# Tunable Capacitor For Superconducting Qubits Using an InAs/InGaAs
Heterostructure

Nicholas Materise1, Matthieu C. Dartiailh2111Present address: Institut Neél,
CNRS, France, William M. Strickland2, Javad Shabani2 and Eliot Kapit1
1Department of Physics, Colorado School of Mines, 1500 Illinois St., Golden,
CO 80401 USA  2Center for Quantum Information Physics, Department of Physics,
New York University, NY 10003, USA

###### Abstract

Adoption of fast, parametric coupling elements has improved the performance of
superconducting qubits, enabling recent demonstrations of quantum advantage in
randomized sampling problems. The development of low loss, high contrast
couplers is critical for scaling up these systems. We present a blueprint for
a gate-tunable coupler realized with a two-dimensional electron gas in an
InAs/InGaAs heterostructure. Rigorous numerical simulations of the
semiconductor and high frequency electromagnetic behavior of the coupler and
microwave circuitry yield an on/off ratio of more than one order of magnitude.
We give an estimate of the dielectric-limited loss from the inclusion of the
coupler in a two qubit system, with coupler coherences ranging from a few to
tens of microseconds.

##  1 Introduction

Tunable couplers for superconducting qubits, previously thought of as long-
term investments in future quantum computers and building blocks towards
demonstrating high fidelity two qubit gates [1, 2], are now center-pieces of
large scale superconducting qubit-based quantum computers. The early quantum
advantage demonstration [3] owes its success, in part, to the two-qubit gate
fidelities across the chip facilitated by fast, tunable couplers. Often
tunable couplers are realized as mutual inductances or effective capacitances
between nearest-neighbor qubits and tuned by flux-biased superconducting
quantum interference devices (SQUIDs), naturally integrating with both fixed
and flux-tunable superconducting qubit fabrication capabilities [4].

Advancements in the growth of superconductor-semiconductor (super-semi)
structures for use in gate-tunable Josephson junctions have led to proposals
[5, 6] and experimental demonstrations of voltage-controlled coupling schemes,
superconducting quantum storage units [7], and readout resonator buses [8].
Unlike their conventional transmon qubit [9] counterparts, whose energies are
either fixed by their shunt capacitors or tuned with magnetic fluxes threading
SQUID loops [10], these hybrid quantum systems consist of epitaxial III-V
semiconductor layers whose properties are tunable with precise composition
control and applied electric fields.

Challenges in optimizing materials and fabrication processes remain to realize
high coherence gatemon [11] qubits and other voltage-tunable super-semi
devices. These gatemon qubits differ from their flux-tunable and fixed
frequency transmon counterparts in that their Josephson junctions are formed
by superconductor-semiconductor-superconductor junctions and their Josephson
energies $E\textsubscript{J}$ are tunable by an external electric potential.
Although achieving coherences of two dimensional electron gas (2DEG)-based
gatemon qubits at parity with conventional transmon-like qubits remains an
open area of research, similar systems acting as low participation couplers
still offer fast, high contrast control with a tolerable reduction in system
coherence. Recent experimental demonstrations of tunable resonators using the
same materials stack, achieved an on/off coupling ratio between resonators of
one order of magnitude, a promising first step towards realizing fast,
voltage-tunable couplers [12].

We propose a voltage-controlled capacitive coupling element between
neighboring superconducting qubits using a III-V semiconductor 2DEG in an
InAs/InGaAs heterostructure. The capacitance of the coupler tunes as a
function of a gate voltage or series of gate voltages applied to the 2DEG,
repelling electrons away from the region underneath the gates. By “parting the
sea of electrons” in the quantum well, the coupler straddles two limits –
fully conducting and fully depleted or insulating. In the intermediate region,
the area of the depleted charges acts as an effective dielectric of some width
$d$, and the capacitance of the coupler decreases with increasing width, as
one might expect a parallel plate capacitor to behave as the separation
between the plates increases. From this simple operational principle and
reduction in sensitivity to bias line fluctuations, we expect such a coupler
to be a drop-in replacement for SQUID-based inductive couplers [13].

Additional capacitors between the coupler and the qubits may minimize unwanted
electric field coupling to other qubits. This is an improvement over SQUID-
based couplers, where stray magnetic fields can lead to classical cross-talk
between qubits [14]. We suspect that the 2DEG coupler may introduce more
charge noise than the inductive couplers through the voltage control lines,
yet transmon qubits, our initial targets for qubit-coupler integration
testing, are exponentially insensitive to this charge noise.

Figure 1: Schematic of two transmon qubits and the 2DEG coupler. Blue regions
(color online) correspond to low electron concentration or effective
dielectrics and red regions correspond to high electron concentration or
effective conductors. We use the labeling of the voltage nodes $V_{i}$
throughout the text, where nodes 1 and 2 correspond to source and drain
terminals, and node 3 refers to the gate terminal.

The structure of the paper is as follows. We start by presenting a conceptual
design of the coupler in Section 2.1. In Section 2.2, we formulate rigorous
numerical models of the 2DEG coupler, starting with COMSOL semiconductor
electron density calculations, followed by additional electrostatic and
frequency-domain COMSOL simulations of the capacitance and admittance
matrices, respectively. That section concludes with a summary of the
dielectric and other loss mechanisms present in the III-V semiconductor and
dielectric materials in the coupler. Section 3.2 details our ANSYS high
frequency simulation software (HFSS) simulations of a prototypical two
transmon qubit circuit coupled by a lumped element capacitor representing the
2DEG coupler. We apply energy participation ratio techniques [15] to extract
the Hamiltonian matrix elements in the dispersive regime, and extend these
calculations to compute the charge-charge interaction matrix elements between
the two transmon qubits. These analyses give similar results when considering
a single lumped element variable capacitor representing the coupler compared
to a full parasitic capacitance model of the coupler from our electrostatic
COMSOL simulations.

##  2 Methods and Modeling

###  2.1 Conceptual Design

Inspired by textbook parallel plate capacitors whose capacitance varies
inversely with plate separation, our coupler design relies on similar carrier
dynamics as field effect transistors to electronically modify the effective
parallel plate capacitor geometry seen by neighboring qubits. We consider a
proximitized semiconductor [16] sandwiched between two transmon-like qubits
with large capacitor plates patterned on top and a metal-oxide gate separating
the two plates. Applying a negative gate voltage decreases the carrier
concentration directly below the gate, modifying the capacitor geometry by
increasing the effective separation of the parallel plates. The high electron
mobility of the carriers in the 2DEG, exceeding 14 000 cm2V-1s-1 at 20 mK,
[17] allows for fast gating, enabling parametric interactions with rapidly
oscillating gate voltages.

