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Real-Time Decoherence of Landau and Levitov Quasiparticles in Quantum Hall Edge Channels 1
D. Ferraro,1,2,3 B. Roussel,1 C. Cabart,1 E. Thibierge,1 G. Fève,4 Ch. Grenier,5 and P. Degiovanni1,*
Université de Lyon, Fédération de Physique A.-M. Ampère, CNRS, Laboratoire de Physique de l’Ecole Normale Supérieure de Lyon, 46 Allée d’Italie, 69364 Lyon Cedex 07, France 2 Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France 3 Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France 4 Laboratoire Pierre Aigrain, Ecole Normale Supérieure, CNRS, Université Pierre et Marie Curie, Université Paris Diderot, 24 Rue Lhomond, 75231 Paris Cedex 05, France 5 Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland (Received 6 April 2014; revised manuscript received 1 September 2014; published 16 October 2014) Quantum Hall edge channels at integer filling factor provide a unique test bench to understand the decoherence and relaxation of single-electron excitations in a ballistic quantum conductor. In this Letter, we obtain a full visualization of the decoherence scenario of energy (Landau)- and time (Levitov)-resolved single-electron excitations at filling factor ν ¼ 2. We show that the Landau excitation exhibits a fast relaxation followed by spin-charge separation whereas the Levitov excitation only experiences spin-charge separation. We finally suggest to use Hong-Ou-Mandel-type experiments to probe specific signatures of these different scenarios. DOI: 10.1103/PhysRevLett.113.166403
PACS numbers: 71.10.Pm, 73.23.-b, 73.43.-f
The recent demonstration of on-demand single-electron sources in quantum Hall edge channels [1–3] or twodimensional electron gas [4,5] has opened a new era of quantum coherent electronics. The celebrated HanburyBrown–Twiss and Hong-Ou-Mandel [6] experiments have been demonstrated at the single-electron level [7,8], thus, opening the way to electron quantum optics [9,10]. However, this emerging field goes beyond a simple analogy with photon quantum optics. First of all, the Fermi statistics of electrons differs from the Bose statistics of photons, thus, leading to the Fermi sea, a state with no analogue in photon quantum optics. Moreover, electrically charged electrons experience Coulomb interactions that, despite screening, induce strong decoherence effects as demonstrated by Mach-Zehnder interferometry experiments [11–15]. Moreover, an in-depth study of electronic relaxation at filling factor ν ¼ 2 has shown that a description of the quantum Hall edge channels in terms of Landau quasiparticle excitations is not valid [16,17]. Although these results have been obtained with stationary sources and by measuring time-averaged quantities, the recent demonstration of the electronic Hong-Ou-Mandel (HOM) experiment [8] calls for a time-resolved approach to single-electron coherence taking into account the finite duration of the electronic excitation. The aim of this Letter is precisely to discuss the decoherence scenario of two single-electron excitations emitted by state of the art sources. First, the Landau quasiparticle corresponds to a Lorentzian wave packet in energy emitted by a quantum dot [1]. Second, the Levitov quasiparticles or levitons [4] are single-electron excitations
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obtained by applying a Lorentzian time-dependent potential with quantized flux [18]. Contrarily to the Landau excitations, which are emitted well above the Fermi energy, levitons are emitted close to the Fermi energy. We show that comparing the real-time aspects of their decoherence brings a better understanding of the underlying mechanisms of electronic decoherence whose specific features could be experimentally tested in HOM experiments. In the ν ¼ 2 quantum Hall edge channel system, strong interchannel interactions [19] lead to fractionalization [20–22]: collective excitations generated by a timedependent voltage pulse such as levitons split into two half-charged pulses, one corresponding to the fast moving eigenmode carrying the electric charge while the other corresponds to the slow neutral mode carrying spin [23]. Recent finite frequency admittance [24] and noise measurements [25] have unraveled these charged and neutral collective modes at ν ¼ 2, confirming the existence of spincharge separation and, thus, indirectly, of fractionalization. But is this simple image still valid when considering an arbitrary single-electron excitation? In particular, can fractionalization be observed for a Landau quasiparticle? Looking at the time-dependent average current, the answer is certainly yes: at strong interchannel coupling, the associated current pulse is expected to split into two fractionally charged current pulses propagating at the slow and fast velocities [20–23]. Recent time-resolved measurements of electrical currents have also confirmed this picture for two counterpropagating edge channels [26,27]. On the other hand, energy relaxation does not show such a splitting: the energy-resolved excitation disappears into a
