# Measurement of tunnel coupling in a Si double quantum dot based on charge
sensing

Xinyu Zhao  Xuedong Hu<EMAIL_ADDRESS>Department of Physics, University at
Buffalo, SUNY, Buffalo, New York 14260, USA

###### Abstract

In Si quantum dots, valley degree of freedom, in particular the generally
small valley splitting and the dot-dependent valley-orbit phase, adds
complexities to the low-energy electron dynamics and the associated spin qubit
manipulation. Here we propose a four-level model to extract tunnel coupling
information for a Si double quantum dot (DQD). This scheme is based on a
charge sensing measurement on the ground state as proposed in the widely used
protocol for a GaAs double dot [DiCarlo et. al., PRL 92. 226801]. Our theory
can help determine both intra- and inter-valley tunnel coupling with high
accuracy, and is robust against system parameters such as valley splittings in
the individual quantum dots.

## Introduction

Tunnel coupling is an essential element in coherent manipulation of electron
qubits in semiconductor quantum dots (QDs) (Zwanenburg  _et al._ , 2013;
Hanson  _et al._ , 2007; Gullans and Petta, 2020; Mills  _et al._ , 2019a;
Petta  _et al._ , 2010; Otxoa  _et al._ , 2019; Kervinen  _et al._ , 2019;
Penthorn  _et al._ , 2019; Joecker  _et al._ , 2019; Zhang  _et al._ , 2019;
Cota and Ulloa, 2018; Martins  _et al._ , 2017; Schoenfield  _et al._ , 2017;
Chen  _et al._ , 2017; Culcer  _et al._ , 2010a, b; Li  _et al._ , 2010,
2017). It allows single-qubit operations on a charge qubit, and exchange gates
for spin qubits (Shi  _et al._ , 2012; Koh  _et al._ , 2012; Kim  _et al._ ,
2014; Cao  _et al._ , 2016; Ferraro  _et al._ , 2014; Taylor  _et al._ , 2005;
Takeda  _et al._ , 2020). Interdot shuttling is also crucial for information
transfer on chip (Ginzel  _et al._ , 2020; Buonacorsi  _et al._ , 2020; Fujita
_et al._ , 2017; Villavicencio  _et al._ , 2013; Zhang  _et al._ , 2013; Zhao
_et al._ , 2016; Zhao and Hu, 2018; Zajac  _et al._ , 2016). With spin and
spin-charge hybrid qubits having been demonstrated as hopeful candidates for
foundational building blocks of future quantum processors (Tyryshkin  _et al._
, 2006; Bluhm  _et al._ , 2011; Petersson  _et al._ , 2012; Tyryshkin  _et
al._ , 2012; Saeedi  _et al._ , 2013; Veldhorst  _et al._ , 2014; Sigillito
_et al._ , 2015), accurately characterizing tunnel coupling between quantum
dots is an imperative task in characterizing these qubits.

A robust approach to detect tunnel coupling in a DQD was developed more than a
decade ago (DiCarlo  _et al._ , 2004) based on measuring the charge
distribution of the DQD in thermal equilibrium as a function of the interdot
detuning, then fits a two-level (2L) model (Shevchenko  _et al._ , 2010) to
obtain the tunnel coupling between the two single-dot ground states. This
measurement technique is particularly successful for a GaAs DQD, where excited
orbital states are generally several meV above the ground state while
experimental temperature is kept at about 100 mK (for a thermal energy of
$\sim 10\mu$eV), so that the 2L model including only the single-dot ground
states works perfectly DiCarlo  _et al._ (2004); Petta  _et al._ (2004).

In recent years, studies of spin qubits have focused on Si QDs because of
their superior coherence properties (Tyryshkin  _et al._ , 2006; Bluhm  _et
al._ , 2011; Petersson  _et al._ , 2012; Pla  _et al._ , 2012; Tyryshkin  _et
al._ , 2012; Saeedi  _et al._ , 2013; Veldhorst  _et al._ , 2014; Sigillito
_et al._ , 2015). The thermal equilibrium charge sensing technique has been
widely used to measure tunnel coupling between the ground orbital states of
two neighboring dots (Eenink  _et al._ , 2019; Sigillito  _et al._ , 2019;
Mills  _et al._ , 2019b; Simmons  _et al._ , 2009; Yoneda  _et al._ , 2020;
Jones  _et al._ , 2019; Ota  _et al._ , 2010; Stehlik  _et al._ , 2012;
Petersson  _et al._ , 2010). However, in Si-based QDs, the valley degree of
freedom introduces extra energy levels a fraction of meV above the ground
states, making the Si DQDs better described as a four-level (4L) system
instead of a 2L system. There are also two relevant tunnel coupling parameters
instead of one. Clearly, a 2L model cannot represent all the relevant
properties of a Si DQD. It is not even clear whether it is capable of
consistently producing accurate measurement of the ground state tunnel
coupling. Although alternative schemes such as spin-cavity coupling have been
employed to successfully measure tunnel coupling (Mi  _et al._ , 2017), the
thermal equilibrium charge sensing technique will still be the most easily
accessible and widely used in the foreseeable future. As such, it is important
to develop an updated procedure of measuring tunnel coupling in a Si DQD that
accounts for the valley dynamics.

Figure 1: The $I_{QPC}(\epsilon)$ curve measured form experiment (red-doted
markers) and the best fit curves. top-left inset: Schematic diagram of triple
dot configuration. QDs “1,2,3” are confined under the plunger gates “P1, P2,
P3”. Dot “1-2” and “2-3” forms two DQD systems. Barrier gates “B1” to “B4”
control the tunneling strength. Dot “Q” plays the role of charge sensor.
Bottom-right inset: Typical energy levels of DQD.

In this paper, we develop a four-level (4L) model for the thermal equilibrium
charge sensing measurement of tunnel coupling for a Si DQD. Specifically, we
develop a numerical 4L fitting procedure for both intra- and inter-valley
tunnel couplings of a Si DQD. We also derive a perturbative 4L fitting
formula, which speeds up the fitting procedure dramatically while maintaining
high degree of accuracy under most conditions. We apply the updated fitting
procedures on multiple sets of data obtained in a linear Si/SiGe triple
quantum dot (schematic diagram in Fig. 1) provided by Adam Mills and Jason
Petta (Mills  _et al._ , 2019b, a), and produce consistent fitting for
sometimes noisy experimental data. We compare the results from these new
fitting procedures with the conventional 2L-based approach, and find
significant differences under common conditions. For example, in a particular
DQD, we observe an average of 46% difference in the intra-valley tunnel
coupling between the 2L fitting and 4L fitting. For a particular set of data,
the two models make totally contradictory predictions on tunnel couplings,
with the 4L prediction more consistent with the experimental procedure. These
examples clearly illustrate the advantages and necessity of the 4L model in
obtaining a reliable estimate of tunnel coupling in a Si DQD. Lastly, we
analyze the robustness of our fitting procedure and identify possible errors.

## Results

### Charge distribution in a four-level model

The six-fold degeneracy of the Si conduction band is lifted by the growth-
direction (nominally the $z$ direction) confinement near an interface, which
leaves two of the bulk valleys with lower energy, denoted as $|z\rangle$ and
$|\bar{z}\rangle$ states. Scattering at the interface further couples
$|z\rangle$ and $|\bar{z}\rangle$ states (Ando  _et al._ , 1982; Culcer  _et
al._ , 2010a, b), leading to the valley eigenstates
$|\pm\rangle=\frac{1}{\sqrt{2}}\left(|z\rangle\pm e^{i\phi}|\bar{z}\right)$,
where the phase $\phi$ is determined by the interface scattering matrix
element. The energy splitting $2|\Delta|$ between $|\pm\rangle$ states is
called the valley splitting, and is typically 0.1 to 0.2 meV in a SiMOS
quantum dot and $\lesssim 0.1$ meV in a Si/SiGe dot. Compared to the few meV
orbital excitation energy in these quantum dots due to in-plane confinement,
valley splitting is much smaller, making it reasonable to neglect intra-valley
orbital excitation but include both valleys when considering charge
distribution in thermal equilibrium at low temperatures.

