to Eq. (26) in the adiabatic

and weak damping limit:

˙ ρC(t) =1

i¯ h[HC,ρC(t)] +

1

4π

?2π

0

LΓC

β(t)[ρC(t)]dβ (31)

Eq. (31) may seem to be more complicated than Eq. (26),

but if we evaluate the term behind the integral in Eq. (31)

we realize that many contributions are cancelled out by

the integration. If we write

?a

then Eq. (31) reduces to the two independent equations

?

˙b(t) = −2i

ρC(t) =

b

b∗1 − a

?

(32)

˙ a(t) = λ sin4θ

2− a

?

?

sin4θ

2+ cos4θ

2

??

?

(33)

¯ hEb − λbsin2θ

2cos2θ

2+12

(34)

with the solutions

a(t) =

?

a(0) −

sin4 θ

2

sin4 θ

2+ cos4 θ

2

?

e−λt(sin4 θ

2+cos4 θ

2),

+

sin4 θ

2

sin4 θ

2+ cos4 θ

2

(35)

b(t) =b(0)e−2iEt

¯ he−λt

?

sin2 θ

2cos2 θ

2+1

2

?

. (36)

Page 5

5

As

C(t)ρC(t)C†(t). As the inversion provides an operational

quantity for inferring the geometric phase by measuring

the relative proportion of ground vs excited states, and

because the terms become rather long, we only write the

inversion w(t), which is

the laststepwe needto evaluateρ(t)=

w(t) =ρ11− ρ22= (2a(t) − 1)cosθ

− 2sinθRe

?

b(t)eiφ(t) cosθ?

(37)

To compare this result with that found by Ellinas et al

[15], we set ρ(0) =1

sinθ

Furthermore we define

2+pσ3with |p| ≤1

2from Eq. (20) and Eq. (21), respectively.

2and substitute for

2and cosθ

K =2Ω2+ ∆2

4Ω2+ ∆2and G =

6Ω2+ ∆2

8Ω2+ 2∆2. (38)

If we furthermore consider the inversion at a time T at

the end of the cyclic evolution we finally get for the in-

version:

w(T) = 2p

??∆¯ h

2E

?2?2Kp + 1

2Kp

e−KλT−

1

2Kp

?

+ cos

?2ET

¯ h

− 2π¯ h∆

2E

?

e−GλT

?Ω¯ h

E

?2?

(39)

which is the same as that derived by [15]. The dynamic and geometric phases are found in the cosine term in this

expression: the difference of the dynamic phases of the eigenstates of the Hamiltonian is given by 2ET/¯ h, and the

difference of the geometric phases of these eigenstates (for φ(T) = 2π) given by 2π¯ h∆/2E. This term is diminished

by a damping factor exp(−GλT) which can influence the observability of the geometric phase effect on the inversion.

The issues of time scales to observe the effect of the geometric phase have been discussed in [15]. However, for the

present, we wish to point out that the result above has been derived here by use of a simple transformation into a

rotating frame. This is to be contrasted with the much more complicated approach of [15], based on calculating the

eigenmatrices of the Liouvillian.

B. Optical Resonance with Dephasing

As in the previous subsection we treat a two level atom driven by a resonant electromagnetic field. This time we

assume the damping is due to dephasing that occurs as a consequence of phase changing collisions, which changes the

relative phase between the excited state and the ground state of the atom (in contrast to strong collisions that change

populations of eigenstates). Since the phase change can vary for each collision we have to consider a one dimensional

manifold of dissipation operators

?1

where λ(α) is the dephasing rate density. Hence we get the master equation

?π

with the Hamiltonian H(t) from the previous subsection with a slowly changing phase φ(t) again. Thus, we get the

same parallel transported eigenstates of the Hamiltonian and we can start by carrying out the same transformation

as in the previous subsection. In the adiabatic and weak damping limit we obtain the transformed master equation

?π

for

Γα=

?

λ(α)

0

0 eiα

?

,−π < α < π, (40)

˙ ρ(t) =

1

i¯ h[H(t),ρ(t)] +

−π

?Γαρ(t)Γ†

α− λ(α)ρ(t)?dα (41)

˙ ρC(t) =1

i¯ h[HC,ρC(t)] +

−π

?ΓC

α(t)ρC(t)ΓC†

α(t) − λ(α)ρC(t)?dα(42)

HC= C†(t)H(t)C(t) =

?E0

0 −E

?

(43)

and

ΓC

α(t) = C†(t)ΓαC(t) =

?

