Probing non-classical light fields with energetic witnesses in Waveguide Quantum Electro-Dynamics
PProbing non-classical light fields with energetic witnessesin Waveguide Quantum Electro-Dynamics
Maria Maffei, Patrice A. Camati, and Alexia Auff`eves Universit´e Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France
We analyze energy exchanges between a qubit and a resonant field propagating in a waveguide.The joint dynamics is analytically solved within a repeated interaction model. The work receivedby the qubit is defined as the unitary component of the field-induced energy change. Using thesame definition for the field, we show that both work flows compensate each other. Focusing on thecharging of a qubit battery by a pulse of light, we evidence that the work provided by a coherent fieldis an upper bound for the qubit ergotropy, while this bound can be violated by non-classical fields,e.g. a coherent superposition of zero- and single-photon states. Our results provide operational,energy-based witnesses to probe the non-classical nature of a light field.
Extending the thermodynamic framework to the quan-tum regime has exciting motivations, e.g. understand-ing how quantum resources like coherence and entangle-ment impact the performance of fundamental protocolslike work extraction [1–3]. This poses the great challengeto propose operational definitions of work, that can giverise to such experimentally testable effects. For thermo-dynamic systems of the classical world, work features anentropy-preserving energy change [4]. In the quantumrealm, such energy exchanges typically take place whena classical entity drives a closed system. However, an-alyzing the nature of energy exchanges among quantumsystems, e.g. between a charger and a battery along awork extraction protocol, is still exploratory and provesto be a very active field of research nowadays [5]. As adistinctive feature, quantum chargers and batteries mayget entangled along the charging protocol. Energy ex-changes are thus irreducibly related to the appearanceof some entanglement entropy, challenging the conceptof work captured by classical intuitions. Most studiesaimed at optimizing the charging/discharging process ofthe battery, focusing on the impact of quantum correla-tions [6, 7], coherence [8], battery’s inner structure [9–12],collective effects [9–11], and feedback control [12].Waveguide quantum electrodynamics (WG-QED)provides an ideal scenery to investigate this newphysics [13–16]. WG-QED features the fundamentalinteractions between a quantum emitter (e.g. a qubit)and the electromagnetic field inside a waveguide, and isexplored on various experimental platforms, e.g. inte-grated photonics [17] and superconducting circuits [18].From a thermodynamic perspective, WG-QED offersthe utmost advantage that the light-matter systemis isolated. Its dynamics is thus generated by sometime-independent Hamiltonian, and was shown to beanalytically solvable for any kind of input field [19–21].Importantly, it allows exploring so-called “autonomousscenarios” where no external action is required, i.e.which do not involve any external source of energy [16].This greatly simplifies the conceptual analysis, whichremains restricted to characterizing the nature of energy
Figure 1. Sketch of the setup. A qubit located at the position x = 0 of a 1DWG interacts with the field a (0 , t ). Input andoutput operators, a in ( t ) (resp. a out ( t )), are defined as theleft and right limits of the field operator a ( (cid:15), t ) for (cid:15) → − (resp. (cid:15) → + ). The field propagates freely for x < x > a in ( t ) = a ( x, t − | x | /v ) (resp. a out ( t ) = a ( x, t + x/v )). exchanges between light and matter.In this Letter, we exploit this conceptually sim-ple framework to define work as the unitary, entropy-preserving component of the energy received by the lightfield (resp. the emitter). This