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PHYSICAL REVIEW A, VOLUME 63, 032106

Stability of quantum states under dissipation

J. C. Retamal1 and N. Zagury21Departamento de Fı´sica, Universidad de Santiago de Chile, Casilla, 307 Santiago-2, Chile

2Instituto de Fı´sica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro 21945-970, Rio de Janeiro, B~Received 24 January 2000; revised manuscript received 25 August 2000; published 14 February 2001!

Using a simple stability argument for the entropy production we find the conditions for a state to remain pureunder certain dissipative dynamics. We also calculate the time scales for the onset of decoherence when theseconditions are not satisfied.

DOI: 10.1103/PhysRevA.63.032106 PACS number~s!: 03.65.Ta

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The understanding of the boundary between the micworld and the macroworld has remained a central problemquantum mechanics@1#. In the microscopic world a systemcan exist in a superposition of different quantum statthereby exhibiting interference. On the other hand, theseperpositions apparently do not manifest themselves inclassical world although the existence of quantum coherein a closed system should occur. This problem was alrediscussed by Schro¨dinger @2# in a famous paper in 1935.

In the last few years there have been important theoreinsights that the interaction of a large system with its enronment plays a major role in the disappearance of quancoherence in the classical domain@3#. Recent progress inmicrowave cavities and electromagnetic traps for ions ledexperimental access to study this fascinating problem inmicroscopic and mesoscopic level. The monitoring of deherence due to the effects of dissipation in a superpositiotwo coherent states@4# was recently reported in a cavitQED microwave experiment@5#. Also the generation of controlled superposition of states of the vibrational motion of tcenter of mass of the ion@6#, as well as entangled statesinternal states of two ions@7,8# and four ions@8# have al-ready been achieved. The use of laser action in these sysopens the possibility of engineering several kinds of dissitive quantum dynamics@9# that may be interpreted as duedifferent kinds of interactions with reservoirs. This engineing of reservoirs allows the study of the process of quantdecoherence under several situations. For example, Turcet al. @10# have presented very recently an experimenstudy of decoherence in a high-temperature amplitude revoir, in a zero-temperature amplitude reservoir and in a hitemperature phase reservoir.

In this paper we will be mostly concerned with some fudamental questions that are still without a complete answ~a! Which are the states that maintain its coherence~remainpure states! under the time evolution of an open system?~b!Given an initial pure state, what is the scale of time whthe decoherence process occurs?

In Sec. I we present the conditions for stability unddissipation. In Sec. II we discuss the onset of decohereFinally, in Sec. III we make some general consideratioabout the measure of decoherence.

I. STABLE STATES

The stability against decoherence may be understoothe process where quantum coherence is preserved alon

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time evolution. In this sense we may say that an initial puquantum state, described by a density operatorr, is ‘‘stable’’if tr r251 for all times. A practical way to measure decherence or the stability of an initial pure state is to use‘‘idempotency defect’’ or ‘‘the first-order entropy,’’

s~ t !512tr r2. ~1!

A classical example of stability under dissipation is givby the time evolution of an initial coherent state of a quatum field mode inside a cavity at zero temperature. InMarkov approximation, the dissipation may be describthrough the master equation@11#:

r5g

2~2crc†2c†cr2rc†c!, ~2!

wherer is the density operator in the interaction represention, c andc† are the annihilation and creation operatorsthe field cavity mode, andg is the field-decay rate. Consideas the initial state, a pure coherent state vector

ua&coh5e2uau2/2(an

An!un&, ~3!

wherea is a complex number andun& is the number state. Iis easy to show that the state at a later timet is still a coher-ent state given byuae2gt/2&coh. That is, coherent states remain pure states and we may say that they are ‘‘stabunder the dynamics given by Eq.~2!. In fact, as we will showbelow, they are the only class of states with this propeThis is not true anymore if the cavity is in equilibrium wita reservoir at a fixed temperatureT.0. In this case the dy-namics is described by

r5g

2~nth11!~2crc†2c†cr2rc†c!

1g

2nth~2c†rc2cc†r2rcc†!, ~4!

where nth is the equilibrium mean photon number at temperatureT. ForTÞ0, it can be shown that there is no ‘‘stabstate’’ and that the time of decoherence scales as 1/(gnth ).

