arXiv:1110.2122v2 [quant-ph] 17 Oct 2012 PACS: Keywords: hr h unu yaia eirusaentepiil u explicitly t importantly, not more are and, stochasti applications, semigroups Itô their dynamical betweassumptions, of quantum interaction use the the the unfam involve where about quite it considerations is derive which measurements, to 9], ways recent [8, semigroups more dynamical quantum of stu by high-temperatur understanding the its in C justifying small the equation, becomes transform Lindblad that the to term a possible of cond is addition system it two-level movement, possibl a Brownian the on to quantum emission associated spontaneous - of durati operator analysis the density with the finite, of measure but elements the instantaneous, and as system 5] treated [4, measured longer applications the recent between in interaction latter, the the For 4]. from [3, processes, process non-unitary and irreversible of treatment ∗ idlda mre aual sacneuneo h Marko the of consequence a system on as naturally emerges Lindbladian students. edrv h onMro atreuto 2 n hnw tra we then and [2] equation master resembl Born-Markov that the them derive between interaction we an with temperature, w systems: two oflo h ruet xoe eei ai nweg fq of Neum knowledge Liouville-von the basic and a operator is density here the the exposed of of arguments concepts validity the the of follow illustration to accessible an provide to lcrncades [email protected] address: Electronic otatn gis t motneadwd ag fapplic of range wide and importance its against Contrasting Marko a for form general most the is [1] equation Lindblad The omk h idldeuto oeudrtnal,ti art this understandable, more equation Lindblad the make To lal,tepeetapoc osntpoetegnrlval general the prove not does approach present the Clearly, 2 36.aFraim 36.zDchrne pnsses q systems; open Decoherence; 03.65.Yz Formalism, 03.65.Ca audd eCêca eBuu nvriaeEtda Paul Estadual Universidade Bauru, de Ciências de Faculdade S alsAeadeBrasil, Alexandre Carlos eprtr hnw aecniee ntepriua aet case particular the fo in equation considera considered have broader Lindblad we and the than semi-groups temperature of dynamical fo th quantum derivation condition by of formal fundamental use caused The a system is main f operator. approximation the instructive, Markov on very the effect is that situation fr the of represent specific degrees Lindblads The bath the form. Jaynes- out the Lindblad tracing resembles and, ha that equation of Born-Markov Hamiltonian bath a interaction with an interacts with system primary specifi a a st consider where We undergraduate situation, to mechanics. quantum accessible on is background basic that - evolutions unitary idldeuto,oe unu systems quantum open equation, Lindblad asdb the by caused S 1 epeetadrvto fteLnba qain-a importa an - equation Lindblad the of derivation a present We nttt eFsc Ge aahn,Uiesdd Estadua Universidade Wataghin", "Gleb Física de Instituto h rnia ytm and system, principal the , 3 nttt eFsc eSoCro,Uiesdd eSoPaulo São de Universidade Carlos, São de Física de Instituto S ipedrvto fteLnba equation Lindblad the of derivation simple A − B interaction. 1 eieFradsFanchini, Fernandes Felipe B hc a eteevrneto h esrmn apparatus, measurement the or environment the be can which , .INTRODUCTION I. h idldeuto a sdi h nrdcino iei time of introduction the in used was equation Lindblad the , erpyia ennsapa eyitmdtn obeginni to intimidating very appear meanings physical heir ispto n eoeec 2 oteqatmmeasurement quantum the to [2] decoherence and dissipation idldeuto onnseilss h nyprerequisi only The non-specialists. to equation Lindblad et nteerylevels. early the in dents no htitrcincagn h rbblte diagona - probabilities the changing interaction that of on nlrsls nteohrhn,i unu pis the optics, quantum in hand, other the On results. final e ntesse n h ee 1] nte deduction, Another [12]. meter the and system the en cst h idldeuto 6.A at ntecs of case the in last, At [6]. equation Lindblad the to ucts n qain ttelvlo e.[3,frexample. for [13], Ref. of level the at equation, ann steoeo h ansCmig oe[] Initially model[2]. Jaynes-Cummings the of one the es dt fteLnba qain u neto ssimply is intention Our equation. Lindblad the of idity tos t rgnlddcin[]ivle h formalism the involves [1] deduction original its ations, ii 2.Teeaeacul fmn plctosof applications many of couple a are These [2]. limit e acls[0 1 r nteseiccs fquantum of case specific the in or, 11] [10, calculus c etaprts hn h esrmn rcs sno is process measurement the Then, apparatus. ment prxmto.Ec idldrpeet h effect the represents Lindblad Each approximation. v e a efudo e.[] hs ehd,their methods, These [2]. Ref. on found be can sed la oms ftesuet n eerhr.Other researchers. and students the of most to iliar inmse qain n ti eyipratfrthe for important very is it and equation, master vian atmmcais nldn aiirt ihthe with familiarity a including mechanics, uantum ler-egt qain[]it idldwt the with Lindblad into [7] equation aldeira-Leggett 2 sa"ui eMsut ih" ar,Brasil Bauru, Filho", Mesquita de "Julio ista dnsi hsc rmteaiswt a with mathematics or physics in udents eottedgeso reo fsystem of freedom of degrees the out ce cepeet t euto nteseiccs of case specific the in deduction its presents icle n eiad eJssNapolitano Jesus de Reginaldo and ri ae tes oraieta the that realize to easy it makes it or umnsfra.W tr ihthe with start We format. Cummings atmsaitclmethods statistical uantum mncoclaosa eotemperature, zero at oscillators rmonic etdhere. reated ae orsodn oavr simple very a to corresponding case, c eo,w bana qaini the in equation an obtain we eedom, in eadn h niomn and environment the regarding tions eCmia,Cmia,Brasil Campinas, Campinas, de l h mrec fteLindbladian the of emergence the r oegnrlcs eursthe requires case general more a r neato ihtebt,and bath, the with interaction e tto o h ramn fnon- of treatment the for tool nt ã als Brasil Carlos, São , ∗ 3 B tzero at The . ng te n l 2