definition matches for-merly proposed approaches in the limit where the fieldbecomes classical [14, 16, 22]. In the general case of a res-onant input field of arbitrary statistics, a first noticeableresult is that the two work flows compensate each otherwhen the initial interaction energy is zero. This allowsdefining a unique work flow, which can be directly mea-sured in the state of the field, after it has scattered offthe qubit. When the WG is in the vacuum, we evidencemeasurable work exchanges of quantum nature along thespontaneous emission process.We then analyze a simple “charging scenario” wherethe qubit (resp. a pulse of light) plays the role of abattery (resp. a charger). We study the behavior of thequbit ergotropy as a function of the light statistics. Ourstudy reveals the existence of a bound relating ergotropyand work, that holds for coherent (e.g. classical) fieldsand gets violated for non-classical ones, e.g. super-positions of zero and one photon pulses. This bound a r X i v : . [ qu a n t - ph ] F e b provides a new witness to probe the non-classicalityof a light field, that is based on purely energetic andobservable quantities. We finally evidence that coherentsuperpositions of zero and one photon states alwaysprovide more ergotropy to the qubit than the equivalentstatistical mixture, providing a new “quantum energeticsignature”[23]. Qubit and field observables.—
We consider a qubit lo-cated at the position x = 0 of a one-dimensional waveg-uide (1DWG) (see Fig. 1). The total, time-independentHamiltonian reads H tot = H q + H f + V . We have intro-duced the bare qubit Hamiltonian H q = ¯ hω σ + σ − , with σ − = | g (cid:105)(cid:104) e | , σ + = | e (cid:105)(cid:104) g | , and σ z = | e (cid:105)(cid:104) e | − | g (cid:105)(cid:104) g | , where | e (cid:105) (resp. | g (cid:105) ) denotes the qubit excited (resp. ground)state. The 1DWG features a reservoir of electromagneticmodes with bare Hamiltonian H f = (cid:80) ∞ k =0 ¯ hω k b † k b k . Theoperator b † k (resp. b k ) creates (annihilates) one photon offrequency ω k ≥ k = ω k v − ≥
0, where v is the group velocity. The coupling Hamiltonian reads V = ¯ hg (cid:80) k ( ib k σ + + h.c.), where h.c. stands for hermi-tian conjugate and g is the coupling constant, assumeduniform over the modes. Throughout the paper, we shalluse the interaction picture with respect to H f + H q .We define the lowering operator at the position x as [24] a ( x, t ) = (cid:114) D ∞ (cid:88) k =0 e − iω k ( t − x/v ) b k = a (0 , t − x/v ) , (1)where D is the mode density, taken as uniform. Equa-tion (1) evidences that the field operators depend on thesingle variable τ = t − x/v and verify the bosonic commu-tation relations [ a ( x, t ) , a † ( x (cid:48) , t (cid:48) )] = [ a (0 , τ ) , a † (0 , τ (cid:48) )] = δ ( τ − τ (cid:48) ), with τ (cid:48) = t (cid:48) − x (cid:48) /v . Conversely, field and qubitstates evolve under the action of the time-dependent cou-pling Hamiltonian in the interaction picture V ( t ) = i ¯ h √ γσ + ( t ) a (0 , t ) + h.c. , (2)where γ = g D and σ + ( t ) = e iω t σ + .Equations (1) and (2) suggest that the field-qubit dy-namics can be modelled as a series of interactions be-tween the qubit and the units of a conveyor belt, thatare shuttled at the velocity v (see Fig. 1). Our approachbears similarities with the so-called collisional or repeatedinteractions model [25], with the substantial differencethat the dynamics is deduced from first principles andprovides full access to the joint qubit-field state. In theabsence of a scatterer (i.e. for x < x > v . It is thus sufficient to de-scribe the field before and after the qubit through theinput and output operators: a in ( t ) = lim (cid:15) → − a ( (cid:15), t ) and a out ( t ) = lim (cid:15) → + a ( (cid:15), t ). These operators satisfy [26] a (0 , t ) = 12 [ a in ( t ) + a out ( t )] . (3) Solving the dynamical equations gives rise to the meaninput-output relation (see Supplemental Material [27]) (cid:104) a out ( t ) (cid:105) = (cid:104) a in ( t ) (cid:105) − √ γ (cid:104) σ − ( t ) (cid:105) . (4)Since Eq. (4) is derived in the interaction picture, it holdson mean values. It should thus not be confused with thetextbook input-output relation written in Heisenbergrepresentation, that holds for the operators [26]. Reduced dynamical equations.—