A minimal model for the low-energy single-electron charge distribution and
dynamics of a Si DQD should thus include the ground orbital state in each dot
(denoted as $|L\rangle$ and $|R\rangle$ for left and right dot), together with
both valley eigenstates, leading to four basis states: $\\{|L,+\rangle$,
$|L,-\rangle$, $|R,+\rangle$, and $|R,-\rangle\\}$. Considering that interdot
barrier is generally a smooth variation in electrical potential at a length
scale much larger than the lattice constant, tunneling is only allowed between
states in the same bulk valley: $\langle L,z|H|R,z\rangle=t_{C}$ is finite
while $\langle L,z|H|R,\bar{z}\rangle=0$. Using the four basis states
$|D,\pm\rangle=\frac{1}{\sqrt{2}}(|D,z\rangle\pm e^{i\phi_{D}}|D,\bar{z})$
($D=L,R$) with local phases $\phi_{D}$, the single-electron Hamiltonian in the
Si DQD can be expressed as

$H=\left[\begin{array}[]{cccc}\epsilon+|\Delta_{L}|&0&t_{+}&t_{-}\\\
0&\epsilon-|\Delta_{L}|&t_{-}&t_{+}\\\
t_{+}^{*}&t_{-}^{*}&-\epsilon+|\Delta_{R}|&0\\\
t_{-}^{*}&t_{+}^{*}&0&-\epsilon-|\Delta_{R}|\end{array}\right]\,.$ (1)

Here $\epsilon$ is the interdot detuning, $|\Delta_{L,R}|$ are the L/R valley
splittings, $t_{\pm}=\frac{1}{2}t_{C}[1\pm e^{-i\delta\phi}]$ are the intra-
and inter-valley (here “valley” means the valley eigenstates $|\pm\rangle$)
tunnel couplings, respectively, and $\delta\phi=\phi_{L}-\phi_{R}$ is the
valley phase difference between the two dots.

Hamiltonian (1) can be numerically diagonalized at any given detuning
$\epsilon$ to obtain the eigenvalues $E_{i}$ and the corresponding eigenstates
$|\Psi_{i}\rangle$ ($i=1,2,3,4$). It can also be diagonalized analytically,
though the general expressions are cumbersome and not transparent. If we treat
inter-valley tunneling as a perturbation, on the other hand, we can obtain
simple analytical expressions for $E_{i}$ and $|\Psi_{i}\rangle$, (see the
“Methods” section), which can be used in a fitting procedure much more
conveniently. Specifically, the left-dot charge distribution for each
eigenstate $|\Psi_{i}\rangle$ takes the form

$\displaystyle P_{L1}$
$\displaystyle=\cos^{2}\frac{\Theta_{1}}{2}\sin^{2}\frac{\theta_{-}}{2}+\sin^{2}\frac{\Theta_{1}}{2}\cos^{2}\frac{\theta_{+}}{2},$
$\displaystyle P_{L2}$
$\displaystyle=\sin^{2}\frac{\Theta_{2}}{2}\cos^{2}\frac{\theta_{-}}{2}+\cos^{2}\frac{\Theta_{2}}{2}\sin^{2}\frac{\theta_{+}}{2},$
$\displaystyle P_{L3}$
$\displaystyle=\cos^{2}\frac{\Theta_{2}}{2}\cos^{2}\frac{\theta_{-}}{2}+\sin^{2}\frac{\Theta_{2}}{2}\sin^{2}\frac{\theta_{+}}{2},$
$\displaystyle P_{L4}$
$\displaystyle=\sin^{2}\frac{\Theta_{1}}{2}\sin^{2}\frac{\theta_{-}}{2}+\cos^{2}\frac{\Theta_{1}}{2}\cos^{2}\frac{\theta_{+}}{2}.$
(2)

Here $\tan\Theta_{1}=\frac{|t_{-}|}{E_{+}+E_{-}+\Delta_{+}}$,
$\tan\Theta_{2}=\frac{|t_{-}|}{E_{+}+E_{-}-\Delta_{+}}$, and
$\tan\theta_{\pm}=\frac{|t_{+}|}{\epsilon\pm\Delta_{-}}$
($\theta_{\pm}\in[0,\pi]$), with
$E_{\pm}=\sqrt{(\epsilon\pm\Delta_{-})^{2}+|t_{+}|^{2}}$ and
$\Delta_{\pm}=\frac{1}{2}(|\Delta_{L}|\pm|\Delta_{R}|)$. The expressions given
in Eq. (2) become exact if $|\Delta_{L}|=|\Delta_{R}|$. A more detailed study
in the “Methods” section shows that the approximation underlying Eq. (2) is
valid in most regions of the parameter space. For example, there is only a 4%
error when $|\Delta_{L}|$ and $|\Delta_{R}|$ are different by 20%.

When the single electron in the DQD is in thermal equilibrium, its density
matrix is given by a thermal state $\rho=\sum_{i}\frac{1}{Z}e^{-\beta
E_{i}}|\Psi_{i}\rangle\langle\Psi_{i}|$, where $\beta=\frac{1}{k_{B}T}$,
$k_{B}$ is the Boltzmann constant, and $Z=\sum_{i}e^{-\beta E_{i}}$ is the
partition function. The total charge occupation in the left dot at temperature
$T$ is then

$P_{L}=\sum_{i}\frac{1}{Z}e^{-\beta E_{i}}P_{Li}\,.$ (3)

Here $P_{L}$ is a function of both tunnel couplings $t_{\pm}$, both valley
splittings $|\Delta_{L}|$ and $|\Delta_{R}|$ (with their phase difference
$\delta\phi$ already contained in $t_{\pm}$), and detuning $\epsilon$. Given
experimentally measured $P_{L}(\epsilon)$, we could thus obtain $t_{\pm}$ via
data fitting. In theory one could obtain the valley splittings from this
fitting procedure as well, though our numerical studies below show that the
results are not particularly sensitive to $|\Delta_{L,R}|$, making the
information obtained from fitting less reliable. Thus we generally treat
valley splittings as known parameters.

There are two major reasons that cause different predictions between 2L and 4L
theories. First, the different number of levels involved means that the
thermal occupations are distributed differently. The impact of this thermal
occupation is typically limited since experimental temperature is usually
about 100 mK and much smaller than valley splittings in the dots. Obvious
exceptions include cases when the valley splittings are very small (for
example $\sim 10\mu$eV, similar to the thermal energy), or when the
temperature is much higher than usual. Second, and more importantly, all
eigenstates $|\Psi_{i}\rangle$ in a Si DQD contain the valley excited states
$|+\rangle$ due to the finite inter-valley tunnel coupling. The involvement of
the excited valley states causes subtle changes to the state compositions,
which then lead to differences in the charge distribution.

In the absence of inter-valley tunneling ($\delta\phi=0$), the two valley
eigenstates decouple into their own subspaces, so that the charge distribution
is reduced to an exact analogy to the two-level case in GaAs when we neglect
the thermal occupation of the excited valley states

$P_{L}=\frac{1}{2}\left[1-\frac{\epsilon-\Delta_{-}}{2E_{+}}\tanh\left(\frac{E_{+}}{2k_{B}T}\right)\right]\,.$
(4)

This is just the fitting formula in Ref. DiCarlo  _et al._ , 2004 with an
$\epsilon$ shift caused by asymmetric valley splittings. If we impose a
further condition that the valley splittings are symmetric ($\Delta_{-}=0$),
the thermal occupation of the excited valley states would have the same left-
right distribution as the ground valley states, so that the 4L theory we
develop here would become identical to the conventional 2L model. In other
words, under the condition

$\delta\phi=0\quad\&\quad|\Delta_{L}|=|\Delta_{R}|$ (5)

Eq. (2) and Eq. (3) would lead exactly to the 2L fitting formula in Ref.
DiCarlo  _et al._ , 2004, as is Eq. (4).

### Extracting tunnel couplings from charge distributions

The functional forms for charge distribution in a Si DQD given by Eqs. (2) and
(3) allow us to obtain tunnel couplings $t_{\pm}$ (or $t_{C}$ and
$\delta\phi$) between the two dots by fitting experimentally measured
$P_{L}(\epsilon)$, similar to the procedure given in Ref. DiCarlo  _et al._ ,
2004.

As we discussed above, the charge distribution $P_{L}$ is a function of
multiple parameters and variables:
$P_{L}=P_{L}(\Delta_{L},\Delta_{R},t_{+},t_{-},\epsilon)$. To obtain more
constrained and reliable knowledge of the tunnel couplings, the valley
splittings $\Delta_{L}$ and $\Delta_{R}$ should be known beforehand, for
example through the spin relaxation hot spot for each dot Yang  _et al._
(2013). If $|\Delta_{L,R}|$ are not known a priori, one can use an estimate
instead, without creating significant errors. A detailed discussion of the
consequences of not knowing these splittings is given in the “Methods”
section.

Experimentally, what is measured is the charge sensor (for example a quantum
point contact, or QPC) current as a function of the interdot detuning:
$I_{QPC}(\epsilon)$. The current is usually assumed to be linearly related to
the charge distribution in the DQD DiCarlo  _et al._ (2004):

$I_{QPC}=I_{0}+\delta I\cdot P_{L}(\epsilon)+\delta I_{noise}\,,$ (6)

where $I_{0}$ is the background current (setting a reference point), $\delta
I$ is the linear conversion ratio, and $\delta I_{noise}$ is the noise in the
$I_{QPC}$ measurement. The first two parameters are part of the fitting
procedure and the impact of $\delta I_{noise}$ will be discussed in the
“Methods” section. In addition, the inter-dot detuning may also have a
background voltage, i.e., $\epsilon_{m}=\epsilon_{0}+\epsilon$, where
$\epsilon_{m}$ is the value measured in the experiment and $\epsilon_{0}$ is a
reference shift.