This concept generalizes to multiple gates, where each region of low electron
concentration corresponds to an effective dielectric and each region with high
electron concentration acts a conductor. The effective capacitance seen by the
two qubits is the series combination of the individual capacitances defined by
alternating effective dielectrics and conductors. Similar gating schemes have
been proposed for nonreciprocal devices [18], tunable quantum buses [8], and
controlled-Z gates [5].

Apart from the aforementioned experimental demonstrations of these devices,
few modeling efforts, if any, have explored the practical considerations of
realizing such couplers. The following numerical simulations aim to address
those concerns by estimating the capacitive tuning range in the presence and
absence of parasitic capacitances, calculating relevant interaction matrix
elements, and providing an upper bound on the losses inherited by the system
from the dielectric materials of the coupler.

###  2.2 Classical Modeling

####  2.2.1 Semiconductor 2DEG Calculations

To estimate the capacitance of the 2DEG coupler, we compute the electron
concentrations in the active region of the device (InGaAs/InAs/InGaAs layers)
using the COMSOL Multiphysics Semiconductor Module [19]. Equilibrium solutions
to the drift-diffusion equations with Fermi-Dirac statistics serve to identify
regions of high depletion under the gate(s) when applying negative voltages on
the order of a few volts, overcoming the work function of the aluminum gate
contact.

We use a layer structure typical of gatemon qubits as in figure 2 and refer to
this structure as the “device stack” [20, 17, 21]. To model the device stack
in COMSOL, we specified the following electronic properties of the
semiconductor materials and the dielectric constant of the gate oxide:
electron and hole effective conduction band masses
$m^{\ast}_{\mathrm{n(p),c}}$, low-field mobilities
$\mu^{\mathrm{lf}}_{\mathrm{n(p)}}$, band gap energies $E\textsubscript{g}$,
conduction band offsets $\Delta E\textsubscript{c}$ between neighboring
semiconductors, dielectric constants $\varepsilon\textsubscript{r}$, and
effective densities of states for the conduction and valence bands
$N\textsubscript{c(v)}$. Taking the electron affinity $\chi$ for InAs as given
by the COMSOL material library, we calculated the remaining affinities using
Anderson’s affinity rule and the conduction band offsets of each material
[22]. Table 1 gives a summary of the material parameters used in these
semiconductor simulations; see A for detailed calculations of the energy gaps,
effective masses, and conduction band offsets for InxGa1-xAs and InxAl1-xAs as
functions of the composition parameter (x).

Figure 2: Schematic of the 2DEG coupler as modeled in COMSOL Multiphysics based on [13]. An aluminum contact deposited on AlxOy defines the gate terminal. We abbreviate the fixed composition ternary III-V alloys In0.81Ga0.19As and In0.81Al0.19As, as InGaAs and InAlAs, respectively. Not shown or modeled is the superlattice graded buffer layer between InP and InAlAs [20, 17, 21]. |  InAs |  InGaAs |  InAlAs |  InP
---|---|---|---|---
$E\textsubscript{g}$ [eV] |  0.354 |  0.473 |  0.752 |  1.344
$\Delta E\textsubscript{c}$ [eV] |  - |  0.200 |  0.201 |  0.12
$\varepsilon_{r}$ |  15.15 |  14.03 |  13.13 |  12.9
$N\textsubscript{c}$ [cm-3] |  6.6E18 |  1.4E17 |  2.1E17 |
$N\textsubscript{v}$ [cm-3] |  8.73E16 |  6.4E18 |  7.8E18 |
$\chi$ [eV] |  4.9 |  4.7 |  4.5 |  4.38
$\mu^{\mathrm{lf}}_{\mathrm{n(p)}}$ [cm2V-1s-1] |  14.4E3 (500) |  14.4E3 (450) |  14.E3 (384) |  5.4E3 (200)
$m^{\ast}_{\mathrm{n(p),c}}$ [$m_{0}$] |  0.023 (1.00) |  0.03 (0.25) |  0.04 (0.31) |  0.08 (0.60)
Table 1: Materials parameters used in the COMSOL Semiconductor Module
calculations. InGaAs and InAlAs abbreviate In0.81Ga0.19As and In0.81Al0.19As.
$m_{0}$ corresponds to the rest mass of an electron (0.511 MeV $c^{-2}$). Out-
of-plane effective electron and hole masses of InAs are set to $m_{0}$ in the
model to simulate 2DEG confinement in the xy-plane. Electron mobilities for
InAs, InGaAs, and InAlAs are all set to the same value as extracted from
measurements of a similar device at millikelvin temperatures [17]. Values not
in parenthesis (in parenthesis) correspond to electron (hole) properties.

We specify the geometry in figure 2 using the native COMSOL CAD editor to
define domains (surfaces or planes) and boundaries (lines or edges), solving
for the electron density in the domains and on the boundaries. Electronic
properties assigned to each domain follow from table 1. We model the terminals
(source – 1, drain – 2, gate – 3, as in figure 1) as Terminal boundary
conditions with voltages $V_{1}$, $V_{2}$, $V_{3}$ and contact work functions
$\Phi\textsubscript{c,1}$, $\Phi\textsubscript{c,2}$,
$\Phi\textsubscript{c,3}$ = 4 V [19].

We selected the density gradient discretization scheme [23] in COMSOL to
approximate the quantum confinement effects in the 2DEG more efficiently than
a self-consistent Schrödinger-Poisson equation calculation. The density
gradients modify the equilibrium electron ($n$) and hole ($p$) concentrations
by [19]

$\displaystyle
n=N\textsubscript{c}F_{1/2}\left(\frac{E\textsubscript{fn}-E\textsubscript{c}+qV^{\mathrm{DG}}_{\mathrm{n}}}{k\textsubscript{B}T}\right)$
(1) $\displaystyle
p=N\textsubscript{v}F_{1/2}\left(\frac{E\textsubscript{v}-E\textsubscript{fp}+qV_{\mathrm{p}}^{\mathrm{DG}}}{k\textsubscript{B}T}\right)$
(2) $\displaystyle N_{\mathrm{c(v)}}=\left(\frac{2m_{\mathrm{n(p)}}^{\ast}\pi
k_{\mathrm{B}}T}{h^{2}}\right)^{3/2},$ (3)

where $E_{\mathrm{c(v)}}$ is a given material’s conduction (valence) band
edge, $E_{\mathrm{fn(p)}}$ are the electron (hole) quasi-Fermi level energies,
$F_{1/2}(\eta)$ is the Fermi-Dirac integral [24], $k_{\mathrm{B}}$ is
Boltzmann’s constant, $T$ is the temperature of the device (approximate
temperature of the mixing chamber stage of typical dilution refrigerators
$\sim$10 mK), and $q$ is the charge of an electron or hole. The quantum
potentials $V_{\mathrm{n(p)}}^{\mathrm{DG}}$ are defined in terms of the
density gradients by [19]

