Classical-to-quantum crossover in electron on-demand emission

TL;DR

This paper presents a theoretical mechanism for generating and tuning single-electron wave packets, enabling experimental exploration of the quantum-to-classical transition in electron emission, with implications for quantum technology.

Contribution

It introduces a method to control the quantum nature of emitted electrons from solid-state sources, facilitating studies of the quantum-to-classical crossover.

Findings

Feasibility of tuning electron wave packets demonstrated

Potential for experimental investigation of quantum-to-classical transition

Implications for development of quantum electron devices

Abstract

Emergence of a classical particle trajectory concept from the full quantum description is a key feature of quantum mechanics. Recent progress of solid state on-demand sources has brought single-electron manipulation into the quantum regime, however, the quantum-to-classical crossover remains unprobed. Here we describe theoretically a mechanism for generating single-electron wave packets by tunneling from a driven localized state, and show how to tune the degree of quantumness. Applying our theory to existing on-demand sources, we demonstrate the feasibility of an experimental investigation of quantum-to-classical crossover for single electrons, and open up yet unexplored potential for few-electron quantum technology devices.

Tables

Table 1. Table 1: Time-energy uncertainty budget in different limits for the specific emission model defined by Eq. ( 15 ).

Quantity	$r\ll 1$	$r\gg 1$	$r=1$
${\Delta }_b\rho \left(x=E/{\Delta }_b\right)$	$r^{-1}\exp \left[-r^{-1}{e}^x+x\right]$	$r^{-1}\Theta \left(x\right)e^{-x/r}$	${\left[2\cosh \left(x/2\right)\right]}^{-2}$
$⟨E⟩/{\Delta }_b$	$-{\gamma }_{\mathrm{Euler}}-\left|\ln r\right|$	$r$	0
$\Delta E/{\Delta }_b$	$\pi /\sqrt{6}$	$r$	$\pi /\sqrt{3}$
$\Delta t_Q/\left(\hslash {\Delta }_b^{-1}\right)$	$1/2$	${\left(8r\right)}^{-1/2}$	$1/\left(2\sqrt{3}\right)$
$\Delta t^{\star }/\left(\hslash {\Delta }_b^{-1}\right)$	$\left[{\left(rs\right)}^{-1}+{\left(r\left|\ln r\right|\right)}^{-1}\right]/\left(2\sqrt{6}\right)$	${\left(2\pi \right)}^{-1}{\left[{\left(r^{-1}-s^{-1}\right)}^2+\left({\pi }^2/6-1\right)r^{-2}\right]}^{1/2}$	${\left(a_2-a_1^2+s^{-2}/12\right)}^{1/2}$ 111Here $a_n\equiv \int {\left[2\cosh \left(x/2\right)\right]}^{-2}{\left(\frac{1}{2\pi }\mathrm{Re}\left\{\psi \left(\frac{1}{2}+i\frac{x}{2\pi }\right)\right\}\right)}^n𝑑x$ which can be evaluated Oloa2016 to $a_1=-\left(1+{\gamma }_{\text{Euler}}\right)/\left(2\pi \right)=-0.25102\dots$, $a_2=\left(1+{\gamma }_{\text{Euler}}+{\gamma }_{\text{Euler}}^2/2\right)/\left(2{\pi }^2\right)-1/48=0.067508\dots$, yielding $a_2-a_1^2=-1/48+{\left(2\pi \right)}^{-2}$.
$\Delta E\Delta t/\hslash$	$\pi \left[{\left(rs\right)}^{-1}+{\left(r\left|\ln r\right|\right)}^{-1}\right]/12$	${\left\{\frac{r}{8}\left[1+\frac{2}{{\pi }^{2}s}\left(\frac{r}{s}-2\right)\right]\right\}}^{1/2}$	$\left(\pi /6\right){\left[\frac{3}{4}+\frac{3}{{\pi }^2}+s^{-2}\right]}^{1/2}$

Equations

$${\sigma }_{\mathrm{c}}(t_{\text{e}})=-d{p}_{\mathrm{d}}(t_{\text{e}})/d{t}_{\text{e}}=\gamma (t_{\text{e}})\exp \left[-{\int }_{t_0}^{t_{\text{e}}}\gamma (t^{\prime })d{t}^{\prime }\right],$$

$${\rho }_{\mathrm{c}}(E)={\sigma }_{\mathrm{c}}(t_{\mathrm{e}}(E))\left|d{t}_{\mathrm{e}}(E)/dE\right|,$$

$$W_{\mathrm{c}}(\tau ,ϵ)={\rho }_{\mathrm{c}}(ϵ)\delta (\tau -t_{\mathrm{e}}(ϵ))={\sigma }_{\mathrm{c}}(\tau )\delta (ϵ-E(\tau )),$$

$$\mathcal{H}(t)={\dot{\epsilon }}_{\mathrm{d}}t|d⟩⟨d|+\sum _k{E}_k|k⟩⟨k|+\sum _k{V}_k|k⟩⟨d|+h.c.$$

