Interferometric Quantum Cascade Systems
IInterferometric Quantum Cascade Systems
Stefano Cusumano, Andrea Mari, and Vittorio Giovannetti
NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy ∗ In this work we consider quantum cascade networks in which quantum systems are connectedthrough unidirectional channels that can mutually interact giving rise to interference effects. Inparticular we show how to compute master equations for cascade systems in an arbitrary interfer-ometric configuration by means of a collisional model. We apply our general theory to two specificexamples: the first consists in two systems arranged in a Mach-Zender-like configuration; the secondis a three system network where it is possible to tune the effective chiral interactions between thenodes exploiting interference effects.
I. INTRODUCTION
Quantum cascade systems (QCSs) describe those phys-ical situations where a first party (the controller ) caninfluence the dynamic of a second party (the idler ) with-out being affected by the latter. The asymmetric char-acter of these couplings originates from the presence ofan environmental medium (e.g. an optical isolator [1] ora bosonic chiral channel) which acts as mediator of theinteractions and which allows for unidirectional propa-gation of pulses from a controller to its associated idler.First interests in these models grew in the 80’s becauseof the necessity of a formalism able to take into accountthe reaction of a quantum system (say an atom or a elec-tromagnetic cavity) to the light emitted by another one[2–8]. In recent years there has been a revival of inter-est towards QCSs, due to the possibility of creating en-tangled states and other tasks for quantum computation[9–12], chiral optical networks [13–15], and in the man-aging of heat transmission [16]; also several experimentalimplementations have been proposed, exploiting, for in-stance, nanophotonic waveguides [17, 18] and spin-orbitcoupling [19].In the QCS models studied so far, the parties compos-ing the systems are typically assumed to be organized toform an oriented linear chain, each acting as controller forthe elements that follow along the line through the me-diation of a single environmental channel. Here insteadwe shall consider more complex configurations where sev-eral subsystems interact, unidirectionally via a networkof mutually intercepting channels as shown in the leftpanel of Fig. 1. In this scenario the QCS couplings whilebeing intrinsically dissipative in nature, can be affectedby interference effects which originate from the propaga-tion of the controlling pulses along the network of con-nections (for instance in the case of figure, the signalsfrom the subsystem S split and recombine before reach-ing subsystem S ). Also, depending on the topology ofthe scheme, controlling signals from different parties (saythe subsystems S and S of the figure) can merge beforereaching a given idler ( S ). The study of such archi-tectures is intriguing as it widens the possibility of en- ∗ [email protected] S S S S S S S S S S E (1) E (2) E (3) E (4) E (1) E (1) E (1) E (1) E (2) E (2) E (2) E (2) E (3) E (3) E (4) E (4) E (1) E (2) E (3) E (4) E (1) E (1) E (1) E (1) E (2) E (2) E (2) E (2) E (3) E (3) E (4) E (4) FIG. 1. Left panel: Pictorial representation of the typicalQCS model we are considering here: a collection S of quan-tum subsystems S , S , · · · , S M (gray circles in the figure)interact unidirectionally by exchanging signals through anoriented network of environmental channels E (1) , E (2) , · · · , E ( K ) which may interfere when intercepting (gray/yellow el-ements). Right panel: collisional model description of thescheme: the propagation of signals along the network is rep-resented in terms of sequence of ordered collisional events in-volving the quantum subsystem and a collection of quantuminformation carriers (black circles). Interference among thesignals arises from collisions between carriers associated withdifferent connecting paths. gineering system-bath coupling in quantum optical sys-tems, which in turn may help in dissipatively preparingquantum many-body states of matter [20–22] with im-portant consequences in the analysis of non-equilibriumcondensed matter physics problems [23–26] and quan-tum information [10, 27–30]. Aim of the present workis to derive a mathematical framework that incorporatethese phenomena in a consistent way. For this purpose weshall adopt the collisional approach to QCS introducedin Refs. [31, 32]. Accordingly each unidirectional channelforming the network of connections is described in termsof a collection of sub-environments (quantum carriers) a r X i v : . [ qu a n t - ph ] M a y that evolve in time stroboscopically through a series oftime-ordered collisions involving the various subsystems– see right panel of Fig. 1. Interference effects are alsodescribed in terms of collisions, this time involving car-riers associated with different channels (e.g. the red andblack carriers of the figure). Similar cascade networkscould also be studied in the Heisenberg picture withinthe so called input-output formalism [6, 7], from whichin principle a master equation could be derived usingquantum stochastic calculus [6, 8]. The collisional modelpresented in this work allows to directly obtain the de-sired master equation and, being based on a simple andoperational model of dissipation, naturally generates aMarkovian completely positive dynamics without the ne-cessity of introducing further hypothesis and approxima-tions typical of other microscopic derivations.Here is the outline of the paper: in Sec. II we reviewbriefly the collisional approach to QCS of Ref. [31, 32]and adapt it for writing the master equation of our model.The resulting expression is then cast in standard Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form [33–36]in Sec. II B. Building from these results, in Sec. III wedescribe the arising of interference effects in the model,by discussing some specific examples. In particular inSec. III A we deal with a Mach-Zender-like interferom-eter, showing how with a phase shift it is possible tomodify the effective temperature felt by the second opti-cal cavity. Then in Sec. III B we turn to a configurationof three cavities where we show how, by appropriatelyexploiting interference effects, it is possible to have a sys-tem with only first-neighbor interactions. The paper thenends with Sec.IV where we draw conclusions and give anoutlook for future works, and with the Appendices wherewe present some technical derivations. II. THE MODEL
In the collisional model approach [37–43] to open quan-tum systems dynamics the environment is represented asa large many-body quantum system whose constituents(quantum information carriers or carriers in the follow-ing) interact with the system of interest via an orderedsequence of impulsive unitary transformations (collisionalevents). This yields a, time-discrete, stroboscopic evolu-tion which can then be turned into a continuous timedynamics by properly sending to infinite the number ofcollisions and to zero the time interval among them whilekeeping constant their product. By means of collisionalmodels it is possible to derive both Markovian [39, 40]and non-Markovian [41–43] master equations. In thispaper we take our steps from the collisional approach toQCS presented in Refs. [31, 32], generalizing it to includenetwork configurations similar to the one presented in theleft panel of Fig. 1.To this aim we consider a system S made out of M (not necessarily identical) subsystems S , S , · · · , S M (e.g. M optical cavities). Similarly to the scheme of S S S S S E (1)1 E (2)1 E (3)1 E (4)1 E (4)2 E (3)2 E (2)2 E (1)2 E (1)3 E (2)3 E (3)3 E (4)3 temporal axis for the subsystems t e m p o r a l a x i s f o r t he en v i r on m en t a l ca rr i e r s E (1) U S , E U S , E U S , E U S , E U S , E U S , E U S , E U S , E U S , E U S , E U S , E U S , E U S , E U S , E U S , E E M (1) E M (2) E M (3) E M (4) E M (4) E M (3) E M (2) E M (1) E M (1) E M (3) E M (4) E M (2) E FIG. 2. Flowchart representation of the couplings in a QCSnetwork in the collisional model approach. The evolution ofthe quantum carriers { E ( k ) n ; k = 1 , , · · · , n = 1 , , · · · } repre-senting the channels evolve in time from top to bottom, whilethe quantum subsystems { S m ; m = 1 , , · · · } evolve from leftto right. The black elements represent collisional events be-tween one of the subsystems and the carriers; the yellow (palegray) elements instead represent the dynamical evolution ofthe carriers among two consecutive collisional events (possi-bly including interactions among carriers of different species).Notice that the upper index and lower index of carrier E ( k ) n re-fer, respectively, to the environmental channel (i.e. the chan-nel E ( k ) in this case) and the time group (i.e. E n ) it belongs to. Fig. 1, they are connected via a network of QCS inter-actions in such a way that for each m = 1 , · · · , M , theelement S m is capable of controlling all the elements S m (cid:48) with m (cid:48) > m without being affected by their dynam-ics, the coupling being provided by a collection of uni-directional environmental channels E (1) , E (2) , · · · , E ( K ) which intercept to form a graph. In what follows each ofthese channels are represented in terms of a long, orderedstring of quantum carriers which act as mediators of theinteractions, propagating along the network and experi-encing impulsive interactions (collisional events) with thesystem elements as sketched on the right panel of Fig. 1.Specifically, for k = 1 , · · · , K , the k -th channel E ( k ) is de-scribed by the carriers { E ( k ) n ; n = 1 , , · · · } , the subscript n indicating the order with which they start interactingwith S . Accordingly we find it convenient to regroupthese elements into sets which includes those that possesthe same value of n independently from the channel theybelong to, e.g. the set E := { E (1)1 , E (2)1 , · · · , E ( K )1 } , theset E := { E (1)2 , E (2)2 , · · · , E ( K )2 } , and so on and so forth.This way, neglecting the time it takes from one carrierto move from one element of S to the next, we can usethe label n as the discrete temporal coordinate of themodel (more on this on the following paragraphs). Inparticular, indicating with ˆ U S m , E n the unitary operatorassociated with the collisional event that couples S m andthe carriers which enters at the n -th temporal step, i.e.the carriers of E n , the causal structure of the model isenforced by imposing that such operator should precedeˆ U S m +1 , E n (meaning that S m +1 sees E n only after it hasinteracted with S m ) and ˆ U S m , E n +1 (meaning that the ele-ment of E n enters the network before those of E n +1 ) – therelative ordering of ˆ U S m +1 , E n and ˆ U