Chemical Physics 268 *2001) 35±53 www.elsevier.nl/locate/chemphys

From completely positive maps to the quantum Markovian semigroup master equation

Daniel A. Lidar a,1, Zsolt Bihary b, K. Birgitta Whaley a,*

a Department of Chemistry, University of California, Berkeley, CA 94720, USA b Department of Chemistry, University of California, Irvine, CA 92697, USA Received 12 January 2001

Abstract A central problem in the theory of the dynamics of open quantum systems is the derivation of a rigorous and computationally tractable master equation for the reduced system density matrix. Most generally, the evolution of an open quantum system is described by a completely positive linear map. We show how to derive a completely positive Markovian master equation *the Lindblad equation) from such a map by a coarse-graining procedure. We provide a novel and explicit recipe for calculating the coecients of the master equation, using perturbation theory in the weak- coupling limit. The only parameter external to our theory is the coarse-graining time-scale. We illustrate the method by explicitly deriving the master equation for the spin-boson model. The results are evaluated for the exactly solvable case of pure dephasing, and an excellent agreement is found within the time-scale where the Markovian approximation is expected to be valid. The method can be extended in principle to include non-Markovian e€ects. Ó 2001 Elsevier Science B.V. All rights reserved.

1. Introduction are always ``open''. The action of the environment is to perform measurements on the system, thus The problem of the emergence of irreversible establishing a preferred ``pointer basis'' and lead- quantum dynamics from closed-system, unitary ing to decoherence [7,8]. Within a Hamiltonian dynamics has occupied the attention of many re- framework, recipes for deriving the associated re- searchers since the birth of quantum mechanics [1± duced-dynamics were known since the early 1960s, 6]. It is generally believed that an acceptable starting with the Zwanzig projection technique: solution is to view every quantum system as cou- one writes down the Heisenberg equation of mo- pled to an environment, i.e., true quantum systems tion for the combined system±environment state, and then projects out the system by tracing out the environment degrees of freedom [9]. This yields an * Corresponding author. Tel.: +1-510-643-6820; fax: +1-510- integro-di€erential equation involving an envi- 643-1255. ronment memory kernel, which must be subjected E-mail address: [email protected] *K.B. Wha- to approximations in order to become useful. The ley). two main techniques available are the derivation 1 Present address: Department of Chemistry, University of Toronto, 80 St. George Street, Toronto, Ont., Canada M5S of master equations by the use of the Born±Mar- 3H6. kov approximation [5], or a representation in

0301-0104/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S0301-0104*01)00330-5 36 D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53 terms of path integrals and in¯uence functionals also provide a recipe for calculating the parame- [6]. Unfortunately, in the former approach it is ters that appear in the equation. Our technique often unclear whether or not complete positivity is involves a coarse-graining procedure which leaves preserved in the sequence of approximations one us with just one phenomenological parameter: the makes [10], while in the latter approach one must coarse-graining time-scale s. The work presented make a semiclassical approximation in order to here is a continuation and generalization of Ref. obtain a tractable theory [11]. [17], where the derivation of the SME from the Complete positivity in reduced dynamics is the OSR was provided for the ®rst time. Here we very common-sensical idea 2 that the open-system verify the validity of the approach in Ref. [17], dynamics must preserve the positivity of a system's by proving now that the resulting SME is, as density matrix *a necessary condition for the required, completely positive *Section 2). Fur- probability interpretation to hold) in the presence thermore, we greatly expand the utility of the of any other non-interacting system. Building on derivation by now also showing explicitly how to this notion, two seminal contributions have been calculate the parameters that appear in the SME made. Kraus established an ``operator-sum repre- *Section 3). The method uses a perturbative ex- sentation'' which describes the most general com- pansion in the system±bath coupling strength. We pletely positive linear map on the density matrix of apply our formalism to the simple example of a a quantum system [14]. This is a formal represen- collection of spins coupled to a boson bath, and tation of the dynamics, which has been used compare the result to the exact solution *Section pro®tably in the quantum information processing 4). We conclude with an overview and assessment community [15], but is impractical to use for dy- of possible extensions *Section 5). namics calculations. To address this, Lindblad has derived the most general completely positive Markovian semigroup master equation for the 2. From the operator-sum representation of reduced dynamics of the density matrix [16]. This master dynamics to the semigroup master equation equation can be integrated and solved to provide the time development of the system density matrix. 2.1. Brief review of the operator-sum representation Both of these results were derived on the basis of axiomatic quantum mechanics. While systemati- The dynamics of a quantum system S coupled cally satisfying, this approach nevertheless pos- to a bath B, which together form a closed system, sesses the disadvantage that the resulting theories evolves unitarily under the combined system±bath are necessarily phenomenological, in the sense that Hamiltonian they contain no recipe for deriving their parame- H ˆ H I ‡ I H ‡ H : 1† ters from ®rst principles [4]. SB S B S B I

Previous formal approaches therefore su€er Here HS, HB and HI are, respectively, the system, from one of two disadvantages. Either the ®nal bath and interaction Hamiltonians, and I is the equations are not necessarily completely positive, identity operator. Assuming that S and B are ini- or they contain parameters which are not derived tially decoupled, so that the total initial density from ®rst principles and must therefore be treated matrix is a tensor product of the system and bath as phenomenological. In this paper we provide a density matrices *q and qB respectively), the system derivation of the semigroup master equation dynamics are described by the reduced density *SME) from the Kraus operator-sum representa- matrix: tion *OSR) which overcomes both of the above q 0†7!q t†ˆTr ‰U q q †UyŠ: 2† problems. Thus, the SME we derive is completely B B positive *i.e., it is of Lindblad form), while we can Here TrB is the partial trace over the bath and   i U ˆ exp À HSBt : 3† 2 See Refs. [12,13] for a debate concerning this assertion. h D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53 37

By usingP a spectral decompositionP for the bath, *13) below), both of these problems are solved, i.e., qB ˆ l }ljlihlj *where l }l ˆ 1), and intro- one obtains a di€erential equation in which there is N ducing a basis fjnignˆ1 for the N-dimensional an explicit separation between terms leading to system Hilbert space H, this can be rewritten in unitary and to non-unitary evolution. This pro- the OSR as [14]: vided the motivation in Ref. [17] to develop an alternative representation of the OSR in a form XK y which approaches the form of the SME as much as q t†ˆ Ai t†q 0†Ai t†; 4† iˆ0 is possible, without yet making any Markovian assumption. We provide only the main steps of where the Kraus operators fAig have matrix ele- this derivation here, and refer the interested reader ments given by to Ref. [17] for full details. p ‰A Š t†ˆ } hmjhljU t†jmijni; i ˆ l; m†: 5† It is convenient for this purpose to introduce a i mn l M ®xed operator basis for A H†. Let fKagaˆ0, with 2 K ˆ I, be such a basis, so that the expansion of K ˆ NB, where NB is the number of bath degrees of 0 freedom. Also, by unitarity of U, one derives the the Kraus operators is given by normalization condition XM XK Ai t†ˆ bia t†Ka: 7† y Ai Ai ˆ IS; 6† aˆ0 iˆ0 Under this expansion, the OSR evolution equa- which guarantees preservation of the trace of q: tion, Eq. *4), becomes P y P y Tr‰q t†Š ˆ Tr‰ i Aiq 0†Ai ŠˆTr‰q 0† i Ai AiŠˆ Tr‰q 0†Š. The Kraus operators belong to the Hil- XM y bert±Schmidt space A H† *itself a Hilbert space) q t†ˆ vab t†Kaq 0†Kb; 8† of bounded operators acting on the system Hilbert a;bˆ0 space, and are represented by N  N matrices, just where v t† is the matrix with elements like q. ab XK à 2.2. Fixed-basis form of the operator-sum represen- vab t†ˆ bia t†bib t†: 9† tation iˆ0