$$|\Psi (t)⟩=c_{\mathrm{d}}(t)|d⟩+\sum _k{\psi }_k(t)e^{-i{E}_{k}t}|k⟩,$$

$$\psi (ϵ)=e^{i{\varphi }_k}\sqrt{\frac{\Gamma (ϵ)}{{\dot{\epsilon }}_d}}\exp \left[-\frac{i}{\hslash {\dot{ϵ}}_d}{\int }_{E_0}^{ϵ}\left[\Sigma (E)-E\right]dE\right],$$

$$E^{\star }={\dot{\epsilon }}_{\mathrm{d}}{t}^{\star }+\mathrm{Re}\Sigma (E^{\star }).$$

$$\begin{gathered} W(\tau ,ϵ)=\frac{1}{2\pi \hslash }\int \sqrt{\rho \left(ϵ+E/2\right)\rho \left(ϵ-E/2\right)}\times \\ \exp \left\{\frac{i}{\hslash }{\int }_{ϵ-E/2}^{ϵ+E/2}[t^{\star }(E^{\prime })-\tau ]d{E}^{\prime }\right\}dE. \end{gathered}$$

$$\phi (\omega )$$

$$\delta t_{\text{sc}}^3(E)$$

$$\delta t_Q^2(E)$$

$$W_{\mathrm{sc}}(\tau ,ϵ)=\rho (ϵ)\mathrm{Ai}\left([t^{\star }(ϵ)-\tau ]/\delta t_{\text{sc}}(ϵ)\right)/\delta t_{\text{sc}}(ϵ),$$

$$W_{\mathrm{G}}(\tau ,ϵ)=\rho (ϵ)\frac{1}{\delta t_{\text{Q}}(ϵ)\sqrt{2\pi }}\exp \left(-\frac{[t^{\star }(ϵ)-\tau {]}^2}{2\delta t_{\text{Q}}^2(ϵ)}\right).$$

$$\Delta t^2=⟨(t^{\star }-⟨t^{\star }⟩{)}^2⟩+⟨\delta t_Q^2⟩,$$

$$\Gamma (E)={\Gamma }_0{\left(1+e^{-E/{\Delta }_{\mathrm{b}}}\right)}^{-1},$$

$$\rho (E)$$

$$t^{\star }$$

$$c_k(t)\equiv {\psi }_k(t)e^{-i{E}_{k}t}=⟨k|U(t,t_0)|d⟩.$$

$$c_k(t)=\frac{V_k}{2\pi \hslash {\dot{\epsilon }}_{\mathrm{d}}}{\int }_{-\infty }^{\infty }\left[\frac{e^{i\varphi (E)-iEt/\hslash }}{E-E_k+i0}{\int }_{-\infty }^E{e}^{-i\varphi (ϵ)+iϵt_0/\hslash }dϵ\right]dE,$$

$$\varphi (E)=\frac{1}{\hslash {\dot{\epsilon }}_{\mathrm{d}}}{\int }_{E_{\text{min}}}^E\left[E^{\prime }-\sum _q\frac{|V_q{|}^2}{E^{\prime }-E_q+i0}\right]d{E}^{\prime }.$$

$$\varphi (E)=\frac{1}{\hslash {\dot{\epsilon }}_{\mathrm{d}}}{\int }_{E_{\text{min}}}^E\left[E^{\prime }-\Sigma (E^{\prime })\right]d{E}^{\prime },$$

$${\int }_{-\infty }^E{e}^{-i\varphi (ϵ)+iϵt_0/\hslash }dϵ=\sqrt{\frac{2\pi }{|{\varphi }^{\prime \prime }(E_0)|}}{e}^{i\pi /4}{e}^{-i\varphi (E_0)+i{E}_0{t}_0/\hslash },$$

$$c_k(t)=\frac{V_k{e}^{i\pi /4}}{\sqrt{2\pi \hslash {\dot{\epsilon }}_{\mathrm{d}}}}{e}^{-i\varphi (E_0)+i{E}_0{t}_0/\hslash }{\int }_{-\infty }^{\infty }\frac{e^{i\varphi (E)-iEt/\hslash }dE}{E-E_k+i0}.$$

$$c_k(t)$$

$$\Sigma (\omega \pm i0)=\frac{\hslash }{2\pi }{\int }_{-\infty }^D\frac{\Gamma (E)}{\omega -E\pm i0}dE$$

$$=s{\Delta }_b\left[\text{uppsi}\left(\frac{1}{2}\mp i\frac{\omega }{2\pi {\Delta }_b}\right)\mp i\frac{\pi }{2}-\ln \frac{D}{2\pi {\Delta }_b}\right],$$

$$⟨E⟩$$

$$\Delta E$$

$$\Delta t_{\mathrm{Q}}=\frac{\hslash }{2}\sqrt{-⟨\frac{d^2\ln \rho (E)}{d{E}^2}⟩}=\frac{\hslash }{2{\Delta }_{\mathrm{b}}}\sqrt{\frac{1}{2r+1}}.$$