S m , E n +1 being insteadirrelevant as they act on different systems and hence com-mute. The unitaries ˆ U S m , E n ’s trigger the dissipative evo-lution of S which is responsible for the QCS dynamics.In our model they are interweaved with Completely Pos-itive and Trace preserving (CPT) super-operators [44]acting on the quantum carriers only, which describe thepropagation of signals along the channels and (possibly)their mutual interactions. In particular, in what followswe shall use the symbol M ( m ) E n to indicate the CPT mapwhich acts on the carries of the set E n after the collisionalevent that couples them with S m and before the one thatinstead couples them with S m +1 – see Fig. 2. A conve-nient way to express the resulting evolution is obtainedby introducing the density matrix ˆ R ( n ) which describesthe joint state of S and of the first n -th carriers of allchannels (i.e. the carriers belonging to the sets E , E , · · · , E n ) after they have interacted. From the above con-struction, the relation between such state and its evolvedcounterpart ˆ R ( n + 1) can then be expressed asˆ R ( n + 1) = C S , E n +1 ( ˆ R ( n ) ⊗ ˆ η E n +1 ) , (1)where ˆ η E n +1 indicates the input state of the elements of E n +1 when they enter the network, while C S , E n +1 is thesuper-operator associated with the collisional events theyparticipate. Explicitly, using the short hand notation ←− Π Mm =1 A ( m ) = A ( M ) A ( M − · · · A (2) A (1) (2)to represent the ordered product of the symbols {A (1) , A (2) , · · · , A ( M ) } this is given by C S , E n = ←− Π Mm =1 [ M ( m ) E n ◦ U S m , E n ] , (3)where “ ◦ ” indicates composition of super-operators andwhere for each m = 1 , · · · , M the symbol U S m , E n indi-cates the super-operator counterpart of the unitary trans-formation ˆ U S m , E n , i.e. U S m , E n ( · · · ) = ˆ U S m , E n ( · · · ) ˆ U † S m , E n . (4)Few remarks are mandatory at this point:i) the density matrices ˆ R ( n ) and ˆ R ( n + 1) operateon different spaces (indeed ˆ R ( n + 1) applies also tothe carriers of the set E n +1 while ˆ R ( n ) does not).What is relevant for us is the fact that by takingthe partial trace over the carriers they give us the temporal evolution of the system of interest at thevarious step of the process. In particularˆ ρ ( n ) := (cid:68) ˆ R ( n ) (cid:69) E = Tr E [ ˆ R ( n )] , (5)is the joint state of the subsystems S at the n -thtime step;ii) as already mentioned in our analysis the time ittakes for a carrier to move from one collision to thenext is assumed to be negligible, only the causal or-dering of these events being preserved. Accordinglyin Fig. 2 time flows from left to right for all the S j synchronously. This assumption is introduced be-cause, differently to the case of a simple linear chainof cascaded systems [7], when dealing with multi-ple channels one cannot eliminate the delay timeby simply shifting the time origin of each subsys-tem. Actually, significant delay times can give riseto non-Markovian effects [42], whose study goes be-yond the goal of the present work.iii) in writing Eq. (1) we are implicitly assuming thatthe input state of the carriers factorizes with re-spect to the grouping E , E , · · · , i.e. no correla-tions are admitted among carriers which enters thescheme at different time steps. Yet, at this level,the model still admits the possibility of correlationsamong carriers of different channels. In what fol-lows we shall however enforce a further constraintthat limits the choices of the input ˆ η E n , see nextpoint and Eq. (12) below.iv) In the original QCS model of Fig. 1 the unidi-rectional channels E (1) , E (2) and E ( K ) form a sta-tionary medium which contributes to the dynam-ics only by allowing signals from one subsystem topropagate to the next one (in other words in theabsence of the interactions with the elements of S they will not present any temporal evolution).To enforce this special character in the collisionalmodel we require it to be translationally invariantwith respect to the index n , e.g. imposing that allthe input states ˆ η E , ˆ η E , · · · , ˆ η E n of the carrierssets E , E , · · · , E n coincide, and that for given m the unitary couplings U S m , E n and the maps M ( m ) E n should be independent from n . This hypothesiscan however be relaxed [32] with the condition thatthe change in the coupling is slow compared to thecharacteristic time scale of the systems S m . A. The continuous time limit
By solving the recursive equation (1) and taking thepartial trace as in Eq. (5) one obtains a collection ofdensity matrices ˆ ρ (0), ˆ ρ (1), · · · , ˆ ρ ( n ), which provides aneffective description of the temporal evolution of the jointstate of the subsystems S , S , · · · , S M in the presenceof a collection of quantum carriers that connects themthrough a network of unidirectional channels. Such stro-boscopic representation of the dynamics can be turnedinto a continuous time description by taking a properlimit in which the number of collisions per second ex-perienced by the element of S goes to infinity [31, 32].Accordingly we write the interaction unitaries asˆ U S m , E n = exp[ − ig K (cid:88) k =1 ˆ H S m ,E ( k ) n ∆ t ] , (6)where g is a coupling constant that we shall use to gaugethe intensity of the system-carrier interactions, ∆ t is theduration of a single collisional event, and whereˆ H S m ,E ( k ) n = (cid:88) (cid:96) ˆ A ( (cid:96),k ) S m ⊗ ˆ B ( (cid:96),m ) E ( k ) n , (7)is the most general Hamiltonian describing the inter-actions between S m and E ( k ) n , with ˆ A ( (cid:96),k ) S m and ˆ B ( (cid:96),m ) E ( k ) n nonzero operators acting locally on such systems respec-tively [36]. Next we take the product g ∆ t to be a smallquantity and expand our equations up to the second or-der in such term. In this regime, upon tracing upon thedegree of freedom of the carriers, Eq. (1) yields the iden-tityˆ ρ ( n + 1) − ˆ ρ ( n )∆ t = − ig (cid:88) m,k,(cid:96) γ ( (cid:96) ) m ( k ) (cid:104) ˆ A ( (cid:96),k ) S m , ˆ ρ ( n ) (cid:105) − + g ∆ t (cid:40) M (cid:88) m =1 L m (ˆ ρ ( n )) + M (cid:88) m (cid:48) = m +1 M − (cid:88) m =1 D m,m (cid:48) (ˆ ρ ( n )) (cid:41) + O ( g ∆ t ) , (8)with L m ( · · · ) = 12 K (cid:88) k,k (cid:48) =1 (cid:88) (cid:96),(cid:96) (cid:48) γ ( (cid:96),(cid:96) (cid:48) ) m ( kk (cid:48) ) (cid:110) A ( (cid:96),k ) S m ( · · · ) ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m − (cid:104) ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m ˆ A ( (cid:96),k ) S m , · · · (cid:105) + (cid:111) , (9)and for m (cid:48) > m , D m → m (cid:48) ( · · · ) = K (cid:88) k,k (cid:48) =1 (cid:88) (cid:96),(cid:96) (cid:48) (cid:110) ζ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) ˆ A ( (cid:96),k ) S m (cid:104) · · · , ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) (cid:105) − − ξ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) (cid:104) · · · , ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) (cid:105) − ˆ A ( (cid:96),k ) S m (cid:111) , (10)where (cid:104) · · · , · · · (cid:105) ± represent the commutator ( − ) andanti-commutator (+) brackets respectively. In the aboveexpressions γ ( (cid:96) ) m ( k ) , γ ( (cid:96),(cid:96) (cid:48) ) m ( kk (cid:48) ) , ζ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) , and ξ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) arecomplex coefficients which depend upon correlation termof the input state of the carriers (see Appendix A for theexplicit definitions).The continuous time limit is finally obtained sendingto infinity n of collisions while the time interval ∆ t of each collision goes to zero and the coupling constant g explodes in such a way that:lim ∆ t → + n ∆ t = t , lim ∆ t → + g ∆ t = γ , (11)with γ being a positive constant which set the time scaleof the model. Notice that the last assumption could leadto problem in the first-order term of the series expan-sion of Eq. (8), whose contribution to the final expres-sion would explode. Such instability is a typical trait inthe derivation of master equations [36] for open quantumsystems. It can be solved by imposing a stability con-dition [31, 32] for the environmental degree of freedomof the system, i.e. by requiring that the input carrierstates ˆ η E n and their evolved counterparts along the net-work should not be influenced (at first order) by the col-lisions with the subsystems. This is consistent with thedescription of the environmental channels as composedby many small sub-environments all in the same refer-ence state that interacts weakly with the subsystems. Inthe standard derivation of master equations such stabilitycondition is usually assumed as well, and it amounts tothe possibility of approximating the joint density matrixas a tensor product between the reduced density matrixof the system and the one of the environment at anytime. In our case this corresponds to nullify the coef-ficients γ ( (cid:96) ) m ( k ) appearing in the rhs of Eq. (8), i.e. byimposing (see Eq. (A10) of the Appendix A) (cid:68) ˆ B ( (cid:96), E ( k ) n ˆ η E n (cid:69) E = 0 , (cid:68) ˆ B ( (cid:96), E ( k ) n M (1) E n (ˆ η E n ) (cid:69) E = 0 , (cid:68) ˆ B ( (cid:96), E ( k ) n [ M (2) E n ◦ M (1) E n ](ˆ η E n ) (cid:69) E = 0 , ... (cid:68) ˆ B ( (cid:96),m ) E ( k ) n [ M ( m − E n ◦ · · · ◦ M (1) E n ](ˆ η E n ) (cid:69) E = 0 , (12)for all k = 1 , · · · , K , for all m = 1 , · · · , M , and for all (cid:96) ( ˆ B ( (cid:96),m ) E ( k ) n being the carriers operators which participate tothe coupling Hamiltonian (7)).By enforcing the condition (12), Eq. (8) finally can becasted in the following differential form ∂ ˆ ρ ( t ) ∂t = γ C (ˆ ρ ( t )) , (13)with C the QCS super-operator C ( · · · ) = M (cid:88) m =1 L m ( · · · ) + M (cid:88) m (cid:48) = m +1 M − (cid:88) m =1 D m → m (cid:48) ( · · · ) . (14)Equation (13) is a Markovian master equation which de-scribes the dynamical evolution of the joint density ma-trix ˆ ρ ( t ) for the system of interest S . The term on therhs is the generator of the dynamics and can be casted inGKSL form [33–36] by properly reorganizing the variouscontributions (see next section). It is however worth an-alyzing the causal structure of the model a bit further bylooking directly at the expression presented in (14). Onthe one hand we have the terms L m which describe lo-cal effects of the interaction between the various elementof S and the environment: they are not capable of cre-ating correlations among the S m ’s and only account fordissipative behaviors. On the other hand the non-localterms D m → m (cid:48) describe the interaction between the m -thand m (cid:48) -th subsystem (with m (cid:48) > m ) originating by thepropagation of the carries from the former to the latter.In principle these are capable of building up correlationsamong the various elements of S . However, at variancewith what would happen with a direct Hamiltonian in-teraction, such couplings are intrinsically asymmetric inagreement with the cascade structure of the network ofconnections. In particular one may observe that by trac-ing over S m (cid:48) the term D m → m (cid:48) (ˆ ρ ( t )) always nullifies, i.e.Tr S m (cid:48) [ D m → m (cid:48) (ˆ ρ ( t ))] = 0 , (15)while this is not necessarily the case when the same termis traced over S m . This implies, for instance, that the re-duced density matrix ˆ ρ ( t ) of the first element of S (theone which in principle controls all the others without be-ing controlled by them) evolves in time without beingaffected by the presence of the latter. Similarly the evo-lution of the first m elements of S does not depend uponthe remaining ones.The derivation of Eq. (13) we have presented hereclosely follows the one of Ref. [31]. The main differencewith the latter is the inner structure of the generators L m and D m,m (cid:48) which in our case includes contributionsfrom multiple unidirectional channels as indicated by thesum over the indexes k and k (cid:48) of Eqs. (9) and (10). As itwill be clear discussing some explicit examples (see nextSection) this is what allows us to account for interfer-ence effects that originate with the signals propagationthrough the network. B. Standard GKSL form and effective Hamiltoniancouplings
