Geometric phase in dephasing systems

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by  X. X. Yi

Geometric phase in dephasing systems
X. X. Yi, L. C. Wang, and W. Wang
Department of physics, Dalian University of Technology, Dalian 116024, China
(Dated: February 1, 2008)
Beyond the quantum Markov approximation, we calculate the geometric phase of a two-level
system driven by a quantized magnetic field subject to phase dephasing. The phase reduces to the
standard geometric phase in the weak coupling limit and it involves the phase information of the
environment in general. In contrast with thegeometric phase in dissipative systems, thegeometric
phase acquired by the system can be observed on a long time scale. We also show that with the
system decohering to its pointer states, the geometric phase factor tends to a sum over the phase
factors pertaining to thepointer states.
5 PACSnumbers: 03.65.Vf,03.65.Yz
0
0
2 Quantum mechanics tell us that physical states are problem beyond the Markov approximation remains un-
equivalent up to a global phase, which in general does touched to our best knowledge. Because the system-
n
a not contain useful information about the described sys- environment interaction HI and the free system Hamil-
J tem and thus can be ignored. This is not the case, how- tonian H commute for dephasing systems, the dynami-
s

7 ever,forasystemtransportedroundacircuitbyvarying cal problem and then the geometric phase of the system
1 the parameters ~s = (s ,s ,...) in its Hamiltonian H(~s). may be solved/calculated precisely. On the other hand,
1 2

As Berry showed[1], the phase canhave a componentof the previous study [17] shows that one can not perform
1
geometric origin called geometric phase with important an arbitrarily long experiment to measure the phase for
v
5 observableconsequences,suchastheAharonov-Bohmef- dissipative systems, i.e., it is not allowed to draw phase
8 fect[2]andthe spin-1 particledrivenby arotatingmag- information out of the system on a long time scale, this
2
0 netic field [1]. The geometric phases that only depend feature of dissipative systems again motivates investiga-
1
on the path followed by the system during its evolution, tion on the problem of geometric phases in dephasing
0
havebeeninvestigatedandtestedina varietyofsettings systems, where in principle one may get analytical re-
5
0 and have been generalized in several directions [3]. The sults for the phase on any time scale.
h/ geometric phases are attractive both from a theoretical In this Letter, we investigate the behavior of the geo-
perspective,andfromthepointofviewofpossibleappli-
p metricphaseofatwo-levelsysteminteractingwithadriv-
- cations, among which geometric quantum computation ingmagneticfieldwhenthissystemisnotonlyquantized
t
n [4, 5, 6, 7] is one of the most importance. butalsosubjectedtodecoherence. Theenvironmentthat
a leads to decoherence may originate from the fluctuation
u As realistic systems always interact with their envi-
in the driving fields, or from the vacuum fluctuations,
q ronment, the study on the geometric phase in open sys-
: tems become interesting. Garrison and Wright [8] were or from the background radiations. We calculate and
v
analyze the effect of dephasing of the driven system on
i the firsttotouchonthis issuebydescribingopensystem
X the geometric phase of the system, our discussions will
evolutionintermsofanon-HermitianHamiltonian. This
r is a pure state analysis, so it did not address the prob- distinguish between two kinds of evolution: (1)The en-
a
vironment undergoes an adiabatic evolution, and (2)it
lem of geometric phases for mixed states. Toward the
evolves as it may.
geometric phase for mixed states in open systems, the
approaches used involve solving the master equation of Let us consider a two-level system driven by a quan-
the system [9, 10, 11, 12, 13], employing a quantum tra- tized magnetic field and subjected to decoherence. The
jectoryanalysis[14,15]orKraussoperators[16],andthe decoherence process is described by the coupling of the
perturbativeexpansions[17,18]. Someinterestingresults two-level system to an environment of harmonic oscilla-
were achieved, briefly summarized as follows: nonhermi- tors with frequencies ω , the Hamiltonian governing
j
{ }
tian Hamiltonianlead to a modificationof Berry’sphase such a system reads [19]
[8, 17], stochastically evolving magnetic fields produce
both energy shift and broadening [18], phenomenolog-
H = H +H +H ,
s I e
ical weakly dissipative Liouvillians alter Berry’s phase
Ω
by introducing an imaginary correction [11] or lead to Hs = h¯ σz +h¯Ωa†a+h¯g(σ+a+σ a†),
2 −
damping andmixingofthe density matrixelements [12].