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wave of electron-hole excitations [28,29]. Here, we reconcile these two apparently discordant points of view by working within the Wigner function approach [30], thus, gaining access to both the time course and energy content of single-electron coherence. To this end, we consider the relaxation of an arbitrary single-electron excitation described by a many-body state R jφe ; Fi ¼ φe ðxÞψ † ðxÞjFidx involving an arbitrary normalized electronic wave packet φe ðxÞ above the Fermi sea jFi. This excitation is injected into an interaction region as depicted in Fig. 1. The quantity of interest is the singleðeÞ electron coherence Gρ;x ðt; t0 Þ ¼ hψ † ðx; t0 Þψðx; tÞiρ at position x and in the time domain [9,31]. The Wigner function defined as [30] Z τ τ ðeÞ ðeÞ W ρ;x ðt; ωÞ ¼ vF Gρ;x t þ ; t − eiωτ dτ ð1Þ 2 2 provides a very convenient real-valued representation for ðeÞ Gρ;x ðt; t0 Þ. It describes the nature of excitations (electron like for ω > 0 and hole like for ω < 0) as well as real-time aspects: both the electron distribution function f e ðωÞ and the average electrical current hiðx; tÞi are marginal distributions of the Wigner function, respectively obtained by averaging over time or integrating over ω [30]. The Wigner function contains a Fermi sea contribution ensuring that W ðeÞ ðt; ωÞ → 1 when ω → −∞. To illustrate the concept, we have depicted on Fig. 2(a) ðeÞ the excess Wigner function ΔW ρ;x ðt; ωÞ defined by subtracting the Fermi sea distribution function for a Landau single-electron excitation of duration τe ¼ γ −1 e injected at an energy ℏωe above the Fermi level. Since ωe =γ e ¼ 10, the excitation is well separated from the Fermi sea. The corresponding current pulse appears in Fig. 4(a) whereas the Lorentzian peak of the excess electron distribution function δf e ðωÞ is shown in Fig. 4(c). By contrast, the excess Wigner function of a Levitov excitation depicted in Fig. 3(a) is not well separated from the Fermi sea. Both of them are purely electronic: they vanish for ω < 0. Let us now send such a coherent single-electron excitation through an interaction region of finite length as
FIG. 1 (color online). A single-electron excitation prepared in a coherent wave packet φe enters a finite-length interaction region. The edge channel comes out entangled with the environmental degrees of freedom. In the case of two edge channels, the second channel plays the role of the environment.
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shown in Fig. 1. Within this region, electrons experience screened Coulomb interactions with neighboring edge channels, other mesoscopic conductors, and gates. In many experimentally relevant cases, such effective interactions can be described in terms of edge-magnetoplasmon scattering [32–34], which are charge density waves along the edge channel. In this simple picture depicted on Fig. 1, an incoming sinusoidal charge density wave at pulsation ω generates excitations at the same frequency in the environmental modes through the capacitive coupling between the edge channel and its electromagnetic environment. In the present Letter, we focus on the case of the ν ¼ 2 edge channel system with short-range interactions, described by a Luttinger liquid Hamiltonian of two coupled copropagating edge channels [19]. Solving the bosonic field equations of motion for the Luttinger liquid model [34] gives the edge-magnetoplasmon scattering matrix (see the Supplemental Material [35] which includes Ref. [36]). The evolution of the many-body state jφe ; Fi can then be completely described in terms of the edge-magnetoplasmon transmission amplitude tðωÞ. This suggests that interactions are most conveniently handled in the frequency domain. Our main result is the nonperturbative computation of the outgoing single-electron coherence in terms of the incoming wave packet φe and of the interactions. Introducing ω and Ω respectively conjugated to t − t0 and ðeÞ ðt þ t0 Þ=2 in Gout ðt; t0 Þ, the outgoing coherence in the frequency domain is obtained as Ω Ω ~GðeÞ ;ω − out ω þ 2 2 Z þ∞ Ω 0 Ω 0 0 ¼ φ~ ω − dω0 ð2Þ Kðω; ω ; ΩÞφ~ e ω þ 2 e 2 −∞ in which the propagator Kðω; ω0 ; ΩÞ encodes the effect of interactions on single-electron excitations and φ~ e ðωÞ denotes the Fourier transform of φe ð−vF tÞ. Contrary to the expressions obtained for energy-resolved singleelectron excitations [28], Eq. (2) and the analytical expressions for Kðω; ω0 ; ΩÞ given in the Supplemental Material [35] (with Ref. [37]) describe both the time and energy dependence of electronic relaxation. They provide the exact solution to the relaxation and decoherence problem for any incoming single electronic wave packet and for generic effective screened Coulomb interactions. Figure 2 presents numerical evaluations of the outgoing Wigner function for an incoming Landau quasiparticle of energy ℏωe and duration τe ¼ γ −1 e for various propagation distances or, equivalently, times of flight. All results are obtained at zero temperature, and videos depicting these decoherence scenarios are provided as Supplemental Material [35]. These results show that the decoherence scenario of the Landau quasiparticle involves two time scales. For an energy-resolved excitation (γ e ≪ ωe ), the
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ðeÞ
FIG. 2 (color online). Excess Wigner function ΔW out ðt; ωÞ at various propagation times for a Landau quasiparticle of lifetime τe emitted at energy ℏωe ¼ 10ℏτ−1 e . The horizontal dashed line at ω ¼ 0 is the Fermi level. Plots correspond to increasing propagating lengths expressed in terms of times of flight: τs for the slow mode and τc ¼ τs =20 corresponds to the fast mode. The time shift τ¯ ¼ ðτc þ τs Þ=2 compensates for the global drift of the excitations. Panel (a) shows the initial excess Wigner function. Panels (b) to (f) shows the excess Wigner function for increasing slow mode times of flight.