While the OSR evolution equation, Eq. *4), is The matrix v is clearly Hermitian, with positive perfectly general, it presents two major dicul- diagonal elements. With some algebraic manipu- ties: *i) It is an evolution equation, rather than a lation [17] one can transform Eq. *8) into: di€erential equation, which expresses q t† in terms oq t† i 1 XM of the initial condition and time-dependent oper- ˆÀ ‰S_ t†; q 0†Š ‡ v_ t† ot h 2 ab ators. Calculating these is equivalent to diagonal- a;bˆ1 izing the entire system±bath Hamiltonian. This is   ‰K ; q 0†Ky Š ‡‰K q 0†; Ky Š : 10† impractical in all but a very few exactly solvable a b a b models. *ii) It is not clear how to separate out the unitary evolution of the system from the possibly where S t† is the hermitian operator de®ned by non-unitary one, which occurs from the coupling ih XM  à of the system to the bath and leads to decoherence. S t†ˆ v t†K À v t†Ky : 11† 2 a0 a 0a a The reason is that in general, each Kraus operator aˆ1 will contain a contribution from both the unitary and the non-unitary components of the evolution. Eq. *10) is the desired result: it represents a ®xed- When one makes the assumption of Markovian basis OSR evolution equation, with a strong re- dynamics, however, as in the SME *Eqs. *12) and semblance to the SME, as we now detail. 38 D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53

2.3. From the ®xed-basis operator-sum representa- and coarse-grain the evolution of the system in tion equation to the semigroup master equation terms of s: qj ˆ q js†; vab;j ˆ vab js†, j an integer. Further, rewriting the OSR Eq. *10) as q t†ˆ ~ 2.3.1. Derivation of the semigroup master equation K t†q ÂÃ0R† and de®ning L t† through K t†ˆ by a coarse-graining procedure T exp t L~ s†ds *T indicates time-ordering) we 0 We recall that in the semigroup approach, un- have der the assumptions of *i) Markovian dynamics, oq t† *ii) initial decoupling between the system and the ˆ L~ t†‰q t†Š: 15† bath, and *iii) the requirement of complete posi- ot tivity, the system evolves according to the SME R R De®ne L~ ˆ j‡1†s L~ s†ds, with sn ˆ t: t L~ s†ds ˆ [4,16]: P j js 0 nÀ1 ~ s jˆ0 Lj. Next we make the assumption that on oq t† i the coarse-graining time-scale s, the evolution ˆ L‰q t†Š  À ‰H ; q t†Š ‡ L ‰q t†Š; 12† ~ ot h S D generatorsÂÃL t† commute in the ``average'' sense ~ ~ that Lj; Lk ˆ 0, 8j; k. Physically, we imagine this 1 XM operation as arising from the ``resetting'' of the L ‰q t†Š ˆ a ‰F ; q t†Fy Š‡‰F q t†; Fy Š†; bath density operator over the time-scale s. This D 2 ab a b a b a;bˆ1 means that s must be larger than any characteristic 13† bath time-scale, and explains the requirement sc  s. Under this assumption, the evolution of where aab is a constant positive semide®nite ma- theQ systemÂà is Markovian when t  s: K t†ˆ trix. This equation bears a clear resemblance to nÀ1 exp sL~ : Further, under the discretization Eq. *10). Analyzing the di€erences between the jˆ0 j of the evolution, this product form of the evolu- SME Eq. *13) and this OSR evolution equation Âà tion implies that q ˆ exp sL~ ‰q Š. In the limit *10) allows one to understand the precise manner j‡1 j j of s  t we expand this exponential, to ®nd that in which the semigroup evolution arises from the OSR evolution under the above-mentioned three q À q j‡1 j ˆ L~ ‰q Š: 16† conditions. An important di€erence between these s j j two equations is the fact that the SME provides a prescription for determining q t† at all times t, This equation is simply a discretization of Eq. *15) given q t0† as an initial condition at any other time under the assumption that s  h, where h is the t > t0 P 0, whereas Eq. *10) determines q t† in time-scale of change for the system density matrix. terms of q 0†, i.e., at the special time t ˆ 0 where Notice in particular that the RHS of Eq. *16) the system and the bath are in a product state. contains the average value of L~ t† over the interval. In Ref. [17] a coarse-graining procedure was Now, from the OSR evolution equation *10), we introduced which allows to transform the exact know the explicit form of L~ t† over the ®rst in- Eq. *10) to the approximate SME. For conve- terval from 0 to s. Discretizing over this interval nience we repeat and clarify the derivation here. we ®nd that We consider three time-scales: *i) the inverse of the bath density of states frequency cuto€ s , *ii) a q À q i 1 XM  c 1 0 ˆÀ ‰hS_ i; q Š‡ hv_ i‰K ; q Ky Š coarse-graining time-scale s which is essentially the s h 0 2 ab a 0 b time-scale for the bath's ``memory'' to disappear a;bˆ1 y ~ *the de®nition will be made more precise below), ‡‰Kaq 0†; KbŠ  L0‰q0Š; 17† and *iii) a system time-scale h which is the typical time-scale for changes in the system density matrix where in the frame rotating with the system Hamiltonian. Z We require that 1 s hiX  X s†ds: 18† sc  s  h; 14† s 0 D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53 39 X à Thus, in the sense of the coarse graining above we v 0†ˆ bia 0†b 0† ~ ab ib have arrived at an explicit form for L0. However, i ~ X deriving an explicit form for L1 and for higher ˆ }lda0db0di; l;l† ˆ da0db0: 22† terms beyond this ®rst interval is impossible be- iˆ l;m† cause Eq. *10) gives the evolution in terms of q 0†. Since we have made the assumption that the bath But in Eq. *19) we are concerned with v_ only ``resets'' over the time-scale s, we expect the bath ab for a; b P 1, so that ®nally, from Eq. *21), the to interact with the system in the same manner submatrix v_ with a; b P 1 is indeed positive over every s-length coarse-grained interval. This is ab semide®nite. The important conclusion is that Eq. equivalent to assuming that L~ ˆ L~ , 8i *which of i 0 *19) is in Lindblad form, i.e., it preserves complete course is the most trivial way of satisfying the positivity. This establishes the validity of our result Markovian evolution condition ‰L~ ; L~ Šˆ0, 8i; j). i j for the SME, and should be contrasted with pro- Then, under the natural identi®cation of the K's jection-operator type derivations of the master with the F's of the SME, and using Eq. *16), one is equation [5,18], which do not necessarily satisfy led to the well known form of the semigroup the complete positivity criterion. equation of motion: oq t† i 1 XM  ˆÀ ‰hS_ i; q t†Š ‡ hv_ i‰K ; q t†Ky Š 2.3.3. Separating out the Hamiltonian ot h 2 ab a b a;bˆ1 We can write Eq. *19) in an alternative form  which distinguishes between the system and bath ‡‰K q t†; Ky Š : 19† a b contributions to the unitary part of the evolution.