In view of studyingthe energy exchanges between qubit and field, we intro-duce the decomposition, valid for any bipartite system, ρ ( t ) = ρ q ( t ) ⊗ ρ f ( t ) + χ ( t ). The reduced density matrix ρ q ( t ) = Tr f { ρ ( t ) } (resp. ρ f ( t ) = Tr q { ρ ( t ) } ) for the qubit(resp. for the field) is computed from the joint quantumstate ρ ( t ) written in the interaction picture, and χ ( t ) isthe correlation matrix. The evolution of the qubit andthe field follow the dynamical equations dρ k dt = − i ¯ h (cid:2) H k ( t ) , ρ k ( t ) (cid:3) − i ¯ h Tr l (cid:54) = k { [ V ( t ) , χ ( t )] } , (5)where k, l ∈ { q, f } , H q ( t ) = i ¯ h √ γ (cid:104) a (0 , t ) (cid:105) σ + ( t ) + h.c.,and H f ( t ) = i ¯ h √ γa (0 , t ) (cid:104) σ + ( t ) (cid:105) + h.c.. Equation (5)evidences that the systems influence each other in twoways. Namely, each system induces an effective drivingterm H k ( t ) in the other system reduced dynamics, whilea non-unitary term results from the build-up of correla-tions between the two systems. It is straightforward toshow that the former term (resp. the latter) is entropy-preserving (resp. is not).We first focus on the qubit evolution. The effectivedrive H q ( t ) can be classically interpreted as resultingfrom the force exerted on the qubit by the local electricfield. This force is proportional to the mean field ampli-tude (cid:104) a (0 , t ) (cid:105) . Employing Eqs. (3) and (4) we write thedrive as H q ( t ) = i ¯ h √ γ (cid:0) (cid:104) a in ( t ) (cid:105) − √ γ (cid:104) σ − ( t ) (cid:105) / (cid:1) σ + ( t ) +h.c., evidencing two components in the force. The first isexerted by the input field initially injected in the 1DWG.The second is exerted by the quantum field radiated bythe qubit itself. Strikingly, the latter remains even inthe absence of input field provided that the mean qubitdipole (cid:104) σ − ( t ) (cid:105) takes non-zero values, i.e., that the qubithas coherences in its bare energy basis. This term wasdubbed radiation reaction force [28, 29] and pinpointedas a fundamental mechanism for spontaneous emission.Conversely, we interpret the effective Hamiltonian H f ( t )as the result of the reciprocal force exerted by the qubiton the field, that is proportional to the mean qubit dipole.An interesting situation is provided by injecting apulse of coherent light in the input port of the 1DWG.We show in [27] that Eq. (5) leads to the same evolutionfor the qubit observables as the Optical Bloch Equations(OBE) at zero temperature. However, the two dynam-ical equations are fundamentally different. Contrary toEq. (5), the driving term of the OBE solely contains theinput field (cid:104) a in ( t ) (cid:105) , while the field radiated by the qubit, √ γ (cid:104) σ − ( t ) (cid:105) , is absorbed in the dissipative part. This isexpected since unlike our framework, the OBE do notprovide any quantum description of the electromagneticfield: it is either treated as a classical drive or as athermal bath. This has important thermodynamicconsequences as we show below. Both descriptions doconverge, however, when the field amplitude is largesuch that |(cid:104) a in ( t ) (cid:105)| (cid:29) γ [27]. This defines the classicallimit of the field, where stimulated emission in thedriving mode largely overcomes spontaneous emission.In this case, the dynamics reduces to classical Rabioscillations and the field-qubit correlations are negligible. Energetic analysis.—