Our fitting procedure thus consists of the following steps. First we use Eq.
(3) or Eq. (4) (for 4L or 2L fitting respectively) to generate a theoretical
curve $I_{th}(\epsilon)$ with a set of candidate fitting parameters such as
$t_{\pm}$. We then calculate the deviation from the experimental data, and
minimize it by varying the fitting parameters. While the three parameters
$I_{0}$, $\delta I$, and $\epsilon_{0}$ are part of the fitting parameter set,
they take up different roles compared to $t_{\pm}$. The tunnel couplings
$t_{\pm}$ determine the “shape” of the curve, while these three parameters
determine the positions of the curve. In particular, $\epsilon_{0}$ determines
the shift in the horizontal (detuning) direction, $I_{0}$ determines the
vertical shift, while $\delta I$ is a scaling factor. None of them contributes
to the shape or curvature of the curve near $\epsilon=0$, which is determined
by $t_{\pm}$. Therefore, they can be obtained separately from the main fitting
parameters $t_{\pm}$. One can follow an adaptive fitting procedure that fit
these two groups of parameters in turn until they converge to steady values,
respectively. A discussion about the fitting inaccuracy caused by errors in
$I_{0},\delta I$, and $\epsilon_{0}$ can be found in the “Methods” section,
particularly in Fig. 7.

### Fitting actual experimental data: an example

With the procedure described above, we examine some experimental data acquired
during the tune-up of a linear array of 9 QDs used to demonstrate charge
shuttling (Mills  _et al._ , 2019b; Zajac  _et al._ , 2016; Mills  _et al._ ,
2019a). The measurements were performed on a triple dot schematically shown in
Fig. 1. It is part of a Si/SiGe 9-dot array with three QPCs as charge sensors
(Mills  _et al._ , 2019b; Zajac  _et al._ , 2016; Mills  _et al._ , 2019a).
The experimental temperature is at $T=50$ mK Mills  _et al._ (2019b) and the
valley splittings $|\Delta_{L}|$ and $|\Delta_{R}|$ are estimated to be around
66-74 $\mu$eV from spin measurements in the same device Mills . For each DQD,
the $I_{QPC}(\epsilon)$ curve are measured with four different barrier gate
voltage $V_{B2}$ (or $V_{B3}$). One set of data, together with our fitting
curve, is shown in Fig. 1. All other data sets and fitting curves are shown in
Fig. 4 in the “Methods” section.

QD 1-2 | (a) | (b) | (c) | (d)
---|---|---|---|---
2L $|t_{+}|$ | $24\pm 0.6$ | $43\pm 1.1$ | $53\pm 1.0$ | $70\pm 2.5$
4L $|t_{+}|$ (N.) | $20\pm 1.2$ | $32\pm 1.9$ | $37\pm 2.6$ | $37\pm 6.0$
4L $|t_{+}|$ (F.) | $20\pm 1.1$ | $33\pm 1.8$ | $37\pm 2.6$ | $38\pm 5.7$
4L $|t_{-}|$ | $39\pm 4.7$ | $64\pm 4.8$ | $76\pm 5.9$ | $112\pm 12$
4L $\delta\phi$ (rad) | $2.2\pm 0.05$ | $2.2\pm 0.03$ | $2.2\pm 0.04$ | $2.5\pm 0.06$
QD 2-3 | (e) | (f) | (g) | (h)
2L $|t_{+}|$ | $22\pm 0.6$ | $41\pm 0.8$ | $44\pm 0.8$ | $\bm{36}\pm 1.3$
4L $|t_{+}|$ (N.) | $22\pm 0.7$ | $41\pm 1.7$ | $44\pm 1.6$ | $26\pm 2.5$
4L $|t_{+}|$ (F.) | $22\pm 0.8$ | $41\pm 1.6$ | $44\pm 1.6$ | $26\pm 2.5$
4L $|t_{-}|$ | $0\pm 9.6$ | $0\pm 14$ | $0\pm 14$ | $62\pm 7.1$
4L $\delta\phi$ (rad) | $0\pm 0.22$ | $0\pm 0.20$ | $0\pm 0.18$ | $2.3\pm 0.04$
Table 1:  Best fitting parameters for tunnel couplings. Data set (a)-(d) are
measured from dot 1-2 and fitted with $|\Delta_{L}|=66$ $\mu$eV,
$|\Delta_{R}|=74$ $\mu$eV. Data set (e)-(h) are measured from dot 2-3 and
fitted with $|\Delta_{L}|=74$ $\mu$eV, $|\Delta_{R}|=74$ $\mu$eV. $|t_{\pm}|$
units are $\mu$eV. (N.) means using numerical diagonalization, (F.) means
using Eq. (2).

The tunnel couplings and other parameters we obtained from data fitting are
summarized in Table 1. In particular, the inter-dot valley phase difference
$\delta\phi$ for QD 1-2 in Table 1 is roughly a constant under different
applied $V_{B2}$, which implies that varying $V_{B2}$ only changes the
interdot barrier height, but does not cause the dots to shift to any
significant degree. Consequently, in the 4L model only $t_{C}$ depends on
$V_{B2}$, while $\delta\phi$ does not.

The tunnel couplings are obtained with different fitting formulas: (A). Eq.
(4), labeled as “2L”, (B). Eq. (3) and (2), labeled as “4L (F.)”, (C). Eq.
(3), with a numerical diagonalization of $H$ to calculate $P_{Li}$, labeled as
“4L (N.)”. According to the fitting results, the analytical formula Eq. (2)
provides almost the same results as numerically diagonalizing the Hamiltonian.
The average difference between 4L $|t_{+}|$ (F.) and 4L $|t_{+}|$ (N.) is only
0.9%, which indicates that Eq. (2) is very accurate here. Additional
discussion about the accuracy of Eq. (2) will be presented in following
subsections as well as in the “Methods” section.

In comparison, the 2L fitting results are quite different from the 4L results:
the average difference between 2L $|t_{+}|$ and 4L $|t_{+}|$ for QD 1-2 is 46%
across the different barrier heights in Table 1, with 2L model consistently
producing larger tunnel splittings. Qualitatively, this deviation is due to
the fact that in the 2L model we are using a single excited level to represent
the effect of three excited levels in the 4L model. As such this single
excited state needs to be above the first excited state but lower than the
third excited state in the 4L model, so that $|t_{+}|$ in the 2L model has to
be larger than that in the 4L model. The error bars are obtained by
numerically generating various stochastic realization of $\delta I_{noise}$
with the same standard deviation as the measured data and then fitting all
realizations. Here we assume the uncertainty mainly comes from the noise in
the $I_{QPC}$ signal as shown in Fig. 1, which is the conclusion from Ref.
(DiCarlo  _et al._ , 2004) as well.

We perform the curve fitting with several combinations of $|\Delta_{L}|$ and
$|\Delta_{R}|$ within the estimated range of 66 to 74 $\mu$eV. The parameters
presented in Table 1 are chosen because they produce the most consistent
fitting results for $\delta\phi$. The fitting results for other
$|\Delta_{L,R}|$ are shown in Fig. 5 in the “Methods” section.

The best fittings for QD 2-3 are shown in Table 1, data set (e)-(h). A notable
contradiction arises for (h). The 2L theory predicts $t_{C}=|t_{+}|=36\mu$eV
for (h), which is smaller than $t_{C}=|t_{+}|=44\mu$eV for (g), even though
the increase in $V_{B3}$ from (g) to (h) should cause the barrier height to
decrease and tunnel coupling to increase (Mills, ). This abnormality does not
show up in the 4L theory, which suggests that
$t_{C}=\sqrt{|t_{+}|^{2}+|t_{-}|^{2}}=67\mu$eV for (h), larger than
$t_{C}=44\mu$eV for (g). However, the 4L result of $\delta\phi$ for (h) is
remarkably different from other fitting results for QD 2-3, as if the dots
have shifted so that at least one of the dots has a significantly different
valley phase. Indeed multiple factors could cause this change in $\delta\phi$.
A physical reason such as possible interface steps in the DQD (Tariq and Hu,
2019) could lead to such a shift. Other reasons, such as the relatively noisy
data (see Fig. 4 in the “Methods” section) or non-linear effect in Eq. (6)
(DiCarlo  _et al._ , 2004), could cause a change in $\delta\phi$, too. Under
imperfect conditions, such as a large $\delta I_{noise}$ for (h), both 2L and
4L theory may fail to provide accurate fitting results, although the 2L
prediction is qualitatively worse since it contradicts the experimental
procedure.