Because Eq. *10) is linear in the vab t† matrix, one 0† can calculate vab t† for the isolated system and 2.3.2. Positivity of the coecient matrix hence de®ne the new terms which come about The positive semide®niteness of the coecient from the coupling of the system to the bath: 0† 1† matrix aab in Eq. *13) is a sucient condition for vab t†ˆvab t†‡vab t†. The terms which corre- the preservation of complete positivity of the sys- spond to the isolated system will then produce a tem dynamics [4]. Thus, to complete the identi®- normal À i=h†‰HS; q t†Š Liouville term in Eq. *19). cation of Eq. *19) as a Lindblad equation, it only Thus Eq. *19) can be rewritten as remains to be shown that v_ab is positive semi- de®nite. To do so let us show ®rst that v itself is ab oq t† i 1 XM positive semide®nite, i.e., that for any vector c, the ˆÀ ‰H ‡hS_ 1†i; q t†Š ‡ hv_ i ot h S 2 ab matrix v satis®es cvcÃt P 0: a;bˆ1   X X y y Ãt à à à ‰Ka; q t†KbŠ ‡‰Kaq t†; KbŠ : 23† cvc ˆ cavabcb ˆ cabiabibcb ab i;ab

2 Identifying v_ with a ,andK with F , this is X X ab ab a a ˆ cÃb P 0; 20† seen to be equivalent to Eqs. *12) and *13), except a ia i a for the presence of the second term in the Liou- villian. This second term S_ 1† , inducing unitary where we used Eq. *9). Next, dynamics on the system, is referred to as the Lamb Z 1 s 1 À Á shift. It explicitly describes the e€ect the bath has hv_ iˆ v_ dt ˆ v s† À v 0† ; 21† ab ab ab ab on the unitary part of the system dynamics, and s 0 s ``renormalizes'' the system Hamiltonian. It is often so that we must show that v 0† does not spoil the implicitly assumed to be present in Eq. *12) [19]. positivity. Now, from Eqs. *3) and *5) A 0†ˆ In summary, we have shown in this section how p p P i } hljU 0†jmiˆ } d I ˆ M b 0†K , so that coarse graining the evolution over the bath mem- l p l lm S aˆ0 ia a bia 0†ˆ }lda0di; l;l† *recall K0 ˆ IS). Thus ory time-scale s allows one to understand the 40 D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53 connection between the OSR evolution and the the system±bath interaction Hamiltonian of Eq. semigroup dynamics. The importance of Eq. *10) *1) in the following perfectly general form: 3 lies in the fact that it allows one to pinpoint the X exact point at which the assumption of Markovian HI ˆ kaSa Ba; 24† dynamics is made. Furthermore, due to the gen- a eral likeness of its form to the SME, it provides where fSag and fBag are the system and bath an easily translatable connection from the non- operators respectively, and fkag are coupling co- Markovian OSR to the Markovian SME. Notice ecients. In the IP we do not have to deal directly also that the assumption of Markovian dynamics with the free system and bath Hamiltonians. introduces an arrow of time in the evolution of the However, as will be seen below, we do recover the system, through the ordering of the environmental Lamb shift. states: The system evolves through time in the di- rection of successive resettings of the bath. Addi- 3.1. The interaction picture tionally, it is important to note that we have shown that our procedure leads to an explicitly Transformation to the IP is accomplished by Lindbladian *completely positive) form of the means of the unitary operator SME, as written in ®nal form in Eq. *23). Finally, we address the question of the inclusion UT ˆ exp ÀitHS† exp ÀitHB†US UB: 25† of non-Markovian e€ects. The approach presented Operators in the IP will be denoted using explicit here also o€ers a route to a systematic inclusion of time-dependence *and where there already was a non-Markovian e€ects, i.e., higher order dynamics time-dependence, with an I subscript). Thus: which include bath memory terms. Such a deri- X y vation of a ``post-Markovian'' master equation is a HI t†ˆUTHIUT ˆ kaSa t† Ba t†; 26† long sought ± after goal of the ®eld of open a quantum systems. Several attempts have been re- where ported, but generally the resulting equations are X y not satisfactory because complete positivity is vi- Sa t†ˆUSSaUS ˆ pab t†Sb; 27† olated [20]. In the context of the present approach, b the formal extension to go beyond the Markovian X regime can be made by replacing the assumption B t†ˆUy B U ˆ q t†B ; 28† ~ a B a B ab b that the evolution generators Lj commute to ®rst- b order *see text below Eq. *15)), by a higher order commutator. The derivation of this commutator with pab 0†ˆqab 0†ˆdab. The density matrix for and the resulting post-Markovian master equation the system and bath combined is denoted qtot t† is left to a future publication. in the Schrodinger picture, and is transformed to y the IP by qtot;I t†ˆUTqtot t†UT. The dynamics of qtot;I t† is governed by the unitary propagator y 3. Explicit derivation of the semigroup master U t†ˆUT exp ÀitHSB†UT, where HSB is the full equation parameters system±bath Hamiltonian: qtot;I t†ˆU t†qtot;I 0† Uy t†. The Schrodinger and IPs coincide at t ˆ 0so

We can now exploit the coarse-grained ®rst- that qtot;I 0†ˆqtot 0†ˆq 0† qB 0†. It is a stan- order *in time) perturbation expansion of the dard exercise to show that [21] OSR, Eq. *10), made in the previous section, in order to derive the explicit form of the parameters 3 and operators appearing in the resulting SME. To Note that fSag and fBag are not assumed to be linear operators, and that any interaction Hamiltonian can be do so, it turns out to be most convenient to work decomposed into a sum of terms acting separately on system in the interaction picture *IP) de®ned with respect and bath. Furthermore, we allow fSag and fBag to be time- to the free system and bath Hamiltonians. Let us dependent. D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53 41  Z  Z Z Z t n X t t t i n† p Ài† U t†ˆT exp À HI s†ds Ai t†ˆ }l dtn dtnÀ1 ... dt1 h 0 n! 0 0 0 a1;...;an X1 Ài†n "#"# ˆ I ‡ Un t†; 29† Yn Yn n!  T k S t † hljT B t † jmi; nˆ1 aj aj j aj j jˆ1 jˆ1 where 35† Z Z Z t t t ÂÃ

