PRL 96, 150403 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending21 APRIL 2006

Geometric Phase Induced by a Cyclically Evolving Squeezed Vacuum Reservoir

Angelo Carollo,1,2 G. Massimo Palma,3,4 Artur Łozinski,3 Marcelo Franca Santos,5 and Vlatko Vedral61Centre for Quantum Computation, DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

2Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, Austria3NEST and Dipartimento Tecnologie dell’Informazione, Universita di Milano, via Bramante 65, I-26013 Crema (CR), Italy

4Dipartimento di Scienze Fisiche ed Astronomiche, Universita di Palermo, via Archirafi 36, I-90123 Palermo, Italy5University of Minas Gerais, Belo Orizonte, Brazil

6School of Physics and Astronomy, University of Leeds, LS2 9JT, United Kingdom(Received 13 July 2005; published 21 April 2006)

0031-9007=

We propose a new way to generate an observable geometric phase by means of a completely incoherentphenomenon. We show how to imprint a geometric phase to a system by adiabatically manipulating theenvironment with which it interacts. As a specific scheme, we analyze a multilevel atom interacting with abroadband squeezed vacuum bosonic bath. As the squeezing parameters are smoothly changed in timealong a closed loop, the ground state of the system acquires a geometric phase. We also propose a schemeto measure such a geometric phase by means of a suitable polarization detection.

DOI: 10.1103/PhysRevLett.96.150403 PACS numbers: 03.65.Vf, 03.67.a, 05.30.d

Whenever a pure quantum state undergoes a paralleltransport along a closed path, it gathers information onthe geometric structure of the Hilbert space in which it lies.In this Letter, we will show that a possible way to generatesuch a parallel transport is by way of an irreversiblequantum evolution. In several models of interaction withthe environment, there are some ‘‘protected’’ subspaces,such as the decoherence-free subspaces, which are leftunaffected [1]. States lying in these subspaces are station-ary; i.e., they do not evolve in time. A typical example isthe ground state of an atomic system, which, trivially,remains unaffected by the coupling with the electromag-netic field. However, there are situations in which theinteraction between a system and an engineered environ-ment can generate nontrivial ground states [2–6]. Forinstance, when a group of atoms collectively interactswith a broadband squeezed vacuum, the highly nonclass-ical correlations which are present in the field are trans-ferred to the atomic system, which relaxes in a complexpure equilibrium state. In such a scenario, the control overthe engineered reservoir allows an indirect control on thestate of the system to which it is coupled [7]. Of particularinterest is the possibility to change in time the reservoirparameters in such a way that the protected system sub-space evolves in a controlled fashion. Here we show that, ifthis change in time is made slowly enough, a state lying insuch a subspace evolves coherently and acquires informa-tion about the geometry of the space explored.

As an explicit example, we consider a suitable multi-level atomic system interacting with a broadband squeezedvacuum. To be more specific, let us consider first a three-level atom whose interaction with an electromagnetic fieldin the rotating wave approximation is described by thefollowing Hamiltonian:

HHSZay!a!d!

Zg!Sya!H:c:d!;

06=96(15)=150403(4)$23.00 15040

where HS P1k1 kjkihkj is the free atomic

Hamiltonian, S j1ih0j j0ih1j is the atomic operatordescribing the absorption of an excitation, and a! is theannihilation operator of the mode with frequency ! (@ 1). The field, which we treat as a reservoir, is assumed to bein a broadband squeezed vacuum state. In mathematicalterms, this is obtained from the ordinary field vacuum stateby means of the unitary operator K

jvacisq Kjvaci; (1)

where

K exp1

2

Zay!ay! H:c:d!

(2)

is a multimode squeezing transformation [1,3], which cor-relates symmetrical pairs of modes around the carrierfrequency , and ei’r is the squeezing parameter,whose polar coordinates ’ 2 f0 . . . 2g and r > 0 arecalled the phase and amplitude of the squeezing,respectively.

The use of the Born Markov approximation, justified bythe broadband nature of the field, leads to the followingmaster equation for the atomic degrees of freedom [1,3]:

ddt

2fRyR RyR 2RRyg; (3)

where 2jgj2, and

R S coshr ei’Sy sinhr: (4)

From (4) follows that the state

j DFi cj 1i ei’sj1i; (5)

with cr coshr=cosh2rp

and sr sinhr=cosh2rp

,satisfies Rj DFi 0. In other words, this state is

3-1 © 2006 The American Physical Society

PRL 96, 150403 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending21 APRIL 2006

unaffected by the environment; i.e., it is decoherence-free.Moreover, j i represents the new ground state, as allthe other states of the atomic system relax to it.