The example given in Fig. 1 and Table 1 clearly shows that our 4L model is a
much better representation of a Si DQD, and provides more reliable knowledge
of the tunnel coupling in the DQD compared to the 2L model that has been
widely used so far. Qualitatively, the 2L theory only includes the intra-
valley tunneling reflected by anti-crossing “A” in Fig. 1, resulting in a
simple form of ground state charge distribution
$P_{L}=\sin^{2}\frac{\theta_{-}}{2}$. The inter-valley tunneling reflected by
anti-crossings “B” and “C” are taken into account in our 4L theory. They
provide a correction to the 2L theory as represented by the factors
$\cos^{2}\frac{\Theta_{1}}{2}$ and $\sin^{2}\frac{\Theta_{1}}{2}$ ($t_{-}$
dependent) in Eq. (2). The details are presented in the “Methods” section and
the magnitude of the 4L corrections from “B” and “C” will be discussed further
in the next subsection.

### Comparison between two-level and four-level models

Our results above show that 2L fitting returns quite different numbers from
the 4L fitting. Nevertheless, it is still valuable to investigate the
differences between the two models in a wide range of experimental conditions
(under various $|t_{\pm}|$, $|\Delta_{L,R}|$, $T$, etc.), to better clarify
their applicability in experimental studies of Si DQDs, which is the subject
of this subsection.

In order to investigate a certain set of parameters ($|t_{\pm}|$,
$|\Delta_{L,R}|$, $T$) other than the measured ones shown in Table 1, we
employ the Hamiltonian (1) to calculate a “pseudo-curve” $I_{QPC}(\epsilon)$
theoretically with given parameters. It represents an expected
$I_{QPC}(\epsilon)$ curve measured in experiments with particular parameters.
Then, we apply the procedure we proposed to fit this “pseudo-curve” and
compare the fitted parameters with the original parameters used to generate
the curve.

Figure 2: Errors for different fitting methods. The parameters to generate the
“pseudo-curve” are $|\Delta_{L}|=0.066$ meV, $|\Delta_{R}|=0.074$ meV.
$t_{C}=0.071$ meV, $\delta\phi=0.7\pi$, and $T=50$ mK unless specified
explicitly in the figures. As shown in the legend, the blue-upper-triangles
are results from the 2L formula (4), the red circles are obtained via
numerical diagonalization of the 4L Hamiltonian, while the black squares are
from the 4L analytical formula of Eq. (2).

We first discuss qualitatively the necessary condition for a 4L model
description of a Si DQD. Consider two well separated dots, when $t_{\pm}$ can
be treated as perturbations in Eq. (1). The eigen-states $|\Psi_{i}\rangle$
are mainly the four unperturbed states $|L,\pm\rangle$, $|R,\pm\rangle$ except
near the anti-crossings. For example, when $\epsilon$ is in the region
$\epsilon_{A}\pm\Delta\epsilon$, where
$\epsilon_{A}=(|\Delta_{L}|-|\Delta_{R}|)/2$ is the center of anti-crossing
“A”, the ground state would be a mixture of $|L,-\rangle$ and $|R,-\rangle$.
The range of this “mixing region” is roughly in the order of intra-valley
tunneling, $\Delta\epsilon\sim|t_{+}|$. Outside the “mixing region”, eigen-
states $|\Psi_{i}\rangle$ are not affected by the this anti-crossing and
remain the unperturbed states. Similarly, anti-crossings “B” and “C” also have
their own “mixing regions”, but mix different states
$|L,+\rangle\rightarrow|R,-\rangle$ or $|L,-\rangle\rightarrow|R,+\rangle$,
with a different magnitude of $\Delta\epsilon\sim|t_{-}|$. The charge
distribution of the ground state is mainly determined by anti-crossing “A”, as
illustrated in Fig. 1. The impacts of “B” and “C” can be regarded as
corrections. Therefore, when “B” and “C” are far away from “A”, the impacts
are limited and the dynamics is roughly a 2L dynamics with a single anti-
crossing “A”. In other words, under the condition

$|\epsilon_{A}-\epsilon_{B}|=|\Delta_{L}|\gg|t_{-}|\quad\&\quad|\epsilon_{A}-\epsilon_{C}|=|\Delta_{R}|\gg|t_{-}|,$
(7)

the 2L theory proposed in Ref. DiCarlo  _et al._ , 2004 is still a good
approximation to describe the ground state charge distribution of a Si DQD.
When the condition (7) is not fulfilled, on the other hand, a 4L model is
necessary.

In Fig. 2 we provide numerical evidences for the condition (7), in the form of
fitting errors’ dependences on various system parameters. Specifically, Fig.
2(a) shows the effects of $\delta\phi$ with a fixed $t_{C}$ and valley
splittings. Clearly, Eq. (7) is fulfilled when $|t_{-}|\approx 0$, which
requires $\delta\phi\approx 0$. In this case, the valley states $|+\rangle$
does not couple to $|-\rangle$, and the 4L system is approximately reduced to
a pair of 2L system. When $\delta\phi$ is finite, on the other hand, the 2L
fitting generally results in significant errors, especially when
$\delta\phi\rightarrow\pi$. At this limit $|t_{+}|\rightarrow 0$, making the
2L model unstable and sensitive to any data noise.

Figure 2 (b), (c), and (d) show the effects of tunnel coupling $t_{C}$, valley
splittings $|\Delta_{L}|$ and $|\Delta_{R}|$, and temperature. The results are
all consistent with condition Eq. (7). Here $t_{C}$ represents tunnel coupling
between bulk valleys, and is not directly measurable. However, larger $t_{C}$
leads to larger $|t_{-}|$ for a given $\delta\phi$, making the 2L theory less
reliable as the condition Eq. (7) is weakened. Similarly, when
$|\Delta_{L,R}|$ is large, the fitting error by 2-level theory is
significantly suppressed, while a smaller $|\Delta_{L,R}|$ leads to
overlapping of mixing regions for anti-crossings A, B, and C, and the 2-level
theory fails. The large errors of the 2L fitting at higher temperatures are
also expected, as there is only one excited state with a simple $\epsilon$
dependence as opposed to three excited states with much more complex
$\epsilon$ dependence.

One advantage of the 4L theory is that it automatically extracts inter-valley
tunnel coupling $t_{-}$ from the ground state charge distribution, as shown
with the cyan triangles in all four subplots of Fig. 2. We also note that the
approximate charge distributions given by the analytical expressions in Eq.
(2) is very accurate except in a very small region when $\delta\phi\approx\pi$
in Fig. 2 (a). Therefore, Eq. (2) are a perfect approximation in most cases,
allowing a much faster fitting calculation compared to a fully numerical
procedure. A more detailed study on the accuracy of Eq. (2) is in the
“Methods” section and Fig. 3.

There are also other practical factors affecting the accuracy of the fitting
such as the inaccurate fitting of the parameters $\delta I$, $I_{0}$,
$\epsilon_{0}$, insufficient information of $|\Delta_{L,R}|$, and signal noise
in $I_{QPC}$. The impacts of all these factors are discussed in the “Methods”
section.

## Discussion

In this paper, we present a four-level model that can extract tunnel coupling
information in a Si double quantum dot via measurement of charge distribution
of the double dot in thermal equilibrium. In essence, we have adapted the
protocol originally proposed and used for GaAs double dot DiCarlo  _et al._
(2004) to a Si DQD by including the valley-orbit coupling and dynamics. We
demonstrate the efficiency and robustness of our model and the associated
fitting procedure by applying it to experimental data collected in a pair of
Si DQD $I_{QPC}(\epsilon)$ (Mills  _et al._ , 2019b). The results clearly
demonstrate the superiority of the 4L model compared to the conventional two-
level model used in the original proposal, with the 2L model produces almost
50% larger tunnel coupling $t_{+}$, not to mention that only by the 4L model
can one extract any information on the inter-valley tunnel coupling $t_{-}$.
In addition to directly diagonalizing the 4L model Hamiltonian in the fitting
procedure, we have also derived a set of approximate formula with the
assumption that $t_{-}$ can be treated perturbatively. Our numerical results
show that the approximate formula perform nearly perfectly in the vast
majority of parameter regimes, with the only exception near the point where
the inter-dot valley phase difference is $\pi$. Lastly, we compare the
performance of the 2L and 4L models, and clarify the condition under which 4L
model is necessary. In short, our 4L model for a Si DQD provides much better
accuracy in extracting the intravalley tunnel coupling $t_{+}$ from a charge
distribution measurement, while carrying the extra benefit of also extracting
inter-valley tunnel coupling $t_{-}$. We hope that the proposed protocol can
help experimentalists to measure tunnel couplings for a Si DQD more accurately
and efficiently.