λ(α)

?

cos2 θ

2+ eiαsin2 θ

2ei(φ(t)cos θ+α

2

isinθsinα

sin2 θ

2ei(−φ(t)cosθ+α

2+ eiαcos2 θ

2)

isinθsinα

2)

2

?

. (44)

Page 6

6

Now everything is much the same as in the previous subsection. Finally we find for the components a and b of the

density operator ρC(t) the decoupled differential equations

˙ a = −4facos2θ

?−2E

2sin2θ

?

2+ 2f cos2θ

sin4θ

2sin2θ

??

2

(45)

˙b = i

¯ h

+ gb

2− cos4θ

2

− fb

?

sin4θ

2+ cos4θ

2

?

. (46)

where

f =

?+π

−π

λ(α)(1 − cosα)dα (47)

and

g =

?+π

−π

λ(α)sinαdα(48)

are properties of the model describing the damping col-

lisions. The solutions of these equations are

a(t) =?a(0) −1

b(t) = b(0)ei(−2E

2

?e−4ftcos2 θ

2sin2 θ

2+1

2, (49)

¯ h

+g(sin4 θ

2−cos4 θ

2))te−(sin4 θ

2+cos4 θ

2)ft.

(50)

To calculate the inversion w(t) we can take Eq. (37)

and substitute a(t) and b(t) with Eq. (49) and Eq. (50),

respectively. Using Eq. (20) and Eq. (21) as well as the

previous definition of K we finally find at time t = T:

??¯ h∆

??

This result is similar to that found in the case of sponta-

neous emission in Eq. (39). There appears in the cosine

term in Eq. (51) the difference of the dynamic phases of

the eigenstates of the Hamiltonian, as given by 2ET/¯ h,

and the difference of the geometric phases of these eigen-

states (for φ(T) = 2π), given by 2π¯ h∆/2E. This term is

also diminished by a damping factor exp(−KfT) which

influences the observability of the geometric phase effect

on the inversion. Both this term and an additional con-

tribution of a shift in the Rabi frequency by g¯ h∆

through the presence of dephasing, though the latter will

only appear if the dephasing rate density λ(α) is not

symmetric.

w(T) =2p

2E

?2

g¯ h∆

e−(Ω¯ h

E)

2fT+1

2

?Ω¯ h

E

?2

??

e−KfT

× cos

2E−2E

¯ h

?

T + 2π¯ h∆

2E

. (51)

2Earise

C.Spin in Magnetic Field with Dephasing

As another example we consider the simple model of

a spin-1

2particle in a magnetic field with constant field

strength, which demonstrates how to remove the dy-

namic phase in a standard model. To induce a geomet-

ric phase we change the direction of the magnetic field

slowly in comparison to E/¯ h. As a source of decoherence

we consider dephasing which is defined by

?

Γα(t)|g(t)? =

|g(t)? and |e(t)? are the parallel transported eigenstates

of the Hamiltonian with spin parallel and antiparallel

to the magnetic field, respectively. Further λ(α) is the

dephasing rate density. Note that dephasing does not

change the energy of the spin-system. It is important to

distinguish between this model and the two level atom

in the previous subsection. Here the Lindblad operators

are defined in the basis of the time-dependent eigenstates

of the Hamiltonian whereas before the Lindblad opera-

tors have been defined in the basis of the excited and the

ground state of the two level atom which are independent

of the properties of the applied laserfield and hence not

the eigenstates of the Hamiltonian. Such dephasing oper-

ators could be realized by random fluctuations of the field

strength of the applied magnetic field. Since the Hamil-

tonian changes in time, the dephasing operators have to

be time-dependent, too. As in Section II we define the

operator

Γα(t)|e(t)? =

λ(α)|e(t)?

?

(52)

λ(α)eiα|g(t)?. (53)

A(t)|e(0? =|e(t)?

A(t)|g(0)? =|g(t)? (54)

and find, from Eq. (13), for ρA= A†(t)ρ(t)A(t)

??E

+1

2

−π

0 eiα

× λ(α)dα

=

¯ h

2iE

¯ hρA

?π

× λ(α)dα.

Here ρA

tion of Eq. (55) is

˙ ρA(t) =1

i¯ h

0

0 −E

?π

?

?1

,ρA(t)

?

ρA(t)

?

2

0

??10

0 e−iα

?

− 2ρA(t)

?

?

0

−2iE

ρA

12(t)

0

21(t)

?

?

