Lindbladian operators, von Neumann entropy and energy conservation in time-dependent quantum open systems

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1 Ldblada operators, vo Neuma etropy ad eergy coservato tme-depedet quatum ope systems Cogje Ou a, Ralph V. Chamberl b, Sumyosh Abe a,c,d a College of Iformato Scece ad Egeerg, Huaqao Uversty, Xame , Cha b Departmet of Physcs, Arzoa State Uversty, Tempe, AZ , USA c Departmet of Physcal Egeerg, Me Uversty, Me , Japa d Isttute of Physcs, Kaza Federal Uversty, Kaza , Russa ABSTRACT The Ldblad equato s wdely employed studes of Markova quatum ope systems. Here, the followg questo s posed: a quatum ope system wth a tme-depedet Hamltoa such as a subsystem cotact wth the heat bath, what s the correspodg Ldblad equato for the quatum state that keeps the teral eergy of the subsystem costat tme? Ths ssue s of mportace realzg quas-statoary states of ope systems such as quatum crcuts ad batteres. As a llustratve example, the tme-depedet harmoc oscllator s aalyzed. It s show that the Ldblada operator s uquely determed wth the help of a Le-algebrac structure, ad the tme dervatve of the vo Neuma etropy s show to be oegatve f the curvature of the harmoc potetal mootocally decreases tme. Keywords: Quatum dsspatve systems, Ldblad equato, Coservato of teral eergy, vo Neuma etropy 1

2 1. Itroducto Quatum ope systems have log bee attractg partcular atteto coecto wth a varety of problems such as errors quatum computato, measuremets, decoherece for mcro-macro trasto, ad foudatos of statstcal mechacs. Accordgly, a lot of effort has bee devoted to the study of outary quatum subdyamcs. The stadard approach s to cosder a solated multpartte system govered by utary dyamcs wth gve teractos, to detfy a objectve subsystem, ad the to elmate the remag evrometal degrees of freedom to obta the subdyamcs of the objectve subsystem. I cases where teracto ad etaglemet betwee the objectve subsystem ad ts evromet are ot strog, t may be possble to detfy a partal Hamltoa of the subsystem, whch however caot fully descrbe the subdyamcs because of the outarty. A questo of terest here s: how s t possble for the teral eergy of such a ope system to be coserved? Ths s relevat to characterzg quas-statoary quatum ope systems that are of cotemporary mportace. A example s foud quatum thermodyamcs, where a costat-teral-eergy process (refereed to as a soeergetc process) s dfferet from a sothermal process, geeral, because of the quatum-mechacal volato of the law of equpartto of eergy that may lead to some exotc propertes of quatum heat eges [1]. A couple of other examples are quatum crcuts [2] ad batteres (see [3] ad the refereces cted there). The questo ca drastcally be smplfed the Markova approxmato, where the equato becomes that of the 2

3 Ldblad type [4-7]. Ad, aother ssue cocerg the quatum subdyamcs s the tme evoluto of the vo Neuma etropy. The purpose of ths paper s to study the problems metoed above. I Sec. 2, coservato of the teral eergy s dscussed for a quatum ope system wth a tme-depedet Hamltoa. There, the equato s derved for the Ldblada operators that coserve the teral eergy. I Sec. 3, the example of the tme-depedet harmoc oscllator s aalyzed detal. It s show that the codto of coservato of the teral eergy uquely determes the Ldblada operators ad the tme dervatve of the vo Neuma etropy s oegatve accordace wth complete postvty of the subdyamcs. Secto 4 s devoted to cocludg remarks. I addto, a succct revew s gve Appedx about tme evoluto of the vo Neuma etropy uder the Ldblad equato. 2. Ldblad equato ad coservato of teral eergy of tme-depedet system Cosder a quatum ope system wth a tme-depedet Hamltoa, H (t). The desty matrx,! (t), descrbg ts state s assumed to obey the Ldblad equato:!!! (t) /!t = [ H (t),! (t)]" # a m Q m Q! (t) +! (t)q m Q " 2Q! (t)q m m, ( ), where Q s ad a m are the operators resposble for outarty of the subdyamcs ad the elemet of a c-umber Hermta matrx, respectvely, ad geeral both of them may also deped explctly o tme. Makg use of the c-umber utary matrx wth the 3