$\displaystyle\nabla\cdot\left(\mathbf{b}_{\mathrm{n}}\nabla\sqrt{n}\right)$
$\displaystyle=\frac{1}{2}\sqrt{n}\ V_{\mathrm{n}}^{\mathrm{DG}}$ (4)
$\displaystyle\nabla\cdot\left(\mathbf{b}_{\mathrm{p}}\nabla\sqrt{p}\right)$
$\displaystyle=\frac{1}{2}\sqrt{p}\ V_{\mathrm{p}}^{\mathrm{DG}},$ (5)

with the density gradient tensors $\mathbf{b}_{\mathrm{n(p)}}$ for electrons
(holes) expressed in terms of the effective mass tensors
$\mathbf{m}_{\mathrm{n(p)}}^{\ast}$

$\displaystyle\mathbf{b}_{\mathrm{n}}=\frac{\hbar^{2}}{12q}\left[\mathbf{m}_{\mathrm{n}}\right]^{-1}$
(6)
$\displaystyle\mathbf{b}_{\mathrm{p}}=\frac{\hbar^{2}}{12q}\left[\mathbf{m}_{\mathrm{p}}\right]^{-1}.$
(7)

Note the distinction between the scalar effective masses
$m_{\mathrm{n(p)}}^{\ast}$ and, the effective mass tensors
$\mathbf{m}_{\mathrm{n(p)}}$. Anisotropy in the effective mass tensors
emulates the quantum confinement effects in the 2DEG, constraining electron
movement to one plane.

For the remaining materials, InAlAs, Al2O3, and air, we used the Electric
Charge Conservation interface, including the following constitutive relations
for each dielectric in terms of its electric permittivity tensor
$\boldsymbol{\varepsilon}$ [19]

$\displaystyle\mathbf{D}=\varepsilon_{0}\boldsymbol{\varepsilon}:\mathbf{E},$
(8)

where $\mathbf{D}$ is the electric displacement field, $\varepsilon_{0}$ is
the permittivity of free space, $\mathbf{E}$ is the electric field, and
$\boldsymbol{\varepsilon}:\mathbf{E}$ is a tensor contraction (matrix vector
product) between $\boldsymbol{\varepsilon}$ and $\mathbf{E}$. Modeling these
regions as pure dielectrics reduces the size of the system of equations
relative to a drift-diffusion calculation applied to the materials that behave
as perfect insulators, e.g. large dielectric layers (InP and InAlAs) and oxide
layers.

Figure 3: Electron concentrations [cm-3] on a base-10 logarithmic scale with
source-drain bias $V_{\mathrm{sd}}$ = +5 mV - (-5 mV) = 10 mV for (a) the
fully conducting $V_{\mathrm{g}}$ = 0 V and (b) fully depleted
$V_{\mathrm{g}}$ = -4 V operating points. We do not solve for $n$ in the
regions where we applied the Electric Charge Conservation equations, i.e. in
the AlxOy, InAlAs, and InP regions, yet the electric fields respect the
boundary conditions set by those layers. The vertical axis is scaled by a
factor of four relative to the horizontal to improve the visibility of the
active region.

####  2.2.2 Electrostatic Capacitance Matrix Calculations

Following the semiconductor calculations described in the previous section, we
developed a simplified model to extract the capacitance and conductance
matrices. This model replaces regions with high electron concentration with
perfect electric conductors and regions with low electron concentration with
perfect dielectrics. The COMSOL Electrostatics Interface defines the Maxwell
Capacitance Matrix $\mathbf{C}$ in terms of the charges and voltages on each
node of an $N$-terminal network as [25]

$\displaystyle\begin{pmatrix}Q_{1}\\\ Q_{2}\\\ \vdots\\\
Q_{N}\end{pmatrix}=\begin{pmatrix}C_{11}&C_{12}&\ldots&C_{1N}\\\
C_{21}&C_{22}&\ldots&C_{2N}\\\ \vdots&\vdots&\ &\vdots\\\
C_{N1}&C_{N2}&\ldots&C_{NN}\end{pmatrix}\begin{pmatrix}V_{1}\\\ V_{2}\\\
\vdots\\\ V_{N}\end{pmatrix},$ (9)

where $Q_{k}$ and $V_{k}$ are the charges and voltages on the $k$-th terminal.
The charges and voltages follow from solutions to Poisson’s equation, and the
charge continuity equation. In two dimensions these equations combine to give
[25]

$\displaystyle-\nabla\cdot d\left(\varepsilon_{0}\nabla
V-\mathbf{P}\right)=\rho,$ (10)

where $V$ is the electric potential, $\mathbf{P}$ is the polarization vector,
$\rho$ is the space charge density, and $d$ is the out-of-plane thickness.
This model gives an approximation of the electron densities from the two-
dimensional semiconductor interface calculations in the previous section, with
$d$ = 5 $\mu$m.

We set the boundaries of domains representing perfect electric conductors as
Floating Potential boundary conditions and define terminals as ohmic contacts
on the corresponding dielectrics in figure 2 [25].

In the fully conducting and depleted limits ($V_{\mathrm{g}}$ = 0 V and
$V_{\mathrm{g}}$ = -4 V), we compute the capacitance matrices in figures 4 (a)
and (b). The matrix elements of interest, $C_{12}=C_{21}$ represent the
effective capacitance between the source and drain terminals. These terminals
form capacitive contacts with any pair of qubits. The tuning ratio $r$, or
on/off contrast of the 2DEG coupler is given by $r$ =
$C_{12,\mathrm{conducting}}$ / $C_{12,\mathrm{depleted}}$ $\approx$ 29 from
the electrostatic solutions in figures 4 (a) and (c).

Figure 4: Capacitance matrix computed with the COMSOL Electrostatics Interface
for (a) the fully conducting and (c) the fully depleted limit of the 2DEG.
Capacitance matrix extracted from the admittance matrix computed with the
COMSOL Electric Currents Interface for (c) the fully conducting and (d) the
fully depleted limit of the 2DEG. Vertical and horizontal axes labels
correspond to terminal indices in figure 1. The tuning ratio between the
conducting and depleted 2DEG is given by
$r=C_{12,\mathrm{conducting}}/C_{12,\mathrm{depleted}}>20$.