Full text

I Introduction

Single photon on-demand sources have key applications in quantum communication and quantum computation as well as in tests of fundamental properties of quantum mechanics. Large efforts have been directed towards realizing fast and efficient sources, emitting photons in quantum mechanically pure states, one by one Migdall et al. (2013); Lounis and Orrit (2005). Single electron on-demand sources in solid state conductors have during the last decade witnessed a similar development Fève et al. (2007); Blumenthal et al. (2007); Kaestner et al. (2008); Pekola et al. (2008); Fujiwara et al. (2008); Hermelin et al. (2011); Jehl et al. (2013); Dubois et al. (2013); Rossi et al. (2014), largely driven by metrological applications of charge quantization Pekola et al. (2013); Kaestner and Kashcheyevs (2015). However, in recent years fundamental electron quantum optics experiments with on-demand sources have been performed, such as Hanbury-Brown-Twiss partitioning of single Bocquillon et al. (2012); Fletcher et al. (2013) and pairs Ubbelohde et al. (2015) of electrons. In particular, indistinguishability and quantum coherence of generated electron excitations have been demonstrated in seminal experiments via Hong-Ou-Mandel interference Bocquillon et al. (2013); Dubois et al. (2013) and Wigner function tomography Grenier et al. (2011); Jullien et al. (2014).

On-demand quantum particle sources also offer a unique possibility to study the emergence of classical properties with unprecedented degree of control. For photons, the conventional description of light in terms of Maxwell equations makes the probability density of coherent states Glauber (1963); Sudarshan (1963) a natural classical limit Mandel (1986). The quantum-classical transition for photon sources has been experimentally demonstrated Bartley et al. (2013); Vered et al. (2015). For electrons, no classical field limit exists and the appropriate classical notion is that of a point particle on a well-defined trajectory. This is conveniently analysed in terms of the Wigner quasi-probability distribution in phase space Wigner (1932); Case (2008), which approaches a delta-function on the classical trajectory as $\hbar\rightarrow 0$ Berry (1977). Such quantum-classical crossover for individual electrons stands unexplored: experiments with quantum coherent sources have focused so far on fixed-shape wave-functions (either Lorentzian Dubois et al. (2013); Jullien et al. (2014) or exponential Fève et al. (2007); Bocquillon et al. (2012) in time), while measurements on tunable-barriers emitters Fletcher et al. (2013); Ubbelohde et al. (2015) have not yet reached the required temporal and spectral resolution Waldie et al. (2015); Kataoka et al. (2016a, b).

In this paper we show how to design a single-electron source that can be easily tuned between classical and quantum emission regimes. The design (shown schematically in Fig. 1)

is based on an exact solution for tunneling emission from a linearly-driven energy level into an empty conduction band, valid for an arbitrary energy dependence of the tunnel coupling density. We quantify the “quantumness” of the source by the spread of the Wigner function of the emitted wave-packet around its guiding trajectory. To illustrate our results and connect with existing experimental realizations Fletcher et al. (2013); Ubbelohde et al. (2015), we consider a simple example of energy dependences of the tunnel coupling that allows tuning of the emitted wave-packet from a semiclassical double-exponential Leicht et al. (2011) via a minimal-uncertainty-product Gaussian Ryu et al. (2016) to a Lorentzian-in-time Keeling et al. (2008); Jullien et al. (2014) shape by a mere change of the driving rate. Our approach allows for a versatile wave-packet shaping, eliminates the need of phase-matched control signals for emission tuning, and opens new design opportunities for on-demand sources in electron quantum optics.

II On-demand tunnelling emission from a driven quantum level

Initially an electron is localized in a ground state $\left\lvert{d}\right\rangle$ of a sufficiently small quantum dot, separated from an empty band by a high tunnel barrier. On-demand emission is initiated by driving the quantum dot potential up until the electron tunnels out due to increase of the tunnel coupling with energy.

II.1 Classical emission model

In a classical description of the emission Leicht et al. (2011); Fletcher et al. (2013); Waldie et al. (2015), tunneling out at a time $t_{\text{e}}$ creates a propagating electron with a well-defined emission energy $E(t)$. For a dot initially populated at time $t_{0}$, the occupation probability $p_{\mathrm{d}}(t)$ obeys a rate equation $dp_{\mathrm{d}}(t)/dt=-p_{\mathrm{d}}(t)\gamma(t)$ with a time-dependent rate $\gamma(t)$. The resulting distributions of emission times and energies,

[TABLE]

are uniquely determined by the externally controlled $\gamma(t)$ and $E(t)$ [via the inverse $t_{\text{e}}(E)$] Waldie et al. (2015).

For emission into a dispersionless one-dimensional channel, at times $t>t_{\text{e}}$ the electron propagates away from the dot at constant speed $v_{0}$, with a simultaneously well defined position $x(t)=v_{0}(t-t_{\mathrm{e}})$ and momentum $p=E(t_{\mathrm{e}})/v_{0}$. The Markov tunneling process generates a statistical ensemble along a line in position-momentum space which, at a time $t_{\mathrm{f}}\gg t_{\mathrm{e}}$, provides a direct imprint of the electron energy dynamics $E(t)$ in the quantum dot before tunneling. Hence it is convenient to use a time $\tau=-x(t_{\mathrm{f}})/v_{0}+t_{\mathrm{f}}$ and energy $\epsilon=pv_{0}$ as the phase space variables characterizing the emitted electron. The corresponding probability density in phase space,

[TABLE]

is a weighted delta-function along a well-defined trajectory $E(t)$.

