Hong-Ou-Mandel heat noise in the quantum Hall regime
Flavio Ronetti, Luca Vannucci, Dario Ferraro, Thibaut Jonckheere, Jérôme Rech, Thierry Martin, Maura Sassetti
HHong-Ou-Mandel heat noise in the quantum Hall regime
Flavio Ronetti,
1, 2, 3, ∗ Luca Vannucci,
1, 2, 4
Dario Ferraro,
1, 5
ThibautJonckheere, J´erˆome Rech, Thierry Martin, and Maura Sassetti
1, 2 Dipartimento di Fisica, Universit`a di Genova, Via Dodecaneso 33, 16146, Genova, Italy CNR-SPIN, Via Dodecaneso 33, 16146, Genova, Italy Aix Marseille Univ, Universit´e de Toulon, CNRS, CPT, Marseille, France CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy
We investigate heat current fluctuations induced by a periodic train of Lorentzian-shaped pulses,carrying an integer number of electronic charges, in a Hong-Ou-Mandel interferometer implementedin a quantum Hall bar in the Laughlin sequence. We demonstrate that the noise in this collisionalexperiment cannot be reproduced in a setup with a single drive, in contrast to what is observedin the charge noise case. Nevertheless, the simultaneous collision of two identical levitons alwaysleads to a total suppression even for the Hong-Ou-Mandel heat noise at all filling factors, despite thepresence of emergent anyonic quasi-particle excitations in the fractional regime. Interestingly, thestrong correlations characterizing the fractional phase are responsible for a remarkable oscillatingpattern in the HOM heat noise, which is completely absent in the integer case. These oscillationscan be related to the recently predicted crystallization of levitons in the fractional quantum Hallregime.
I. INTRODUCTION
The recent progress in generating and controlling co-herent few-particle excitations in quantum conductorsopened the way to a new research field, known as elec-tron quantum optics (EQO) [1, 2]. The main purposeof EQO is to reproduce conventional optics experimentsusing electronic wave-packets propagating in condensedmatter systems instead of photons travelling along wave-guides.In this context, a remarkable effort has been put forthby the condensed matter community to implement on-demand sources of electronic wave-packets in mesoscopicsystems. After seminal theoretical works and ground-breaking experimental results, two main methods to re-alize single-electron sources assumed a prominent role inthe field of EQO [3–7]. The first injection protocol relieson the periodic driving of the discrete energy spectrumof a quantum dot, which plays the role of a mesoscopiccapacitor [8–10]. In this way, it is possible to achievethe periodic injection of an electron and a hole along theballistic channels of a system coupled to this mesoscopiccapacitor through a quantum point contact (QPC) [11–14].A second major step has been the recent realizationof an on-demand source of electron through the appli-cation of a time-dependent voltage to a quantum con-ductor [5, 6, 15–19]. The main challenge to face, in thiscase, has been that an ac voltage would generally exciteunwanted neutral electron-hole pairs, thus spoiling at itsheart the idea of a single-electron source. The turningpoint to overcome this issue was the theoretical predic-tion by Levitov and co-workers that a periodic train of ∗ ronetti@fisica.unige.it quantized Lorentzian-shaped pulses, carrying an integernumber of particles per period, is able to inject mini-mal single-electron excitations devoid of any additionalelectron-hole pair, then termed levitons [20–22]. Indeed,this kind of single-electron source is simple to realizeand operate, since it relies on usual electronic compo-nents, and potentially provides a high level of miniatur-ization and scalability. For their fascinating properties[23], levitons have been proposed as flying qubits [24] andas source of entanglement [25–28] with appealing appli-cations for quantum information processing. Moreover,quantum tomography protocols able to reconstruct theirsingle-electron wave-functions have been proposed [29–31] and experimentally realized [32].While the implementation of single-electron sourceshas not been a trivial task, the condensed matter ana-logues of other quantum optics experimental componentscan be found in a more natural way. The wave-guides forphotons can be replaced by the ballistic edge channelsof mesoscopic devices, such as quantum Hall systems.Moreover, the role of electronic beam splitter, whichshould mimic the half-silvered mirror of conventional op-tics, can be played by a QPC, where electrons are re-flected or transmitted with a tunable probability, whichis typically assumed as energy independent. By combin-ing these elements with the single-electron sources pre-viously described, interferometric setups, originally con-ceived for optics experiments, can be implemented also inthe condensed matter realm [33, 34]. One famous exam-ple is the Hanbury-Brown-Twiss (HBT) interferometer[35], where a stream of electronic wave-packets is excitedalong ballistic channels and partitioned against a QPC[12]. The shot noise signal, generated due to the gran-ular nature of electrons [36, 37], was employed to probethe single-electron nature of levitons in a non-interactingtwo-dimensional electron gas [15, 38]. Its extension tothe fractional quantum Hall regime was considered in a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Ref. [39], where it was shown that levitons are minimalexcitations also in strongly correlated edge channels.A fundamental achievement of EQO has been the im-plementation of the Hong-Ou-Mandel (HOM) interfer-ometer [40], where electrons impinge on the opposite sideof a QPC with a tunable delay [6, 38, 41]. By perform-ing this kind of collisional experiments, it is possible togather information about the forms of the impinging elec-tronic wave-packets and to measure their degree of indis-tinguishability [14, 16, 42]. For instance, when two indis-tinguishable and coherent electronic states collide simul-taneously (zero time delay) at the QPC, charge currentfluctuations are known to vanish at zero temperature,thus showing the so called Pauli dip [6, 38, 43]. Thisdip can be interpreted in terms of anti-bunching effectsrelated to the Fermi statistics of electrons. HOM experi-ments can thus be employed to test whether decoherenceand dephasing, induced by electron-electron interactions,reduce the degree of indistinguishability of colliding elec-trons [31, 44–47].As discussed above, the main driving force behindEQO has been to properly revise quantum optics exper-iments focusing on charge transport properties of single-electron excitations. Nevertheless, some recent ground-breaking experiments has spurred the investigation alsoin the direction of heat transport at the nanoscale [48–53]. In this context, the coherent transport and manip-ulation of heat fluxes have been reported in Josephsonjunctions [54–56] and quantum Hall systems [57–59]. In-triguingly, the quantization of heat conductance has beenobserved in integer [60] and fractional quantum Hall sys-tems [61–63], which were already known for the extremelyprecise quantization of their charge conductance. In thisway, ample and valuable information about these pecu-liar states of matter, which was not accessible by chargemeasurement, is now available with