single-electron coherence relaxes close to the Fermi level after a time of flight proportional to ω−1 e . Then, after a time of flight proportional to the wave packet duration ðeÞ −1 γ −1 e ≫ ωe , ΔW out ðt; ωÞ splits into two parts progressing at the velocities of the two edge-magnetoplasmon eigenmodes, thus, giving birth to collective excitations close to the Fermi sea. The first phenomenon is associated with energy relaxation probed by Le Sueur et al. [17] whereas the second one corresponds to the expected fractionalization arising from the charged or neutral mode separation probed by Bocquillon et al. [24]. This decoherence scenario should be compared to the one for the Levitov quasiparticle shown in Fig. 3: the leviton fractionalizes into two half-levitons, which are Lorentzian current pulses carrying a charge −e=2. The only time scale appearing is the time needed to fractionalize a leviton [23]. The difference between these scenarios comes from the nature of the incoming many-body states. The leviton is an
edge-magnetoplasmon coherent state or, equivalently, a quasiclassical charge density wave [23]. Since the interaction region acts as a frequency-dependent beam splitter for the edge magnetoplasmons, the leviton many-body state does not entangle with its environment: it is a pointer state [38] that does not experience decoherence. The Wigner function changes shown in Fig. 3 come from electron-hole pair generation in this pure many-body state. On the other hand, the many-body state corresponding to the Landau quasiparticle is a superposition of such pointer states. Entanglement with the second edge channel leads to its decoherence. Depending on the copropagation distance, the outgoing many-body state is a partly decohered mixture of coherent states, each of them corresponding to a localized electronic excitation dressed by a cloud of electron-hole pairs (see the Supplemental Material [35]). This extrinsic decoherence leads to the rapid decay of the Wigner function at the initial injection energy ℏωe.
ðeÞ
FIG. 3 (color online). Excess Wigner function ΔW out ðt; ωÞ of a single electronic leviton of width τ0 for various propagation times; τs (respectively τc ) denotes the time of flight of the spin (respectively charged) mode within the interaction region, and τc ¼ τs =20 and τ¯ ¼ ðτs þ τc Þ=2. Panel (a) shows the initial excess Wigner function whereas panels (b) and (c) shows the same quantity for increasing slow mode times of flight.
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Therefore, the rapid electronic relaxation shown on Fig. 2 appears as the electronic analogue of the decay of interference fringes expected for the Wigner function of the superposition of two coherent states of an electromagnetic mode recently observed in cavity QED experiments [39]. Here, it arises from the decoherence of a mesoscopic superposition of quasiclassical charge density waves. This process takes place over a shorter time than the evolution of each of these quasiclassical states, which corresponds to spin-charge separation. Once extrinsic decoherence has taken place, the outgoing many-body state is an incoherent mixture of fractionalized localized electronic excitations. We confirm this scenario by computing both the current pulse and the electron distribution function corresponding to the Wigner functions of Fig. 2: the decay of the quasiparticle peak takes place at short times [Fig. 4(c)] while the current pulses are almost unseparated [Fig. 4(a)] and no hole excitations are created, thus, confirming that it is a purely extrinsic decoherence effect. As the two half-charge current pulses split [Fig. 4(b)], hole excitations are created [Fig. 4(d)]. For typical duration of the wave packet generated by single-electron sources [8] τe ≃ 60 ps and typical velocity of the slow mode [24] vs ≃ 4.6 × 104 m s−1 , energy relaxation takes place over a propagation distance smaller than vs τe ≃ 2.8 μm. Therefore, without any decoherence or relaxation preserving design [15,40], the single-electron coherence measured after a few μm propagation corresponds to collective excitations close to the Fermi sea such as the one depicted in Fig. 2(f). This decay of singleelectron coherence is responsible for the reduction of contrast in Mach-Zenhder interferometry experiments [19,41]. Moreover, interactions also lead to a reduction of the Pauli dip in HOM experiments [42], as recently observed [8].