+

−π

0(e−iα− 1)ρA

12(t)

(eiα− 1)ρA

21(t)0

?

(55)

ij(t) denotes the components of ρA(t). The solu-

ρA

11(t) = ρA

11(0)

12(0)e−i(2E

(56)

ρA

12(t) = ρA

¯ h+g)te−ft

(57)

Page 7

7

with f and g defined in Eqs. (47) and (48), respec-

tively.As the last step we need to calculate ρ(t) =

A(t)ρA(t)A†(t). If the evolution of the Hamiltonian is

cyclic we have A(t) =diag(eiϕ,e−iϕ) where ϕ is the geo-

metric phase for |e?. The geometric phase is half of the

solid angle enclosed by the path which |e(t)? drives on

the Bloch sphere [1]. This is equivalent to half of the solid

angle enclosed by the path determined by the direction

of the magnetic field. Hence we finally get for the com-

ponents of the density operator after the Hamiltonian

undergoes a closed loop

ρ11(T) = ρ11(0)

ρ12(T) = ρ12(0)e−i(2ET

¯ h

+gT−2ϕ)e−fT

(58)

In the latter equation we can see a phase change due to

the energy difference of the system, an additional phase

change due to the dephasing and the geometric phase.

Furthermore we see how the absolute value of the off-

diagonal element of the density operator decreases expo-

nentially in time because of the dephasing.

Our task now is to remove the dynamic phase. We

do a σx-transformation in our system and then in the

time interval [T,2T] we drive the direction of the mag-

netic field around the same loop as before but backwards:

?B(T +t) =?B(T −t). The components of the density op-

erator ρ′(T) after the σx-transformation are

ρ′

11(T) = ρ22(T) = 1 − ρ11(0)

ρ′

12(T) = ρ21(T) = ρ∗

12(0)ei(2ET

¯ h

+Tg−2ϕ)e−Tf

(59)

When we drive the magnetic field backwards, then the

parallel transported eigenstates are |e(T+t)? = |e(T−t)?

and |g(T + t)? = |g(T − t)?.

Now we define the operator

A′(T + t)|e(T)? = |e(T + t)? = |e(T − t)?

A′(T + t)|g(T)? = |g(T + t)? = |g(T − t)?

which parallel transports the eigenstates of the Hamilto-

nian H(T +t). Again we transform the density operator

ρ′A(T +t) = A′†(T +t)ρ′(T +t)A′(T +t) and find for the

components of ρ′A(2T) Eq. (58)

(60)

ρ′A

11(2T) = ρ′

11(T) = 1 − ρ11(0)

12(T)e−i(2E

ρ′A

12(2T) = ρ′

¯ h+g)Te−Tf= ρ12(0)∗e−2iϕe−2Tf

(61)

where in the last step Eq. (59) is used. Now we need

to find ρ′(2T) = A′(2T)ρ′A(2T)A′†(2T). From Eq. (60),

and because now we drive the loop backwards and hence

get the same geometric phase up to a sign, it follows that

A′(2T) = A†(T) =diag(e−iϕ,e+iϕ)

After another σx-transformation we finally get the den-

sity operator

?

ρ(2T) =

ρ11(0)ρ12(0)e4iϕe−2Tf

1 − ρ11(0)ρ12(0)∗e−4iϕe−2Tf

?

. (62)

Hence we see that not only the dynamic phase of the

Hamiltonian is removed, but also the phase shift through

dephasing. What stays is twice the difference of the geo-

metric phases of the ground state and the excited state.

This geometric effect appears in the off-diagonal compo-

nents of the density operator and is damped out expo-

nentially in time through dephasing.

IV.GENERALIZATIONS

A.Non-Cyclic Evolution

To consider a non-cyclic evolution we first outline

Pati’s analysis [11].If H(T) = H(0), Pati compared

the phase of the parallel transported eigenstate of the

Hamiltonian at time t = T, |n(T)? with the phase of

the eigenstate at time t = 0, |n(0)?. If the Hamilto-

nian does not undergo a closed loop, i.e. H(T) ?= H(0),

then |n(T)? is not |n(0)? up to a geometric phase. Com-

paring the phases of states which differ not only by a

phase is not straightforward. Pati introduced a reference

section

?

quirement to make?

?

to non-cylic evolution.

We use this idea and generalize it to open systems.