Un t† dtn dtnÀ1 ... dt1T and we used Sai ti†; Baj tj† ˆ 0 to separate the 0 0 0 n† time-ordering operations. Ai is proportional to  fgH t † ...H t † : 30† n I 1 I n †ka , so that in the weak-coupling case of ka  1, we can truncate the expansion at small n. The Dyson time-ordered product is de®ned with respect to any set of operators Oi ts† as [21]: 3.3. First-order case T fgO t † ...O t † ˆ O t † ...O t †, where 1 1 n n s1 s1 sn sn t > t > ÁÁÁ> t . The system density matrix in First we note that from Eq. *34), with K ˆ I : s1 s2 sn p 0 S the IP is obtained, as usual, by tracing over the bi0 t†ˆ }ldlm. Now, let us calculate the expres- bath, which leads to the OSR: sion for n ˆ 1. In this case there is no need to worry about time ordering, and we have: ÂÃXK Z q t†ˆTr q t† ˆ A t†q 0†Ay t†; 31† p X t I B tot;I i i A 1† t†ˆÀi } dt k S t †hljB t †jmi iˆ0 iˆlm l 1 b b 1 b 1 b 0 X where the Kraus operators are now de®ned in the p ac ˆÀit }l SakbhljBcjmiCb t† IP: abc p X A t†ˆ } hljU t†jmi: 32† i l  bia t†Ka; 36† a Repeating the derivation of Sections 2.2 and 2.3 where the second line follows using Eqs. *27) and we thus obtain the very same form for the SME as *28) the third from the ®xed-basis operator ex- in Eq. *19), but now it is a SME for the interaction pansion in Eq. *7), and we de®ned representation, qI t†. Finally, the transformation Z t back to the Schrodinger picture is accomplished by bc 1 Ca t† dt1 pab t1†qac t1†: 37† t 0 y q t†ˆUSq t†U : 33† I S This dimensionless quantity thus depends en- tirely on the transformation to the IP i.e., it con- tains no information on the system±bath coupling, 3.2. Perturbation theory expansion of the Kraus but only on the internal system and bath dynam- operators ics. Next, let us identify Ka ˆ Sa *the system op- erators) and assume that our basis is trace Our next task is to calculate the Kraus opera- orthogonal: tors. We do so by using the expansion for U t† and Âà y Eqs. *29) and *32). We have then: Tr SaSb ˆ dab=Na; 38† where N is a normalization constant. Then p X1 a A t†ˆ } d I ‡ A n† t†; 34† X i l lm S i p aa00 nˆ1 bia t†ˆÀit }l ka0 hljBa00 jmiCa0 t† a P 1: a0a00 where 39† 42 D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53 P à Using these results and vab t†ˆ iˆlm bia t†bib t† As for the decoherence term, we ®nd we can reconstruct the v matrix: v t†ˆ P 00 aab ˆhv_abi l }l ˆ 1, and for a P 1: X à y aa00 bb00 à X X ˆ s ka0 k 0 hBa00 B 00 i C 0 s† C 0 s†† : à p b b B a b va0 t†ˆ bia t†bi0 t†ˆ }lbll;a t† a0a00b0b00 iˆlm l X 46† aa00 ˆÀit ka0 hBa00 iBCa0 t†; 40† a0a00 Note that both the Lamb shift parameters /a and the decoherence parameters aab formally depend where we used on the coarse-graining time s. Since s is a ``dum- X my'' di€erentiation parameter in our theory, the hXi  Tr‰q XŠˆ } hljXjli; 41† B B l dependence upon it should disappear in an explicit l calculation. We deal with this in the example de®ning the bath-averaged expectation value of an studied in Section 4. arbitrary operator X. Finally, for both a; b P 1 X 3.4. Second-order case à vab t†ˆ bia t†bib t† iˆlm Expanding the Kraus operators to second-order X à yields: 2 à y aa00 bb00 ˆ t ka0 k 0 hBa00 B 00 i C 0 t† C 0 t† ; p Z Z b b B a b } X t t a0a00b0b00 A 2† t†ˆÀ l k k dt dt iˆlm a1 a2 2 1 2 0 0 42† a1;a2  T ‰ŠhS t †S t † ljT ‰ŠjB t †B t † mi; where in the last line we used the completeness a1 1 a2 2 a1 1 a2 2 P p 2 X relation jmihmjˆI . }lt m B ˆÀ S S 2 a1 a2 Now, as shown in Eqs. *21) and *22): a a X 1 2 a1a2;c1c2 1 À Á v s†  kb kb hljBc Bc jmiC t†; ab 1 2 1 2 b1b2 aab ˆhv_abiˆ vab s† À vab 0† ˆ ; 43† s s b1b2;c1c2 47† unless both a ˆ b ˆ 0 *in which case hiv_00 ˆ 0). Together with the SME Eq. *19) we thus have all where Z Z the ingredients. In particular, we can calculate the 1 t t Âà b1b2;c1c2 _ C t† dt2 dt1Tpa b t1†pa b t2† S Lamb shift term in Eq. *19) from Eq. *11): a1a2 t2 1 1 2 2 ÂÃ0 0 i X  Tq t †q t † : 48† hS_ iˆ hv_ iS Àhv_ iÃSy a1c1 1 a2c2 2 2 a0 a a0 a a We need to compare this expression to the ex- X 1 à y pansion of the Kraus operators in terms of the PM ˆ /aSa ‡ /aSa; 44† 2 ®xed basis, Ai t†ˆ aˆ0 bia t†Ka *Eq. *7)). To do a so we must now extend the ®xed basis set so that it where includes product terms: X XM X X aa00 /  k 0 hB 00 i C s† 45† b t†K ˆ b t†S ‡ b t†S S : a a a B a0 ia a i;a1 a1 i;a1a2 a1 a2 0 00 a a aˆ0 a1ˆ0 a1;a2ˆ0 49† is a correlation function which contributes to the Lamb shift. Note that unlike in Eq. *23), S_ does Comparing this expansion to Eq. *47) we can read not contain the system Hamiltonian, as indeed it o€ the second-order time-dependent coecients should not in the IP. b as i;a1a2 D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53 43 p } t2 X two-level systems *qubits) coupled via a phase- b ˆÀ l k k hljB B jmi i;a1a2 2 b1 b2 c1 c2 damping *non-dissipative) interaction to a boson b1b2;c1c2 bath. The Hamiltonians in this spin-boson model  Ca1a2;c1c2 t†; 50† b1b2 are [22]: 1 X provided that the basis of system operators is H ˆÀ hxi ri ; 51† closed under multiplication *or more generally: is S 2 0 z i   trace-orthonormal for all products of basis ele- X 1 ments). The decoherence and Lamb shift para- HB ˆ hxk Nk ‡ ; 52† 2 meters can then be calculated as in the ®rst-order Xk À Á i i ià y case. HI ˆ rz kkbk ‡ kk bk ; 53† i;k

3.5. Putting it all together i i where hx0 are the qubit energies, kk are coupling coecients, b and by are the kth bath mode an- In conclusion, we have derived an explicit form k k nihilation and creation operators obeying the for the SME of Eq. *19) *with S replacing K ). To Âà a a boson commutation relation b ; by ˆ 1d , and ®nd the full SME for a given problem, it is nec- k l kl N ˆ by b is the number operator. Comparing to essary to: k k k Eq. *24) we read o€ the system operators as S ˆ ri , and the bath operators as B ˆ b . Below 1. Identify the system operators fS g in the in- a z a k a we deal with the required modi®cations to our teraction Hamiltonian *and recall that these treatment of the indices in order to account for operators must be trace orthogonal in our for- these assignments. malism). We assume that the boson bath is in thermal 2. Solve for the time-dependent system and bath equilibrium at temperature T ˆ 1= †k b *k is the operators in the IP *Eqs. *27) and *28)), and B B Boltzman constant). Thus the bath density matrix thus ®nd C from Eq. *37). P is q ˆ 1=Z†eÀbHB ˆ 1=Z† eÀbEl jlihlj; where 3. Calculate the expectation values of the bath op- B l l ˆfn ; n ; ...; n ; ...g are the numbers of quanta erators. The results of this step will depend on 1 2 k in all bath modes, E is the energy of the ®eld at a the initial state of the bath *e.g., thermal equi- l given occupation l, and Z T †ˆTr‰exp ÀbH †Š ˆ librium, coherent state, squeezed state, etc.). Q B eÀbhxk =2= †1 À eÀbhxk is the canonical partition 4. Use the results of the previous steps to calculate k function. Some useful results for the average the Lamb shift term S_ , the decoherence ma- number of quanta in the kth bath mode and the trix v_ , and ®nally, to write down the SME. ab averages of the creation and annihilation opera- tors are: Since this SME is of the Lindblad form [16] it is guaranteed to preserve positivity of the density y y 1 hbkbliB ˆhbkbl iÀdk;l ˆ dk;l ; matrix. Systematic corrections may be derived by ebhxk À 1 54† continuing the expansion in Eq. *34) to higher y y y hbkiB ˆhbkiB ˆhbkbl iB ˆhbkbliB ˆ 0: orders in n. We now proceed to calculate the various quantities appearing in the SME. 4. Example: spin-boson model 4.1.1. Form of the interaction representation oper- 4.1. Pure dephasing of multiple qubits ators Formally, we need to solve Eqs. *27) and *28) As a concrete and simple example of the pro- for the time-dependent system and bath opera- cedure described above, we consider the form of tors. In the present simple example, however it i the SME derived for a collection of independent is clear that since rz commutes with the system 44 D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53