4 elemets u ml, we express the c-umber matrx as a m =! u ml! l u * l, where! l s l real. Accordgly, defg, L l =! u ml Q m we rewrte the equato as follows: m!!! (t)!t ( ). (1) = [ H (t),! (t)]" #! L! (t) +! (t)l " 2! (t) Ths lear equato preserves the ormalzato codto, tr! (t) = 1, ad s kow to geerate a completely postve dyamcal sem-group f! s are oegatve [4-7]. The oegatvty codto s requred order to corporate ay possble quatum state. The teral eergy s gve by E = H (t)! tr( H (t)! (t)). We are terested a physcal stuato where E s coserved tme. Therefore, we have $ tr!! H (t)! (t) " H (t) #! L! (t) +! (t)l & " 2! (t) %!t ( ) ' ) = 0, (2) ( where Eq. (1) ad tr( H (t) [ H (t),! (t)]) = 0 have bee used. Ths codto s satsfed f the followg equato holds:!! H (t)!t = "! ( L H (t) + H (t)l # 2 L H (t) ). (3) Thus, gve H (t), we requre s to be determed by ths geeral codto. Mathematcally, Eq. (3) may also cota a term that has a vashg expectato value. A example s A! A, where A s a certa observable. However, we do ot 4

5 cosder such a case here sce we wsh the Hamltoa to be depedet of the quatum state. From Eq. (3), t s clear that at least oe of s should ot commute wth the Hamltoa. Also, the case whe the Hamltoa ad s belog to the trace class, t follows from Eq. (3) that the sum of the egevalues of the Hamltoa s costat tme f! " L, # $ = 0 (.e., s are ormal). It s emphaszed that Eq. (3), whch may determe s, does ot deped o the detals of the teracto betwee the objectve subsystem ad ts evromet, mplyg how coservato of the teral eergy sets a strget codto o the master equato. 3. A example: Tme-depedet harmoc oscllator Let us apply the geeral dscusso developed above to a smple but llustratve example of the harmoc oscllator wth ut mass ad tme-depedet sprg coeffcet, k (t). The Hamltoa s H (t) = p 2 / 2 + k (t) x 2 / 2. It s coveet to troduce the operators: K 1 = p 2 / 2, K 2 = x 2 / 2, K 3 = (x p + p x) / 2. These satsfy the commutato relatos:! " K 1, K 2 # $ = %!K 3,! " K 2, K 3 # $ = 2!K 2,!K " 3, K 1 # $ = 2!K 1, whch show that K a s form a Le algebra somorphc to su(1,1). The Hamltoa s the expressed as 5

6 H (t) = K 1 + k (t)k 2. (4) As see below, s ca be chose to be Hermta ad, accordgly, Eq. (3) s rewrtte as follows:!! H (t)!t = &! ", " #, H (t) $ $ # %%. (5) I addto, t turs out to be suffcet to cosder oly oe operator, say L 1! L. That s,! 1!! (t) (! 2 =! 3 =!!! = 0). Because of the Le-algebrac structure, L has the form: L = c 1 K 1 + c 2 K 2 + c 3 K 3, (6) where c a s are real c-umbers. The, Eq. (5) s calculated to be!k (t)k 2 =!2"! (t) k (t)c 2 2 {( 1! c 1 c 2 + 2c 3 ) K 1! k(t)c 1 c 2! c 2 2 ( 2! 2k(t)c 3 )K 2 + ( k(t)c 1 + c 2 )c 3 K 3}, (7) where! k (t)! d k (t) / d t. Therefore, we have the followg coupled equatos: k (t)c 2 1! c 1 c 2 + 2c 2 3 = 0, k! (t) = 2!! (t) k(t)c 1 c 2! c 2 2 ( 2! 2k(t)c 3 ), ( k(t)c 1 + c 2 )c 3 = 0. The oly otrval soluto of these equatos s: c 1 = c 3 = 0, c 2 2 =! k! (t) /[ 2!! (t)]. Sce c 2 ca always be absorbed the defto of! (t), we may set t equal to uty. Therefore, we have L = K 2 = x 2 / 2,! (t) =!! k (t) / (2"). Cosequetly, we 6

7 obta the followg master equato:!!! (t)!t k = [ H (t),! (t)]+! (t) 8" " # x 2, "# x 2,! (t) $ % $ %. (8) I order for ths dyamcs to be completely postve, the codto!k (t)! 0 (9) should hold. Equato (8) should be compared wth the oe dscussed Ref. [8] for formulatg quatum dyamcs of macroscopc objects. There, the operator correspodg to L s lear wth respect to the posto operator, whereas L Eq. (8) s quadratc. Fally, let dscuss how the vo Neuma etropy S [!] =! tr (! l!) (10) evolves tme uder the master equato Eq. (8). Equato (A.1) Appedx ca be wrtte terms of! 1 "! = tr l! {( )( L 2! # L! L) } as d S d t =! k! (t) ". (11) " 2 Sce L s Hermta, t mmedately follows from Eq. (A.3) Appedx that! " 0. (12) 7

