Não Markovianidade em modelos colisionais Gaussianos

Transcrição

1 Uiversidade de São Paulo Istituto de Física Não Markoviaidade em modelos colisioais Gaussiaos Rolado Ramirez Camasca Orietador: Prof. Dr. Gabriel Teixeira Ladi Dissertação de mestrado apresetada ao Istituto de Física da Uiversidade de São Paulo, como requisito parcial para a obteção do título de Mestre em Ciêcias. Baca Examiadora: Prof. Dr. Gabriel Teixeira Ladi - Istituto de Física da Uiversidade de São Paulo Prof. Dr. - Prof. Dr. - São Paulo 2020

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3 Uiversity of São Paulo Physics Istitute No-Markoviaity i Gaussia Collisioal Models Rolado Ramirez Camasca Supervisor: Prof. Dr. Gabriel Teixeira Ladi Dissertatio submitted to the Physics Istitute of the Uiversity of São Paulo i partial fulfillmet of the requiremets for the degree of Master of Sciece. Examiig Committee: Prof. Dr. Gabriel Teixeira Ladi - Physics Istitute of the Uiversity of São Paulo Prof. Dr. - Prof. Dr. - São Paulo 2020

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5 Erest Hemigway oce wrote: The world is a fie place ad worth fightig for. I agree with the secod part. Se7e (1995) i

6 To my gradfather... ii

7 Abstract Memory effects i the dyamics of ope systems have bee the subject of sigificat iterest i the last decades. The methods ivolved i quatifyig this effect, however, are ofte difficult to compute ad may lack aalytical isight. With this i mid, we study collisioal models where o-markoviaity is itroduced by meas of additioal iteractios betwee eighborig evirometal uits. We show that the dyamics ca be cast i terms of a Markovia Embeddig of the covariace matrix, which yields closed form expressios for the memory kerel that govers the dyamics, a quatity that ca seldom be computed aalytically. The same is also possible for a divisibility mootoe, based o the complete positivity of itermediate maps. By focusig o cotiuous-variables Gaussia dyamics, we are able to aalytically study models of arbitrary size. We aalyze i detail two types of iteractios, a beam-splitter implemetig a partial SWAP ad a two-mode squeezig, which etagles the acillas ad, at the same time, feeds excitatios ito the system. By aalyzig the memory kerel ad divisibility for these two represetative scearios, our results help to shed light o the itricate mechaisms behid memory effects i the quatum domai. Keywords: Ope Quatum Systems; Collisioal Model; No-Markoviaity; Memory Kerel; Map Divisibility. iii

8 Resumo Os efeitos da memória a diâmica dos sistemas abertos têm sido objeto de grade iteresse as últimas décadas. Os métodos evolvidos a quatificação desse efeito, o etato, muitas vezes são difíceis de calcular e podem carecer de uma visão aalítica. Com isso em mete, estudamos modelos colisioais em que a ão markoviaidade é itroduzida por meio de iterações adicioais etre uidades ambietais vizihas. Mostramos que a diâmica pode ser moldada em termos de um Embeddig Markoviao da matriz de covariâcia, que produz expressões de forma fechada para o kerel de memória que govera a diâmica, uma quatidade que raramete pode ser calculada aaliticamete. O mesmo também é possível para uma divisibilidade moótoa, baseada a positividade completa dos mapas itermediários. Ao focarmos a diâmica gaussiaa de variáveis cotíuas, podemos estudar aaliticamete modelos de tamaho arbitrário. Aalisamos em detalhes dois tipos de iterações, um beam splitter implemetado um SWAP parcial e uma compressão de dois modos, que emaraha as acillas e, ao mesmo tempo, alimeta excitações o sistema. Ao aalisar o kerel de memória e a divisibilidade para esses dois ceários represetativos, ossos resultados ajudam a laçar luz sobre os itricados mecaismos por trás dos efeitos da memória o domíio quâtico. Keywords: Sistemas quâticos abertos; Modelos colisioais; Não-Markoviaidade; Kerels de memoria; CP-Divisibilidade. iv

9 Cotets 1 Itroductio 1 2 Ope Quatum Systems States ad Dyamics Memory Kerel Microscopic derivatio: Qubit iteractig with bosoic eviromet Lidblad Equatio Cotiuous Variable Formalism Coheret States Gaussia States ad Maps Multi-mode States Gaussia Dyamics Collisioal Models Motivatio Formal Framework Gaussia Collisioal Model Markovia Embeddig Example Dyamics No-Markoviaity Quatum o-markoviaity Mutual Iformatio Memory Kerel CP-Divisibility Memory effects i Gaussia Collisioal Models Gaussia Mutual Iformatio Memory Kerel Geeral derivatio of the Memory Kerel Memory Kerel for the BS dyamics Memory Kerel for the TMS dyamics Gaussia CP-Divisibility BS dyamics TMS dyamics v

10 Cotets 7 Coclusios 67 A Stability Theory 70 B Memory Kerel for the BS dyamics 72 vi

11 List of Figures 1.1 No-Markovia collisioal models. (a) First few steps of the dyamics. The system-acilla iteractios SE are iterspersed by acilla-acilla iteractios E E +1, which propagate iformatio forward, makig the dyamics o-markovia i a fully cotrollable way. (b) Basic structure of the Markovia embeddig dyamics, which is a map from the Hilbert space of SE to that of SE +1. (c) Mutual iformatio. Previous iteractios SE ad E E +1 correlate the system S ad the acilla E +1 eve before explicitly iteractig. (d) The memory kerel quatifies how differet istats of the past affect the evolutio at preset times. (e) CPdivisibility. The maps i gray, from time 0 to t or t m are, by costructio, CPTP. But the itermediate map from t to t m > t may ot ecessarily be Schematics of a ope quatum system. The system S is iteractig with the eviromet while the whole bipartite system ρ SE is cosider as closed Qubit system S iterchagig excitatios with ifiite bosoic eviromets. The evirometal uits are modeled as harmoic oscillators with frequecy ω k iteractig oly with the system proportioal to the couplig stregth g k Husimi-Q fuctio of the thermal Gibbs state with occupatio umber = First few steps of the collisioal model dyamics. The system-acilla iteractios SE are iterspersed by acilla-acilla iteractios E E +1, which propagate iformatio forward, makig the dyamics o-markovia i a fully cotroballe way Basic structure of the Markovia embeddig dyamics (4.20), which is a map from the Hilbert space of SE to that of SE Secod momets as a fuctio of time, computed from Eq. (4.20) for the BS dyamics (4.21) with λ s = 0.5 ad differet values of λ e (with λ e > 0 o the left colum ad λ e < 0 o the right colum). The acillas are assumed to start i the vacuum, ad the system i a thermal state with a a 0 = 20. (a,b) Number of excitatios i the system a a. (c,d) Number of excitatio i the th acilla b b. (e,f) Correlatio fuctio betwee the system ad the th acilla a b vii

12 List of Figures 4.4 Secod momets as a fuctio of time, computed from Eq. (4.20) for the TMS dyamics (4.23) with λ s = 0.1 ad differet values of ν e (with ν e < νe crit o the left colum ad ν e νe crit o the right colum, where νe crit = sih 1 (1)). The acillas are assumed to start i the vacuum, ad the system i a thermal state with a a 0 = 20. (a,b) Number of excitatios i the system a a. (c,d) Number of excitatios i the th acilla b b. (e,f) Correlatio fuctio betwee the system ad the th acilla a b Quatum Mutual Iformatio (5.1) for the BS (a,b) with λ s = 0.5 (λ e > 0 i (a) ad λ e < 0 i (b)), ad TMS (c,d) dyamics with λ s = 0.1 (ν e < νe crit i (a) ad ν e νe crit i (b)). Other parameters are the same as Fig. 4.3 ad Fig Compariso betwee Markovia ad o-markovia dyamics ad role of the Mutual Iformatio. I blue circles we show the early dyamics of a a vs. for the BS dyamics with (a) λ e = 1.1 ad (b) λ e = 0.3, with fixed λ s = 0.5 [c.f. Fig. 4.3(a)]. The correspodig Markovia case (λ e = 0) is show i orage triagles. These curves are to be compared with the MI (5.1), show by gree squares i the two cases [Fig. 6.1(a)]. The heights of each curve were adjusted for better visibility The memory Kerel for the BS dyamics, Eq. (4.21). I this case the oly o-zero etry i Eq. (6.2) is κ 11, the term proportioal to the idetity. The plots are for λ s = 0.5 (upper pael) ad λ s = 0.05 (lower pael), with λ e > 0 (left) ad λ e < 0 (right) Diagrams for the memory kerel of the BS dyamics. Each plot shows κ 11 i the (λ s, λ e ) plae for a differet value of, from = 0 to = The memory Kerel for the (stable) TMS dyamics, Eq. (4.23) with λ s = 0.1 ad differet values of λ e. Each curve correspods to a differet etry of Eq. (6.2); amely, κ 11, κ 1,σ z, κ σ z,1 ad κ σ z,σ z The MK for Q 2 ad P 2, Eq. (6.24), for the TMS dyamics. Other parameters are the same as Fig Diagrams for the memory kerel coefficiet κ q [Eq. (6.24)] of the TMS dyamics, i the (λ s, ν e ) place, for = 0,..., Similar to Fig. 6.7, but for κ p Similar to Fig. 6.6, but for values of ν e close to, ad larger tha, νe crit = Example of the divisibility criteria for the BS dyamics. The plots show N m i the (, m) plae, with the size of each poit reflectig the magitude of N m. All curves are for λ s = 1.1 ad (a) λ e = 0.75, (b) 0.9, (c) 1.1 ad (d) CP-divisibility measure N +1, [Eq. (6.31)] i the (λ s, λ e ) plae, for the BS dyamics. Each plot correspods to a differet values of : i the first 2 lies, rages from 1 to 8 i steps of 1. I the remaiig lies, = 9, 10, 20, 21, 30, 31, 40, 41, 50, 51 ad 100, CP divisibility measure N +1, as a fuctio of, for the BS dyamics with λ s = 0.8 ad λ e = 0.9, 1.3, 0.5, 0.8. Complemets Fig. (6.11).. 63 viii

13 List of Figures 6.13 CP-divisibility measure, N m,1 [Eq. (6.31)] i the (λ s, λ e ) plae, for the BS dyamics. Each plot correspods to a differet values of m: i the first 2 lies, m rages from 2 to 9 i steps of 1. I the 3rd ad 4th, from m = 10 to 24 i steps of 2. lies, Regios i the (λ s, λ e ) plae where the BS dyamics is ot CP-divisible for at least oe choice of (, m) CP-divisibility measure, N +1, [Eq. (6.31)] i the (λ s, ν e ) plae, for the TMS dyamics. Each plot correspods to a differet values of, from 1 to 9 i steps of ix

14 Chapter 1 Itroductio The theory of ope quatum systems is the backboe of moder research i quatum mechaics ad its applicatios. Some examples iclude the decay of ustable states i uclei, trasport of electros through quatum dots, molecular etworks, aophotoic structures, amog others. This is because, i real life pheomea, the systems are ever isolated, but are costatly iteractig with its surroudig eviromet. I some cases, whe the effects of the eviromet are egligible, the closed system descriptio is accurate, described by the well-kow Liouvillia equatio. Nevertheless, more ofte tha ot, it is ot possible. Obtaiig the most geeral microscopic descriptio of the ope system dyamics is a very difficult problem, oe that is yet to be solved. Oe istace we ca get a microscopic derivatio is whe we assume that the systemeviromet iteractio is weak but also that the bath correlatio fuctios decay quickly. This is kow as the Bor-Markov approximatio [1 3]. Typically, whe the system iteracts with the eviromet, iformatio about the former is traslated to the latter. Whe the eviromet is very large ad complex, this iformatio may ever retur. I this case the dyamics is called Markovia. This is characterized by the Lidblad equatio give by [4]: d dt ρ S = i [H, ρ S] + k ( g k L k ρ S L k 1 }) {L k 2 L k, ρ S, (1.1) where ρ S is the system s desity matrix, H is the Hamiltoia, ad L k are arbitrary operators. I geeral, however, there may be a partial backflow of iformatio which charac- 1

15 Chapter 1. Itroductio terizes the o-markovia evolutio [5]. The dyamics of the ope quatum systems i the o-markovia regime ca geerally be writte as [6, 7]: dρ t S dt = i[h S, ρ S ] + 0 K t t [ρ S (t )] dt, (1.2) where K t t, called the memory kerel (MK), is a liear superoperator codesig all the iformatio o how the evolutio of ρ S at time t depeds o its past values. We see that the Markovia case (1.1) is recovered whe K t t δ(t t ). The MK has bee itesively studied i recet years, as it provides clear isights oto the ier workigs of o-markoviaity [8 13]. From the poit of view of causality, this backflow quatifies the ability of the dyamics to commuicate past iformatio to the future [14]. No- Markoviaity therefore touches at the core of iformatio processig where the otio of iformatio flow is fudametal. Cosiderable attetio was give i recet years o how to characterize ad quatify o-markoviaity i the quatum domai [15, 16]. Due to the richess ivolved, however, there is o sigle approach capable of capturig its full essece. The most importat otio is that of map divisibility: o-markoviaity requires that the uderlyig dyamical map should ot be divisible [17, 18]. The otio of iformatio flow, o the other had, relies o iformatio-theoretic quatifiers ad is thus ot uiquely defied. The most widely used measures ivolve the trace distace [17 20] betwee differet iitial states or etaglemet [21] betwee the system ad a acilla. Several other quatifiers have also bee explored [22 30]. However, aalyzig o-markoviaity for geeral eviromets is i geeral a extremely difficult task. First, the calculatios quickly become impractical whe the size of the bath is large. Ad secod, realistic baths ofte have may additioal features which ted to mask the effects oe is iterested i. This motivates the search for cotrollable models, where the degree of o-markoviaity ca be fiely tued. Oe way to accomplish this, which has see a eormous surge i popularity i recet years, are through the so-called collisioal models [31 42]. Usually, a commo cofiguratio to study o-markoviaity is to cosider the system S coupled to N evirometal uits E all at the same time (e.g. the Caldeira-Legett model [43]). I the collisioal models setup istead oly a small part of the eviromet is iteractig with the sys- 2