The paper is structured as follows: in the Sec. II we derive the Born-Markov by tracing out the degrees of freedom of system B, starting from the Liouville-von Neumann equation; in Sec. III we derive the Lindblad equation; and in Sec. IV we present the conclusion.

II. THE BORN-MARKOV MASTER EQUATION

Let us consider a physical situation where a principal system S, whose dynamics is the object of interest, is coupled with another quantum system B, called bath. Here, HS and HB are, respectively, the Hilbert spaces of principal system S and bathB; the global S + B will be represented by the tensor-product space HS ⊗ HB. The total Hamiltonian is

Hˆ (t)= HˆS ⊗ 1ˆB + 1ˆS ⊗ HˆB + αHˆSB, (1) where HˆS describes the principal system S, HˆB describes the bath B, HˆSB is the Hamiltonian for the system-bath interaction and 1ˆB and 1ˆS are the corresponding identities in the Hilbert spaces. Here, we will considerate HˆS and HˆB both time-independent. For the sake of simplicity, let us ignore the symbol ⊗ and write

Hˆ (t)= HˆS + HˆB + αHˆSB . (2)

Here, α is a real constant that provides the intensity of interaction between the principal system and the bath. Writing ρˆSB for the global density operator (S + B), the Liouville-von Neumann equation will be:

d i ρˆ = − Hˆ + Hˆ + αHˆ , ρˆ . (3) dt SB ~ S B SB SB h i It is convenient to write Eq. (3) in the interaction picture of HˆS + HˆB. With the definitions of the new density operator and Hamiltonian:

i ˆ ˆ i ˆ ˆ ~ (HS +HB )t − ~ (HS +HB )t Hˆ (t)= e HˆSB e (4) and

i ˆ ˆ i ˆ ˆ ~ (HS +HB )t − ~ (HS +HB )t ρˆ(t)= e ρˆSB (t) e , (5) the new equation forρˆ(t) will be

d i ρˆ(t)= − α Hˆ (t) , ρˆ(t) . (6) dt ~ h i Here and in the following, we will use the time argument explicited ((t)) to indicate the interaction-picture transfor- mation. We want to find the evolution for ρˆS (t)= trB {ρˆSB (t)} where, according Eq. (5),

i ˆ ˆ i ˆ ˆ − ~ (HS +HB )t ~ (HS +HB )t ρˆSB (t)= e ρeˆ . (7)

Equation (6) is the starting point of our iterative approach. Its time derivative yields

i t ρˆ(t)=ˆρ (0) − α Hˆ (t′) , ρˆ(t′) dt′. (8) ~ ˆ0 h i Replacing Eq. (8) into Eq. (6), we have

t d i ˆ 1 2 ˆ ˆ ′ ′ ′ ρˆ(t)= − α H (t) , ρˆ(0) − 2 α H (t) , H (t ) , ρˆ(t ) dt . (9) dt ~ ~ ˆ0 h i  h i  3