We first notice that the qubit-field system is isolated, therefore the total energy U tot ( t ) = Tr { H tot ρ ( t ) } is conserved along the evolution.It splits into three terms. We define the qubit’s (resp.the field’s) internal energies as U q ( t ) = Tr { H q ρ ( t ) } (resp. U f ( t ) = Tr { H f ρ ( t ) } ), while the coupling energy equals V ( t ) = Tr { V ρ ( t ) } . Below we focus on two general cases:either the input field is in the vacuum, or it is quasi-resonant with the qubit transition and oscillates in phasewith the qubit dipole. We show in [27] that both situa-tions lead to a constantly zero coupling term, V ( t ) = 0.In turn, this implies that ˙ U q ( t ) = − ˙ U f ( t ), where ˙ U k stands for the rate of energy change. This situation con-veys the intuitive picture of two systems exchanging en-ergy through time. We now focus on the nature of theseenergy exchanges.Using Eq. (5), we derive the equations ruling the evo-lution of the qubit and field internal energy rates:˙ U k ( t ) = − i ¯ h Tr k (cid:8) [ H k , H k ( t )] ρ k ( t ) (cid:9) − i ¯ h Tr { H k [ V ( t ) , χ ( t )] }≡ ˙ W k ( t ) + ˙ Q k ( t ) , (6)where k ∈ { q, f } . We have denoted by ˙ W k (resp. ˙ Q k ) theenergy flow induced by the unitary (resp. non-unitary)part of the reduced dynamical equation. For both sys-tems, ˙ W k features an entropy preserving energy change.When the field becomes classical, the quantum correla-tions vanish and ˙ W k is the only remaining energy com-ponent. In this case, stimulated emission prevails andqubit-field energetic exchanges are restricted to work ex-changes. In what follows, we shall therefore dub ˙ W k thework flow.Conversely, the term ˙ Q k features a local change of en-ergy that is accompanied by an entropy change. Further-more, ˙ Q k neither relates to the internal energy change ofa thermal bath [30] nor the energy input by a classicalmeasurement [31]–hence it is not eligible as “heat”. Wefurther refer to ˙ Q k as the correlation energy flow. Weshow in [27] that in the range of situations studied in thepaper, ˙ W q ( t ) + ˙ W f ( t ) = 0, yielding ˙ Q q ( t ) + ˙ Q f ( t ) = 0.Remarkably, the qubit and the field not only exchangeequal amounts of energy but also equal amounts of work.This leads us to define a unique flow ˙ W ( t ) = ˙ W q ( t ) = − ˙ W f ( t ), that features the work received by the qubit andprovided by the field.Employing Eqs. (3) and (4) to expand ˙ W as a functionof the input field and qubit observables we obtain:˙ W ( t ) = ¯ hω (cid:0) √ γ Re[ (cid:104) σ − ( t ) (cid:105)(cid:104) a in ( t ) (cid:105) ∗ ] − γ |(cid:104) σ − ( t ) (cid:105)| (cid:1) . (7)The first term features work exchanges by stimulatedemission, and is the only work flow captured by theOBE [22]. The latter is remarkable since it remains inthe absence of an input field, (cid:104) a in ( t ) (cid:105) = 0, i.e. during thespontaneous emission. On the qubit side, it features the“self-work” of the radiation reaction force. This self-workis always negative since the qubit can only provide en-ergy to the vacuum. On the field side, it corresponds tothe “spontaneous work” already evidenced in [16], thatwas shown to strongly depend on the initial coherence ofthe qubit.It is interesting to rewrite the work flow as a functionof the field observables only:˙ W ( t ) = − ¯ hω (cid:0) |(cid:104) a out ( t ) (cid:105)| − |(cid:104) a in ( t ) (cid:105)| (cid:1) . (8)Equation (8) reveals that the work flow is the energychange of the field coherent component [32], which isaccessible through homodyne or heterodyne measure-ment schemes [14, 33]. It evidences that the amount ofexchanged work is fully encoded in the field state, fromwhich it can be directly measured – in other words,work is an observable. Conversely, the energy change ofthe field incoherent component directly reflects that thequbit and the field got correlated during the interaction. Probing non-classical fields with energetic witnesses.—