The decomposition of the coupling Hamiltonians pre-sented in Eq. (7) is clearly not unique. Alternatives canbe obtained by replacing the ˆ A ( (cid:96),k ) S m ’s (resp. the ˆ B ( (cid:96),m ) E ( k ) n ’s)with proper linear combinations of the same objects forinstance by expanding them into an operator basis. Themaster equation (12) clearly does not depend on thischoice as it derives from a pertubative expansion on thecoupling parameter g which enters in the model as a mul-tiplicative factor of ˆ H S m ,E ( k ) n , and from the stability con-ditions (12), which are explicitly invariant under linearcombinations of the ˆ B ( (cid:96),m ) E ( k ) n ’s. In this section we shall in-voke this freedom assuming the ˆ A ( (cid:96),k ) S m ’s and the ˆ B ( (cid:96),m ) E ( k ) n ’sto be self-adjoint (a possibility which is allowed by the fact that ˆ H S m ,E ( k ) n has to be self-adjoint as well). Thisworking hypothesis is not fundamental but, as pointedout in Refs. [31, 32], turns out to be useful as it makesexplicit some structural properties of the resulting super-operators, ensuring for instance the identities γ ( (cid:96),(cid:96) (cid:48) ) m ( kk (cid:48) ) = (cid:104) γ ( (cid:96) (cid:48) ,(cid:96) ) m ( k (cid:48) k ) (cid:105) ∗ , (16) ξ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) = (cid:104) ζ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) (cid:105) ∗ , (17)as evident from Eqs. (A12)-(A14) of the Appendix. Ouraim is to exploit these properties to generalize the anal-ysis of Ref. [12] by casting the QCS super-operator (14)into an explicit standard GKSL form [36], i.e. as thesum of an effective Hamiltonian term plus a collection ofpurely dissipative contributions C ( · · · ) = − i [ ˆ H, ( · · · )] (18)+ (cid:88) i L ( i ) ( · · · ) ˆ L ( i ) † − (cid:104) ˆ L ( i ) † ˆ L ( i ) , ( · · · ) (cid:105) + , with ˆ H being self-adjoint and with the ˆ L ( i ) ’s being a col-lection of operators acting on S . In Ref. [12] this trickwas used to show that a collection of two-level atoms cou-pled in QCS fashion via an unidirectional optical fiber,initialized at zero temperature, can be described as orig-inating from an effective two-body coupling Hamiltonianwith chiral symmetry.We start by focusing on the local contributions ofEq. (13). Indicating with j the joint index ( (cid:96), k ), Eq. (16)implies that, for each m assigned, the matrix Θ jj (cid:48) of ele-ments γ ( (cid:96),(cid:96) (cid:48) ) m ( kk (cid:48) ) / jj (cid:48) = Θ ∗ j (cid:48) j . Further-more, by direct inspection of Eq. (A12) one can easilyprove that, being ˆ B ( (cid:96),m ) E ( k ) n self-adjoint, such matrix is alsosemi-positive definite. Accordingly Eq. (9) can be ex-pressed as a purely dissipative term L m ( · · · ) = (cid:88) s λ s (cid:110) ( s ) S m ( · · · )ˆΛ ( s ) † S m − (cid:104) ˆΛ ( s ) † S m ˆΛ ( s ) S m , · · · (cid:105) + (cid:111) , (19)where { λ s } s are the eigenvalues of Θ j,j (cid:48) and where wehave introduced the operatorsˆΛ ( s ) S m = (cid:88) k,(cid:96) v ( (cid:96),k ) ,s ˆ A ( (cid:96),k ) S m , (20)with v j,s being the unitary matrix which allows us todiagonalize Θ j,j (cid:48) , i.e. Θ j,j (cid:48) = (cid:80) s v j,s λ s v ∗ s,j (cid:48) . In the ab-sence of the coupling contributions D m,m (cid:48) , Eq. (13) willhence reduce to the standard form (18) with ˆ H = 0 andwith the dissipative operators ˆ L ( i ) being identified with √ λ s ˆΛ ( s ) S m .Consider next the non-local contributions of Eq. (13).Due to their peculiar structure they cannot directly pro-duce terms as those on the right hand side of Eq. (18).We notice however that for all m (cid:48) > m one can writeˆ A ( (cid:96),k ) S m (cid:104) · · · , ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) (cid:105) − = − (cid:104) ˆ A ( (cid:96),k ) S m ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) , · · · (cid:105) − + ˆ A ( (cid:96),k ) S m · · · ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) − (cid:104) ˆ A ( (cid:96),k ) S m ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) , · · · (cid:105) + , and (cid:104) · · · , ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) (cid:105) − ˆ A ( (cid:96),k ) S m = − (cid:104) ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) ˆ A ( (cid:96),k ) S m , · · · (cid:105) − − ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) · · · ˆ A ( (cid:96),k ) S m + 12 (cid:104) ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) ˆ A ( (cid:96),k ) S m , · · · (cid:105) + , (21)which simply follow from the fact that ˆ A ( (cid:96),k ) S m , ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) op-erate on different quantum systems and hence commute.Replacing these identities into Eq. (10) we can write D m → m (cid:48) ( · · · ) = − i (cid:104) ˆ H m,m (cid:48) , ( · · · ) (cid:105) − + ∆ L m,m (cid:48) ( · · · ) . (22)In this expression the first contribution is an effectiveHamiltonian term withˆ H m,m (cid:48) = K (cid:88) k,k (cid:48) =1 (cid:88) (cid:96),(cid:96) (cid:48) ξ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) − ζ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) i ˆ A ( (cid:96),k ) S m ⊗ ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) = K (cid:88) k,k (cid:48) =1 (cid:88) (cid:96),(cid:96) (cid:48) Im[ ξ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) ] ˆ A ( (cid:96),k ) S m ⊗ ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) , (23)where in the second line we used (16). The second con-tribution on the right hand side of (22) instead featuresthe super-operator∆ L m,m (cid:48) ( · · · ) = (cid:88) m ,m = m,m (cid:48) K (cid:88) k ,k =1 (cid:88) (cid:96) ,(cid:96) ∆ D ( (cid:96) ,(cid:96) ) m ,m ( k k ) × (cid:26) A ( (cid:96) ,k ) S m ( · · · ) ˆ A ( (cid:96) ,k ) S m − (cid:104) ˆ A ( (cid:96) ,k ) S m ˆ A ( (cid:96) ,k ) S m , ( · · · ) (cid:105) + (cid:27) (24)with coefficients∆ D ( (cid:96) ,(cid:96) ) m ,m ( k k ) = 12 ζ ( (cid:96) ,(cid:96) ) m m ( k k ) for m < m m = m ξ ( (cid:96) ,(cid:96) ) m m ( k k ) for m > m . (25)One may notice that indicating with j the joint index( (cid:96), k, m ), then from Eq. (16) it follows that the ma-trix ∆Ω j,j (cid:48) of elements ∆ D ( (cid:96) ,(cid:96) ) m ,m ( k k ) is Hermitian, i.e.∆ D ( (cid:96) ,(cid:96) ) m ,m ( k k ) = (cid:104) ∆ D ( (cid:96) ,(cid:96) ) m ,m ( k k ) (cid:105) ∗ . Yet there is noguarantee that ∆Ω j,j (cid:48) is semi-positive definite (an ex-plicit counter-example will be presented in the next sec-tion) thus preventing one from directly expressing (24)as a sum of dissipative contributions by diagonalizationof ∆Ω j,j (cid:48) as we did for the local terms of C . However by replacing Eq. (22) into (14) we arrive to C ( · · · ) = − i [ ˆ H, ( · · · )] + M (cid:88) m ,m =1 K (cid:88) k ,k =1 (cid:88) (cid:96) ,(cid:96) D ( (cid:96) ,(cid:96) ) m ,m ( k k ) × (cid:26) A ( (cid:96) ,k ) S m ( · · · ) ˆ A ( (cid:96) ,k ) S m − (cid:104) ˆ A ( (cid:96) ,k ) S m ˆ A ( (cid:96) ,k ) S m , ( · · · ) (cid:105) + (cid:27) (26)where now ˆ H is the effective Hamiltonianˆ H = M (cid:88) m (cid:48) = m +1 M (cid:88) m =1 ˆ H m,m (cid:48) , (27)and where the coefficients D ( (cid:96) ,(cid:96) ) m ,m ( k k ) are obtained fromthose of Eq. (25) by using the elements γ ( (cid:96),(cid:96) (cid:48) ) m ( kk ) to fill thezero’s on the m , m diagonal, i.e. D ( (cid:96) ,(cid:96) ) m ,m ( k k ) = ∆ D ( (cid:96) ,(cid:96) ) m ,m ( k k ) for m (cid:54) = m γ ( (cid:96) ,(cid:96) ) m ( k ,k ) / m = m . (28)To complete the derivation of Eq. (18) one should provethe non-negativity of the matrix Ω j,j (cid:48) = D ( (cid:96) ,(cid:96) ) m ,m ( k k ) ( j being once more the joint index ( (cid:96), k, m )). This isshown explicitly in App. B. Indicating hence with κ i ( ≥
0) the eigenvalues of Ω j,j (cid:48) and with w j,i the elementsof the unitary matrix that diagonalizes it (i.e. Ω j,j (cid:48) = (cid:80) s w j,i κ i w ∗ i,j (cid:48) ) we can finally identify the operators ˆ L ( i ) of Eq. (18) withˆ L ( i ) = √ κ i (cid:88) (cid:96),k,m w ( (cid:96),k,m ) ,i ˆ A ( (cid:96),k ) S m . (29)A final remark before concluding the section: as al-ready mentioned in deriving the above results we find itconvenient to assume the operators ˆ A ( (cid:96),k ) S m and ˆ B ( (cid:96),m ) E ( k ) n tobe self-adjoint. Yet the analysis presented here is stillvalid even when this assumption does not hold - simplysome of the structural properties of the involved mathe-matical objects are less explicit. In particular the eigen-values of the matrices (25) and (28) can be shown tobe independent from the decomposition adopted in writ-ing (7) (the associated matrices being related by similar-ity transformations). III. INTERFERENCE EFFECTS
Here we present a couple of examples of QCSs whichenlighten the arising of interference effects during thepropagation of signals on a network of unidirectional con-nections and how they can be used to externally tune thecouplings among the various subsystems. E (1) E (2) BS BS PS S S E (1)1 E (2)1 E (2)2 E (1)2 E (1)3 E (2)3 S S E (1) E (1) E (2) E (2) … PS BS BS FIG. 3. Left panel: A sketch of the QCS scheme discussedin Sec. III A. S and S are two quantum system connectedvia two unidirectional bosonic channels E (1) and E (2) whichare interweaved to form a Mach-Zehnder interferometer ( BS , BS being beam splitters and P S being a phase-shifter ele-ment). Right panel: causal flowchart of the couplings of themodel in the collisional approach.