However,almostallthesestudiesareperformedfordissi- HI = (σ+a+σ a†) ¯hλj(b†j +bj),
pativesystems,andthusthe representationsareapplica- − Xj
ble for systems whose energy is not conserved. For open He = h¯ ωjb†jbj, (1)
systems with conserved energy (dephasing systems), the Xj
2
wherea, a arebosonoperatorsforthe drivingfield, σ , to consider the universe (system+environment) to un-
† +
σ , and σ are pauli operators for two relevant internal dergo an adiabatic evolution, in this case the states of
z
−
atomic levels |ei and |gi, and b†j, bj are the creation and the environment never evolve during the evolution. Be-
annihilation operators of the environment bosons. The cause the relaxation of the system due to its coupling
system hamiltonian H characterizes Jaynes-Cummings to the environment is independent of the dressed state
s
dynamics without neither energy relaxation nor phase indication n, the acquired geometric phase of the open
dephasing, while terms H and H describe a bosonic system are the same as in the case without the environ-
e I
environment and its coupling to the Jaynes-Cummings ment. The second situation is more practical, in which
system. The choice of the coupling between the system weassumetheenvironmentinitiallyisinitsgroundstate
and the environment determines effects of the environ- 0 = λj√n+1 c andevolvegovernbythe
ment. For example, the choice of the system operators j| jibj j|± ωj iB±,j
HQamiltonianQEq. (1). This initial state means that the
that do not change the quantum number of the driving
environment is initialized in the vacuum of modes b ,
j
field when they operate on the dressed state would re- { }
or in coherent state λj√n+1 c of modes B .
sult in relaxationswithin the dressedstates indicated by |± ωj iB±,j { ±,j}
Thegeometricphaseoftheuniverseinthissituationmay
the quantum number n, but not energy relaxations be-
be calculatedby removingthe accumulationof these dy-
tween states with different n. The system-environment
namical phase from the total phase, i.e.,
coupling chosen in our model is exactly such a choice,
and thus, we may rewrite the Hamiltonian Eq. (1) in T ∂
terms of the dressed states as φg(n)=arg Ψ(0)Ψ(T) +i dt Ψ(t) Ψ(t) , (5)
h | i Z h |∂t| i
0
H = Hn, it is easy to demonstrated that for closed systems φg(n)
Mn reducestotheAharonov-Anandanformulaforcyclicevo-
Hn = E+(n) +(n) +(n) lutions [20] and to the Berry phase for adiabatic and
| ih |
cyclic evolutions [1]. The initial state together with the
+ E−(n)|−(n)ih−(n)|+Xj ¯hωjb†jbj time evolution operator determine the path followed by
the system, in this sense the geometric phase might de-
+ √n+1( +(n) +(n)
| ih | pend on the initial condition. If we choose Ψ(0) =
| i
− |−(n)ih−(n)|)Xj ¯hλj(b†j +bj), (2) |st+ate(n,)ait⊗timQejt|ΛωthjjiecBu+n,jiv(eΛrsje=evoλlvje√snto+1) as the initial
wsthaetersewEi±th(nin)d=ica2tnio2+n1¯hnΩar±e ¯hg√n+1, and the dressed |Ψ(t)i=|+(n)i⊗Yj |Λωjje−iωjticB+,j. (6)
1
(n) = (g,n+1 e,n ). (3) In contrast with classically driving field, in this study
|± i √2 | i±| i the driving field is quantized. In order to generate a
phase change in the state of the field, we borrow the
Here n stand for the Fock states of the driving field.