interesting implica-tions also for quantum computation [64–66]. New in-triguing challenges posed by extending concepts like en-ergy harvesting [67–73], driven heat and energy transport[74–78], energy exchange in open systems [79, 80] andfluctuation-dissipation theorems [81–84] to the quantumrealm resulted in a great progress of the field of quantumthermodynamics.A new perspective on EQO has been also triggered bythe rising interest for heat transport properties of single-electron excitations. Mixed-charge correlators [85–87]and heat fluctuations [88, 89] produced by single-electronsources were investigated and, in particular, it was shownthat levitons are minimal excitations also for heat trans-port [90]. In addition, heat current has revealed a usefulresource for the full reconstruction of a single-electronwave-function [91].Here, we address the problem of the heat noise gen-erated by levitons injected in a HOM interferometer inthe fractional quantum Hall regime. We consider a fourterminal quantum Hall bar in the Laughlin sequence [92],where a single channel arises on each edge. Two terminalsare contacted to time-dependent voltages, namely V L and V R . Tunneling processes of quasi-particles are allowed bythe presence of a QPC connecting the two edge states.In this case, charge noise generated in the HOM setupis identical to the one generated in a single-drive setupdriven by the voltage V L − V R . Interestingly, we provethat this does not hold true anymore for heat noise, sinceit is possible to identify a contribution to HOM heat noisewhich is absent in a single-drive interferometer drivenby V L − V R . In addition, we prove that the HOM heatnoise always vanishes for a zero delay between the driv-ing voltage, both for integer and fractional filling factors.Finally, we focus on the case of Lorentzian-shaped volt-age carrying an integer number of electrons and we showthat the HOM heat noise displays unexpected side dipsin the fractional quantum Hall regime, which have noparallel in the integer regime. Intriguingly, the numberof these side dips increases with the number of levitonsinjected per period. This result is consistent with therecently predicted phenomenon of charge crystallizationof levitons in the fractional quantum Hall regime [93].The paper is organized as follows. In Sec. II, we intro-duce the model and the setup. Then, we evaluate chargeand heat noises in Sec. III. In Sec. IV, we present ourresults focusing on the peculiar case of levitons. Finally,we draw the conclusions in Sec. V. Three Appendices aredevoted to the technical aspects. II. MODEL
A quantum Hall bar in a four terminal geometry is de-picted in Fig. 1. In the Laughlin sequence ν = n +1 ,with integer n ≥
0, a single chiral mode arises on eachedge [92, 94]. In the special case of integer quantumHall effect at ν = 1 ( n = 0), the system is com-posed by ordinary fermions and the chiral edge statesare one-dimensional Fermi liquids. This descriptionfails for other filling factors, where the excitations arequasi-particles with fractional charge − νe (with e > R/L ( x ), which satisfy commutation relations (cid:2) Φ R/L ( x ) , Φ R/L ( y ) (cid:3) = ± iπ sign ( x − y ). The free Hamil-tonian of these edge modes is (we set (cid:126) = 1 throughoutthe paper) [95] H = v π (cid:90) dx (cid:88) r = R,L ( ∂ x Φ r ( x )) , (1)where v is the velocity of propagation of right and leftmoving bosonic modes.Terminals 1 and 4 are assumed to be connected to exter-nal time-dependent drives, while the remaining terminalsare used to perform measurements. The charge densities,defined as ρ R/L ( x ) = ± e √ ν π ∂ x Φ R/L ( x ) , (2) Figure 1. (Color online) Four-terminal setup for Hong-Ou-Mandel interferometry in the FQH regime. Contact 1 and 4are used as input terminals, while contact 2 and 3 are theoutput terminals where current and noise are measured. are capacitively coupled to the gate potentials V R/L ( x, t )through the following gate Hamiltonian [96, 97] H g = (cid:90) dx {V R ( x, t ) ρ R ( x ) + V L ( x, t ) ρ L ( x ) } . (3)The spatial dependence of the potentials is restricted tothe region containing the semi-infinite contacts 1 ( R )and 4 ( L ) by putting V R ( x, t ) = Θ( − ( x + d )) V R ( t )and V L ( x, t ) = Θ( x − d ) V L ( t ) (with d > V R/L ( t ) = V R/L,dc + V R/L,ac ( t ) are periodic voltages,where V R/L,dc are time-independent dc components and V R/L,ac are pure periodic ac signals with period T = πω ,such that (cid:82) T dt T V R/L ( t ) = V R/L,dc . We remark thatsuch modelization of the electromagnetic coupling be-tween gate voltages and Hall bar occurs for gauge fixingwith zero vector potential.Since backscattering between the two edges is exponen-tially suppressed, we introduce a quantum point contact(QPC) at x = 0, as shown in Fig. 1, in order to allowfor tunneling events between right- and left-moving ex-citations. We assume the QPC is tuned to a very lowtransparency, i.e. in the weak backscattering regime,where the tunneling of fractional quasi-particles is theonly relevant process [98–100]. The corresponding addi-tional term in the Hamiltonian is H t = Λ Ψ † R (0)Ψ L (0) + H.c. , (4)where we introduced the quasi-particle fields representedby the bosonization identity [36, 101, 102]Ψ R/L ( x ) = F R/L √ πa e − i √ ν Φ R/L ( x ) , (5)with F R/L the so-called Klein factor, necessary for theproper anti-commutation relations, and a the short-length cut-off. III. NOISES IN THE DOUBLE-DRIVECONFIGURATION
The random partitioning, due to the poissonian tun-neling at the QPC, generates fluctuations in the currents flowing along the quantum Hall bar. In this Section, wederive the expressions for charge and heat current noisein the double-drive configuration introduced in Sec. II,focusing on the regions downstream of the voltage con-tacts, namely − d < x < d . A. Charge noise
We start by recalling the calculations for charge noise[3, 15, 39]. Charge current operators entering reservoirs2 and 3 (located in x = − d and x = d , respectively) canbe expressed, due to chirality of Laughlin edge states, interms of charge densities in Eq. (2) j / ( t ) = ± vρ R/L ( ± d, t ) . (6)The zero frequency cross-correlated charge noise is S C = (cid:90) T dt T (cid:90) + ∞−∞ dt (cid:48) [ (cid:104) j ( t (cid:48) ) j ( t ) (cid:105) − (cid:104) j ( t (cid:48) ) (cid:105) (cid:104) j ( t ) (cid:105) ] , (7)where the thermal average is performed over the initialequilibrium density matrix, in absence of tunneling anddriving voltage. In the weak backscattering regime, stan-dard perturbative approach in the tunneling Hamiltonianwill be used. The total time evolution of charge currentoperators with respect to H + H g + H t can be then con-structed in terms of powers of Λ and reads j / ( t ) = j (0)2 / ( t ) + j (1)2 / ( t ) + j (2)2 / ( t ) + O ( | Λ | ) , (8)with j (0)2 / ( t ) = ± vρ (0) R/L ( ± d, t ) , (9) j (1)2 / ( t ) = ± iv t (cid:90) −∞ dt (cid:48) (cid:104) H t ( t (cid:48) ) , ρ (0) R/L ( ± d, t ) (cid:105) , (10) j (2)2 / ( t ) = ± ( i ) v t (cid:90) −∞ dt (cid:48) t (cid:48) (cid:90) −∞ dt (cid:48)(cid:48) (cid:104) H t ( t (cid:48)(cid:48) ) , (cid:104) H t ( t (cid:48) ) , ρ (0) R/L ( ± d, t ) (cid:105)(cid:105) , (11)where the tunneling Hamiltonian H t ( t ) and the chargedensities ρ (0) R/L ( x, t ) evolve in the interaction picture withrespect to H + H g . In order to make explicit the form of ρ (0) R/L ( x, t ) it is sufficient to solve the equations of motionfor the bosonic fields Φ R/L with respect to H + H g , i.e.in the absence of tunneling. The solutions readΦ R/L ( x, t ) = φ R/L ( x, t ) − e √ ν (cid:90) t ∓ xv − dv dsV R/L ( s ) , (12)where φ R/L ( x, t ) = φ R/L ( x ∓ vt ) are the chiral bosonicfields at equilibrium (zero applied drive).By exploiting the commutator (cid:104) H t ( t (cid:48) ) , ρ (0) R/L ( x, t ) (cid:105) = − δ (cid:16) t (cid:48) − (cid:16) t ∓ xv (cid:17)(cid:17) ˙ N R/L ( x, t ) , (13)where˙ N R ( x, t ) = iνe ΛΨ † R ( x − vt, L ( x − vt,