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Since the experimental signal in a Hong-Ou-Mandel interferometer is directly related to the overlap of the two incoming excess Wigner functions [30], HOM experiments ðeÞ can efficiently probe ΔW out ðt; ωÞ with subnanosecond time resolution. First, the recently predicted side peak structures ðeÞ in the HOM signal [42] reflect the splitting of ΔW out ðt; ωÞ with increasing propagation distance, as can be seen in Figs. 2(d)–2(f). But more generally, using, for example, narrow Lorentzian pulses [43] and a dc bias, a HOM-like experiment would probe the precise time and frequency ðeÞ dependence of ΔW out ðt; ωÞ. Antilevitons (Lorentzian pulses carrying charge þe) could probe both the time and freðeÞ quency dependence of ΔW out ðt; ωÞ for ω < 0, and the corresponding HOM signal may exhibit interaction-induced electron-hole coherence contributions [45]. Finally, the extrinsic decoherence manifests itself through a striking phenomenon: since it suppresses contributions arising from the initial coherence φe ðtþ Þφe ðt− Þ at different times tþ ≠ t− , ðeÞ ΔW out ðt; ωÞ only depends on jφe ðtÞj2 . Consequently, it only depends on the shape of the incoming current pulse and not on its injection energy. This striking feature can be easily tested in a HOM experiment. At fixed γ e , the HOM curve should not depend on the injection energy of the electron when ωe ≳ γ e . However, the precise shape of this curve depends on the effective interactions [24]. To conclude, by computing the Wigner function of a single-electron excitation in the ν ¼ 2 edge channel system, we have shown that, although current pulses always fractionalize into two half-charged pulses for large enough τs , the Landau and Levitov quasiparticle decoherence scenarios deeply differ because of many-body extrinsic decoherence effects. To get an insight on this many-body effect, we propose to use HOM interferometry to probe the incoherent mixture of fractionalized localized electronic excitations induced by Coulomb interactions on a Landau quasiparticle. This would shed light on the decoherence mechanisms and effective interactions at a much lower experimental cost than the generic tomography protocols based on HBT [46] or Mach-Zenhder interferometry [47]. We thank J. M. Berroir, E. Bocquillon, V. Freulon, and B. Plaçais as well as C. Bauerle for useful discussions. We also thank T. Jonckheere, T. Martin, J. Rech, and C. Wahl for discussions about their work and for very useful remarks on our manuscript. This work is supported by the Agence Nationale pour la Recherche ANR Grant “1shot” (No. ANR-2010-BLANC-0412).
FIG. 4 (color online). The average current hiðtÞi in units of −e=τe [(a) and (b)] and the excess electron distribution function ðeÞ δfe ðωÞ [(c) and (d)] corresponding to ΔW out ðt; ωÞ presented on Fig. 2. The initial hiðtÞi and δfe ðωÞ values appear as solid line black curves.
*
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[25] H. Inoue, A. Grivnin, N. Ofek, I. Neder, M. Heiblum, V. Umansky, and D. Mahalu, Phys. Rev. Lett. 112, 166801 (2014). [26] H. Kamata, N. Kumada, M. Hashisaka, K. Muraki, and T. Fujisawa, Nat. Nanotechnol. 9, 177 (2014). [27] E. Perfetto, G. Stefanucci, H. Kamata, and T. Fujisawa, Phys. Rev. B 89, 201413(R) (2014). [28] P. Degiovanni, Ch. Grenier, and G. Fève, Phys. Rev. B 80, 241307(R) (2009). [29] S. Takei, M. Milletarì, and B. Rosenow, Phys. Rev. B 82, 041306(R) (2010). [30] D. Ferraro, A. Feller, A. Ghibaudo, E. Thibierge, E. Bocquillon, G. Fève, Ch. Grenier, and P. Degiovanni, Phys. Rev. B 88, 205303 (2013). [31] G. Haack, M. Moskalets, and M. Büttiker, Phys. Rev. B 87, 201302(R) (2013). [32] I. Safi and H. Schulz, in Quantum Transport in Semiconductor Submicron Structures (Kluwer Academic Press, Dordrecht, The Netherlands, 1995), p. 159. [33] I. Safi, Eur. Phys. J. D 12, 451 (1999). [34] P. Degiovanni, Ch. Grenier, G. Fève, C. Altimiras, H. le Sueur, and F. Pierre, Phys. Rev. B 81, 121302(R) (2010). [35] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.113.166403 for details. [36] P. Degiovanni et al., See Supplemental Material of Ref. [34]. [37] I. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products, 5th ed. (Academic Press, New York, 1994). [38] W. H. Zurek, S. Habib, and J. P. Paz, Phys. Rev. Lett. 70, 1187 (1993). [39] S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, Nature (London) 455, 510 (2008). [40] C. Altimiras, H. le Sueur, U. Gennser, A. Cavanna, D. Mailly, and F. Pierre, Phys. Rev. Lett. 105, 226804 (2010). [41] C. Neuenhahn and F. Marquardt, Phys. Rev. Lett. 102, 046806 (2009). [42] C. Wahl, J. Rech, T. Jonckheere, and T. Martin, Phys. Rev. Lett. 112, 046802 (2014). [43] Minimizing the partition noise [44] or controlling the harmonics of the drive ensures that undistorted levitons arrive at the QPC. [44] J. Dubois, T. Jullien, Ch. Grenier, P. Degiovanni, P. Roulleau, and D. C. Glattli, Phys. Rev. B 88, 085301 (2013). [45] T. Jonckheere, J. Rech, C. Wahl, and T. Martin, Phys. Rev. B 86, 125425 (2012). [46] Ch. Grenier, R. Hervé, E. Bocquillon, F. D. Parmentier, B. Plaçais, J. M. Berroir, G. Fève, and P. Degiovanni, New J. Phys. 13, 093007 (2011). [47] G. Haack, M. Moskalets, J. Splettstoesser, and M. Büttiker, Phys. Rev. B 84, 081303 (2011).
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Real-Time Decoherence of Landau and Levitov Quasiparticles in Quantum Hall Edge Channels 1
D. Ferraro,1,2,3 B. Roussel,1 C. Cabart,1 E. Thibierge,1 G. Fève,4 Ch. Grenier,5 and P. Degiovanni1,*
Université de Lyon, Fédération de Physique A.-M. Ampère, CNRS, Laboratoire de Physique de l’Ecole Normale Supérieure de Lyon, 46 Allée d’Italie, 69364 Lyon Cedex 07, France 2 Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France 3 Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France 4 Laboratoire Pierre Aigrain, Ecole Normale Supérieure, CNRS, Université Pierre et Marie Curie, Université Paris Diderot, 24 Rue Lhomond, 75231 Paris Cedex 05, France 5 Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland (Received 6 April 2014; revised manuscript received 1 September 2014; published 16 October 2014) Quantum Hall edge channels at integer filling factor provide a unique test bench to understand the decoherence and relaxation of single-electron excitations in a ballistic quantum conductor. In this Letter, we obtain a full visualization of the decoherence scenario of energy (Landau)- and time (Levitov)-resolved single-electron excitations at filling factor ν ¼ 2. We show that the Landau excitation exhibits a fast relaxation followed by spin-charge separation whereas the Levitov excitation only experiences spin-charge separation. We finally suggest to use Hong-Ou-Mandel-type experiments to probe specific signatures of these different scenarios. DOI: 10.1103/PhysRevLett.113.166403
PACS numbers: 71.10.Pm, 73.23.-b, 73.43.-f
The recent demonstration of on-demand single-electron sources in quantum Hall edge channels [1–3] or twodimensional electron gas [4,5] has opened a new era of quantum coherent electronics. The celebrated HanburyBrown–Twiss and Hong-Ou-Mandel [6] experiments have been demonstrated at the single-electron level [7,8], thus, opening the way to electron quantum optics [9,10]. However, this emerging field goes beyond a simple analogy with photon quantum optics. First of all, the Fermi statistics of electrons differs from the Bose statistics of photons, thus, leading to the Fermi sea, a state with no analogue in photon quantum optics. Moreover, electrically charged electrons experience Coulomb interactions that, despite screening, induce strong decoherence effects as demonstrated by Mach-Zehnder interferometry experiments [11–15]. Moreover, an in-depth study of electronic relaxation at filling factor ν ¼ 2 has shown that a description of the quantum Hall edge channels in terms of Landau quasiparticle excitations is not valid [16,17]. Although these results have been obtained with stationary sources and by measuring time-averaged quantities, the recent demonstration of the electronic Hong-Ou-Mandel (HOM) experiment [8] calls for a time-resolved approach to single-electron coherence taking into account the finite duration of the electronic excitation. The aim of this Letter is precisely to discuss the decoherence scenario of two single-electron excitations emitted by state of the art sources. First, the Landau quasiparticle corresponds to a Lorentzian wave packet in energy emitted by a quantum dot [1]. Second, the Levitov quasiparticles or levitons [4] are single-electron excitations