####  2.2.3 Electric Currents Admittance Matrix Calculations

To extract the conductance matrix and verify the capacitance matrix of the
device under high frequency excitation agrees with the electrostatic result,
we use the AC/DC Electric Currents Interface to compute the admittance matrix
$\mathbf{Y}$ defined in terms of the $N$ terminal voltages $V_{k}$ and
currents $I_{k}$ [25]

$\displaystyle\begin{pmatrix}I_{1}\\\ I_{2}\\\ \vdots\\\
I_{N}\end{pmatrix}.=\begin{pmatrix}Y_{11}&Y_{12}&\ldots&Y_{1N}\\\
Y_{21}&Y_{22}&\ldots&Y_{2N}\\\ \vdots&\vdots&\ &\vdots\\\
Y_{N1}&Y_{N2}&\ldots&Y_{NN}\end{pmatrix}\begin{pmatrix}V_{1}\\\ V_{2}\\\
\vdots\\\ V_{N}\end{pmatrix}$ (11)

In the frequency domain, the voltages and currents become phasors of the form
$\widetilde{V}_{k}e^{i\omega t}$ and $\widetilde{I}_{k}e^{i\omega t}$, with
the admittance matrix given by

$\displaystyle\mathbf{Y}=\mathbf{G}+i\omega\mathbf{C},$ (12)

where $\mathbf{G}$ and $\mathbf{C}$ are the conductance and capacitance
matrices, $i=\sqrt{-1}$, and $\omega$ is the angular frequency [25]. Both
matrices are symmetric for our linear, three-terminal device in figure 4.

The Electric Currents Interface in COMSOL computes the currents and voltages,
in (11) by solving the time harmonic equations [25]

$\displaystyle\nabla\cdot\left(\boldsymbol{\sigma}:\mathbf{E}+\mathbf{J}_{e}\right)+i\omega\rho=0,$
(13) $\displaystyle\nabla\cdot\mathbf{D}=\rho,$ (14)

where $\mathbf{E}$ is the electric field, $\mathbf{J}_{e}$ is the externally
generated electric current density, $\boldsymbol{\sigma}$ is the electrical
conductivity tensor, and $\rho$ is the electric space charge density. We
compute the currents at each terminal, given voltage source excitations, as a
function of frequency $\omega$ in the band of 1-10 GHz relevant to
superconducting qubit and resonator frequencies, and extracted the conductance
matrix as the real, frequency-independent part of $\mathbf{Y}$ and the
capacitance matrix as the derivative of the imaginary part of $\mathbf{Y}$
from (12)

$\displaystyle G_{ij}=\mathrm{Re}\\{Y_{ij}(\omega)\\}$ (15) $\displaystyle
C_{ij}=\mathrm{Im}\left\\{\frac{\rmd Y_{ij}(\omega)}{\rmd\omega}\right\\},$
(16)

where $\rmd Y_{ij}(\omega)/\rmd\omega$ is a constant in our case, as we omit
the junction inductance $L_{\mathrm{J}_{0}}$ leading to discontinuities at
resonance frequencies proportional to $(C_{ij}L_{\mathrm{J}_{0}})^{-1/2}$
[26].

##  3 Results

Figure 4 compares the electrostatic and frequency domain capacitance matrices,
with the two calculations in good agreement. We summarize the conductance and
resistance matrix elements in table 2, with the columns corresponding to the
same terminal indices as the rows, labeled 1–3 from left to right. The AC/DC
2DEG coupler on/off ratio is given by
$r=C_{12,\mathrm{conducting}}/C_{12,\mathrm{depleting}}\approx 25$, agreeing
with the result from the electrostatic calculations.

Terminal Index |  $G_{ij}$ [S]
---|---
1 (d) | 2.63 | -2.63 | -1.11E-10
2 (d) | -2.63 | 2.63 | -1.11E-10
3 (d) | -1.11E-10 | -1.11E-10 | 2.22E-10
1 (c) | 1.35E5 | -1.35E5 | -6.75E-16
2 (c) | -1.35E5 | 1.35E5 | -6.75E-16
3 (c) | -6.75E-16 | -6.75E-16 | 1.53E-15
Terminal Index |  $R_{ij}=[\mathbf{G}^{-1}]_{ij}$ [$\Omega$]
1 (d) | -4.505E9 | -4.505E9 | -4.505E9
2 (d) | -4.505E9 | -4.505E9 | -4.505E9
3 (d) | -4.505E9 | -4.505E9 | 0
1 (c) | -8.395E14 | -8.395E14 | -7.407E14
2 (c) | -8.395E14 | -8.395E14 | -7.407E14
3 (c) | -7.407E14 | -7.407E14 | 0
Table 2: Conductance and resistance matrix elements extracted from the
admittance matrix using the COMSOL Electric Currents Interface in (c) the
fully conducting and (d) the fully depleted limit of the 2DEG.

###  3.1 Coupler Loss Estimates

We give bounds on the losses introduced by the 2DEG coupler from experimental
measurements of the high participating gate dielectrics and InGaAs upper
layer, along with the other layers in the device stack. In table 3, we compute
the electric field participation ratios $p_{j}$ following the procedure
developed by [27, 28]. The relaxation time $T_{1}$ at a given angular
frequency $\omega$, as a function of the dielectric material properties and
geometric factors, reads [28]

$\displaystyle T_{1}^{-1}$
$\displaystyle=\frac{\omega}{Q}=\omega\sum_{j}\frac{p_{j}}{Q_{j}}+\Gamma_{0}$
(17) $\displaystyle Q_{j}^{-1}$ $\displaystyle=\tan\delta_{j}$ (18)
$\displaystyle p_{j}$
$\displaystyle=W_{\mathrm{e}}^{-1}t_{\mathrm{oxide}}\varepsilon_{1,j}\int_{S_{j}}\left|\mathbf{E}\right|^{2}dS$
(19) $\displaystyle W_{\mathrm{e}}$ $\displaystyle=\int_{V}|\mathbf{E}|^{2}dV$
(20)

where $W_{e}$ is the electric field energy density stored in the volume of the
entire geometry $V$, $Q_{j}$ are the quality factors, $\tan\delta_{j}$ are the
loss tangents, $\varepsilon_{1,j}$ are the real parts of the dielectric
function, and $t_{\mathrm{oxide}}$ is the thickness of the participating lossy
surface, assumed to be 3 nm for all materials [28]. The participation ratios
give the fraction of the electrical energy stored in a given surface $S_{j}$
relative to the total electrical energy stored in the entire volume of the
device. The last term in (17), $\Gamma_{0}$, includes all other loss
mechanisms contributing to $T_{1}$ besides dielectric loss [28]. Note, these
participation ratios differ from those in subsequent calculations involving
energy participation ratios referenced to a given mode rather than a
particular surface.