For the Born approximation, Eq. (9) is enough. Then, we take the partial trace of the bath degrees of freedom,

t d i ˆ 1 2 ˆ ˆ ′ ′ ′ ρˆS (t)= − αtrB H (t) , ρˆ(0) − 2 α trB H (t) , H (t ) , ρˆ(t ) dt . (10) dt ~ ~ ˆ0 nh io  h i 

By the definition in Eq. (4), Hˆ (t) depends on HˆSB, and HˆSB can always be defined in a manner in which the first term of the right hand side of Eq. (10) is zero. Hence, we obtain

t d 1 2 ˆ ˆ ′ ′ ′ ρˆS (t)= − 2 α trB H (t) , H (t ) , ρˆ(t ) dt . (11) dt ~ ˆ0  h i  Integrating Eq. (11) from t to t′ yields

′ ′ 1 t t ρ t ρ t′ α2 dt′tr Hˆ t′ , Hˆ t′′ , ρ t′′ dt′′ , ˆS ( ) − ˆS ( ) = −~2 B ( ) ( ) ˆ( ) ˆt (" ˆ0 #) h i ′ which shows that the difference between ρˆS (t) and ρˆS (t ) is of the second order of magnitude in α and, therefore, we can write ρˆS (t) in the integrand of Eq. (11), obtaining a time-local equation for the density operator, without violating the Born approximation:

t d 1 2 ˆ ˆ ′ ′ ρˆS (t)= − 2 α trB H (t) , H (t ) , ρˆ(t) dt . (12) dt ~ ˆ0  h i  The α constant was introduced in Eq. (2) only for clarifying the order of magnitude of each term in the iteration and, now, it can be supressed, that is, let us take α =1 (full interaction). Thus, let us write:

t d 1 ˆ ˆ ′ ′ ρˆS (t)= − 2 trB H (t) , H (t ) , ρˆ(t) dt . (13) dt ~ ˆ0  h i  For this approximation, we can write ρˆ(t)=ˆρS (t) ⊗ ρˆB inside the integral and obtain the equation that will be used in the next calculations (again, for the sake of simplicity, let us ignore the symbol ⊗):

∞ d 1 ′ ˆ ˆ ′ ρˆS (t)= − 2 dt trB H (t) , H (t ) , ρˆS (t)ˆρB , (14) dt ~ ˆ0 nh h iio where we assume that the integration can be extended to infinity without changing its result. Equation (14) is the Born-Markov master equation[2].

III. LINDBLAD EQUATION

A. The master equation commutator

Let us consider that system-bath interaction is of the following form,

† † HˆSB = ~ SˆBˆ + Sˆ Bˆ , (15)   where Sˆ is a general operator that acts only on the principal system S, and Bˆ is an operator that acts only on the bath B. Now, we consider that Sˆ commutes with HˆS, i. e.,

S,ˆ HˆS = 0, h i resulting in

Sˆ (t) = Sˆ (16) 4

(Sˆ is not affected by the interaction-picture transformation). Let us consider the bath hamiltonian defined by a bath of bosons,

ˆ ~ † HB = ωkaˆkaˆk (17) k X † where aˆk e aˆk are the annihilation and creation bath operators, the ωk are the characteristic frequencies of each mode, and the Bˆ operator on Eq. (15) defined by

ˆ ∗ B = gkaˆk, (18) k X where gk are complex coefficients representing coupling constantes. Then, in the interaction picture,

i ˆ i ˆ Bˆ (t) = e ~ HB tBeˆ − ~ HB t. (19)

Expanding each exponential and using the commutator relations, Eq. (19) will result in

ˆ ∗ −iωk t B (t)= gkaˆke . (20) k X The interaction (15) with the definition (18) resembles the Jaynes-Cummings one, who represents a single two-level atom interacting with a single mode of the radiation field [2, 11]. ′ With this, the commutator in Eq. (14), Hˆ (t) , Hˆ (t ) , ρˆS (t)ˆρB , will be evaluated. Firstly, h h ii ′ † † ′ Hˆ (t) , Hˆ (t ) , ρˆBρˆS (t) = ~ SˆBˆ (t)+ Sˆ Bˆ (t) , Hˆ (t ) , ρˆBρˆS (t)

h h ii h † ′ h ii = ~ SˆBˆ (t) , Hˆ (t ) , ρˆBρˆS (t)

h † h ′ ii + ~ Sˆ Bˆ (t) , Hˆ (t ) , ρˆBρˆS (t) . (21) h h ii The gradual expansion of each term in Eq. (21) will result in