A. Example 1: Mach-Zehnder model
As a first example we analyze the scheme of Fig. 3where M = 2 quantum systems S and S which can beidentified either with monochromatic quantum electro-dynamical (QED) cavities of frequency ω or with twotwo-level atoms of energy gap (cid:126) ω , interacting via K = 2unidirectional (chiral) optical channels E (1) and E (2) thatare interweaved to form a Mach-Zehnder interferometer.Specifically the environment E (1) , which we assume tobe in a thermal state of temperature T , interacts withthe first subsystem S via a standard excitation-hoppingterm. The output from S is then mixed with the sec-ond environment E (2) (initialized at temperature T ) in afirst beam splitter BS , and then the two signals followtwo paths accumulating a phase shift P S , before mixingonce again in the second beam splitter BS . Finally theoutput from one of the two ports is sent to the secondsubsystem S .In the collisional approach we shall represent E (1) ( E (2) ) as a collection of independent monochromaticoptical quantum carriers { E (1) n ; n = 1 , , · · · } (resp. { E (2) n ; n = 1 , , · · · } ) described by the annihilation op-erators { ˆ b E (1) n ; n = 1 , · · · } (resp. { ˆ b E (2) n ; n = 1 , · · · } )each initialized into Gibbs states of temperature T (resp. T ), i.e. the Gaussian stateˆ η E (1) n := exp[ − β ˆ b † E (1) n ˆ b E (1) n ]Tr[exp[ − β ˆ b † E (1) n ˆ b E (1) n ]] , (30)with β = (cid:126) ω/k B T (resp. ˆ η E (2) n with β = (cid:126) ω/k B T ).Accordingly the input states ˆ η E n of Eq. (1) are now ex-pressed as ˆ η E n = ˆ η E (1) n ⊗ ˆ η E (2) n . (31)The interactions between such elements and S , S willfollow the causal structure depicted on the right panel of Fig. 3. In particular we assume no direct couplingsbetween { ˆ b E (2) n ; n = 1 , · · · } and the cavities, i.e.ˆ H S m ,E (2) n = 0 , (32)and take the Hamiltonian (7) which describes the inter-action with the modes { ˆ b E (1) n ; n = 1 , · · · } asˆ H S m ,E (1) n = ˆ a † m ˆ b E (1) n + ˆ b † E (1) n ˆ a m , (33)where, for m = 1 ,
2, ˆ a m , ˆ a † m are the annihilation andcreation operators of the cavity S m , or in case where S , S correspond to two-level quantum systems, to the as-sociated lowering and raising Pauli operators. Finallywe have to specify the structure of the CPT map M (1) E n which is responsible for the evolution of the carriers E ( k ) n between their collisions with S and S (see Fig. 2) andpossibly for the emergence of interference effects in themodel. In the case we are studying it is given by theconcatenation of three unitary terms, ˆ V BS ˆ V P S ˆ V BS , thefirst and the third being associated respectively with thebeam-splitter transformations BS and BS that couplethe two channels, the second with the phase shift trans-formation P S acting on the carriers of E only, i.e. M (1) E n ( · · · ) = ˆ V BS ˆ V P S ˆ V BS ( · · · ) V † BS ˆ V † P S ˆ V † BS . (34)Specifically, indicating with (cid:15) j the transmissivity of BS j ,the action of ˆ V BS j is fully determined by the identitiesˆ V † BS j ˆ b E (1) n ˆ V BS j = √ (cid:15) j ˆ b E (1) n − i (cid:112) − (cid:15) j ˆ b E (2) n , (35)ˆ V † BS j ˆ b E (2) n ˆ V BS j = − i (cid:112) − (cid:15) j ˆ b E (1) n + √ (cid:15) j ˆ b E (2) n , (36)while the action of ˆ V P S by the identityˆ V † P S ˆ b E (1) n ˆ V P S = e − iϕ ˆ b E (1) n , (37)ˆ V † P S ˆ b E (2) n ˆ V P S = ˆ b E (2) n . (38)It is worth observing that, in the limit where (cid:15) = (cid:15) = 1(i.e. no mixing between E and E ) and T = 0, the modeljust described reproduce the one analyzed in Ref. [12] for M = 2 two-level atoms.We first observe that with the above choices the sta-bility condition (12) is fulfilled. Indeed from Eq. (32)follows trivially γ ( (cid:96) ) m (2) = 0. Instead from Eq. (33) we cantake ˆ A ( (cid:96),k ) S m = δ k, (cid:26) ˆ a † m for (cid:96) = 1ˆ a m for (cid:96) = 2, (39)ˆ B ( (cid:96),k ) S m = δ k, (cid:40) ˆ b E (1) n for (cid:96) = 1ˆ b † E (1) n for (cid:96) = 2, (40)so that γ (1)1(1) = [ γ (2)1(1) ] ∗ = (cid:68) ˆ b E (1) n ˆ η E n (cid:69) E = (cid:68) ˆ b E (1) n ˆ η E (1) n (cid:69) E = 0 , (41)which trivially follow from the fact that the annihilationoperator admits zero expectation value on Gibbs states.Analogously we have γ (1)2(1) = [ γ (2)2(1) ] ∗ = (cid:68) ˆ b E (1) n M (1) E n (ˆ η E n ) (cid:69) E = (cid:68) ˜ M (1) E n (ˆ b E (1) n ) ˆ η E n (cid:69) E = c ( ϕ ) (cid:68) ˆ b E (1) n ˆ η E (1) n (cid:69) E + s ( ϕ ) (cid:68) ˆ b E (2) n ˆ η E (2) n (cid:69) E = 0 , (42)where ˜ M (1) E n is the complementary counterpart of M (1) E n fulling the property˜ M (1) E n (ˆ b E (1) n ) := ( V † BS ˆ V † P S ˆ V † BS )ˆ b E (1) n ( ˆ V BS ˆ V P S ˆ V BS )= c ( ϕ )ˆ b E (1) n + s ( ϕ )ˆ b E (2) n , (43)with c ( ϕ ) = e − iϕ √ (cid:15) (cid:15) − (cid:112) (1 − (cid:15) )(1 − (cid:15) ) ,s ( ϕ ) = − ie − iϕ (cid:112) (1 − (cid:15) ) (cid:15) − i (cid:112) (cid:15) (1 − (cid:15) ) . (44)In a similar way we can evaluate the coefficients γ ( (cid:96),(cid:96) (cid:48) ) m ( kk (cid:48) ) , ζ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) , and ξ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) that define the super-operators (9) and (10). First of all we notice that fromEq. (32) it follows that only the terms with k = k (cid:48) = 1can have non vanishing values. Next, indicating with N k = ( e β k − − , (45)the mean photon numbers of the k -th thermal bath, weobserve that for the local terms of S the following iden-tities hold: γ (1 , kk (cid:48) ) = (cid:2) γ (2 , kk (cid:48) ) (cid:3) ∗ = δ k, δ k (cid:48) , (cid:68) ˆ b E (1) n ˆ η E (1) n (cid:69) E = 0 ,γ (2 , kk (cid:48) ) = δ k, δ k (cid:48) , (cid:68) ˆ b † E (1) n ˆ b E (1) n ˆ η E (1) n (cid:69) E = δ k, δ k (cid:48) , N ,γ (1 , kk (cid:48) ) = δ k, δ k (cid:48) , (cid:68) ˆ b E (1) n ˆ b † E (1) n ˆ η E (1) n (cid:69) E = δ k, δ k (cid:48) , ( N + 1) , where δ k,k (cid:48) indicates the Kronecker delta and where weused known properties of the second order expectationvalues of the Gibbs states. Accordingly the associatedsuper-operator (9) becomes L ( · · · ) = ( N + 1) (cid:16) ˆ a ( · · · )ˆ a † − (cid:104) ˆ a † ˆ a , · · · (cid:105) + (cid:17) + N (cid:16) ˆ a † ( · · · )ˆ a − (cid:104) ˆ a ˆ a † , · · · (cid:105) + (cid:17) , (46)which is already in the standard GKSL form (19) andwhich describes a thermalization process where S ab-sorbs and emits excitations from a thermal bath at tem-perature T . Similarly the local terms for the S gives γ (1 , kk (cid:48) ) = (cid:2) γ (2 , kk (cid:48) ) (cid:3) ∗ = δ k, δ k (cid:48) , (cid:68) ˆ b E (1) n M (1) E n (ˆ η E n ) (cid:69) E = δ k, δ k (cid:48) , (cid:68) (cid:16) c ( ϕ )ˆ b E (1) n + s ( ϕ )ˆ b E (2) n (cid:17) ˆ η E n (cid:69) E = 0 , (47) and