The aim of the present work is to analyze the stabilityquantum coherence and the time scales in which a gi

©2001 The American Physical Society06-1

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J. C. RETAMAL AND N. ZAGURY PHYSICAL REVIEW A 63 032106

quantum state becomes unstable. It is possible to showder the assumptions of Markovian dynamics, the comppositivity and initial decoupling between the system andreservoir@12#, and the evolution of the system in the inteaction picture is described by a Lindblad form

r5(i j

g i j

2~2 f ir f j

†2 f i†f jr2r f i

†f j !. ~5!

Herer is the reduced density operator in the interaction rresentation andf i are operators that act on the system. Tstructure of the above equation is particularly interestingthe case of ions in a trap. As shown by Poyatos, Cirac,Zoller @9#, different reservoirs, corresponding to sevekinds of combinations of annihilation operator,a and cre-ation operatorsa† associated to the vibrational motion, cabe modeled by properly adjusting the interaction of the iowith external laser fields. They have shown how to engindissipative dynamics of the form of Eq.~5! for a single op-eratorf, such asak,(a2a)(a2b),ma1na†,a†a, etc.

The experimental access to generate different dissipadynamics, through laser pulses, should lead to a contromonitoring of the disappearance of quantum coherence wthe system is in contact with different reservoirs. An impotant example would be the study of decoherence in a suposition of two states with a Wigner distribution with peawell separated in phase space. This study is related todisappearance of coherence in macroscopic objects. Mvated by these ideas we will consider in our analysis, withloss of generality, that the dynamics is governed by Eq.~5!for a single operatorf, which may be any linear combinatioof the operatorsaj ,(a†)m, (a†a)k, andI.

We first show that the only states that could remain punder the dynamics described by Eq.~5! are the eigenstateof the operatorf. Consider the rate of change in the idemptency defects(t) by calculating its first derivative, sometimes called the ‘‘entropy production’’

s~ t !52g@ tr f †f r t22tr f †r t f r t#. ~6!

If at an arbitrary timet0 the system is in a pure statuC(t0)&, this means thatr t0

2 5r t05uC(t0)&^C(t0)u and we

may write

s~ t !52g~ tr f †f r t2tr f †r ttr f r t!52g~^ f †f &2^ f †&^ f &!.~7!

Therefore,s(t0)50 only if uC(t0)& is an eigenstate off. Weimmediately exclude the possibility of dynamical stabiliwhen f does not have an eigenstate as, for example, whef5(a†)k. In these cases, which describe gain in the systthe entropy always increases. Kimet al. @13# analyzed thecoherence loss in terms of a perturbation expansion ofreduced density operator and found that for any initial ftorized density operator,s(t) increases at least ast2, whichseems to be in contradiction with the above result. The Liblad equation, however, is obtained after a coarse grainrivative over intervals of time much greater than the corretion time of the reservoir.

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Trivial examples of stable states are the eigenstatesHermitian operators and normalizable eigenstates assocwith the eigenvalue zero of any operatorf @9,14#. Theseeigenstates remain constant in time as can be easily shby evaluating directly the right-hand side of Eq.~5! to findthat r50. As examples we mention the Fock statesun& iff 5a†a and the states

(n50

k21

cnue2p in/kb&coh, ~8!

wherecn are arbitrary complex numbers iff 5ak2b.When f 5a2l it is easy to show that if the initial state i

the coherent stateua1l&coh, it evolves in time to the coherent stateuae2gt/21l&coh, that is, an eigenstate ofa2l witheigenvalueae2gt/2. In fact, for f 5a2l, we have

r5g

2D~l!†~2ara†2a†ar2 ra†a!D~l! ~9!

where D(l)5el* a2la†is a displacement operator andr

5D(l)†rD(l). As l does not depend ont, it is clear that

r5D~l!uae2g/2t&coh ae2g/2tucohD~l!†

5uae2g/2t1l&coh ae2g/2t1lucoh ~10!

is the solution of Eq.~9! since uae2g/2t&coh ae2g/2tucoh is asolution of Eq.~2!. Another example is the eigenstates of toperator S(reiu)aS†(reiu)5cosh(r)a1eiu sinh(r)a†. If theinitial state is an eigenstate with eigenvaluea

S~z!ua&coh5e(1/2)(z* a22za†2)ua&coh, ~11!

where z5reiu, it is easy to show that it evolves to thsqueezed stateS(z)uae2g/2t&coh using a similar procedure ain the above case for the operatora2l.