† ′ † † ′ † ′ SˆBˆ (t) , Hˆ (t ) , ρˆBρˆS (t) = ~ SˆBˆ (t) , SˆBˆ (t )+ Sˆ Bˆ (t ) , ρˆBρˆS (t)

h h ii h † h † ′ † ′ ii = ~SˆBˆ (t) SˆBˆ (t )+ Sˆ Bˆ (t ) ρˆBρˆS (t)

† h † ′ i † ′ − ~SˆBˆ (t)ˆρBρˆS (t) SˆBˆ (t )+ Sˆ Bˆ (t )

† ′ † h ′ † i − ~ SˆBˆ (t )+ Sˆ Bˆ (t ) ρˆBρˆS (t) SˆBˆ (t)

h † ′ i † ′ † + ~ρˆBρˆS (t) SˆBˆ (t )+ Sˆ Bˆ (t ) SˆBˆ (t) (22) h i and

† ′ † † ′ † ′ Sˆ Bˆ (t) , Hˆ (t ) , ρˆBρˆS (t) = ~ Sˆ Bˆ (t) , SˆBˆ (t )+ Sˆ Bˆ (t ) , ρˆBρˆS (t)

h h ii h† h † ′ † ′ ii = ~Sˆ Bˆ (t) SˆBˆ (t )+ Sˆ Bˆ (t ) ρˆBρˆS (t)

† h † ′ i † ′ − ~Sˆ Bˆ (t)ˆρBρˆS (t) SˆBˆ (t )+ Sˆ Bˆ (t )

† ′ † h ′ † i − ~ SˆBˆ (t )+ Sˆ Bˆ (t ) ρˆBρˆS (t) Sˆ Bˆ (t)

h † ′ i † ′ † + ~ρˆBρˆS (t) SˆBˆ (t )+ Sˆ Bˆ (t ) Sˆ Bˆ (t) , (23) h i or, expanding Eqs. (22) and (23) and grouping the similar terms in S and B, we have: 5

† ′ † † ′ † † ′ SˆBˆ (t) , Hˆ (t ) , ρˆBρˆS (t) = ~SˆSˆρˆS (t) Bˆ (t) Bˆ (t )ˆρB + ~SˆSˆ ρˆS (t) Bˆ (t) Bˆ (t )ˆρB h h ii † † ′ † † ′ − ~SˆρˆS (t) SˆBˆ (t)ˆρBBˆ (t ) − ~SˆρˆS (t) Sˆ Bˆ (t)ˆρBBˆ (t ) † ′ † † ′ † − ~SˆρˆS (t) SˆBˆ (t )ˆρBBˆ (t) − ~Sˆ ρˆS (t) SˆBˆ (t )ˆρBBˆ (t) † ′ † † ′ † + ~ρˆS (t) SˆSˆρˆBBˆ (t ) Bˆ (t)+ ~ρˆS (t) Sˆ SˆρˆBBˆ (t ) Bˆ (t) (24) and

† ′ † † ′ † † ′ Sˆ Bˆ (t) , Hˆ (t ) , ρˆBρˆS (t) = ~Sˆ SˆρˆS (t) Bˆ (t) Bˆ (t )ˆρB + ~Sˆ Sˆ ρˆS (t) Bˆ (t) Bˆ (t )ˆρB h h ii † † ′ † † ′ − ~Sˆ ρˆS (t) SˆBˆ (t)ˆρBBˆ (t ) − ~Sˆ ρˆS (t) Sˆ Bˆ (t)ˆρBBˆ (t ) 2 † † ′ † † ′ − ~ SˆρˆS (t) Sˆ Bˆ (t )ˆρBBˆ (t) − ~Sˆ ρˆS (t) Sˆ Bˆ (t )ˆρBBˆ (t) † † ′ † † ′ + ~ρˆS (t) SˆSˆ ρˆB Bˆ (t ) Bˆ (t)+ ~ρˆS (t) Sˆ Sˆ ρˆB Bˆ (t ) Bˆ (t) . (25)

B. The partial trace

Now we are in a position to trace out the bath degrees of freedom in Eqs. (24) and (25). As we can verify with Eq. (20),

′ † † ′ ′ trB Bˆ (t) Bˆ (t )ˆρB = trB Bˆ (t) Bˆ (t )ˆρB =0, ∀t,t . n o n o With this, then,

† ′ † † ′ trB SˆBˆ (t) , Hˆ (t ) , ρˆBρˆS (t) = ~SˆSˆ ρˆS (t) trB Bˆ (t) Bˆ (t )ˆρB

nh h iio † n † ′ o − ~SˆρˆS (t) Sˆ trB Bˆ (t)ˆρBBˆ (t )

