Multiple interference in the time domain

In order to measure the function F defined in the previous section we need an more complex setup. Indeed, two signals come out of the N-block system and we have to measure current and visibility of both signals. This mean that one has to modify the setup of Fig. 4.6 such that the amplitude that goes out from the N-block system in the channel o interferes with a signal of known phase ϕ⁰ before going out from the whole device from contact 4. By measuring the transmission amplitudes T₃(ϕ) and

T₄(ϕ⁰) = 1

2(hRi_δ+ 1) +| hri_δ|cos (arg(hri_δ)−ϕ⁰), (4.15) with R = |r|², and their visibilities V₃ and V₄ in contact 3 and 4 we can reconstruct the functionF. Denoting ¯T₃ and ¯T₄ the mean values of the transmission probabilities in 3 and 4, we can write

F =V₃²T¯₃²+V₄²T¯₄². (4.16) This relation allows us to understand the behavior of theF as a function of. The visibility V₃ necessarily starts from zero, as T = 0 for = 0, as shown in Fig. 4.7. To the contrary, the visibilityV₄necessarily starts from one, asT = 0 for= 0 andR = 1−T. At small but non-zero one has a reduction of the F due to the non linear dependence on the currents and visibilities in Eq. (4.16). Hence the minimum in the visibility, as shown in Figs. 4.3 and 4.4. As →1, in the case of IFM one of the two addenda in Eq. (4.16) survives, and F →1 asymptotically, otherwise it monotonically decreases to zero.

4.4 Multiple interference in the time domain

It is possible to realize an IFM scheme based on the Mach-Zehnder (MZ) interferometer in the integer quantum Hall architecture at filling factor ν = 1, experimentally realized in Ref. [13–19]. To this end one needs to employ a quantized electron emitter [20–23], as illustrated in Fig. 4.8.

The periodic time-dependent potentialV(t) applied to a small circular cavity produces, on a linear extended channel connected to contact 1, an ac current composed by a very well time-resolved electron and hole pair [20–23]. A quantum point contact (QPC1), driven by a time-dependent external potential U₁(t), connects the linear edge to the MZ and lets only electrons to be transmitted in the interferometer, while reflecting holes in the lead 1. Suppose that after an electron has been injected into the MZ, the QPC1 closes and completely detaches the MZ from lead 1. The transmitted electron travels with a precise velocity v_F along the edge e_bl in the MZ until it encounters the first beam splitter BSL.

There it is split into two packets that follow two different edgese_tr ande_brof equal lengthL and will collide at BSR after a timeL/v_F. The two packets then mix together at BSR and then follow the edgese_bl and e_tl of length L. The sequence repeats itself many times, with the electronic wavepacket being split and reunited many times at the beam splitters BSL and BSR. Due to the fact that the drift velocity on the edge is constant and by engineering the arms of the MZ to have all equal lengths, we can map the time propagation in the MZ into a spatial concatenation of BSs and phase shifters PSs. Via properly tuning the degree

90 Chapter 4. Electronic implementations of Interaction-Free Measurements

BST

BSB ν = 1

1 2

3 4

e^{i ϕ} 3’ ν = 2 N blocks

N blocks BST 1

BSB

3’

3 3’

4 2 e^{i ϕ}

Figure 4.6: Schematic representation of the proposal for an experimental realiza-tion of an N-block noise-sensitive electron channel embedded in a Mach-Zehnder interferometer. Electrons entering the a Hall bar from contact 1 split at the beam splitter BST. The electrons transmitted will traverse theN-block system and even-tually go out from contact 4 or impinge onto BSB. The latter mix with those initially reflected at BST and interfere. The result of the interference can be collected in contact 3 or 3’. In the yellow areas the filling factor isν = 1 and in the rest of the Hall bar the filling factor is ν = 2. The coherence of the outgoing signal can be directly addressed by measurement of the visibility of current in contact 3 versus the tunable phase ϕ acquired during the propagation by the electron reflected at BST. Inset: Schematics of the main picture.

of channel admixture at the BSs, we make the electron appear after N rounds entirely in front of QPC2, that can be opened in a time-resolved way, such that the electron can be collected into lead 2.