0) + H.c. , (14)˙ N L ( x, t ) = − iνe ΛΨ † R ( x + vt, L ( x + vt,
0) + H.c. , (15)Eqs. (10) and (11) can be further recast as j (1)2 / ( t ) = ˙ N R/L ( ± d, t ) , (16) j (2)2 / ( t ) = i t − dv (cid:90) −∞ dt (cid:48)(cid:48) (cid:104) H t ( t (cid:48)(cid:48) ) , ˙ N R/L ( ± d, t ) (cid:105) . (17)In these expressions, we introduced the time evolutionof quasi-particle fields with respect to H + H g , whichcan be obtained from Eq. (12) using the bosonizationidentityΨ R,L ( x, t ) = F R/L √ πa e − i √ νφ R/L ( x,t ) e iνe (cid:82) t ∓ xv − dv dt (cid:48) V R/L ( t (cid:48) ) , (18)The current noise can be obtained from Eqs. (9), (10):the only non-vanishing contribution to second order in Λcomes from j (1)2 ( t + τ ) j (1)3 ( t ), with terms j (0)2 ( t + τ ) j (2)3 ( t )and j (2)2 ( t + τ ) j (0)3 ( t ) averaging to zero.By introducing the correlator ( k B = 1) P g ( t (cid:48) − t ) = (cid:104) e i √ gφ R/L (0 ,t (cid:48) ) e − i √ gφ R/L (0 ,t ) (cid:105) == (cid:20) πθ ( t (cid:48) − t )sinh ( πθ ( t (cid:48) − t )) (1 + iω c ( t (cid:48) − t )) (cid:21) g , (19)with θ the temperature and ω c = v/a the high energycut-off, one finds ( λ = Λ2 πa ) S C = − νe ) | λ | (cid:90) T dt T (cid:90) + ∞−∞ dt (cid:48) ×× cos (cid:40) νe (cid:90) t (cid:48) t V − ( τ ) dτ (cid:41) P ν ( t (cid:48) − t ) , (20)where V − = V R − V L .Even though this charge noise is generated in a double-drive configuration, it is interesting to point out that itactually depends only on the single effective drive V − ( t ).The configuration with a single drive is usually termedin literature Hanbury-Brown-Twiss (HBT) setup [12, 35,38, 103].Therefore, the charge noise presented in Eq. (20) is thesame as the one generated in a single-drive configuration,where reservoir 4 is grounded ( V L ( t ) = 0) and reservoir1 is contacted to the periodic voltage V − ( t ), such that S C ( V R , V L ) = S C ( V − , . (21) Here, the arguments in brackets indicate the voltage ap-plied to reservoirs 1 and 4, respectively.One might consider Eq. (21) as a consequence of a triv-ial shift of both voltages by a value corresponding to V L . Nevertheless, such a result cannot be obtained bymeans of a gauge transformation (see Appendix A). Inthis sense, Eq. (21) implies that the charge noise inci-dentally acquires the same expression in these two phys-ically distinct experimental setups. As will be clearer inthe following, for the charge case this is a consequence ofthe presence of a single local (energy independent) QPC.Generally, we expect that the double-drive and the single-drive ( V R ( t ) = V − ( t ) and V L ( t ) = 0) configurations re-turn different outcomes for other physical observables,such as heat noise, as discussed in the next part. B. Heat noise
In the following, we evaluate the correlation noise ofheat current between terminal 2 and 3 in the double-driveconfiguration. The heat current operators of terminal 2and 3 can be expressed in terms of heat density operators[104] Q R/L ( x, t ) = v π (cid:0) ∂ x Φ R/L ( x, t ) (cid:1) , (22)as J / ( t ) = ± v Q R/L ( ± d, t ) , (23)due to the chirality of Laughlin edge states.Then, we can define the cross-correlated heat noise S Q = (cid:90) T dt T (cid:90) dt (cid:48) {(cid:104)J ( t (cid:48) ) J ( t ) (cid:105) − (cid:104)J ( t (cid:48) ) (cid:105) (cid:104)J ( t ) (cid:105)} , (24)Analogously to charge current, one can expand heat cur-rent operators in power of the tunneling amplitude Λ,thus obtaining J / ( t ) = J (0)2 / ( t ) + J (1)2 / ( t ) + J (2)2 / ( t ) + O (cid:16) | Λ | (cid:17) , (25)where J (0)2 / ( t ) = ± v Q (0) R/L ( ± d, t ) , (26) J (1)2 / ( t ) = ± iv t (cid:90) −∞ dt (cid:48) (cid:104) H t ( t (cid:48) ) , Q (0) R/L ( ± d, t ) (cid:105) , (27) J (2)2 / ( t ) = ± i v t (cid:90) −∞ dt (cid:48) t (cid:48) (cid:90) −∞ dt (cid:48)(cid:48) (cid:104) H t ( t (cid:48)(cid:48) ) , (cid:104) H t ( t (cid:48) ) , Q (0) R/L ( ± d, t ) (cid:105)(cid:105) . (28)In the above equations we have denoted with Q (0) ( x, t ),the time evolution of heat density in the absence of tun-neling, which can be obtained from the time evolution ofbosonic fields in Eq. (12) and reads Q (0) R/L ( x, t ) = v π (cid:104) (cid:0) ∂ x φ R/L ( x, t ) (cid:1) + ± e √ ν∂ x φ R/L ( x, t ) V R/L (cid:16) t ∓ xv (cid:17) + e νv V R/L (cid:16) t ∓ xv (cid:17) (cid:105) . (29)The following commutator (cid:104) H t ( t (cid:48) ) , Q (0) R/L ( x, t ) (cid:105) = − iδ (cid:16) t (cid:48) − (cid:16) t ∓ xv (cid:17)(cid:17) ˙ Q R/L ( x, t ) , (30)where˙ Q R ( x, t ) = v Λ (cid:16) ∂ x Ψ † R ( x, t ) (cid:17) Ψ L ( x, t ) + H.c. , (31)˙ Q L ( x, t ) = − v ΛΨ † R ( x, t ) ( ∂ x Ψ L ( x, t )) + H.c. , (32) can be used to recast Eqs. (27) and (28) J (1)2 / ( t ) = ± ˙ Q R/L ( ± d, t ) , (33) J (2)2 / ( t ) = ± i t − dv (cid:90) −∞ dt (cid:48)(cid:48) (cid:104) H t ( t (cid:48)(cid:48) ) , ˙ Q R/L ( ± d, t ) (cid:105) . (34)The perturbative expansion of heat current operator inEq. (25) allows to express heat correlation noise to lowestorder as S Q = S (02) Q + S (20) Q + S (11) Q + O (cid:16) | Λ | (cid:17) , (35)where S ( ij ) Q = (cid:90) T dt T (cid:90) dt (cid:48) (cid:110) (cid:104)J ( i )2 ( t (cid:48) ) J ( j )3 ( t ) (cid:105) − (cid:104)J ( i )2 ( t (cid:48) ) (cid:105)(cid:104)J ( j )3 ( t ) (cid:105) (cid:111) . (36)Now, we can perform standard calculations, whose de-tails are given in Appendix B, in order to evaluate allthe terms appearing in Eq. (35). By using the resultof this calculation, it is possible to check whether an ex-pression analogous to Eq. (21) holds true also for heatnoise. Interestingly, one finds that S Q ( V R , V L ) = S Q ( V − ,
0) + ∆ S Q ( V R , V L ) , (37)thus showing that, in contrast with the charge sector,heat fluctuations generated in the double-drive or in thesingle-drive configurations are different. The two contri-butions in Eq. (37) are S Q ( V − ,