As anticipated, the key idea is to smoothly change thesqueezing parameter of the field in order to ‘‘adiabatically’’drag a state initially prepared in j DF0i into j DFti,where t is the time dependent squeezing parameter. Wewill show the existence of an ‘‘adiabatic’’ limit such thatthe transition probability of j DFi to the orthogonalsubspace vanishes as the rate of change of becomessufficiently small. Furthermore, we will show that, after acyclic evolution of , the state j DFi acquires a geometricphase. It is worth stressing that this procedure, althoughreminiscent of the usual adiabatic evolution, is a differentphysical phenomenon. The usual adiabatic approximationrefers to a coherent evolution, obtained by tuning theparameters of the system Hamiltonian, while the ‘‘steeringprocess’’ discussed here is achieved by manipulating theenvironment. The essential difference is that in the lattercase the system state can be adiabatically controlled en-tirely by means of an incoherent phenomenon and noHamiltonian term contributes to its time evolution. Toshow how this incoherent adiabatic steering process cantake place, consider the time dependent version of Eq. (3),where Rt is explicitly dependent on time through t. Itis useful to express the equation of motion in the referenceframe where DF is time independent. To this end, considerthe following unitary transformation:

O crei’=2 0 srei’=2

0 1 0srei’=2 0 crei’=2

0B@

1CA; (6)

from the basis j1i, j0i, j1i to the time dependent basis j~1i,j~0i, j~1i, where j DFi coincides with j~1i Under thischange of frame, the equation of motion becomes

d~dt

~

2 ~Ry ~R ~~ ~Ry ~R 2 ~R ~ ~Ry iG; ~; (7)

where, in this new frame, ~ OtOyt, ~R

OtRtOyt=cosh2rp

, ~ cosh2r, and G idO=dtOy is a Hamiltonian term arising from the changeof picture. Moreover, in this frame the Lindbladian term~Ry ~R assumes a simple diagonal form:

~R y ~R j~1ih~1j j~0ih~0j: (8)

The main advantage of this transformation is that itallows one to formulate clearly the adiabatic condition,since the rate of change of the environment parameters iscontained in the operator G. The limit that we are inter-ested in is the one in which the dominant contribution inEq. (7) comes from the incoherent terms, i.e., jGj ~.

An interesting case is the one in which the squeezingamplitude is kept constant while its phase is slowlychanged from 0 to 2. This adiabatic evolution can be

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easily achieved by tuning, for example, the carrier fre-quency 2 of the squeezed state slightly off resonancefrom the two photon transition j1i $ j1i. By introducingthis detuning (assuming ), the master equationobtained has the form of Eqs. (3) and (4), where ’ isreplaced by ’t ’0 t. Hence, a sufficiently smallvalue of determines the required adiabatic evolution.Under this condition, the operator G assumes the form

G _’t2

0 0 0 0 0

0@

1A; (9)

where 1cosh2r and sinh2r

cosh2r . We show that, when _’is small enough, the state j 1i j DFi is adiabaticallydecoupled from its orthogonal subspace and a cyclic evo-lution in ’ results in a geometric phase acquired by j DFidepending only, in this case, on the amount of squeezing r.Note, however, that, since the steering process is essen-tially incoherent, any phase information acquired by asuperposition of DF and a state belonging to the orthogo-nal subspace is inevitably lost, as the latter is subject todecoherence. The only way to retain such information is toconsider an auxiliary level jai, unaffected by the noise,playing the role of a reference state for an interferometricmeasurement. For simplicity, assume that jai is unaffectedby the environment during the whole evolution and, hence,is time independent. As a consequence, the action of theunitary transformation O on jai is trivial, and Eq. (7)remains essentially unchanged.

