Many-body manifestation of interaction-free measurement: The Elitzur-Vaidman bomb

We consider an implementation of the Elitzur-Vaidman bomb experiment in a dc-biased electronic Mach-Zehnder interferometer with a leakage port on one of its arms playing the role of a “lousy bomb.” Many-body correlations tend to screen out manifestations of interaction-free measurement. Analyzing the correlations between the current at the interferometer's drains and at the leakage port, we identify the limit where the originally proposed single-particle effect is recovered. Specifically, we find that in the regime of sufficiently diluted injected electron beam and short measurement times, effects of quantum-mechanical wave-particle duality emerge in the cross-current correlations.

Figure 1

Illustration of the Mach-Zehnder interferometers (MZIs) under study. Chiral channels are represented by solid lines leading from the sources $(S_1$, biased, and $S_2$, grounded) to the grounded drains $(D_1$ and $D_2)$. Interedge tunneling takes place at intersection points. (a) A standard MZI with arms 1 and 2 of lengths $l_1$ and $l_2$, respectively. (b) The dangling end at $C$ (leading to $D_3)$ serves as an absorber replacing the lousy bomb.

Figure 2

The many-body cross-current correlator as a function of $\alpha$ [see (11)]. Here, we have taken $|t_C{|}^2=0.3,|r_A{|}^2={\left|r_B\right|}^2=0.5$, and ${\mathrm{\Phi }}_{\mathrm{AB}}+{\varphi }_T^{†}+{\varphi }_C^{†}=\pi$. For a temperature of $T=10$ mK the parameter $\alpha$ corresponds to realistic bias values of up to $\sim 54\mu \mathrm{V}$. The different plots correspond to different values of $N=1,...,10$. As a function of $\alpha$ for a fixed $T,\tau$ should be changed in order to keep $N$ constant, i.e., $\tau =N\frac{2\pi \hslash }{eV}=N\frac{\hslash \beta }{\alpha }\sim 7.63823\times {10}^{-10}(N/\alpha )$ (seconds). We mark by circles the point $\alpha \equiv N$ as the threshold for which our assumption $\tau \gg L/v$ breaks for existing electronic interferometers [30]. (a) The case of $\mathrm{\Delta }\tilde{L}\to 0$. (b) The case of $\mathrm{\Delta }\tilde{L}=0.03$, where dephasing affects both the classical and quantum correlators (due to varying interference lengths per wave number). Nonetheless, the single-particle limit remains unaffected, as expected from Eq. (6).