II.2 Quantum emission model

A relatively slow drive compared to the emission time-scales can be linearized in time as $\dot{\varepsilon}_{\mathrm{d}}t$. The corresponding Hamiltonian of the quantum emission model is

[TABLE]

Here $\left\lvert{d}\right\rangle(\left\lvert{k}\right\rangle)$ denotes the level (a band state) and $V_{k}$ is the time-independent amplitude for tunneling between the level and a band state with energy $E_{k}$. For energies $E_{k}<E_{0}=\dot{\varepsilon}_{\mathrm{d}}\,t_{0}$ the amplitude $V_{k}\rightarrow 0$. The Shrödinger equation for a single-particle state,

[TABLE]

with initial conditions $c_{\rm d}(t_{0})=1$, $\psi_{k}(t_{0})=0$, and Hamiltonian (4), can be reduced to a single integro-differential equation for the dot amplitude $c_{d}(t)$ Basko (2016). The corresponding conduction band amplitudes are $\psi_{k}(t)=-(i/\hbar)V_{k}\int_{t_{0}}^{t}e^{iE_{k}t^{\prime}/\hbar}c_{d}(t^{\prime})\,dt^{\prime}$. Once the emission is complete, $t>t_{\mathrm{f}}$, $c_{d}(t)\to 0$ and $(dE_{k}/dk)^{-1/2}\psi_{k}(t)\to\psi(E_{k})$, independent of time. The state (5) then describes an electron wave packet freely propagating away from the dot, uniquely determined by the emission protocol via $\dot{\varepsilon}_{d}$, $V_{k}$. The observable energy and time distributions are $\rho(\epsilon)=|\psi(\epsilon)|^{2}$ and $\sigma(\tau)=(2\pi\hbar)^{-1}\lvert\int e^{-iE\tau/\hbar}\psi(E)\,dE\rvert^{2}$ (shifted to origin by $-t_{\text{f}}$), respectively. Note that $\sigma(\tau)$ can be seen as a distribution of single-electron waiting times Brandes (2008) relative to external trigger (first passage).

Equation (4) describes a multi-level Landau-Zener problem, first solved by Demkov and Osherov (DO) Demkov, Y and Osherov, V (1968) for $|\psi_{k}|^{2}$ and later by Macek and Cavagnero (MC) Macek and Cavagnero (1998) for $\psi_{k}(t)$. Taking the continuum-limit of the DOMC solution, as derived in Appendix A, we arrive at

[TABLE]

where $\Gamma(E)=(2\pi/\hbar)\sum_{k}|V_{k}|^{2}\delta(E_{k}-E)$ is the tunnel coupling density, $\Sigma(E)=(\hbar/2\pi)\int\Gamma(E)/(E-E^{\prime}+i0)\,dE^{\prime}=\operatorname{Re}\Sigma(E)-i\hbar\Gamma(E)/2$ coincides with the retarded self-energy of the quantum dot state $\left\lvert{d}\right\rangle$ due to coupling to the lead, and $\phi_{k}=\arg V_{k}+5\pi/4+{\dot{\varepsilon}}_{\mathrm{d}}t_{0}^{2}/\hbar$.

II.3 Quantum-classical correspondence

The energy spectrum $\rho(E)$ computed from Eq. (6) has the same form as $\rho_{\mathrm{c}}(E)$ in Eq. (2) if we use a “naive” identification of the classical parameters, $E(t)=\dot{\varepsilon}_{d}t$ and $\gamma(t)=\Gamma\left(E(t)\right)$, based on the bare (non-renormalized) values for the quantum model. This corresponds to the well-known interpretation of the DO solution Demkov, Y and Osherov, V (1968) as a sequence of independent level-crossing events, each having a (small) probability of adiabatic transition from $E(t)$ to $E_{k}$ dictated by the two-level Landau-Zener formula, $1-\exp[-2\pi|V_{k}|^{2}/(\pi\dot{\varepsilon}_{d})]$. The exact distribution of times, $\sigma(t)$, however, involves the energy-dependent level renormalization $\operatorname{Re}\Sigma(E)$, and thus converges to $\sigma_{c}(t)=\dot{\varepsilon}_{d}\rho_{c}(E(t))$ only in the very restrictive perturbative limit of $\operatorname{Re}\Sigma(E)\to 0$.

Here we propose the following non-perturbative definition of the classical trajectory $(t^{\star},E^{\star})$ for tunneling emission:

[TABLE]

A unique function $t^{\star}(E^{\star})$ is always defined by Eq. (7). The inverse, $E^{\star}(t^{\star})$ can be interpreted as the fully dressed adiabatic energy of the state $\left\lvert{d}\right\rangle$. Note that in case of strong dispersion, $d\operatorname{Re}\Sigma(E)/dE>1$, $E^{\star}(t^{\star})$ becomes multivalued.

