Decoherence of charge density waves in beam splitters for interacting quantum wires

Simple intersections between one-dimensional channels can act as coherent beam splitters for noninteracting electrons. Here we examine how coherent splitting at such intersections is affected by interparticle interactions, in the special case of an intersection of topological edge states. We derive an effective impurity model which represents the edge-state intersection within Luttinger liquid theory at low energy. For Luttinger $K=1/2$, we compute the exact time-dependent expectation values of the charge density as well as the density-density correlation functions. In general, a single incoming charge density wave packet will split into four outgoing wave packets with transmission and reflection coefficients depending on the strengths of the tunneling processes between the wires at the junction. We find that when multiple charge density wave packets from different directions pass through the intersection at the same time, reflection and splitting of the packets depend on the relative phases of the waves. Active use of this phase-dependent splitting of wave packets may make Luttinger interferometry possible. We also find that coherent incident packets generally suffer partial decoherence from the intersection, with some of their initially coherent signal being transferred into correlated quantum noise. In an extreme case, four incident coherent wave packets can be transformed entirely into density-density correlations, with the charge density itself having a zero expectation value everywhere in the final state.

Figure 1

Schematic representation of the intersection. Two one-dimensional quantum wires effectively intersect at a common point (left), although microscopically (right) the “intersection” may really only be a close approach which allows particles to tunnel between the wires within a small region. The continuous wires labeled 1 and 2 are therefore the “northwest-to-northeast” and “southwest-to-southeast” angled lines, as indicated. The coordinate $s$ within each wire is chosen to run in the directions indicated by the bent dashed lines with arrowheads. Fermions within each wire can move in both positive and negative $s$ directions. Those moving in the positive $s$ direction in each wire will be denoted as right movers with ‘$`R$” subscripts, while left movers going in the negative $s$ directions will have ‘$`L$” subscripts.

Figure 2

Relevant two-particle tunneling processes at the intersection. Thick black curves denote the two wires; wide arrows indicate simultaneous tunneling of two particles from one wire to the other. (Top) Two conjugate processes in ${\widehat{T}}_B$. (Bottom) Two conjugate processes in ${\widehat{T}}_F$.

Figure 3

Expectation value of the local charge density along the two intersecting wires, for various pairs of incident charge density wave packets, all in the illustrative case ${\mathcal{R}}^2={\mathcal{T}}^2=1/2$. The height of the ripples is the local charge density. All packets are identical except for wave phase; in particular their patterns of charge density are either equal or opposite in sign. The two packets in each pair are timed to overlap at the intersection. Plots labeled “Incident” show the charge density at a time shortly before the packets have reached the intersection; plots labeled “Emitted” show the outgoing packets shortly after the packets have traversed the intersection. In (a) and (b), incident packets come from opposite ends of wire 1, with phases equal (a) or opposite (b). In (c) and (d), the incident packets come from the left in both wires, with phases equal (c) or opposite (d). The relative phase of the two packets determines whether or not they will split as they pass through the intersection.

Figure 4

Diagonally crossing axes show the expectation value of the charge density as in Fig. 3, again for the illustrative case ${\mathcal{R}}^2={\mathcal{T}}^2=1/2$. (a) shows a single incident wave packet dividing at the intersection into four outgoing packets. (b) shows four converging packets, identical except that the packets coming from the right have opposite phase; the packets of charge density expectation value annihilate each other at the intersection. In both panels the horizontal blue axis shows the density-density correlation function between the two wires, at the points joined by the vertical lines. For the single incident packet in (a), some of the incident energy and information is emitted in correlated quantum noise that is generated as the packet crosses the intersection. For the four incident packets in (b), the entire incoming signal is transformed into quantum noise with zero expectation value but nonzero two-point correlation function.