# Radio frequency driven superconducting diode and parity conserving Cooper
pair transport in a two-dimensional germanium hole gas

Marco Valentini1,∗, Oliver Sagi1, Levon Baghumyan1, Thijs de Gijsel1,2, Jason
Jung2, Stefano Calcaterra3,
Andrea Ballabio3, Juan Aguilera Servin1, Kushagra Aggarwal1,4, Marian Janik1,
Thomas Adletzberger1,
Rubén Seoane Souto5,6, Martin Leijnse7, Jeroen Danon8, Constantin Schrade5,
Erik Bakkers2,
Daniel Chrastina3, Giovanni Isella3, Georgios Katsaros1,∗

(1Institute of Science and Technology Austria, Am Campus 1, 3400
Klosterneuburg, Austria.
2 Department of Applied Physics, Eindhoven University of Technology,
Eindhoven, The Netherlands.
3 L-NESS, Physics Department, Politecnico di Milano, via Anzani 42, 22100,
Como, Italy.
4 Department of Materials, University of Oxford, Oxford OX1 3PH, United
Kingdom.
5 Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen,
2100 Copenhagen, Denmark.
6 Instituto de Ciencia de Materiales de Madrid, Consejo Superior de
Investigaciones Científicas (ICMM-CSIC), Madrid, Spain.
7 NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund,
Sweden.
8 Department of Physics, Norwegian University of Science and Technology,
NO-7491 Trondheim, Norway.
∗ Corresponding authors<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>.
)

## Abstract

Superconductor/semiconductor hybrid devices have attracted increasing interest
in the past years. Superconducting electronics aims to complement
semiconductor technology, while hybrid architectures are at the forefront of
new ideas such as topological superconductivity and protected qubits. In this
work, we engineer the induced superconductivity in two-dimensional germanium
hole gas by varying the distance between the quantum well and the aluminum. We
demonstrate a hard superconducting gap and realize an electrically and flux
tunable superconducting diode using a superconducting quantum interference
device (SQUID). This allows to tune the current phase relation (CPR), to a
regime where single Cooper pair tunneling is suppressed, creating a
$\boldsymbol{\sin\left(2\varphi\right)}$ CPR. Shapiro experiments complement
this interpretation and the microwave drive allows to create a diode with
$\boldsymbol{\approx 100\%}$ efficiency. The reported results open up the path
towards monolithic integration of spin qubit devices, microwave resonators and
(protected) superconducting qubits on a silicon technology compatible
platform.

Figure 1: Proximity induced superconductivity in Planar Ge. a Heavy hole (HH)
[light hole (LH)] band energies (black trace) [(grey trace)] along the growth
direction $z$ simulated using NextNano. HHs are accumulated at the upper QW
interface, as shown by the pink trace representing the HH wavefunction density
plotted in arbitrary units. b Hole mobility $\mu_{\textrm{h}}$ [upper panel]
and mean free path $l_{\textrm{h}}$ [lower panel], extracted from Hall bar
measurements, as a function of carrier density $n_{\textrm{h}}$ for samples
with ${\textrm{Si}}_{0.3}{\textrm{Ge}}_{0.7}$ spacer thickness (D) of 5nm (D5)
and 8nm (D8). c TEM image of the upper part of the Ge/SiGe heterostructure.
The left inset shows EDX data which confirm the absence of Al both in the
spacer and in the Ge QW. The scale bar corresponds to 20 nm. d Cross-section
sketch of a JoFET device; the dashed rectangle corresponds to the inset in c.
The gate voltage $V_{\textrm{g}}$ is used for varying the hole carrier density
in the underlying Ge QW. e Top-view sketch of a JoFET device with the circuit
for the 4-probe measurement. The width ($W$) and the channel length ($L$) are
indicated. f Voltage drop $V_{\textrm{JJ}}$ measured as a function of the
applied current $I_{\textrm{JJ}}$ for D5 [upper panel] and for D8 [lower
panel]. Lighter colors indicate lower values of $V_{\textrm{g}}$ [higher
carrier density] and darker colors indicate higher values of $V_{\textrm{g}}$
[lower carrier density]. Traces are equally spaced for both panels. g
$I_{\textrm{sw}}R_{\textrm{N}}$ product as a function of $V_{\textrm{g}}$ for
D5 and D8 and for Josephson junctions with different dimensions as indicated
in the inset. $W$ is reported in units of $\unit{\mu m}$, while $L$ is in
units of $\unit{nm}$.

## Main

III-V semiconductors have become the materials of choice for realizing high-
quality hybrid devices, due to the possibility of growing epitaxial Al on top
of them [1]. Gate-tunable superconducting and Andreev spin qubits [2, 3, 4, 5,
6], parametric amplifiers [7], highly efficient Cooper pair splitters [8, 9,
10] and a minimal Kitaev chain [11] are prominent examples of what has been
achieved in the past decade. In addition, non-reciprocal devices, such as
superconducting diodes have attracted a lot of interest [12], especially in
Josephson junctions in the presence of a Zeeman field [13, 14, 15, 16]. Diodes
can be also realized in a superconducting quantum interference device (SQUID)
geometry by exploiting a magnetic flux to achieve time-reversal breaking [17,
18, 19]. Such SQUIDs can be also used as a building block to create a
protected superconducting qubit by engineering a $\sin\left(2\varphi\right)$
current phase relation (CPR) [20, 21, 22, 23, 24, 25].