From now on we focus on a specific scenario in which apulse of light propagating in the waveguide scatters off aqubit initially prepared in the ground state. We analyzethe transient excitation of the qubit by the pulse as thecharging and discharging of a battery (the qubit) by thefield (the charger). The figure of merit to optimize isthe so-called ergotropy E of the qubit [34], which hasthe simple form E ( t ) = ¯ hω [ r ( t ) + z ( t )] /
2, where r ( t ) = (cid:112)(cid:80) i (cid:104) σ i ( t ) (cid:105) , with i ∈ { x, y, z } , is the radius of the qubitBloch vector, and z ( t ) = (cid:104) σ z ( t ) (cid:105) (see [27]). By definition,the ergotropy quantifies the maximal amount of workthat can be extracted by unitary operations. One easilyshows that it equals the qubit energy when it is in a purestate, and vanishes for thermal states characterized by r = − z . Such zero-ergotropy states define passive states.Conversely, active states are defined by E >
0, which isthe case e.g. for states with z > Q ( t ) = Figure 2. Qubit battery charged by a single-photon pulsebeing the mode-matched inverted exponential pulse. (a)Temporal profile of the input pulse. The field state reads | (cid:105) = (cid:82) −∞ dtα ( t ) a † (0 , t ) | (cid:105) where α ( t ) = √ γe γt/ − iω t and γ is the spontaneous emission rate. (b) Energetic analysis ofthe charging-discharging process of the qubit battery: internalenergy U (dotted black), received work W (solid blue), corre-lation energy Q (dashed red) and ergotropy E (dashed-dottedgreen) as a function of time. Inset: Qubit states (i), (ii), and(iii) in the Bloch representation for the charging process attimes γt ∈ {− , − , } , respectively. γ ( |(cid:104) σ − (cid:105)| − (cid:104) σ + σ − (cid:105) ) ≤
0, see [27]. This implies the fol-lowing chain of inequalities: W ( t ) ≥ U ( t ) ≥ E ( t ) ≥ W ( t ) ≡ (cid:82) t dt (cid:48) ˙ W ( t (cid:48) ) [resp. Q ( t ) ≡ (cid:82) t dt (cid:48) ˙ Q ( t (cid:48) )]. Im-portantly, this shows that a coherent field cannot providemore ergotropy than work to a qubit: below we refer tothis bound as the classical ergotropy bound. The presentsituation is reciprocal of the situation studied in [16],where an initially excited qubit (the charger) is chargingthe waveguide battery. The classical ergotropy boundcompletes the result derived in [16], where the amountof work received by the waveguide field was shown to beupper bounded by the initial ergotropy of the qubit.The situation drastically changes when the input fieldis non-classical, i.e. its statistics is not coherent. A criti-cal case is provided by a single photon pulse, whose tem-poral shape is a mode-matched inverted exponential assketched in Fig. 2(a), see also [27]. Such fields have beentheoretically [21, 35] and experimentally [36–38] shown tolead to complete population inversion of the qubit. Thisappears on Fig. 2(b) where the qubit energetic proper-ties have been plotted as a function of time. In the restof the paper, the origin of time is set by the comple-tion of the charging process, i.e. when the qubit en-ergy and ergotropy reach their maximum. Single-photoninput states do not create any coherence on the qubitstate, see inset of Fig. 2(b). Consequently, no work is Figure 3. Qubit battery charged by a coherent superpositionof the vacuum and a single-photon pulse being the mode-matched inverted exponential pulse | ψ θ (cid:105) (see text), with θ = 0(light-blue), π/ π/ π/ π (red). (a) Qubit energy. (b) Ratios W / U = c θ (solid lines)and E / U (dotted lines). exchanged W ( t ) = 0, the qubit energy being solely fu-eled by the correlation energy Q ( t ). Population inversionprovides ergotropy to the qubit, such that E ( t ) ≥ W ( t ).The violation of the classical ergotropy bound is an en-ergetic witness for the non-classicality of the field. Itcompletes other criteria based on Wigner function nega-tivities [32, 39], and relies on operational quantities thatare experimentally accessible.Remarkably, the relation E ( t ) ≥ W ( t ) remains validfor any coherent superposition of zero and one photonstates as charging fields. Such quantum states can be pro-duced and probed experimentally, see e.g. [17]. We de-note them as | ψ θ (cid:105) = c θ | (cid:105) + s θ | (cid:105) , with c θ = cos( θ/
