Mitigating decoherence in hot electron interferometry

Due to their high energy, hot electrons in quantum Hall edge states can be considered as single particles that have the potential to be used for quantum optics-like experiments. Unlike photons, however, electrons typically undergo scattering processes in transport, which results in a loss of coherence and limits their ability to show true quantum behaviour. Here we study the decoherence of hot electrons in a Mach-Zehnder interferometer, and show that by optimising experimental parameters and employing energy filtration it is possible to minimise decoherence and achieve visibilities over $95\%$ in an interferometer with $10\mu$m arms. This represents a significant improvement over Fermi-level electron quantum optics, and proposes hot-electron charge pumps as an ideal platform for realising coherent nanoelectronic devices.

The realisation of quantum optics experiments with electrons is a long-standing pursuit of the mesoscopic physics community [1,2]. In a typical architecture, quantum Hall edge (QHE) channels form the electronic analogue of photonic waveguides and quantum point contacts act as beamsplitters. A number of classic optical experiments have been realised in this way: Hanbury-Brown-Twiss [3] and Hong-Ou-Mandel [4] experiments, as well as the focus of this work, the Mach-Zehnder interferometer (MZI) [5][6][7][8][9].
Recent advances in electron pumps [19], however, have led to technology able to inject single high-energy electrons into edge channels well above the Fermi sea at a high rate and accuracy [20][21][22][23][24][25][26][27][28]. These hot electrons are well separated, both energetically and spatially, from those in the Fermi sea, and it has been suggested that this leads to a significant reduction in the Coulomb scattering of injected electrons [29]. Indeed, in Ref. [30] it was observed above a certain energy, electron transport was close to ballistic, indicating a suppression of Coulomb interactions.
Within this new hot-electron regime, however, phonon emission becomes a significant relaxation process, as has been discussed theoretically [31,32] and observed experimentally [30,33]. Moreover, quantum optics-like experiments require electrons to maintain their quantum coherence throughout the device and, in an interferometer, the emission of a phonon reveals "which-way" informa-  1. (a) Schematic hot-electron MZI with a charge pump source and an electron wave packet moving in a QHE channels that is split and recombined by QPC beamsplitters. Before arriving at the detector, the current Iout is partially blocked by a potential barrier of height E b . This energy filtration removes decohered electrons and enhances visibility. (b) Dispersion relation of outermost QHE-channel electrons with indication of dominant relaxation processes. At low injection energy, E0, electrons close to the Fermi sea relax mostly through plasmon creation. At higher energies, the electron-electron interactions are suppressed due to the large separation between guide centre yG (transverse average position) and the Fermi sea. At these high energies, phonon emission becomes the dominant decay mechanism. tion, destroying this coherence.
In this letter, we calculate relaxation and decoherence rates of a hot electron in a MZI geometry, see Fig. 1a, and find their dependence on parameters such as injection energy and magnetic field. This gives a theoretical underpinning to the suppression of electron-electron interactions and the dominance of phonon channels for hot electrons. We then consider decoherence and show that, through a careful choice of experimental parameters and arXiv:2003.06574v1 [cond-mat.mes-hall] 14 Mar 2020 Mean free paths of hot electrons (defined as the ratio of mean injection velocity over the relevant decay rate, l0 = v0/Γ), for the indicated inelastic processes as a function of magnetic field B. Results are shown for mean injection energies of E0 = 50, 100, 150meV above the band bottom with an arm length of 10µm shown for comparison. At low energy, the LADP and TAPZ phonons are dominant. As injection energy is increased, LO phonons become more significant at the low magnetic field. Only at the very lowest magnetic field and energy, does the plasmon rate become comparable with acoustic phonon ones. Parameters were a transverse confinement strength ωy = 2.7meV, z-confinement length a = 3nm [34] and a Fermi-energy of 10meV.
also the addition of an energy filter, we can minimise decoherence and maximise interferometer visibility. By so doing, we find that visibilites of 95% are attainable in hot-electron interferometers with arm lengths of 10µm. The techniques that we outline here to preserve quantum coherence thus pave the way for scalable coherent nanoelectronic devices with single-electron sources.