16 Chapter 1. Itroductio tem at each istat of time. Afterwards, this part leaves ad returs to the bath where it will collide with the other bath uits ad thermalize, forgettig all iformatio about the system. We model this by replacig the ope dyamics of a system by a series of sequetial iteractios betwee the system S ad small evirometal uits E 1, E 2, E 3... (heceforth referred to as acillas). All acillas are prepared i the same state ad each iteractio oly lasts for a fixed time, after which they ever iteract agai. This therefore leads to a stroboscopic dyamics for the system. Although seemigly artificial, collisioal models actually faithfully describe some experimetally relevat situatios, for istace whe a system is coupled to a 1D waveguide [44 47]. The dispersio relatio of 1D waveguides allows oe to discretize the field operator ito time bis, so that the iteractio at each time iterval oly ivolves oe bi operator. The picture that emerges is the exactly that of a collisioal model. Moreover, this model is i geeral o-markovia by costructio, which depeds o the iput state of the electromagetic field, as well as o the ature of the iteractio. The specific coditios determiig whether the esuig dyamics will be Markovia or ot are discussed i detail i a recet review o the subject [48]. The advatage of collisioal models is that o-markoviaity ca be itroduced i a fully cotrollable maer. There are two mai ways to do so. The first is to cosider that the acillas already start correlated [49 53]. The other oe is to assume iformatio is trasmitted betwee them durig the process [54 62]. I this dissertatio, we shall focus o the secod case. That is, we cosider a sceario where eighborig acillas E E +1 iteract with each other i betwee the iteractios SE ad SE +1. (see Fig. 1.1(a)). This additioal iteractio sigals iformatio from the past to the future, so that whe the SE +1 iteractio arrives, the acilla E +1 will already cotai some iformatio about the system. I particular, we focus o cotiuous-variable collisioal models, udergoig Gaussiapreservig dyamics [63 71]. The advatages that come with the Gaussia toolbox allows us to costruct a complete framework for the study of o-markoviaity, which: (i) ecompass a broad rage of scearios; (ii) ca be easily use to compute the mutual iformatio; (iii) allows for the explicit costructio ad computatio of the memory kerel ad (iv) provides access to a CP-divisibility mootoe, which ca be directly compared 3

17 Chapter 1. Itroductio (a) S E1 E2 E3 E1 S E1 (b) S E2 E2 E3 S E3 E1 E2 (c) (d) E3 (e) Figure 1.1: No-Markovia collisioal models. (a) First few steps of the dyamics. The system-acilla iteractios SE are iterspersed by acilla-acilla iteractios E E+1, which propagate iformatio forward, makig the dyamics o-markovia i a fully cotrollable way. (b) Basic structure of the Markovia embeddig dyamics, which is a map from the Hilbert space of SE to that of SE+1. (c) Mutual iformatio. Previous iteractios SE ad E E+1 correlate the system S ad the acilla E+1 eve before explicitly iteractig. (d) The memory kerel quatifies how differet istats of the past affect the evolutio at preset times. (e) CP-divisibility. The maps i gray, from time 0 to t or tm are, by costructio, CPTP. But the itermediate map from t to tm > t may ot ecessarily be. with the memory kerel. The framework is also ameable to aalytical calculatios ad extremely efficiet from a umerical perspective. Thus, despite beig restricted to Gaussia iteractios, it offers multiple advatages over more geeral maps. We also provide a complete umerical library for efficietly simulatig Gaussia collisioal models i Pytho [72]. All plots were geerated with this code. The dissertatio is divided as follows. We begi i Chapter 2 by itroducig importat cocepts of ope quatum systems ad review the differet methods to describe the ope system s dyamics. Particularly, we cosider a qubit iteractig with a bosoic eviromet to exemplify the approximatios eeded to obtai a microscopic derivatio. Next, we study the cotiuous variable formalism i Chapter 3. Specifically, we examie Gaussia states ad maps as they will become corerstoe for this work. The followig chapters are the most importat of the dissertatio. They represet the origial cotributio from the authors to the field of o-markoviaity i collisioal models. The geeral framework is developed i Chapter 4. We show that the full o-markovia dyamics 4

18 Chapter 1. Itroductio ca be cast as a Makorvia embeddig, ivolvig a Markovia map at a higher dimesio (Fig. 1.1(b)). We the specialize this to the case of Gaussia models, where the embeddig is writte as a set of matrix-differece equatios with clear physical iterpretatio. Throughout this chapter our expositio will be example-orieted, with a focus o two specific types of iteractios. The framework, however, is geeral ad we will specify, i each part, how to properly make this geeralizatio. Followig, i Chapter 5, we review o-markoviaity i the quatum domai as well as the quatifiers we will use for our collisioal model. Armed with all these results, i Chapter 6 we provide a full characterizatio of the memory effects usig the mutual iformatio (Fig. 1.1(c)), the memory kerel (Fig. 1.1(d)) ad the map divisibility (Fig. 1.1(e)). Fially, i Chapter 7 we draw our coclusios ad our perspectives o future works which ca be doe usig this robust framework. 5

19 Chapter 2 Ope Quatum Systems 2.1 States ad Dyamics Quatum mechaics is the fudametal theory that describes the behavior of ature at the micro scale [73 78]. However, it is ofte assumed that the quatum system is isolated, overlookig the effects of the iteractio with its surroudigs. Such effects ca lead to dramatic chages i the system s dyamics such as the dissipatio of the quatum correlatios, makig them ecessary for a accurate depictio of real pheomea. The field that studies this kid of pheomeo is called ope quatum systems [1 3]. The descriptio of ope quatum systems starts by defiig what the object of study is: the states. I quatum mechaics, we were iterested i kets ψ i which live i a Hilbert space H. Aalogously, ope quatum systems is iterested i the desity matrix ρ = c i ψ i ψ i which lives i L(H), the state space of liear operators from H to H. This desity matrix ρ satisfies the followig properties: Hermicity: ρ = ρ, Positivity: ρ 0, Normalizatio: Tr(ρ) = 1. Besides these properties, we also wat to kow how a state ρ evolves i time. For that purpose, we look at the evolutio of isolated systems which evolve through Schöridger s 6

20 Chapter 2. Ope Quatum Systems ρ SE : System + Eviromet H S H E S ρ S : System H S ρ E : Eviromet H E Figure 2.1: Schematics of a ope quatum system. The system S is iteractig with the eviromet while the whole bipartite system ρ SE is cosider as closed. equatio: t ψ(t) = i H(t) ψ(t). (2.1) The aalogue evolutio i the desity matrix formalism is kow as vo Neuma s equatio: dρ(t) dt = i [H(t), ρ(t)]. (2.2) This result holds oly for closed systems. The ext step is to obtai a descriptio of the ope system s dyamics. Oe way to achieve this is to cosider a bipartite system ρ SE which cotemplates both the system S ad a eviromet E, as a closed system evolvig uder vo Neuma s equatio, as depicted i Fig Tracig out the eviromet E will the lead to a effective descriptio of the system s desity matrix ρ S dyamics. Ufortuately, this is a complicated task, ofte relyig o multiple approximatio schemes to obtai aalytical results. We study this procedure i detail i the ext sectios. 2.2 Memory Kerel We follow the semial works of Nakajima ad Zwazig throughout this subsectio [6, 7]. We cosider the complete desity matrix ρ SE of a system S plus a eviromet E iitially ucorrelated ρ SE (t 0 ) = ρ S (t 0 ) ρ E (t 0 ) evolvig uder a Hamiltoia H SE = 7

21 Chapter 2. Ope Quatum Systems H S + H E + V : dρ SE (t) dt = i [H SE, ρ SE (t)]. (2.3) It is coveiet to formulate (2.3) i the iteractio picture ρ I SE = eih 0t ρ SE e ih 0t with respect to H 0 = H S + H E : dρ I SE (t) dt = i [ VI (t), ρ I SE(t) ] := L t ρ I SE(t), V I = e ih 0t V e ih 0t, (2.4) where we defied the superoperator L t. From here o, we will loose the superscript I i ρ I SE for otatioal coveiece. Our goal is to obtai the master equatio for the system ρ S (t) = Tr E (ρ SE (t)). To that ed, we defie a pair of orthogoal projectio operators P ad Q Pρ SE = Tr E (ρ SE ) ρ E, Qρ SE = ρ SE Tr E (ρ SE ) ρ E, (2.5) where ρ E is a arbitrary evirometal desity matrix, ot ecessarily related to the state of the eviromet. Beig projectio operators, they satisfy PQ = QP = 0 ad P +Q = I which follow from the defiitio. Applyig P to Eq. (2.4) we obtai d dt Pρ SE(t) = PL t ρ SE (t). (2.6) We ca the itroduce the idetity I = P + Q to get a differetial equatio for Pρ SE (t) d dt Pρ SE(t) = PL t ( P + Q ) ρse (t) = PL t Pρ SE (t) + PL t Qρ SE (t). (2.7) Similarly, we ca obtai a differetial equatio for Qρ SE (t) applyig Q to Eq. (2.3) d dt Qρ SE(t) = QL t Qρ SE (t) + QL t Pρ SE (t). (2.8) We ow have two coupled differetial equatios. We solve Eq. (2.8) for Qρ SE (t), Qρ SE (t) = G(t, t 0 )Qρ SE (t 0 ) + t t 0 dt G(t, t )QL t Pρ SE (t ), (2.9) 8

22 Chapter 2. Ope Quatum Systems where G(t, t ) is the Gree s fuctio G(t, t 0 ) = T exp ( t t 0 dt QL t ). Isertig (2.9) i (2.7) leads to the Nakajima-Zwazig master equatio d dt Pρ ( t SE(t) = PL t PρSE (t)+g(t, t 0 )Qρ SE (t 0 )+ dt G(t, t )QL t Pρ SE (t ) ). (2.10) t 0 This is a reduced equatio for Pρ SE (t) after itegratig out the eviromet. Most importatly, this equatio is exact. We ca further reduce this expressio by fixig the arbitrary ρ E to be the eviromet s iitial state ρ E(t 0 ). This selectio implies that Qρ SE (t 0 ) = 0, leadig to d t dt ρ S(t) ρ E (t 0 ) = PL t Pρ SE (t)+ t 0 dt Tr E ( ( Lt G(t, t )QL t ρs (t ) ρ E (t 0 ) )) ρ E (t 0 ). (2.11) Fially, if [ρ E (t 0 ), H E ] = 0 we ca redefie the iteractio such that PL t Pρ SE (t) = 0, thus arrivig to: d t dt ρ ( S(t) = dt K t,t ρ S (t ), K t,t = Tr E Lt G(t, t )QL t(... ρ E (t 0 )) ), (2.12) t 0 where K t,t is the memory kerel, a liear superoperator codesig all the iformatio o how the evolutio of ρ S (t) depeds o its past values. 2.3 Microscopic derivatio: Eq. (2.12) still ivolves a superoperator, which is quite geerally difficult to compute aalytically. Approximatios are the eeded to obtai a more maageable equatio, oe which will be more useful uder certai coditios. We start by re-scalig the iteractio term V I to αv I by a dimesioless small parameter α. The assumptio that α is small is usually called the Bor approximatio. Expadig the memory kerel to secod order, we get ( K t,t = α 2 Tr E Lt QL t P ) + O ( α 3), (2.13) 9

23 Chapter 2. Ope Quatum Systems which leads to a equatio of motio of secod order for ρ S (t) d t dt ρ S(t) = α 2 t 0 dt Tr E ( Lt L t ρ S (t ) ρ E (t 0 ) ). (2.14) Puttig the explicit expressios for the superoperator L t we get d t dt ρ S(t) = dt Tr E [V I (t), [V I (t ), ρ S (t ) ρ E (t 0 )]]. (2.15) t 0 Next, we proceed with the Markovia approximatio. We assume that the eviromet s correlatios decays much faster tha the system s so that the state of the system at time t oly depeds o the preset state ρ S (t). To that ed, we make the previous equatio time local by replacig ρ S (t ) by ρ S (t): d t dt ρ S(t) = dt Tr E [V I (t), [V I (t ), ρ S (t) ρ E (t 0 )]]. (2.16) t 0 This is called the Redfield equatio. Still this equatio is ot yet Markovia sice the time evolutio depeds upo the iitial time t 0. We ca make the substitutio t by t t i Eq. (2.16) ad itegrate up to ifiity. This is possible as we are assumig that the time scale over which the state of the system varies is large compared to the time scale of the decay of the eviromet s correlatios. Without loss of geerally, we ca set t 0 = 0 ad obtai d dt ρ S(t) = 0 dt Tr E [V I (t), [V I (t t ), ρ S (t) ρ E (0)]]. (2.17) The result is a itegro-equatio for ρ S (t) which we ca calculate by pluggig i the iteractio potetial i the iteractio picture. Eve so, we will ofte eed to add more approximatios to obtai a differetial equatio for the ρ S (t). 2.4 Qubit iteractig with bosoic eviromet Oe iterestig model which simulates the effects a eviromet ca have o the system is a two-level system iteractig with a bosoic bath. For example, this describes a atom 10