0031-9007=14=113(16)=166403(5)
obtained by applying a Lorentzian time-dependent potential with quantized flux [18]. Contrarily to the Landau excitations, which are emitted well above the Fermi energy, levitons are emitted close to the Fermi energy. We show that comparing the real-time aspects of their decoherence brings a better understanding of the underlying mechanisms of electronic decoherence whose specific features could be experimentally tested in HOM experiments. In the ν ¼ 2 quantum Hall edge channel system, strong interchannel interactions [19] lead to fractionalization [20–22]: collective excitations generated by a timedependent voltage pulse such as levitons split into two half-charged pulses, one corresponding to the fast moving eigenmode carrying the electric charge while the other corresponds to the slow neutral mode carrying spin [23]. Recent finite frequency admittance [24] and noise measurements [25] have unraveled these charged and neutral collective modes at ν ¼ 2, confirming the existence of spincharge separation and, thus, indirectly, of fractionalization. But is this simple image still valid when considering an arbitrary single-electron excitation? In particular, can fractionalization be observed for a Landau quasiparticle? Looking at the time-dependent average current, the answer is certainly yes: at strong interchannel coupling, the associated current pulse is expected to split into two fractionally charged current pulses propagating at the slow and fast velocities [20–23]. Recent time-resolved measurements of electrical currents have also confirmed this picture for two counterpropagating edge channels [26,27]. On the other hand, energy relaxation does not show such a splitting: the energy-resolved excitation disappears into a
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wave of electron-hole excitations [28,29]. Here, we reconcile these two apparently discordant points of view by working within the Wigner function approach [30], thus, gaining access to both the time course and energy content of single-electron coherence. To this end, we consider the relaxation of an arbitrary single-electron excitation described by a many-body state R jφe ; Fi ¼ φe ðxÞψ † ðxÞjFidx involving an arbitrary normalized electronic wave packet φe ðxÞ above the Fermi sea jFi. This excitation is injected into an interaction region as depicted in Fig. 1. The quantity of interest is the singleðeÞ electron coherence Gρ;x ðt; t0 Þ ¼ hψ † ðx; t0 Þψðx; tÞiρ at position x and in the time domain [9,31]. The Wigner function defined as [30] Z τ τ ðeÞ ðeÞ W ρ;x ðt; ωÞ ¼ vF Gρ;x t þ ; t − eiωτ dτ ð1Þ 2 2 provides a very convenient real-valued representation for ðeÞ Gρ;x ðt; t0 Þ. It describes the nature of excitations (electron like for ω > 0 and hole like for ω < 0) as well as real-time aspects: both the electron distribution function f e ðωÞ and the average electrical current hiðx; tÞi are marginal distributions of the Wigner function, respectively obtained by averaging over time or integrating over ω [30]. The Wigner function contains a Fermi sea contribution ensuring that W ðeÞ ðt; ωÞ → 1 when ω → −∞. To illustrate the concept, we have depicted on Fig. 2(a) ðeÞ the excess Wigner function ΔW ρ;x ðt; ωÞ defined by subtracting the Fermi sea distribution function for a Landau single-electron excitation of duration τe ¼ γ −1 e injected at an energy ℏωe above the Fermi level. Since ωe =γ e ¼ 10, the excitation is well separated from the Fermi sea. The corresponding current pulse appears in Fig. 4(a) whereas the Lorentzian peak of the excess electron distribution function δf e ðωÞ is shown in Fig. 4(c). By contrast, the excess Wigner function of a Levitov excitation depicted in Fig. 3(a) is not well separated from the Fermi sea. Both of them are purely electronic: they vanish for ω < 0. Let us now send such a coherent single-electron excitation through an interaction region of finite length as
FIG. 1 (color online). A single-electron excitation prepared in a coherent wave packet φe enters a finite-length interaction region. The edge channel comes out entangled with the environmental degrees of freedom. In the case of two edge channels, the second channel plays the role of the environment.