## Methods

### Theoretical charge distribution

In our approximate treatment, we consider inter-valley tunnel coupling as a
perturbation, while include intra-valley tunnel couplings in the unperturbed
Hamiltonian. In essence we take a DQD with a completely smooth interface as
our starting point. The Hamiltonian (1) can thus be split into two parts

$H=H_{0}+H_{1},$ (8)

where

$H_{0}=\left[\begin{array}[]{cccc}\epsilon+|\Delta_{L}|&0&t_{+}&0\\\
0&\epsilon-|\Delta_{L}|&0&t_{+}\\\ t_{+}^{*}&0&-\epsilon+|\Delta_{R}|&0\\\
0&t_{+}^{*}&0&-\epsilon-|\Delta_{R}|\end{array}\right],$ (9)
$H_{1}=\left[\begin{array}[]{cccc}0&0&0&t_{-}\\\ 0&0&t_{-}&0\\\
0&t_{-}^{*}&0&0\\\ t_{-}^{*}&0&0&0\end{array}\right].$ (10)

The eigen-energies and eigen-states of $H_{0}$ are

$E_{1,\pm}=\pm\Delta_{+}-E_{\pm}$ (11) $E_{2,\pm}=\pm\Delta_{+}+E_{\pm}$ (12)

where $E_{\pm}=\sqrt{(\epsilon\pm\Delta_{-})^{2}+|t_{+}|^{2}}$ and
$\Delta_{\pm}=\frac{1}{2}(|\Delta_{L}|\pm|\Delta_{R}|)$, and the corresponding
eigen vectors are

$|\psi_{1,\mp}\rangle=\cos\frac{\theta_{\mp}}{2}|R,\mp\rangle-e^{-i\delta\phi/2}\sin\frac{\theta_{\mp}}{2}|L,\mp\rangle$
(13)
$|\psi_{2,\mp}\rangle=e^{i\delta\phi/2}\sin\frac{\theta_{\mp}}{2}|R,\mp\rangle+\cos\frac{\theta_{\mp}}{2}|L,\mp\rangle$
(14)

where $\tan\theta_{\mp}=\frac{|t_{+}|}{\epsilon\mp\Delta_{-}}$
($\theta_{\mp}\in[0,\pi]$).

When the inter-valley tunneling $|t_{-}|$ is finite, the Hamiltonian $H$ can
be rewritten in the new basis
$\\{|\psi_{1,\mp}\rangle,|\psi_{2,\mp}\rangle\\}$. The matrix representation
of $H_{0}$ becomes diagonal and the matrix elements of $H_{1}$ can be obtained
as, for example,

$\langle\psi_{1,-}|H_{1}|\psi_{1,+}\rangle=-i|t_{-}|\sin\left(\frac{\theta_{-}}{2}-\frac{\theta_{+}}{2}\right)$
(15)
$\langle\psi_{1,-}|H_{1}|\psi_{2,+}\rangle=t_{-}^{*}\cos\left(\frac{\theta_{-}}{2}-\frac{\theta_{+}}{2}\right)$
(16)

When $|\Delta_{L}|=|\Delta_{R}|$, $\theta_{-}=\theta_{+}$. As a result,
$\cos\left(\frac{\theta_{-}}{2}-\frac{\theta_{+}}{2}\right)=1$ and
$\sin\left(\frac{\theta_{-}}{2}-\frac{\theta_{+}}{2}\right)=0$. In the new
basis $\\{|\psi_{1,\mp}\rangle,|\psi_{2,\mp}\rangle\\}$, the rotated full
Hamiltonian $\tilde{H}$ can be written as

$\tilde{H}=\left[\begin{array}[]{cccc}-\Delta_{+}-E_{-}&0&0&t_{-}^{*}\\\
0&-\Delta_{+}+E_{-}&t_{-}^{*}&0\\\ 0&t_{-}&\Delta_{+}-E_{+}&0\\\
t_{-}&0&0&\Delta_{+}+E_{+}\end{array}\right]$ (17)

The eigen-energies are then

$E_{1}=\frac{1}{2}\left(E_{+}-E_{-}-\sqrt{(E_{+}+E_{-}+2|\Delta_{+}|)^{2}+|t_{-}|^{2}}\right)$
(18)

$E_{2}=\frac{1}{2}\left(E_{-}-E_{+}-\sqrt{(E_{+}+E_{-}-2|\Delta_{+}|)^{2}+|t_{-}|^{2}}\right)$
(19)

$E_{3}=\frac{1}{2}\left(E_{-}-E_{+}+\sqrt{(E_{+}+E_{-}-2|\Delta_{+}|)^{2}+|t_{-}|^{2}}\right)$
(20)

$E_{4}=\frac{1}{2}\left(E_{+}-E_{-}+\sqrt{(E_{+}+E_{-}+2|\Delta_{+}|)^{2}+|t_{-}|^{2}}\right)$
(21)

and the corresponding eigen-states are

$|\Psi_{1}\rangle=e^{i\phi}\cos\frac{\Theta_{1}}{2}|\psi_{1,-}\rangle-\sin\frac{\Theta_{1}}{2}|\psi_{2,+}\rangle$
(22)

$|\Psi_{2}\rangle=-e^{i\phi}\sin\frac{\Theta_{2}}{2}|\psi_{2,-}\rangle+\cos\frac{\Theta_{2}}{2}|\psi_{1,+}\rangle$
(23)

$|\Psi_{3}\rangle=e^{i\phi}\cos\frac{\Theta_{2}}{2}|\psi_{2,-}\rangle+\sin\frac{\Theta_{2}}{2}|\psi_{1,+}\rangle$
(24)

$|\Psi_{4}\rangle=e^{i\phi}\sin\frac{\Theta_{1}}{2}|\psi_{1,-}\rangle+\cos\frac{\Theta_{1}}{2}|\psi_{2,+}\rangle$
(25)

where $\tan\Theta_{1}=\frac{|t_{-}|}{E_{+}+E_{-}+\Delta_{+}}$ and
$\tan\Theta_{2}=\frac{|t_{-}|}{E_{+}+E_{-}-\Delta_{+}}$. The left-dot charge
distribution $|\langle L,-|\Psi_{i}\rangle|^{2}+|\langle
L,+|\Psi_{i}\rangle|^{2}$ for these four eigen-states are in the form of Eq.
(2).

### Accuracy of the approximate solution

Figure 3: Accuracy of approximate diagonalization under different conditions.
The valley splittings are $|\Delta_{L}|=0.045$ meV, $|\Delta_{R}|=0.055$ meV.

The eigen-states (22-25) are obtained when $|\Delta_{L}|=|\Delta_{R}|$.
Practically, $|\Delta_{L}|$ is usually not identical to $|\Delta_{R}|$.
However the charge distributions in Eq. (2) remain a good approximation. This
is because in most cases the nearby dots have similar valley splittings
$|\Delta_{L}|\approx|\Delta_{R}|$, which makes
$\sin\left(\frac{\theta_{-}}{2}-\frac{\theta_{+}}{2}\right)\approx 0$. As
such, the term $\langle\psi_{1,-}|H_{1}|\psi_{1,+}\rangle$ that we neglected
is generally a small correction compared to
$\langle\psi_{1,-}|H_{1}|\psi_{2,+}\rangle$. Even if $|\Delta_{L}|$ and
$|\Delta_{R}|$ is quite different, our numerical results in Fig. 3 suggest
only a small error in Eq. (2).

In Fig. 3, we plot the factor
$\sin\left(\frac{\theta_{-}}{2}-\frac{\theta_{+}}{2}\right)$ under different
QD parameters. It shows that in most region, the factor we have neglected in
Eq. (15) is quite small. Only in the very special case when the phase
difference $\delta\phi\rightarrow\pi$ and the detuning $\epsilon\rightarrow
0$, the factor $\sin\left(\frac{\theta_{-}}{2}-\frac{\theta_{+}}{2}\right)$ is
notable. Otherwise, Eq. (2) is always a good approximation. Besides, for any
$\delta\phi$ and $|\Delta_{L,R}|$, the notable deviation always appears near
$\epsilon=0$. If we consider the average of
$\sin\left(\frac{\theta_{-}}{2}-\frac{\theta_{+}}{2}\right)$ over all
$\epsilon$ (because the fitting depends on the charge distribution over all
$\epsilon$, not just at $\epsilon=0$), the average deviation is always small.
For example, when there is a 20% difference between the two valley splittings,
$|\Delta_{-}|/|\Delta_{+}|=0.2$, the overall magnitude (average value over
different $\epsilon$) of the factor
$\sin\left(\frac{\theta_{-}}{2}-\frac{\theta_{+}}{2}\right)$ is only 4% of the
factor $\cos\left(\frac{\theta_{-}}{2}-\frac{\theta_{+}}{2}\right)$. Namely,
the terms we dropped are indeed negligible even when there is a notable
difference between the two valley splittings.

### Details of data fitting

Figure 4: Curve fitting for actual data measured from experiment. (a), (b) and
main plot (d) are obtained from the left two dots (dot 1 and 2) with different
barrier gate voltage. (e)-(h) in the right-bottom corner are measured from
right two dots (2 and 3). Data set (c) is absent since it is presented in Fig.
1. Figure 5: Fitting results of $\delta\phi$ by using various combination of
$|\Delta_{L}|$ and $|\Delta_{R}|$.