† n ′ † o − ~Sˆ ρˆS (t) Strˆ B Bˆ (t )ˆρBBˆ (t)

† n ′ † o + ~ρˆS (t) Sˆ Strˆ B ρˆBBˆ (t ) Bˆ (t) (26) n o and

† ′ † † ′ trB Sˆ Bˆ (t) , Hˆ (t ) , ρˆBρˆS (t) = ~Sˆ SˆρˆS (t) trB Bˆ (t) Bˆ (t )ˆρB

nh h iio † n † ′ o − ~Sˆ ρˆS (t) Strˆ B Bˆ (t)ˆρBBˆ (t )

† n † ′ o − ~SˆρˆS (t) Sˆ trB Bˆ (t )ˆρBBˆ (t)

† n † ′ o + ~ρˆS (t) SˆSˆ trB ρˆBBˆ (t ) Bˆ (t) , (27) n o where, if we use the ciclic properties of the trace,

† ′ † † † ′ trB SˆBˆ (t) , Hˆ (t ) , ρˆBρˆS (t) = ~ SˆSˆ ρˆS (t) − Sˆ ρˆS (t) Sˆ trB Bˆ (t) Bˆ (t )ˆρB

nh h iio h † †i n ′ † o + ~ ρˆS (t) Sˆ Sˆ − SˆρˆS (t) Sˆ trB Bˆ (t ) Bˆ (t)ˆρB (28) h i n o and

† ′ † † † ′ trB Sˆ Bˆ (t) , Hˆ (t ) , ρˆBρˆS (t) = ~ Sˆ SˆρˆS (t) − SˆρˆS (t) Sˆ trB Bˆ (t) Bˆ (t )ˆρB

nh h iio h † † i n † ′ o + ~ ρˆS (t) SˆSˆ − Sˆ ρˆS (t) Sˆ trB Bˆ (t ) Bˆ (t)ˆρB . (29) h i n o 6

The terms represented by Eqs. (28) and (29) allow us to return to the Eq. (21):

′ 2 † † † ′ trB Hˆ (t) , Hˆ (t ) , ρˆBρˆS (t) = ~ SˆSˆ ρˆS (t) − Sˆ ρˆS (t) Sˆ trB Bˆ (t) Bˆ (t )ˆρB

nh h iio 2 h † †i n ′ † o + ~ ρˆS (t) Sˆ Sˆ − SˆρˆS (t) Sˆ trB Bˆ (t ) Bˆ (t)ˆρB

2 h † † i n † ′ o + ~ Sˆ SˆρˆS (t) − SˆρˆS (t) Sˆ trB Bˆ (t) Bˆ (t )ˆρB

2 h † † i n † ′ o + ~ ρˆS (t) SˆSˆ − Sˆ ρˆS (t) Sˆ trB Bˆ (t ) Bˆ (t)ˆρB . (30) h i n o

C. The expansion of the integrand of the master equation

With the results of the preceding paragraphs, the integrand in Eq. (14) becomes:

′ 2 † † † ′ trB Hˆ (t) , Hˆ (t ) , ρˆBρˆS (t) = ~ SˆSˆ ρˆS (t) − Sˆ ρˆS (t) Sˆ trB Bˆ (t) Bˆ (t )ˆρB

nh h iio 2 h † †i n ′ † o + ~ ρˆS (t) Sˆ Sˆ − SˆρˆS (t) Sˆ trB Bˆ (t ) Bˆ (t)ˆρB

2 h † † i n † ′ o + ~ Sˆ SˆρˆS (t) − SˆρˆS (t) Sˆ trB Bˆ (t) Bˆ (t )ˆρB

2 h † † i n † ′ o + ~ ρˆS (t) SˆSˆ − Sˆ ρˆS (t) Sˆ trB Bˆ (t ) Bˆ (t)ˆρB . (31) h i n o For convenience, let us define the functions

t ′ † ′ F (t) = dt trB Bˆ (t) Bˆ (t )ˆρB , ˆ0 t n o ′ † ′ G (t) = dt trB Bˆ (t ) Bˆ (t) ρB . (32) ˆ0 n o Then,

t ∗ ′ ′ † F (t) = dt trB Bˆ (t ) Bˆ (t)ˆρB , ˆ0 t n o ∗ ′ † ′ G (t) = dt trB Bˆ (t) Bˆ (t )ˆρB . ˆ0 n o Replacing Eq. (31) in Eq. (14) yields:

d ρˆ (t) = − SˆSˆ†ρˆ (t) − Sˆ†ρˆ (t) Sˆ G∗ (t) − ρˆ (t) Sˆ†Sˆ − Sˆρˆ (t) Sˆ† F ∗ (t) dt S S S S S h† † i h † † i − Sˆ SˆρˆS (t) − SˆρˆS (t) Sˆ F (t) − ρˆS (t) SˆSˆ − Sˆ ρˆS (t) Sˆ G (t) . (33) h i h i Actually, the usual Lindblad equation emerges when G(t)=0 and F (t) = F ∗(t). In the following, we make some specifications about the environment to discuss these approximations in detail.