Inelastic processes:-We begin by discussing the relative strengths of the various inelastic mechanisms at work here, see Fig. 1b, by calculating total scattering rates with Fermi's golden rule for screened-Coulomb and Fröhlich Hamiltonians for outermost edge-state electrons only [34]. The results, expressed as mean free paths and compared with a putative MZI arm length of 10µm, are shown in Fig. 2.
Electron-electron interactions, which in 1D are dominated by the excitation of plasma modes [15,17,18], are seen to be negligible for the relevant energies ( 50meV) and fields ( 5T) here. This can be understood by looking at the quasi-1D Coulomb interaction, V (q) ∼ K 0 (|qδ G |), where K 0 (x) is the modified Bessel function of the second kind [17,35], q is the momentum exchange and δ G is the difference in guide centre of the interacting electrons. At high injection energy, the separation between the hot electron and those in the Fermi sea (see Fig. 1b) is large and the Bessel function rapidly decreases, thus resulting in low interaction rates [34].
With electron-electron interactions suppressed, phonon emission comes to dominate, and here there are several contributing processes, Fig. 2. The electron may emit a longitudinal-optical (LO) phonon, which are approximately dispersionless with energy 36meV in GaAs [36]. The relaxation rate due to these phonons increases with higher injection energy and decreases with magnetic field [31]. The hot electron may also emit longitudinal acoustic phonons via either of the deformation potential (LADP) or piezoelectric field interactions (LAPZ), or transverse phonons via piezoelectric interaction (TAPZ). Figure 2 shows that, whereas the LAPZ interaction is always negligible, LADP and TAPZ interactions are not. Indeed at high magnetic field they are the dominant inelastic processes, and this results in a transition in the dominant scattering mechanism from optical to acoustic phonons as magnetic field increases. Despite their similar mean free paths, energy relaxation due to acoustic phonons is dominated by the LADP interaction because the typical energy loss per phonon is much greater that for TAPZ phonons [32].
These results are consistent with the observations in Ref. [30], where little evidence of electron-electron interactions were observed for energies above 50meV from the band bottom. Moreover, our results suggest that the deviations from ballistic transport of electrons in Ref. [30] might be explained by acoustic phonon emission rather than electron-electron interactions.
Decoherence:-We next consider the effect of phonons on the MZI, which for simplicity we assume to be symmetric. From the above considerations, we neglect electron-electron interactions and work in a regime where LO emission is suppressed (but see later on this point). As both LADP and TAPZ rates are comparable and since either process has the capacity to provide 'which-way' information, we include all acoustic-phonon processes. We assess the effect of decoherence with the help of a quantum master equation with rates derived from a Frohlich Hamiltonian modified to include emission from interferometer arms separated by a distance D [34]. This allows us to calculate the time dependence of the probability densities P αβ (E) ≡ k δ (E − E k ) Tr a αk ρa † βk , where ρ is the density matrix of the hot electron and a αk is an electron annihilation operator of an edge-channel electron with wavenumber k in arm α = 1, 2. In the symmetric case, we have probability density P 11 (E) = P 22 (E) and, with the initial conditions we consider here, the inter-arm coherence P 12 (E) = P 21 (E) and Im [P 12 ] = 0.
The results of our master equation calculation are shown in Fig. 3, where the initial state of the electron (after the first beamsplitter) is a Gaussian wave packet, of width σ in energy space, [37,38] that is fully coherent between the two arms. Assuming zero dispersion across the wave packet (justified since σ E 0 ), we obtain the position of the center of the wavepacket along the interferometer arm as is the electron velocity at the injection energy. Fig. 3 shows that, as the wave packet travels, the population density P 11 (E) relaxes and also broadens slighty due to the stochastic nature of phonon emission [32]. In contrast, the coherence P 12 (E) shows no significant drift or diffusion, but rather shrinks in amplitude as time goes on.