24 Chapter 2. Ope Quatum Systems iteractig with the may modes of a radiatio field. The Hamiltoia of this model is: H = 1 2 ω 0σ z + k ω k b k b k + k g k (σ + b k + σ b k ), (2.18) where we are modelig the eviromet as ifiite harmoic oscillators, all with differet frequecies ω k. The iteractio betwee our qubit system ad the eviromet is of the form of a typical exchage: creates a excitatio o the system/bath by destroyig oe o the bath/system with stregth g k. S Figure 2.2: Qubit system S iterchagig excitatios with ifiite bosoic eviromets. The evirometal uits are modeled as harmoic oscillators with frequecy ω k iteractig oly with the system proportioal to the couplig stregth g k. We move to the iteractio picture to get V I (t): V I (t) = e ih 0t V e ih 0t = k g k ( e i k t σ + b k + e i kt σ b k), (2.19) where k = ω 0 ω k, ad compute the commutator term iside the trace of Eq. (2.17). d dt ρ S(t) = 0 dt Tr E ( VI (t)ρ S (t)ρ E V I (t t ) ρ S (t)ρ E V I (t t )V (t) ) +h.c. (2.20) where h.c. stads for hermitia cojugate. We focus o the first term iside the itegral 11

25 Chapter 2. Ope Quatum Systems Tr E V I (t)ρ S (t)ρ E V I (t t ): g k g q (e i( k+ q)t i qt σ + ρ S σ + b q b k + e i( k q)t+i qt σ + ρ S σ b qb k k,q e i( k q)t i qt σ ρ S σ + b q b k + e i( k+ q)t+i qt σ ρ S σ b qb k ). To compute the correlatios, we eed to kow the state of the eviromet. We assume that the bath is i a thermal state, meaig that: b q b k = b qb k = 0, b qb k = b k b q δ kq = δ kq ωk, (2.21) where ωk is the bosoic occupatio distributio 1/(e βω k 1). This leads to: Tr E V I (t)ρ S ρ E V I (t t ) = k g 2 k( e i k t ωk σ + ρ S σ +e i kt ( ωk +1)σ ρ S σ + ). (2.22) It is useful to defie a quatity called the spectral desity of the bath: J(ω) = 2π k g 2 kδ(ω ω k ), (2.23) such that the first term i Eq. (2.20) ow looks like: 0 dt 0 dω 2π J(ω)( ) i(ω0 ω)t e ω σ + ρ S σ + e i(ω 0 ω)t ( ω + 1)σ ρ S σ +. (2.24) To compute the itegral we use the kow idetity: 0 dt 0 ω)t ei(ω = 1 2π 2 δ(ω 0 ω) i ( ) 1 2 P, (2.25) ω 0 ω where P is the Cauchy pricipal value. This term oly accouts to a rescalig of the frequecy ω 0 to ω 0 + i the uitary dyamics. This et effect is called the Lamb shift. O the other had, the first term will lead to ew terms i the evolutio of ρ S which are of dissipative ature. Focusig o this term we the get: 0 dω 2 J(ω)δ(ω 0 ω) ( ) J(ω 0 )( ) ω σ + ρ S σ +( ω +1)σ ρ S σ + = ω0 σ + ρ S σ +( ω0 +1)σ ρ S σ

26 Chapter 2. Ope Quatum Systems A similar precedure ca be doe to the other terms i Eq. (2.20), resultig i equal terms up to some permutatios of the operators. Pluggig them back, we get: d dt ρ S(t) = γ ( σ + ρ S σ 1 2 {σ σ +, ρ S } ) + γ( + 1) ( σ ρ S σ {σ +σ, ρ S } ). (2.26) Goig back to the Schrödiger picture simply reitroduces the uitary cotributio ad we thus fially arrive at d dt ρ S(t) = i [ ] ω0 + σ z, ρ S (t) + D(ρ S (t)), (2.27) 2 where the first term is the familiar uitary term ad D(ρ S (t)) is the dissipative cotributio foud i Eq. (2.26). By compariso with the closed system s evolutio, the effects of tracig out the eviromet results i ew additioal terms. Still, several approximatios ad assumptios were eeded to arrive to this equatio, most of which do ot hold i real experimets. 2.5 Lidblad Equatio Suppose the evolutio of the system is time local, d dt ρ S(t) = R t ρ S (t), (2.28) for some superoperator R t. We the ask: What is the geeral structure of R t required to esure that startig with a desity matrix ρ(0), the evolved state ρ(t) is always a geuie desity matrix for all later times t? Let us start with a simpler questio. Suppose the solutio is give by a liear map of the form: ρ S (t) = V t (ρ S (0)), (2.29) for some superoperator V t. The questio ow becomes: What coditios shall V t fulfill i order for ρ S (t) be a desity matrix? The aswer is that the map should be a quatum operatio, oe that is completely positive trace preservig (CPTP) [79]. This ca be 13

27 Chapter 2. Ope Quatum Systems represeted by the operator-sum represetatio V t (ρ S (0)) = k M k ρ S (0)M k, M k M k = 1, (2.30) k where the set of M k are called the Kraus operators [80]. Returig ow to the origial questio, if (2.29) is to be the solutio of (2.28) the V t must satisfy the semi-group property V t2 V t1 = V t2 +t 1. (2.31) The semi-group property, together with the restrictio that V t must be a quatum operatio, completely determies the basic structure of R t. The geerator of ay quatum operatio satisfyig the semigroup property must have the form [4] d dt ρ S = i [H, ρ S] + k ( g k L k ρ S L k 1 }) {L k 2 L k, ρ S, (2.32) where H is a Hermitia operator, L k are arbitrary operators ad g k 0. This is called Lidblad s theorem ad master equatios havig this structure are called Lidblad equatios. We ca see that our microscopic derivatio of the system s desity evolutio Eq. (2.27) already has this structure. We proceed to formally proof Lidblad s theorem. The semigroup property esures that the evolutio of the desity matrix for a ifiitesimal t is ρ S (t + t) = k M k ( t)ρ S (t)m k ( t), (2.33) ad expadig ρ S (t + t) to first order i t, we fid ρ S (t + t) = ρ S (t) + t d dt ρ S(t) + O ( t 2). (2.34) This fixes the correct scalig of the M k o t. We choose the k = 0 term to be proportioal to the idetity which will guaratee that i the limit of t 0 we get the correct result. Similarly, the other terms are proportioal to t to esure that products 14

28 Chapter 2. Ope Quatum Systems M k (...)M k are of order t. That is, we take M 0 = I + G t, M k = tl k, k 0, (2.35) where G ad L k are arbitrary operators. Normalizatio of the Kraus operators leads to M k M k= ( I + G t )( I + G t ) + t L k L k, (2.36) k 0 k = I + (G + G ) t + t k 0 L k L k + O ( t 2). (2.37) We ca parametrize G = K ih where K ad H are both Hermitia. It follows from the ormalizatio coditio K = 1 L k 2 L k. (2.38) k 0 We ow substitute our results ito Eq. (2.33) to fially get: ρ S (t + t)= ( I + G t ) ρ S ( I + G t ) + t k 0 L k ρ S L k, (2.39) = ρ S (t) + t ( Gρ S + ρ S G ) + t k 0 L k ρ S L k, (2.40) = ρ S (t) i t[h, ρ S] + k 0 ( L k ρ S L k 1 }) {L k 2 L k, ρ S. (2.41) Rearragig ad takig the limit t 0 we get: ρ S (t + t) ρ S (t) t = d dt ρ S = i [H, ρ S] + k 0 ( L k ρ S L k 1 }) {L k 2 L k, ρ S, (2.42) which is exactly Eq. (2.32). 15

29 Chapter 3 Cotiuous Variable Formalism 3.1 Coheret States A useful set of physical states are the coheret states. This basis allows the descriptio of several experimets, most promietly i quatum optics where the electromagetic field is described by quatized harmoic oscillators [2, 81]; it allows for the descriptio of quatum may-body pheomea by meas of the coheret path itegral [82 86]; ad, specially i quatum iformatio, they represet a corerstoe i all facets of iformatio processig [63, 79, 87, 88]. The coheret states α are defied as the eigestates of the ahilatio operator a: a α = α α, α = e αa α a 0, (3.1) where we will heceforth refer to 0 as the vacuum state. Although ot crucial, it will be useful to state some properties of the coheret states [2, 73, 82]: Takig the hermitia cojugate of Eq. (3.1), we get the bra α which is a left eigestate of the creatio operator a, α a = α α, (3.2) where α is the complex cojugate of α. 16

30 Chapter 3. Cotiuous Variable Formalism The actio of the creatio operator a o the coheret state yields a α = α α. (3.3) The overlap betwee two coheret states is give by µ α = e µ α 1 2 µ α 2. (3.4) I particular, the orm of a coheret state is α α 2 = 1. Most importatly, the coheret states form a overcomplete set of states: d 2 α α α = 1, (3.5) π where d 2 α = d Re α d Im α. The trace of a desity matrix ρ i the coheret basis is give by d 2 α Tr(ρ) = α ρ α. (3.6) π The expectatio value of ay operator O i the coheret basis is d 2 α O = Tr(ρO) = α ρo α. (3.7) π The coheret basis is particularly useful whe represetig the desity matrix i quatum phase space, the quatum aalogue of classical phase space. Amog the differet represetatios, we choose the Husimi-Q fuctio [89]. This is defied as the expectatio value of the desity matrix ρ i the coheret basis α Q(α, α ) = 1 α ρ α. (3.8) π As a example, let us take ρ to be the thermal Gibbs state ρ = e βωa a Tr ( ). (3.9) e βωa a 17

31 Chapter 3. Cotiuous Variable Formalism Its Husimi fuctio will be Q(α, α ) = 1 e βω π e βω α α = =0 1 α 2 π( + 1) e +1, (3.10) where = (e βω 1) 1 is the bosoic occupatio umber. We plot i Fig. 3.1 the Gibbs state with ñ = 20. We ca also use the Husimi fuctio to calculate the expectatio value of ati-ormally ordered operators. That is operators of the form a k (a ) l = d 2 α α k (α ) l Q(α, α ), (3.11) which reduces to a itegral i the complex plae of α. 3.2 Gaussia States ad Maps The Gibbs state is a example of a Gaussia state. These are characterized by their represetatio i quatum phase space where its Husimi-Q fuctio is a Gaussia fuctio of the coheret variables Q(α, α ) = 1 π Θ exp ( 1 2 α Θ 1 α ), α = (α, α ), (3.12) where Θ is called the covariace matrix ad Θ is the determiat of such matrix Im(α) Re(α) Figure 3.1: Husimi-Q fuctio of the thermal Gibbs state with occupatio umber =

32 Chapter 3. Cotiuous Variable Formalism Related to Gaussia states is the cocept of Gaussia maps [67 69, 71, 90]. Gaussia maps are trasformatios that preserves Gaussiaity. That is, that take Gaussia states ito Gaussia states. Dyamically, this is achieved by imposig certai restrictios i the Lidblad equatio (2.32): The Hamiltoia H must be at most quadratic i a ad a. Thus, the most geeral Gaussia preservig Hamiltoia has the form H = ωa a + γa a + γ aa + fa + f a. (3.13) The Lidblad geerators L k must be liear o a ad a. As a example the thermal bath geerator D(ρ) = γ ( a ρa 1 2 { aa, ρ }) + γ( + 1) ( aρa 1 2 { a a, ρ }), (3.14) is a Gaussia preservig map. Gaussia states ad Gaussia maps are a importat tool sice they simplify dramatically a potetially usolvable problem. While, i geeral, cotiuous variable systems require a ifiite dimesioal Hilbert space, Gaussia states are fully determied by the first ad secod momets. Moreover, usig Gaussia maps allows the dyamics to be described by a fiite set of parameters. The reaso is that for a Gaussia map, the equatios for the first ad secod momets are closed, therefore reducig the complexity of the problem which otherwise would be itractable. Eve though we could, i priciple, tackle the oe-mode problem umerically, the approach becomes obsolete as we start cosiderig multi-mode states. Gaussiaity becomes esetial to obtai aalytical solutios. 3.3 Multi-mode States We cosider N bosoic modes with algebra give by [ ] ai, a j = δij, [ ai, a j ] = 0, (3.15) 19

