Coupling Quantum States through a Continuum: A Mesoscopic Multistate Fano Resonance

We demonstrate a fully tunable realization of a multistate Fano resonance, in which a pair of remote quantum states experience an effective coupling due to their mutual overlap with a continuum. Our mesoscopic implementation of this system exploits the ability of the semiconductor nanostructures known as quantum point contacts (QPCs) to serve, in the low-density limit close to pinch-off, as an on-demand localized state. By coupling the states formed on two separate QPCs, through a two-dimensional electron gas that serves as a continuum, we observe a robust effective interaction between the QPCs. To explain this result, we develop a theoretical formulation, based on the ideas of the Schrieffer-Wolff transformation, which is able to reproduce our key experimental findings. According to this model, the robust character of the interaction between the two remote states arises from the fact that the interaction is essentially mediated by a large number of degenerate continuum states. While the continuum is often viewed as a source of decoherence, our experiment therefore instead suggests the possibility of using this medium to support the interaction of quantum states, a result that may allow new approaches to coherently couple nanostructures in extended geometries.

Popular Summary

Examples of resonances abound in classical physics, from the oscillations of a loaded spring to the collapse of the Tacoma Narrows Bridge. In quantum mechanics, the principle of wave-function superposition leads to new manifestations of resonant behavior. This is demonstrated no more strikingly than by a class of phenomena called Fano resonances. A Fano resonance, in its simplest and most fundamental form, involves the interference of a discrete quantum state with a continuum of states in the same system. Although originally discovered in atomic scattering, in 1935 by Hans Beutler, and interpreted by Ugo Fano in the same year, Fano resonances have since been demonstrated in many other areas of physics, including optics, plasmonics, matter-wave scattering in ultracold atom systems, and charge transport through mesoscopic quantum dots. It is this last area that provides the specific context for this paper. Numerous works in the past have focused on the Fano resonance that arises from the interference of a single discrete state with a continuum. In this paper, we give a new and significant twist to this phenomenon by demonstrating, experimentally and for the first time, a novel multistate Fano resonance involving two single-electron states localized on two spatially remote quantum point contacts and a continuum of states of an intervening two-dimensional (2D) electron gas.

The existence of single-electron states localized on a single quantum point contact (or “bound states”) was already demonstrated in earlier experimental work, through a Fano resonance generated by nonlocal coupling between a single such state and a point-contact detector. In this work, we extend the earlier approach to generate and demonstrate a multistate Fano resonance that involves two bound states spatially separated by a distance of several hundred nanometers but interacting with each other through a 2D electron gas. The energy of these single-electron bound states can be controlled by suitable tuning of the voltages applied to the quantum-point-contact gates. By performing this tuning such that the two states are close to each other in energy and by measuring the charge transport through the device using the detector, we show that the two states experience a strong and coherent interaction—manifested as anticrossing of the gate-voltage-dependent positions of the two individual Fano resonances associated with the two different localized states. This interaction is, in fact, much stronger than that exhibited by two quantum dots that are so close that their discrete states overlap with each other.

Interpreting this counterintuitive observation with theoretical modeling, we show that the strong coherent interaction between the two remote states emerges from the fact that the interaction is mediated by all of the states of the intervening 2D electron gas. Therefore, our work suggests a new approach to use a continuum to engineer coherent, nonlocal coupling between nano structures in extended mesoscopic systems.

Figure 1

Schematic illustration of the Fano resonance exhibited by a detector QPC that is coupled through a region of two-dimensional electron gas to a bound state in another QPC [22, 23, 24, 25, 26]. An electrochemical-potential difference between source and drain is used to drive electrons (thick red line) through the detector. A correlation then arises between the bound state and the detector, due to tunneling of electrons (thin blue line) to and from the bound state, and is resonantly enhanced when the bound-state energy matches the Fermi level in the reservoir.

Figure 2

(a) Schematic illustration of the manner in which we use a multi-QPC device to implement remotely interacting BSs. ${\mathrm{BS}}_1$ and ${\mathrm{BS}}_2$ are formed in the swept and control QPCs, respectively, and are coupled to each other, and to the detector, via intervening regions of 2DEG (thin blue lines). (b) Scanning electron micrographs of our multi-QPC device are used to illustrate the manner in which we implemented the system in (a) experimentally. Uncolored gates were held at ground potential in the experiment and so had no influence on the underlying 2DEG.

Figure 3

Measurements of the detector conductance [$G_d(V_s)$] for Configuration I of Fig. 2b, at 4.2 K and for several different values of the control-QPC gate voltage ($V_c$, indicated in each panel). Also shown in the top-left (bottom-right) panel, as the black (green) line, is the corresponding variation of the swept-QPC (control-QPC) conductance $(G_{s,c})$, along with fits [25] (open circles) to the FR formula of Eq. (1). Top-left panel: $q=-22$ and $\Gamma =0.082\mathrm{V}$. Bottom-right panel: $q=-7$ and $\Gamma =0.225\mathrm{V}$.