Other sources of loss relevant to III-V semiconductor materials, but not
considered in this study, include piezoelectricity [29], non-equilibrium
quasiparticles [30], cosmic ray muon flux [31], and, to a lesser extent, stray
magnetic fields [32].

Depleted |  $t_{j}$ [nm] | $p_{j,\mathrm{norm}}$ | $\tan\delta_{j}^{\ast}$ |  $T_{1}$ [$\mu$s]
---|---|---|---|---
InGaAs (Top) | 10 | 4.19E-2 | 4.1E-4 | 1.85E+0
InAs | 4 | 1.03E-2 | 4.1E-4 | 7.53E+0
InGaAs (Bottom) | 4 | 7.85E-3 | 4.1E-4 | 9.89E+0
InAlAs | 100 | 2.92E-2 | 4.1E-4 | 2.67E+0
Al2O3 [33] | 50 | 9.04E-1 | 5E-3 | 6.87E-3
InP | 3.5E3 | 6.97E-3 | 4.1E-4 | 8.95E+0
Total | - | 1 | 7.3E-3 | 6.81E-3
Conducting |  $t_{j}$ [nm] | $p_{j,\mathrm{norm}}$ | $\tan\delta_{j}^{\ast}$ |  $T_{1}$ [$\mu$s]
InGaAs (Top) | 10 | 1.01E-8 | 4.1E-4 | 7.69E+6
InAs | 4 | 3.73E-9 | 4.1E-4 | 2.08E+7
InGaAs (Bottom) | 4 | 4.03E-9 | 4.1E-4 | 1.93E+7
InAlAs | 100 | 1.10E-9 | 4.1E-4 | 7.06E+7
Al2O3 [33] | 50 | 9.9999E-1 | 5E-3 | 6.24E-3
InP | 3.5E3 | 1.41E-5 | 4.1E-4 | 4.43E+3
Total | - | 1 | 7.3E-3 | 6.24E-1
Table 3: Participation ratios $p_{j}$, dielectric loss tangents
$\tan\delta_{j}$, layer thicknesses $t_{j}$, and estimated dielectric-loss-
limited $T_{1,j}$. All $T_{1,j}$ times are referenced to a qubit frequency of
$\omega/2\pi$ = 5 GHz and $\tan\delta_{j}^{\ast}$ indicates that in the
absence of reliable loss tangent data for the individual InAs, InGaAs, InAlAs,
and InP layers, we used the low power loss extracted from measurements of an
Al patterned CPW resonator on the full III-V stack modeled in this work and
measured at 100 mK as an estimate [34].

###  3.2 Integration with Circuit QED

####  3.2.1 Two Qubit Coupler

In figure 5 we have a microwave circuit model of two transmon qubits coupled
by a lumped impedance
$Z_{\mathrm{JJ}_{\mathrm{c}}}(\omega)=1/\left(1/R+i\omega C\right)$, where $R$
and $C$ take the values of $R_{12}$ and $C_{12}$ in either the fully
conducting or fully depleted limits of the 2DEG coupler. In the conducting
limit, where some current can flow across the coupler and act like a Josephson
junction, one might consider adding an inductance to the coupler lumped
element model. Taking $R_{12}$ to be the normal resistance of a Josephson
junction and computing the junction inductance with the Ambegaokar-Baratoff
formula [35], the junction inductances would be very small, on the order of a
few aH to tens of fH, resulting in high coupler mode frequencies, far outside
of the frequency band of the finite element electromagnetic field solver,
Ansys HFSS. For this reason and expected small modifications to qubit-qubit
interactions, we omit these inductances in our model and use Ansys HFSS to
compute the lowest electromagnetic eigenmodes of the device with the two
transmon qubits, indexed by $j$, defined as parallel $LC$ lumped elements,
$Z_{\mathrm{JJ}_{\mathrm{q}},j}=1/\left(1/(i\omega L_{\mathrm{q},j})+i\omega
C_{\mathrm{q},j}\right).$ In the following section, we use these eigenmode
solutions to estimate the Hamiltonian matrix elements corresponding to qubit-
qubit mode and qubit-coupler mode coupling strengths. We will differentiate
between this modal coupling from direct capacitive coupling in the final part
of this section, where we calculate the direct charge-charge interaction
matrix elements.

Figure 5: False color geometry of two transmon qubits with the capacitive
coupler in between used in the HFSS simulations. Lumped impedances defined in
the gold regions of the insets, represent the linear response of the Josephson
junctions and capacitive coupling element in the HFSS model.

####  3.2.2 Energy Participation Ratios and Quantization

To extract the coupling matrix elements between the qubits in our microwave
device layout, we employ the energy participation ratio (EPR) method developed
by Minev [36]. This approach goes beyond the larger family of black box
quantization methods [26, 37], where the Hamiltonian describes a collection of
Josephson junction-based qubits interacting with any number of harmonic modes
separates into linear and nonlinear terms.

One can relate the modal decomposition of the classical electromagnetic
response, e.g. impedance, admittance, or electromagnetic energies, with the
linear parts of the Hamiltonian. Additional inputs describing the Josephson
junction energy scales, $E_{\mathrm{J}}$ and $E_{\mathrm{C}}$, related to the
inductive and capacitive energies of the junction, account for the nonlinear
terms. The total Hamiltonian, accounting for $M$ modes, in the dispersive
regime and under the rotating wave approximation, reads

$\displaystyle H$ $\displaystyle=H_{\mathrm{lin}}+H_{\mathrm{nl}}$ (21)
$\displaystyle H_{\mathrm{lin}}/\hbar$
$\displaystyle=\sum_{m=1}^{M}\omega_{m}a_{m}^{\dagger}a_{m}$ (22)
$\displaystyle H_{\mathrm{nl}}/\hbar$
$\displaystyle=-\sum_{m=1}^{M}\left(\Delta_{m}a_{m}^{\dagger}a_{m}+\frac{1}{2}\alpha_{m}a_{m}^{\dagger\
2}a_{m}^{2}\right)$ (23) $\displaystyle+\frac{1}{2}\sum_{m\neq
n}\chi_{mn}a_{m}^{\dagger}a_{m}a_{n}^{\dagger}a_{n},$

where the Lamb shifts $\Delta_{m}$, cross-Kerr coefficients $\chi_{mn}$, and
anharmonicities $\alpha_{m}$ are given by [36]

$\displaystyle\Delta_{m}$ $\displaystyle=\frac{1}{2}\sum_{n=1}^{M}\chi_{mn}$
(24) $\displaystyle\chi_{mn}$ $\displaystyle=-\sum_{j\in
J}\frac{1}{2}\frac{\hbar\omega_{m}\omega_{n}}{4E_{\mathrm{J}_{j}}}$ (25)
$\displaystyle\alpha_{m}$ $\displaystyle=\frac{1}{2}\chi_{mm}.$ (26)

Table 4 gives a summary of the cross- and self-Kerr (anharmonicities)
coefficients along with the three lowest electromagnetic modes of the device
as extracted with the pyEPR Python package [36, 15]. The eigenfrequencies and
quality factors in the first two columns of table 4 follow from the HFSS
eigenmode solutions.