This behaviour can be understood with an extension of Ref. [32] to include inter-arm coherences. In this approach, elements P αβ (E) each obey a Fokker-Planck equatioṅ For the diagonal elements, A αα = 0, v αα = − d Γ( ) represents the drift velocity of the electron wave packet in energy space and D αα = 1/2 d 2 Γ( ) is its diffusion coefficient. Here, Γ( ) is the relaxation rate resolved as a function of energy loss, , which is approximately independent of initial energy over the energy scales considered here [32]. Meanwhile, the coefficients in the equation for the decay rates of the inter-arm coherences. Approximate expressions for the rates can be found using the saddlepoint method, which reveals that the ratio of population- all acoustic phonon types. For realistic devices, D l Ω , such that Γ( ) Γ ( ) and therefore v 12 ≈ D 12 ≈ 0, while A 12 ≈ d Γ( ). Thus, the inter-arm coherence is effectively stationary with an amplitude that decays as P 12 (t) ≈ P 12 (0) exp (−A 12 t), with A 12 the total population relaxation rate.
The visibility of interference fringes is directly related to the density matrix elements P αβ . Indeed, in the symmetric case we find that the visibility at a particular energy is simply given by the ratio V(E) = P 12 (E)/P 11 (E) [34]. Fig. 3 shows that this energy-resolved visibility is close to 1 at the high-energy end of the wavepacket, but suppressed at low-energy because the electrons here have an increased likelihood that they have undergone scattering.
Visibility enhancement:-We consider detection of interference signal past the second quantum point contact by the mean current past a potential barrier of height E b that blocks electrons with outgoing energy E < E b . We model this with a filtration function F (E − E b ), centred on an energy E b where F (x) = 1 for x 0 and F (x) = 0 for x 0. The total visibility from the current detected past the barrier is [34]. For simplicity, we assume a sharp barrier, such that F (x) = θ(x), the Heaviside step function. Since lower energy electrons will be more likely to have emitted phonons and hence have lost coherence, the barrier thus preferentially filters out decoherent electrons, meaning raising the barrier height increases the visibility at the detector. The trade-off for this is that the overall current is reduced.
We consider four strategies for the placement of this barrier and choice of injection energy and magnetic field strength, the visibilities for which are shown in Fig. 4. In scheme (a) we set the barrier in the window E 0 − ω LO +4σ E b E 0 −4σ such that electrons which have emitted LO phonons and therefore definitely contribute nothing to the visibility, are blocked. For each value of B, we set the injection energy to minimise the total amount of scattering of all the decay processes that we consider. This effectively corresponds to the crossing of the LO phonon/plasmon rates with the LADP rates shown in Fig. 2. This gives a visibility up to 94.2% at 4T, although this rapidly drops with increasing B. The loss of current in the low magnetic field regime, where the visibility is highest is low ( 10% loss, see Fig. 4 inset). In scheme (b) we choose E 0 and B parameters as above but raise the potential barrier to the injection energy E b = E 0 . This filters out not only those electrons that have emitted LO phonons but also those in the lower energy portion of the wave packet in Fig. 3. Since the electrons that pass the filter are more coherent (higher energy-resolved visibility) this increases V tot up to a value of 96.2% at low magnetic field strength but with signal losses of order 60%.
In scheme (c) we set the barrier as in (a) but choose to accept higher loss of signal due to LO phonons. This allows us to operate at higher energies where acoustic phonons are more suppressed. The results in Fig. 4 are obtained with signal loss of 90%, which increases the visibility up to 96.9% at 4T. This visibility decreases more rapidly than scheme (b) and as such becomes worse at high magnetic field strengths. The final scheme (d) is a combination of schemes (b) and (c), where we set E b = E 0 and choose an injection energy to give a LO loss of 90%. This gives a very high visibility, up to 98%, but results in a signal with around 95% loss. While there is only a small difference in the best visibilities from this scheme with (c), it performs significantly better at high B (∼ 86% rather than ∼ 77% at B = 15T).