D. The bath specification

Furthermore, for the initial state of the thermal bath, we consider the vacuum state:

ρˆB = (|0i |0i ...) ⊗ (h0| h0| ...) . (34)

The evaluation of the F (t) and G (t) functions defined in Eq. (32) are done considering the Bˆ (t) and ρˆB definitions in Eqs. (20) and (34). By Eq. (20), Bˆ† (t) is 7

ˆ† † iωkt B (t)= gkaˆke . (35) k X Then, the partial trace in F (t) and G (t) can be evaluated:

† ′ † ′ trB Bˆ (t) Bˆ (t )ˆρB = trB Bˆ (t) Bˆ (t ) (|0i |0i ...) ⊗ (h0| h0| ...) (36) n o n o and

† ′ † ′ trB Bˆ (t ) Bˆ (t)ˆρB = trB Bˆ (t ) Bˆ (t) (|0i |0i ...) ⊗ (h0| h0| ...) . (37) n o n o If we use some bath state basis{|bi}, Eqs. (36) and (37) become

† ′ † ′ trB Bˆ (t) Bˆ (t )ˆρB = hb| Bˆ (t) Bˆ (t ) (|0i |0i ...) ⊗ (h0| h0| ...) |bi b n o X = (h0| h0| ...) |bi hb| Bˆ (t) Bˆ† (t′) (|0i |0i ...) b X = (h0| h0| ...) |bi hb| Bˆ (t) Bˆ† (t′) (|0i |0i ...) b X = (h0| h0| ...) Bˆ (t) Bˆ† (t′) (|0i |0i ...) (38) and

† ′ † ′ trB Bˆ (t ) Bˆ (t)ˆρB = hb| Bˆ (t ) Bˆ (t) (|0i |0i ...) ⊗ (h0| h0| ...) |bi b n o X = (h0| h0| ...) |bi hb| Bˆ† (t′) Bˆ (t) (|0i |0i ...) b X = (h0| h0| ...) |bi hb| Bˆ† (t′) Bˆ (t) (|0i |0i ...) b X = (h0| h0| ...) Bˆ† (t′) Bˆ (t) (|0i |0i ...) . (39)

Let us expand Bˆ† (t) and Bˆ (t) using Eqs. (20) and (35):

′ † ′ ˆ ˆ† ′ ∗ −iωkt ′ iωk t trB B (t) B (t )ˆρB = (h0| h0| ...) gkaˆke gk aˆk′ e (|0i |0i ...) k k′ n o X ′ X ′ † ∗ ′ −i(ωkt−ωk t ) = gkgk e (h0| h0| ...)ˆakaˆk′ (|0i |0i ...) (40) k,k′ X and

′ † ′ ˆ† ′ ˆ ′ iωk t ∗ −iωkt trB B (t ) B (t) ρB = (h0| h0| ...) gk aˆk′ e gkaˆke (|0i |0i ...) k′ k n o X ′ X ′ † ∗ ′ −i(ωkt−ωk t ) = gkgk e (h0| h0| ...)ˆak′ aˆk (|0i |0i ...)=0 (41) k,k′ X † Hence, we can rewrite Eq. (40) with the aˆk operators on the left of the aˆk operators. We know that

† † ′ aˆkaˆk′ = δk,k +ˆak′ aˆk. (42) 8

Then

′ ′ ′ ′ † ˆ ˆ† ′ ∗ ′ −i(ωkt−ωk t ) ′ ∗ ′ −i(ωkt−ωk t ) trB B (t) B (t )ˆρB = gkgk e δk,k + gkgk e (h0| h0| ...)ˆak′ aˆk (|0i |0i ...) k,k′ k,k′ n o X X 2 ′ −iωk(t−t ) = |gk| e . (43) k X Therefore, from Eqs. (41) and (43), it follows that t 2 ′ ′ −iωk(t−t ) F (t) = |gk| dt e , ˆ0 k X G (t) = 0. (44)

E. Transition to the continuum

In the expression of F (t) in Eq. (44), if we adopt the general definition of the density of states as