To explore the quantum-to-classical crossover, we consider the energy-time Wigner function $W(\tau,\epsilon)=(2\pi\hbar)^{-1}\!\int\psi^{\ast}(\epsilon-E/2)\,\psi(\epsilon+E/2)e^{-iE\tau/\hbar}dE$, which can be written as Note (2)

[TABLE]

The relation to the classical limit is elucidated by a saddle-point-type approximation to $W$ via a power series expansion of $\ln\rho(E)$ and $t^{\star}(E)$ around $E=\epsilon$. This gives $W\!\approx\!\rho(\epsilon)\int e^{\varphi(\omega)}d\omega/(2\pi)$ with

[TABLE]

and

[TABLE]

The Wigner function computed from Eq. (9) is centered on the trajectory line $\tau=t^{\star}(\epsilon)$ in energy-time space. Explicit analytic evaluation of the corresponding $W(\tau,\epsilon)$ is possible in two limiting cases. In the first case, $|\delta t_{\text{sc}}(\epsilon)|\gg|\delta t_{Q}(\epsilon)|$, the integral over $E$ in Eq. (8) is cut by fast phase oscillations due to $t^{\star}(E)$, so that the third order in $\omega$ dominates over the second order in Eq. (9). This gives

[TABLE]

where $\mathrm{Ai}$ is the Airy function. The limit of $W\to W_{\text{sc}}$ corresponds to the semi-classical limit defined by Berry for finite quantum systems Berry (1977). Equation (12) reveals limited quantum fringes on the scale of $|\delta t_{\text{sc}}(\epsilon)|$ on the concave side of the classical trajectory Berry (1977).

In the other analytic limit of the saddle-point approximation, $|\delta t_{\text{sc}}(\epsilon)|\ll|\delta t_{Q}(\epsilon)|$, the guiding trajectory is sufficiently straight to be broadened in the temporal direction by the Fourier transform of $\sqrt{\rho(E)}$. Omitting the term containing $\delta t_{\text{sc}}(\epsilon)$ but keeping $\delta t_{Q}(\epsilon)$ in Eq. (9) amounts to a local Gaussian expansion, $\ln\sqrt{\rho(\epsilon\!+\!E/2)\rho(\epsilon\!-\!E/2)}\approx\ln\rho(\epsilon)-[E\,\delta t_{Q}(\epsilon)/\hbar]^{2}/2$, with at most linear $t^{\star}(E)$, which gives

[TABLE]

In the formal limit of $\hbar\to 0$ both $\delta t_{\text{sc}}(\epsilon),\delta t_{\text{Q}}(\epsilon)\to 0$ and the classical expression for the Wigner function, Eq. (3), is recovered with $t_{\text{e}}(E)=t^{\star}(E)$, $E(t)=E^{\star}(t)$, and $\rho_{c}(E)=\rho(E)$, thus validating our classical trajectory definition (7).

To quantify the contribution of quantum coherence to the overall spread of an emitted wave-packet, we express the second moment, $\Delta t$, of the time distribution in terms of energy averages, $\langle\mathcal{F}(E)\rangle\equiv\int\mathcal{F}(E)\rho(E)\,dE$, as

[TABLE]

The clear separation of $t^{\star}$ contribution motivates us to define a quantumness measure $0\leq\theta\leq 1$ as the fraction of Fourier broadening in the total temporal width, $\theta=\langle{\delta t_{Q}^{2}}\rangle/\Delta t^{2}$.

The semiclassical limit (12) applies only if $|\delta t_{\text{sc}}(\epsilon)|\gg|\delta t_{\text{Q}}(\epsilon)|$ which implies $\theta\ll 1$. The quantum limit, $\theta\to 1$, corresponds to instantaneous emission, $t^{\star}(\epsilon)\approx\text{const}$, with the emission time uncertainty $\Delta t$ minimized down to the Heisenberg limit for a given (e.g., measured) energy spectrum $\rho(E)$. If the latter is globally Gaussian, then $\delta t_{\text{Q}}(\epsilon)=\hbar/(2\Delta E)=\text{const}$ and the Heisenberg uncertainty product, $\Delta E\,\Delta t$, in case of fully quantum emission ($\theta=1$), reaches the Kennard bound Kennard (1927) of $\hbar/2$. For a Gaussian $\rho(E)$ and at most linear trajectory equation $E^{\star}(t)$, the time distribution $\sigma(\tau)$ is also necessarily Gaussian, with the measure of quantumness $\theta=(\hbar/2)^{2}/(\Delta E\,\Delta t)^{2}=1-\mathrm{R}^{2}$ reduced by the amount of time-energy correlations. Here $\mathrm{R}$ is the classical Pearson’s correlation coefficient which is well-defined for $W_{G}>0$. Note that our quantumness measure $\theta$, which can be maximised by $W\geq 0$ (for example, $W=W_{G}$), provides a different non-classicality criterion compared to the Wigner function negativity Kenfack and Życzkowski (2004), recently adapted to wavepackets emitted from single particle sources Haack et al. (2013); Ferraro et al. (2013).