Conclusions:-In this letter we have presented an overview of the mechanisms by which hot electrons from Population and coherence densities, P11(E) and P12(E), as well as the energy-resolved visibility of a hot electron injected at E0 = 100meV as a Gaussian wave packet of width σ = 1meV. The four panels show results at the times indicated when the centre of the electron wave packet has travelled a distance of x = 0, 3, 7, and 10µm from the first beamsplitter. At x = 0, the electron is fully coherent and P12 = P11. But as the electron traverses the interferometer, decoherence reduces P12 relative to P11, and suppresses the energy-resolved visibillity at the low-energy side of the wave packet. Parameters as in Fig. 2 with B = 12T. dynamical quantum dot charge pumps undergo relaxation and decoherence. Electron-electron interactions play only a very small role in hot-electron regime, and electrons that have undergone LO phonon emission are energetically isolated and easily filtered out. Thus, it is the emission of acoustic phonons that emerges as the dominant factor in limiting coherence in hot-electron quantum optics experiments.
Concerning the MZI, we have identified a series of strategies based on parameter selection and energyfiltering that can be used to enhance the interferometer visibility. Through altering the relative strength of optical to acoustic phonon emission and blocking electrons below a certain energy, these schemes allow us to sacrifice signal for visibility in various ways, and we have shown that visibilities of ∼ 98% are achievable in this way. One further way in which the visibility might be enhanced (or losses reduced, for the same visibility) is to modify the transverse confinement of the electrons. In particular, decreasing the confinement in the plane of the two-dimensional electron gas is known to decrease acoustic phonon rates [32] whilst leaving those of LO phonons largely unaltered. Increasing the in-plane confinement has a more complicated effect but generally, it further increases visibility at the expense of greater signal loss from LO-phonon emission.
While undesirable, a high signal loss may be acceptable in hot-electron quantum optics due to the high rate of Total MZI visibility for the four parameter and energy-filtration schemes described here, with the inset showing the corresponding fraction of electrons arriving at the detector. Whilst the total visibility drops as a function of magnetic field due to the increasing acoustic-phonon emission rate, it can be significantly enhanced by filtering out the lower-energy more incoherent electrons. This filtration moves to higher energy and hence becomes more effective as we step through schemes a to d. Parameters as in Fig. 2 with MZI arm length of 10µm.
operation of the electron pumps [20]. It should also be noted that we have based these results on a MZI with arm length of 10µm, which is roughly twice the length of those within previous experiments [5][6][7][8], and despite this, we still predict a significantly higher visibility than those previously observed. Our results hence illustrate the potential of hot-electrons as a platform for coherent electronic devices, as well as providing insight into the optimal conditions in which to conduct such experiments.
Acknowledgements: This research was supported by EPSRC Grant No. EP/P034012/1. M.K. was supported by the UK Department for Business, Energy, and Industrial Strategy, and by 17FUN04 SEQUOIA. This project has received funding from the European Metrology Programme for Innovation and Research (EMPIR) program, cofinanced by the Participating States, and from the European Unions Horizon 2020 Research and Innovation Program.

Model
In this letter we consider electrons confined in QHE states with transport in the x direction. We assume strong confinement in the z-direction and weaker harmonic confinement in the y direction with frequency ω y . The relevant quantum numbers are the wavenumber in transport direction, k, Landau level m and quantum number n describing the z-direction confinement. The Hamiltonian describing the single particle states is with energies, measured relative to the bottom of the lowest subband, given by [39] E nmk = n + m Ω + 1 2 with effective mass m * e and elementary charge e. The energy n is from the energy level in the z-direction with 1 = 0, while Ω is the effective confinement frequency given by Ω = ω 2 y + ω 2 c , with cyclotron frequency ω c = |eB/m * e |. The guide centre y G (k) describes the mean y co-ordinate of a confined electron and is given by (4) Within this study, we consider only electrons in the outermost subband (n = 1, m = 0), as most of the dynamics are expected to occur here [32]. The wavefunction describing such a state is with where we have assumed a triangular well confinement in the z-direction [40] and H 0 (x) is the 0 th -order Hermite polynomial. We model electron-electron interactions with the Hamiltonian with annihilation (creation) operators a (a † ) transferring electrons with wavevectors k and p to k + q and p − q. This is mediated by the quasi-1D screened Coulomb potential W c (q, ω) [15,17,18,35] and V c is the bare quasi-1D Coulomb potential, while (q, ω) is the dielectric function. Here K 0 is the zeroth order modified Bessel function of the second kind and we take q to be the momentum exchange in the x-direction. The frequency ω corresponds to the energy exchanged in this process.