In the “Results” section, we show all the fitting results in Table 1 and the
raw data of data set (c) in Fig. 1. Here, we show all the other 7 sets of raw
data and the best fitting curves. The raw data are extracted from the readout
of the QPC sensor directly and the best fitting curves are shown in Fig. 4.
For panels (a), (b) and (d), the data are measured from QD 1-2, with different
barrier gate voltage $V_{B2}$ (which tunes $t_{C}$). The valley splitting
$|\Delta_{L}|$ and $|\Delta_{R}|$ are not actually measured directly in the
experiment, and are estimated to be around 66-74 $\mu$eV Mills . We perform
the curve fitting with several groups of $|\Delta_{L}|$ and $|\Delta_{R}|$
ranging from 66-74 $\mu$eV, as shown in Fig. 5. The data presented in Tables 1
is picked because the fitting results of $\delta\phi$ are relatively
consistent. We choose this criterion because the only tuned parameter in the
experiment is $t_{C}$, which would generally not affect $\delta\phi$ when it
is not varied too significantly. Interestingly, Fig. 5 shows that other
estimates of $|\Delta_{L}|$ lead to very similar results on $\delta\phi$. The
fitting results of $\delta\phi$ is always around 2.2 rad. The value of
$\delta\phi$ fitted from data set (d) has a small difference from the results
from other data sets (a), (b) and (c). We believe this is mostly because data
set (d) has larger noise in the raw data of $I_{QPC}$, which is apparent in
Fig. 4. In short, our fitting procedure does not seem to be overly sensitive
to the choices of the valley splittings, as long as they are not too different
across the two dots.

We highlight data set (d) in Fig. 4 because it has the largest $t_{C}$, making
the difference between 2L fitting curve and 4L fitting curve clearly
observable with bare eyes. One can easily see the 4L curve fits better to the
raw data. The curve obtained by using Eq. (2) almost coincide with the curve
obtained by fully numerical diagonalization, illustrating the robustness of
our approximate expressions. Besides, we would also like to emphasize that the
actual 2L and 4L fitting results for (d) are quite different (almost 100%
according the results in Table 1), much larger than it seems from the two
curves.

Similarly, the best curve fittings of the data measured from QD 2-3 are shown
in Fig. 4 (e)-(h). The most interesting result is the last panel (h). It is
discussed that the 2L theory predicts $t_{C}$(h) to be smaller than
$t_{C}$(g), while the 4L theory suggests that $t_{C}$(h) is larger than
$t_{C}$(g). Experimentally, it is expected that the true value of $t_{C}$ in
panel (h) should be larger because the gate voltage $V_{B3}$ used to tune
$t_{C}$ between dot 2-3 is increased from (g) to (h) when the experiment is
performed. Here, Fig. 4 (h) shows this set of data is obviously measured with
a notably larger noise than other sets. This large noise leads to more
significant error for the fitting results. However, we also see that even with
such a large noise the 4L fitting still make a prediction which does not
contradict with the experimental setup.

### Difference between 2L and 4L fitting curves

Figure 6: Intuitive difference of two fitting formulas. Parameters are
$t_{C}=50\mu$eV, $|\Delta_{L}|=74\mu$eV. $|\Delta_{R}|$ and $\delta\phi$ are
given in each subplot. $\epsilon$ is ranging from $-0.2$ meV to $0.2$ meV.

The difference between the 2L fitting curve and the 4L fitting curve in Fig. 1
looks insignificant. This is mostly because the particular parameters in the
experiment happen to produce similar 2L and 4L curves, even though the
corresponding tunnel coupling strengths are quite different. In Fig. 6, we
show that the differences between a 2L and a 4L fitting curves can both be
minimal and be dramatic. For example, very different valley splittings
$|\Delta_{L,R}|$ in the two dots makes the 4L curve lose its symmetry around
zero detuning, while a 2L curve is always symmetric. The two fitting curve can
also be easily distinguished when the condition (7) is not fulfilled. Even
when two curves look very similar, they may yield very different fitting
results. For instance, the curves in Fig. 4 (d) look quite similar, but there
is nearly 90% difference between the 2L and 4L fitting results on $|t_{+}|$ in
Table 1.

### Estimation of fitting parameters $I_{0}$, $\delta I$, and $\epsilon_{0}$

Figure 7: Fitting error caused by wrong estimation of $I_{0}$, $\delta I$, and
$\epsilon_{0}$. Parameters are the same as Table 1, QD 1-2.

In the “Results” section, we describe our fitting procedure by splitting the
fitting parameters into two groups: (1) $I_{0}$, $\delta I$, and
$\epsilon_{0}$, which determine the position of the fitting curve; and (2)
$t_{+}$ and $t_{-}$, which determine the shape of the curve. In our protocol
we fit the two groups of parameters in turn until the results converge.
Practically, we only perform the iterative fitting 2 rounds because a bad
estimate on $I_{0}$, $\delta I$, and $\epsilon_{0}$ does not result in too
much error on the final fitting results of $t_{\pm}$. In Fig. 7, we plot the
fitting error caused by a wrong estimate of $I_{0}$, $\delta I$, and
$\epsilon_{0}$. Fig. 7 is plotted in a relative wide range. However,
practically the errors cannot be too large, otherwise the best fitting curve
will have a notable shift. Therefore, the iterative fitting converge very fast
and 2 rounds of fitting is generally enough.

### fitting error caused by incomplete knowledge of system parameters

Valley parameters such as $|\Delta_{L}|$, $|\Delta_{R}|$ are crucial in
describing a Si DQD. As shown in Eq. (2) and (3), the fitting protocol in our
model requires a preliminary measurement on the valley splitting of the two
dots. Practically, the valley splittings may be unknown or only roughly
estimated. In the example in Table 1, the valley splittings are indeed
estimated bu not measured. It is thus important to know the impact on the
accuracy of $t_{\pm}$ by inexact knowledge of $|\Delta_{L,R}|$.

In Fig. 8 we plot the error caused by incomoplete knowledge of valley
splitting $|\Delta_{L,R}|$. The numerical data used in fitting are generated
with $|\Delta_{L,R}|=74$ $\mu$eV. We then vary $|\Delta_{L}|$ or
$|\Delta_{R}|$ on purpose to examine the sensitivity of our protocol to this
systematic error.

The numerical results show that a moderately off estimate of $|\Delta_{L,R}|$
will not lead to sizable errors in 4L fitting unless $|\Delta_{L,R}|$ is
significantly underestimated. In Fig. 8 (a), when $|\Delta_{L,R}|\approx 37$
$\mu$eV (50% underestimated), the 4L fitting error is still only about 10%.
Furthermore, an overestimated $|\Delta_{L,R}|$ will result in even smaller
error comparing to an underestimated $|\Delta_{L,R}|$, and the 4L fitting
error is always smaller than the 2L fitting. The right panel (b) shows the
case that only one valley splitting ($|\Delta_{L}|$) is known inaccurately.
Similar to panel (a), the 4L fitting error is always smaller than the 2L
fitting. We note that the fitting accuracy of inter-valley tunnel coupling
|$t_{-}$| is more sensitive to the knowledge of $|\Delta_{L}|$, though
information on $t_{-}$ is not accessible at all to a 2L model, since valley
physics is not included there.

Figure 8: Fitting error by different methods, the “pseudo-curve” data is
obtained with $t_{C}=0.071$ meV, $|\Delta_{L}|=|\Delta_{R}|=0.074$ meV, and
$\delta\phi=0.7\pi$. Inaccurate $|\Delta_{L,R}|$ indicate a wrong
estimation/measurement on $|\Delta_{L,R}|$. In (a), we assume
$|\Delta_{L}|=|\Delta_{R}|$. In (b), we assume $|\Delta_{R}|$ is measured
accurately, only $|\Delta_{L}|$ has an error.

Another source of error in tunnel couplings is the measurement of the current
$I_{PQC}$, which always contains some noise $\delta I_{noise}$ as shown in Eq.
(6) and illustrated in Fig. 1. Here, we use a stochastic function $\delta
I_{noise}$ to simulate the uncertainty in current measurement. The stochastic
function is characterized by the mean $\langle\delta I_{noise}\rangle=0$ and
the standard deviation $\sigma(\delta I_{noise})$, which indicates the
strength of the noise. Numerical results as presented in Table 2 show that the
noise $\delta I_{noise}$ has a much larger impact on the performance of 2L
fitting comparing to the 4L fitting. By using the 4L model, even when the
relative strength of the noise reaches 5%, the fitting error is still under
10%, while the 2L fitting always produces a significant error over 35%. The
errors from 4L fitting do increase rapidly with an increasing $\sigma(\delta
I_{noise})$, while errors in the 2L fitting remains large and do not change
dramatically as $\sigma(\delta I_{noise})$ increases.

$\frac{\sigma(\delta I_{noise})}{\delta I}$ | 0.01 | 0.02 | 0.03 | 0.04 | 0.05
---|---|---|---|---|---
Error on $|t_{+}|$ (2L) | 36.87% | 36.53% | 36.18% | 36.82% | 37.18%
Error on $|t_{+}|$ (4L) | 1.56% | 4.30% | 4.98% | 6.25% | 8.19%
Error on $|t_{-}|$ (4L) | 1.06% | 2.81% | 3.54% | 4.16% | 4.68%
Table 2: Fitting error caused by noise on $I_{QPC}$ measurement.