III Example: onset of tunnelling density over a finite energy range

We illustrate the above general results by considering an emitter with the following coupling density,

[TABLE]

which describes a gradual increase of tunneling from zero to a saturation rate $\Gamma_{0}$ over a characteristic energy scale $\Delta_{b}$. Emission regimes for the density in Eq. (15) are determined by two dimensionless parameters, level rise “rapidity” $r=\dot{\varepsilon}_{\mathrm{d}}/(\Gamma_{\mathrm{0}}\Delta_{\mathrm{b}})$, and barrier “sharpness” $s=\hbar\Gamma_{\mathrm{0}}/(2\pi\Delta_{\mathrm{b}})$. Equation (6) gives the energy distribution

[TABLE]

and Eq. (7) gives the guiding trajectory

[TABLE]

where $\uppsi$ is the digamma function and $D\gg\Delta_{b},E$ is the upper cut-off energy of the band. A complete phase diagram in terms of $\theta(s,r)$, with qualitatively distinct emission regimes marked A, B, and C, and selected examples of $W(\tau,\epsilon)$ are plotted in Fig. 2.

The energy distribution (16) is controlled by the rapidity $r$ alone, with emission below ($r\!\ll\!1$, cases b, c in Fig. 2), above ($r\!\gg\!1$, cases e,f) or at the edge $E\!=\!0$ ($r\!=\!1$, case d). The sharpness $s$ controls the shape of the classical trajectory (17) and quantum broadening effects.

For $s\ll 1$, the stationary phase approximation for $W$ applies regardless of $r$, and the level renormalization is reduced to a constant shift [logarithmic term in Eq. (17)], hence the classical emission model is valid. In Fig. 2, this corresponds to region A of the phase diagram (a) and a representative Wigner function (b). We find (see Appendix A) $\Delta E\Delta t/\hbar\sim s^{-1}\max(r,r^{-1})\gg 1$ as expected. For $r,s\ll 1$ [region $A_{\text{D}}$ in Fig. 2(a)] the energy distribution peaks at $E_{\text{e}}=\Delta_{b}\ln r$ and both $\rho(E)$ and $\sigma(t)$ are well-approximated by the double exponential form Leicht et al. (2011), ie. $\rho(E)=-(d/dE)\exp\left[-e^{(E-E_{\mathrm{e}})/\Delta_{\mathrm{b}}}\right]$.

For $s\gtrsim 1$, a classical-to-quantum crossover can be realized by tuning $r$. As $r$ is increased towards $1$, the guiding trajectory bends, and the Wigner function develops fringes [see Fig. 2(c)], in accordance with Eq. (12). The classical model (1) and (2) is still applicable for $r\lesssim 0.1$ and large $s$, but with parameters, renormalised according to Eq. (17): $E(t)=E^{\star}(t)$ and $\gamma(t)=\Gamma[E(t)]dE(t)/d(\dot{\varepsilon}_{\text{d}}t)$.

As the rapidity is tuned to $r\!=\!1$, the energy spectrum becomes symmetric. The corresponding arrival time distribution broadening depends on $s$, and the measure of quantumness equals to $\theta(s,r\!=\!1)\!=\!s^{2}/(1\!+\!1.054\,s^{2})$. For $s\gg 1$, the emitter generates quasi-Gaussian, Heisenberg-limited wave-packets with $\Delta E\,\Delta t\to\sqrt{(4+\pi^{2})/48}\approx 0.538\hbar$, which is close to the tightest possible simultaneous localization in time and energy, see Fig. 2(d) and region B in Fig. 2(a). The corresponding width $\Delta E=\pi\sqrt{3}\Delta_{b}\gg\hbar\Gamma_{0}$ is set by the energy-dependence (but not the absolute value) of $\Gamma(E)$.

Finally, the limit of $\Delta_{b}\to 0$ [$s$, $r\!\to\!\infty$, region C in Fig. 2(a)] corresponds to a sudden onset of emission at a constant rate $\Gamma_{0}$ and is equivalent to the zero-temperature limit of a linearly driven small mesoscopic capacitor Keeling et al. (2008). The energy spectrum is a simple exponential with $\Delta E\!=\!\dot{\varepsilon}_{\mathrm{d}}/\Gamma_{\mathrm{0}}$ while the time distribution crosses over from a Lorentzian at $1\ll r\ll s$ [see Fig. 2(e) and region $C_{\text{L}}$ in Fig. 2(a)] via an oscillating regime at $s\sim r$ [similar to Fig. 2(f)] to an exponential at $1\ll s\ll r$ [region $C_{\text{E}}$ in Fig. 2(a)], in exact accord with Ref. Keeling et al., 2008. Although the overall shape of both $\sigma(t)$ and $\rho(E)$ is exponential for $r\gg s\gg 1$ and thus consistent with the classical relation (2), the quantumness measure $\theta$ drops from $1$ to [math] only at $r\gtrsim s^{2}$, see the boundary between regions $A_{\text{E}}$ and $C_{\text{E}}$ in Fig. 2(a). This is because the quantum contribution to the second moment $\Delta t$ is sensitive to the tails of $\sigma(t)$, and the latter are broadened by a small but non-zero $\Delta_{b}$ (such a regime is beyond the dispersionless model of Ref. Keeling et al., 2008). The product of uncertainties always remains large for large rapidities: $\Delta E\Delta t/\hbar\sim\max(\sqrt{r},r/s)\gg 1$ for $r\gg 1$. Note that in contrast to temperature in Fermi-sea-triggered emitters Fève et al. (2007); Keeling et al. (2008); Bocquillon et al. (2014), finite $\Delta_{b}$ allows for coherent shaping of wave-packets, exemplified by regime $B$ discussed above.

