Transmission lines and resonators based on quantum Hall plasmonics: Electromagnetic field, attenuation, and coupling to qubits

Quantum Hall edge states have some characteristic features that can prove useful to measure and control solid state qubits. For example, their high voltage to current ratio and their dissipationless nature can be exploited to manufacture low-loss microwave transmission lines and resonators with a characteristic impedance of the order of the quantum of resistance $h/e^2\sim 25\mathrm{k}\mathrm{\Omega }$. The high value of the impedance guarantees that the voltage per photon is high, and for this reason, high-impedance resonators can be exploited to obtain larger values of coupling to systems with a small charge dipole, e.g., spin qubits. In this paper, we provide a microscopic analysis of the physics of quantum Hall effect devices capacitively coupled to external electrodes. The electrical current in these devices is carried by edge magnetoplasmonic excitations and by using a semiclassical model, valid for a wide range of quantum Hall materials, we discuss the spatial profile of the electromagnetic field in a variety of situations of interest. Also, we perform a numerical analysis to estimate the lifetime of these excitations and, from the numerics, we extrapolate a simple fitting formula which quantifies the $Q$ factor in quantum Hall resonators. We then explore the possibility of reaching the strong photon-qubit coupling regime, where the strength of the interaction is higher than the losses in the system. We compute the Coulomb coupling strength between the edge magnetoplasmons and singlet-triplet qubits, and we obtain values of the coupling parameter in the order of 100 MHz; comparing these values to the estimated attenuation in the resonator, we find that for realistic qubit designs the coupling can indeed be strong.

Figure 1

Top view of the edge of a two-dimensional material (red half-plane) in the quantum Hall regime surrounded by an insulator (green region) with dielectric constant ${\epsilon }_S$. The QH material supports edge excitations that are driven by an external potential $V_a$ applied to a capacitively coupled ideal electrode (gray area). The edge magnetoplasmon propagates chirally at the boundary of the QH material with a velocity $\sim c_S\alpha$, where $c_S$ is the speed of light in the medium and $\alpha \approx 1/137$ is the fine structure constant.

Figure 2

Cross-section of a gated quantum Hall device. A QH material (red line) is coupled capacitively to a back gate at distance $d_B$ and to two top gates placed at a distance $d_{1,2}$ and separated by a distance $L_B$ in the $y$ direction. The back gate is grounded while the $i\mathrm{th}$ top gate is driven by a voltage $V_i$, measured with respect to ground. For simplicity, all the gates extend indefinitely in the $x$-direction, while the QH material occupies only the half-plane $x>0$. The $y$ direction is divided into the three regions ${\mathcal{R}}_i$ characterized by the different gating configuration; ${\mathcal{R}}_{1,2}$ extends to $y\to \mp \infty$, respectively. The dielectric background is assumed to have a homogeneous and isotropic dielectric constant ${\epsilon }_S$. An homogeneous magnetic field $B$ is applied in the $z$ direction.

Figure 3

Driving term $V_a(0,y,0,\omega )$ as a function of $y$. We show a comparison between the numerical solution of the Laplace equation for the electrostatic configuration in Fig. 2 (black line) and the step function approximation discussed in the text (red line). The value of $V_a(0,y,0,\omega )$ in ${\mathcal{R}}_1$ used for the step function is defined in Eq. (21). For the plot, we fix the ratio $d_B/d_1=3$ and neglect the effect of the second top electrode; the latter approximation is justified when the two electrodes are far away from each other $d_{1,2}\ll L_B$.

Figure 4

Top view of a QH nanowire of width $W$. The nanowire is drawn in light red, while the gray areas indicate the position of the metal electrodes. The vertical cross-section of the device is shown in Fig. 2 and the $y$ direction is divided into three regions ${\mathcal{R}}_i$ characterized by a different gating structure. The length of the nanowire is $y_2^t-y_1^t=L_1+L_2+L_B$, where $y_i^t$ are the end points of the QH material in the $y$ direction and $L_i$ is the length of ${\mathcal{R}}_i$. The intra- and interedge Coulomb couplings are parametrized by the velocities $v^{F,I}$, respectively. The interedge component $v^I$ causes Coulomb drag. We sketch schematically the charge profile in the $x$ direction of the two EMP eigenmodes $g^{\pm }$, moving with velocities $\pm v^C=\pm \sqrt{{\left(v^F\right)}^2-{\left(v^I\right)}^2}$, respectively. A positive charge (red) localized mostly at one edge drags a $r$ times smaller negative charge (blue) at the opposite edge; in the plot the sign of $g^{\pm }$ is chosen to satisfy Eq. (46).