## DATA AVAILABILITY

Data and an example of fitting code are available from the authors on
reasonable request.

## ACKNOWLEDGEMENTS

This work is supported by the Army Research Office, Project No. W911NF1710257.
The authors thank Adam Mills and Jason Petta for providing original data and
helpful discussions.

## AUTHOR CONTRIBUTIONS

X.Z. and X.H. contributed to theoretical/numerical/physical analysis and
prepared the manuscript.

## Additional Information

Competing Interests: The authors declare no competing interests.

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# Observations of Bell Inequality Violations with causal isolation between
source and detectors

Marc Fleury

###### Abstract

We report the experimental observations of Bell Inequality Violations (BIV) in
entangled photons causally separated by a rotating mirror. A Foucault mirror
gating geometry is used to causally isolate the entangled photon source and
detectors. We report an observed BIV of CHSH-$S=2.30\pm 0.07>2.00$. This
result rules out theories that explain correlations with traveling
communication between source and detectors.

##  I Motivation and Hypothesis

###  I.1 On Bell Inequality Violations and Local Realism

It is commonly accepted that the Aspect [1] and Zeilinger [2] experiments
convincingly closed the locality loophole for Bell Inequality Violations (BIV)
experiments. They do so by causally isolating the acts of measurements at two
distinct points called Alice and Bob. Modern BIV experiments close the
locality loophole in increasingly dramatic fashion. See [3] for a submarine
fiber implementation, [4] for an orbital station implementation, and [5] for
measurement settings triggered by distant stars. The experiments all focus on
isolating the measuring stations from one another: Alice and Bob are the ones
causally separated, by land, sea, satellites and deep space. While the
experiments undoubtedly achieve observer to observer isolation we will argue
in section I.2 that these experiments do not isolate the emitting atoms from
the observers. We will argue that in all these geometries, there is an ever
present line of sight between the measuring apparatus and the source. We will
motivate that such influence of the measuring stations on the source can
account for BIVs in a classic local realistic ways in section I.3. It follows
that we will lose BIV by removing the line of sight between source and
detection. A Foucault mirror design gates and effectively removes the line of
sight between observers and emitters. With this gate in place we perform a
CHSH measure of BIV in a 2-channel setup. In section III we report on the
observation of CHSH $S=2.30>2$. This rules out travelling influences between
source and detectors, including super-luminal, as candidates for the Bell
effect in section IV.

###  I.2 On experimental locality: Observer to observer vs observer to
emitting atoms

Figure 1: schematic representation of the isolation achieved in 4 channel Bell
experiments with AOM and EOM. The isolation between Alice and Bob is complete.
However there is no isolation between the four measures at Alice and Bob and
the emitting atoms.

We will now review the isolation experimentally achieved thus far in the
current state of the art experiments. We will observe a static line of sight
between emitting atoms and observing ones. Bell famously prescribed ”randomly
setting the measure during the flight of the photons” as a way to close the
locality loophole. Modern experiments causally isolate the observers between
themselves. In the Aspect and Weihs-Zeilinger experiments, we never set the
direction of the polarization analyzers during the flight of photons. This is
in fact mechanically impossible [6]. Instead, the analyzers are all preset and
we dynamically route the photons to a randomly chosen analyzer. This is
achieved by randomly switching the path of the photons while in flight. We
then claim physical analogy to ”setting the measures” but the measures were
chosen more than they were set. The settings are in fact all preset, it is
only the choice of the measure which is dynamic and random. To effect this
random optical switch, the original experiments by A.A. used Acousto-Optic
Modulators (AOMs) operating in the MHz domain [1], the modern state-of-the-art
experiments in the vein of Weihs and Zeilinger, use Electro-Optic Modulators
(EOMs) operating in the GHz domain [2] [7]. These AEOM devices randomly switch
the path of the photons and therefore randomly choose which one of the preset
polarization analyzers will perform the actual measure. Both experiments are
based on a similar implementation concept of the optical switches and
measures: they preset all polarization analyzers upfront and then choose one
by randomly modulating the index of refraction of some media, either
acoustically or electrically. In the EOM experiments, the emitting atoms
inside the BBO crystal have a permanent line of sight to all four polarizers
with static settings. The EOM modulate the phases, the AOM modulate
amplitudes. The AOMs behave more like switching mirrors with a 4:1 reflection
contrast. Therefore, a line of sight from the detectors to the emitting
cascading atoms is also always present in the Aspect experiments. The presence
of the AOM in the path does not remove this line of sight, it modulates the
amplitudes of signals coming through it. State-of-the-art AOMs never realize a
true zero in amplitude transmission, the residual signal is rather large.
There is always a line of sight. Therefore we conclude that the atoms in the
sources of both geometries (SPCD/BBO and cascading atoms) are connected to all
four static measurements at all times. The emission of the entangled photons
happens with full visibility of the measurements at a fixed angle in both
types of experiments. See fig. 1 for a schematic rendering of isolation
achieved with these AEOMS. This line of sight was hiding in plain sight, so to
speak.

###  I.3 Hypothesis: on a background influence between emitting atoms and
measuring atoms

With regards to Bell’s theorem, such a background field would allow for Bell
type correlations in simple local realistic ways. Such a de-localized
background field trivially breaks the isolation assumption of the theorem
according to [8]. This is not in violation of, but rather in accord with,
Bell’s theorem. [9] considers [8] wrong claiming it implies a super-luminal
communication in the background. We will now briefly review the literature for
BIVs and show that there is little agreement as to what is causing the Bell
correlations. For example, other modern local realist approach hypothesize
non-linear amplifications of the so-called zero-point field modes combined
with a hypothetical threshold detection in Avalanche Photo Detectors (APDs)
[10] [11]. Others, more generically, focus on Bohr contextuality pointing out
that the measures always interfere with the measured, a point made clear since
the Stern-Gerlach days. Both the photons in flight and the emitting source
atoms are subject to dynamic influences [12] [13]. Others argue the physical
fact that Bell type correlations emerge from classical Electro-Magnetism [14].
Counting coincidences in a Bell game is after all measuring intensity at two
different points. This intensity interferometer (as opposed to amplitude) is
known as a Hanbury Brown-Twiss (HBT) interferometer [15]. Others still within
the HBT interferometer approach, reconstruct the proper Bell intensity
correlations with a classical EM local realist model [16]. They hypothesize
circularly polarized single photons coming out of the SPDC process in the BBO
crystals. By definition of circular polarization. this introduces a quarter-
wave phase delay between the horizontal and vertical modes of the BBO
crystals. This is in clear opposition to the understanding of BBO crystal
emissions in [6]. There the horizontal and vertical modes are understood to be
in phase which results in a ”quantum superposition” of the linear modes. This
is called a singlet or triplet state depending on the sign of the delay. In
fact, as reported in Weihs’ PhD thesis (German only) one fine-tunes the exit
phases of the BBO modes so as to be in phase, a ”sine-qua-non” condition of
the observation of BIVs and a big part of the experimental approach. This
results in a ’quantum superposition’ by erasing the which-path information
contained in the phases. Finally and as if to further confuse everyone in the
ivory tower of B(e)aBel(l) [17], some recently report observations of Bell-
type correlations using uncorrelated laser sources. [18].

Hypothesis: If a field is responsible for the Bell effect, then when we remove
said field, we can anticipate a loss of Bell effect. (No line of sight? no
BIV!). To test this background influence hypothesis and without further
speculation as to the underlying physics involved, we now introduce a Foucault
mirror designed to disrupt the EM line of sight between emitting atoms and
measurement apparatus. With this contraption we remove any EM background field
between observers and emitters. How does this affect a Bell-CHSH measure? We
anticipate a possible loss of BVI.

##  II Experimental Setup

###  II.1 A 2-channel photon Entanglement measure using a rotating mirror

To test this hypothesis, we introduced a spinning Foucault mirror in a classic
2 channel Bell experiment. See fig. 2. This is a table-top 2-channel
experiment showing a Bell-CHSH Inequality Violation as opposed to the
4-channel detection loophole experiments reviewed above. We remove the line of
sight by introducing a rotating Foucault mirror, inspired from the eponymous
1850 measure of the speed of light. This connects the components only for a
brief amount of time and the source is optically isolated from the measures
for the vast majority of the time. Contrast this to the AEOMs experiments,
where the source is continuously connected to the detectors. Here we only
focus on the fiber communication, we do not care about physical line-of-flight
isolation, in fact everything fits on an table top optical bread board. This
type of causal isolation of the source is new and therefore, to the best of
our knowledge, an experimental first.