The whole information about the geometrical phase andthe coherence retained by the system during its evolution isthenrecorded in the phase and amplitude of the density ma-trix term a h1j~jai, whose evolution is describedby the following set of coupled differential equations:

_a ihajGj~1i ia a_’2;

_a ~

2a ih~1jGjai

1

2~ i _’a ia

_’2;

where a h1j~jai. Assume that initially the excitedstates j1i and j0i of the system are not populated; hence,a0 0, and the coherence a evolves as

at a01

i _’et

i _’et;

where ~=4 12

~2=4 i~ _’ _’2

q.

In the limit _’, we obtain for the coherence a

at a01 ei’=2 _’t~t

ei’=2 _’t~=212t; (10)

3-2

2Ω

a

a

1

0

-1

a 2Ωa1

a1

a2

a2

1

0

-1

1

-1

(a) (b)

FIG. 1. (a) Schematic representation of the four systems con-sidered. The energy gap between states j1i and j0i andbetween j0i and j1i is . The transitions between these levelsare coupled to the modes a! of the reservoir. The referencestate jai is decoupled from the reservoir. (b) Five-level system,transitions 1$ 0 and 0$ 1 are coupled to modes a1! and10 $ 0 and 0$ 10 are coupled to modes a2! of the reser-voir.

PRL 96, 150403 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending21 APRIL 2006

where ’ 2=2 _’=~2. We are interested in a cyclicevolution, corresponding to T 2= _’. By retainingonly the leading terms in _’=, the total evolution at timeT is given by

aT a0ei2 _’=; (11)

where we have substituted ~ with =. Finally, going backto the original frame by means ofOyT, the correspondingcoherence at h DFjtjai is given by:

aT a0ei12 _’=: (12)

For example, a state initially prepared in j ~ 0i 1=

2pj DF0i jai, after closing the loop, evolves into

T12

_’

j ~ Tih ~ Tj

2_’~j DFih DFj; (13)

where j ~ Ti 1=2pei1j DF0i jai.

It is clear from this expression that, in the limit _’= 1, the dominant contribution to the time evolution

is just a phase factor ei, with 1 . Thisproves that, in the adiabatic approximation, the systempreserves its coherence. In fact, according to Eq. (12),the amplitude damping of a occurs only when we takeinto account the first order contribution in , which showsan exponential decay rate of the order of 2 _’=. Thisproves that for small _’ the system admits an adiabaticlimit, in which the subspace HDFt spanned by j DFtiand jai is adiabatically decoupled from its orthogonalsubspace H?t. For this reason, HDFt is decoupledfrom the effects of the decoherence, which affect onlystates lying in its orthogonal subspace.

Within this approximation, then, a state prepared in thespace HDF0 is adiabatically transported rigidly inside theevolving subspace HDFt. As a result of this adiabaticsteering, when the system is brought back to its initialconfiguration, the coherence a acquires a phase 1 . This phase can be interpreted as the geometricphase accumulated by the state j DFti. By using thecanonical formula for the Berry phase, it easy to see thatthe geometric phase of j DFti is given by

g iIh DFjdj DFi i

Z 2

0h DFj

dd’j DFid’

1 :

As expected, the value of depends only on the squeezingand vanishes as the squeezing tends to zero. Moreover,notice that the phase is purely geometrical; i.e., there isno dynamical contribution arising from an existingHamiltonian, since, in the absence of any steering process,the states inside HDF have a trivial dynamics. This makesthe measurement of this phase a relatively easy task. Usual

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procedures to measure geometric phases make use ofsuitably designed techniques to eliminate dynamical phasecontributions, such as spin-echo [8] or parallel transportconditions [9]. In this setup, the geometric phase is the onlycontribution to the phase accumulated by j DFi, and,hence, it is straightforward to measure by a suitable inter-ferometric setup.

A simple scheme to measure the geometric phase ob-tained by such a steering process can be realized with asimple variation of our system. Let us consider the five-level atomic system shown in Fig. 1(b). It essentiallyconsists of two replicas of the three-level system discussedabove, with the level j0i in common. The important ingre-dient is that transitions j0i $ j1i and j1i $ j0i arecoupled with modes of the reservoir which are differ-ent from those coupled to the transitions j0i $ j10i andj10i $ j0i. A simple way to achieve this is to choose, forexample, polarization selective transitions, say, left-circularly polarized modes for the former transitions andright-circularly polarized for the latter ones. The completeHamiltonian of such a system is

H HS Xi1;2

Zayi !ai!d!