i i i Hamiltonian, rz t†ˆrz 0†rz. Further, it is an By linearity, the required modi®cation is clearly ix t elementary exercise to show that bk t†ˆbke k . that Eq. *39) should be replaced with X Therefore the interaction Hamiltonian in the IP is: p aa00 b t†ˆÀit } ‰k 0 hljB 00 jmiC t† X lm;a l a a a0 0 00 i i ixk t ià Àixk t y a a HI t†ˆ r k e bk ‡ k e b †: 55† z k k k à y aa00 à i;k ‡ ka0 hljBa00 jmi Ca0 t†† Š a P 1 57† *where it is assumed that the expansion coecients

of Sa, the pab t†, are real, as in the example treated 4.1.2. Calculation of the Lamb shift and decoher- here). Using Eqs. *40) and *45) this leads to Lamb ence parameters shift parameters of the form: In the derivation of Section 3.3 the interaction X Hamiltonian was expressed as a sum over the aa00 /a ˆ 2 Re‰ka0 hBa00 iBCa0 s†Š ˆ 0; 58† single index a, in order to reduce the clutter of a0a00 indices to a minimum. However, as seen from the interaction Hamiltonian of Eq. *53), in reality we since using Eq. *54) expectation values of creation need more indices. In particular, we need to clarify and annihilation operators between number states the indexation of Cbc of Eq. *37): each of the in- vanish. As for the decoherence part, using Eq. *54) a y y again we ®nd that hB 00 B 00 i and hB B i vanish, dices a, b and c may now correspond to a qubit a b B a00 b00 B position *denoted i or j), a Pauli matrix index so that using Eqs. *42) and *46), the decoherence *denoted n ˆ x; y; z), or a bath mode *denoted k or parameters are: l). Qubits variables can have both position and X à y aa00 bb00 à hv_ iˆs k 0 k ‰hB 00 B i C s†C s† Pauli indices, but bath variables have only a mode ab a b0 a b00 B a0 b0 index. We will use a comma to separate qubit and a0a00b0b00 y aa00 à bb00 bath variables, as in a ˆ †in; k , when all three in- ‡hB B 00 i C s† C s†Š; 59† a00 b B a0 b0 dices are needed. When one of the indices is ir- relevant it will simply be dropped. To separate or, using the results for the spin-boson case: bc X groups of indices, such as the bc in Ca , we will use i0 j0à y in;k00 jn;l hv_ iˆs k 0 k 0 ‰hbk00 b 00 i C 0 0 s† a semicolon. For example, C is short for Cbc in;jn1 k l l B i z;k iz;k a i0j0;k0l0;k00l00 0 0 0 with a ˆ †iz; k , b ˆ jn; k †, c ˆ j n ; l†, where k , 00 00 00 1 jn1;l à y in;k à jn1;l 0  Cj0z;l0 s†† ‡hbk00 bl00 iB Ci0z;k0 s†† Cj0z;l0 s†Š j and n1 are irrelevant. With these preparations we X are now ready to calculate Cbc. By comparing ˆ sd d ki kjÃsinc2 †x s=2 a nz n1z k k k i i ixk t k rz t†ˆ rz, bk t†ˆbke to Eqs. *27) and *28) and using the correct index convention, we have that bhxk 0  coth ii ixk t 2 pzn t†ˆdii0 dnz and qkk00 t†ˆdkk00 e . Therefore: zz ˆ aij : 60† Z t i0n;k00 1 ii0 C t†ˆ dt p t †q 00 t † iz;k 1 zn 1 kk 1 Our ®nal result for the SME in the IP can thus t 0 x t be written as ixk t=2 k ˆ d 0 d 00 d e sinc ; 56† ii kk nz 2 oq t† 1 X À Á I ˆ azz ‰ri ; q t†rjŠ ‡‰ri q t†; rjŠ ; ot 2 ij z I z z I z where sinc †x sin †x =x. i;j Before proceeding to calculate the Lamb shift 61† and decoherence parameters, we should note that where in the de®nition of Sa and Ba in Eq. *24), each Sa is coupled to a B with the same index a, whereas in s X bhx a azz ˆ ki kjÃsinc2 †x s=2 coth k : 62† ij 2 k k k the present case each Sa *i.e., rz) is coupled to both h 2 y k Ba *i.e., bk) and Ba. Let us brie¯y again suppress for clarity the ij; kl indices of the present example. and we reintroduced h. D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53 45

As commented above, the dependence on s 4.2. Model with dissipation should disappear after all is done. Let us see how this comes about within a simple continuum We now generalize our model to include dissi- 2 model. If we assume that jkkj only depends on xk, pative terms. On the other hand, to keep the we can rewrite this expression as an integral over analysis tractable, we will consider the case of a x: single spin coupled to a boson bath. We keep the Z system and bath Hamiltonians of Eqs. *51) and 1 1 xs a s†ˆ dxg x†ki x†kjà x†ssinc2 *52). The new interaction Hamiltonian is: ij 2 h 0 2 X À Á à y bhx HI ˆ rz kkzbk ‡ kkzbk ‡ r‡  coth ; 63† 2 kÀ Á À Á à y à y kk‡bk ‡ kk‡bk ‡ rÀ kkÀbk ‡ kkÀbk where g x† is the density of states. In any rea- à sonable physical model the density of states has a *where kk‡ ˆ kkÀ). Transforming to the IP we ®nd: 2 r t†ˆr eix0at and b t†ˆb eixk t where a ˆ z, Æ, cut-o€ frequency xc. Now, ssinc †xs=2 is a a a k k and x ˆ 0, x ˆÇx : As above, this translates function that peaks strongly at 0,R has a width 1=s 0z 0Æ 0 and its integral is ®nite: 1=p† 1 dxssinc2 xs= into diagonal p and q *recall Eqs. *27) and *28)): 0 2†ˆ1, i.e., ix0at ixk t pab t†ˆdabe qkk0 t†ˆdkk0 e ; 67†

1 2 d~ s; x† ssinc †xs=2 and we ®nd for aab: p 64† s X ~ a s†ˆ k kà hby b i C x ‡ x † lim d s; x†ˆd x†: ab 2 ka kb k k B 0a k sˆ1 h k à y But s, the coarse-graining time-scale, must indeed  C Àx0b À xk†‡kka0 kkb0 hbkbkiB be large compared to the time-scale of the bath  C x0a À xk†C Àx0b ‡ xk†: 68† sc ˆ 1=xc *recall Eq. *14)), so in this limit we can y perform the integral and we ®nally get: Here kka0 is the coupling coecient for ra, we al- y y ready dropped the vanishing hbkbkiB and hbkbkiB p bhx azz ˆ lim g x†ki x†kj x†Ã coth : 65† terms, and ij 2 Z h xˆ0 2 1 s C x†ˆ eixt dt ˆ eixs=2 sinc †xs=2 : 69† Thus, the dependence on s has indeed disappeared. s 0 We now apply this result to the case of a single In particular, for the diagonal terms we obtain: two-level atom coupled to a harmonic bath. In s X the case of phonons and electromagnetic radia- a s†ˆ jk j2hbyb i jC x ‡ x †j2 aa 2 ka k k B 0a k tion, the interaction couples to the amplitude of h k 1=2 y the oscillators: x ˆ h= ††2mx b ‡ b†, so that 2 y 2 ‡jk 0 j hb b i jC x À x †j : 70† jk x†j2 / 1=x. At the relevant low-frequency re- ka k k B 0a k gime we can equivalently use the high-temperature For azz this yields the same result as above. For the result: new decoherence parameters a‡‡ and aÀÀ we ®nd: kT s X hbybi ˆhbbyi  : 66† a s†ˆ jk j2 hby b isinc2 x À x †s=2† B B hx ‡‡ 2 k‡ k k k 0 h k y 2 For a three-dimensional crystal *or radiation ®eld) ‡hbkbkisinc xk ‡ x0†s=2†† g x†/x2. Collecting terms, we see that the limit s X a s†ˆ jk j2 hby b isinc2 x ‡ x †s=2† yielding azz is well de®ned. Decoherence depends ÀÀ h2 kÀ k k k 0 quadratically on the coupling, and linearly on k y 2 temperature and on the density of low-frequency ‡hbkbkisinc xk À x0†s=2††; phonons. 71† 46 D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53  or, using the integral form: dq cq À bq À 2a ‡ b‡c†q ˆ 11 00 2 01 : 76† Z dt À 2a ‡ b‡c†q bq À cq p 1 2 10 00 11 a s†ˆ dxg x†jk x†j2 ‡‡ 2 ‡ h 0 The o€-diagonal elements decay exponentially, y ~ y ~ À1  hb bidx Àx0†‡hbb id x ‡ x0†† with a rate sdec ˆ 2a ‡ b ‡ c†=2††. The diagonal Z 1 elements approach the thermal equilibrium values p 2 a s†ˆ dxg x†jk x†j ther ther ÀÀ 2 À q00 and q11 , where h 0 y ~ y ~ ther  hb bidx ‡x0†‡hbb id x À x0††: q c aÀÀ 00 ˆ ˆ ˆ ebhx0 : 77† ther 72† q11 b a‡‡