2) and s θ = sin( θ/ θ ∈ [0 , π ] and | (cid:105) is a single-photonpulse of arbitrary temporal shape. We show in [27] that W ( t ) = | c θ | U ( t ), leading to 0 ≤ W ( t ) ≤ E ( t ) ≤ U ( t ).This behavior is pictured in Fig. 3 for different values of θ . Finally, we focus on the impact of quantum coher-ence on the fueling ability of the field. Namely, wehave compared the qubit ergotropy E provided by thecoherent superposition | ψ θ (cid:105) and by the statistical mix-ture ρ θ = c θ | (cid:105)(cid:104) | + s θ | (cid:105)(cid:104) | , both carrying the sameenergy s θ ¯ hω . Figure 4(a) [resp. 4(b)] depicts the evo-lution of the qubit energetic properties and of the Blochvector when the impinging field is a coherent superpo- Figure 4. Impact of the field quantum coherence on the charg-ing process. Energetic analysis of the charging and discharg-ing process for the charging field being | ψ θ (cid:105) (a), and ρ θ (b),with θ = 5 π/ U (dotted black), W (solid blue), Q (dashedred), and E (dashed-dotted green). Inset: Qubit states (i),(ii), and (iii) in the Bloch representation for the chargingprocess at times γt ∈ {− , − , } , respectively. (c) Maxi-mum ergotropy (ergotropy at t = 0) changing θ for a qubitcharged by | ψ θ (cid:105) (solid line) and ρ θ (dashed line). sition (resp. a mixture) for θ = 5 π/
8. In the formercase, both work and correlation energy are provided tothe qubit, increasing its energy and ergotropy. The workflow gives rise to a rotation of the Bloch vector aroundthe axis defined by the effective drive H q ( t ), whose phaseis set by the phase of the coherent superposition | ψ θ (cid:105) .Conversely in the latter, only the correlation energy flowacts as a source of ergotropy. This source operates onlyif partial population inversion can be induced along theinteraction. Finally, the maximum ergotropy of the qubitis plotted in Fig. 4(c) as a function of θ for the pure andmixed fueling pulses. The effect of coherence clearly ap-pears on the figure, providing another striking feature ofhow quantum resources impact thermodynamic mecha-nisms. Outlooks.—
Our work brings new evidences thatquantum resources impact the energetic behavior ofquantum systems. Originally, they take the form ofoperational energetic witnesses, that complement thewell-known quasi-probability distributions of quantumoptics. The experiments we propose are feasible onstate-of-the-art platforms of circuit QED or integratedphotonics. Beyond the WG-QED scenery, our frame-work opens new avenues to extend the concept of workin the quantum world, building on the existence of a well-defined classical limit for the driving field. Ourapproach can be further used to analyze energy flowsinduced by other kinds of interactions, e.g. modelingmeasurement processes.
Acknowledgments.—
The authors warmly thank G.Landi for his precious comments. This work was sup-ported by the Foundational Questions Institute Fund(Grant number FQXi-IAF19-05), the Templeton WorldCharity Foundation, Inc (Grant No. TWCF0338) andthe ANR Research Collaborative Project “Qu-DICE”(ANR-PRC-CES47). [1] M. Esposito, U. Harbola, and S. Mukamel, Rev. Mod.Phys. , 1665 (2009).[2] M. Campisi, P. H¨anggi, and P. Talkner, Rev. Mod. Phys. , 771 (2011).[3] F. Binder, L. A. Correa, C. Gogolin, J. Anders, andG. Adesso, Thermodynamics in the quantum regime , Fun-damental Theories of Physics, Vol. 195 (Springer, 2018).[4] H. B. Callen,
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