33 Chapter 3. Cotiuous Variable Formalism ad quadratures by q i = 1 2 (a i + a i ), p i = i 2 (a i a i). (3.16) We would like to merge all these relatios ito a sigle coditio. For this reaso, we defie two 2N dimesioal vectors X = ( a 1, a 1, a 2, a 2,... a N, a N), Y = ( q1, p 1, q 2, p 2,... q N, p N ). (3.17) These two vectors are equivalet represetatios of the N bosoic modes. It will be coveiet to write dow the basic results for both of them. The coectio betwee these two vectors is give by Y = Λ X, Λ = N i= i 2 i 2, (3.18) where Λ is a uitary trasformatio aalogue to (3.16). Additioally, the multi-mode commutatio relatios (3.15) are ow codesed i the vector commutators [ N X, X T] = 1 0, 0 1 i=1 [ N Y, Y T] = 0 1, (3.19) 1 0 i=1 where the vector commutator is defied as [ A, B T] T = A B ( B T A) T. Equatios (3.17)-(3.19) establishes the algebra of the group of operators. As metioed i the previous sectio, we are iterested i the first ad secod momets. We defie the first momets x ad y related to X ad Y as simply x i = X i ad y i = Y i. More iterestig, we defie the covariace matrices (CM) as Θ = 1 2 { X i, X j } Xi X j, σ = 1 2 {Y i, Y j } Y i Y j, (3.20) where by costructio Θ is Hermitia, ad σ is real ad symmetric. For example, for two 20

34 Chapter 3. Cotiuous Variable Formalism modes the covariace matrices would look like: q {q 2 1, p 1 } q 1 q 2 q 1 p 2 1 σ = {q 2 1, p 1 } p 2 1 p 1 q 2 p 1 p 2 q 2 q 1 q 2 p 1 q {q, (3.21) 2, p 2 } 1 p 2 q 1 p 2 p 1 {q 2 2, p 2 } p 2 2 ad a 1a a 2 1 a 1 a 1 a 2 a 1 a 2 a 1a 1 a 1a a 2 1a 2 a 1a 2 Θ = a 1a 2 a 1 a 2 a 2a a 2 2 a 2 a 1a 2 a 1 a 2 a 2a 2 a 2a , (3.22) where is implicitly assumed that a i = 0. Notice that the covariace matrix is structured i 2 2 blocks where the diagoal blocks are the CMs of modes 1 ad 2, while the offdiagoal blocks represet their correlatios. These two matrices are also related via the uitary trasformatio (3.18), amely σ = Λ Θ Λ. (3.23) 3.4 Gaussia Dyamics We ow focus o the dyamical evolutio of Gaussia states subject to the Lidblad equatio: d ρ = i[h, ρ] + D(ρ), (3.24) dt where H is a Gaussia Hamiltoia ad D is a Gaussia preservig dissipator. Give the master equatio, we ca obtai the evolutio of ay observable by takig the expectatio value of such operator: d O = i [H, O] + Tr(OD(ρ)). (3.25) dt 21

35 Chapter 3. Cotiuous Variable Formalism Usig the cyclic property of the trace, we ca write Tr ( O ( LρL 1 2 { L L, ρ })) = L OL 1 2 { L L, O } := D(O). (3.26) Hece, we ca rewrite Eq. (3.25) as: d dt O = i [H, O] + D(O). (3.27) So far we have t use the fact that the evolutio is described by Gaussia maps. Imposig Gaussiaity results i the equatios for the first ad secod momets to close. That meas that the expected values of operators like a i or a i a j a i a j will oly deped o the elemets of the vector x ad the covariace matrix Θ. I cotrast, i the o-gaussia sceario the equatios for these momets will deped also o the higher momets, leadig to a ifiite hierarchy of coupled equatios. Thus, the Gaussia dyamics will be ecapsulated i the evolutio of the vector of averages ad the covariace matrix. Takig the evolutio of the vector of averages x ad y ito Eq. (3.27) we get d dt x = W x f, d y = M y g, (3.28) dt where W ad M deped o the choice of Hamiltoia H while f ad g o the choice of dissipator D. Similarly, we get for the covariace matrices: d dt Θ = W Θ + ΘW + F, d dt σ = Mσ + σm T + G, (3.29) where the matrices F ad G deped oly o the dissipator D whereas the matrices W ad M are the same as Eq. (3.28). Let us illustrate the procedure with two simple examples that we will further explore i Chapter 4. Suppose we have two bosoic modes a 1 ad a 2 iteractig via a beam-splitter Hamiltoia H BS = ig(a 1a 2 a 2a 1 ). (3.30) This will create/destroy a particle i oe mode ad destroy/create a particle i the other 22

36 Chapter 3. Cotiuous Variable Formalism mode, coservig always the total umber of particles N = a 1a 1 +a 2a 2. We use Eq. (3.27) to get the first momets evolutio d a 1 dt d a 1 dt d a 2 dt d a 2 dt = i [ H BS, a 1 ] = g a2, (3.31) = i [ ] H BS, a 1 = g a 2, (3.32) = i [ H BS, a 2 ] = g a1, (3.33) = i [ ] H BS, a 2 = g a 1. (3.34) The matrix W associated to this beam-splitter Hamiltoia is give by 0 0 g g W BS =. (3.35) g g 0 0 The solutio for the first momets vector x(t) is thus x(t) = V BS (t) x 0, (3.36) where the V BS matrix itroduced is cos (gt) 0 si (gt) 0 0 cos (gt) 0 si (gt) V BS (t) := e WBSt =. (3.37) si (gt) 0 cos (gt) 0 0 si (gt) 0 cos (gt) It will be useful to re-state this matrix i terms of 2 2 blocks cos (gt)i V BS (t) = si (gt)i, (3.38) si (gt)i cos (gt)i where I is the 2 2 idetity matrix. 23

37 Chapter 3. Cotiuous Variable Formalism We ca also obtai the y(t) solutio by the iterdepedece relatio (3.18), cos (gt)i y(t) = S BS (t) y 0, S BS (t) = Λ V BS (t) Λ = si (gt)i si (gt)i. (3.39) cos (gt)i Likewise, the solutio for the covariace matrices are: Θ(t) = V BS (t) Θ 0 V BS (t), σ(t) = S BS(t) σ 0 S T BS(t). (3.40) Particularly, we will be iterested i obtaiig the solutio of oe of the modes (the system) by tracig out the other modes (eviromet). This is achieved by lookig at the system s block term i Eq. (3.40). As a example, let us take the first mode as the system ad the secod mode as the eviromet. Tracig out the secod mode, we get: σ 1 (t) = cos 2 (gt) σ 1 (0) + si 2 (gt) σ 2 (0). (3.41) This equatio ca be also expressed as: σ 1 (t) = Xσ 1 (0)X T + Y, (3.42) where X = cos(gt) I, ad Y = si 2 (gt) σ 2 (0). This is the Gaussia aalogue of Eq. (2.30) ad it is kow as a Gaussia CPTP map. Aother Gaussia Hamiltoia we will be iterested i is the two-mode squeezig H SQ = iµ(a 1a 2 a 2 a 1 ). (3.43) I oppositio to the beam-splitter, this Hamiltoia does ot coserve the total umber of particles N: it creates/destroys a particle i both modes with stregth µ. 24

38 Chapter 3. Cotiuous Variable Formalism The equatios of motio for the first momets are: d a 1 dt d a 1 dt d a 2 dt d a 2 dt = i [ H SQ, a 1 ] = µ a 2, (3.44) = i [ H SQ, a 1] = µ a2, (3.45) = i [ H SQ, a 2 ] = µ a 1, (3.46) = i [ H SQ, a 2] = µ a1. (3.47) The matrix W associated with this Hamiltoia is µ 0 0 µ 0 W SQ =. (3.48) 0 µ 0 0 µ Similar solutios (3.36)-(3.40) are obtaied for the two-mode squeezig provided we chage V BS with the correspodig V SQ matrix cosh (µt)i V SQ (t) := e WSQt = sih sih (µt)σ x (µt)σ x, (3.49) cosh (µt)i ad chage S BS with S SQ cosh (µt)i S SQ (t) = Λ V SQ (t) Λ = sih sih (µt)σ z (µt)σ z, (3.50) cosh (µt)i where σ x ad σ z are the familiar 2 2 Pauli Matrices. Fially, we ca get the CPTP Gaussia map (3.42) for the two-mode squeezig: σ 1 (t) = cosh 2 (µt) σ 1 (0) + sih 2 (µt) σ z σ 2 (0) σ z, (3.51) where X = cosh (µt) I, ad Y = sih 2 (µt) σ z σ 2 (0) σ z. 25

39 Chapter 4 Collisioal Models This chapter cotais oe of the mai origial results developed i this dissertatio. It presets the theoretical model to be studied i these last chapters. The expositio here follows closely Sec. II of the submitted arxiv preprit [91]. 4.1 Motivatio As we have see previously, obtaiig a microscopic descriptio of the ope quatum system s dyamics is a really complicated task, ofte relyig o approximatio schemes to get aalytical results. We particularly aalyzed the case of a qubit iteractig with a bosoic eviromet ad realized how may approximatios were made to obtai the system s evolutio. This motivates the search for alterative models, where aalytic equatios ca be obtaied without resourcig to such severe approximatios. Oe way to accomplish this, which has see a eormous surge i popularity i recet years, are through the so-called collisioal models [31 42, 91]. The basic idea is to replace the ope dyamics of a system by a series of sequetial iteractios betwee the system S ad small evirometal uits E 1, E 2, E 3... (also referred to as acillas). All iteractios lasts for a fixed time after which they ever iteract agai. This therefore leads to a stroboscopic dyamics for the system. The collisioal model we are iterested i this work is the oe depicted i Fig That is, we cosider a sceario where eighborig acillas E E +1 iteract with each other i betwee the iteractios SE ad SE +1. This additioal iteractio sigals 26

40 Chapter 4. Collisioal Models S S E1 E2 E3 E1 E2 E3 S S E1 E2 E3 E1 E2 E3 Figure 4.1: First few steps of the collisioal model dyamics. The system-acilla iteractios SE are iterspersed by acilla-acilla iteractios E E +1, which propagate iformatio forward, makig the dyamics o-markovia i a fully cotroballe way. iformatio from the past to the future, so that whe the SE +1 iteractio arrives, the acilla E +1 will already cotai some iformatio about the system. 4.2 Formal Framework Here we cosider the collisioal model sceario preseted i Fig A system S is put to iteract sequetially with a arbitrary umber of eviromet acillas E 1, E 2, E 3... The acillas are idepedet ad idetically prepared, each with the iitial desity matrix ρ E. The iteractio betwee S ad E is described by a uitary U. After this, S ad E ever iteract agai. After the collisio SE but before the SE +1, we put acillas E ad E +1 to iteract with each other by meas of aother uitary V,+1. Sice E already iteracted with S, it cotais some iformatio about each other, obtaied from E. Past iformatio about S ca thus backflow at SE +1. Let ρ 0 = ρ S ρ E1 ρ E2... deote the iitial state of the composite system SE 1 E 2... We cout time i iteger steps, such that at time the collisios SE ad E E +1 already took place. That is, at time the system has already iteracted with its correspodig 27

41 Chapter 4. Collisioal Models acilla E ad this acilla has already passed dow its iformatio to the ext oe. The map takig the composite system SE 1 E 2... from 1 to reads ρ = V,+1 U ρ 1 U V,+1. (4.1) To avoid cofusio we heceforth use superscripts to deote time so that ρ refers to the global state of SE 1 E 2... at time. The map (4.1) ivolves oly SE E +1. All acillas E m with m + 2 did ot yet participate i the process ad therefore remai i a product state with everythig else. I additio, the acillas with m < will ever participate agai ad hece ca be traced out (discarded). The process (4.1) ca thus be equivaletly writte as ρ SE E +1 = V,+1 U ( ρ 1 SE ρ E+1 ) U V,+1, (4.2) where ρ 1 SE is the state of SE at time 1 ad ρ E+1 = ρ E refers to the iitial state of E +1. This also holds for the first step, provided oe recalls that ρ 0 SE 1 = ρ 0 S ρ E 1. After the iteractio (4.2), oe may trace out E, leadig to ( ( ) ) ρ SE +1 = Tr E V,+1 U ρ 1 SE ρ E+1 U V,+1 := Φ(ρ 1 SE ). (4.3) This ca ow be fed agai to Eq. (4.2) to evolve to the ext step. This equatio also defies the quatum chael Φ(.), which is a map from the Hilbert space of SE to that of SE +1. Moreover, sice we are assumig that the uitaries U ad V,+1 are the same for all collisios, the map Φ itself is actually idepedet of ; the oly depedece is i the iput ρ 1 SE. Crucially, we see that the map Φ(.) is both time-local ad completely positive sice it is writte as a Stiesprig dilatio [92]. Hece, it represets a etirely Markovia evolutio. Eq. (4.3) is kow as a Markovia embeddig of the dyamics [59]: it expresses a o-markovia evolutio as a Markovia oe at the expese of workig with maps that act betwee differet Hilbert spaces ad also have a higher dimesio. It is coveiet to defie the more compact otatio ϱ = ρ SE +1 of SE +1 at time. for the joit state The etire dyamics ca the be captured by the stroboscopic, 28

42 Chapter 4. Collisioal Models Figure 4.2: Basic structure of the Markovia embeddig dyamics (4.20), which is a map from the Hilbert space of SE to that of SE +1 Markovia, CPTP evolutio: ϱ +1 = Φ(ϱ ), (4.4) ad the correspodig sequece of states ϱ 0, ϱ 1, ϱ 2,... that it geerates. At each step, the reduced state of the system is always available as ρ S = Tr E +1 ϱ. The Markovia embeddig (4.4) will be cetral to this work. Sice iitially the system is ucorrelated from all acillas, it is possible to defie a CPTP map takig ρ 0 S ρ S : ρ S = E (ρ 0 S) = Tr E+1 Φ (ρ 0 S ρ E1 ). (4.5) Eve though this map is CPTP, the map takig ρ m S to the o-markovia dyamics. ρ S will ot geerally be CPTP due 4.3 Gaussia Collisioal Model We are iterested i obtaiig aalytical results. To accomplish this, we therefore specialize ow to the case of cotiuous-variable systems udergoig Gaussia-preservig dyamics reviewed i Chapter 3. Our expositio, i what follows, will be example orieted. However, the fial results will be quite geeral [Eqs. (4.19), (4.20) ad (4.22)]. 29