With the appearance of the system's unitary time- The exponential rate of convergence of the diag- onal elements, i.e., the dissipation rate, is sÀ1 ˆ scale *1=x0) we have to rede®ne our coarse- diss graining procedure. We can consider two opposite aÀÀ ‡ a‡‡. Within the framework of our model, limits, where either *i) the system energy, or *ii) the both rates depend linearly on temperature and interaction energy is dominant. These two limits quadratically on the corresponding coupling correspond respectively to *i) the system's internal strengths. The important di€erence between them unitary evolution being fast *so we are actually is the presence of azz in the dephasing rate. The coarse graining this out as well), and *ii) the sys- parameters a‡‡ and aÀÀ depend on the bath's tem's unitary evolution being slow. density of states at x0. Dissipation therefore can be quite slow in a number of important cases, for example when there is a gap in the phonon spec- 4.2.1. The fast-system limit trum, or when x is actually greater than the cuto€ In this case x s  1. The d functions are cen- 0 0 frequency. In these cases only much weaker multi- tered at x and at Àx , much further from zero 0 0 phonon processes cause dissipation. than their width. The ones at Àx thus do not 0 The parameter a , on the other hand, depends contribute *o€-resonance), so, similarly to a we zz zz on the density of low-frequency phonons. This can ®nd: be small only in very special circumstances *e.g., 2p super¯uidity, or a discrete phonon density of states a ˆ lim g x†jk x†j2hbybi; 73† ‡‡ 2 ‡ as would be found in a quantum dot [23]) and its h xˆx0 vanishing indeed usually causes macroscopic quantum-e€ects. In typical situations the rate of 2p a ˆ lim g x†jk x†j2hbbyi: 74† dephasing will be greater than the rate of dissipa- ÀÀ 2 À h xˆx0 tion. The important general conclusion is the fol- The o€-diagonal Lindblad parameters aab vanish, lowing: If our coarse graining includes the *fast) as they involve the product of two d functions that system as well, then the density matrix rapidly are centered further apart than their widths. The decoheres into the system's energy eigen-basis [24]. coecient matrix is diagonal in the fast-system Then, *typically slower) it converges into the limit. If we assume for further simplicity that thermalized density matrix *which is of course also k‡ x†ˆkÀ x†, then the diagonal parameters a‡‡ diagonal in the system's energy eigen-basis). See and aÀÀ only di€er in the bath expectation values Ref. [25] for a more detailed discussion of these at x0. It then follows that di€erent regimes.

bhx0 aÀÀ ˆ a‡‡e : 75† 4.2.2. The slow-system limit

Let us now consider brie¯y the resulting IP In this case x0s  1. We consider only the ze- master equation. Using the notation a ˆ azz, b ˆ roth approximation, i.e., set x0 ˆ 0. Using Eq. a‡‡, c ˆ aÀÀ for simplicity, we obtain: *68) we obtain: D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53 47 X s à y à y 2 thermal bath is that the time-dependence of the aab s†ˆ kkak hb bki‡k kkbhbkb i†jC xk†j ; 2 kb k ka k ÀC T ;t† h k o€-diagonal terms is proportional to e , with or, integrating out the d function again, using Eq. 2t2 X hx C ˆ jk j2sinc2 †x t=2 coth k : 82† *66) for the present low-frequency limit, and as- 2 k k 2k T h k B suming real kka for simplicity: 2p g x† This exact result holds for arbitrary times t and for aab ˆ kT lim ka x†kb x†ˆaaab both ®nite and in®nite baths. 2 xˆ0 hx h On the other hand, recall that the SME result 1=2 78† 2p g x† 2 for the single qubit case was *Eq. *62)): aa  kT lim ka x† : h2 xˆ0 hx s X hx a ˆ jk j2sinc2 †x s=2 coth k : 83† Unlike in the fast-system case, the o€-diagonal zz 2 k k 2k T h k B elements of the coecient matrix do not vanish.

Instead, in the slow-system limit aab is a projection, This is the dephasing rate for a single qubit satis- i.e., aab is an outer product of the vector of com- fying the Lindblad master equation ponents fa g with itself. This allows us to write the a dq SME using just one Lindblad operator: ˆ 1a ‡‰Šr q; r ‰Š†r ; qr ; 84† dt 2 zz z z z z X G  a r : 79† a a whence the o€-diagonal q01 / exp À2azzt†. a How do these results relate to one another? We This coarse-grained interaction operator is just a have in the Markovian case: linear combination of the system operators as they ! 2p X hx appear in the interaction Hamiltonian, but with qSME / exp À t jk j2d~ s; x † coth k ; 01 2 k k 2k T the dependence on the bath degrees of freedom h k B already averaged out. Using G, the SME becomes: 85† oq t† ˆ 1 ‰G; q t†GyŠ‡‰Gq t†; GyŠ†: 80† whereas in the exact case: 2 ot ! 2p X hx Diagonalizing G and transforming q into G's ei- qexact / exp À t jk j2d~ t; x † coth k : 01 2 k k 2k T genbasis then leads to uncoupled equations for the h k B components of the transformed q. Therefore, in 86† the slow-system limit, the density matrix becomes diagonal in the eigenbasis of the course-grained While super®cially the similarity between these interaction Hamiltonian *i.e., G), and for the rate results is striking, there is nevertheless a crucial of this decoherence we ®nd: di€erence: the exact solution has recurrences, since   À Á its time-dependence is periodic *for a ®nite bath), À1 2p g x† 2 2 2 sdec ˆ kT lim 2kz x† ‡ k‡ x†‡kÀ x† : whereas the SME result is a purely exponential h2 xˆ0 x decay. Thus they describe very di€erent behaviors. 81† Indeed, for small t *xkt  1) the exact result de- cays as exp Àt2† *quantum Zeno e€ect [26,27]), while the Markovian result always decays as 4.3. Comparison of the Markovian result to exact exp Àt†. This is of course not a surprise: the solution of spin-boson model for pure dephasing Markovian result cannot describe the dynamics for times shorter than the coarse-graining time- The spin-boson model is exactly solvable in the scale, s. pure dephasing limit, and we present the detailed Let us now turn to see the limit in which the two solution in Appendix A. The result for an initial solutions do agree. To prevent recurrences in the 48 D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53 exact solution we once again replace the sum over the SME solutions of coarse are just straight lines, modes by an integral, to obtain: as they all describe simple exponential decays. It is clear that the SME solutions cannot account for  Z 2p the initial transition period, but for suciently qSME / exp À t dxg x†jk x†j2 large s *in units of the bath cuto€ time 1=x ) the 01 h2 c  SME result approximates the exact solution very hx  d~ s; x† coth well at large times, in accordance with Eq. *87). 2kBT Let us summarize: the Markovian approxima-  Z tion we introduced gives reliable results for times 2p greater than the coarse-graining time-scale, which qexact / exp À t dxg x†jk x†j2 01 h2 in turn must be greater than the bath cuto€ time. It  hx does not account for the initial *transitional) time  d~ t; x† coth : 2k T evolution, and it should be applied in cases of an B in®nite bath with continuous spectrum.