A dephasing source affecting the propagation of the electron in the edgee_tr can be simply described by randomly shifting the phase of the corresponding wavepacket. The electron will have very high probability to be in front of QPC1. By opening the latter one can collect a coherent electron in contact 1, thus performing an IFM of the dephasing source. This protocol should operate in a cyclic way. The periodic potential V(t) produces an electron at time t₊ and a hole at time t− in front of the QPC1 every cycle, with 0 ≤ t₊ ≤ T/2 and T/2 ≤ t₋ ≤ T, and with T the period of the cycle. The QPC1 will be open during

4.4 Multiple interference in the time domain 91

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

N = 20 N = 50 N = 100 N = 150

ε

V

Figure 4.7: Visibility VT of the current in contact 3 versus the strength of the dephasing field, for several number of blocks N. In the coherent case = 0 the current in contact 3 is zero, for which alsoVT. Increasingthe visibility approaches one. We set γ =π/N.

first half cycle, to let the electron in, and closed for the second half cycle. The two QPCs open and close simultaneously. An electron emitted in front of the QPC1 at time t+ will appear in front of one of the two QPCs after a time t₊+N∆t, with ∆t≡2L/v_F, having performed N rounds. With the choice t₊ = (1/2−1/M)T and ∆t = (m+ 1/M)T, with m, M = 1,2,3. . ., an electron makes Nmax =M/2 rounds inside the MZ before finding the QPCs open. Depending on the interference conditions the electron will be found either in front of QPC1, in case of no dephasing, or in front of QPC2, in presence of a dephasing source.

In the case for which no dephasing field is present, = 0, it is possible to tune the MZ such that after N_max rounds the electron is exactly in front QPC2 and can be collected in contact 2. The current in contact 1 will be given by a train of holes, whereas the current in contact 2 by a train of electrons. In the case of maximal dephasing,= 1, no current is expected in contact 2, and a train of coherent electron-hole pairs per cycle is expected in contact 1.

The energy level spacing inside the MZ can be estimated as ∆E ∼ h/∆t = _(m+1/M)T^h . For large M it is determined by the size of m, that can be chosen to be large enough for a continuum approximation of the level spacing to be valid. This picture allows us to describe the physics in the Landauer-B¨uttiker formulation of quantum transport[34, 35], with no needs of the Floquet treatment of the time-dependent problem. We introduce the electron annihilation operators {ˆe_tr,eˆ_br,eˆ_bl,eˆ_tl}that annihilate an electron on the edge

92 Chapter 4. Electronic implementations of Interaction-Free Measurements

BSL BSR

V(t)

U (t)

U (t)

ε

2

1 QPC2

QPC1

e tr e tl

e br e bl

Figure 4.8: Mapping of concatenation in space to the time domain in a Mach-Zehnder interferometer. A time-dependent voltage generates a current of well sep-arated electron and holes. The QPC1 let the electrons enter the Mach-Zehnder, perform N rounds in the interferometer and then collect them back into contact 1 in the case a dephasing field of strength affects the dynamics of the channel e_tr. In the coherent case = 0 the electrons are collected in contact 2.

states {e_tr, e_br, e_bl, e_tl}. In order to obtain the transport regime described in the previous section we have to tune the beam splitters BSL and BSR such that

S_BSL =S_BSR =

cos(γ/2) isin(γ/2) isin(γ/2) cos(γ/2)

, (4.17)

with (ˆe_tr,eˆ_br)^T =S_BSL(ˆe_bl,eˆ_tl)^T and (ˆe_bl,ˆe_tl)^T =S_BSR(ˆe_tr,eˆ_br)^T, with the particular choice γ = π/N_max. Concerning with the dynamical phase acquired by propagating along the edge channels, arms of equal lengthLgive rise to no phase shift between the arms, and the condition for the working point φ = 0 depends only upon proper tuning of the magnetic field.