0) = | λ | (cid:90) T dt T (cid:90) dt (cid:48) (cid:110) cos (cid:32) νe (cid:90) t (cid:48) t dτ V − ( τ ) (cid:33) (cid:60) (cid:2) P ν ( t (cid:48) − t ) ∂ t P ν ( t (cid:48) − t ) (cid:3) ++ νevπ (cid:90) dt (cid:48)(cid:48) V − ( t (cid:48) ) K ( t (cid:48) , t, t (cid:48)(cid:48) ) sin (cid:32) νe (cid:90) t (cid:48) t dτ V − ( τ ) (cid:33) (cid:61) [ ∂ t (cid:48)(cid:48) P ν ( t (cid:48)(cid:48) − t )] (cid:111) , (38)∆ S Q ( V R , V L ) = ν e | λ | (cid:90) T dt T (cid:90) dt (cid:48) cos (cid:32) νe (cid:90) t (cid:48) t dτ V − ( τ ) (cid:33) (cid:16) α RL ( t, t (cid:48) ) (cid:60) [ P ν ( t (cid:48) − t )] + β RL ( t, t (cid:48) ) (cid:61) [ P ν ( t (cid:48) − t )] (cid:17) , (39)where we defined the following functions K ( t (cid:48) , t, t (cid:48)(cid:48) ) = (cid:90) dτ P ( t (cid:48) − τ ) (Θ( τ − t (cid:48)(cid:48) ) − Θ( τ − t )) == πθv sinh ( πθ ( t − t (cid:48)(cid:48) ))sinh ( πθ ( t (cid:48) − t )) sinh ( πθ ( t (cid:48) − t (cid:48)(cid:48) )) , (40) α RL ( t, t (cid:48) ) = ( V R ( t ) V L ( t (cid:48) ) − V L ( t ) V R ( t (cid:48) )) , (41) β RL = vπ (cid:90) dt (cid:48)(cid:48) K ( t (cid:48)(cid:48) , t, t (cid:48) ) V R ( t (cid:48)(cid:48) ) [ V L ( t (cid:48) ) − V L ( t )] . (42) The result of Eq. (37) arises because heat noise is sen-sitive to the energy distribution of the injected parti-cles, thus leading to different outcomes in the single- anddouble-drive configurations. In this light, we expect thisto hold true for general energy-dependent phenomena oc-curring at the QPC. For instance, any similarity betweencharge noises generated in the two setups discussed pre-viously would disappear for more complicated tunnelinggeometry, such as multiple QPC or extended contacts,where transmission functions become energy-dependent[105–108].Eq. (37) further indicates that the double-drive and thesingle-drive configurations are completely distinct setupsand that the relation in Eq. (21) is solely a contingenteffect of the single local QPC geometry.It is useful to express heat correlation noise in energyspace, by introducing the following Fourier series νeV R/L ( t ) = (cid:88) k c k,R/L e ikωt , (43) e − iνe (cid:82) t dτV − ( τ ) = (cid:88) l ˜ p l e − i ( l + q R − q L ) ωt , (44) where we defined also the number of particles excited by V R/L along the system in a period q R/L = νe π (cid:90) T dt V R/L ( t ) = νeV R/L,dc ω , (45)and the Fourier transform of P g ( t ) in Eq. (19)˜ P g ( E ) = (cid:90) dt P g ( t ) e iEt == (cid:18) πθω c (cid:19) g − e E θ Γ( g ) ω c (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:18) g − i E πθ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (46)By exploiting these results, the two contributions to S Q become S Q ( V − ,
0) = − | λ | (cid:88) l (cid:110) ν π θ + (1 + ν ) (( l + q R − q L ) ω ) ν | ˜ p l | ˜ P ν (( l + q R − q L ) ω )+ − (cid:88) k (cid:54) =0 ( c k,R − c k,L ) (cid:0) ˜ p l − k ˜ p ∗ l − ˜ p l ˜ p ∗ l + k (cid:1) ( l + q R − q L ) ω coth kω θ (cid:16) ˜ P ν (( l + q R − q L ) ω ) − ˜ P ν ( − ( l + q R − q L ) ω ) (cid:17) (cid:111) , (47)∆ S Q ( V R , V L ) = | λ | (cid:88) k,p,l ( c k,R c p,L − c k,L c p,R ) ˜ p l + k + p ˜ p ∗ l W ( l + q R − q L ) ,k,p + W ( l + q R − q L ) ,p,k , (48)where the coefficients W l,k,p encodes all the effects due to temperature and interaction on ∆ S Q and reads W l,k,p = ω c π (cid:90) dE π (cid:110) ˜ P ( E ) ˜ P ( kω − E ) (cid:104) ˜ P ν − ( E − lω ) + ˜ P ν − ( − E − ( l + k + p ) ω ) + ˜ P ν − ( − E + lω ) (cid:105) ++ ˜ P ν − ( E + ( l + k + p ) ω ) (cid:111) − (cid:16) ˜ P ν (( l + k ) ω ) + ˜ P ν ( − ( l + k ) (cid:17) . (49)Let us observe that the contribution ∆ S Q exists only inthe double-drive configurations. Indeed, in the config-uration with a single drive, where V L = 0, one obtainsthat c k,L = 0 for each k , and the contribution in Eq. (48)vanishes. C. Hong-Ou-Mandel noises
Among all the possible choices for the configuration in-volving the two voltages V R and V L , one of the most in-teresting, even from the experimental point of view, is theHong-Ou-Mandel (HOM) setup, where two identical volt-age drives are applied to reservoirs 1 and 4 and delayedby a constant time t D . This experimental configurationcorresponds to set V R ( t ) = V ( t ) and V L ( t ) = V ( t + t D )in Eq. (20), with V ( t ) a generic periodic drive. In thissituation the charge excited by each drive along the edgechannels are equal, such that q R = q L = q .For notational convenience, we define the single-drive heat noise and the HOM charge and heat noises as S sdQ = S Q ( V − ( t ) , , (50) S HOMC/Q = S C/Q ( V ( t ) , V ( t + t D )) . (51)According to Eq. (37) and using the above definitions,the HOM heat noise can be expressed as S HOMQ = S sdQ + ∆ S Q . (52)From the existing literature [3, 39, 93], it is well estab-lished that charge HOM noise reduces to its equilibriumvalue for null time delay. Before entering into the detailsof our discussion, we would like to prove analytically thatthe same holds true for HOM heat noise S HOMQ , indepen-dently of the choice of any parameter. The photo-assistedamplitude in Eq. (44) reduces to ˜ p l = δ l, and the Fouriercoefficients c k, − vanish for all k . Let us start by lookingat the single-drive contribution. By substituting this an-alytical simplification in Eq. (47), we obtain S sdQ ( t D = 0) = − | λ | ν π θ ν ≡ S vacQ , (53)which is independent of the injected particles and corre-spond simply to the equilibrium noise S vacQ due to ther-mal fluctuations. This can be clearly understood giventhe fact that V − ( t ) = 0 for t D = 0 and the single-drivecontribution corresponds to the noise generated in a driv-eless configuration.Concerning the remaining part in Eq. (52), one has for t D = 0∆ S Q = | λ | (cid:88) k c k,R c − k,R ( W ,k, − k + W , − k,k ) , (54)where W ,k, − k = ˜ P ν ( kω ) − ˜ P ν ( − kω )2 . (55)From Eq. (55), we can clearly deduce that W ,k, − k = −W , − k,k , which enforces the vanishing of ∆ S Q in Eq.(54). This is enough to prove that HOM heat noise al-ways reaches its equilibrium value at t D = 0, such that S HOMQ ( t D = 0) = S sdQ ( t D = 0) = S vacQ . (56)Let us note that this is not a trivial result since S HOMQ does not depend effectively on the single drive V − as S sdQ ,but on both V R and V L and even at t D = 0 the systemis still driven by these two voltages. IV. RESULTS AND DISCUSSIONS
In this section, we discuss the results concerning theheat correlation noises in the HOM interferometer. Inparticular, we focus our discussion on a specific drivingvoltage, namely a periodic train of Lorentzian pulses V Lor ( t ) = V π + ∞ (cid:88) k = −∞ WW + ( t − k T ) . (57)A Lorentzian-shaped drive, which satisfy the additionalquantization condition νe (cid:90) T dt V Lor ( t ) = 2 πq, (58)with q an integer number, constitutes the optimal driv-ing able to inject clean pulses devoid of any additionalelectron-hole pairs. The minimal excitations thus emit-ted into the quantum Hall channels are the aforemen-tioned levitons [20, 22]. The Fourier coefficients for thisspecific drive are given in Appendix C.In the HOM setup previously described, a state com-posed by q L = q R = q levitons [109] is injected by each driven contact and collide at the QPC, separated by acontrollable time delay.In analogy with the previous literature on charge noise,we introduce the following ratio [30, 39, 44, 46] R HOMC/Q = S HOMC/Q − S vacC/Q S RC/Q − S vacC/Q , (59)where we subtracted the equilibrium noise S vacC/Q and wenormalize with respect to S RC/Q ≡ S