Figure 5

Eigenvalues of Eq. (54): comparison between the numerical results and the perturbative solution (58). In the insets, we plot the percentage error in making the approximation. In (a), we plot the eigenfrequency without considering screening gates, while in (b), we include a top gate at distance $d=10l^{\prime }$ from the EMP.

Figure 6

Near- and far-field approximations of the function $g_0(x,q)$, defined by the limit $z\to 0$ of Eq. (61). The black dots are obtained by solving the integral numerically, while the solid red and blue lines are the asymptotic solution in the near- and far-field limits, respectively. For illustrative purposes, we used a rather high value of $q{l}^{\prime }=0.1$.

Figure 7

Electromagnetic field in the plane $y=0$ generated by a QH material in the $(x,y)$ plane and without electrodes. In (a), we show the results obtained by the potential and current density in Eq. (60), and in (b), we show the analogous far field obtained from Eq. (10). In the plots, we choose $q{l}^{\prime }=0.01$ and the axis are in units $l^{\prime }=l/c_0$ with $c_0=0.53$. The orange (black) stream lines represent the electric (magnetic) field, while in the background, we plotted the scalar potential $V$. The thick red lines indicate the position of the QH material, and their opacity is weighted by the conductivity profile.

Figure 8

Attenuation of the EMPs: $\mathrm{Im}\left(\omega \right)$. The red lines are computed numerically by diagonalizing (72), while the blue lines are obtained by using the complex-valued length $l$ (74) in the far-field EMP eigenfrequency (7), with the appropriate velocity dependent on the electrostatic configuration. In the insets, we plot the percentage error made in $\mathrm{Re}\left(\omega \right)$ by this approximation. In (a) and (b), we plot $\mathrm{Im}\left(\omega \right)$ as a function of the wave vector $q{l}^{\prime }$ at a fixed value of the diagonal conductivity. In (a), we do not include metal gates and use (6). The solid, dashed and dotted lines are obtained for ${\sigma }_{xx}/{\sigma }_{xy}={10}^{-4},{\sigma }_{xx}/{\sigma }_{xy}={10}^{-3}$, and ${\sigma }_{xx}/{\sigma }_{xy}=2\times {10}^{-3}$, respectively. In (b), we show the effect of a top gate and use (25a). We consider ${\sigma }_{xx}/{\sigma }_{xy}={10}^{-3}$ and plot with a solid line the value of $\mathrm{Im}\left(\omega \right)$ including a top gate at distance $d=10l^{\prime }$; the dashed line is shown for comparison and is obtained without gates. In (c), we plot $\mathrm{Im}\left(\omega \right)$ as a function of the ratio of diagonal to off-diagonal conductivity ${\sigma }_{xx}/{\sigma }_{xy}$ and we use $q{l}^{\prime }={10}^{-3}$.

Figure 9

QH resonator and ST qubit. To define the ST qubit, we consider two electrons in a double dot potential perpendicular to the edge of a QH resonator of perimeter $L_y$. The two dots are separated by a distance $2a$ and an external electric field $E_0$ is applied in the direction connecting the dots; the eigenstates of each dot are characterized by a confinement length $l_T$, see Eq. (80). The QH edge state extends into the bulk for a length $l^{\prime }=0.75l_B$ (49) and its center of mass (red solid line) is at a distance $\tau$ from the center of mass of the double dot; $\tau$ accounts for the shift $x_0=1.15l_B$ (50) of the EMP charge density into the bulk of the QH material. An homogeneous magnetic field $B$ is applied in the direction perpendicular to the plane.

