Motivated by recent experiments [29, 30], we now consider time-dependent ex-citations of the Fermi sea. In the experiments, Lorentzian-shaped voltage pulses applied to an ohmic contact led to the creation of clean single particle excitations (levitons), following a theoretical proposal by Levitov and co-workers [26, 27, 28].
Similar excitations can be generated by applying a slow linear drive to a quantum capacitor [24, 146]. A third strategy, which applies to our tight-binding chain, is to modulate the phase of the tunneling amplitude between neighboring sites as we show in App. A.3. In this case, the time-dependent part of the Hamiltonian reads
Hˆ0(t) = −tQPC(|1, Lih1, R|+h.c.)−¯tX
{j,α}
eiφjα(t)|j, αihj−1, α|+h.c.
, (3.36) where the rst term corresponds to the (static) coupling of the leads due to the QPC, and the second term describes the creation of levitons at site j in lead α = L, R.
The sum runs over the sites{j, α} where we wish to create levitons. As we show in App. A.3, the phase of the tunneling amplitude should be chosen as
φjα(t) = 2 arctan
t−tjα τ
+π, (3.37)
0.45
Figure 3.7: Creation of levitons. We show the time-evolution of the particle density n(x) along a tight-binding chain with a QPC placed at x = 0. A leviton and an anti-leviton are created by modulating the phase of the hopping amplitude in the middle of the left lead according to Eq. (3.37). Panel a) shows the leviton and the anti-leviton emerging out of the Fermi sea. In panel b) the two quasi-particles can be seen propagating in opposite directions. The red and green lines indicate their Lorentzian density proles. In panel c), the leviton scatters on the QPC tuned to half transmission, while the anti-leviton is reected at the end of the chain. Panel d) shows the transmitted and reected parts of the leviton, which again have Lorentzian density proles. The unit of time isτ0 =~/¯t.
where tjα is the emission time and τ determines the width of the Lorentzian wave packet that is produced. With this phase, a right-moving leviton is created together with a left-moving anti-leviton (a hole), but without additional electron-hole pairs.
Changing the sign of the phase would produce the same quasi-particles moving in the opposite directions.
Figure 3.7 shows the particle densityn(x)along the chain at dierent times. The QPC is positioned atx= 0withx <0(x >0) corresponding to the left (right) lead.
The system is initialized in equilibrium at half lling at the time t0 min{tjα}, long before any excitation is applied. As the pulse is applied, a leviton and an anti-leviton are generated in the left lead as seen in panel (a). The excitations propagate in opposite directions with the Fermi velocityvF = 2¯t/~. In a continuum description, the wave function of the leviton reads [28]
ψ±(x, t) = ip vFτ /π
x−x0 ±vF(t−te) +ivFτ, (3.38) wherex0 is the position at which the leviton is created, and the sign corresponds to a leviton moving to the left (−) or to the right (+). With this wave function, the corresponding particle densityn(x) becomes Lorentzian, which agrees well with our results in Fig. 3.7.
As the leviton scatters on the QPC, it is partitioned into a transmitted and a re-ected wave packet. The QPC is tuned to half transmission, such that transmission
20 0 20 40 60 80 100
Figure 3.8: a) Entanglement entropy for a single leviton impinging on a QPC. The leviton is created at time tjα = 0 at site j = 50 in the left lead (α = L). The entanglement entropy increases nearly byln 2. The colored lines show the cumulant series with an increasing cut-o K. b) Entanglement entropy in a fermionic Ou-Mandel experiment with levitons. The entanglement entropy exhibits a Hong-Ou-Mandel-like suppression as a function of the time delay ∆te between the arrival times at the QPC. We show the exact result together with the cumulant series using an increasing cut-oK. The black line indicates the expected increase of the second cumulant according to Eq. (3.39). The inset focuses on the dierences between the curves.
and reection occur with equal weight. In Fig. 3.8a we show the time evolution of the entanglement entropy during the scattering process. Only the rst few cumu-lants are needed to obtain the entanglement entropy from the series in Eq. (3.5), and already with K = 6 the agreement with the full result is very good. For a binomial process with fty percent success probability, the entropy should increase by log 2. Our results are close to this value, although slightly lower due to the exponential distribution of the leviton in the energy domain [28],ψ(ε)∝e−ετΘ(ε). As a result, dierent components of the wave packet are scattered with dierent transmission amplitudes, leading to a smaller increase in the entropy. We note that the entan-glement generated here comes from a superposition of states with dierent particle numbers in each lead and thus may not be accessible.
Following Ref. [29] we now consider the situation where one leviton is created in each lead and brought into collision at the QPC in a fermionic Hong-Ou-Mandel experiment [130, 25]. In Fig. 3.8b we show the entanglement entropy generated by interfering two levitons on the QPC as a function of the time delay ∆te between the arrival times at the QPC. The results are divided by the entanglement entropy generated by the scattering of just a single leviton on the QPC. Together with the full result, we show the entanglement entropy obtained using the cumulant series with an increasing cut-o.