γ (2 , kk (cid:48) ) = δ k, δ k (cid:48) , (cid:68) (cid:16) c ∗ ( ϕ )ˆ b † E (1) n + s ∗ ( ϕ )ˆ b † E (2) n (cid:17) × (cid:16) c ( ϕ )ˆ b E (1) n + s ( ϕ )ˆ b E (2) n (cid:17) ˆ η E n (cid:69) E = δ k, δ k (cid:48) , N ( ϕ ) , (48) γ (1 , kk (cid:48) ) = δ k, δ k (cid:48) , (cid:68) (cid:16) c ( ϕ )ˆ b E (1) n + s ( ϕ )ˆ b E (2) n (cid:17) × (cid:16) c ∗ ( ϕ )ˆ b † E (1) n + s ∗ ( ϕ )ˆ b † E (2) n (cid:17) ˆ η E n (cid:69) E = δ k, δ k (cid:48) , ( N ( ϕ ) + 1) . (49)where we introduced N ( ϕ ) = | c ( ϕ ) | N + | s ( ϕ ) | N = N + ( N − N ) | c ( ϕ ) | , (50)Replacing all this into Eq. (9) we hence get the followingsuper-operator L ( · · · ) = ( N ( ϕ ) + 1) (cid:16) ˆ a ( · · · )ˆ a † − (cid:104) ˆ a † ˆ a , · · · (cid:105) + (cid:17) + N ( ϕ ) (cid:16) ˆ a † ( · · · )ˆ a − (cid:104) ˆ a ˆ a † , · · · (cid:105) + (cid:17) , (51)which represents a thermalization process induced by aneffective bath whose temperature is intermediate betweenthe one of E and E and depends by the mixing of thesignals induced by their propagation through the Mach-Zehnder.Consider next the non-local contribution D , of themaster equation. In this case we get ζ (1 , , kk (cid:48) ) = (cid:104) ξ (2 , , kk (cid:48) ) (cid:105) ∗ = δ k, δ k (cid:48) , (cid:68) ˆ b E (1) n M (1) E n (ˆ b E (1) n ˆ η E n ) (cid:69) E = δ k, δ k (cid:48) , (cid:68) ( c ( ϕ )ˆ b E (1) n + s ( ϕ )ˆ b E (2) n ) ˆ b E (1) n ˆ η E n (cid:69) E = 0 ,ζ (2 , , kk (cid:48) ) = (cid:104) ξ (1 , , kk (cid:48) ) (cid:105) ∗ = δ k, δ k (cid:48) , (cid:68) ˆ b † E (1) n M (1) E n (ˆ b † E (1) n ˆ η E n ) (cid:69) E = δ k, δ k (cid:48) , (cid:68) ( c ∗ ( ϕ )ˆ b † E (1) n + s ∗ ( ϕ )ˆ b † E (2) n ) ˆ b † E (1) n ˆ η E n (cid:69) E = 0 , (52)and ζ (1 , , kk (cid:48) ) = (cid:104) ξ (2 , , kk (cid:48) ) (cid:105) ∗ = δ k, δ k (cid:48) , (cid:68) ˆ b † E (1) n M (1) E n (ˆ b E (1) n ˆ η E n ) (cid:69) E = δ k, δ k (cid:48) , (cid:68) ( c ∗ ( ϕ )ˆ b † E (1) n + s ∗ ( ϕ )ˆ b † E (2) n ) ˆ b E (1) n ˆ η E n (cid:69) E = δ k, δ k (cid:48) , c ∗ ( ϕ ) N ,ζ (2 , , kk (cid:48) ) = (cid:104) ξ (1 , , kk (cid:48) ) (cid:105) ∗ = δ k, δ k (cid:48) , (cid:68) ˆ b E (1) n M (1) E n (ˆ b † E (1) n ˆ η E n ) (cid:69) E = δ k, δ k (cid:48) , (cid:68) ( c ( ϕ )ˆ b E (1) n + s ( ϕ )ˆ b E (2) n ) ˆ b † E (1) n ˆ η E n (cid:69) E = δ k, δ k (cid:48) , c ( ϕ ) ( N + 1) , (53)so that D → ( · · · ) = N (cid:110) c ∗ ( ϕ ) ˆ a † (cid:104) · · · , ˆ a (cid:105) − − c ( ϕ ) (cid:104) · · · , ˆ a † (cid:105) − ˆ a (cid:111) + ( N + 1) (cid:110) c ( ϕ ) ˆ a (cid:104) · · · , ˆ a † (cid:105) − − c ∗ ( ϕ ) (cid:104) · · · , ˆ a (cid:105) − ˆ a † (cid:111) . (54)One notices that at variance with the contribution (46)which fully define the dynamics of S , both the localterm (51) of S and the coupling super-operator (54)are modulated by the phase ϕ . In particular by set-ting the transmissivities of BS and BS at 50% (i.e. (cid:15) = (cid:15) = 0 . c ( ϕ ) will acquire an oscil-lating behavior nullifying for ϕ = ± π (specifically we get c ( ϕ ) = − ie − iϕ/ sin( ϕ/ ϕ we can hence modify the cascade coupling between S and S .Following the derivation of Sec. II B we can finallywrite the QCS super-operator in the GKSL form (22).In particular in this case the effective Hamiltonian ap-pearing in Eq. (10) is given byˆ H , = − i (cid:16) c ( ϕ ) ˆ a † ˆ a − c ∗ ( ϕ ) ˆ a † ˆ a (cid:17) (55)= − i | c ( ϕ ) | (cid:16) e i arg[ c ( ϕ )] ˆ a † ˆ a − e − i arg[ c ( ϕ )] ˆ a † ˆ a (cid:17) , which by absorbing the phase arg[ c ( ϕ )] into (say) ˆ a ex-hibits the same chiral symmetry under exchange of S and S (i.e. ˆ H , = − ˆ H , ) observed in Ref. [12]. Thesuper-operator ∆ L , of Eq. (10) instead in this case isgiven by∆ L , ( · · · ) = N c ∗ ( ϕ ) (cid:18) ˆ a † ( · · · )ˆ a − (cid:104) ˆ a † ˆ a , ( · · · ) (cid:105) + (cid:19) +( N + 1) c ( ϕ ) (cid:18) ˆ a ( · · · )ˆ a † − (cid:104) ˆ a † ˆ a , ( · · · ) (cid:105) + (cid:19) + h.c. (56)which, remembering (39), can be expressed as in (24)with ∆ D ( (cid:96) ,(cid:96) ) m ,m ( k k ) = δ k, δ k (cid:48) , ∆ D ( (cid:96) ,(cid:96) ) m ,m (1 , , (57)where for m , m = 1 , (cid:96) , (cid:96) = 1 ,
2, ∆ D ( (cid:96) ,(cid:96) ) m ,m (1 , is the 4 × N c ∗ ( ϕ ) 00 0 0 ( N + 1) c ( ϕ ) N c ( ϕ ) 0 0 00 ( N + 1) c ∗ ( ϕ ) 0 0 the top-left and bottom right 2 × m = m = 1 and m = m = 2 respectively. Asanticipated in the previous section, while being Hermi-tian, this is in general not positive semi-definite (indeedit admits eigenvalues ± N | c ( ϕ ) | and ± ( N + 1) | c ( ϕ ) | ).On the contrary the matrix (28) which describe the sumof ∆ L , with the local terms L of Eq. (46) and L ofEq. (50) is given by N N c ∗ ( ϕ ) 00 N + 1 0 ( N + 1) c ( ϕ ) N c ( ϕ ) 0 N ( ϕ ) 00 ( N + 1) c ∗ ( ϕ ) 0 N ( ϕ ) + 1 and has eigenvalues κ , ± = 12 (cid:16) N + N ( ϕ ) + 2 (58) ± (cid:112) ( N − N ) + 4( N + 1) | c ( ϕ ) | (cid:17) ,κ , ± = 12 (cid:16) N + N ( ϕ ) (59) ± (cid:113) ( N − N ) + 4 N | c ( ϕ ) | (cid:17) , which are non-negative for all possible choices of N , N ≥ | c ( ϕ ) | ∈ [0 , L (1 , +) = (cid:112) k , + w , + ˆ a + ˆ a (cid:112) | w , + | , (60)ˆ L (1 , − ) = (cid:112) k , − w , − ˆ a † + ˆ a † (cid:112) | w , − | , (61)ˆ L (2 , +) = (cid:112) k , + w , + ˆ a + ˆ a (cid:112) | w , + | , (62)ˆ L (2 , − ) = (cid:112) k , − w , − ˆ a † + ˆ a † (cid:112) | w , − | , (63)with w , ± = 12( N + 1) c ∗ ( ϕ ) (cid:104) N − N ( ϕ ) (64) ± (cid:112) ( N − N ( ϕ )) + 4( N + 1) | c ( ϕ ) | (cid:105) ,w , ± = 12 N c ( ϕ ) (cid:104) N − N ( ϕ ) (65) ± (cid:113) ( N − N ( ϕ )) + 4 N | c ( ϕ ) | (cid:105) . It is worth noticing that in the already cited limit of (cid:15) , = 1 and T = 0 reproducing the model in [12], wehave that only the eigenvalue k , + = 2 is different fromzero, so that one has only one collective jump operatorˆ L (1 , +) = ˆ a + ˆ a . B. Example 2: controlling the topology of thenetwork via interference