43 Chapter 4. Collisioal Models We assume the system S is described by bosoic ahilatio operator a ad correspodig quadrature Q = (a + a )/ 2 ad P = i(a a)/ 2. Similarly, the acillas are described by bosoic ahilatio operators b 1, b 2,... with correspodig quadratures q, p. The geeralizatio to multimode system is straightforward. We take the systemacilla iteractio U i Eq. (4.2) to be a simple beam-splitter-type uitary U = e λs(a b b a), (4.6) described by a parameter λ s. Oe ca view (4.6) as a iteractio with a Hamiltoia ig(a b b a) that lasts for a time τ such that gτ = λ s. Sice we are oly iterested i the stroboscopic dyamics, we ca omit the iteral details for simplicity. As for the E E +1 collisio uitary V,+1, we shall explore two possibilities. The first is agai a beam-splitter map V,+1 = e λe(b b +1 b +1 b), (4.7) with iteractio stregth λ e. We shall heceforth refer to this as the BS dyamics. I additio, we shall also look at a two-mode squeezig iteractio (TMS), Ṽ,+1 = e νe(b b +1 b +1b ), (4.8) with stregth ν e ad view (4.8) as comig from a Hamiltoia ig(b b +1 b +1 b ) for a time τ such that gτ = ν e. The reaso behid this choice is related to the fact that twomode squeezig iteractios geerate stroger forms of correlatios (e.g. etaglemet) betwee the acillas. The uitaries (4.6)-(4.8) are Gaussia preservig. If we assume that the iitial state is Gaussia, the dyamics will the be completely characterized by the first ad secod momets. We assume for simplicity that the first momets are iitially zero so that they will remai so throughout. The covariace matrix of the iitial state is block-diagoal, of the form σ 0 = diag(θ 0, ɛ, ɛ,...), (4.9) where each block is 2 2: θ 0 is the arbitrary iitial CM of the system ad ɛ is the iitial CM of the acillas (which are all the same, sice we are assumig the acillas are iid). 30

44 Chapter 4. Collisioal Models The global dyamics of SE 1 E 2... is uitary. As a cosequece, the map (4.1) is traslated ito a symplectic evolutio for the CM: σ = S,+1 S σ 1 S T S T,+1, (4.10) where S ad S,+1 are the symplectic matrices associated with the uitaries U ad V,+1. The symplectic matrix associated to the beam-splitter iteractio (4.6) is remarkably simple because all etries are proportioal to the 2 2 idetity (3.39) [this is partially because of the choice of phase i the expoet of (4.6)]. For istace, the iteractio S 2 betwee the S ad E 2 reads x 0 y S 2 = y 0 x 0..., (4.11) where each etry is a 2 2 matrix, with x = cos(λ s ) ad y = si(λ s ). The same structure also holds for the BS uitary V,+1 betwee E E +1 [Eq. (4.7)], except that ow the positio of the o-zero etries chages. For istace, z w 0... S 1,2 = 0 w z 0..., (4.12) where z = cos(λ e ) ad w = si(λ e ). The TMS iteractio (4.8) is slightly more complicated sice some etries are proportioal to the idetity, while others are proportioal to 31

45 Chapter 4. Collisioal Models the Pauli matrix σ z as see i (3.50). For istace, z wσ z 0... S 1,2 = 0 wσ z z 0..., (4.13) with z = cosh(ν e ) ad w = sih(ν e ). The BS dyamics is completely characterized by the pair (λ s, λ e ), while the TMS dyamics is characterized by (λ s, ν e ). O top of that, oe also has the choice of acilla iitial state ɛ. More geeral Gaussia maps will cotiue to have a similar structure. The symplectic S will have the form A 0 B S 2 = C 0 D 0..., (4.14) for block matrices A, B, C, D. The matrices S for other values of are obtaied by simply placig A, B, C, D at the correct positios. Note also that the coditio that S must be symplectic imposes costraits o A, B, C, D which, however, are ot particularly illumiatig. Similarly, the E E +1 iteractio reads E F 0... S 1,2 = 0 G J 0..., (4.15) for block matrices E, F, G, J. Note that these two expressios also aturally cotemplate the case where either the system or each acilla are, idividually, composed of multiple 32

46 Chapter 4. Collisioal Models modes (which would simply affect the size of the matrices A,..., J). 4.4 Markovia Embeddig The biggest advatage of Gaussia collisioal models, as we will ow show, is that the full evolutio ca be coverted ito a simple system of matrix differece equatios for oly a hadful of etries of the full CM σ. As already discussed below Eq. (4.2), the step from σ 1 to σ ivolves oly S, E ad E +1. At time 1 the acilla E +1 is still ucorrelated from the rest, whereas S ad E are already correlated because of the previous step. Thus, the tripartite CM of SE E +1, at time 1, will have the block structure where ɛ 1 σ 1 SE E +1 = θ 1 ξ 1,T ξ 1 0 ɛ 1 0, (4.16) 0 0 ɛ is the state of acilla E at time 1, which is o loger the origial value ɛ because it already iteracted with E 1 i the previous step. Moreover, ξ 1 are the correlatios betwee SE that were developed i the previous step. We the apply the map (4.10) to Eq. (4.16), usig the matrices i Eqs. (4.11)-(4.13) σ SE E +1 = S,+1 S ( σ 1 SE E +1 ) S T S T,+1. (4.17) This will lead to a matrix σ with may o-zero etries. However, as far as the dyamics of S is cocered, oly three etries are eeded: the state of the system θ, the state ɛ +1 of acilla E +1 ad the correlatios ξ+1 betwee S ad E +1. To gai ituitio, let us first aalyze the BS case, which is simple sice all blocks i Eq. (4.12) are proportioal to the idetity. Usig Eqs. (4.11) ad (4.12) i (4.17), oe 33

47 Chapter 4. Collisioal Models fids the followig system of matrix differece equatios: θ = x 2 θ 1 + y 2 ɛ 1 + xy(ξ 1 + ξ 1,T ), [ ɛ +1 = z 2 ɛ + w 2 x 2 ɛ 1 + y 2 θ 1 xy(ξ 1 + ξ 1,T ] [ ξ+1 = w xy(θ 1 ɛ 1 ) + y 2 ξ 1,T x 2 ξ 1 ). ], (4.18) This provides a eat illustratio of the map Φ( ) i Eq. (4.3): the quatities o the lefthad ad right-had side refer to differet acillas: for istace, ɛ +1 is the state of acilla E +1 at time, whereas ɛ 1 is the state of E at time 1. Of course, oe could also compute ɛ, but this is ot ecessary for describig the dyamics of S. The system of matrix differece equatios (4.18) cotais the miimum amout of iformatio required to fully accout for the dyamics of S. These equatios ca also be recast i a more compact form i terms of the Markovia embeddig (4.4). We defie the reduced CM of SE +1 at time as γ+1 γ = θ ξ+1, (4.19) ξ,t +1 ɛ +1 where the otatio γ will be used to simplify the expressios. Eq. (4.18) ca the be writte compactly as γ +1 = Xγ X T + Y, (4.20) where the time idex was shifted by 1. Here X ad Y are 4 4 matrices with block form x X = yw y 0 0, Y =, (4.21) wx 0 z 2 ɛ where, agai, each block is proportioal to the idetity. Eq. (4.20) beautifully illustrates the otio of Markovia embeddig. It has the structure of a typical Gaussia CPTP map (3.40), beig Markovia (time-local) by costructio. However, this Markovia dyamics takes place at the larger space of the system plus oe acilla (which oe, i specific, chages at each collisio). Thus, we have embedded 34

48 Chapter 4. Collisioal Models the o-markovia dyamics ito a Markovia dyamics at a larger space. Notice how the size of the space is directly related to the fact that we chose E to oly iteract with its earest eighbor E +1. That is, we fixed the memory legth to be 1, which defies the size of the miimal space required for the embeddig [59]. The matrices (4.21) refer to the beam-splitter uitary (4.7). The geeralizatio to the arbitrary Gaussia iteractios (4.14) ad (4.15) is similar, albeit more cumbersome. The result is A X = GC B 0 0, Y =. (4.22) GD 0 JɛJ T For istace, i the case of the TMS iteractio, Eq. (4.13), oe has G = wσ z ad J = z, i additio to A = D = x, B = y ad C = y (which come from S i (4.11)). Oe the fids that x X = y wσ z y wxσ z, Y = 0 0. (4.23) 0 z 2 ɛ The blocks i X are therefore o loger proportioal to the idetity, but some are proportioal to σ z. To summarize, the geeral o-markovia dyamics will be described by the embeddig (4.20), with γ defied i (4.19), ad with X ad Y give by (4.22). This framework therefore provides a quite geeral platform, eablig oe to study a broad rage of situatios. 4.5 Example Dyamics Eqs (4.20)-(4.23) are the first mai results of this work. They provide a compact ad efficiet way of describig the o-markovia dyamics of a bosoic mode i terms of a simple matrix differece equatio for the augmeted CM γ. The reduced state of the system is always readily accessible from the first 2 2 block [Eq. (4.19)]. Before proceedig to quatify the o-markoviaity of the process, we first illustrate the typical behavior of the BS ad TMS maps, by plottig the average system occupatio a a as a fuctio of time for differet values of the E E +1 iteractio stregth λ e (for the BS case) or ν e (for the TMS case). We choose the system to start i a thermal state with 35

49 Chapter 4. Collisioal Models occupatio umber a a 0 = 20, while the acillas start i the vacuum, ɛ = I 2 /2. The results are summarized i Fig. 4.3 for the BS evolutio ad i Fig. 4.4 for the TMS evolutio. The BS dyamics is sesitive to the relative sigs betwee λ s ad λ e (ad, cosequetly, of y = si(λ s ) ad w = si(λ e )). This is a iterferece effect, which occurs due to the fact we are combiig two beam-splitters [Eqs. (4.6) ad (4.7)]. A similar effect was also observed i Refs [61, 62]. We emphasize this i Fig. 4.3 by comparig λ e > 0 ad λ e < 0, with λ s > 0. I both cases we see that for small λ e the system s excitatios [Fig. 4.3(a,b)] ted to decay mootoically, which is what oe would expect of a Markovia BS iteractio with a vacuum bath. For larger λ e, o the other had, the occupatios preset oscillatios. Sice the iteractio coserves the umber of quata, these revivals i excitatios must ecessarily be due to a backflow caused by the o-markovia behavior. That is, some of the excitatios that leave the system towards E are trasferred from E to E +1 ad the make it back ito the system i the SE +1 iteractio. The ature of these oscillatios, however, is differet whether λ e > 0 or λ e < 0, beig fast i the former ad slow i the latter. Irrespective of the value of λ e, however, after a ifiite time the system will always thermalize to the acilla s state, which i this case meas a a = 0 [the oly exceptio is at λ e = ±π/2, which is somewhat pathological]. Despite its resemblace with the system s behavior, the acilla s excitatios [Fig. 4.3(c,d)] will have some uique features. First, the excitatios spike at the first steps ad the ted to decay mootoically. There are o lost particles yet ad the acilla gets the maximum amout from the system. Ad secod, the umber of excitatios oscillates i the opposite maer to the system s oscillatios. This is due to the coservatio of total umber of quata: the amout that the system loses, the acilla gais it. Fially, we ca also look at the correlatios build betwee the system ad the th acilla [Fig. 4.3(e,f)]. The modulus of the correlatios will be iflueced by the umber of total excitatios. This is related to the fact that, i order to correlate the system with the acilla, there eeds to be itermediary acilla s excitatios actig as messegers for the parties. Results for the TMS iteractio are show i Fig I this case the relative sigs are immaterial, but the dyamics becomes more sesitive o the magitude of ν e, sice z ad w are hyperbolic fuctios. The TMS iteractio etagles E E +1, eve if both 36

50 Chapter 4. Collisioal Models are iitially i the vacuum. As a cosequece, it also spotaeously create excitatios, so that the umber of quata is ot preserved. At each E E +1 collisio the et umber of excitatios therefore icreases. Part of these excitatios are lost whe the acillas are discarded ad part flow to the system. As a cosequece, depedig o the rate at which excitatios are created, the dyamics ca be either stable or ustable. This occurs at the critical poit ν crit e = sih 1 (1) , which is whe w = 1, thus markig the situatio where the total umber of excitatios grow uboudedly [c.f. Eq. (4.23)]. Whe ν e < ν crit e the dyamics will be stable ad both, the system ad the acilla, will coverge to a ucorrelated steady-state value a a = sih 2 ν e (1 sih 2 ν e ) 1 idepedetly of λ s [Fig. 4.4(a,c,e)]. Coversely, for ν e νe crit, the dyamics becomes ustable. Therefore, the umber of excitatios ad correlatios diverge [Fig. 4.4(b,d,f)]. These asymptotic values ca be uderstood from argumets of stability theory, as show i Appedix A. 37