2 J (ω)= |gl| δ (ω − ωl) , (45) l X then the sum over k can be replaced by an integral over a continuum of frequencies:

∞ t ′ F (t) = dωJ (ω) dt′e−iω(t−t ). ˆ0 ˆ0 Let us introduce the new variable τ = t − t′, dτ = −dt′, with

t 0 t dt′ = − dτ = dτ, ˆ0 ˆt ˆ0 yielding ∞ t F (t) = dωJ (ω) dτe−iωτ . ˆ0 ˆ0

F. The Markov approximation

In the Markov approximation, the limit t → ∞ is taken in the time integral, as we have mentioned regarding Eq. (14), that is, we take ∞ −iωτ 0 dτe . ´ As the integrand oscillates, we will use the device: ∞ ∞ dτe−iωτ = lim dτe−iωτ−ητ ˆ0 η→0+ ˆ0 1 = lim η→0+ η + iω η − iω = lim 2 2 η→0+ η + ω η iω = lim 2 2 − lim 2 2 η→0+ η + ω η→0+ η + ω 1 = πδ (ω) − iP , ω 9 where P stands for the Cauchy principal part. Then,

∞ ∞ J (ω) F = π dωJ (ω) δ (ω) − iP dω . ˆ0 ˆ0 ω

G. The final form

For a general density of states, F yields

γ + iε F = , (46) 2 where

∞ γ ≡ 2π dωJ (ω) δ (ω) , (47) ˆ0 ∞ J (ω) ε ≡ −2P dω . (48) ˆ0 ω

As we have verified that G =0, let us replace Eq. (46) in Eq. (33):

d γ − iε γ + iε ρˆ (t) = − ρˆ (t) Sˆ†Sˆ − Sˆρˆ (t) Sˆ† − Sˆ†Sˆρˆ (t) − Sˆρˆ (t) Sˆ† dt S S S 2 S S 2 γh i h i = − ρˆ (t) Sˆ†Sˆ − Sˆρˆ (t) Sˆ† + Sˆ†Sˆρˆ (t) − Sˆρˆ (t) Sˆ† 2 S S S S ε h i + i ρˆ (t) Sˆ†Sˆ − Sˆρˆ (t) Sˆ† − Sˆ†Sˆρˆ (t) − Sˆρˆ (t) Sˆ† . 2 S S S S h i If the density of states is chosen to yield ε = 0 (a Lorentzian, for example, where we can extend the lower limit of integration to −∞), the final result is:

d 1 ρˆ (t)= γ Sˆρˆ (t) Sˆ† − Sˆ†S,ˆ ρˆ (t) . (49) dt S S 2 S  n o Let us, then, return to the original picture. Since

i ˆ i ˆ ~ HS t − ~ HS t ρˆS (t)= e ρˆSe , (50) then

d i i ˆ i ˆ i ˆ dρˆS i ˆ i i ˆ i ˆ ρˆ (t) = e ~ HS tHˆ ρˆ e− ~ HS t + e ~ HS t e− ~ HS t − e ~ HS tρˆ Hˆ e− ~ HS t dt S ~ S S dt ~ S S i ˆ dρˆS i ˆ i i ˆ i ˆ = e ~ HS t e− ~ HS t + e ~ HS t Hˆ , ρˆ e− ~ HS t. (51) dt ~ S S h i Performing the same operation on the right-hand side of Eq. (49) gives

1 i ˆ 1 i ˆ Sˆρˆ (t) Sˆ† − Sˆ†S,ˆ ρˆ (t) = e ~ HS t Sˆρˆ Sˆ† − Sˆ†S,ˆ ρˆ e− ~ HS t. (52) S 2 S S 2 S  n o  n o Replacing Eqs. (51) and (52) in Eq. (49), we obtain:

dρˆ i 1 S = − Hˆ , ρˆ + γ Sˆρˆ Sˆ† − Sˆ†S,ˆ ρˆ . (53) dt ~ S S S 2 S h i  n o 10

IV. CONCLUSION

In summary, in this paper we consider an interaction that resembles the Jaynes-Cummings interaction [2], Eq. (15), between a bath and a system S, assuming that the operator Sˆ commutes with the system Hamiltonian, HˆS, at zero temperature. We substituted Eq. (15) in the Born-Markov master equation (14) and took the partial trace of the degrees of freedom of B. The T =0 hypothesis is necessary to simplify the calculations, making them more accessible to the students, simplifying the treatment of Eqs. (36) and (37), and avoiding complications such as the Lamb shift in Eq. (46). The Markov approximation in Sec. III-F was also vital to obtain the final result, Eq. (53). All these simplifications limit the validity of our derivation to more general cases, but it provides a detailed illustration of the physical meaning of each term appearing in the Lindblad equation. Equation (53) is commonly presented with several Sˆ operators, usually denoted by Lˆ, in a linear combination of Lindbladian operators. The Lˆ operators are named Lindblad operators and, in the general case, the Lindblad equation takes the form:

dρˆ i 1 S = − Hˆ , ρˆ + γ Lˆ ρˆ Lˆ† − Lˆ†Lˆ , ρˆ . (54) dt ~ S S j S j 2 j j S j h i X  n o If we consider only the first term on the right hand side of Eq. (54), we obtain the Liouville-von Neumann equation. This term is the Liouvillian and describes the unitary evolution of the density operator. The second term on the right hand side of the Eq. (53) is the Lindbladian and it emerges when we take the partial trace - a non-unitary operation - of the degrees of freedom of system B. The Lindbladian describes the non-unitary evolution of the density operator. By the interaction form adopted here, Eq. (15), the physical meaning of the Lindblad operators can be understood: they represent the system S contribution to the S − B interaction - remembering once more that the Lindblad equation was derived from the Liouville-von Neumann one by tracing the bath degrees of freedom. This conclusion is also achieved with the more general derivation [1, 2]. It is important to emphasize that, due to our simplifying assumptions, the summation appearing in Eq. (54) was not obtained in our derivation of Eq. (53). If the Lindblad operators Lˆj are Hermitian (observables), the Lindblad equation can be used to treat the measure- ment process. A simple application in this sense is the Hamiltonian HˆS ∝ σˆz (σˆz is the 2-level z-Pauli mattrix) when we want to measure one specific component of the spin (Lˆ ∝ σˆα, α = x,y,z, without the summation) [3, 5]. If the Lindblads are non-Hermitian, the equation can be used to treat dissipation, decoherence or decays. For this, a simple ˆ ˆ σˆx−iσˆy example is the same Hamiltonian HS ∝ σˆz with the Lindblad L ∝ σˆ− (σˆ− = 2 ), where γ will be the spontaneous emission rate [6].

Acknowledgments

C. A. Brasil acknowledges support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) project number 2011/19848-4, Brazil. F. F. Fanchini acknowledges support from FAPESP and Conselho Nacional de Desenvolvimento Científico e Tec- nológico (CNPq) through the Instituto Nacional de Ciência e Tecnologia - Informação Quântica (INCT-IQ), Brazil. R. d. J. Napolitano acknowledges support from CNPq, Brazil.

[1] Göran Lindblad, “On the Generators of Quantum dynamical semigroups”, Commun. Math. Phys. 48, 119 (1976). [2] Heinz-Peter Breuer and Francesco Petruccione, The theory of open quantum systems (Oxford University Press, Oxford, 2002). [3] Ian Percival, diffusion (Cambridge University Press, Cambridge, 1998). [4] Carlos A. Brasil and Reginaldo de J. Napolitano, “The master equation for the reduced open-system dynamics, including a Lindbladian description of finite-duration measurement”, Eur. Phys. J. Plus 126, 91 (2011). [5] Carlos A. Brasil, Leonardo A. de Castro and Reginaldo de J. Napolitano, “Protecting a quantum state from environmental noise by an incompatible finite-time measurement”, Phys. Rev. A 84, 022112 (2011). [6] Michael A. Nielsen and Isaac L. Chuang, Computação quântica e informação quântica (Bookman, Porto Alegre, 2005). [7] Amir O. Caldeira and Anthony J. Leggett, Physica A 121, 587 (1983). [8] Edward Brian Davies, Quantum theory of open systems (Academic Press, London, 1976). [9] Robert Alicki and Karl Lendi, Quantum dynamical semigroups and applications (Springer-Verlag, Berlin, 2007). 11

[10] Stephen L. Adler, “Derivation of the Lindblad generator structure by use of the Itô stochastic calculus”, Phys. Lett. A 265, 58 (2000). [11] Stephen L. Adler, “Corrigendum to: ’Derivation of the Lindblad generator structure by use of the Itô stochastic calculus’ ”, Phys. Lett. A 267, 212 (2000). [12] Asher Peres, “Classical interventions in quantum systems. I. The measuring process”, Phys. Rev. A 61, 022116 (2000). [13] Claude Cohen-Tannoudji, Bernard Diu and Franck Lalo, (Wiley, New York, 1977). [11] Wolfgang P. Schleich, in Phase Space (Wiley, Berlin, 2001).