First one has to calculate the density operator ρA(T) in

the rotating axis representation as in Section II. Instead

of transforming back to the original picture we transform

to the picture given by the reference section introduced

in [11]. We define the operator˜A(T) by

|n(t)? which is supported by eigenstates of the

Hamiltonian H(t). The phase of

|n(t)? in phase with |n(0)? as defined

by means of the work of [3], i.e. ?n(0)?

is defined to be the generalization of the geometric phase

?

|n(t)? is fixed by the re-

|n(t)? = 0. Then

|n(T)? and |n(T)? differ only by a phase and this phase

?

|n(T)? =˜A(T)|n(0)?(63)

The density operator in this new picture is

ρ˜ A(T) =˜A†(T)A(T)ρA(T)A†(T)˜A(T)

=diag(eiϕ1,...,eiϕN)ρA(T)diag(e−iϕ1,...,e−iϕN)

(64)

and˜A†(T)A(T) = diag(eiϕ1,...,eiϕN) is the generalized

holonomy transformation with respect to the reference

section

?

B. Non-Abelian Holonomies

|n(T)?.

Again we consider the master equation Eq. (10) with

an adiabatically changing Hamiltonian. For simplicity

we restrict to a cyclic Hamiltonian H(T) = H(0).

Until now we assumed the eigenenergies of the Hamil-

tonian to be non-degenerate. However, if the eigenvalues

Page 8

8

are degenerate we expect to get non-Abelian holonomies

[6, 8] as the generalization of the geometric phase. In

this case we have the eigenvalue equation

H(t)|nm(t)? = En(t)|nm(t)?

in which n = 1,··· ,N ; m = 1,··· ,Mnand Mnis the

degree of degeneracy of the subspace of the Hamiltonian

with energy En. The Mnare required to be constant in

time, i.e. we do not allow any level crossings of the Hamil-

tonian H(t). The |nm(t)? are now assumed to satisfy the

modified parallel transport condition [8]

(65)

?nm(t)|d

dt|nm′(t)? = 0∀ m,m′= 1,··· ,Mn. (66)

Now we can define the operator A(t) by

A(t)|nm(0)? = |nm(t)?(67)

and with its help we transform the master equation in

the rotating axis representation to remove the time de-

pendence of the eigenspaces of the Hamiltonian analo-

gous to Section II. Doing this we get the new master

equation Eq. (12) for ρA(t) = A†(t)ρ(t)A(t) as before.

Again it can be shown that the terms ρA(t)A†(t)˙A(t)

and A†(t)˙A(t)ρA(t) can be neglected if the |nm(t)? sat-

isfy the modified parallel transport condition Eq. (66) in

the adiabatic and weak damping limit. This justifies the

condition Eq. (66). Since the proof for this is much the

same as for non-degenerate Hamiltonians as done in the

Appendix, we do not carry out the proof in this case.

Now we have to solve (16) which represents the dynam-

ics. Finally we have to transform back to the original

picture:

ρ(T) = A(T)ρA(T)A†(T).(68)

We obtain the non-Abelian holonomy A(T) ∈ U(M1) ⊗

··· ⊗ U(MN) simliar to the way obtained the geometric

phase for non-degenerate systems.

V.CONCLUSIONS AND FUTURE

DIRECTIONS

We have introduced the rotating axis transformation,

in which parallel transport of the eigenstates of the

Hamiltonian plays an important role, to study the geo-

metric phase for an adiabatically evolving multilevel sys-

tem. This transformation was shown to be particularly

useful in simplifying the calculation of open-system evo-

lution, described by a master equation of the Lindblad

form, as it allows an easy separation of dynamic and ge-

ometric phases.These advantages were illustrated by

applying it to optical resonance with spontaneous emis-

sion, where we obtain known results but more easily. The

method was then used to quickly and easily study the ef-

fects of the geometric phase in a number of new problems.

In one application we show explicitly how to remove the

dynamic phase.

Although, in our applications, we concentrated on

Abelian holonomies for nondegenerate systems, the gen-

eralization to non-Abelian holonomies for degenerate

Hamiltonians and to non-cyclic evolution is straight for-

ward.

Acknowledgments

BCS acknowledges financial support from an Aus-

tralian DEST IAP grant to support participation in

the European Fifth Framework project QUPRODIS and

from Alberta’s informatics Circle of Research Excellence

(iCORE) as well as valuable discussions with S. Ghose

and K.-P. Marzlin.

APPENDIX A: NEGLECTING A†(t)˙A(t) IN THE

ADIABATIC APPROXIMATION

Here we prove that the terms containing A†(t)˙A(t) in

Eq. (12) can be neglected in the adiabatic approximation.

