Detector models

The measurement of WTDs is a subject of current research. In our tight-binding approach as well as in the scattering approach [54], the Fermi sea is excluded from the calculation. Depending on the actual measurement device, uctuations in the Fermi sea that happen on the time scaleε⁻¹_F could have an impact on the WTD.

For quantum dot systems, experiments monitoring the charge state of the system in real time have been performed by analyzing the conductance of a nearby QPC [36, 48, 51, 124, 125, 126, 127]. The time traces measured this way allow for a determination of the WTD for these systems. Interesting short-time eects, however, might be missed due to the nite bandwidth of the QPC.

In this section we investigate two models of imperfect detectors and their eect on the WTD. We consider a nite response time of the detector as well as a nite detector eciency. In addition, we discuss the eect of dephasing on the WTD for a DQD. These dierent aspects of the measurement process might have to be combined in order to explain actual data.

(0,1) (1,1) (1,0) (0,0)

a)

ΓL ΓR

Γd

n

I(n)

b)

Γ_L

Γ_L Γ_R

Γ_R

Γd

Γ_d

Figure 4.9: Finite bandwidth model. a) A single level quantum dot is capacitively coupled to a biased QPC. The currentI through the QPC adapts with the rateΓd to changes in the numbern of electrons on the dot. b) Schematic representation of the possible transitions in the setup. A change in the occupation of the dot can either be followed by the detector adapting to the change with rate Γd, or by a transition of the system back into the previous state, in which case the event is not recorded by the detector.

Finite bandwidth model

The nite bandwidth model has been developed to take the nite response time of a detector into account [164]. It has played an important role in the interpretation of real-time counting measurements [51, 124, 126].

Here we consider the example of a setup in which the occupation of a quantum dot is monitored by a nearby QPC as shown in Fig 4.9a. The QPC transmission depends on the dot occupation through capacitive coupling, so that by monitoring the time-dependent current through the QPC the occupation of the quantum dot can be measured in real time [165]. However, changes in the system might be missed if they happen on a faster timescale than the response time of the detector. It is important to consider this eect in order to achieve a qualitative agreement between experimental observations and theory.

We treat the nite bandwidth model within the master equation approach for the single level quantum dot discussed above. The density matrix has four non-zero elements,ρˆS = (ˆρ00,ρˆ10,ρˆ01,ρˆ11)^T, whereρˆnsn_ddenotes a state withnsparticles in the system while the detector indicates nd particles. Transitions in the system happen with the rates Γ_L and Γ_R, and the detector reacts with a rate Γ_d to changes in the system. A schematic representation of the possible transitions in the setup is shown in Fig 4.9b.

The transformed Liouvillian reads

where ρˆ₀₁ →ρˆ₀₀ is the only transition where the detector recognizes a transfer of a particle to the right lead.

From Eq. (4.35), we nd the WTD in Laplace space as Wf(z) = Y domain is a sum of four exponential decays,

W(τ) = ΓLΓRΓd

We plot the result in the time domain in Fig. 4.10a. For a perfect detector (Γd → ∞) we recover Eq. (4.37), while the WTD becomes atter as the detectors response becomes slower.

For double dot systems the form of the transformed Liouvillian in the nite band-width model depends on how the QPC couples to the levels, whether superposition states can be observed, and the strength of Coulomb interactions. Considering these details, the approach presented here can be generalized.

Finite detector eciency

In Sec. 4.3 we have seen that the ITP is obtained from the MGF, χ(λ, t) =X

n

P(n, t)e^iλn, (4.49)

in the limit λ →i∞. This amounts to assuming that the detector is perfect, i.e. it detects every single particle transfer of relevance for the WTD.

0 5 10 15 20

Figure 4.10: Detector models for a single-level quantum dot. a) In the nite band-width model, the detector reacts with a rate Γ_d to changes in the system. For decreasing Γd, the WTD becomes at as detection events become rare. b) A detec-tor with nite eciency randomly misses events with the error probability p. For increasing p, the WTD approaches a Poissonian distribution.

Here we take into account that a real detector might randomly miss an event.