Qubit Index |  $\omega/2\pi$ [GHz] |  Q |  $\chi/2\pi$ [MHz]
---|---|---|---
1 (d) |  6.0228 |  1.7E7 |  129 |  1.02
2 (d) |  8.6135 |  5.1E9 |  1.02 |  0.002
1 (c) |  6.0228 |  4.5E8 |  223 |  67.1
2 (c) |  8.6135 |  1.3E9 |  67.1 |  223
Table 4: Extracted matrix elements from energy participation ratio
calculations for values of $R_{12}$, $C_{12}$ in the fully depleted (d) and
fully conducting (c) limits of the 2DEG. Diagonal entries of $\chi$ are scaled
by 1/2 to denote the anharmonicities from (26). Figure 6: Electric field
magnitude (dB scale to enhance color contrast) for the first two eigenmode
solutions computed with HFSS. (a) 6.0228 GHz and (b) 8.6135 GHz qubits in the
fully depleted limit of the coupler.

####  3.2.3 Extraction of the Exchange Interaction

To compute the charge-charge interaction strength between the transmon qubits
in our HFSS model, we consider the capacitance matrix associated with a
persistent current or flux qubit following the derivation by Orlando [38]. For
details on the derivation of the capacitance matrix, see B. The Hamiltonian
for the coupled two transmons, written in terms of the Josephson junction
phases $\varphi_{j}$ and node charges $q_{j}$, is given by

$\displaystyle H$
$\displaystyle=\frac{1}{2}\mathbf{Q}^{T}\mathbf{C}^{-1}\mathbf{Q}+U(\mathbf{\varphi})$
(27) $\displaystyle U(\mathbf{\varphi})$
$\displaystyle=\sum_{j}E_{\mathrm{J}_{j}}\left(1-\cos\varphi_{j}\right)$ (28)
$\displaystyle\mathbf{C}$ $\displaystyle=\begin{pmatrix}C_{1}+C_{3}&-C_{3}\\\
-C_{3}&C_{2}+C_{3}\end{pmatrix}.$ (29)

In (27), the charge-charge matrix elements are one half the entries of the
inverse of the capacitance matrix. We numerically inverted $\mathbf{C}$ in
(29) using values for $C_{1}$, $C_{2}$ obtained from (48) and
$C_{3}=C_{12}(V_{\mathrm{g}})$ in the depleting and conducting limits. Table 5
summarizes these results, recovering an on/off interation ratio of more than
one order of magnitude, $r_{\mathrm{int}}\approx$14.

Qubit Index |  $e^{2}[\mathbf{C}^{-1}]_{ij}$ [MHz]
---|---
1 (d) |  -198 |  -2.22
2 (d) |  -2.22 |  -198
1 (c) |  -161 |  -31.6
2 (c) |  -31.6 |  -161
Table 5: Simplified capacitance model charge-charge interaction matrix elements in the (d) depleting and (c) conducting limits of the 2DEG coupler. Node Index |  $e^{2}[\mathbf{C}^{-1}]_{ij}$ [MHz]
---|---
1 (d) |  -195 |  -2.30 |  -49.2
2 (d) |  -2.30 |  -195 |  -49.2
3 (d) |  -49.2 |  -49.2 |  -1.44E4
1 (c) |  -125 |  -24.7 |  -37.5
2 (c) |  -24.7 |  -125 |  -37.5
3 (c) |  -37.5 |  -37.5 |  -1.17E4
Table 6: Parasitic capacitance charge-charge interaction matrix elements in
the (d) depleting and (c) conducting limits of the 2DEG coupler.

We emphasize here that the off-diagonal charge-charge interaction matrix
elements give a more accurate description of the coupling between the qubits
mediated by the 2DEG coupler than the EPR calculations of the cross-Kerr
coefficients.

A more detailed treatment of the coupler including the parasitic capacitances
from the Maxwell capacitance matrices in figure 4 gives a three-by-three
coupling capacitance matrix in the Lagrangian of the form in (53). Table 6
summarizes the modified charge-charge interaction matrix elements in the full
parasitic capacitance model. In the parasitic capacitance model, we find an
on/off interaction ratio of $r_{\mathrm{int}}\approx 11$.

###  3.3 Estimation of Coupler Coherence Limit

To estimate the total coherence limit of our coupler in the two qubit device
in figure 6, a back-of-the-envelope calculation of the energy stored in the
coupler surface, in either qubit mode, gives an electric field participation
ratio on the order of 10-3, resulting in a coherence limit of a few to tens of
$\mu$s, when considering the loss to be dominated by the gate dielectric and
the top InGaAs layer. This is consistent with previous studies of transmon
qubits whose participations are near unity in their given mode [26, 15] and
gives us further confidence that a coupler of a similar geometry could support
fast parametric operations with moderate coherence.

##  4 Discussion

From our COMSOL simulations of the 2DEG semiconductor physics, electrostatic
and electric current analyses, we modeled a tunable capacitor with an order of
magnitude on/off contrast. The numerical results agree with the schematic
picture of modulating a parallel plate geometry by gating a high mobility
2DEG. At the level of estimating the lumped capacitance and resistance inputs
to HFSS, our models incorporate 2D semiconductor behavior in greater detail
than previous mixed experimental/computational reports [17, 13]. Two models of
the coupler, with and without the parasitic capacitances extracted from
COMSOL, give similar on/off interaction ratios and similar absolute
interaction strengths from a few MHz to tens of MHz, on the same order of
magnitude as flux-tunable couplers [14, 3].

Although the qubit-qubit matrix elements between the two models, the stray
charge-charge matrix elements, $C^{-1}_{13}$ and $C^{-1}_{23}$ in the
parasitic model, are on the same order of magnitude as the qubit-qubit
interaction matrix element. This highlights the importance of simulating the
full capacitance matrix of multi-terminal coupler devices, motivating the
choice of single gate over multi-gate coupler designs [39].