C/Q ( V R , V R ( t ) = V Lor ( t ) and V L ( t ) = 0. The expressions for S vacC and S RC are well-known and have been derived inprevious paper [3, 39, 90]. The expression for S RQ can beobtained from our results in Sec. III B and reads S RQ = − | λ | (cid:88) l (cid:110) ν π θ + (1 + ν ) ( lω ) ν | p l | ˜ P ν (( l + q ) ω )+ − (cid:88) k (cid:54) =0 c k (cid:0) p l − k p ∗ l − p l p ∗ l + k (cid:1) ( l + q ) ω ˜ P ( kω )2 kω ×× (cid:16) ˜ P ν (( l + q ) ω ) − ˜ P ν ( − ( l + q ) ω ) (cid:17) (cid:111) , (60)where c k = νe (cid:82) T dt T V Lor ( t ) e ikωt are the Fourier co-efficients for a single Lorentzian voltage and p l = (cid:82) T dt T e − iνe (cid:82) t dτV lor ( τ ) e i ( l + q ) ωt (see Appendix C).In addition, we define an analogous ratio for single-driveheat noise as R sdQ = S sdQ − S vacQ S RQ − S vacQ , (61)in order to asses its relative contribution to the overallHOM heat noise.Let us notice that, according to Eqs (53) and (56) bothratios, R HOMQ and R sdQ vanish for t D = 0. In the spe-cific case of levitons, which are single-electron excita-tions, at ν = 1 the physical explanation for the totaldip at t D = 0 involves the anti-bunching effect of iden-tical fermions: electron-like excitations colliding at theQPC at the same time are forced to escape on oppositechannels, thus leading to a total suppression of fluctua-tions at t D = 0 and generating the so called Pauli dip[12, 38, 42]. For fractional filling factors, it is remarkablethat this total dip is still present despite the presence ofanyonic quasi-particles in the system, which do not obeyFermi-Pauli statistics [16, 39]. Anyway, this single QPCgeometry does not allow for the braiding of one quasi-particle around the other, thus excluding any possibleeffect due to fractional statistics.In the following, we exploit the full generality of ourderivation by performing the analysis for different val-ues of q .We start by considering the regime where thermal andquantum fluctuations are comparable.As a beginning, we focus on the relevant case of q = 1, Figure 2. (Color online) HOM heat ratio R HOMQ (upper pan-els) and single-drive heat ratio R sdQ (lower panels) as a func-tion of the time delay t D for q = 1 and temperatures θ = 0 . ω (solid lines) and θ = 0 . ω (dashed lines). The integer case (leftpanel) and the fractional case for ν = (right panel) are com-pared. The other parameters are W = 0 . T and ω = 0 . ω c . where states formed by a single leviton are injected fromboth sources [91]. The collision of identical single-levitonstates is very interesting because previous work on fluc-tuations of charge current proved that in this case theratio of HOM charge noise is independent of filling fac-tors and temperatures, acquiring an universal analyticalexpression [15, 39]. In order to perform a similar com-parison for the heat noise, we present in Fig. 2 the HOMheat ratio (upper panels) considering two temperatures θ = 0 . ω (solid line) and θ = 0 . ω (dashed lines) forboth the integer and fractional case. Contrarily to thecharge case, these curves are all clearly distinct. Thismeans that this universality does not extend also to heatfluctuations. This fact can be explained by the depen-dence of heat HOM noise on the energy distribution ofparticles injected by the drives, which in turn is signifi-cantly affected by the temperature and by the strength ofcorrelations encoded in the filling factor ν . In particular,as the temperature is further increased, the thermal fluc-tuations tend to hide the effect of the voltages, resultingin a reduction of R HOMQ for both filling factors.Interestingly, we also note that the single-drive ratio canswitch sign as t D is tuned, independently of the fillingfactor. Since S RQ is independent of t D , the change of signof R sdQ is entirely due to S sdQ itself. This is a remarkable difference with respect to the charge noise generated inthe same configurations, since charge conservation fixesthe sign of current-current correlations. On the contrary,it should be pointed out that the sign of heat noise is notconstrained by any conservation law [83].In Fig. 3, we start looking at the collision of states com- Figure 3. HOM heat ratio R HOMQ (solid lines) and HOMcharge ratio R HOMC (dashed lines) as a function of the timedelay t D for q = 2 and q = 4. The integer case (upper panels)and the fractional case for ν = (lower panels) are compared.Black vertical lines demonstrate the exact correspondence ofside peaks appearing in charge and heat ratio. The otherparameters are W = 0 . T , θ = 0 . ω and ω = 0 . ω c . posed by multiple levitons and compare HOM charge andheat ratios (solid and dashed lines, respectively) for q = 2and q = 4. In the fermionic case, presented in the twoupper panels, both charge and heat ratio show a singlesmooth dip at t D = 0, without additional side features.Interestingly, heat fluctuations are enhanced with respectto charge: in particular, heat HOM ratios saturate totheir asymptotic value for smaller values of time delaycompared to charge ratio. Again, the enhancement ofheat fluctuations can be related to the fact that heat isnot constrained by any conservation law, in contrast tothe case of charge.Very remarkably, the curves for the HOM ratio in thefractional case display instead some unexpected sidepeaks and dips in addition to the central dip. In partic-ular, the number of these maxima and minima increases Figure 4. (Color online) HOM heat ratio R HOMQ as a functionof the time delay t D for q = 1, q = 2, q = 3, q = 4. The integercase (dashed lines) and the fractional case for ν = (solidlines) are compared. The other parameters are W = 0 . T , θ = 10 − ω and ω = 0 . ω c . for states composed with more levitons. A recent pa-per by the authors explained this intriguing result forcharge HOM noise in terms of a crystallization processinduced by strong correlation on the charge density of q levitons , i.e. a re-arrangement of the density into anoscillating and ordered pattern with a number of peaksrelated to q [93]. Black vertical lines in the lower panelof Fig. 3 demonstrate the exact correspondence of sidepeaks appearing in charge and heat ratio as a functionof time delay. Based on this argument, we can infer thatthe HOM heat noise is affected by the crystallization in-duced in the propagating levitons, thus giving rise to thefeatures observed in the lower panel of Fig. 3. While theoscillating pattern of R HOMQ remarkably matches withthat of R HOMC , the amplitude oscillations are widely en-hanced for heat fluctuations, in particular for the peaksoccurring at small values of time delay.We conclude by noticing that strong correlation of thefractional regime can increase the value of the HOM heatratio even above 1. Once again, since this is not the casefor the single-drive contribution, this is due to the pres-ence of ∆ S Q , which is peculiar to collision between levi-tons incoming from different reservoirs.Now, we consider the regime of very low temperature θ (cid:28) ω , where the quantum effects should be largely en-hanced with respect to the thermal fluctuations. Having established from the previous discussion the connectionbetween ∆ S Q and S HOMQ in the fractional regime, we fo-cus only on the HOM heat ratio R HOMQ .The plots for R HOMQ in the integer and in the fractionalcase are compared in Fig. 4 for different values of q . Inthe integer case, a single smooth dip is present for all thevalues of q , confirming the phenomenology described forthe finite temperature case. For the strongly correlatedcase, at q = 1 one observes a smooth profile, except for asmall decrease close to t D = 0 .