We evaluate the decay rates via Fermi's golden rule. The direct interaction of a quasi 1D system is highly supressed for a high energy injected electron. This is due to the only allowed interaction of this kind being a swap of the injected quasiparticle with one within the Fermi sea, requiring q to be large. Furthermore, the spatial separation between a hot electron and the Fermi sea is large. As such, the interaction element v(γ) is extremely small, resulting in a negligibly small rate. The main source of relaxation through the Coulomb interaction is hence through plasmonic excitations. For the high energy regime that we consider here, the momentum transfer q is small [41], although the separation here is still large. This reuslts in a more significant interaction, though generally still weak compared to phononic processes, as shown in Fig. 2 in the main body.
To model the phonon relaxation processes within the system, we take a Fröhlich Hamiltonian [42,43] of in the interaction picture, where a hot electron is transferred from the QHE state k to k and creates/absorbs a phonon of type γ ∈ {LO, LADP, LAPZ, TAPZ}, with momentum exchange q with matrix element Λ γ kk [32]. We then extend the above to describe an electron within the interferometer. We model this situation by assuming that the two arms of the interferometer are separated only in the y-direction (perpendicular to propagation) by a distance D, which we will approximate as constant. This results in a phase difference in the phonon field seen by the electron in each of the two arms. The resulting Hamiltonian reads This is a reasonable model to assume as only for D ∼ l Ω will we see deviations from this, and these occur only for a short time in the overall evolution. In the limiting case D = 0 and that we consider only one arm, the standard Fröhlich Hamiltonian is restored, resulting in the same form as in Eq. (10). We proceed to calculate the relevant decay rates in the same way as in Ref. [32].

Interferometer visibility
In order to quantify the sensitivity of the interferometer, we consider the visibility, defined as with I max/min the maximum/minimum signal intensity measured in an output port. Suppose a single particle of energy E is injected into an interferometer, constructed with a pair of identical 50:50 beamsplitters, with density matrix ρ = |1 1|, with |α corresponding to a particle in arm x. After passing through the first beamsplitter and traversing the length of the interferometer before arriving at the second beamsplitter, the density matrix describing the particle is where ϕ is the phase of arm 2 relative to arm 1 and describes the amount of decoherence between the two arms that has occurred (ε = 1 for no decoherence, ε = 0 for complete decoherence). Accordingly, ε can be expressed as ε = ρ αβ /ρ αα for {α, β} ∈ [1, 2]. After passing through the second beamsplitter, the density matrix becomes The maximum and minimum intensities can be determined by looking at the diagonal elements of Eq. (14). Computing this and substituting into Eq. (12), we find that V = ε for both arms in an interferometer with 50:50 beamsplitters.
Considering an electron like those that we consider in this letter, the particle does not have a finite energy. However, for each value of E the energy resolved visibility is still defined as with P αβ (E) the probability density of population/coherence with arm indicies α, β, either from a full solution of the master equation or from the Fokker-Planck terms in Eq. (1). We may also consider the total visibility V tot , which we define as the weighted sum of all the energy-resolved visibilities with the corresponding population density at that point In the proposed optimisation schemes, we cut off part of the injected Gaussian wavepacket by placing a potential barrier of height E b in order to maximise the total visibility of the signal. In such a case, the calculation of the total visibility requires a renormalisation. Accordingly, the total visibility becomes where F (E − E b ) is an increasing function in the interval [0, 1] that acts as a filtration function due to the potential barrier. In the limit that the potential barrier is a sharp barrier with maximum perfect efficiency, the filtration function is the Heavisdie step function. In such a case, the total visibility becomes We assume this quantity when calculating V tot in our results. In obtaining optimisation schemes for the total visibilities described in the main body, we determine an optimum injection energy as a function of magnetic field based on the criteria of each proposal, as plotted in Fig. 5. The general result here is that we should operate at higher injection energies when using high magnetic fields.