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shown in Fig. 1. Within this region, electrons experience screened Coulomb interactions with neighboring edge channels, other mesoscopic conductors, and gates. In many experimentally relevant cases, such effective interactions can be described in terms of edge-magnetoplasmon scattering [32–34], which are charge density waves along the edge channel. In this simple picture depicted on Fig. 1, an incoming sinusoidal charge density wave at pulsation ω generates excitations at the same frequency in the environmental modes through the capacitive coupling between the edge channel and its electromagnetic environment. In the present Letter, we focus on the case of the ν ¼ 2 edge channel system with short-range interactions, described by a Luttinger liquid Hamiltonian of two coupled copropagating edge channels [19]. Solving the bosonic field equations of motion for the Luttinger liquid model [34] gives the edge-magnetoplasmon scattering matrix (see the Supplemental Material [35] which includes Ref. [36]). The evolution of the many-body state jφe ; Fi can then be completely described in terms of the edge-magnetoplasmon transmission amplitude tðωÞ. This suggests that interactions are most conveniently handled in the frequency domain. Our main result is the nonperturbative computation of the outgoing single-electron coherence in terms of the incoming wave packet φe and of the interactions. Introducing ω and Ω respectively conjugated to t − t0 and ðeÞ ðt þ t0 Þ=2 in Gout ðt; t0 Þ, the outgoing coherence in the frequency domain is obtained as Ω Ω ~GðeÞ ;ω − out ω þ 2 2 Z þ∞ Ω 0 Ω 0 0 ¼ φ~ ω − dω0 ð2Þ Kðω; ω ; ΩÞφ~ e ω þ 2 e 2 −∞ in which the propagator Kðω; ω0 ; ΩÞ encodes the effect of interactions on single-electron excitations and φ~ e ðωÞ denotes the Fourier transform of φe ð−vF tÞ. Contrary to the expressions obtained for energy-resolved singleelectron excitations [28], Eq. (2) and the analytical expressions for Kðω; ω0 ; ΩÞ given in the Supplemental Material [35] (with Ref. [37]) describe both the time and energy dependence of electronic relaxation. They provide the exact solution to the relaxation and decoherence problem for any incoming single electronic wave packet and for generic effective screened Coulomb interactions. Figure 2 presents numerical evaluations of the outgoing Wigner function for an incoming Landau quasiparticle of energy ℏωe and duration τe ¼ γ −1 e for various propagation distances or, equivalently, times of flight. All results are obtained at zero temperature, and videos depicting these decoherence scenarios are provided as Supplemental Material [35]. These results show that the decoherence scenario of the Landau quasiparticle involves two time scales. For an energy-resolved excitation (γ e ≪ ωe ), the
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ðeÞ
FIG. 2 (color online). Excess Wigner function ΔW out ðt; ωÞ at various propagation times for a Landau quasiparticle of lifetime τe emitted at energy ℏωe ¼ 10ℏτ−1 e . The horizontal dashed line at ω ¼ 0 is the Fermi level. Plots correspond to increasing propagating lengths expressed in terms of times of flight: τs for the slow mode and τc ¼ τs =20 corresponds to the fast mode. The time shift τ¯ ¼ ðτc þ τs Þ=2 compensates for the global drift of the excitations. Panel (a) shows the initial excess Wigner function. Panels (b) to (f) shows the excess Wigner function for increasing slow mode times of flight.
single-electron coherence relaxes close to the Fermi level after a time of flight proportional to ω−1 e . Then, after a time of flight proportional to the wave packet duration ðeÞ −1 γ −1 e ≫ ωe , ΔW out ðt; ωÞ splits into two parts progressing at the velocities of the two edge-magnetoplasmon eigenmodes, thus, giving birth to collective excitations close to the Fermi sea. The first phenomenon is associated with energy relaxation probed by Le Sueur et al. [17] whereas the second one corresponds to the expected fractionalization arising from the charged or neutral mode separation probed by Bocquillon et al. [24]. This decoherence scenario should be compared to the one for the Levitov quasiparticle shown in Fig. 3: the leviton fractionalizes into two half-levitons, which are Lorentzian current pulses carrying a charge −e=2. The only time scale appearing is the time needed to fractionalize a leviton [23]. The difference between these scenarios comes from the nature of the incoming many-body states. The leviton is an
edge-magnetoplasmon coherent state or, equivalently, a quasiclassical charge density wave [23]. Since the interaction region acts as a frequency-dependent beam splitter for the edge magnetoplasmons, the leviton many-body state does not entangle with its environment: it is a pointer state [38] that does not experience decoherence. The Wigner function changes shown in Fig. 3 come from electron-hole pair generation in this pure many-body state. On the other hand, the many-body state corresponding to the Landau quasiparticle is a superposition of such pointer states. Entanglement with the second edge channel leads to its decoherence. Depending on the copropagation distance, the outgoing many-body state is a partly decohered mixture of coherent states, each of them corresponding to a localized electronic excitation dressed by a cloud of electron-hole pairs (see the Supplemental Material [35]). This extrinsic decoherence leads to the rapid decay of the Wigner function at the initial injection energy ℏωe.
ðeÞ
FIG. 3 (color online). Excess Wigner function ΔW out ðt; ωÞ of a single electronic leviton of width τ0 for various propagation times; τs (respectively τc ) denotes the time of flight of the spin (respectively charged) mode within the interaction region, and τc ¼ τs =20 and τ¯ ¼ ðτs þ τc Þ=2. Panel (a) shows the initial excess Wigner function whereas panels (b) and (c) shows the same quantity for increasing slow mode times of flight.