Our coherence estimates further highlight the that incorporating our coupler
design with existing transmon qubit designs comes at a modest reduction in
system coherence. With coupler coherences limiting the system coherence to
tens of $\mu$s and expected improvements in the base coherences of the coupler
materials, we are optimistic that future couplers using a similar operational
principle as 2DEG coupler may incur a lower system coherence penalty at the
same low participation as modeled here.

##  5 Conclusion

We simulated a 2DEG-based, voltage-controlled tunable coupler compatible with
superconducting qubits. With an estimated tuning ratio of one order of
magnitude and a high impedance of the off state, our proposed design is an
excellent candidate for coupling superconducting qubits. HFSS simulations and
subsequent capacitance matrix inversion analysis suggest that the coupling
matrix elements exhibit the same range of tunability, and estimates of the
loss suggest that the gate oxides limit the lifetime of the coupler to nearly
ten $\mu$s and the top InGaAs layer limit is several tens of $\mu$s. Low loss
gate dielectrics such as tantalum oxide [40] and hexagonal boron nitride [41],
along with improvements in the fabrication of the III-V stack may increase
these coherence limits in the near term. Subsequent design iterations may look
to reduce the parasitic capacitances with geometric optimization techniques to
maximize contrast and minimize stray interactions [42]. These results, coupled
with the benefit of exponential suppression of charge noise over first order
sensitivity flux noise in SQUID-based couplers, give us confidence that
voltage-controlled coupling elements of the form developed here have the
potential to supplant their inductive counterparts as the coupler-of-choice in
superconducting qubit systems.

## Acknowledgments

We acknowledge funding from NSF Grant PHY-1653820 and Army Research Office
Grant Nos. W911NF-18-1-0125 and W911NF-18-1-0067. N. M. acknowledges funding
from the Graduate Fellowships for STEM Diversity and W. M. S. acknowledges
funding from the Army Research Office QuaCGR Fellowship. We would like to
thank Bradley Llyod, Chien Liu, Paul Niyonkuru, Alan Phillips, John Rose,
Meenakshi Singh, and David Young for many insightful discussions.

##  Appendix A III-V Ternary Alloy Parameter Calculations

Following the standard linear and quadratic interpolation schemes for III-V
ternary alloys AxB1-xC, with composition parameter $x$ and in terms of
experimentally measured values of their binary constituents, AB and BC, we
have the lattice constant $a$, energy gap $E$, and effective mass at the
$\Gamma$ point $m^{\Gamma\ast}$ as [43]

$\displaystyle a_{\text{A\textsubscript{x}B\textsubscript{1-x}C}}$
$\displaystyle=xa\textsubscript{AC}+(1-x)a_{\text{BC}},$ (30) $\displaystyle
E_{\text{A\textsubscript{x}B\textsubscript{1-x}C}}$
$\displaystyle=xE_{\text{AC}}+(1-x)E_{\text{BC}}+x(1-x)E_{\text{AB}},$ (31)
$\displaystyle m^{\Gamma\ast}_{A\textsubscript{x}B\textsubscript{1-x}C}$
$\displaystyle=xm^{\Gamma}_{\text{AC}}+(1-x)m^{\Gamma}_{\text{BC}}+x(1-x)m^{\Gamma}_{\text{AB}}.$
(32)

Similarly, the hole effective masses follow from a quadratic interpolation
scheme of the AB, AC binary components as computed from a spherical band
approximation of the valence band edge [43]

$\displaystyle
m_{\mathrm{p,dos}}=\left(m_{\mathrm{lh}}^{3/2}+m_{\mathrm{hh}}^{3/2}\right)^{2/3},$
(33) $\displaystyle
m_{\mathrm{p,c}}=\frac{m_{\mathrm{lh}}^{5/2}+m_{\mathrm{hh}}^{5/2}}{m_{\mathrm{p,dos}}},$
(34) $\displaystyle
m_{\mathrm{p,c,A\textsubscript{x}B\textsubscript{1-x}C}}=xm_{\mathrm{p,c,AC}}+(1-x)m_{\mathrm{p,c,BC}},$
(35) $\displaystyle
m_{\mathrm{p,dos,A\textsubscript{x}B\textsubscript{1-x}C}}=xm_{\mathrm{p,dos,AC}}+(1-x)m_{\mathrm{p,dos,BC}}.$
(36)

We recognize that the spherical band approximation may not apply to the III-V
materials in our study, but it gives an estimate for density of states and
conduction band effective masses that are inputs to the COMSOL Semiconductor
Module materials models.

To estimate the conduction band offsets between the InxAl1-xAs and InxGa1-xAs
layers, we followed another interpolation scheme that computes the absolute
conduction band edges $E_{\mathrm{c}}$ using experimentally measured
parameters of InAs, AlAs, and GaAs[44]

$\displaystyle E_{\mathrm{c}}$
$\displaystyle=E\textsubscript{v,avg}+\frac{\Delta_{0}}{3}+E_{\mathrm{g}}+\Delta
E_{\mathrm{c}}^{\mathrm{hy}},$ (37) $\displaystyle\Delta E_{\mathrm{c}}$
$\displaystyle=E_{\mathrm{c}}^{B}-E_{\mathrm{c}}^{A},$ (38)

where $E_{\mathrm{v,avg}}$ is the average valence band edge, $\Delta_{0}$ is
the spin-orbit splitting in the absence of strain, $E_{\mathrm{g}}$ is the
band gap energy, and $\Delta E_{\mathrm{c}}^{\mathrm{hy}}$ is the shift of the
conduction band edge due to hydrostatic strain. These parameters are
calculated from the following expressions with coefficients $C_{ij}$ read-off
from Table 3 compiled by Krijn [44]

$\displaystyle E\textsubscript{v, avg}$
$\displaystyle=\sum_{i=1}^{2}C_{i0}(E\textsubscript{v,avg})x^{i},$ (39)
$\displaystyle\Delta_{0}$
$\displaystyle=\sum_{i=1}^{2}C_{i0}(\Delta_{0})x^{i},$ (40)
$\displaystyle\Delta E_{\mathrm{c}}^{\mathrm{hy}}$ $\displaystyle=\frac{\Delta
a(x)}{a(x)}\sum_{i=0}^{1}C_{i0}(\Delta E_{\mathrm{c}}^{\mathrm{hy}})x^{i},$
(41) $\displaystyle\Delta a(x)$ $\displaystyle=a_{0}-a(x).$ (42)

##  Appendix B Charge–Charge Interaction Matrix Element Derivation

Figure 7: Coupler circuit models. (a) The 2DEG coupler compact representation
with a single gate and a pair of transmon qubits compared with (b) the
simplified circuit used in the derivation of the charge-charge interaction
matrix in the main text. (c) Parasitic capacitance circuit model of the
coupler and two transmons; capacitances taken from figure 4.

