Quantum statistics and self-interference in extended colliders

Abstract

Collision of quantum particles remains an effective way of probing their mutual statistics. Colliders based on quantum point contacts in quantum Hall edge states have been successfully used to probe the statistics of the underlying quantum particles. Notwithstanding the extensive theoretical work focusing on point-like colliders, when it comes to experiment, the colliders are rarely point-like objects and can support a resonant level or multiple tunneling points. We present a study of a paradigmatic extended (non-point-like) fermionic collider (and an extension to bosonic colliders). As with particle interferometers, in an extended collider there is an infinite number of trajectories for any single or multi-particle event. Self-interference of the former can lead to an apparent bunching of fermions when we compare the cross-current correlator with a classical benchmark representing two colliding beams. In view of this apparent bunching behavior of fermions, we identify an experimentally accessible current correlator which reveals the true mutual statistics of fermions.

Figures (6)

• Figure 1: (a) Extended collider comprising two sources ($S_{1}$, $S_{2}$) and two detectors ($D_{1}$, $D_{2}$). A dilute beam having large separation $L_{\lambda}$ between wave packets of width $\ell$, emitted from $S_{1}$, can give rise to a quasi-bound state at the collision area, a chiral quantum anti-dot (QAD) of circumference $L$. An incoming beam can tunnel into the QAD. As a result the outgoing beam comprises infinitely many wavelets of ever decreasing amplitude [determined by the junction tunneling amplitudes, Eq. \ref{['eq:smatrix']}], spatially separated by $L$. (b) Processes that contribute to the amplitude to tunnel from $S_{1}$ to $D_{2}$. The arrows in each segment of the QAD represent the number of times this segment was visited before being emitted from the QAD in the direction of $D_{2}$. The processes involve half-integer number of windings, depicted by the subscript in the amplitude $A^{(1\to2)}_{n+\frac{1}{2}}$. Summing up these partial amplitudes facilitates evaluation of the "anti-bunching probability" $P(11)_{\text{F/B}}$ for fermions/bosons, Sec. III of SM, Ref. Note2.
• Figure 2: Apparent and truly statistical measures of anti-bunching. (a) Difference between the anti-bunching probabilities $P(11)_{\text{F}}$, Eq. \ref{['eq:p11bf']} of fermions and $P(11)_{\text{Cl}}$ of classical particles as functions of the wave packet width and the edge-QAD reflection amplitude $r$, Eq. \ref{['eq:smatrix']}. Use of the classical particles as a benchmark shows an apparent bunching of fermions (blue region). We take $k_f = 0$ and $\theta = 0$. (b) Anti-bunching probabilities $P(11)$ for a model square wave packet with $\frac{\ell}{L}=2.5$, where $\ell=v/eV$Note4 with $V$ as the bias voltage at the dilutor QPC and $v$ is the velocity, Sec. IV of SM Note2. Using $P(11)_{\text{Cl}}$ for classical particles as a benchmark ($\mathcal{B}_{1}$) shows the apparent bunching of fermions near $r = 1$. The classical wave benchmark ($\mathcal{B}_{2}$) on the other hand reflects the true quantum statistics of fermions and bosons.
• Figure S1: (a) Trajectories of the particle from the source $S_{1}$ to $D_{1}$. The amplitude for the process $A(1\rightarrow 1)$ is a sum of amplitudes of each of the diagram. (b) Trajectories of the particle from the source $S_{1}$ to $D_{2}$. The amplitude for the process $A(1\rightarrow 2)$ is a sum of amplitudes of each of the diagram. Amplitude of each of the trajectory can be written by using the junction $S-$matrix, Eq. (1).
• Figure S2: The pair-correlation function $g^{(2)}(t_{1},t_{2})$ [see Eq. (\ref{['eq:g2']})] for fermions/bosons that arrive simultaneously at the collider but are detected at times $t_{1},t_{2}$ in the same drain. The wave packets are emitted simultaneously at $t=0$ and the collider is at a distance $5L/6$ from the sources and the detectors. We use $\frac{\ell}{L}=2.5$, $k_f = 0$, $r=0.95$, $\theta =0$ and a finite temperature $\beta^{-1} = eV/75$, see Sec. \ref{['sec:finitetempwp']}. (a) Fermions. The function $g^{(2)}$ is generally non-vanishing in an extended collider (hence a characteristic feature), unlike in the case of a point-like collider. Vanishing of $g^{(2)}$ at $t_{1}=t_{2}$ validates exclusion property of fermions. (b) Bosons. Non-vanishing along $t_{1}=t_{2}$ shows that bosons can be found in the same location at the same time.
• Figure S3: Plot for $g^{(2)}_{\text{Irr}}-$function for a pair of incoming particles from the sources $S_{1}$ and $S_{2}$. (a) $g^{(2)}_{\text{Irr}}-$function for a pair of (a1) fermions and (a2) bosons without a delay in the arrival times. (b) $g^{(2)}_{\text{Irr}}-$function for a pair of (b1) fermions and (b2) bosons with a delay $L/v$ in the arrival times.