5. Intriguingly, the oscil-lations observed in Fig. 3 for q > q , in addition to the centralone and can also reach negative values. V. CONCLUSION
In this work, we investigated charge and heat currentfluctuations in an HOM interferometer in the fractionalquantum Hall regime. Here, two identical leviton ex-citations impinge at a QPC with a given time delay.We started by evaluating zero-frequency cross-correlatedcharge and heat noises in the presence of two genericdriving voltage V L and V R . We demonstrated that heatnoise in this double-drive configuration depends on both V + = V L + V R and V − = V L − V R and, thus, cannotbe reproduced in a single-drive setup driven by the volt-age V − only. In particular, this implies that single-driveconfiguration and HOM interferometer implemented withvoltage sources are two physically distinct experimentalconfigurations. Moreover, we proved that the HOM heatratio vanishes for a null time delay for both integer andfractional filling factors, despite the presence, in the lat-ter case, of emergent fractionally charged quasi-particles.Finally, we investigated the form of HOM heat ratio fordifferent regimes of temperatures. Interestingly, unex-pected side dips emerged only in the fractional regimewhich can be related to the crystallization mechanismrecently predicted for levitons [93]. ACKNOWLEDGMENTS
L.V. and M.S. acknowledge support from CNR SPINthrough Seed project “Electron quantum optics withquantized energy packets”. This work was granted ac-cess to the HPC resources of Aix-Marseille Universit´efinanced by the project Equip@Meso (Grant No. ANR-10-EQPX-29-01). It has been carried out in the frame-work of project “1shot reloaded” (Grant No. ANR-14-CE32-0017) and benefited from the support of theLabex ARCHIMEDE (Grant No. ANR-11-LABX-0033),all funded by the “investissements d’avenir” French Gov-ernment program managed by the French National Re-search Agency (ANR). The project leading to this pub-lication has received funding from Excellence Initiative0of Aix-Marseille University - A*MIDEX, a French “in-vestissements d’avenir” programme.
Appendix A: Coupling to the gate
In this Appendix, we show that there is no gauge trans-formation able to link the equations of motion for theconfigurations with two driving voltages V R and V L andthe configuration with a single drive V − = V R − V L , pre-sented in the main text.In the double-drive setup a voltage drive is applied bothto right-moving and left-moving excitations. We considera situation in which the vector potentials A R/L ( x, t ) areabsent. The Lagrangian density is L = 14 π (cid:26) − ∂ x Φ R ( x, t ) (cid:2) ∂ t Φ R ( x, t ) + v∂ x Φ R ( x, t ) (cid:3) + ∂ x Φ L ( x, t ) (cid:2) ∂ t Φ L ( x, t ) − v∂ x Φ L ( x, t ) (cid:3)(cid:27) ++ e √ ν π (cid:20) ∂ x Φ R ( x, t ) V R ( x, t ) − ∂ x Φ L ( x, t ) V L ( x, t ) (cid:21) . (A1)The Euler-Lagrange equations ∂ t δ L δ∂ t Φ α + ∂ x δ L δ∂ x Φ α − δ L δ Φ α = 0 (A2)with α = R, L , give rise to the following equation ofmotions for the bosonic fields:( ∂ t + v∂ x )Φ R ( x, t ) = e √ ν V R ( x, t ) (A3)( ∂ t − v∂ x )Φ L ( x, t ) = e √ ν V L ( x, t ) . (A4)In order to model the system presented in Sec. II, theform for the voltage drives is V R ( x, t ) = f R ( x ) V R ( t ) (A5) V L ( x, t ) = f L ( x ) V L ( t ) (A6)where f R/L ( x ) are time-independent, while V R/L ( t ) arespace-independent. In this case equation of motions forthe double-drive setup are( ∂ t + v∂ x )Φ R ( x, t ) = e √ νf R ( x ) V R ( t ) (A7a)( ∂ t − v∂ x )Φ L ( x, t ) = e √ νf L ( x ) V L ( t ) . (A7b)We also consider a single-drive setup with an effectivevoltage drive V R ( x, t ) = f R ( x )[ V R ( t ) − V L ( t )] on theright side, and the left side grounded [ V L ( x, t ) = 0]. Westill consider that the magnetic potential is zero on bothedges. It is immediate to show that the equation of mo-tions are now( ∂ t + v∂ x )Φ R ( x, t ) = e √ νf R ( x )[ V R ( t ) − V L ( t )] (A8a)( ∂ t − v∂ x )Φ L ( x, t ) = 0 (A8b)
1. Applying gauge transformations to the HOMsetup
Here we show that a gauge transformation that oper-ates in the following way on the voltage drives (cid:40) V R ( x, t ) = f R ( x ) V R ( t ) V L ( x, t ) = f L ( x ) V L ( t ) −→ (cid:40) V (cid:48) R ( x, t ) = f R ( x )[ V R ( t ) − V L ( t )] V (cid:48) L ( x, t ) = 0 , (A9)does not transform Eqs. (A7) into Eqs. (A8), but leavesthem unchanged.We recall that a general gauge transformation that leavesinvariant an electromagnetic field is given by V (cid:48) R/L ( x, t ) = V R/L ( x, t ) − ∂ t χ R ( x, t ) , (A10) A (cid:48) R/L ( x, t ) = A R/L ( x, t ) + ∂ x χ R ( x, t ) , (A11)with χ R/L ( x, t ) a scalar function.In our particular case, voltage potentials are required totransform as V (cid:48) R ( x, t ) = f R ( x ) V R ( x ) − ∂ t χ R ( x, t ) = f R ( x )[ V R ( t ) − V L ( t )](A12) V (cid:48) L ( x, t ) = f L ( x ) V L ( x ) − ∂ t χ L ( x, t ) = 0 (A13)for the right-moving and left-moving sector respectively.The transformation is evidently implemented by thechoice χ R ( x, t ) = f R ( x ) (cid:90) t dτ V L ( τ ) (A14a) χ L ( x, t ) = f L ( x ) (cid:90) t dτ V L ( τ ) (A14b)Since these equations involve spatial-dependent func-tions, we expect that non-zero magnetic potentials arise1as a consequence of the gauge transformation. In the newgauge we get non-zero magnetic potentials given by (in our initial gauge choice A R/L = 0) A (cid:48) R ( x, t ) = ∂ x f R ( x ) (cid:90) t dτ V L ( τ ) (A15) A (cid:48) L ( x, t ) = ∂ x f L ( x ) (cid:90) t dτ V L ( τ ) (A16)and the Lagrangian density now reads L (cid:48) = 14 π (cid:26) − ∂ x Φ R ( x, t ) (cid:2) ∂ t Φ R ( x, t ) + v∂ x Φ R ( x, t ) (cid:3) + ∂ x Φ L ( x, t ) (cid:2) ∂ t Φ L ( x, t ) − v∂ x Φ L ( x, t ) (cid:3)(cid:27) ++ e √ ν π (cid:26) ∂ x Φ R ( x, t ) f R ( x )[ V R ( t ) − V L ( t )] + (cid:2) ∂ t Φ R ( x, t ) ∂ x f R ( x ) − ∂ t Φ L ( x, t ) ∂ x f L ( x ) (cid:3) (cid:90) tt dτ V L ( τ ) (cid:27) (A17)where the last term accounts for the presence of A (cid:48) R ( x, t )and A (cid:48) L ( x, t ). We now look for the equation of motions inthis new configuration. From Euler-Lagrange equationsone gets( ∂ t + v∂ x )Φ R ( x, t ) == e √ νf R ( x )[ V R ( t ) − V L ( t )] + e √ νf R ( x ) V L ( t ) == e √ νf R ( x ) V R ( t ) (A18)( ∂ t − v∂ x )Φ L ( x, t ) = e √ νf L ( x ) V L ( t ) . (A19)Note that we have not recovered the equation of motionsfor the single drive setup, Eqs. (A8), as one may naivelyexpect. On the contrary, we have found the equations ofmotion for the double-drive setup, Eqs. (A7). Appendix B: Heat noise
In this Appendix, we give more details about the calcu-lation of heat noise presented in Sec. III. Before startingwith the derivation of heat noise, we would give someformulas that would be useful in the following parts.