The proof will be analogous to the proof of the adiabatic

theorem given in [27]. For simplicity we assume H(t) in

Eq. (12) to be diagonal which can always be achieved by

a proper time independent transformation and hence is

no restriction. We start by transforming Eq. (12) in the

interaction picture. We define

ρH(t) = e−i?t

ΓH

(A†˙A)H(t) = e−i?t

and get by use of Eq. (12) the master equation in the

interaction picture

0HA(t)dtρA(t)ei?t

0HA(t)dtΓA

0HA(t)dtA†(t)˙A(t)ei?t

0HA(t) dt

α(t) = e−i?t

α(t)ei?t

0HA(t) dt

0HA(t)dt

˙ ρH= ρH(A†˙A)H− (A†˙A)HρH+1

2

k

?

α=1

LΓH

α[ρH] (A1)

The formal solution of this equation is

ρH(t) =ρH(0) +

?t

0

?

ρH(A†˙A)H− (A†˙A)HρH

?

+1

2

k

?

α=1

LΓH

α[ρH](s)ds.(A2)

Within the integral, there are some contributions of the

form of a product of a slowly varying function and a

fast oscillating function. These contributions are known

to become small when the frequency of the oscillating

function increases in comparison with the time derivative

of the slowly varying function. To see this we make use

Page 9

9

of the result, following [27],

?t

0

f(s)eiωsds =

1

iω

?

[f(s)eiωs]t

0−

?t

0

f′(s)eiωsds

?

ω

f′→∞

−→ 0 (A3)

To make use of this we write Eq. (A2) in components

?

ρH

ij(t) = ρH

ij(0) +

?t

0

ρH

ik(A†˙A)H

kj− (A†˙A)H

ikρH

kj

+1

2

k

?

α=1

2ΓH

αikρH

klΓH†

αlj− ΓH†

αikΓH

αklρH

lj

− ρH

ikΓH†

αklΓH

αlj

?

(s)ds.(A4)

Here and later, summations are implied over all indices

except of i and j. The components of˙A and Γ are as-

sumed to be small in comparison with ωij = Ei− Ej

(adiabaticity and weak damping, respectively) and hence

we see in Eq. (A4) that all components of ˙ ρHare small

in comparison with ωij. The components of (A† ˙A) are

slowly varying and hence the off-diagonal components,

(A† ˙A)H

ij, i ?= j are oscillating with frequency ωijand can

be neglected. Because of the parallel transport condition

the diagonal elements (A† ˙A)H

nnare null as we can see:

(A†˙A)H

nn=(A†˙A)nn

=?n(0)|A†˙A|n(0)? = ?n(t)|d

dt|n(t)? = 0

The last equality is true because we assumed the |n(t)?

to be parallel transported. So we have proved that we

can neglect A† ˙A in Eq. (12).

Furthermore we can rewrite Eq. (A4) as

ρH

ij(t) =ρH

ij(0) +

?t

αklρH

0

?

1

2

k

?

α=1

2ΓH

αikρH

klΓH∗

αjl

− ΓH∗

αkiΓH

lj− ρH

ikΓH∗

αlkΓH

αlj

?

(s)ds. (A5)

The star denotes complex conjugation. The functions

ΓH

jlare oscillating with frequency ωik−ωjl. Hence we

can achieve a significant simplification if the differences

of all eigenfrequences, ωik− ωjl are not vanishing (are

big in comparison with˙H and Γα) which is always the

case if we consider a two-level system. Then the condition

ωik−ωjl= 0 implies i = k ,j = l or i = j ,k = l and hence

only corresponding parts will contribute to the integral:

ikΓH∗

ρH

ij(t) =ρH

ij(0) +

?t

αiiρH

0

?

1

2

k

?

αii+ 2ΓH

α=1

2δijΓH

αikρH

kkΓH∗

αik

− 2δijΓH

iiΓH∗

αiiρH

ijΓH∗

?

αjj

− ΓH∗

αkiΓH

αkiρH

ij− ρH

ijΓH∗

αljΓH

αlj

(s)ds.(A6)

Here we see that only the absolute value of the off-

diagonal elements of ΓH

Eq. (12).

α and hence ΓA

αcontribute in

We can further note that if we set i = j, we find that

the diagonal elements of the density operator are coupled

only to diagonal elements, whereas for i ?= j, we find that

ρH

ijis coupled only to itself.

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