The ITP measured by the detector then reads

Π_d(τ) = χ(λ=iγ, t) =P(0, τ) +pP(1, τ) +p²P₂(τ) +. . . , (4.50) wherep=e^−γ is the error probability. The WTD obtained from this result reads

Wd(τ) = (1−p)Tr[Je^M(iγ)tJρˆS]

Tr[JρˆS] , (4.51)

or in Laplace space

Wfd(z) = (1−p)Tr[J {z− M(iγ)}⁻¹JρˆS]

Tr[JρˆS] . (4.52)

For the single-level quantum dot discussed above this gives Wfd(z) = (1−p)ΓLΓR

(z+ ΓL)(z+ ΓR)−pΓLΓR

, (4.53)

and in the time domain

Wd(τ) = (1−p)ΓLΓRe^−Γτsinh(∆τ)

∆ , (4.54)

where Γ = (ΓL + ΓR)/2 and ∆ = p

(ΓL−ΓR)²+ 4pΓLΓR/2. In the limit p → 0, Eqs. (4.37) and (4.38) are recovered.

0 1 2 3 4 5

Figure 4.11: Detector models for a serial double dot. a) Finite detector eciency:

as the error probabilitypgrows, the oscillations lose their visibility as and the WTD approaches a Poissonian distribution. b) Dephasing: The dephasing rateΓ_φ damps coherent oscillations while the weight of the distribution remains at short times.

In Fig. 4.10b we plot the WTD for dierent values of p. In contrast to the nite bandwidth model, the detector misses events randomly, not necessarily at short times. The maximum of the distribution thus remains at the same position, but the tail of the distribution at long times gains weight asp grows.

Equation (4.53) is related to the result for a perfect detector, Eq. (4.38), by Wfd(z) = TWf(z)

1−RWf(z), (4.55)

with R = p and T = 1−p. This means that the process satises the renewal as-sumption stating that subsequent waiting times are uncorrelated [154]. Knowing the error probabilityp, the WTD that would have been measured by a perfect detector can thus be obtained by inverting Eq. (4.55),

Wf(z) = Wfd(z)

1−p+pWfd(z). (4.56)

In Fig. 4.11a we show the WTD obtained by a detector with nite eciency for a DQD without Coulomb interaction between the levels. As p increases, coherent oscillations are gradually washed out and the WTD is shifted towards larger times.

Dephasing model

For multi-level systems, uctuations of the surrounding charges and the inuence of the measurement device cause superposition states to decay. This eect is often included via a phenomenological dephasing rate Γφ. For specic situations this rate may be derived from a microscopic model [165].

The rate Γφ is added to the elements of the transformed Liouvillian that govern the decay of the superposition states. For the serial DQD discussed in Sec. 4.3, this means that Γφ is added to the last two diagonal elements of the transformed Liouvillian.

Neglecting Coulomb interactions between the levels, the WTD in Laplace space becomes,

Wf(z) = ΓLΓR(z+ 2Γ)²(z+ ˜Γ)Ω² (z+ ΓL)(z+ ΓR)Γn

z(z+ 2Γ)h

(z+ ˜Γ)²+ε²i

+ 4Ω²(z+ ˜Γ)²o. (4.57) In Fig. 4.11b we plot the WTD of a serial double dot for dierent values of Γφ. Similar to the case of a nite detector eciency, the oscillation amplitude reduces with growing dephasing rate. The weight, however, remains at short times.

Summary

In this section, we have reviewed the calculation of electronic WTDs for unidi-rectional transport in systems described by Markovian master equations [52]. We found excellent agreement with our tight-binding approach for systems with a low transmission. We have investigated the eect of an imperfect detector on the WTD, and found that a nite detector response time reduces the increase of the WTD at short times, while a nite detection eciency causes the WTD to approach a Poisso-nian distribution. Furthermore, we have found that coherent oscillation in the WTD for a serial DQD are damped due to a nite dephasing rate. We stress that already today the WTD can be extracted from real-time measurements of the charge state of quantum dot systems. Our results may thus serve as a guide for the interpretation of experimental data obtained this way.