1. Useful formulas
In the following, we derive some results that would beuseful for the evaluation of heat current fluctuations. Inparticular, our goal is to evaluate the following averagevalues (for simplicity, we drop all the low indices R or L ) C ( t , t , t ) = (cid:104) ∂ t φ ( t ) e i √ νφ ( t ) e − i √ νφ ( t ) (cid:105) , (B1) C ( t , t , t ) = (cid:104) e i √ νφ ( t ) e − i √ νφ ( t ) ∂ t φ ( t ) (cid:105) , (B2) D ( t , t , t ) = (cid:104) ( ∂ t φ ( t )) e i √ νφ ( t ) e − i √ νφ ( t ) (cid:105) , (B3) D ( t , t , t ) = (cid:104) e i √ νφ ( t ) e − i √ νφ ( t ) ( ∂ t φ ( t )) (cid:105) , (B4)where the thermal average is performed over the initialequilibrium density matrix, in absence of tunneling and driving voltage and bosonic fields evolve according to theedge Hamiltonian H . In order to evaluate C and C , westart by considering the following general average value E ( (cid:15) , (cid:15) , (cid:15) ; t , t , t ) = (cid:104) e − i(cid:15) φ ( t ) e − i(cid:15) φ ( t ) e − i(cid:15) φ ( t ) (cid:105) , (B5)which is connected to C and C by this relation C ( t , t , t ) = i∂ t (cid:26) lim (cid:15) → ∂ (cid:15) E ( (cid:15) , (cid:15) , (cid:15) ; t , t , t ) (cid:27) (cid:15) = −√ ν(cid:15) = √ ν , (B6) C ( t , t , t ) = i∂ t (cid:26) lim (cid:15) → ∂ (cid:15) E ( (cid:15) , (cid:15) , (cid:15) ; t , t , t ) (cid:27) (cid:15) = −√ ν(cid:15) = √ ν . (B7)By using [110] (cid:104) e χ ( t ) e χ ( t ) e χ ( t ) (cid:105) = e (cid:80) i =1 (cid:104) χ ( t i ) (cid:105) e (cid:80) i 2. Calculations of heat noise Our starting point is the perturbative expression ofheat noise given in the main text (see Eq. (25)) S Q = S (02) Q + S (20) Q + S (11) Q + O (cid:16) | Λ | (cid:17) . (B29) Firstly, we derive the term S (11) Q , which reads S (11) Q = (cid:90) T dt T (cid:90) dt (cid:48) (cid:110) (cid:104) ∂ t (cid:48) Ψ † R (0 , t (cid:48) )Ψ L (0 , t (cid:48) ) ∂ t Ψ † L (0 , t )Ψ R (0 , t ) (cid:105) ++ (cid:104) Ψ † L (0 , t (cid:48) ) ∂ t (cid:48) Ψ R (0 , t (cid:48) )Ψ † R (0 , t ) ∂ t Ψ L (0 , t ) (cid:105) (cid:111) , (B30)3since (cid:104)J (1)2 / ( t ) (cid:105) = 0 (see Eq. (33) in the main text). Werecall that the time evolution of quasi-particle fields isΨ R,L ( x, t ) = F R/L √ πa e − i √ νφ R/L ( x,t ) e iνe (cid:82) t ∓ xv − dvt dt (cid:48) V R/L ( t (cid:48) ) . (B31)We can further express the average in the above equationas S (11) Q = 2 | λ | (cid:90) T dt T (cid:90) dt (cid:48) (cid:110) cos (cid:18) νe (cid:90) tt (cid:48) dt (cid:48)(cid:48) V R ( t (cid:48)(cid:48) ) − V L ( t (cid:48)(cid:48) ) (cid:19) ∂ (cid:48) t P ν ( t (cid:48) − t ) ∂ t P ν ( t (cid:48) − t )+ (B32)+ νeV R ( t (cid:48) ) sin (cid:18) νe (cid:90) tt (cid:48) dt (cid:48)(cid:48) V R ( t (cid:48)(cid:48) ) − V L ( t (cid:48)(cid:48) ) (cid:19) ∂ t P ν ( t (cid:48) − t ) + νeV L ( t ) sin (cid:18) νe (cid:90) tt (cid:48) dt (cid:48)(cid:48) V R ( t (cid:48)(cid:48) ) − V L ( t (cid:48)(cid:48) ) (cid:19) ∂ t (cid:48) P ν ( t (cid:48) − t )+ − ν e V R ( t (cid:48) ) V L ( t ) cos (cid:18) νe (cid:90) tt (cid:48) dt (cid:48)(cid:48) V R ( t (cid:48)(cid:48) ) − V L ( t (cid:48)(cid:48) ) (cid:19) P ν ( t (cid:48) − t ) (cid:111) , where the function P g ( t ) is defined in Eq. (B12) and λ ≡ Λ2 πa . The integration by parts of second and third line ofEq. (B32) provides some useful eliminations, providing the final expression for this contribution S (11) Q = 2 | λ | (cid:90) T dt T (cid:90) dt (cid:48) (cid:110) cos (cid:18) νe (cid:90) tt (cid:48) dt (cid:48)(cid:48) ( V R ( t (cid:48)(cid:48) ) − V L ( t (cid:48)(cid:48) )) (cid:19) ∂ (cid:48) t P ν ( t (cid:48) − t ) ∂ t P ν ( t (cid:48) − t )+ (B33) − ν e ( V R ( t (cid:48) ) V R ( t ) + V L ( t (cid:48) ) V L ( t )) cos (cid:18) νe (cid:90) tt (cid:48) dt (cid:48)(cid:48) ( V R ( t (cid:48)(cid:48) ) − V L ( t (cid:48)(cid:48) )) (cid:19) P ν ( t (cid:48) − t ) (cid:111) . (B34)We focus on the remaining contributions, starting from S (02) Q : the calculations for the other term would be anal-ogous. By plugging Eqs. (26) and (28) in the definition of S (02) Q , one finds S (02) Q = − i | λ | π (cid:90) dt (cid:90) T dt (cid:48) T (cid:90) dt (cid:48)(cid:48) θ ( t (cid:48) − t (cid:48)(cid:48) ) (cid:110) (cid:104) ( ∂ t φ R (0 , t )) (cid:104) Ψ † R (0 , t (cid:48)(cid:48) )Ψ L (0 , t (cid:48)(cid:48) ) , ∂ t (cid:48) Ψ † L (0 , t (cid:48) )Ψ R (0 , t (cid:48) ) (cid:105) (cid:105) + − νeV R ( t ) (cid:104) ∂ t φ R (0 , t ) (cid:104) Ψ † R (0 , t (cid:48)(cid:48) )Ψ L (0 , t (cid:48)(cid:48) ) , ∂ t (cid:48) Ψ † L (0 , t (cid:48) )Ψ R (0 , t (cid:48) ) (cid:105) (cid:105) + − (cid:104) ( ∂ t φ R (0 , t )) (cid:105)(cid:104) (cid:104) Ψ † R (0 , t (cid:48)(cid:48) )Ψ L (0 , t (cid:48)(cid:48) ) , ∂ t (cid:48) Ψ † L (0 , t (cid:48) )Ψ R (0 , t (cid:48) ) (cid:105) (cid:105) (cid:111) . (B35)The averages involving the commutators can be per-formed by using the expression in Eq. (18) for the time evolution of quasi-particle fields and by resorting to theformulas in Eqs (B10), (B14),(B25) and (B26) derived inthe Appendix B 1. Indeed, one finds4 S (02)( Q ) = − i | λ | π (cid:90) dt (cid:90) T dt (cid:48) T (cid:90) dt (cid:48)(cid:48) Θ ( t (cid:48) − t (cid:48)(cid:48) ) (cid:110) − (cid:104) ν∂ t (cid:48) K ( t, t (cid:48) , t (cid:48)(cid:48) ) cos (cid:32) νe (cid:90) t (cid:48) t (cid:48)(cid:48) dτ V − ( τ ) (cid:33) ( P ν ( t (cid:48)(cid:48) − t (cid:48) ) − P ν ( t (cid:48) − t (cid:48)(cid:48) )) ++ νeV L ( t (cid:48) ) K ( t, t (cid:48) , t (cid:48)(cid:48) ) ( P ν ( t (cid:48)(cid:48) − t (cid:48) ) − P ν ( t (cid:48) − t (cid:48)(cid:48) )) sin (cid:32) νe (cid:90) t (cid:48) t (cid:48)(cid:48) dτ V − ( τ ) (cid:33) (cid:105) ++ K ( t, t (cid:48) , t (cid:48)(cid:48) ) (cid:104) νeV R ( t ) sin (cid:32) νe (cid:90) t (cid:48) t (cid:48)(cid:48) dτ V − ( τ ) (cid:33) ∂ t (cid:48) [ P ν ( t (cid:48)(cid:48) − t (cid:48) ) − P ν ( t (cid:48) − t (cid:48)(cid:48) )] + − ν e V R ( t ) V L ( t (cid:48) ) cos (cid:32) νe (cid:90) t (cid:48) t (cid:48)(cid:48) dτ V − ( τ ) (cid:33) [ P ν ( t (cid:48)(cid:48) − t (cid:48) ) − P ν ( t (cid:48) − t (cid:48)(cid:48) )] (cid:105)(cid:111) , A similar calculation leads to the expression for the last contribution, given by S (20)( Q ) = − i | λ | π (cid:90) dt (cid:90) T dt (cid:48) T (cid:90) dt (cid:48)(cid:48) Θ ( t (cid:48) − t (cid:48)(cid:48) ) (cid:110)(cid:104) ν∂ t (cid:48) K ( t, t (cid:48) , t (cid:48)(cid:48) ) cos (cid:32) νe (cid:90) t (cid:48) t (cid:48)(cid:48) dτ V − ( τ ) (cid:33) ( P ν ( t (cid:48)(cid:48) − t (cid:48) ) − P ν ( t (cid:48) − t (cid:48)(cid:48) )) ++ νeV L ( t (cid:48) ) K ( t, t (cid:48) , t (cid:48)(cid:48) ) ( P ν ( t (cid:48)(cid:48) − t (cid:48) ) − P ν ( t (cid:48) − t (cid:48)(cid:48) )) sin (cid:32) νe (cid:90) t (cid:48) t (cid:48)(cid:48) dτ V − ( τ ) (cid:33) (cid:105) ++ K ( t, t (cid:48)(cid:48) , t (cid:48) ) (cid:104) νeV L ( t ) sin (cid:32) νe (cid:90) t (cid:48) t (cid:48)(cid:48) dτ V − ( τ ) (cid:33) ∂ t (cid:48) [ P ν ( t (cid:48)(cid:48) − t (cid:48) ) − P ν ( t (cid:48) − t (cid:48)(cid:48) )] ++ 4 ν e V R ( t ) V L ( t (cid:48)(cid:48) ) cos (cid:32) νe (cid:90) t (cid:48) t (cid:48)(cid:48) dτ V − ( τ ) (cid:33) [ P ν ( t (cid:48)(cid:48) − t (cid:48) ) − P ν ( t (cid:48) − t (cid:48)(cid:48) )] (cid:105)(cid:111) . By summing up the two contributions, one can see that the first lines cancel out and the remaining two lines add upin a way that allows to get rid of the function Θ( t (cid:48) − t (cid:48)(cid:48) ), thus obtaining S (02)( Q ) + S (20)( Q ) = − i | λ | π (cid:90) dt (cid:90) T dt (cid:48) T (cid:90) dt (cid:48)(cid:48) (cid:110) K ( t, t (cid:48)(cid:48) , t (cid:48) ) (cid:104) νeV − ( t ) sin (cid:32) νe (cid:90) t (cid:48) t (cid:48)(cid:48) dτ V − ( τ ) (cid:33) ∂ t (cid:48) [ P ν ( t (cid:48)(cid:48) − t (cid:48) ) − P ν ( t (cid:48) − t (cid:48)(cid:48) )] + − ν e V R ( t ) V L ( t (cid:48)(cid:48) ) cos (cid:32) νe (cid:90) t (cid:48) t (cid:48)(cid:48) dτ V − ( τ ) (cid:33) [ P ν ( t (cid:48)(cid:48) − t (cid:48) ) − P ν ( t (cid:48) − t (cid:48)(cid:48) )] (cid:105)(cid:111) . Now, summing all the contributions according to Eq.(37), it is possible to obtain the result presented in the main text, which reads S Q ( V R , V L ) = S Q ( V − , 0) + ∆ S Q ( V R , V L ) , (B36)with S Q ( V − , 0) = | λ | (cid:90) T dt T (cid:90) dt (cid:48) (cid:110) cos (cid:32) νe (cid:90) t (cid:48) t dτ V − ( τ ) (cid:33) (cid:60) (cid:2) P ν ( t (cid:48) − t ) ∂ t P ν ( t (cid:48) − t ) (cid:3) ++ νevπ (cid:90) dt (cid:48)(cid:48) V − ( t (cid:48) ) K ( t (cid:48) , t, t (cid:48)(cid:48) ) sin (cid:32) νe (cid:90) t (cid:48) t dτ V − ( τ ) (cid:33) (cid:61) [ ∂ t (cid:48)(cid:48) P ν ( t (cid:48)(cid:48) − t )] (cid:111) , (B37)∆ S Q ( V R , V L ) = ν e | λ | (cid:90) T dt T (cid:90) dt (cid:48) cos (cid:32) νe (cid:90) t (cid:48) t dτ V − ( τ ) (cid:33) (cid:16) α RL ( t, t (cid:48) ) (cid:60) [ P ν ( t (cid:48) − t )] + β RL ( t, t (cid:48) ) (cid:61) [ P ν ( t (cid:48) − t )] (cid:17) , (B38)where we defined the following functions α RL ( t, t (cid:48) ) = ( V R ( t ) V L ( t (cid:48) ) − V L ( t ) V R ( t (cid:48) )) , (B39) β RL = vπ (cid:90) dt (cid:48)(cid:48) K ( t (cid:48)(cid:48) , t, t (cid:48) ) V R ( t (cid:48)(cid:48) ) [ V L ( t (cid:48) ) − V L ( t )] . (B40) Appendix C: Fourier coefficients This Appendix is devoted to the Fourier analysis ofthe Lorentzian periodic signal V Lor ( t ) and of the phase5 e − iνe (cid:82) tt dt (cid:48) V Lor ( t (cid:48) ) , where V Lor ( t ) = V π + ∞ (cid:88) k = −∞ WW + ( t − k T ) , (C1)where T is the periodic, V the amplitude and W the halfwidth at half maximum.The coefficients for the Fourier series of the expression νeV Lor ( t ) = (cid:80) k c k e ikωt are c k = qωe − π W T | k | , (C2)with q = νe π (cid:82) T dtV Lor ( t ) = νeV ω .We also note that, for the time delayed voltage V Lor ( t + t D ), the coefficients become c (cid:48) k = c k e − ikωt D .The Fourier series e − iνe (cid:82) t dt (cid:48) ( V Lor ( t (cid:48) ) − V ) = (cid:80) l p l e − ilωt allows to deal with the time-dependent problem as a su-perposition of time-independent configurations, with en-ergy shifted by an integer amount of energy quanta ω .For the Lorentzian case, it is convenient to switch to acomplex representation in terms of the variable z = e iωt .After some algebra and introducing γ = e − πη one finds[3, 4] p l = 12 πi (cid:73) | z | =1 dz z l + q − (cid:18) − zγz − γ (cid:19) q . 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