In this section we discuss how interference can be usedto effectively modify the topology of the QCS interactionnetwork by selectively activating/deactivating some ofthe couplings which enter the scheme. In particular we fo-cus on the case of three quantum systems, dubbed Q , Q and Q connected as schematically shown in Fig. 4. Thisis basically the same configuration discussed in Sec. III Awhere Q and Q take the positions of S and S respec-tively, while Q is placed inside the Mach-Zehnder inter-ferometer. Accordingly the model exhibits direct QCSconnections among first neighboring elements (i.e. thecouple Q and Q and the couple Q and Q ), whilethe QCS coupling among Q and Q is mediated by two0 Q Q Q PS BS BS E (1) E (2) E (1) E (1) E (2) E (2) Q E (1)1 E (1)2 E (1)3 E (2)3 E (2)2 E (2)1 Q Q PS BS BS FIG. 4. Left panel: A sketch of the QCS scheme discussedin Sec III B. Q , Q , Q are the quantum system elementswhich are connected by the QCS network formed by the uni-directional bosonic channels E (1) and E (2) . As in the case ofSec. III A they are interweaved by two beam splitters and aphase shifter. Right panel: causal flowchart of the couplingsof the system in the collisional model. channels which interfere. The dynamics of the model canbe derived following the same line of the previous section– see Appendix C for the explicit calculations.Expressed as in Eq. (14) the resulting master equationexhibits the following local contributions: L ( · · · ) = ( N + 1) (cid:16) ˆ a ( · · · )ˆ a † − (cid:104) ˆ a † ˆ a , · · · (cid:105) + (cid:17) + N (cid:16) ˆ a † ( · · · )ˆ a − (cid:104) ˆ a ˆ a † , · · · (cid:105) + (cid:17) , (66) L ( · · · ) = (cid:16) ¯ N + 1 (cid:17) (cid:16) ˆ a ( · · · )ˆ a † − (cid:104) ˆ a † ˆ a , · · · (cid:105) + (cid:17) + ¯ N (cid:16) ˆ a † ( · · · )ˆ a − (cid:104) ˆ a ˆ a † , · · · (cid:105) + (cid:17) , (67) L ( · · · ) = (cid:16) N ( ϕ ) + 1 (cid:17)(cid:16) ˆ a ( · · · )ˆ a † − (cid:104) ˆ a † ˆ a , · · · (cid:105) + (cid:17) + N ( ϕ ) (cid:16) ˆ a † ( · · · )ˆ a − (cid:104) ˆ a ˆ a † , · · · (cid:105) + (cid:17) , (68)with N ( ϕ ) defined as in Eq. (50) and ¯ N being theaverage photon number of the environments perceivedby Q , i.e.¯ N = (cid:15) N + (1 − (cid:15) ) N = N + (cid:15) ( N − N ) . (69)Notice that the local terms of Q and Q coincide respec-tively with those of S and S of the previous section andthe L doesn’t depend upon the phase ϕ .The non-local contributions of the model are insteadgiven by two first-neighboring elements, connecting thecouples Q , Q and Q Q , plus a second-neighboring con-tribution, connecting Q and Q . The first two are givenby D → ( · · · ) = √ (cid:15) N (cid:16) ˆ a † (cid:104) · · · , ˆ a (cid:105) − + (cid:104) ˆ a † , · · · (cid:105) − ˆ a (cid:17) + √ (cid:15) (cid:16) N + 1 (cid:17)(cid:16) ˆ a (cid:104) · · · , ˆ a † (cid:105) − + (cid:104) ˆ a , · · · (cid:105) − ˆ a † (cid:17) , (70) and D → ( · · · ) = M ∗ ( ϕ )ˆ a † (cid:104) · · · , ˆ a (cid:105) − + M ( ϕ ) (cid:104) ˆ a † , · · · (cid:105) − ˆ a + ( M ( ϕ ) + λ ( ϕ ))ˆ a (cid:104) · · · , ˆ a † (cid:105) − + ( M ∗ ( ϕ ) + λ ∗ ( ϕ )) (cid:104) ˆ a , · · · (cid:105) − ˆ a † , (71)where we introduced the functions M ( ϕ ) = √ (cid:15) c ( ϕ ) N + i √ − (cid:15) s ( ϕ ) N ,λ ( ϕ ) = √ (cid:15) c ( ϕ ) + i √ − (cid:15) s ( ϕ ) , (72)with c ( ϕ ) and s ( ϕ ) as in Eq. (44). The third term insteadis given by D → ( · · · ) = N (cid:110) c ∗ ( ϕ ) ˆ a † (cid:104) · · · , ˆ a (cid:105) − + c ( ϕ ) (cid:104) ˆ a † , · · · (cid:105) − ˆ a (cid:111) + ( N + 1) (cid:110) c ( ϕ ) ˆ a (cid:104) · · · , ˆ a † (cid:105) − + c ∗ ( ϕ ) (cid:104) ˆ a , · · · (cid:105) − ˆ a † (cid:111) , (73)and formally coincides with the element D → ( · · · ) of theprevious section which connected S and S . The aboveexpressions make it clear that the various coupling termshave different functional dependences upon the phase pa-rameter ϕ . To better appreciate this it is useful to focuson the zero temperature regime (i.e. N = N = 0), andto assume the beam splitters to have 50% transmissivi-ties (i.e. (cid:15) = (cid:15) = 1 / ϕ , i.e. L m ( · · · ) = ˆ a m ( · · · )ˆ a † m − (cid:104) ˆ a † m ˆ a m , · · · (cid:105) + , m = 1 , , D → ( . . . ) = 1 √ (cid:16) ˆ a (cid:104) · · · , ˆ a † (cid:105) − + (cid:104) ˆ a , · · · (cid:105) − ˆ a † (cid:17) , D → ( . . . ) = 1 √ (cid:16) e − iϕ ˆ a (cid:104) · · · , ˆ a † (cid:105) − + e iϕ (cid:104) ˆ a , · · · (cid:105) − ˆ a † (cid:17) , D → ( . . . ) = − i sin ϕ (cid:16) e − i ϕ ˆ a (cid:104) · · · , ˆ a † (cid:105) − + e i ϕ (cid:104) ˆ a , · · · (cid:105) − ˆ a † (cid:17) . (74)The above equations make it explicit that the param-eter ϕ contributes to the system dynamics in two dif-ferent ways. First it introduces a non-trivial relativephase between Q , Q and Q which, at variance withthe two body problem of the previous section cannot beremoved by simply redefining their corresponding anni-hilation/creation operators. Second it induces a selec-tive modulation of the intensity of the Q Q interactions.These facts are reflected into the structure of the effec-tive Hamiltonian (27) stemming from the the reshaping1 Q Q Q Q Q Q Q Q Q ' = 0 ' = ⇡/ ' = ⇡ FIG. 5. Pictorial representation of the QCS interactionsamong Q , Q and Q . Left panel: interaction scheme for ϕ = 0, where there are only interactions between first neigh-bors. Central panel: for ϕ = π/ Q interacts also with Q and their interaction is of the same strength as the firstneighbor ones. Right panel: for ϕ = π , not only there is aninteraction between Q and Q , but it is even stronger thanthe first neighbor ones. of the ME in Lindblad form, i.e.ˆ H , = − i √ (cid:16) ˆ a ˆ a † − ˆ a † ˆ a (cid:17) , (75)ˆ H , = − i √ (cid:16) e i ϕ ˆ a ˆ a † − e − i ϕ ˆ a † ˆ a (cid:17) , (76)ˆ H , = − i ϕ (cid:16) e i ϕ + π ˆ a ˆ a † − e − i ϕ + π ˆ a † ˆ a (cid:17) , (77)see Eqs. (C9) – (C11) of Appendix C. Accordingly wesee that acting on ϕ the topology of the system interac-tions can be modified, moving from the case where theinteractions among Q and Q is null (e.g. ϕ = 0) oramplified ( ϕ = π ) with respect to their Q Q and Q Q counterparts, whose associated intensities are instead in-dependent from ϕ – see Fig. 5. IV. CONCLUSIONS
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Appendix A: Derivation of the master equation
The second order expansion of Eq. (4) with respect tothe product g ∆ t is U S m , E n = I S m , E n + ( g ∆ t ) U (cid:48) S m , E n + ( g ∆ t ) U (cid:48)(cid:48) S m , E n + O ( g ∆ t ) , (A1) with I S m , E n being the identity super-operator and U (cid:48) S m , E n ( · · · ) = − i K (cid:88) k =1 (cid:104) ˆ H S m ,E ( k ) n , ( · · · ) (cid:105) − , (A2) U (cid:48)(cid:48) S m , E n ( · · · ) = M (cid:88) k,k (cid:48) =1 (cid:110) H S m ,E ( k ) n ( · · · ) H S m ,E ( k (cid:48) ) n (A3) − (cid:104) H S m ,E ( k ) n H S m ,E ( k (cid:48) ) n , ( · · · ) (cid:105) + (cid:111) . By replacing these expressions into Eq. (14) we then ob-tain the expansion of the super-operator C S , E n ,i.e. C S , E n = C S , E n + ( g ∆ t ) C (cid:48)S , E n + ( g ∆ t ) C (cid:48)(cid:48)S , E n + O ( g ∆ t ) , (A4)where C S , E n = M ( M ← E n , C (cid:48)S , E n = M (cid:88) m =1 M ( M ← m ) E n ◦ U (cid:48) S m , E n ◦ M ( m − ← E n , C (cid:48)(cid:48)S , E n = C (cid:48)(cid:48) ( a ) S , E n + C (cid:48)(cid:48) ( b ) S , E n , (A5)and C (cid:48)(cid:48) ( a ) S , E n = M (cid:88) m =1 M ( M ← m ) E n ◦ U (cid:48)(cid:48) S m , E n ◦ M ( m − ← E n , C (cid:48)(cid:48) ( b ) S , E n = M (cid:88) m (cid:48) = m +1 M − (cid:88) m =1 (cid:110) M ( M ← m (cid:48) ) E n ◦ U (cid:48) S m (cid:48) , E n ◦M ( m (cid:48) − ← m ) E n ◦ U (cid:48) S m , E n ◦ M ( m − ← E n (cid:111) , (A6)where we defined M ( m ← m ) E n := ←− Π m m = m M ( m ) E n for m ≥ m , I for m < m , (A7)to indicate the ordered product of the maps M ( m ) E n , M ( m +1) E n , · · · , M ( m ) E n – see also definition (2). Insert-ing all this into Eq. (1) and taking the partial trace withrespect to the carriers then allows us to write the follow-ing equationˆ ρ ( n + 1) − ˆ ρ ( n )∆ t = g (cid:68) C (cid:48)S , E n +1 ( ˆ R ( n ) ⊗ ˆ η E n +1 ) (cid:69) E + g ∆ t (cid:68) C (cid:48)(cid:48)S , E n +1 ( ˆ R ( n ) ⊗ ˆ η E n +1 ) (cid:69) E + O ( g ∆ t ) , (A8)which by explicit evaluation of the various terms reducesto Eq. (8) of the main text. Indeed the first order termin g of this expression can be written as (cid:68) C (cid:48)S , E n +1 ( ˆ R ( n ) ⊗ ˆ η E n +1 ) (cid:69) E (A9)= − i (cid:88) m,k,(cid:96) (cid:68) ˆ B ( (cid:96),m ) E ( k ) n +1 M ( m − ← E n +1 (ˆ η E n +1 ) (cid:69) E (cid:104) ˆ A ( (cid:96),k ) S m , ˆ ρ ( n ) (cid:105) − , and coincides with the first order contribution of Eq. (8)with γ ( (cid:96) ) m ( k ) = (cid:68) ˆ B ( (cid:96),m ) E ( k ) n +1 M ( m − ← E n +1 (ˆ η E n +1 ) (cid:69) E . (A10)Similarly the second order term of (A8) is given by twocontributions:3 (cid:68) C (cid:48)(cid:48) ( a ) S , E n +1 ( ˆ R ( n ) ⊗ ˆ η E n +1 ) (cid:69) E = 12 M (cid:88) m =1 (cid:88) k,k (cid:48) (cid:88) (cid:96),(cid:96) (cid:48) γ ( (cid:96),(cid:96) (cid:48) ) m ( kk (cid:48) ) (cid:110) A ( (cid:96),k ) S m ˆ ρ ( n ) ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m − (cid:104) ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m ˆ A ( (cid:96),k ) S m , ˆ ρ ( n ) (cid:105) + (cid:111) (A11) (cid:68) C (cid:48)(cid:48) ( b ) S , E n +1 ( ˆ R ( n ) ⊗ ˆ η E n +1 ) (cid:69) E = M (cid:88) m (cid:48) = m +1 M − (cid:88) m =1 (cid:88) k,k (cid:48) (cid:88) (cid:96),(cid:96) (cid:48) (cid:110) ζ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) ˆ A ( (cid:96),k ) S m (cid:104) ˆ ρ ( n ) , ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) (cid:105) − − ξ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) (cid:104) ˆ ρ ( n ) , ˆ A ( (cid:96) (cid:48) ,k (cid:48) ) S m (cid:48) (cid:105) − ˆ A ( (cid:96),k ) S m (cid:111) with coefficients γ ( (cid:96),(cid:96) (cid:48) ) m ( kk (cid:48) ) = (cid:68) ˆ B ( (cid:96) (cid:48) ,m ) E ( k (cid:48) ) n +1 ˆ B ( (cid:96),m ) E ( k ) n +1 M ( m − ← E n +1 (ˆ η E n +1 ) (cid:69) E , (A12) ζ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) = (cid:68) ˆ B ( (cid:96) (cid:48) ,m (cid:48) ) E ( k (cid:48) ) n +1 M ( m (cid:48) − ← m ) E n +1 (cid:16) ˆ B ( (cid:96),m ) E ( k ) n +1 M ( m − ← E n +1 (ˆ η E n +1 ) (cid:17)(cid:69) E , (A13) ξ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) = (cid:68) ˆ B ( (cid:96) (cid:48) ,m (cid:48) ) E ( k (cid:48) ) n +1 M ( m (cid:48) − ← m ) E n +1 (cid:16) M ( m − ← E n +1 (ˆ η E n +1 ) ˆ B ( (cid:96),m ) E ( k ) n +1 (cid:17)(cid:69) E . (A14) Appendix B: Positivity of the matrix D ( (cid:96) ,(cid:96) ) m ,m ( k k ) As anticipated in Sec. II B one can show that the ma-trix Ω of elements Ω j,j (cid:48) = D ( (cid:96) ,(cid:96) ) m ,m ( k k ) ( j being the jointindex ( (cid:96), k, m ) and D ( (cid:96) ,(cid:96) ) m ,m ( k k ) as in Eq. (28)) is non-negative, i.e. that for all row vectors (cid:126)q of complex ele- ments q j the following inequality applies (cid:126)q Ω (cid:126)q † := (cid:88) j,j (cid:48) q j Ω j,j (cid:48) q ∗ j (cid:48) ≥ . (B1)Indeed from Eq. (A13)-(A14) it follows that2 (cid:126)q Ω (cid:126)q † = (cid:88) m q ( (cid:96),k,m ) q ∗ ( (cid:96) (cid:48) ,k (cid:48) ,m ) γ ( (cid:96),(cid:96) (cid:48) ) m ( kk (cid:48) ) + (cid:88) m (cid:48) >m (cid:104) q ( (cid:96),k,m ) q ∗ ( (cid:96) (cid:48) ,k (cid:48) ,m (cid:48) ) ζ ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) + h.c. (cid:105) (B2)= (cid:88) m (cid:68) ˆ Q ( m ) †E n +1 ˆ Q ( m ) E n +1 M ( m − ← E n +1 (ˆ η E n +1 ) (cid:69) E + (cid:88) m (cid:48) >m (cid:104)(cid:68) ˆ Q ( m (cid:48) ) †E n +1 M ( m (cid:48) − ← m ) E n +1 (cid:16) ˆ Q ( m ) E n +1 M ( m − ← E n +1 (ˆ η E n +1 ) (cid:17)(cid:69) E + h.c. (cid:105) , where in the first line we use Eq. (17) and, for the easyof notation, the convention of sum over repeated indexes,while in the second line we introduce the operatorsˆ Q ( m ) E n +1 = (cid:88) (cid:96),k q ( (cid:96),k,m ) ˆ B ( (cid:96),m ) E ( k ) n +1 . (B3)To proceed further we invoke the Stinespring decompo-sition [44] to write M ( m ) E n ( · · · ) = Tr A n [ V ( m ) E n A ( · · · ⊗ | (cid:105) A (cid:104) | )] , (B4) V ( m ) E n A ( · · · ) := V ( m ) E n A ( · · · ) V ( m ) † E n A , (B5) with | (cid:105) A being a (fixed) reference state of an ancillarysystem A and V ( m ) E n A being a unitary transformation thatcouples it with E n . Accordingly from Eq. (A7) it followsthat for all m ≥ m one has M ( m ← m ) E n = Tr A n [ V ( m ← m ) E n A ( · · · ⊗ | (cid:105) A (cid:104) | )] , (B6) V ( m ← m ) E n A ( · · · ) := V ( m ← m ) E n ( · · · ) V ( m ← m ) † E n , (B7)with V ( m ← m ) E n = V ( m ) E n A V ( m − E n A · · · V ( m ) E n A . (B8)Hence Eq. (B2) now rewrites as2 (cid:126)q Ω (cid:126)q † = (cid:88) m (cid:68) ˆ Q ( m ) †E n +1 ˆ Q ( m ) E n +1 · V ( m − ← E n +1 A (ˆ η E n +1 ⊗ | (cid:105) A (cid:104) | ) (cid:69) EA + (cid:88) m (cid:48) >m (cid:104)(cid:68) ˆ Q ( m (cid:48) ) †E n +1 · V ( m (cid:48) − ← m ) E n +1 A (cid:16) ˆ Q ( m ) E n +1 · V ( m − ← E n +1 A (ˆ η E n +1 ⊗ | (cid:105) A (cid:104) | ) (cid:17)(cid:69) EA + h.c. (cid:105) = (cid:88) m (cid:68) ˜ V ( m − ← E n +1 A ( ˆ Q ( m ) †E n +1 ˆ Q ( m ) E n +1 ) · (ˆ η E n +1 ⊗ | (cid:105) A (cid:104) | ) (cid:69) EA + (cid:88) m (cid:48) >m (cid:104)(cid:68) ˜ V ( m − ← E n +1 A (cid:16) ˜ V ( m (cid:48) − ← m ) E n +1 A ( ˆ Q ( m (cid:48) ) †E n +1 ) · ˆ Q ( m ) E n +1 (cid:17) · (ˆ η E n +1 ⊗ | (cid:105) A (cid:104) | ) (cid:69) EA + h.c. (cid:105) , (B9)4where we used the ciclicity of the trace, where ˜ V ( m ← m ) E n A is the conjugate transformation of V ( m ← m ) E n A , i.e. themapping ˜ V ( m ← m ) E n A ( · · · ) := V ( m ← m ) † E n ( · · · ) V ( m ← m ) E n , (B10)and where we introduced the symbol “ · ” to indicate the regular product between operators whenever needed to avoidpossible misinterpretations. Now observe that˜ V ( m − ← E n +1 A ( ˆ Q ( m ) †E n +1 ˆ Q ( m ) E n +1 ) = ˜ V ( m − ← E n +1 A ( ˆ Q ( m ) †E n +1 ) · ˜ V ( m − ← E n +1 A ( ˆ Q ( m ) E n +1 ) = ˆ T ( m ) †E n +1 A ˆ T ( m ) E n +1 A , (B11)˜ V ( m − ← E n +1 A (cid:16) ˜ V ( m (cid:48) − ← m ) E n +1 A ( ˆ Q ( m (cid:48) ) †E n +1 ) · ˆ Q ( m ) E n +1 (cid:17) = ˜ V ( m (cid:48) − ← E n +1 A ( ˆ Q ( m (cid:48) ) †E n +1 ) · ˜ V ( m − ← E n +1 A ( ˆ Q ( m ) E n +1 ) = ˆ T ( m (cid:48) ) †E n +1 A ˆ T ( m ) E n +1 A , (B12)where we used the fact that for for all m > m > m integer one has V ( m ← m ) E n A = V ( m ← m ) E n A V ( m ← m ) E n A , (B13)and introduced the operatorsˆ T ( m ) E n +1 A = ˜ V ( m − ← E n +1 A ( ˆ Q ( m ) E n +1 ) . (B14)Replacing all these into Eq. (B9) and re-organizing thevarious terms finally yields the thesis, i.e.2 (cid:126)q · Ω · (cid:126)q † = M (cid:88) m,m (cid:48) =1 (cid:68) ˆ T ( m (cid:48) ) †E n +1 A ˆ T ( m ) E n +1 A · (ˆ η E n +1 ⊗ | (cid:105) A (cid:104) | ) (cid:69) EA ≥ . (B15) Appendix C: Derivation of the three body QCSmaster equation