Starting from the two transmon circuit coupled by a voltage-controlled
Josephson junction (our 2DEG coupler) in figure 7 (b), with phases
$\varphi_{1},\varphi_{2},\varphi_{3}$ referring to the left, right, and
coupling junctions, respectively, we have [38]

$\displaystyle\varphi_{1}-\varphi_{2}+\varphi_{3}=-\frac{2\pi}{\Phi_{0}}\Phi_{\mathrm{ext}},$
(43)

where $\Phi_{\mathrm{ext}}$ is the flux threading the loop formed by the three
Josephson junctions as in a typical flux qubit circuit. For a finite
$\Phi_{\mathrm{ext}}$, the potential energy $U$ is given by

$\displaystyle U(\mathbf{\varphi})$
$\displaystyle=\sum_{j}E_{\mathrm{J}_{j}}(1-\cos\varphi_{j})$ (44)
$\displaystyle=E_{\mathrm{J}_{1}}(1-\cos\varphi_{1})+E_{\mathrm{J}_{2}}(1-\cos\varphi_{2})$
$\displaystyle+E_{\mathrm{J}_{3}}(1-\cos(\varphi_{2}-\varphi_{1}-2\pi\Phi_{\mathrm{ext}}/\Phi_{0})),$

with the signs on the phases following figure 7 (b), preserving the
conventions chosen in. [38] We order the phases in a single column vector as

$\displaystyle\mathbf{\varphi}=\begin{pmatrix}\varphi_{1}\\\
\varphi_{2}\end{pmatrix}.$ (45)

Setting $\Phi_{\mathrm{ext}}$ = 0, we compute the kinetic energy $T$ by using
the Josephson equation relating the voltages at nodes with $k=\left\\{1,\
2\right\\}$, $V_{k}=(\Phi_{0}/2\pi)\dot{\varphi}_{k}$ and the definition of
$T$ in terms of $\dot{\varphi}_{k}$

$\displaystyle T$
$\displaystyle=\frac{1}{2}\left(C_{1}V_{1}^{2}+C_{2}V_{2}^{2}+C_{3}V_{3}^{2}\right)$
(46)
$\displaystyle=\frac{1}{2}\left(\frac{\Phi_{0}}{2\pi}\right)^{2}\left(C_{1}\dot{\varphi}^{2}_{1}+C_{2}\dot{\varphi}^{2}_{2}+C_{3}\left(\dot{\varphi}_{2}-\dot{\varphi}_{1}\right)^{2}\right)$
$\displaystyle=\frac{1}{2}\left(\frac{\Phi_{0}}{2\pi}\right)^{2}\dot{\mathbf{\varphi}}^{T}\mathbf{C}\dot{\mathbf{\varphi}},$

and reading off the capacitance matrix

$\displaystyle\mathbf{C}=\begin{pmatrix}C_{1}+C_{3}&-C_{3}\\\
-C_{3}&C_{2}+C_{3}\end{pmatrix}.$ (47)

Relating the total capacitances (both the intrinsic junction and external
capacitance, commonly referred to as $C_{\Sigma}$ [45]) shunting the
junctions, $C_{1},\ C_{2}$, to the anharmonicities extracted from the EPR
calculations, we have, from the asymptotic expressions derived by Koch et al
[9]

$\displaystyle C_{k}$
$\displaystyle=\frac{e^{2}}{2E_{\mathrm{C}}}\simeq-\frac{e^{2}}{2\alpha_{k}}$
(48)

and we take $C_{3}$ = $C_{12}(V_{\mathrm{g}})$, the gate voltage-dependent
capacitance across the 2DEG coupler. The classical Lagrangian $\mathcal{L}$
and Hamiltonian $\mathcal{H}$ associated with the kinetic and potential
energies above, then read [38]

$\displaystyle\mathcal{L(\boldsymbol{\varphi},\boldsymbol{\dot{\varphi}})}$
$\displaystyle=T-U$ (49)
$\displaystyle=\frac{1}{2}\left(\frac{\Phi_{0}}{2\pi}\right)^{2}\dot{\boldsymbol{\varphi}}^{T}\mathbf{C}\dot{\boldsymbol{\varphi}}-\sum_{j}E_{J_{j}}(1-\cos\varphi_{j})$
$\displaystyle\mathcal{H}$
$\displaystyle=\mathbf{P}^{T}\boldsymbol{\dot{\varphi}}-\mathcal{L}$ (50)
$\displaystyle=\frac{1}{2}\mathbf{Q}^{T}\mathbf{C}^{-1}\mathbf{Q}+U(\boldsymbol{\varphi})$
$\displaystyle P_{j}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\dot{\varphi_{j}}}=\left(\frac{\Phi_{0}}{2\pi}\right)^{2}\sum_{k}C_{jk}\boldsymbol{\dot{\varphi}}_{k},\
\mathbf{Q}=\frac{2\pi}{\Phi_{0}}\mathbf{P}$ (51)

We take the form of the quantized Hamiltonian to be the same as the classical
one in (50) with classical variables promoted to operators, and identify the
charge-charge matrix elements as $e^{2}[\mathbf{C}^{-1}]_{ij}/2$. Similarly,
we write the Lagrangian and identify the capacitance matrix corresponding to
the parasitic capacitance model given by the circuit in Fig. 7 (c) as

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}\left(\frac{\Phi_{0}}{2\pi}\right)^{2}\left[(C_{1}+C_{11})\dot{\varphi}^{2}_{1}+(C_{2}+C_{22})\dot{\varphi}^{2}_{2}\right.$
(52)
$\displaystyle+\left.C_{33}\dot{\varphi}^{2}_{3}+C_{12}(\dot{\varphi}_{2}-\dot{\varphi}_{1})^{2}\right.$
$\displaystyle+\left.C_{13}(\dot{\varphi}_{3}-\dot{\varphi}_{1})^{2}+C_{23}(\dot{\varphi}_{2}-\dot{\varphi}_{3})^{2}\right]-U(\varphi)$
$\displaystyle\mathbf{C}$
$\displaystyle=\begin{pmatrix}\widetilde{C}_{11}&-C_{12}&-C_{13}\\\
-C_{12}&\widetilde{C}_{22}&-C_{23}\\\
-C_{13}&-C_{23}&\widetilde{C}_{33}\end{pmatrix},$ (53)

where $\widetilde{C}_{11}=C_{1}+C_{11}+C_{12}+C_{13}$,
$\widetilde{C}_{22}=C_{2}+C_{22}+C_{12}+C_{23}$, and
$\widetilde{C}_{33}=C_{13}+C_{23}+C_{33}$.

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