IV Feasibility and generalizations

The single-particle approach adopted in our model for the electron emission is justified for experimental realisations where electron is emitted well above the Fermi energy Fletcher et al. (2013); Ubbelohde et al. (2015); van Zanten et al. (2016). The strong coupling regime (ie. essential renormalization and non-classical emission) is reached via the competition of the tunnel coupling strength with the characteristic scale for its variability in energy (e.g., $\Delta_{b}$ in the example of Sec. III). Both may still be significantly smaller than other energies scales relevant for the localized state physics, such as level spacing, Coulomb charging energy, Kondo scale, superconducting gap etc.

Experiments with single-electron emission from tunable-barrier quantum dots coupled to ballistic edge channels in GaAs Leicht et al. (2011); Fletcher et al. (2013); Ubbelohde et al. (2015); Kataoka et al. (2016a) have recently demonstrated Waldie et al. (2015) $\Delta E\,\Delta t\leq(1.0\,\text{meV})(2.7\,\text{ps})\sim 4.2\hbar$ with contributions to $\Delta t$ due to classical time-energy correlations Kataoka et al. (2016b), putting the quantum limit within reach. In addition to lithographic and electrostatic confinement, individual impurities Roche et al. (2013) or superconductors van Zanten et al. (2016); Basko (2016) may be used to tailor $\Gamma(E)$.

Our general analysis of quantumness in terms of Eqs. (6)–(14) does not rely on the explicit DO solution (6), and applies to an arbitrary coherent wave-packet $\psi(E)=\sqrt{\rho(E)}\exp[i\int^{E}t^{\star}(E^{\prime})dE^{\prime}/\hbar]$ with $\rho(E)$ and $t^{\star}(E)$ derived from a microscopic quantum model, appropriate for a particular barrier and protocol design, e.g., a tuneable barrier with known $\epsilon(t)$ and $V_{k}(t)$ Battista and Samuelsson (2012); Gurvitz (2015); Basko (2016) or a real-space model beyond single-level approximation Ryu et al. (2016). Generalization of the quantumness criterion for on-demand single-particle excitations to mixed states Ferraro et al. (2013) and many-body systems Calzona et al. (2016); Litinski et al. (2016) is a promising avenue for further research.

V Conclusions

We propose the use of a statically structured tunnel coupling density to control the time and energy distribution of coherent electrons emitted on demand. Using an exact non-Markovian solution for spontaneous electron emission from an initially localized pure state into a one-dimensional ballistic channel, we have theoretically demonstrated the feasibility of a crossover from semiclassical to quantum-limited wavepacket emission which can be realized as a function of the driving rate alone using a suitable $\Gamma(E)$.​​ This opens new possibilities for engineering solid-state electron wavepackets that have a broad application potential from basic studies of entanglement in solid state Splettstoesser et al. (2009); Sherkunov et al. (2012); Bocquillon et al. (2013); Ubbelohde et al. (2015) to electronic quantum technology, such as ultafast voltage sampling Johnson et al. (2016).

Acknowledgements.

We acknowledge discussions with G. Fève, J. Rammer, J. Erdmanis, E. Locane, and F. Viñas. This work has been supported by the Latvian Council of Science (V.K., grant no. 146/2012), Swedish Science Foundation (P.S) and NanoLund (V.K.).

Appendix A Continuous limit of DO-MC solution

Here we derive Eq. (6) for the asymptotic amplitude $\psi(E_{k})=(dE_{k}/dk)^{-1/2}\lim\limits_{t\to\infty}\psi_{k}(t)$ for electron emission from a localized state $\left\lvert{d}\right\rangle$ at $t_{0}\to-\infty$ into of a normalized quasi-continuous scattering state $\left\lvert{k}\right\rangle$ with energy $E_{k}$, as defined by Eqs. (4) and (5).

The wave function $\left\lvert{\Psi(t)}\right\rangle$ defined by Eq. (5) in the main text can be written in terms of a time evolution operator $U(t,t_{0})$, with $t\geq t_{0}$ which has been computed by MC for a discrete set of levels with arbitrary $V_{k}$ and $E_{k}$ (multi-level Landau-Zener problem). We are interested in the solution of the initial value problem with $\left\lvert{\Psi(t_{0})}\right\rangle=\left\lvert{d}\right\rangle$, so the required amplitude is

[TABLE]

This quantity is given by Eq. (47) of MC paper, which in our notation reads

[TABLE]

where the complex phase function is defined by

[TABLE]

Taking the continuum limit, the sum over $q$ in turns to an integral over $E$ and $\phi(E)$ becomes

[TABLE]

where $\Sigma(E)$ is the retarded self energy defined in the main text. Note that the value of $c_{k}(t)$ is independent of the lower limit $E_{\text{min}}$ of the energy integral in (21) as long as $E_{\text{min}}$ is smaller than any relevant $E_{k}$, i.e., $\Gamma(E)=0$ and $\phi(E)$ is real for $E<E_{\text{min}}$. Both $\Sigma(E)$ and $\phi(E)$ are holomorphic in the upper half-plane of complex $E$.