Here we report the explicit calculation of the modeldescribed in Fig. 4. Following the flowchart representa-tion presented in the right panel of the figure we writethe Hamiltonians (7) asˆ H Q m ,E (1) n = ˆ a † m ˆ b E (1) n + ˆ a m ˆ b † E (1) n , (C1)ˆ H Q m ,E (2) n = 0 , (C2)where now, for m = 1 , ,
3, ˆ a m and ˆ a † m are the lower-ing and raising operators of the system Q m while ˆ b E ( k ) n and ˆ b † E ( k ) n are the bosonic operators associated with thequantum carriers of the unidirectional channel E ( k ) (no-tice that no direct coupling is assigned between the Q m ’sand E (2) ). The free dynamics of the environmental ele-ments are instead defined by two distinct maps: the map M (1) E n ( · · · ) associated with the beam-splitter BS thatcharacterizes the evolution of the quantum carriers afterthe interactions with Q and before the interactions with Q ; and the map M (2) E n ( · · · ) associated with the beam-splitter BS and the phase shift element P S which in-stead acts after the collisional events with Q and before those involving Q . Adopting the same convention usedin Eqs. (35)-(38) they can be expressed as M (1) E n ( · · · ) = ˆ V BS ( · · · ) ˆ V † BS , (C3) M (2) E n ( · · · ) = ˆ V BS ˆ V P S ( · · · ) ˆ V † P S ˆ V † BS . (C4)With this choice and assuming then the same initial con-ditions of Eq. (31) one can verify that stationary condi-tion still holds for the same reasons of Sec. III A, so wewon’t repeat the calculations of the coefficients γ ( (cid:96) ) m ( k ) . Bythe same token it follows that the local term L is identi-cal to the one in Eq. (46), because the collisional schemeis identical up to this point. Similarly the computationof the coefficients γ ( (cid:96),(cid:96) (cid:48) )3( kk (cid:48) ) , associated with the local termof Q , and the computation of ζ ( (cid:96),(cid:96) (cid:48) )1 , kk (cid:48) ) and ξ ( (cid:96),(cid:96) (cid:48) )1 , kk (cid:48) ) as-sociated with the QCS coupling connecting Q with Q coincide with the corresponding elements of S and S ofthe model of Sec. III A, yielding the expressions reportedin Eqs. (68) and (73) of the main text. What is left ishence the computation of the terms associated with Q ,i.e. L , D → and D → . Regarding the first we noticethat exploiting (35) and invoking the definition of ¯ N presetend in Eq. (69), the coefficients γ ( (cid:96),(cid:96) (cid:48) )2( kk (cid:48) ) can be ex-pressed as γ (1 , kk (cid:48) ) = [ γ (2 , kk (cid:48) ) ] ∗ = δ k δ k (cid:48) (cid:68) ˆ b E (1) n M (1) E n (ˆ η E n ) (cid:69) = 0 ,γ (1 , kk (cid:48) ) = δ k δ k (cid:48) (cid:68) ˆ b † E (1) n ˆ b E (1) n M (1) E n (ˆ η E n ) (cid:69) = δ k δ k (cid:48) (cid:68)(cid:16) √ (cid:15) ˆ b † E (1) n + i √ − (cid:15) ˆ b † E (2) n (cid:17) × (cid:16) √ (cid:15) ˆ b E (1) n − i √ − (cid:15) ˆ b E (2) n (cid:17) ˆ η E n (cid:69) = δ k δ k (cid:48) ¯ N , (C5) γ (2 , kk (cid:48) ) = δ k δ k (cid:48) (cid:68) ˆ b E (1) n ˆ b † E (1) n M (1) E n (ˆ η E n ) (cid:69) = δ k δ k (cid:48) (cid:68)(cid:16) √ (cid:15) ˆ b E (1) n − i √ − (cid:15) ˆ b E (2) n (cid:17) × (cid:16) √ (cid:15) ˆ b † E (1) n + i √ − (cid:15) ˆ b † E (2) n (cid:17) ˆ η E n (cid:69) = δ k δ k (cid:48) ( ¯ N + 1) , (C6)5which give Eq. (67). The expression (70) for D → in-stead follows from the identities ζ (1 , kk (cid:48) ) = [ ξ (2 , kk (cid:48) ) ] ∗ = δ k δ k (cid:48) (cid:68) ˆ b E (1) n M (1) E n (ˆ b E (1) n ˆ η E n ) (cid:69) = 0 ,ζ (2 , kk (cid:48) ) = [ ξ (1 , kk (cid:48) ) ] ∗ = δ k δ k (cid:48) (cid:68) ˆ b † E (1) n M (1) E n (ˆ b † E (1) n ˆ η E n ) (cid:69) = 0 ,ζ (1 , kk (cid:48) ) = [ ξ (2 , kk (cid:48) ) ] ∗ = δ k δ k (cid:48) (cid:68) ˆ b † E (1) n M (1) E n (cid:16) ˆ b E (1) n ˆ η E n (cid:17)(cid:69) = δ k δ k (cid:48) (cid:68)(cid:16) √ (cid:15) ˆ b † E (1) n + i √ − (cid:15) ˆ b † E (2) n (cid:17) ˆ b E (1) n ˆ η E n (cid:69) = δ k δ k (cid:48) √ (cid:15) N , (C7) ζ (2 , kk (cid:48) ) = [ ξ (1 , kk (cid:48) ) ] ∗ = δ k δ k (cid:48) (cid:68) ˆ b E (1) n M (1) E n (cid:16) ˆ b † E (1) n ˆ η E n (cid:17)(cid:69) = δ k δ k (cid:48) (cid:68)(cid:16) √ (cid:15) ˆ b E (1) n − i √ − (cid:15) ˆ b E (2) n (cid:17) ˆ b † E (1) n ˆ η E n (cid:69) = δ k δ k (cid:48) √ (cid:15) (cid:16) N + 1 (cid:17) , (C8)while finally (71) for D → ( · · · ) follows from ζ (1 , kk (cid:48) ) = [ ξ (2 , kk (cid:48) ) ] ∗ = δ k δ k (cid:48) (cid:68) ˆ b E (1) n M (2) E n (cid:16) ˆ b E (1) n M (1) E n ˆ η E n (cid:17)(cid:69) = 0 ,ζ (2 , kk (cid:48) ) = [ ξ (1 , kk (cid:48) ) ] ∗ = δ k δ k (cid:48) (cid:68) ˆ b † E (1) n M (2) E n (cid:16) ˆ b † E (1) n M (1) E n ˆ η E n (cid:17)(cid:69) = 0 , and ζ (1 , kk (cid:48) ) = [ ξ (2 , kk (cid:48) ) ] ∗ = δ k δ k (cid:48) (cid:68) ˆ b † E (1) n M (2) E n (cid:16) ˆ b E (1) n M (1) E n ˆ η E n (cid:17)(cid:69) = δ k δ k (cid:48) (cid:68)(cid:104) c ∗ ( ϕ )ˆ b † E (1) n + s ∗ ( ϕ )ˆ b † E (2) n (cid:105) × (cid:104) √ (cid:15) ˆ b E (1) n − i √ − (cid:15) ˆ b E (2) n (cid:105) ˆ η E n (cid:69) = δ k δ k (cid:48) M ∗ ( ϕ ) ,ζ (2 , kk (cid:48) ) = [ ξ (1 , kk (cid:48) ) ] ∗ = δ k δ k (cid:48) (cid:68) ˆ b E (1) n M (2) E n (cid:16) ˆ b † E (1) n M (1) E n ˆ η E n (cid:17)(cid:69) = δ k δ k (cid:48) (cid:68)(cid:104) c ( ϕ )ˆ b E (1) n + s ( ϕ )ˆ b E (2) n (cid:105) × (cid:104) √ (cid:15) ˆ b † E (1) n + i √ − (cid:15) ˆ b † E (2) n (cid:105) ˆ η E n (cid:69) = δ k δ k (cid:48) ( M ( ϕ ) + λ ( ϕ )) , where we adopted the definitions (72).The matrix D ( (cid:96),(cid:96) (cid:48) ) mm (cid:48) ( kk (cid:48) ) for this system can then be cast in the following form N √ (cid:15) N c ∗ ( ϕ ) N N + 1 0 √ (cid:15) ( N + 1) 0 c ( ϕ )( N + 1) √ (cid:15) N (cid:15) N + (1 − (cid:15) ) N M ∗ , ( ϕ ) 00 √ (cid:15) ( N + 1) 0 (cid:15) ( N + 1) + (1 − (cid:15) )( N + 1) 0 M , ( ϕ ) + λ ( ϕ ) c ( ϕ ) N M , ( ϕ ) 0 N ( ϕ ) 00 c ∗ ( ϕ )( N + 1) 0 M ∗ , ( ϕ ) + λ ∗ ( ϕ ) 0 N ( ϕ ) + 1 , which upon diagonalization yields the following effectiveHamiltonians contributionsˆ H , = − i √ (cid:15) (cid:16) ˆ a ˆ a † − ˆ a † ˆ a (cid:17) , (C9)ˆ H , = − i (cid:16) λ ∗ ( ϕ )ˆ a ˆ a † − λ ( ϕ )ˆ a † ˆ a (cid:17) , (C10)ˆ H , = − i (cid:16) c ∗ ( ϕ )ˆ a ˆ a † − c ( ϕ )ˆ a † ˆ a (cid:17) ..