8 Therefore, wth Eq. (9), we fd that d S / d t! 0. Ths result may have a smple physcal terpretato. Equato (9) mples that the harmoc potetal s wdeg (.e., expaso), ad accordgly the eergy spectrum s beg lowered. To coserve the teral eergy, the oscllator has to absorb the eergy from the evromet, e.g., heat eergy, f the evromet s the heat bath. 4. Cocludg remarks We have dscussed quatum ope systems, whose Hamltoas are depedet o tme explctly but wth teral eerges beg coserved. We have derved the codto o the Ldblada operators that must be satsfed such a stuato. We have aalyzed detal the tme-depedet Harmoc oscllator as a llustratve example ad have show how the correspodg Ldblada operator ca be obtaed from the codto of coservato of the teral eergy. We have also show that the tme dervatve of the vo Neuma etropy s oegatve accordace wth the tme depedece of the oscllator Hamltoa. Ackowledgmets The work of CO was supported by the grats from Fuja Provce (No. 2015J01016, No. JA12001, No. 2014FJ-NCET-ZR04) ad from Huaqao Uversty (No. 8

9 ZQN-PY114). He also thaks Toka-Doghua Educatoal ad Cultural Exchage Foudato for provdg hm wth the scholarshp ad Me Uversty for the hosptalty exteded to hm. SA would lke to thak the Hgh-Ed Foreg Expert Program of Cha for support ad the warm hosptalty of Huaqao Uversty. Hs work was also supported part by a Grat--Ad for Scetfc Research from the Japa Socety for the Promoto of Scece (No ) ad by the Program of Compettve Growth of Kaza Federal Uversty from the Mstry of Educato ad Scece of the Russa Federato. Note added. Recetly, the cocept of weak varats has bee proposed ad studed for tme-depedet quatum dsspatve systems [9]. From ths vewpot, the Hamltoa Eq. (4) ca be regarded as a weak varat assocated wth the Ldblad equato (8). Appedx Here, let us dscuss how the vo Neuma etropy Eq. (10) evolves tme uder the Ldblad equato. It tured out that ths ssue has already bee studed almost 30 years ago [10]. However, t seems coveet for the reader to preset a succct revew of the dscusso. Usg the ormalzato codto o the desty matrx ad Eq. (1) as well as the basc propertes of the trace operato, we fd 9

10 d S d t = 2!! ", (A.1)! where! s gve by [ ] L { #! (t) "! (t) }. (A.2)! = tr l! (t) $ % & The purpose here s to show that the quatty Eq. (A.2) satsfes the followg equalty:! " # $ L, % & ' tr {# L, % }. (A.3) $ &! (t) To do so, frst let us perform the stataeous dagoalzato of the desty matrx at tme t:! (t) =! p (t) (t) u (t), where { (t) } s a certa complete orthoormal system satsfyg (t) u j (t) =! j ad! (t) u (t) = I wth I beg the detty matrx. It s assumed here that the desty matrx s postve defte,.e., p (t)!(0, 1) [that should satsfy the ormalzato codto:! p (t) = 1 ]. Substtutg the dagoalzed form of the desty matrx to Eq. (A.2), we have ( )! = " p l p L + " # p j l p, j ( ) u j 2. (A.4) Now, from the equalty l A! A "1 (A > 0) ad detfcato A = p / p j, t follows that! p j l p "! p j l p j + p j! p. Therefore, we fd that 10

11 ( )! " # p l p L + # $ p j l p j + p j $ p, j ( ) u j 2 ( ) =! p l p L "! p j l p j, j ( ) u j u j +! p j " p, j ( ) u j u j =! p " # L, $ %, (A.5), whch proves Eq. (A.3). It should be oted that the etropy rate gve Eq. (A1) s the sum of! s weghted by! s ad therefore ts sg s ot mmedately determed by the sg of each!. I a recet work [11], Eqs. (A.1) ad (A.3) have bee geeralzed for the Réy etropy. Therefore, ow they are see to be the smple lmtg cases of the results gve there. Refereces [1] C. Ou, S. Abe, EPL 113 (2016) [2] A. K. Rajagopal, Phys. Lett. A 246 (1998) 237. [3] F. C. Bder, S. Vjaampathy, K. Mod, J. Goold, New J. Phys. 17 (2015)

12 [4] G. Ldblad, Commu. Math. Phys. 48 (1976) 119. [5] V. Gor, A. Kossakowsk, E. C. G. Sudarsha, J. Math. Phys. 17 (1976) 821. [6] G. Mahler, Quatum Thermodyamc Processes: Eergy ad Iformato Flow at the Naoscale, Pa Staford, Sgapore, [7] T. Baks, L. Susskd, M. E. Pesk, Nucl. Phys. B 244 (1984) 125. [8] G. C. Ghrard, A. Rm, T. Weber, Phys. Rev. D 34 (1986) 470. [9] S. Abe, Phys. Rev. A 94 (2016) [10] F. Beatt, H. Narhofer, Lett. Math. Phys. 15 (1988) 325. [11] S. Abe, Phys. Rev. E 94 (2016)

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