Figure 4

The left panel shows the variation of $G_d(V_s)$ at 4.2 K, for Configuration I of Fig. 2b. From the lowest curve up, $V_c$ is incremented from $-0.96\mathrm{V}$ to $-2.21\mathrm{V}$ in 50-mV steps, and from $-2.31\mathrm{V}$ to $-2.91\mathrm{V}$ in 100-mV steps. Curves are shifted up in increments of $0.1\times 2e^2/h$. The right panel shows the variation of $G_d$($V_s$) at 4.2 K, for Configuration II of Fig. 2b. From the lowest curve up, $V_c$ is incremented from $-0.50\mathrm{V}$ to $-2.30\mathrm{V}$ in 50-mV steps. Curves are shifted up in increments of $0.05\times 2e^2/h$. The red dotted lines in both panels denote the missing portions of the resonance spectrum.

Figure 5

(a) $G_c(V_s,V_c)$ for Configuration I of Fig. 2b at 4.2 K. Data are for the range of $V_{s,c}$ where the control QPC is extremely close to pinch-off. (b) $G_s(V_s,V_c)$ for Configuration I for the same gate-voltage range and temperature as in (a). White dotted lines in (a) and (b) denote the same variation of $V_s-V_c$. The black dotted line at the front of the contour in (a) is a guide to show the line shape of $G_c(V_s)$. (c) Evolution of $R_1$ and $R_2$ for Configuration I as a function of $V_c$ and $V_s$. Black symbols were obtained using the control QPC as a detector; red symbols were inferred directly from the data of Fig. 4. The plot shown in the inset was obtained by using the lever arm [26] of the two BSs to convert variations of $V_{c,s}$ to corresponding BS-energy shifts.

Figure 6

(a) Measurements of the double-peak structure at different temperatures. The data are for Configuration I of Fig. 2b and a slowly varying background, with an average value of $3.5\times 2e^2/h$, has been subtracted from $G_d$. Successive curves are shifted upward in increments of $0.02\times 2e^2/h$. Curves with open symbols denote curves for which the peak separation increases with decreasing $T$. From bottom to top, respectively, temperatures are: 40, 36, 34, 32, 30, 28, 26, 24, 22, 20, 18, 15, 13, 11, 9, 8, 7, 6, 5, and 4.2 K. (b) Data from (a), plotted as a contour, to show appearance of peak repulsion (see arrow) below 10 K. (c) The main panel plots the $V_s$ position of the two peaks $R_1\mathrm{\text{and}}{R}_2$ from (a) as a function of $T$. The inset plots the energy splitting inferred from the separation between the actual position of $R_2$ and the linear extrapolation indicated in the main figure by the dotted line. (See text for further details.) (d) The main panel plots the $V_s$ position of the two peaks $R_1\mathrm{\text{and}}{R}_2$, obtained with $V_s\mathrm{\text{and}}{V}_c$ configured so that the two resonances are far from their avoided crossing. The inset illustrates the temperature dependence of the two resonances under these conditions. The results of two separate measurements are combined at each temperature, since each measurement exhibits just one resonance (due to ${\mathrm{BS}}_1$ or ${\mathrm{BS}}_2$).

Figure 7

The upper row of panels shows the calculated LDOS for (a) ${\mathrm{BS}}_1$, (b) ${\mathrm{BS}}_1$ and ${\mathrm{BS}}_2$, and (c) ${\mathrm{BS}}_2$ as functions of the gate voltages $V_s$ and $V_c$ (expressed as BS-energy shifts, in meV). As in the experiment, ${\mathrm{BS}}_1$ is defined by $V_s$ alone while ${\mathrm{BS}}_2$ is formed by $V_s$ and $V_c$. The lower row of panels shows the correction to the detector conductance, calculated from Eq. 35 in the Supplemental Material [37]. (d) Contribution from first term. (e) Contribution from both terms. (f) Contribution from the second term. A description of the parameter values used in these calculations is provided in the Supplemental Material [37].

Figure 8

The upper-left panel shows the data of Fig. 4 (Configuration I), replotted as a 3D contour, while the upper-right panel shows a similar replotting of Fig. 7e. The lower four panels use electron micrographs of the device to indicate its configuration, for each of the four resonances (I through IV) identified in the upper 3D contours.

Figure 9

(a) Total electron population of the two BSs and its variation with $V_{c,s}$. (b) Corresponding populations ($N_1\mathrm{\text{and}}{N}_2$) of the two separate BSs (${\mathrm{BS}}_1\mathrm{\text{and}}{\mathrm{BS}}_2$, respectively) for the same variation of $V_{c,s}$.

Figure 10

(a) Schematic potential diagram showing two BSs formed on QPCs that are separated by an intervening 2DEG. The coupling of each BS to this reservoir is denoted by the matrix element $v_{\mathbf{k}\sigma 1(2)}$. (b) The system shown in (a) is replaced here with an equivalent setup consisting of two BSs that are directly coupled through a thin tunnel barrier, with an effective tunneling rate of $W_{12}$.