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Therefore, the rapid electronic relaxation shown on Fig. 2 appears as the electronic analogue of the decay of interference fringes expected for the Wigner function of the superposition of two coherent states of an electromagnetic mode recently observed in cavity QED experiments [39]. Here, it arises from the decoherence of a mesoscopic superposition of quasiclassical charge density waves. This process takes place over a shorter time than the evolution of each of these quasiclassical states, which corresponds to spin-charge separation. Once extrinsic decoherence has taken place, the outgoing many-body state is an incoherent mixture of fractionalized localized electronic excitations. We confirm this scenario by computing both the current pulse and the electron distribution function corresponding to the Wigner functions of Fig. 2: the decay of the quasiparticle peak takes place at short times [Fig. 4(c)] while the current pulses are almost unseparated [Fig. 4(a)] and no hole excitations are created, thus, confirming that it is a purely extrinsic decoherence effect. As the two half-charge current pulses split [Fig. 4(b)], hole excitations are created [Fig. 4(d)]. For typical duration of the wave packet generated by single-electron sources [8] τe ≃ 60 ps and typical velocity of the slow mode [24] vs ≃ 4.6 × 104 m s−1 , energy relaxation takes place over a propagation distance smaller than vs τe ≃ 2.8 μm. Therefore, without any decoherence or relaxation preserving design [15,40], the single-electron coherence measured after a few μm propagation corresponds to collective excitations close to the Fermi sea such as the one depicted in Fig. 2(f). This decay of singleelectron coherence is responsible for the reduction of contrast in Mach-Zenhder interferometry experiments [19,41]. Moreover, interactions also lead to a reduction of the Pauli dip in HOM experiments [42], as recently observed [8].
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Since the experimental signal in a Hong-Ou-Mandel interferometer is directly related to the overlap of the two incoming excess Wigner functions [30], HOM experiments ðeÞ can efficiently probe ΔW out ðt; ωÞ with subnanosecond time resolution. First, the recently predicted side peak structures ðeÞ in the HOM signal [42] reflect the splitting of ΔW out ðt; ωÞ with increasing propagation distance, as can be seen in Figs. 2(d)–2(f). But more generally, using, for example, narrow Lorentzian pulses [43] and a dc bias, a HOM-like experiment would probe the precise time and frequency ðeÞ dependence of ΔW out ðt; ωÞ. Antilevitons (Lorentzian pulses carrying charge þe) could probe both the time and freðeÞ quency dependence of ΔW out ðt; ωÞ for ω < 0, and the corresponding HOM signal may exhibit interaction-induced electron-hole coherence contributions [45]. Finally, the extrinsic decoherence manifests itself through a striking phenomenon: since it suppresses contributions arising from the initial coherence φe ðtþ Þφe ðt− Þ at different times tþ ≠ t− , ðeÞ ΔW out ðt; ωÞ only depends on jφe ðtÞj2 . Consequently, it only depends on the shape of the incoming current pulse and not on its injection energy. This striking feature can be easily tested in a HOM experiment. At fixed γ e , the HOM curve should not depend on the injection energy of the electron when ωe ≳ γ e . However, the precise shape of this curve depends on the effective interactions [24]. To conclude, by computing the Wigner function of a single-electron excitation in the ν ¼ 2 edge channel system, we have shown that, although current pulses always fractionalize into two half-charged pulses for large enough τs , the Landau and Levitov quasiparticle decoherence scenarios deeply differ because of many-body extrinsic decoherence effects. To get an insight on this many-body effect, we propose to use HOM interferometry to probe the incoherent mixture of fractionalized localized electronic excitations induced by Coulomb interactions on a Landau quasiparticle. This would shed light on the decoherence mechanisms and effective interactions at a much lower experimental cost than the generic tomography protocols based on HBT [46] or Mach-Zenhder interferometry [47]. We thank J. M. Berroir, E. Bocquillon, V. Freulon, and B. Plaçais as well as C. Bauerle for useful discussions. We also thank T. Jonckheere, T. Martin, J. Rech, and C. Wahl for discussions about their work and for very useful remarks on our manuscript. This work is supported by the Agence Nationale pour la Recherche ANR Grant “1shot” (No. ANR-2010-BLANC-0412).
FIG. 4 (color online). The average current hiðtÞi in units of −e=τe [(a) and (b)] and the excess electron distribution function ðeÞ δfe ðωÞ [(c) and (d)] corresponding to ΔW out ðt; ωÞ presented on Fig. 2. The initial hiðtÞi and δfe ðωÞ values appear as solid line black curves.
*
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