Xi1;2

Zgi!S

yi ai! H:c:d!; (14)

whereHS P1k1 kjkihkj jk

0ihk0j, S1 j1ih0j j0ih1j and S2 j10ih0j j0ih10j, and ai! is the anni-hilation operator of the mode with the energy ! andpolarization i 2 f1; 2g. Assume broadband squeezed vac-uum states for the set of modes a1! and modes a2!with different squeezing parameters 1 r1ei’1 and 2 r2ei’2 :

jvac1; 2isq K11K22jvaci; (15)

where Kii are the analogs of the operator (2) acting on

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PRL 96, 150403 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending21 APRIL 2006

the set of modes ai. Under the same assumptions whichlead to Eq. (3), we obtain the master equation:

ddt

Xi

i2fRyi Ri R

yi Ri 2RiR

yi g; (16)

where i 2jgij2 and Rii Si coshri

ei’iSyi sinhri. This system admits a two-dimensionaldecoherent-free subspace, spanned by states j 1i andj 2i whose definition is the analog of state j DFi ofEq. (5). We assume again time dependent squeezing pa-rameters ti, and again we examine the time dependence ofthe system in a rotating frame, i.e., a frame where the statej iti appears stationary. This leads to the following mas-ter equation for the five-level system in the rotating frame:

d~dt

Xi

~i2 ~Ryi ~Ri~ ~ ~Ryi ~Ri 2 ~Ri ~ ~Ryi

iXi

Gi; ~; (17)

where Gi idO=diOy _i, Ot being the unitary trans-formation producing the change of frame. Assume again,for simplicity, that the parameters r1 and r2 are keptconstant and that ’1 ’2 ’. Under this assumption,the master equation can be exactly solved. The solution isanalogous to the one obtained for the system previouslyanalyzed. Suppose that the system is initially prepared in acoherent superposition of state j 10

1i and j 202i, for

example: j 0i 1=2pj 1

01i j 2

02i. At a

later time, one has

1 2t

1

2exp

i2 1

_’2

2

1

2~1

0222~2

_’2

t;

(18)

with i 1= cosh2ri and i sinh2ri= cosh2ri.When the parameter ’ closes a loop, at t T 2= _’,the coherence has gained a phase

i2 1 2 1; (19)

which is the difference between the geometric phasesi 1 i acquired by the states j ii, respectively. As inthe previous scheme, the visibility is reduced by a factorwhich is linear in the ‘‘adiabatic parameters’’ _’=i, whichguarantees the existence of the adiabatic limit. The advan-tage of this modified scheme is that the value of thegeometric phases can be readily measured from the polar-ization of the light emitted when the system relaxes. Infact, if the value of the squeezing parameters ri is switchedsuddenly to zero, the states j ii are no longer decoherence-free and decay to a superposition of the ground states j1iand j10i. This dissipation process is accompanied by twophoton emissions into the reservoir. Because of the struc-ture of the interaction (14) with the reservoir, the photon

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emitted due to the transitions j1i ! j0i and j10i ! j0i ispolarized according to the geometric phase accumulatedbetween j 1i and j 2i. For example, if a1! and a2! areright- and left-circularly polarized modes, respectively, thefirst dissipation process produces the linearly polarizedphoton:

j 1i ei’1’2j 2i ! jRi ei’1’2jLi: (20)

The detection of the polarization of the emitted photonmakes possible a direct measurement of the geometricphase.

We have presented a scheme to generate a geometricphase via a completely incoherent control procedure. Thisscheme is conceptually different from the usual coherentadiabatic control. The latter is realized through a smoothevolution of suitable Hamiltonians, whereas here the adia-batic steering is the effect of an externally controlledenvironment. The phase generated is purely geometricaland, therefore, experimentally detectable without resortingto techniques for the elimination of dynamical contribu-tions. Because of its very nature, this scheme is immunefrom unwanted environmental effects. Moreover, like anygeometric effects, it presents an inherent degree of robust-ness against uncertainties in the control parameters.

This work was supported in part by the EU under grantIST-TOPQIP, ‘‘Topological Quantum InformationProcessing’’ (Contract No. IST-2001-39215). V. V. alsoacknowledges support from EPSRC and the BritishCouncil in Austria. A. C. acknowledges support fromMarie Curie RNT Project CONQUEST.

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