Figure 10

Coupling constants ${\gamma }_{1,2}$. For the plots, we use $2a=30\mathrm{nm},\mathrm{\Omega }=3.85\mathrm{meV},B=0.44\mathrm{T}$, and $L_y=27.7\mu \mathrm{m}$; we consider $l^{\prime }=0.75l_B$ and $x_0=1.15l_B$, as discussed in Sec. 2b. We also include the effect of a top gate placed at distance $d=0.6\mu \mathrm{m}$ from the EMP. In (a), we consider $E_0=0$. The blue line is obtained using Eq. (86), and the red line is obtained using Eq. (93). In the inset, we plot the dimensionless parameter $\zeta$, defined in Eq. (C18), which quantifies the coupling between the computational and the noncomputational subspace of the double dot. In (b), we show the dependence of the coupling ${\gamma }_2$ on a homogeneous electric field $E_0$. The length $b_0$ is defined in Eq. (91) and it is proportional to $|E_0|$; to include negative values of $E_0$, we rescale the horizontal axis by $\mathrm{sign}\left(E_0\right)$. The blue and red lines are obtained respectively by using Eqs. (89) and (94). The solid and dashed lines are obtained by using $\tau ={\tau }_{\text{s}}=x_0+a+l_T\approx 77\mathrm{nm}$, and $\tau =2{\tau }_{\text{s}}\approx 154\mathrm{nm}$. These values of $\tau$ are marked in (a) by using an orange dot and a black circle, respectively.

Figure 11

Ratio $\mathrm{\Gamma }$ of the coupling constant ${\gamma }_1$ and the attenuation of the QH resonator as a function of the magnetic field $B$ and of the harmonic confinement strength $\mathrm{\Omega }$. In (a), we consider two dots very close to each other, with $2a=30\mathrm{nm}$ and we include a top gate at distance $d=0.6\mu \mathrm{m}$ from the system. In (b), we consider two dots $2a=100\mathrm{nm}$ apart and we included a top gate at distance $d=2.2\mu \mathrm{m}$. The orange and black dots in (a) and (b) mark the parameters used in Fig. 10 $(B=0.44\mathrm{T}$ and $\mathrm{\Omega }=3.85\mathrm{meV})$ and in Fig. 12 $(B=90\mathrm{mT}$ and $\mathrm{\Omega }=0.12\mathrm{meV})$, respectively. In both figures, we consider a resonator of length $L_y=27.7\mu \mathrm{m}$ and we place the qubit at a distance $\tau ={\tau }_s=x_0+a+l_T$ from the QH edge $\left(x_0=1.15l_B\right)$. The value of ${\tau }_s$ depends on $B$ and $\mathrm{\Omega }$; for the parameters corresponding to the orange (black) point, it is ${\tau }_s=77\mathrm{nm}\left({\tau }_s=242\mathrm{nm}\right)$. The regions of parameters marked by red dashed lines are characterized by an exchange coupling outside the microwave domain, i.e., $J_{\text{ex}}>15\mathrm{GHz}$.

Figure 12

Ratio $\mathrm{\Gamma }$ of the coupling constant ${\gamma }_1$ to the attenuation of the QH resonator as a function of perimeter of the QH resonator $L_y$. For the plot, we use $2a=100\mathrm{nm},d=2.2\mu \mathrm{m},\tau =242\mathrm{nm},B=90\mathrm{mT}$, and $\mathrm{\Omega }=0.12\mathrm{meV}$. In the inset, we show the dependence of the resonance frequency on $L_y$. The black dots mark the special value $L_y=27.7\mu \mathrm{m}$: for this value of $L_y$, the value of $\mathrm{\Gamma }$ here is equal to the value marked by the black dot in Fig. 11.

Figure 13

Quantum velocities as a function of the Fermi energy ${\epsilon }_F$. The blue and red lines are the quantum velocities associated to the first and second Landau level, respectively.

Figure 14

(a) Comparison between the exact wave functions in Eq. (A3b) (red lines) and the normalized and shifted gaussian in Eq. (A5) (blue lines). We used ${\epsilon }_F=1.4\hslash {\omega }_c$ for the solid lines and ${\epsilon }_F=0.6\hslash {\omega }_c$ for the dashed lines. The thick gray line indicates the physical edge of the QH material. (b) Fitting parameters $l^{\prime }$ (blue line) and $x_0$ (red line) in units $l_B$ as a function of the Fermi energy ${\epsilon }_F$.