Figure 2: schematic layout. SPDC source down in the middle. Alice and Bob
photons are transported over 200 m Single-Mode Fiber. We compensate with 3 x
bat ears. ($\tfrac{1}{4}$, $\tfrac{1}{2}$, $\tfrac{1}{4}$) wave delays both at
the in and out of both fibers for a total of 4 bat ears. Both paths are
bounced off of the Foucault mirror which creates a line arc over an aperture
at the detectors. We then filter through static computerized polarizers, into
collectors and into the Avalanche Photo Detectors (APD) counters from qutools.

An entanglement source commercially available from quTools, GmbH, allows for a
classic 2-channel singlet experiment showing Bell-CHSH violations. A rotating
mirror ($1000$Hz, $34$ facets) purchased from Cambridge Technologies (model
SA34 P/N:1-1-3304-001-00) completes the setup. The geometrical arc described
by the beams reflected off each of the rotating mirror facets covers the slits
over Alice and Bob. See fig. 2. The result is a gate: only when the rotating
beams cover the openings over the slits are the gates opened and the
components connected.

#### Time of aperture:

We can adjust the time this window stays open and during which the paths are
connected. The time of aperture is determined by the aperture width of the
slits $A=10^{-3}$m, the rotational velocity $w=10^{3}$Hz, and the physical
distance between the slits and the mirror on our bread board $R=34\cdot
10^{-2}$m:

$T_{on}=\frac{A}{2\pi Rw}=4.7\cdot 10^{-7}s$ (1)

Since we use the same mirror for both paths, the shutter effects at both slits
are in phase. The slits are alight at the same time and we thus detect
coincidences in the APDs. Note that two different mirrors would not be in
phase and we would not detect coincidences in the counters because the APDs
would not be alight at the same time, or at least within the $20$ns
coincidence window programmed in hardware.

#### Causal Isolation:

During the time of aperture, photons will travel a distance of $140\,$m. We
introduce $200$m of single-mode fibers between source and mirror as in fig. 2.
This achieves isolation of detectors and crystal when separated by the fibers.
In our table top setup, we only consider fiber isolation. The detectors are
physically sitting next to each other on our bread board. They are within
$10cm$ physical distance but their fiber optical paths are separated by $400$m
($2\times 200$m) of single-mode transport. Detector to detector isolation is a
byproduct of source to detector fiber isolation as we double the distance of
fiber transport needed to $400$m.

#### Degradation:

Luminosity and time of aperture are proportional. The shorter the time gate,
the shorter the exposure to the light and the shorter the collection of
photons. This is true of single counts. The coincidence count is proportional
to the single count, and not a quadratic of the single count. As we have
observed, the mirror shines the separate individual slits at the same time,
the gate effects at distant slits are in sync. This is the reason we detect
coincidences in the first place. This yields a linear relationship between
coincidences and singles, mainly due to the quantum efficiency of the
detectors. The floor in our experimental setup is set by the dark count of
singles and the dark count of coincidences. With rotation, the photons are
spread over $2\,$ lines that repeat $1000\,$ times per second. We calculate
the signal degradation as:

$D_{th}=\frac{A\cdot N}{2\pi R}=0.016$ (2)

with the aperture size $A=10^{-3}\,$m, the radius mirror to slit $R=0.34\,$m
and the number of facets $N=34$.

##  III Results

#### Dark counts and detection window:

Our APDs have individual dark count rates of $1300\,$s-1 and $600\,$s-1, and
the dark count coincidence rate is approx $300\,$hr-1. The time for the
detection window of coincidences in APDs is $20\,$ns and is set in hardware by
quTools. We do not control this time window parameter. See tab. 1.

#### Degradation:

The results of measurements for determining the luminosity and signal
degradation due to the rotating Foucault mirror can be seen in tab. 1. The
dark counts of the detectors are measured in order to be subtracted from the
results in further calculations.

$D_{ex}=\frac{\mathrm{Signal\;with\;rotation}}{\mathrm{Signal\;without\;rotation}}=0.018\;$
(3)

The first observation is that we retain a clear coincidence signal, well above
the noise level. The degraded coincidence signal in our experiments is on the
order of $30,000\,$hr-1 and the noise for said coincidences is measured at
$300\,$hr-1. We have a clear $100:1$ contrast in the signal-to-noise ratio.
The ratio between singles and coincidences is due to the quantum efficiency of
the detectors. The higher value of the experimentally observed signal
degradation could be due to the non-linear quantum efficiency of the
detectors: The higher the measured count rate, the lower the quantum
efficiency.

Table 1: Results of signal luminosity and degradation measurements, in photons per second. | Single 0 (/s) | Single 1 (/s) | Coinc. (/s)
---|---|---|---
Dark Counts | 1300 | 600 | 0.08
No rotation | 33894 | 20329 | 389
With rotation | 2301 | 1098 | 7
Degradation $D_{ex}$ | $0.031\pm 0.003$ | $0.025\pm 0.004$ | $0.018\pm 0.008$

#### Bell CHSH observation:

We have observed BIV of Bell-CHSH $S=2.30\pm 0.07$. See tab. 2.

Table 2: CHSH inequality: observed. $S=2.302\pm 0.071$. There are 16 measurement settings for the polarizers. In blue the individual counts and in grey to the right the accidental coincidence counts (noise because of single counts). For example, the first cell is 0:22.5, the coincidence count is 226 photons and the noise is 5 photons. The integration time is 60,000ms (1 minute per measure, 16 measures, 16 minutes total). The 1-minute per-measure setting is the maximum setting we can program within the quTools environment.  Angle in ∘ | 0 | 45 | 90 | 135
---|---|---|---|---
22.5 | 226-5 | 85-4 | 42-4 | 184-4
67.5 | 70-5 | 34-4 | 182-4 | 239-5
112.5 | 46-5 | 187-5 | 227-4 | 70-4
157.5 | 198-5 | 217-5 | 88-4 | 34-4

##  IV Conclusion

We demonstrated that the AEOMs Bell experiments close the locality loopholes
between observers but do not causally isolate the sources from the observers
in classic 4-channel loophole experiments. We introduced a spinning mirror to
create such isolation in a 2-channel Bell experiment. By intermittently
removing this hypothetical background we anticipated a loss of the Bell effect
(loss of BVI). We reported instead on observed Bell violations.
Regarding a hypothetical background mediating an influence from the
measurement apparatus to the emitting atoms and which is responsible for the
Bell effect; we will prove that the influence cannot be a traveling wave. To
prove this statement, we first observe that the time the gate is open is
$4.7\cdot 10^{-7}$s and that the distance covered by photons in the single-
mode fibers during this time is approximately $140$m. Let us assume, for the
sake of argument, that the influence from the detector to the source is
instantaneous. The BBO crystal would start emitting under the influence of the
detector as soon as the gate opens. The photons informed by this instantaneous
influence would then still need to cover the $200$m of single-mode fiber that
separates them from the detectors. By the time the photons reach the mirror,
the slit is not in view anymore. These photons are not detected. We then
observe that this is also true of a slower influence signal: the influence
will simply take longer to reach the source and delay the return trip by as
much. We therefore conclude that the influence cannot be that of a travelling
wave, not even super-luminal, not even an instantaneous wave. QED.

##  V Discussion

For discussion we will offer the following logical derivation. Either:

  * •

1/ There is no background influence, or

  * •

2/ There is a background influence, and either

    * –

2a/ it is a traveling local wave, or

    * –

2b/ not (standing wave or deBroglie hypothesis)

The logical structure of the above is a list enumeration of the possible
scenarios: $\nexists$ background $\lor(\exists$ background $\land($ standing
$\lor\neg$standing)). This is declarative, there are no physics yet.

One can argue in scenario 1 that there is no background field it just does not
exist or something else is responsible for the Bell effect. All we have done
here is reduce luminosity and further validate orthodoxy. In scenario 2 we
still consider the existence of a background influence. The conclusion of this
paper was to experimentally rule out option 2a.

Therefore we are left with option 2b. The wave responsible for Bell is NOT a
traveling wave. What form could this wave be? Standing waves come to mind. The
author looks to the Hydrodynamic Walkers for visual inspiration [19]. A
droplet is bouncing and creating impact waves. Behind are standing waves. The
sum of these standing waves is the wake. A chaotic form of time-averaged wake
emerges [20]. This is such an example of memory we derive from the dynamic
fields of the hydrodynamic walkers. These are the Faraday wave-fields which
are manifested in the hydrodynamic walkers [21]. This is also in alignment
with the deBroglie’s intuition of matter modelled with standing waves about
singular points. In the Hydrodynamic Quantum Analog field (HQA) of the
walkers, the walkers follow a cavity geometry which is represented by a
ghostly but emerging mean field which includes a form of echo-location. This
mean field of standing waves represents a memory of past events, the wake, and
usually includes information from the enclosing cavity, in the form of
standing waves of the cavity, and this resonant memory guides the particles.

##  VI Acknowledgments

We wish to thank Emmanuel Fort, Herman Batelaan, Gregor Weihs, and Louis
Vervoort for in depth feedback on this paper, Henning Weier and Nico Klein of
QuTools for technical assistance. This work was self funded.

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