II. THE ONSET OF INSTABILITIES

To find out if the eigenstates off are the only states tharemain pure under the dynamics described by Eq.~5! andsimultaneously the time scale of the onset of instabilitieswill consider the change ins(t) by calculating its deriva-tives.

Assume that we have an initial state that is an eigensof f, so that the entropys(0)50 and the entropy productions(0)50. To show thats(t)50 for any time, the derivativesof any order should be zero~we are assuming thats(t) is ananalytical function oft). Let us denote bysn andrn the nthderivative ofs(t) and r(t) calculated att50, respectively.The first few derivatives of the entropy can be easily writtas

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STABILITY OF QUANTUM STATES UNDER DISSIPATION PHYSICAL REVIEW A63 032106

s1522 trrr1 ,

s2522 tr~r121rr2!,

s3522 tr~3r1r21rr3!,

s4522 tr~3r2214r1r31rr4!. ~12!

The explicit form of these derivatives as a function of toperatorsr, f and f † is, in general, very complicated. Belowwe give the expressions for the first two derivatives,

s1522g tr~r f †r f 2r2f †f !,

s252g2 tr@4r f †f r f †f 12r2~ f †f !223r f r f †2f

23r f †r f †f 212r f 2r f †22r f †r f f †f 2r f r f †f f †#.

~13!

Using the fact that the initial state is a pure state and iseigenstate of f , we set r2(0)5r(0) and f uC f(t0)&5buC f(t0)& in Eq. ~13!. In this case the first and seconderivatives are zero and the third derivative may be obtaiafter a lengthy but straightforward calculation

s35g3ubu2~^A2&2^A&2!, ~14!

whereA5@ f , f †#. That is, the third derivative of the idempotency defects will vanish at t50 either if b50 or if thevarianceDA25^A2&2^A&2 of the commutatorA is zero. IfbÞ0 the initial state is a simultaneous eigenfunction ofAand f. A sufficient condition for this to be possible is th@A, f #50, which happens either whenf is Hermitian inwhich caseA50, or whenf 5ma1na†1lI , which impliesthatA52 Re(mn* )I . It is easy to show that for all ’’stable’’states discussed above we have eitherb50 or DA250 as itshould be, since all derivativessn(0) are zero. For otheoperatorsf we do not have stable states with eigenvaluesbÞ0. However if we start with an initial eigenstate of onethese operators, the idempotency defect increases, for st, at most ast3,

s~ t !'1

3! S t

tdD 3

. ~15!

We should remark that this short-time behavior of the linentropy is a nontrivial property of the Lindblad structure thwe are concerned with in this work. The value oftd5@1/s3(0)#1/3 gives a time scale for the onset of decoherenfor states that initially are eigenstates off. For example, letus consider the case off 5a2. The eigenstates ofa2 belong-ing to the eigenvalueb5a2 are any linear combination oua& and u2a&:

uc~0!&5c1ua&coh1c2u2a&coh, ~16!

where

uc1u21uc2u212 Re~c1c2* !e22uau251. ~17!

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In this case the third derivative ofs(t) at t50 is given by

s3516g3uau6@124 Re~c1* c2!e22uau218uau2Re~c1* c2!

3e22uau2$122 Re~c1* c2!e22uau2%#, ~18!

so that foruau2@1,

td5S 1

16D1/3 1

guau2. ~19!

Thus for largeuau the values oftd for any combination ofua& andu2a& coincide. In particular this shows that for thkind of reservoir,td is the same for a coherent stateua&coh(c250) and for an even or an odd coherent state,

ua&coh6u2a&coh

~262e22uau2!1/2. ~20!