The only di€erence is the appearance of s and t in the widths of the d~ functions. Now, our coarse- 4.4. The Lamb shift graining procedure was de®ned such that s 

1=xc *recall the discussion surrounding Eq. *64)), Finally, in the exact solution for multiple qubits and in this limit d~ s; x†ˆd x†. For times t > s, there is also a non-vanishing Lamb shift, which d~ t; x†ˆd x† also holds, so we can summarize the arises as a consequence of the Hamiltonian not condition for the exact and Markovian solutions commuting with itself at di€erent times [28]. The to agree as: Lamb shift does vanish for a single qubit in the exact solution of the pure dephasing spin-boson t > s  1=x : 87† c model *see Appendix A and Ref. [28]). The Lamb To illustrate this let us consider the Debye model shift also vanished in our Markovian calculation. as a simple example. Then: This discrepancy is not only due to the fact that we considered a single qubit: the more fundamental  2 reason is that we only carried out our Markovian x for x < xc g x†/ : calculations to ®rst-order in perturbation theory, 0 for x P xc where time ordering did not play a role. However, when we consider the multiple-qubit case in sec- As before, let the coupling coecient k depend ond-order perturbation theory *recall Section 3.4) on x only due to amplitude-coupling: jk x†j2 / there is a Lamb shift. This arises because of terms xÀ1. In the high-temperature limit coth hx= like ri rjbyb . Physically, this is a phonon-induced, 2k T †/xÀ1, so that in all we have z z k k B indirect, exchange-interaction between the two  Z  xc spins. It is quadratic in k, linear in temperature, SME 2 q01 / exp À Cts dxsinc †xs=2 88† and acts to pull the spin-energies towards an av- 0 erage value.  Z  xc exact 2 2 q01 / exp À Ct dxsinc †xt=2 ; 89† 0 5. Conclusions where C is the temperature-dependent coupling strength, with dimensions of frequency. Fig. 1 A central task of modern condensed phase shows the argument of the exponential, C t†, for chemistry and physics is the quantitative descrip- the exact solution and for the SME results, cor- tion of open quantum systems. These are systems responding to di€erent values of the coarse- that are coupled to an external uncontrollable en- graining time-scale, s. The curves corresponding to vironment *bath), a coupling which generally leads D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53 49

Fig. 1. Comparison of exact solution of the spin-boson model for single-qubit pure dephasing to the result obtained from the Markovian master equation. Straight lines correspond to the Markovian solution, which intersects the exact solution *thick line) at t ˆ s, as seen from Eqs. *88) and *89). The density of states of the boson bath is represented by the Debye model here. Data is plotted for C ˆ 0:05 and xc ˆ 1.

to decoherence. In this paper we provided such a known in many-body physics could be employed quantitative description, by deriving a practical here as well. To test the validity of our theory, we way to calculate the coecients in the quantum compared it here to an exactly solvable model, Markovian semigroup master equation *com- namely, the spin-boson Hamiltonian with pure monly known as the Lindblad equation). Our phase-damping. For times longer than the coarse- starting point was the exact Kraus operator sum graining time, the agreement was found to be representation, which presents the evolution of an excellent already at the level of ®rst-order pertur- open quantum system as a general, completely bation theory. positive, linear map. By coarse-graining this evo- lution over a time-scale typical of the bath *the inverse of the bath density-of-states frequency- Acknowledgements cuto€), we showed how the Lindblad equation can be derived, and how its coecients can be sys- It is a pleasure to acknowledge very insightful tematically calculated using perturbation theory in discussions with Dave Bacon. This work was the system±bath coupling strength. This resolves supported in part by the National Security Agency an important shortcoming in the theory of open *NSA) and Advanced Research and Development quantum systems: so far no practical general Activity *ARDA) under Army Research Oce method was known which takes as input an in- *ARO) contract number DAAG55-98-1-0371. teraction Hamiltonian, and then produces the Lindblad equation together with all its coecients. The complexity of our method is determined by Appendix A. Analytical solution of the spin-boson the diculty of calculating certain time-ordered model for pure dephasing integrals, which of course increases with higher orders of perturbation theory. In principle, this is We present here the analytical solution of the equivalent to the calculation of standard Feynman spin-boson model for pure dephasing. The deri- diagrams, and thus the arsenal of techniques vation is based on Ref. [28]. 50 D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53  Z  Starting from the IP Hamiltonian: i t U t†ˆT exp À HI s†ds X  ÀÁ à h 0 à y H t†ˆ ri ki eÀixk ta ‡ ki eixk ta ; A:1† "# I z k k k k XN i;k ˆ T lim exp Hn Dt Dt!0 nˆ0 Y   we want to ®nd the system density matrix 1 2 ˆ lim exp À ‰Hn; Hn0 Š †Dt Dt!0 2 Âà n

2 aÃa†, ‰a; ayŠˆ1, and the Baker±Hausdorf formula X x t À sin x t X ˆ k k ki ri 1: exp A ‡ B†ˆexp À‰A; BŠ=2† exp A† exp B† *again, 2 k z k †hxk i valid if ‰‰A; BŠ; AŠˆ‰‰A; BŠ; BŠˆ0). Now let R t†ai t†ay À ai t†Ãa and consider A:9† ÂÃik k k k k i exp rz Rik t† : Note that f is an operator acting just on the system, and is a simple phase for the case of a X1 R2n X1 R2n‡1 exp‰rz RŠˆIS ‡ rz single qubit. Since the ak operators commute for 2n†! 2n ‡ 1†! di€erent modes we have as our ®nal simpli®ed nˆ0 nˆ0 result for the evolution operator: ˆ IS cosh R ‡ rz sinh R Y  À Áà if t† i i i à y 1 U t†ˆe exp rz ak t†ak À ak t† ak : ˆ I ‰D a†‡D Àa†Š S 2 i;k ‡ r 1‰D a†ÀD Àa†Š A:10† z 2 ˆj0ih0j D a†‡j1ih1j D Àa†: A.2. Calculation of the density matrix A:17†

Now recall the de®nition of the coherent states. This is an important result since it shows that These are eigenstates of the annihilation operator: depending on whether the ®eld is coupled to the ajaiˆajai: A:11† qubit j0i or j1i state, the ®eld acquires a di€erent displacement. This is the source of the dephasing They are minimum-uncertainty states in a har- the qubits undergo, since when acting on a su- monic potential, etc. As is well known, perposition state of a qubit, the qubit and ®eld become entangled: X1 n 2 a jaiˆeÀjaj =2 p jni; A:12† exp ‰ r RŠ aj0i‡bj1i†jbiˆaj0i D †ja bi‡bj1i nˆ0 n! z D Àa†jbiˆe abÃÀaÃb†=2aj0i ja ‡ bi where jni are number *Fock) states. The com- À abÃÀaÃb†=2 pleteness relation for the coherent states is ‡ e bj1i jb À ai: 52 D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53