{| i} idea in Ref.[21] to introduce the phase shift operator
The Hamiltonian H describes a driven harmonic oscil-
n U(ψ) = exp( iψa a) and adiabatically apply it to the
lator, the driving terms depend on the dressed states, Hamiltonian −of the†system. Changing ψ = Ω¯ t slowly
but they are independent of the dressed state indication ·
from 0 to 2π (the corresponding time from 0 to T = 2π)
n. With these properties, we may simplify the problem Ω¯
the geometric phase generated is calculated by Eq. (5)
and restrictour study within the dressedstates with the
as follows,
same indication n. The eigenstates of the Hamiltonian
Hn take the following form, φ+(n) = arg[ e−|Λωjj|2(1−e−iωjT)]
g
e (n) = +(n) n , Yj
| + i{nj} | i⊗Yj | jiB+,j + [ωjTe−|Λωjj|2 ∞ m(Λj/ωj)2m]
m!
|e−(n)i{nj} = |−(n)i⊗Yj |njiB−,j, (4) + Xγj (n), mX=0 (7)
+
where |njiB±,j denotes the Fock state for new environ- where γ+(n) = (2n +1)π is the Berry phase acquired
ment mode B = (b λ √n+1/ω ) (n) (n), when the universe remains unchanged. For a continuous
,j j j j
± ± | ± ih± |
id.ere.,ssBed±†,jsBta±te,js|.njiBT±h,ej =cornrejs|npjoinBd±i,njg, deeigpeennedninerggioens tahree sthpeecatbruomveoefxpthreesesinovnisroisnmreepnlatacledmboydeasn, tinhteegsuramlionvveorlvjining
¯hg√n+1+ j¯hωjnj and −¯hg√n+1+ j¯hωjnj, re- the spectral density with a cutoff frequency ωc
spectively. PWe distinguish between twoPsituations to ω
ρ(ω)=ε( )2,0 ω ω , (8)
study the geometric phase of the system, the first is λ ≤ ≤ c
ω
3
this spectrum density is of Ohmic type. Eq. (5) and the phases are not zero, which means that the vacuum
Eq.(8) together yield driving fieldintroduces a correctionin geometricphases,
this expression is relevant when systems are driven by
φ+(n)=γ (n)+εωc2(n+1)T + (n+1)ε[cos(ω T) 1]. fields with few photons. For weak system-environment
g + 2 T c − coupling [22], Eq. (7)( Eq. (11), in the same way) may
(9)
be expanded in powers of Λ /ω , up to the second order
j j
If we choose the eigenstate Ψ(0) = (n)
| i |− i⊗ j|− of Λj/ωj, Eq.(7) follows,
Λj c from another set of eigenstates Eq. (4)Qas the
ωjiB−,j
initial condition, the time evolution of the universe can
Λ
be expressed as, φg(n)=γ+(n)+ ωjT( j)2. (12)
ω
Xj j
Λ
|Φ(t)i=|−(n)i⊗ |− ωje−iωjticB−,j. (10)
Yj j The explanation of this result is very simple, as the
system-environmentcouplingisveryweak,themostcon-
Inthesameway,wecangetthegeometricphasepertain-
tribution to the geometric phase come from the system-
ing to this evolution loop,
drivingfieldcoupling. The samedependence ofφ (n)on
g
Λ (i.e., Λ2, no contribution proportional to Λ ) can be
φ−g(n)=φ+g(n)−γ+(n)+γ−(n), (11) fojund in Rjef. [17]. j
namely, the contributions from the system-environment Uptonow,wehavecalculatedthegeometricphasefor
coupling are the same for the both pathes. Here γ (n) theuniverse,explicitexpressionsforthegeometricphase
−
is the Berry phase attaining to the dressed state (n) were obtained, these expressions are of relevance for the
|− i
in the case without environment, it is easy to prove that caseinwhichsystemsarecoupledtoanenvironmentthat
it takes the same expression as γ (n). describesparameterfluctuations. Forinstance,imperfect
+
The last two terms in equation Eq. (9) result from dipole transitions between states g and e due to fluc-
| i | i
the system-environmentcouplings,they vanishwhenthe tuation of the driving laser intensity may be modelled
couplings tend to zero (in the equations, ω 0), thus by the coupling in Eq. (1), in this situation the environ-
c
→
the expressionsreturnto the geometricphasespresented ment and the driving field are the same. This is not the
in Ref. [21]. The path the system followed changes the case, however, when the environment is an independent
system-enviromentcoupling, and the environment is im- system (say, black body radiations), we have to trace