In the limit of $t_{0}\rightarrow-\infty$ the integral in the bracket in Eq. (19) can be evaluated exatly by the method of stationary phase. The stationary point energy $E_{0}$ is the solution to $d[\operatorname{Re}\phi(E)-Et_{0}/\hbar]/dE=0$ which is given by $t_{0}=[E_{0}-\operatorname{Re}\Sigma(E_{0})]/\dot{\varepsilon}_{\mathrm{b}}=t^{\star}(E_{0})$. Taking into account that for $t_{0}\rightarrow-\infty$ we have $\Sigma(E_{0})\rightarrow 0$ and hence $E_{0}\rightarrow\dot{\varepsilon}_{\mathrm{d}}t_{0}$, the saddle point evaluation gives

[TABLE]

where $\phi^{\prime\prime}(E_{0})=d^{2}\phi(E)/dE^{2}|_{E=E_{0}}\to 1/(\hbar\dot{\varepsilon}_{\mathrm{d}})$. We thus have the Heisenberg picture solution for the asymptotic initial condition $\left\lvert{\Psi(t_{0}\!\to\!-\infty)}\right\rangle=\left\lvert{d}\right\rangle$,

[TABLE]

In the limit of $t\rightarrow\infty$ we can perform the remaining integral by contour integration. We shift the contour down into the lower half of the complex energy plane, so that it runs parallel to the real axis with a small negative imaginary part $-i|\eta|$ to the integration variable $E$. The value of $\eta$ is chosen such that the pole at $E=E_{k}-i0$ is enclosed but none of the poles of $e^{i\phi(E)}$ are. For $t\rightarrow\infty$, the value of the integrand along the shifted contour is exponentially suppressed as $e^{-|\eta|t}$ and can be neglected. The integral in Eq. (23) thus evaluates to $-2\pi e^{i\phi(E_{k})-iE_{k}t/\hbar}$, and the expression for $c_{k}(t)$ becomes

[TABLE]

Taking into account the definitions (21) and (18), one recognizes (24) as Eq. (6) of the main text. Note that up to the initial phase factor, the amplitude $\psi_{k}$ is the scattering matrix element for $\left\lvert{d}\right\rangle\to\left\lvert{k}\right\rangle$ transition with the time-dependent scattering potential defined by $V_{k}$ and $\dot{\varepsilon}_{\mathrm{d}}(t-t_{0})$.

Appendix B Analytic results for the specific emission model

Here we provide a derivation of the time and energy distribution parameters for the barrier with energy dependent rate $\Gamma(E)=\Gamma_{\mathrm{0}}[1+\exp(E/\Delta_{\mathrm{b}})]^{-1}$ presented in Eq. (15) of the main text. In particular, we derive explicit results for different limits of the rapidity $r=\dot{\varepsilon}_{\mathrm{d}}/(\Gamma_{\mathrm{0}}\Delta_{\mathrm{b}})$ and the sharpness $s=\hbar\Gamma_{\mathrm{0}}/(2\pi\Delta_{\mathrm{b}})$.

The self-energy function $\Sigma(z)$ for this special case is

[TABLE]

with exponential accuracy for $D-\omega\gg\Delta_{b}$. Equation (25) leads directly to the trajectory equation given by  (17) of the main text.

The emission energy distribution $\rho(E)$ is given by computing $|\psi(E)|^{2}$ to which only the imaginary part of $\Sigma(E)$ contributes. The specific form (15) of the latter can be integrated analytically which yields Eq. (16). The first two central moments of the energy distribution can also be computed explicitly,

[TABLE]

where $\gamma_{\mathrm{Euler}}=0.577\ldots$ is the Euler constant and $\uppsi_{1}(z)=d\uppsi(z)/dz$ is the trigamma function.

For computing the time uncertainty $\Delta t$, we note that the Fourier transform of $\sigma(\tau)=\int W(\tau,\epsilon)\,d\epsilon$ is by definition the generating function for the moments, $\langle t^{n}\rangle_{t}$, where $n=0,1,2\ldots$ and $\langle\cdots\rangle_{t}=\int\cdots\sigma(t)\,dt$. With this argument, the general expression (8) for the Wigner function leads to Eq. (14) where $(\Delta t)^{2}\equiv\langle t^{2}\rangle_{t}-\langle t\rangle_{t}^{2}=(\Delta t^{\star})^{2}+(\Delta t_{\mathrm{Q}})^{2}$ is expressed via $(\Delta t^{\star})^{2}=\langle{[t^{\star}(E)-\langle{t^{\star}(E)}\rangle]^{2}}\rangle$ and $(\Delta t_{\mathrm{Q}})^{2}=\langle{\delta t_{Q}(E)^{2}}\rangle$.

For the specific $\Gamma(E)$ given by Eq. (15), the quantum contribution to time-broadening can be evaluated analytically for any $r$,

[TABLE]

The semiclassical contribution $\Delta t^{\star}$ involves integrals $\langle(\operatorname{Re}\Sigma(E)-E)^{n}\rangle_{E}$ for $n=1,2$ which we could not perform analytically for arbitrary $r,s$. Asymptotic limits of the moments of time and energy distributions for $r\to 0,1$, and $\infty$ are summarized in Table 1. These results have been used in the description of different emission regimes in the main text.