This means that for times smaller thantd the entropy in-creases in the same way regardless of the distance betthe two ‘‘hills’’ in the Wigner representation of these stateIt does not mean, however, that for longer times the behaof s(t) will be the same. In Fig. 1 we shows(t) for uau53 in the cases whereuc(0)&5ua& and uc(0)&5c(ua&1u2a&). We observe that the linear entropies differ at lartimes, the even coherent state evolving to the vacuum swhile the coherent state evolves to a statistical mixture ofvacuum stateu0& and the stateu1&. This may be explained bythe fact that the even coherent state is a linear combinatioFock states with even numbers of quanta while the cohestateua& is a linear combination of Fock states with even aodd numbers of quanta. As the system can exchangeeven numbers of quanta with the reservoir, the even cohestate will evolve to the vacuum state, while the coherent swill evolve to a statistical mixture with almost equal weighof n51 and n50 Fock states. An example of a physicsystem where this time of decoherence could be measuran ion in a trap in contact with an engineered reservoir@9#with f 5a2. For example, for an ion oscillating with a frequencyn/2p'(20230) MHz and subject to side band transitions such that the Lamb-Dicke parameter ish'0.15, it iseasy to find attainable values of laser intensities suchg'(1988219 880) s21. In this case we would get, foruau53, td'(2.2222) ms. These times are smaller than the iverse of the decay rate due to present experimental cstraints, 0.17 ms of the ‘‘natural’’ reservoir@10# and shouldbe easily observable.

The time scale for decoherence of any initial eigensta

(n50

k21

cnue2p in/ka&coh ~21!

of f 5ak can also be easily found whenuau@1. In this case

A5 (l 50

k21k! 2

l ! 2~k2l !!a†l al . ~22!

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J. C. RETAMAL AND N. ZAGURY PHYSICAL REVIEW A 63 032106

FIG. 1. Linear entropy as a function of thadimensional timegt for f 5a2, for a coherentstate and a Schrodinger cat state of amplitudea53.

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Therefore, to the leading order inuau we have, from Eq.~14!,

s3'g3uau2kk4~^a†(k21)a(k21)a†(k21)a(k21)&

2^a†(k21)a(k21)&2!. ~23!

From this expression it is not difficult to show that foruau2@1

td'1

$k4~k21!2%1/3

1

guau2(k21), ~24!

which shows thattd has a behavior that depends on the nolinearity of the interaction. The greater the nonlinearity, tsmaller is the decoherence time.

Our results above refer to a situation where initially whave a single eigenstate of the operatorf. The decoherencetime scale may be completely different if the initial state islinear combination of two eigenstates of the operatorf be-longing to different eigenvalues. In this cases(0)Þ0 and thedecoherence time scale is given byt51/s(0). As an ex-ample considerf 5ak and an initial stateuc(0)&5N(ua&coh1ub&coh). In this case it is easy to show that

s~0!'guak2bku2

2, ~25!

when ua2bu2@1. Until now we were mostly concernewith operators that would have normalizable eigenstatesthat there would be a chance of obtaining stable states.

It is worth mentioning that for other operators we malso obtain decoherence rates easily. Let us consider athe finite temperature reservoir of Eq.~4!. The rate of linearentropy change for an initial coherent stateuz& going to athermal state can be obtained by directly calculation usEq. ~12!

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s~0!52gn. ~26!

We can also obtain this time scale from the exact analytresult fors(t). This can be obtained by considering the timevolution of a coherent state in a thermal reservoir@15#

r t5D~zt!r th~ nt!D†~zt!, ~27!

wherezt5ze2gt/2 andr th(nt) represents a thermal state wiaverage number of thermal excitationsnt5n(12e2gt).From this result we easily obtain

s~ t !52n~12e2gt!

112n~12e2gt!. ~28!

This expression shows that, ifn@1/2, we may define twotime scales, one associated with the dissipation processg21

and another associated with decoherence (2ng)21!g21.For n!1/2 the decoherence and dissipation scales beccomparable.

In the case of an even coherent state Eq.~20! evolvingunder the same dynamics, we obtain for the decay rate oflinear entropy

s~0!52g~2n11!uau2~12e22uau2!

~11e22uau2!12gn. ~29!

The calculation of these decoherence scales has beenformed in previous works@16,17#, by considering the quasiprobability distributions. Here we show that a consideraamount of information is obtained just considering the bhavior of s(t) around the initial time.

Another interesting situation studied in the literature is tsqueezed vacuum state reservoir. In this case the Lindform is given by@18#

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STABILITY OF QUANTUM STATES UNDER DISSIPATION PHYSICAL REVIEW A63 032106

r5g

2~N11!~2ara†2a†ar2ra†a!

1g

2N~2a†ra2aa†r2raa†!

1g

2M ~2a†ra†2a†a†r2ra†a†!

1g

2M* ~2ara2aar2raa!, ~30!

whereM and N are the squeezed reservoir parameters@18#

satisfying the inequalityuM u2<N(N11). In this case it isnot difficult to prove that

st52g~2N11!~ tr a†ar22tr ara†r!