51 Chapter 4. Collisioal Models a a (a) a a (b) b b (c) b b (d) a b (e) a b (f) Figure 4.3: Secod momets as a fuctio of time, computed from Eq. (4.20) for the BS dyamics (4.21) with λ s = 0.5 ad differet values of λ e (with λ e > 0 o the left colum ad λ e < 0 o the right colum). The acillas are assumed to start i the vacuum, ad the system i a thermal state with a a 0 = 20. (a,b) Number of excitatios i the system a a. (c,d) Number of excitatio i the th acilla b b. (e,f) Correlatio fuctio betwee the system ad the th acilla a b. 38

52 Chapter 4. Collisioal Models a a a b b b (a) (c) (e) a a b b a b (b) (d) (f) Figure 4.4: Secod momets as a fuctio of time, computed from Eq. (4.20) for the TMS dyamics (4.23) with λ s = 0.1 ad differet values of ν e (with ν e < νe crit o the left colum ad ν e νe crit o the right colum, where νe crit = sih 1 (1)). The acillas are assumed to start i the vacuum, ad the system i a thermal state with a a 0 = 20. (a,b) Number of excitatios i the system a a. (c,d) Number of excitatios i the th acilla b b. (e,f) Correlatio fuctio betwee the system ad the th acilla a b. 39

53 Chapter 5 No-Markoviaity 5.1 Quatum o-markoviaity I Chapter 2, we reviewed the dyamics of ope quatum systems, i which we describe the system s evolutio by a Lidblad master equatio (2.32). We foud out i Sec. 2.3 that a microscopic derivatio was possible provided we resort to some approximatio schemes, for example the Bor-Markov approximatio. However, these approximatios caot be employed to may processes i ope quatum systems ad thus, the system s dyamics caot be described by a divisible dyamical map. Typically, this is because the eviromet s correlatio time scale is ot small compared to the system s decoherece time, rederig the Markov approximatio impossible. Examples are the cases of strogly correlated systems, low temperatures, fiite reservoirs, or large iitial correlatios for which some experimetal demostratios ca be foud i [93 97]. These processes are called o-markovia. No-Markoviaity first arose i the theory of classical stochastic processes [98 101]. Its classical defiitio is based o classical probability theory where we say that a process is o-markovia if the coditioal probability of the future states of the process depeds o the sequece of evets that preceded it. Quatum theory, however, caot be formulated as a statistical theory o classical probability space [1, 102]. Therefore, the cocept of quatum o-markoviaity caot be based o the classical otios oly. O the oe had, there s the otio of iformatio flow betwee the system ad the surroudig eviromet. I the quatum realm, iformatio leaks are much more effi- 40

54 Chapter 5. No-Markoviaity ciet, so that whe a system iteracts with a eviromet, iformatio about the former is ievitably trasferred to the latter. Whe the eviromet is very large ad complex, this iformatio may ever retur. I geeral, however, there may be a partial backflow of iformatio, which characterizes a o-markovia evolutio. From the poit of view of causality, this backflow quatifies the ability of the dyamics to commuicate past iformatio to the future. O the other had, there s the otio of map divisibility. For istace, cosiderig the case of collisioal models studied i Chapter 4, the map from ρ 0 S ρ S i Eq. (4.5) is always CPTP by costructio. Oe may also defie more geeral maps E m takig the desity matrix ρ m S E 1 to ρ S (m < ). Assumig that the iverse exists, they are defied as E E 1 m. Albeit mathematically well defied, these maps are i geeral ot CP. Markovia maps, coversely, are CP by costructio. This defies the otio of CP-divisibility, which provides a widely used criteria for characterizatio of o-markoviaity: a map is CP-divisible whe the itermediate maps E m are CP. Cosiderable attetio was give i recet years o how to characterize ad quatify o-markoviaity i the quatum domai (for recet reviews, see [15, 16]). Due to the richess ivolved, however, there is o sigle approach capable of capturig its full essece. Several quatifiers have bee studied: dyamical divisibility [22, ], trace distace [17, 19, 20], mutual iformatio [23], relative etropy [24, 25], memory kerel [8 10], amog others [26 30]. I this work, we will be particularly iterested o the oes best suited for the collisioal model setup described i Chapter 4. We will explore them i detail i the followig sectios. 5.2 Mutual Iformatio The first approach to quatum o-markoviaity to be discussed is based o the idea that memory effects i the ope quatum system s dyamics are liked to the exchage of iformatio betwee the system ad its eviromet. While i a Markovia process the ope system cotiuously loses iformatio to the eviromet, a o-markovia process is characterized by a flow of iformatio from the eviromet back ito the system. I this way, quatum o-markoviaity is associated with a otio of quatum memory, amely, iformatio which has bee trasferred to the eviromet, i the form 41

55 Chapter 5. No-Markoviaity of system-eviromet correlatios or chages i the evirometal states, ad is later recovered by the system. No-Markoviaity ad backflow of iformatio must be related to correlatios that develop betwee system ad bath. However, whe a system iteracts with multiple modes of the bath at the same time, there are may differet correlatios oe may cosider, betwee the system ad all possible parts of the bath. Ad it is ot clear which of these correlatios are relevat for the o-markovia evolutio. For istace, the correlatio with a part of the bath the system will ever iteract agai is irrelevat, as far as o- Markoviaity is cocered. But i the stadard sceario, it is i geeral ot possible to idetify which are the relevat correlatios. I a collisioal model picture, o the other had, this is uambiguous: the relevat correlatios are those betwee S ad acilla E +1 at time (see Fig. 4.1). These are the correlatios that acilla E trasferred to E +1 after its iteractio. Hece, they represet the oly possible source of iformatio backflow at each collisio. Coveietly, this is also exactly what the Markovia embeddig (4.4) offers. A useful measure of correlatios, for istace, is the quatum mutual iformatio (MI), defied as I (SE +1 ) = S(ρ S) + S(ρ E +1 ) S(ϱ ), (5.1) where S(ρ) = tr(ρ l ρ) is the vo Neuma etropy. The states ρ S ad ρ E +1 are both computed from ϱ by takig the appropriate partial trace. Thus, by moitorig ϱ as a fuctio of time, oe has direct access to the relevat measure of correlatio. Of course, I (SE +1 ) is ot the oly relevat measure of correlatio. Differet choices, from two-poit fuctios, to quatifiers of etaglemet ad quatum discord, may also be of iterest. The relevat poit is that ay such measure will, ecessarily, be cotaied i ϱ. 5.3 Memory Kerel A much older otio of o-markoviaity is that of a memory kerel, as preset already i the semial works of Nakajima ad Zwazig [6, 7]. The basic idea is that the ope 42

56 Chapter 5. No-Markoviaity dyamics of a system s desity matrix ρ S ca, quite geerally, be writte as dρ t S dt = i[h S, ρ S ] + 0 K t t [ρ(t )] dt, (5.2) where K t t, called the memory kerel (MK), is a liear superoperator codesig all the iformatio o how the evolutio of ρ at time t depeds o its past values. The MK has bee studied itesively i recet years [8 13], as it provides clear isights oto the ier workigs of o-markoviaity. It ca also be give a operatioal iterpretatio, i terms of the so-called trasfer tesors [106], rederig it accessible to experimets [107]. However, beig a superoperator, it is geerally difficult to compute it aalytically. We also metio i passig that, the broader otio of process tesor, which icludes also all possible iput ad output operatios performed i the system [ ]. I the origial formulatio of Nakajima ad Zwazig [6, 7], the memory kerel (MK) was represeted as a sigle-parameter cotiuous superoperator K t that quatifies the memory that is retaied about the system s cofiguratio a time t i the past [c.f. Eq. (5.2)]. The collisioal model aalog of that will be a superoperator K, labeled by the discrete time idex. Thus, the stroboscopic aalog of Eq. (5.2) should be of the form ρ S = 1 m=0 K m (ρ m S ). (5.3) For istace, ρ 3 S = K 1(ρ 2 S ) + K 2(ρ 1 S ) + K 3(ρ 0 S ). The term K 1(ρ 1 S ) describes the shortterm memory from the very last step, while K (ρ 0 S ) describes the log-term memory all the way from the iitial state. A explicit formula for the MK ca be costructed usig a reasoig similar to that used for trasfer tesors i Ref. [106]: Startig with the reduced map E i Eq. (4.5), we defie the MK recursively from K = E 1 m=1 K m E m. (5.4) That this is ideed the correct formula ca ow be verified by substitutig it back i Eq. (5.3). For istace, K 1 = E 1, K 2 = E 2 K 1 E 1, K 3 = E 3 K 2 E 1 K 1 E 2, ad so o. Eq. (5.4) provides a algorithmic method for computig the MK. However, this requires 43

57 Chapter 5. No-Markoviaity heavy umerics, eve i simple cases. For istace, K 3 = E 3 E 2 E 1 E 1 E 2 + E 1 E 1 E 1, ad the complexity of the formulas oly grow from there. Oe of the mai results i this work will be to show that, for the case of Gaussia collisios, it is possible to write dow a closed ad compact formula for K (Sec. 6.2), which provides valuable isight ito the ier workigs of the memory kerel. 5.4 CP-Divisibility Lastly, let us retur to the idea of the divisibility of the map. Give that the iverse map E 1 exists for all times t > 0, we ca defie the family of itermediate maps E m from state ρ m S to state ρ S (m < ) E m = E E 1 m. (5.5) The existece of the iverse is what allows the otio of divisibility. Eve though E ad E m are completely-positive by costructio, the itermediate map E m will ot ecessarily be. This is because the iverse map E 1 m ot to be positive. of the completely-positive map E m eed Now, if these maps (5.5) are CP for all m, the evolutio is Markovia. For o- Markoviaity to occur there must exist a m value to ot be completely positive. Hece, by measurig how much the itermediate map E m departs from the CP map, we are measurig the degree of o-markoviaity of the time evolutio. Testig CP divisibility, however, is ot always easy. For istace, it may it require aalyzig the distace betwee differet pairs of iitial coditios ρ 0 S [16]. Accordig to the data processig iequality, these distaces are always cotractive for CP maps. Violatios of cotractivity are thus idetified as violatios of divisibility. This, however, requires a maximizatio over all possible iitial coditios. I Sec. 6.3 we discuss the Gaussia versio of this cocept ad show that the maximizatio is replaced by a alterative coditio, that provides a clea ad easily applicable formula for quatifyig CP-divisibility. 44

58 Chapter 6 Memory effects i Gaussia Collisioal Models This chapter cotiues with the remaiig origial results obtaied i this dissertatio. The expositio here follows closely Sec. III ad IV of the submitted arxiv preprit [91]. 6.1 Gaussia Mutual Iformatio The Gaussia framework developed i Sec. 4.3 makes the mutual iformatio (5.1) readily accessible from the CM γ i Eq. (4.19). Correlatios are related to the off-diagoal blocks ξ+1 (the MI would be zero if γ was block-diagoal). The three etropies i Eq. (5.1) ca be readily computed i terms of the symplectic eigevalues of γ [63]. Illustrative results are show i Fig. 6.1, for the same collectio of parameters as Fig. 4.3 ad Fig As a saity check, the MI is idetically zero whe λ e = ν e = 0. It also teds to be larger for short times, tedig to zero as grows. The oly exceptio is the ustable dyamics i Fig. 6.1(d), where the MI grows uboudedly. The oscillatory patters i a a are also preset i the MI. The mutual iformatio will also be highly depedet o the total umber of excitatios, as itermediary acilla excitatios are eeded for iformatio backflow (See also Sec. 4.5). To better uderstad the role of the MI i the o-markovia dyamics we preset i Fig. 6.2 a compariso betwee the occupatio umber a a of Fig. 4.3(a) ad the MI of Fig. 6.1 for the BS dyamics. We focus o early times (small ) ad also compare a a 45

59 Chapter 6. Memory effects i Gaussia Collisioal Models (S : E + 1) (a) (S : E + 1) (b) (S : E + 1) (c) (S : E + 1) (d) Figure 6.1: Quatum Mutual Iformatio (5.1) for the BS (a,b) with λ s = 0.5 (λ e > 0 i (a) ad λ e < 0 i (b)), ad TMS (c,d) dyamics with λ s = 0.1 (ν e < νe crit i (a) ad ν e νe crit i (b)). Other parameters are the same as Fig. 4.3 ad Fig a a a a mark (S : E ) (a) (b) Figure 6.2: Compariso betwee Markovia ad o-markovia dyamics ad role of the Mutual Iformatio. I blue circles we show the early dyamics of a a vs. for the BS dyamics with (a) λ e = 1.1 ad (b) λ e = 0.3, with fixed λ s = 0.5 [c.f. Fig. 4.3(a)]. The correspodig Markovia case (λ e = 0) is show i orage triagles. These curves are to be compared with the MI (5.1), show by gree squares i the two cases [Fig. 6.1(a)]. The heights of each curve were adjusted for better visibility. 46