The evolution operator can be written as to calculate the evolution of each of the four pure Y  ÀÁ states jxihyj separately. Thus U t†ˆeif t† j0i h0j D ai ‡j1i h1j  à i k i y i;k q t†ˆTrB U t†jxihyj q 0†U t† À Áà x;y " B D À ai : A:18† Y k if t† ˆ TrB e ‰j0ih0j D †‡jak 1ih1j Now assume that the boson bath is in thermal k equilibrium: D Àak†Šjxihyj Y Y  y qB;m j0ih0j D †‡jal 1ih1j 1 ÀbH q ˆ e B m l B Z # "# à À1 y Àif y t† Y eÀbhxk =2 D Àal† e : ˆ 1 À eÀbhxk k  ! The terms in the three products match one-to-one X 1  exp À b hx N ‡ for equal indices, so we can write everything as a k k 2 k product over a single index k. Using Tr A B†ˆ Y 1 TrA  TrB to rearrange the order of the trace and ˆ exp Àbhx N †; A:19† hN i k k the products, and Dy †ˆÀa D †a , we have: k k q t†ˆd d eif t†j0ih0jeÀif y t† where the mean boson occupation number is x;y x;Y0 y;0  à TrB D †ak qB;kD À ak† 1 k hNkiˆ : A:20† ebhxk À 1 ‡ d d eif t†j0ih1jeÀif y t† Yx;0 y;1 Âà As shown in Ref. [5, pp. 122±123], this can be TrB D †ak qB;kD †ak transformed into the coherent-state representa- k if t† Àif y t† tion, with the result: ‡ dx;1dy;0e j1ih0je Y Y  à TrB D À ak†qB;kD Àak† qB ˆ qB;k A:21† k k if t† Àif y t† ‡ dx;1dy;1e j1ih1je where Y  à Tr D À a †q D †a : ! B k B;k k Z 2 k 1 2 jakj qB;k ˆ d ak exp À jakihakj: phNki hNki Consider the TrB terms: for j0ih0j and j1ih1j by cycling in the trace the displacement operators A:22† Âà cancel and TrB qB;k ˆ 1. Thus, as expected the Now consider the system density matrix. Let diagonal terms do not change *apart from the q ˆjxi hyj where x; y ˆf0; 1g. Since we are Lamb shift due to f t†). As for the o€-diagonal xi;yi i dealing with qubits the system density matrix is a terms *evaluating the trace in any complete basis): sum of all possible tensor products of single qubit Z !  à 2 1 2 jbkj pure states, i.e., of terms of the form qfx ;y g  Tr D Æ 2a †q ˆ d b exp À i i B k B;k phN i k hN i qx ;y ÁÁÁ qx ;y . Thus it can be expanded as k k 1 1 X N N X  hnjD Æ2ak†jbkihbkjni q 0†ˆ cfxi;yigqfx ;y g: A:23† i i n fxi;yig Z ! 1 jb j2 ˆ d2b exp À k RecallÂà that we set out to evaluate q t†ˆ k y phNki hNki TrB U t†q 0† qB 0†U t† . For simplicity let us now consider the case of a single qubit. It suces ÂhbkjD Æ2ak†jbki: D.A. Lidar et al. / Chemical Physics 268 -2001) 35±53 53

Now: [2] J.A. Wheeler, W.H. Zurek *Eds.), Quantum Theory and Measurement, Princeton University Press, Princeton, NJ, hbjD Æ2a†jbiˆexp ‰Æ Àabà aÃb†ŠhbjÆa ‡ bi 1983.  [3] E.B. Davies, Quantum Theory of Open Systems, Academic ˆ exp ‰Æ Àabà aÃb†Š exp bà Æ2a ‡ b† Press, London, 1976.  [4] R. Alicki, K. Lendi, Quantum Dynamical Semigroups and 1   À jbj2 ‡jÆ2a ‡ bj2 Applications, Lecture Notes in Physics, vol. 286, Springer, 2 Berlin, 1987.   [5] C. Gardiner, Quantum Noise, Springer Series in Synerget- 2 à à ˆ exp À 2jaj Æ 2 Àab a b† : ics, vol. 56, Springer, Berlin, 1991. [6] U. Weiss, Quantum Dissipative Systems, World Scienti®c, Thus: Singapore, 1993. Z [7] W.H. Zurek, Phys. Rev. D 24 *1981) 1516.  à 2 1 2 [8] D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, TrB D Æ 2ak†qB;k ˆ exp À2jak j † d bk phNk i H.D. Zeh *Eds.), Decoherence and the Appearance of a ! Classical World in Quantum Theory, Springer, Berlin, 2 À Á jbk j à à 1996.  exp À Æ 2 ak bk À ak bk hNk i [9] R.W. Zwanzig, Statistical Mechanics of Irreversibility,  Lectures in Theoretical Physics, vol. 3, Interscience, New 2 exp À 2jakj h York, 1961, p. 106. ˆ phN i phN i k [10] D. Kohen, C.C. Marston, D.J. Tannor, J. Chem. Phys. 107 k i *1997) 5236. 2  exp À 4jakj hNk i [11] N. Makri, Comp. Phys. Commun. 63 *1991) 389.    [12] P. Pechukas, Phys. Rev. Lett. 73 *1994) 1060. 1 ˆ exp À 4ja j2 hN i ‡ [13] R. Alicki, Phys. Rev. Lett. 75 *1995) 3020, Comment on k k 2 " ``Reduced Dynamics Need Not Be Completely Positive''; à 2 Reply by P. Pechukas, Phys. Rev. Lett. 75 *1995) 3020. k eixk t À 1† k ˆ exp À 4 [14] K. Kraus, States, E€ects and Operations, Fundamental hxk Notions of Quantum Theory, Academic, Berlin, 1983. #   [15] M.A. Nielsen, I.L. Chuang, Quantum Computation and 1 1  ‡ Quantum Information, Cambridge University Press, Cam- ebhxk À 1 2 bridge, UK, 2000. " [16] G. Lindblad, Commun. Math. Phys. 48 *1976) 119. 1 À cos x t ˆ exp À 4jk j2 k ; [17] D. Bacon, D.A. Lidar, K.B. Whaley, Phys. Rev. A 60 k 2 †hxk *1999) 1944. # [18] W.T. Pollard, A.K. Felts, R.A. Friesner, in: I. Prigogine, S.A. Rice *Eds.), The Red®eld equation in condensed-  coth bhxk =2 : phase quantum dynamics, Advances in Chemical Physics, vol. XCIII, Wiley, New York, 1996, p. 77. Thus decay of the o€-diagonal terms goes as [19] P. Beck, K. Lendi, Phys. Rev. A 47 *1993) 346. eÀC T ;t†, with [20] T. Yu, L. Diosi, N. Gisin, W.T. Strunz, Phys. Lett. A 265 *2000) 331. X 1 À cos x t hx [21] N. March, W.H. Young, S. Sampanthar, The Many-Body C ˆ 4 jk j2 k coth k ; A:24† k 2 2k T Problem in Quantum Mechanics, Dover, New York, 1995. k †hxk B [22] A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.A.P. Fisher, A. Garg, W. Zwerger, Rev. Mod. Phys. 59 *1987) 1. which is equivalent to the result that appeared in [23] T. Takagahara, J. Lumin. 70 *1996) 129. Eq. *82) above. [24] J.P. Paz, W.H. Zurek, Phys. Rev. Lett. 82 *1999) 5181. [25] G.M. Palma, K.-A. Suominen, A.K. Ekert, Proc. Roy. Soc. London Ser. A 452 *1996) 567. [26] B. Misra, E.C.G. Sudarshan, J. Math. Phys. 18 *1977) 756. References [27] W.M. Itano, D.J. Heinzen, J.J. Bollinger, D.J. Wineland, Phys. Rev. A 41 *1990) 2295. [1] E. Schrodinger, Naturwissenschaften 23 *1935) 807 trans- [28] L.-M. Duan, G.-C. Guo, Decoherence of quantum regis- lated in Ref. [2]. ters, LANL Report no. quant-ph/9703036.