If the levitons arrive simultaneously, they anti-bunch such that one leviton leaves
the QPC in each direction after the scattering event. In this case, there are essentially no charge uctuations and almost no entanglement entropy is generated. In contrast, for large time dierences, |∆te| τ, the levitons scatter independently of each other and the entanglement entropy equals twice the entropy generated by a single scattering event. The nal state is a coherent superposition of states with zero, one, and two levitons in one lead. The state with one leviton in each lead is time-bin entangled with nite accessible entanglement entropy.
To understand the shape of the curve in between these limiting situations, we consider the increase of the second cumulant following a Hong-Ou-Mandel experiment with levitons. It can be written as [147]
C2HOM = 2C21(1− C), (3.39) whereC21 is the increase for a single leviton and
C =| hψ−|ψ+i |2 = 1
1 + ∆t2τe2 (3.40)
is the overlap of the leviton wave functions, takingx0∓vFt = 0 in Eq. (3.38).
This expression is in good agreement with our results for the second cumulant, and it essentially determines the shape of the entanglement entropy as a function of
∆te with only minor corrections due to higher cumulants.
Summary
We have investigated the entanglement and Rényi entropies for two Fermi seas connected via a barrier with a time-dependent transmission. Such a system can be implemented in mesoscopic physics by connecting two electrodes via a QPC, or in an optical lattice with cold fermionic atoms. Using exact expressions for the entropies in terms of the cumulants of the FCS, we have shown how the entanglement and Rényi entropies in a quantum many-body system can be deduced from measurements of the charge uctuations between the two reservoirs. In particular, for a quantum switch operated under suitable experimental conditions, a logarithmic growth of the entanglement entropy, as predicted by conformal eld theory, can be inferred from only the rst few cumulants. Motivated by recent experiments, we have evaluated the entanglement entropy generated by partitioning clean single-particle excitations (levitons) on a QPC as well as by interfering two levitons on the QPC, tuned to half transmission. In this case, we identify a Hong-Ou-Mandel-like suppression of the entanglement entropy as a function of the dierence of arrival times at the QPC.
The results presented here may serve as a guideline for future experiments aimed at measuring the entanglement entropy in a solid-state environment.
Chapter 4
Waiting time distributions in electronic transport
The waiting time distribution (WTD)W(τ)is the probability density of observing a timeτ between two successive events of interest. It has been considered for various stochastic processes, for instance in quantum optics [148, 149] and single-molecule chemistry [150, 151]. In electronic transport, the WTD has been suggested as a means to characterize short-time dynamics of transmission processes. In this context, it was rst investigated for systems described by Markovian master equations [52, 152, 153], and later on for coherent [54, 55], as well as driven systems [53, 154, 155]. Other works considered WTDs in superconducting hybrid structures [156], in the transient regime [157], and for spin-polarized systems [158].
In the following we present our contributions to the eld. In Sec. 4.1 we give a short introduction to WTDs for stationary processes. In Sec. 4.2 we present a formalism to obtain WTDs or non-interacting fermions on a tight-binding chain, which we compare to other approaches for a number of model systems. In Sec. 4.3 we introduce the master equation approach to WTDs and discuss dierent detector models for quantum dot structures. Finally, we present a generalization of the master equation approach to non-Markovian dynamics (Sec. 4.4) and illustrate our results with the example of a dissipative double quantum dot. Throughout this chapter we focus single-channel systems and neglect the spin degree of freedom. WTDs for multi-channel and spinful systems have been investigated in Refs. [55, 154].
4.1 Waiting time distributions for stationary pro-cesses
In electronic transport, W(τ) is the probability of observing a waiting time τ between successive transmission events. This is illustrated in Fig. 4.1, where the time-resolved current trace of an imaginary process is shown. If the time resolution is suciently large, the WTD can be extracted from such a trace.
I(t)
τ
t
1
Figure 4.1: Time resolved current trace showing single-particle events as delta peaks.
The WTD is the distribution of waiting timesτ between successive events.
The WTD is related to the probability of observing no transmissions during the time interval [t0, t0+τ], which is called the idle time probability (ITP). If the process under consideration is stationary, the ITP is independent of t0 because of translational invariance in time. It can then be expressed in terms of the WTD as [54, 55]
is the mean waiting time. The square brackets in Eq. (4.1) give the probability that no transmissions are observed during the time interval [te, t0 +τ], after the last transmission occurred atte. This expression is integrated over te using that for stationary processes transmission events are uniformly distributed with weighthτi−1. SinceW(τ) is normalized, Eq. (4.1) can be rewritten as
Π(τ) = 1
which shows explicitly that the ITP is independent oft0. From Eq. (4.3) follows the important result
W(τ) =hτi d2
dτ2Π(τ). (4.4)
Furthermore, by applying the normalizationR∞
0 dτW(τ) = 1to Eq. (4.4) we nd the mean waiting time as
hτi=− 1
Π(0)˙ . (4.5)
Equations (4.4) and (4.5) provide the basis to obtain the WTD from the ITP for stationary processes. In addition, for unidirectional processes, the ITP is given in terms of the FCS by
Π(τ) = P(n = 0, τ). (4.6)