Decoherence and relaxation of a single electron in a one-dimensional conductor

We study the decoherence and relaxation of a single elementary electronic excitation propagating in a one-dimensional chiral conductor. Using two-particle interferences in the electronic analog of the Hong-Ou-Mandel experiment, we analyze quantitatively the decoherence scenario of a single electron propagating along a quantum Hall edge channel at filling factor 2. The decoherence results from the emergence of collective neutral excitations induced by Coulomb interaction and leading, in one dimension, to the destruction of the elementary quasiparticle. This study establishes the relevance of electron quantum optics setups to provide stringent tests of strong interaction effects in one-dimensional conductors described by the Luttinger liquids paradigm.

Figure 1

Hong-Ou-Mandel interferometry. Left panel, sketch of two-particle interferences. Right panel, modified scanning electronic microscope picture of the sample. The electron gas is represented in blue, the edge channels by blue (outer channel) and green (inner channel) lines, and the metallic gates are in gold. The emitters are placed at inputs 1 and 2 of the QPC used as an electronic beam splitter. Single-electron emission by source $i$ on the outer channel is triggered by the square voltage $V_{\text{exc},i}$ of amplitude 0.7 K. The dot-to-edge transmission of source $i$ is tuned by the gate voltage $V_{\text{g},i}$. The central QPC is set to partition ($R=0.5$) the outer channel using the gate voltage $V_{\text{qpc}}$. Interaction regions of length $l\approx 3\mu \text{m}$ are represented by light blue boxes. Average ac current measurements are performed on the splitter output 4 in order to characterize the source parameters (in particular ${\tau }_e$). Low-frequency noise measurements $\mathrm{\Delta }{S}_{33}$ are performed on output 3.

Figure 2

Electronic Hong-Ou-Mandel experiment. HOM trace $\mathrm{\Delta }q$ as a function of the time delay between the sources $\tau$ for three values of the emission time ${\tau }_e=30$ ps (red squares), ${\tau }_e=100$ ps (blue squares), and ${\tau }_e=180$ ps (black dots). The plain lines represent exponential fits, $\mathrm{\Delta }q\left(\tau \right)=1-\gamma e^{-\left|\tau \right|/{\tau }_e}$.

Figure 3

Escape time asymmetry and energy emission fluctuations. $\mathrm{\Delta }q\left(\tau \right)$ for asymmetric (${\tau }_{e,1}=160, {\tau }_{e,2}=60$ ps, black dots) escape times. The black line is an exponential fit with ${\tau }_e=160$ ps for $\tau \le 0, {\tau }_e=60$ ps for $\tau \ge 0$. Inset, $\mathrm{\Delta }q\left(\tau \right)$ with (black dots) and without (red squares) external noise applied on the static potential of dot 1. The noise amplitude corresponds to a blurring of 400 mK of the dot emission energy. The black line is a fit with an exponential decay.

Figure 4

Contrast versus emission time. Evolution of contrast $\gamma$ as a function of emission time ${\tau }_e$ for $\nu =2$ (blue dots) and $\nu =3$ (red squares). The plain lines correspond to the fits by the phenomenological model $\gamma \left({\tau }_e\right)=1/(1+2{\tau }_e/{\tau }_c)$.

Figure 5

Destruction of the elementary quasiparticle. Wigner representations $\mathrm{\Delta }{W}^{\left(e\right)}(\overline{t},\omega )$ of the excess single-electron coherence at $T=0$ K for different propagation lengths ${\tau }_s=0$, 28, and 70 ps. The time axis are shifted by time $\overline{\tau }=l/v_{\rho }$ to account for the propagation time on length $l$. For ${\tau }_s=0, \mathrm{\Delta }{W}^{\left(e\right)}(\overline{t},\omega )$ represents the state emitted in the outer edge channel (blue line) described by Eq. (6), with ${\omega }_e=0.7$ K and ${\tau }_e=60$ ps. For ${\tau }_s=28$ and 70 ps, short-range Coulomb interactions between the outer and inner (green line) edge channels are taken into account.

Figure 6

Data/model comparison. Upper-left panel, $\mathrm{\Delta }q\left(\tau \right)$ for various emission times. Theory accounting for Coulomb interaction is represented by the dotted line ($T=0$ K) and dashed line ($T=0.1$ K). Lower-left panel, $\mathrm{\Delta }q\left(\tau \right)$ for asymmetric emission times. Theory predictions accounting for Coulomb interaction ($T=0.1$ K) are represented by dashed lines. Predictions of the noninteracting model in blurred black. Upper-right panel, contrast $\gamma$ versus emission time ${\tau }_e$ (in log-linear scale). The dotted ($T=0$) and dashed ($T=100$ mK) lines represent theory predictions accounting for Coulomb interaction. Lower-right panel: (a) data, ${\tau }_e=40$ ps, 400 mK gate noise on dot 1; (b) data, ${\tau }_e=40$ ps without gate noise; (c) theory, $T=0.1$ K, ${\omega }_1=0.7$ K, ${\omega }_2=0.3$ K, ${\tau }_e=40$ ps; (d) theory, $T=0.1$ K, ${\omega }_1={\omega }_2=0.7$ K; (e) noninteracting model, ${\omega }_1=0.7$ K, ${\omega }_2=0.3$ K, ${\tau }_e=40$ ps; (f) noninteracting model, ${\omega }_1={\omega }_2=0.7$ K, ${\tau }_e=40$ ps.