60 Chapter 6. Memory effects i Gaussia Collisioal Models with the correspodig Markovia dyamics (λ e = 0). The differece betwee the o- Markovia (blue circles) ad Markovia (orage triagles) dyamics reflects the extet to which the backflow of iformatio affects the evolutio. This, as ca be see i the figure, is directly correlated with the MI (gree squares) of the previous step. That is, a large MI i a give step implies a large differece betwee the blue ad orage curves i the followig oe. This is particularly clear i Fig. 6.2(a) ad serves to illustrate how the correlatios built betwee SE +1, at step, affect the future iteractio betwee S ad E +1 at the ext step. 6.2 Memory Kerel The Memory Kerel (MK) discussed i Eqs. (5.2) ad (5.3), ad Sec. 5.3, is perhaps the most physically trasparet way of aalyzig o-markoviaity. Startig from ay global map betwee system ad bath, oe ca always write dow a differetial equatio for the reduced desity matrix ρ S of the system. This equatio, however, will i geeral be time-o-local; i.e., it will be a itegro-differetial equatio of the form (5.2), where K t t [ρ S (t )] describes how dρ S (t)/dt depeds o ρ S (t ) i previous times t < t. The MK therefore cotais all the iformatio about the dyamics, with o-markoviaity beig related to its overall depedece o t t : the slower the decay of K t t with t t, the loger the memory ad hece the more o-markovia is the dyamics. The Markovia case is recovered whe K t t δ(t t ). The memory kerel K t t is a superoperator actig o the full Hilbert space of the system. Computig it is thus, i geeral, a very difficult task. Withi our framework, however, oe may equivaletly formulate a memory kerel actig oly i the system s CM θ. This ca be accomplished startig from Eq. (4.20) ad writig dow a differece equatio for θ oly. As we will demostrate below, this equatio will have the form (cotrast with (2.12) ad (5.3)): 1 θ +1 = x 2 θ + K r 1 (θ r ) + G, (6.1) r=0 47

61 Chapter 6. Memory effects i Gaussia Collisioal Models where G is a cotributio that depeds oly o the iitial state of the acillas 1 ad K is the memory kerel. The way we defie the MK is such that K 0 measures how the step from θ to θ +1 is affected by θ 1 ad K 1 measures how it is affected by θ 0. K is still a superoperator, but oe which acts o the space of 2 2 CMs. Oe ca write it more explicitly i terms of a Kraus operator-sum represetatio [88, 111] K (θ) = ij κ ijm i θm T j, (6.2) where κ ij are coefficiets that deped o time ad {M i } are a complete set of 2 2 matrices; a coveiet choice is the set of Pauli matrices {I 2, σ z, σ +, σ }. A geeral recipe to compute the coefficiets κ ij i Eq. (6.2) is give below i Eq. (6.21). Crucially, as we show, it depeds oly o the matrix X of the Markovia embeddig (4.20). The memory itself is cotaied i the depedece of κ ij o. The depedece o i, j determies how differet elemets of θ r affect θ. For istace, as we will show below, i the case of the BS map [Eq. (4.21)], the oly o-zero coefficiet will be the oe proportioal to I 2 θi 2 = θ, which we refer to as κ 11; that is, the memory kerel is actually a c-umber, K (θ) = κ 11θ. This implies that the MK is the same for all etries of θ ad each etry (θ ) ij is oly affected by the correspodig etry (θ r ) ij at past times. Coversely, i the TMS map there will be four o-zero coefficiets, correspodig to combiatios of M 1 = I 2 ad M 2 = σ z ; we refer to them as κ 11, κ 1,z, κ z,1 ad κ z,z. This meas that the memory kerel of (θ ) 11 will be differet from that of (θ ) 2,2 ad so o (each etry will have its ow memory kerel). Fially, a memory kerel cotaiig a depedece o σ ± would imply that (θ ) 11 would deped o the past values of other etries, such as (θ r ) 12 ad (θ r ) Geeral derivatio of the Memory Kerel We ow carry out the derivatio of the memory kerel for the Gaussia collisioal model. We cosider here a more geeral sceario, which relies oly o the structure of the Markovia embeddig i Eq. (4.20). We also assume that the system ad acillas are each 1 I priciple oe could also iterpret G as beig part of the memory kerel. However, we have opted ot to do so, sice G refers oly to the states of the acillas, while the actual depedece o the state of the system, which is what oe expects from a memory kerel to model, is fully cotaied i K. 48

62 Chapter 6. Memory effects i Gaussia Collisioal Models composed of a arbitrary umber of modes N S ad N E [Eqs. (4.21) ad (4.23) are recovered for N S = N E = 1]. More specifically, we take the matrices X ad Y to have the followig block structure, X 11 X X =, Y =, (6.3) X 21 X 22 0 Y 22 where e.g., X 11 ad X 22 are of size 2N S ad 2N E respectively. This therefore cotemplates both multimode system ad acillas, as well as collisios with loger memory. For istace, if E collides with E +1 ad E +2, the we would have N S = 1 ad N E = 2. Our derivatio follows the geeral approach of Nakajima ad Zwazig [6, 7], but adapted to the preset cotext. We begi by otig the followig property [112]: the solutio of a geeric differece equatio of the form ψ( + 1) = αψ() + g(), (6.4) is give by 1 ψ() = α ψ(0) + α r 1 g(r). (6.5) This solutio holds for arbitrary objects ψ, provided α is a liear operator. It therefore holds whe ψ is a vector ad α is a matrix, or whe ψ is a matrix ad α is a superoperator. Thus, for istace, the solutio of Eq. (4.20) is 1 γ = X γ 0 (X T ) + X r 1 Y (X T ) r 1. (6.6) r=0 Here the otatio γ, to deote the time idex, becomes a bit ambiguous sice X is the matrix X to the power. But there is o room for cofusio, sice X will be the oly quatity where the superscript does ot refer to the time. We ow itroduce the vectorizatio operatio [113], which trasforms a matrix A ito r=0 49

63 Chapter 6. Memory effects i Gaussia Collisioal Models a vector A = vec(a) by stackig its colums. For istace, vec a c a b c =. (6.7) d b d Oe may verify that, for ay three matrices A, B, C, vec(abc) = (C T A)vec(B). (6.8) With this, the matrix differece equatio (4.20) is coverted ito a vector differece equatio γ +1 = (X X) γ + Y. (6.9) We also itroduce projectio matrices oto the subspaces of system ad acilla, I 2NS P S =, P E =, (6.10) I 2NE which are of size 2N S + 2N E. I the larger space relevat for vectorizatio there are four possible projectios, P S (...)P S, P S (...)P E ad so o. These operatios chop the covariace matrix γ i 4 blocks, as i Eq. (4.19). Our iterest is i P S (γ )P S, as it cotais the system CM θ. We therefore also itroduce P = P S P S, (6.11) together with its complemet Q = 1 P. Note, though, that Q P E P E. We ow multiply Eq. (6.9) by P ad use that P + Q = 1, together with the fact that P Y = 0 [c.f. Eq. (6.3)]. We the get P γ +1 = P (X X)P γ + P (X X)Q γ. (6.12) 50

64 Chapter 6. Memory effects i Gaussia Collisioal Models Similarly, multiplyig Eq. (6.9) by Q we fid Q γ +1 = Q(X X)Q γ + Q(X X)P γ + Y. (6.13) Now comes the crucial idea of the Nakajima ad Zwazig method [6, 7]. We iterpret Eqs. (6.12) ad (6.13) as two coupled equatios for the variables P γ ad Q γ. Sice our iterest is i P γ, we first solve Eq. (6.13), assumig a give P γ, ad the substitute the result i Eq. (6.12). Eq. (6.13) is of the form (6.4) with α = Q(X X) ad g() = Q(X X)P γ + Y. Eq. (6.5) the gives 1 Q γ = (Q(X X)) Q γ 0 + (Q(X X)) [Q(X r 1 X)P γ r + Y r=0 Pluggig this i Eq. (6.12) we the arrive at where P γ +1 = P (X X)P γ + 1 r=0 ]. ˆK r 1 P γ r + G, (6.14) ˆK r 1 = P (X X)[Q(X X)] r 1 Q(X X), (6.15) is the memory kerel i vectorized form (i.e., as a matrix of size (2N S +2N E ) 2 ). The term G, o the other had, is a fuctio that depeds oly o the iitial state of the acillas ad reads 1 G = P (X X)[Q(X X)] Q γ 0 + P (X X)[Q(X X)] r 1 Y. What is left is to rewrite Eq. (6.14) as a equatio for the evolutio of the system s CM θ oly. We itroduce the (2N S ) 2 (2N S +2N E ) 2 rectagular matrix π defied such that π γ = θ. For istace, i the case N S = N E = 1, the matrix π will be 4 16, of r=0 51

65 Chapter 6. Memory effects i Gaussia Collisioal Models the form (for more ituitio o this matrix, see Appedix B) π = (6.16) We also otice that P = π T π ad ππ T = I (2NS ) 2. Multiplyig Eq. (6.14) o the left by π we the get θ +1 = (X 11 X 11 ) θ + 1 r=0 ˆK r 1 θ r + G, (6.17) where we also used the fact that π(x X)π T = X 11 X 11. Here G = π G is agai a term that depeds oly o the iitial coditios of the acillas, whereas ˆK = π ˆK π T = π(x X) [ Q(X X) ] +1 π T, is the memory kerel, ow expressed as a matrix of size (2N S ) 2 (2N S ) 2 actig o θ r. This ca also be writte more symmetrically, by exploitig the fact that Q 2 = Q. We ca the arrage it as ˆK = π(x X)Q [ Q(X X)Q ] Q(X X)π T. (6.18) The extra Q s outside the square brackets are placed simply to esure the result also holds for = 0. This is the fial form of the MK. Crucially, otice how it depeds oly o the matrix X of the Markovia embeddig (4.20). To obtai a matrix differece equatio for θ we must uvec Eq. (6.17); that is, apply the iverse map of (6.7). Uvecig the first term is trivial sice, by Eq. (6.8), uvec [ (X 11 X 11 ) θ ] = X 11 θ X T 11. The memory kerel (6.18), o the other had, caot be uvecked as a sigle product of 52

66 Chapter 6. Memory effects i Gaussia Collisioal Models Aθ B. Istead, it is coveiet to express it as ˆK = ij κ ijm j M i, (6.19) where κ ij are real coefficiets ad {M i } are a set of operators spaig the vector space of 2N S -dimesioal real matrices. Decomposed i this form, the uvecked versio of the memory kerel will the be, from (6.8), K (θ) = ij κ ijm i θm T j. (6.20) Fially, the form of the coefficiets κ ij ca be foud if we assume that the M i form a orthogoal basis with respect to the Hilbert-Schmidt orm (A B) = tr(a T B) (which is the case of the Pauli basis, for istace). Multiplyig Eq. (6.19) by M j M i ad tracig the yields, by orthogoality, κ ij = tr [ (Mj T Mi T ) ˆK ] tr(mi TM i) tr ( Mj TM ). (6.21) j This, together with Eq. (6.18), is all that is required to compute the memory kerel. With all these defiitios, oe may ow fially uvec Eq. (6.17), leadig to 1 θ +1 = X 11 θ X11 T + K r 1 (θ r ) + G, (6.22) r=0 where G = uvec( G ) = uvec(π G ) is, agai, a term depedig oly o the iitial states of the acillas Memory Kerel for the BS dyamics We ow illustrate the memory kerel for the two maps cosidered i Sec. 4.3, startig with the BS dyamics. I geeral, the structure of the memory kerel will be quite complicated. For the BS dyamics [Eq. (4.21)], however, the oly o-zero coefficiet i Eq. (6.21) is κ 11, the term proportioal to the idetity. I this case the memory kerel is therefore rather simple, as it is just a c-umber multiplyig all etries of θ r. A more compact formula for 53

67 Chapter 6. Memory effects i Gaussia Collisioal Models the MK i this case is give i Appedix B. Results for the BS dyamics are show i Fig The upper pael correspods to λ s = 0.5, which is similar to Fig As ca be see, for λ e > 0 (Fig. 6.3(a)) the memory kerel s decay is oscillatory, with a expoetial evelope. For λ e < 0, oscillatios are also observed, but these are rather differet i ature ad more asymmetrical with respect to the horizotal axis. Whe λ s = 0.05 the situatio chages (Figs. 6.3(c) ad (d)). The dyamics of a a is still quite similar to that of λ s = 0.5, show i Fig. 4.3, except that the time-scales become much loger. But i the MK oe sees somethig etirely differet. I particular, oe fids that while κ 11 cotiues to oscillate whe λ e > 0, it ow becomes exclusively egative for λ e < 0. I this case therefore, all past values of θ r ted to cotribute egatively to the evolutio. Negative values i the memory kerel are rather importat, as they are associated with faster covergece. The reaso is that the CM is a positive matrix ad the first (a) (b) (c) (d) Figure 6.3: The memory Kerel for the BS dyamics, Eq. (4.21). I this case the oly o-zero etry i Eq. (6.2) is κ 11, the term proportioal to the idetity. The plots are for λ s = 0.5 (upper pael) ad λ s = 0.05 (lower pael), with λ e > 0 (left) ad λ e < 0 (right). 54

68 Chapter 6. Memory effects i Gaussia Collisioal Models term i (6.2) is always positive. The egativities observed i Fig. 6.3 therefore represet a accelerated draiig of excitatios from the system. This sheds light o some of the behaviors previously observed for the umber operator (Fig. 4.3) ad mutual iformatio (Fig. 6.1). It is possible to codesed a lot of iformatio about the memory kerel by plottig κ 11 i the (λ s, λ e ) plae, for differet values of. This is show i Fig Each plot correspods to a differet value of, from 0 up to 8. The depedece o the relative sigs of λ s ad λ e is clearly visible, as is the overall dampig of the memory with icreasig. Figure 6.4: Diagrams for the memory kerel of the BS dyamics. Each plot shows κ 11 i the (λ s, λ e ) plae for a differet value of, from = 0 to = 8. 55