possible to returnto its initial state due to its huge vari- out the environment in order to calculate the dynami-
ety offreedom,the geometricphasethen depends onthe cal information for the system, thus the evolution of the
timeT whenwedrawoutphaseinformationfromthesys- system is no longer unitary. For non-unitary evolution,
tem. Intheclassicallimitn ,thecontributionsfrom the geometric phase can be calculated as follows. First,
→∞
the system-environmentcoupling tend to infinity caused solvetheeigenvalueproblemforthereduceddensityma-
by relative strong coupling between the system and the trix ρ(t) and obtain its eigenvalues ε (t) as well as the
k
driving field, hence it becomes undefined in the sense of corresponding eigenvectors ψ (t) ; secondly, substitute
k
| i
interferometry. It is interesting to note that for n = 0 ε (t) and ψ (t) into
k k
| i
T
Φg(n)=arg( εk(0)εk(T) ψk(0)ψk(T) e− 0 hψk(t)|∂/∂t|ψk(t)idt). (13)
h | i R
Xk p
Here,Φ (n)isthe geometricphaseforthesystemunder- density matrix ρ(t),
g
going non-unitary evolution [23]. The geometric phase
Eq. (13) is gauge invariant and can be reduced to the c 2 c c F(t)
well-knownresultsinthe unitaryevolution,thusitis ex- ρ(t)=(cid:18)c+c|∗+F|∗(t) ∗+c− 2 (cid:19), (14)
perimentally testable. Now, we exploit the expression − | −|
to calculate the geometric phase for the driven two-level
where the initial state of (c + (n) + c (n) )
system. To this aim, we first write down the reduced +| i −| − i ⊗
0 is assumed for the universe, and F(t) =
j| jibj
eQxp[ i(E (n) E (n))t] exp[ η (t)] with η (t)=
−h¯ + − − · − j j j
4 Λj 2(1 cosω t). SimplealgebraPgivestheeigenvalues
|h¯ωj| − j
4
and the corresponding eigenvectors of ρ(t),
1 1
ε±(t) = 2 ± 2q(|c+|2−|c−|2)2+4|c∗+c−F(t)|2, [[21]] YM..AVh.aBroernroyv, PanrodcD. R. B.Soohcm.,LPonhydso.nRAev3.9121,54,54(8159(8149)5.9).
ψ (t) = X (t) +(n) +Y (t) (n) , (15)
| ± i ± | i ± |− i [3] Geometric phase in physics, Edited by A. Shapere and
F. Wilczek ( World Scientific,Singapore, 1989).
whereX (t)=c c F(t)/ c c F(t)2+(ε (t) c 2),
± ∗+ − | ∗+ − | ± −| +| [4] P.ZanardiandM.Rasetti,Phys.Lett.A264,94(1999).
and Y (t) = 1 X (t)p2. Eq. (15) and Eq.(13) to- [5] J.A.Jones, V.Vedral,A.Ekert,andG.Castagnoli, Na-
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[12] D.GamlielandJ.H.Freed,Phys.Rev.A39,3238(1989).
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Summarizing, the geometric phases for a dephasing V. Vedral, Phys. Rev. Lett. 90,160402(2003); ibid 92,
020402(2004).
system (open system) have presented and discussed.
[16] K. P. Marzlin, S. Ghose, and B. C. Sanders, e-
The open system has been demonstrated by a driven
print:quant-ph/0405052(Phys. Rev. Lett., to be pub-
two-level system coupling to an environment of har-
lished).
monic oscillators. The results show that there are no [17] R. S. Whitney, and Y. Gefen, Phys. Rev. Lett. 90,
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in the phase is path dependent when the constraint on
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1593(1987).
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states with the open system decohering to its pointer [22] Theweak couplingmeansthatΛj/ωj <<1,thesystem-
states, it is a reflection of the decoherence in geometric environmentcouplings fall in thisregime when thereare
phases. few quanta in the driving field, and λj is adjusted small
Comments from Dr. Peter Marzlin and simulating with respect to ωj. This is possible for the engineered
environment in trapped ion system, see for example, C.
discussions with Dr. Robert Whitney are gratefully J. Myatt, et al.,Nature 403, 1269(2000).
acknowledged. This work was supported by EYTP of [23] D. M. Tong, E. Sj¨oqvist, L. C. Kwek, C. H. Oh, Phys.
M.O.E, and NSF of China Project No. 10305002. Rev. Lett.93, 080405 (2004).