22gNst12gN12gM ~ tr a2r22tr arar!

12gM* ~ tr a† 2r22tr a†ra†r!.

For a coherent state the last two terms are zero so thatst hasa behavior similar to that of a thermal reservoir, the thermnumber of photon being replaced by the excitation numbeNof the squeezed reservoir. For an initial even coherent sthe entropy production is given by

s052guau2F ~2N11!~12e22uau2!

~11e22uau2!12 ReM G12gn.

~31!

For M5uM ueip anduau2@1, this last equation may be written as

s052guau2$2~N2M !11%12gn. ~32!

which coincides with the result obtained in Ref.@17#.

III. GENERAL CONSIDERATIONS

We have chosen the particular functional operators(t)512tr r2 to characterize the decoherence evolution. Thia choice that depends only on the system in consideraOther invariants, likeI n(t)512tr rn, could have also beenused to characterize the tendency for a system to evolvestatistical mixture. If the initial state is an eigenstate of toperatorf , it easy to show that the first and second derivtives of I n(t) are zero while the third derivative is given b

d3I n~0!

dt35

n

2g3ubu2~^A2&2^A&2!. ~33!

That is, a resultn/2 times larger than that obtained in thcase ofn52. Of course, these choices of measuring deherence, although basis independent, are arbitrary.

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A better measure of the effect of decoherence shouldways be related to the question of how the result of a givmeasurement is affected if the system presents decoherin its time evolution.

The density matrix can always be written in the Schmbasis as(l i(t)ul i(t)&^l i(t)u. Thea priori probability that ameasure of the observableO gives the valueo8 is

P~o8,t !5(1

N

l i~ t !u^o8ul i~ t !&u2, ~34!

where Ouo8&5o8uo8&. If there is no decoherence and thinitial state is ul1(0)&, r(t)5ul1(t)&^l1(t)u and P(o8,t)5u^o8ul1(t)&u2. If there is decoherence, more than one stwill contribute in the sum of Eq.~34! and we have

P~o8,t !5(1

N

l i~ t !u^o8ul i~ t !&u2. ~35!

This result shows that the consequences of decoherencegiven experiment depend not only on how much the denmatrix operator deviates from the pure stateul1(t)&^l1(t)u,but also on how large are the projections of the state veuo8& on the Schmidt statesul i(t)&.

A universal measure of decoherence may only make sein the neighborhood oft50 and when we are dealing witobservables such thatPo85 z^o8ul1(0)& z2 are different fromzero. In this case the deviation of the measurable observais related to deviations in the density operator, which cobe defined, for example, as

12tr rul1~ t !&^l1~ t !u512l1~ t !, ~36!

which is, for small deviations of a pure state@l1(t)'1#,equal toI n(t)/n. In this sense any invariantI n(t) would givea suitable measure of decoherence.

Summarizing, we have studied the problem of stabilityquantum states under dissipation in a class of Lindbladsipative dynamics by considering the behavior of first-ordentropy around the initial time. Stable states may be chaterized through the stability of linear entropy. We have fouthe nontrivial result that the instabilities ofs(t) are at least ofthe third order in time if the initial state is an eigenstatethe operatorf. However, linear combinations of nondegeneate eigenstates off are unstable and their linear entropy icreases linearly in time.

ACKNOWLEDGMENTS

This work was partially supported by the VITAE Foundation, Fondecyt~Chile! under Grant No. 1990838, DYCYTUniversidad de Santiago de Chile, and the Brazilian agenConselho Nacional de Desenvolvimento Cientı´fico e Tecno-logico ~CNPq!, Fundac¸ao Jose´ Bonifacio, and the Programade Apoio a Nu´cleos de Exceleˆncia ~PRONEX!.

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J. C. RETAMAL AND N. ZAGURY PHYSICAL REVIEW A 63 032106

@1# For a review, see W. H. Zurek, Phys. Today44 ~10!, 36~1991!.

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@16# M. S. Kim and V. Buzek, Phys. Rev. A46, 4239~1992!.@17# M. S. Kim and V. Buzek, Phys. Rev. A47, 610 ~1993!.@18# D. F. Walls and G. J. Milburn,Quantum Optics~Springer-

Verlag, Berlin, 1994!.

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