69 Chapter 6. Memory effects i Gaussia Collisioal Models Particularly iterestig, this map is able to very clearly pipoit the regios have egative memory kerels, somethig which is foud to be highly o-trivial Memory Kerel for the TMS dyamics Next we tur to the TMS case (4.23). I this case it is foud that there are, i total, K (θ) = κ 11θ + κ 1zθσ z + κ z1σ z θ + κ zzσ z θσ z. (6.23) These quatities are plotted i Fig. 6.5 for the stable dyamics (ν e < ν crit e ), with λ s = 0.1. All four coefficiets are foud to decay i time i a oscillatory fashio. The physics of each coefficiet, however, is ot ecessarily trasparet. I order to (a) , z (b) z, (c) z, z (d) Figure 6.5: The memory Kerel for the (stable) TMS dyamics, Eq. (4.23) with λ s = 0.1 ad differet values of λ e. Each curve correspods to a differet etry of Eq. (6.2); amely, κ 11, κ 1,σ z, κ σ z,1 ad κ σ z,σ z. 56

70 Chapter 6. Memory effects i Gaussia Collisioal Models q (a) p (b) Figure 6.6: The MK for Q 2 ad P 2, Eq. (6.24), for the TMS dyamics. Other parameters are the same as Fig gai better ituitio, let us focus o the diagoal etries of θ. I this case oe fids that ( ) K (θ) = ( κ 11 + κ 1z + κ z1 + κzz) θ 11 := κ q θ11, 11 ( ) K (θ) = ( κ 11 κ 1z κ z1 + κzz) θ 22 := κ pθ (6.24) The coefficiets κ q ad κ p therefore describe the idividual memory kerels of Q 2 ad P 2, which are differet i the TMS dyamics. These two cotributios are show i Fig. 6.6, for the same parameters as i Fig We also preset diagrams i the (λ s, ν e ) plae i Figs. 6.7 ad 6.8. The plots i Fig. 6.6 reveal a extremely iterestig asymmetry betwee the two quadratures. We see that the memory associated with Q 2 is oscillatory, whereas that associated with P 2 is always egative ad decays mootoically. This asymmetry is a cosequece of our choice of two-mode squeezig i the TMS iteractio (4.8). Figs. 6.7 ad 6.8, however, show that the situatio is more itricate. Ideed, for fixed (λ s, ν e ), κ q is foud to oscillate with. But for κ p this is ot ecessarily the case. Fially, i Fig. 6.9 we compare the previous result with the case of ν e i the viciity, ad larger tha, ν crit e = ; i.e., i the situatio where the dyamics diverges. As ca be see, i this case both κ q ad κ p diverge as well (otice the differet scale of the horizotal axis). This is therefore cotrary to our usual otio of memory: It meas that 57

71 Chapter 6. Memory effects i Gaussia Collisioal Models Figure 6.7: Diagrams for the memory kerel coefficiet κ q [Eq. (6.24)] of the TMS dyamics, i the (λ s, ν e ) place, for = 0,..., 3. Figure 6.8: Similar to Fig. 6.7, but for κ p (a) 0.04 (b) q p Figure 6.9: Similar to Fig. 6.6, but for values of ν e close to, ad larger tha, ν crit e = the system retais a stroger memory from evets i the distat past, tha those i the recet oe. I other words, the relative importace of past evets accumulate. 58

72 Chapter 6. Memory effects i Gaussia Collisioal Models 6.3 Gaussia CP-Divisibility Eve though the MK explicitly shows the depedece o previous states, this aloe does ot ecessarily imply a o-markovia dyamic [11]. It is therefore importat to cotrast the MK with a actual test of o-markoviaity. Here we focus o CP-divisibility of itermediate maps, itroduced i Sec The formulatio for Gaussia dyamics, at the level of the covariace matrix, i Refs. [104, 105]. Ay Gaussia CPTP map, as see i Sec. 3.4, must have the form θ X θx T +Y, where X ad Y are matrices satisfyig [63, 114] M[X, Y] := 2Y + iω ix ΩX T 0, (6.25) with Ω = iσ y the symplectic form. Here M 0 meas the matrix must be positive semidefiite. I our case, the evolutio of the system s CM, from time 0 to, must therefore also be of this form: θ = X θ 0 X T + Y. (6.26) The matrices X ad Y ca be read from the (1, 1) block of the geeral solutio (6.6) ad are idepedet of the iitial state θ 0 ; viz., X = (X ) 11, (6.27) 1 ] Y = (X ) 12 ɛ(x T ) 12 + [X r 1 Y (X T ) r 1, (6.28) 11 r=0 where the subscripts i, j refer here to specific blocks. This easiess i readig of the correspodig map matrices is aother sigificat advatage of the Markovia embeddig represetatio (4.20). To probe whether the dyamics is divisible, we cosider the map takig the system from to m >. Assumig that X ad Y are ivertible, which is true i our case, this will have the form [104] θ m = X m θ X T m + Y m, (6.29) where X m = X m X 1, Y m = Y m X m Y X T m. (6.30) 59

73 Chapter 6. Memory effects i Gaussia Collisioal Models The dyamics is the cosidered divisible whe the itermediate maps (6.29) are a proper CPTP Gaussia map. That is, whe M[X m, Y m ] 0 [Eq. (6.25)]. The above criteria ca be used ot oly as a dichotomic measure of divisibility, but also as a figure of merit [104]. This is accomplished by defiig N m = k m k m ( ) k, {m k } = eigs M[X m, Y m ]. (6.31) 2 This quatity is always o-egative ad the map is divisible iff N m 0 for all m,. Otherwise, the magitude of N m quatifies the extet to which divisibility is broke for that choice of m,. m m (a) (c) m m (b) (d) Figure 6.10: Example of the divisibility criteria for the BS dyamics. The plots show N m i the (, m) plae, with the size of each poit reflectig the magitude of N m. All curves are for λ s = 1.1 ad (a) λ e = 0.75, (b) 0.9, (c) 1.1 ad (d)

74 Chapter 6. Memory effects i Gaussia Collisioal Models BS dyamics We begi our ivestigatio of N m by focusig o the BS dyamics [Eq. (4.21)]. A example of the behaviour of (6.31) is show i Fig. 6.10, where we plot N m i the (, m) plae, with fixed λ s = 1.1 ad differet values of λ e. The magitude of N m is represeted by the size of each poit. These diagrams are iterpreted as follows. We start with Fig. 6.10(a). I this case we see that, for = 1, N m is o-zero oly for m = 2 ad m = 4, beig smaller i the latter. For = 3 the map is always divisible. Ad for = 3, it is ot divisible oly for m = 4 ad 6. These irregularities are a cosequece of the oscillatory character of the parameters appearig, e.g., i Eq. (4.21). Still cocerig Fig. 6.10(a), we see otwithstadig that as gets large, the map teds to be Markovia for all m. As we icrease λ e, however, as i Figs. 6.10(b) ad (c), we see that overall the regios where N m > 0 ted to icrease. They icrease both as a fuctio of, as well as a fuctio of m for fixed. Whe λ e < 0, however, strage thigs happe [Fig. 6.10(d)]. I this case we fid that there ca be highly irregular values of (, m) which yield o-zero N m which, i fact, ca reach sigificatly large values. For istace, the largest value plotted i Fig. 6.10(d) is for = 13, m = 14 ad has the value N For = 16, m = 17, however, oe fids N (ot show). This is to be cotrasted with Fig. 6.10(a), whose largest value is N = We preset these results simply to emphasize that N m ca oscillate violetly. The reaso is due to the term X 1 values of λ s, λ e ad. i Eq. (6.30), which ca blow up for certai Next we tur to the divisibility of a sigle collisio; that is, with m = + 1. Plots of N +1, i the (λ s, λ e ) plae are show i Fig The overall behaviour is foud to alterate with eve ad odd. For eve, the map is always divisible for λ e > 0 ad potetially o-divisible withi certai regios of λ e < 0. Coversely, for odd, oe fids that divisibility breaks dow i sigificat portios of the (λ s, λ e ) plae. A additioal illustratio of the complex depedece of N +1, o λ s, λ e, is provided i Fig. 6.12, where we plot N +1, as a fuctio of for selected values of λ s ad λ e. From this figure, both the eve/odd behavior, as well as the dramatic variatios i the (λ s, λ e ) plae ca be more clearly appreciated. The behavior of N +1, i Fig is exacerbated close to the special poits λ s(e) = 61

75 Chapter 6. Memory effects i Gaussia Collisioal Models Figure 6.11: CP-divisibility measure N +1, [Eq. (6.31)] i the (λ s, λ e ) plae, for the BS dyamics. Each plot correspods to a differet values of : i the first 2 lies, rages from 1 to 8 i steps of 1. I the remaiig lies, = 9, 10, 20, 21, 30, 31, 40, 41, 50, 51 ad 100,

76 Chapter 6. Memory effects i Gaussia Collisioal Models π/2. For istace, i the viciity of λ s = π/2, the dyamics is o-divisible eve for ifiitesimally small λ e. This occurs because λ s = π/2 correspods to the full SWAP, where the CM of the system is completely trasferred to the acilla. As a cosequece, whe the ext acilla arrives to iteract with the system, it will always cotai a sigificat amout of iformatio about it. We therefore expect that i the limit the diagrams i Fig should coverge to arrow lies goig through these special poits (although, ufortuately, we caot actually verify this sice the simulatio cost becomes prohibitive for extremely large ). We may also study similar diagrams for collisios that are more broadly spaced i time. I Fig we preset results for N 1,1+m for differet values of m (we focus o eve values, m = 2, 4,...). This therefore describes the log-term memory of the map, cocerig the first collisio. Two features stad out from this figure. First, as oe would expect, the overall regio i the (λ s, λ e ) plae where the map is CP-divisible teds to shrik with icreasig m. However, the regios aroud λ s = ±π/2 ted to be remarkably + 1, + 1, (a) (c) , + 1, (b) (d) Figure 6.12: CP divisibility measure N +1, as a fuctio of, for the BS dyamics with λ s = 0.8 ad λ e = 0.9, 1.3, 0.5, 0.8. Complemets Fig. (6.11). 63

77 Chapter 6. Memory effects i Gaussia Collisioal Models persistet, remaiig highly o-divisible eve for large m. The results i Figs ad 6.13 refer to divisibility for specific times (, m). We ca also combie all data ad ask, for which regios i the (λ s, λ e ) plae, the BS dyamics is divisible for all (, m). This is show i Fig As expected, for most choices of parameters, the map will ot be CP-divisible for some (, m). Notwithstadig, there are regios where the map is always divisible. These regios ted to be cocetrated close to λ e = 0 (or λ e = π, which is equivalet). Ad they exist eve for large values of λ s. A direct compariso with the memory kerel, Sec. 6.2, is ot geerally possible sice Figure 6.13: CP-divisibility measure, N m,1 [Eq. (6.31)] i the (λ s, λ e ) plae, for the BS dyamics. Each plot correspods to a differet values of m: i the first 2 lies, m rages from 2 to 9 i steps of 1. I the 3rd ad 4th, from m = 10 to 24 i steps of 2. lies, 64

78 Chapter 6. Memory effects i Gaussia Collisioal Models Figure 6.14: Regios i the (λ s, λ e ) plae where the BS dyamics is ot CP-divisible for at least oe choice of (, m). both refer to differet physical aspects of the problem. But if we focus o N +1,, the some compariso is possible. Recall that the MK describes how the dyamics from + 1 is affected by previous times. Thus, regios where the memory kerel is large ted to be accompaied by regios where N +1, > 0. This is ideed the case, as ca be see by comparig Fig with TMS dyamics The situatio for the TMS dyamics is dramatically differet. Diagrams for N +1, i the (λ s, ν e ) plae are show i Fig for differet values of. I cotrast to the BS maps, ow most of parameter space is o-divisible. Moreover, the regio where it is odivisible icreases for loger times. Ad fially, what is perhaps the least ituitive, the regios where the map is o-divisible are deser for small, istead of large, ν e (although the values of N +1, are smaller correspodigly smaller). This is a cosequece of the fact that the TMS dyamics spotaeously creates excitatios i the system, which implies that for large ν e a substatial amout of oise is itroduced, makig the map more likely to be divisible. If ν e = 0 the map is, of course, divisible by costructio. 65

79 Chapter 6. Memory effects i Gaussia Collisioal Models Figure 6.15: CP-divisibility measure, N +1, [Eq. (6.31)] i the (λ s, ν e ) plae, for the TMS dyamics. Each plot correspods to a differet values of, from 1 to 9 i steps of 1. However, the results i Fig show that for arbitrarily small, but o-zero ν e, the map is already o-divisible, albeit with a small N +1,. As with the BS dyamics, oe could also combie all these diagrams to ask whether there are regios i the (λ s, ν e ) where the map is always divisible, for all (, m). The aswer to this questio is, i this case, egative: for the TMS dyamics the dyamics is ever divisible, except for the trivial lie ν e = 0. This represets a major differece i compariso with teh BS dyamics ad, oce agai, is ultimately a property of the etaglig ature of the two-mode squeezig iteractio (4.8). 66