One drawback of III-V materials is their non-zero nuclear spin, which, through
hyperfine interaction, drastically reduces the electron spin coherence time,
limiting therefore the use of hybrid devices in combination with the spin
degree of freedom [5, 6]. Germanium, on the other hand, is a material which
allows proximity induced superconductivity and has shown great potential for
spin qubit devices [26]. Induced superconductivity in germanium was first
demonstrated in 0D and 1D systems [27, 28]. A few years later,
superconductivity was also induced in a two-dimensional Ge hole gas [29, 30].
Recent works demonstrated how induced superconductivity can be improved in
planar germanium, either by using a double superconducting stack [31] or by
annealing platinum contacts [32]. Here, using a shallow quantum well (QW) we
establish Ge/SiGe heterostructures as an alternative platform to III-V
materials for hybrid devices and microwave experiments, opening therefore the
path to the coexistence of semiconductor and superconducting qubits.

Figure 2: Superconducting gap tunability. a Sketch of the proximity effect. Al
has a superconducting parent gap $\Delta$ and it is coupled to the Ge hole
gas. The coupling $t$, and therefore the induced gap $\Delta^{*}$, depends on
the distance between the Al and Ge, i.e. on D. b Top-view sketch of the device
layout used to perform tunneling spectroscopy. The part of the Ge QW (right
side) not covered by Al is tuned to be fully conductive and behaves like a
normal metal reservoir. The two split gates are used for creating a tunnel
barrier by applying voltages $V_{\textrm{dg1}}$ and $V_{\textrm{dg2}}$. The
accumulation gate which covers the sample without Al on top is not depicted in
the sketch. c Side-view sketch of b. The green profile is a sketch of the
tunnel barrier for holes formed at the border between the conductive Ge and
the hybridized Ge. d [g] $dI/dV$ as a function of $V$ and
$V_{\textrm{dg}}=V_{\textrm{dg1}}+V_{\textrm{dg2}}$ plotted in logarithmic
scale for D8 [D5]. Data for lower $V_{\textrm{dg}}$ are shown in Fig. ED1. e
[h] Line-cuts taken from d [g] at different $V_{\textrm{dg}}$ (see small solid
circles) demonstrating a hard gap for sample D5. f [i] Line-cuts taken from d
[g] plotted in a normalized scale, in which the measured $dI/dV$ is divided by
the normal state conductance $G_{\textrm{N}}=$(dI/dV)$/G_{\textrm{normal}}$.
The traces are shifted vertically by 0.25 $G_{\textrm{N}}$ with respect to
each other.

## Material characterization and Josephson junctions

Compressively strained Ge QWs have been deposited on relaxed, linearly graded
buffers with 70% Ge content. The 18 nm thick QWs are separated from the top
surface by a spacer of thickness D. The built-in in-plane compressive strain
leads to charge confinement in the heavy-hole band (see Fig. 1a). Mobility
($\mu_{\textrm{h}}$) and mean free path ($l_{\textrm{h}}$) as a function of
the carrier density $n_{\textrm{h}}$ are displayed in Fig. 1b for a QW with
$\approx\qty{5}{nm}$ ${\textrm{Si}}_{0.3}{\textrm{Ge}}_{0.7}$ spacer (D5 -
red) and $\approx\qty{8}{nm}$ (D8 - blue), respectively. At high density, D5
(D8) shows $\mu_{\textrm{h}}\approx\qty{10000}{{cm}^{2}/Vs}$
($\qty{30000}{{cm}^{2}/Vs}$) and $l_{\textrm{h}}\approx\qty{250}{nm}$
($\qty{700}{nm}$).

For creating hybrid superconductor-semiconductor devices a thin film of
aluminum ($\approx 8-10\ \unit{nm}$) is deposited ex situ and at low
temperature on top of the ${\textrm{Si}}_{0.3}{\textrm{Ge}}_{0.7}$ spacer (see
Methods for the growth, the aluminum deposition and fabrication details). The
Al has a polymorphic structure and it is not grown epitaxially on top of the
${\textrm{Si}}_{0.3}{\textrm{Ge}}_{0.7}$ spacer. Importantly, energy
dispersive X-ray (EDX) data do not reveal interdiffusion of Al inside the
${\textrm{Si}}_{0.3}{\textrm{Ge}}_{0.7}$ spacer and Ge QW, see Fig. 1c.

In order to check if the superconducting properties can leak into the Ge hole
gas, Josephson field effect transistors (JoFETs) were fabricated (Fig. 1d).
Representative $V_{\textrm{JJ}}$ vs $I_{\textrm{JJ}}$ traces, measured in a
four-terminal current-biased configuration (Fig. 1e), for D5 [red, upper plot]
and D8 [blue, lower plot] are shown in Fig. 1f. The devices switch from
superconducting to the dissipative regime at the gate tunable switching
current $I_{\textrm{sw}}$. A common figure of merit used for estimating the
quality of the proximity effect is the product between $I_{\textrm{sw}}$ and
the normal state resistance $R_{\textrm{N}}$. Fig. 1g reports this product as
a function of the gate voltage $V_{\textrm{g}}$ for 6 different junctions with
different dimensions. At high negative values of gate voltages,
$I_{\textrm{sw}}R_{\textrm{N}}$ spans from slightly below $\qty{100}{\mu eV}$
to above $\qty{400}{\mu eV}$ depending on D and on the JoFET dimensions. These
values are favourably compared to previous results obtained with Ge
heterostructures hybridized by Al [29, 30] and they are on par with more
mature material systems [33].

In sample D5, the proximity effect is expected to be more effective because
the Al is closer to the Ge hole gas. However, the results reported in Fig. 1g
surprisingly suggest the opposite. One factor that could play a role is the
fact that D8, especially at high density, is in the short ballistic regime
($L<l_{\textrm{h}},\xi_{\textrm{N}}$), where $I_{\textrm{c}}R_{\textrm{N}}$ is
expected to be equal to $\pi\Delta/e$ [34]; $I_{\textrm{c}}$ is the critical
current and $\xi_{\textrm{N}}=\frac{\hbar^{2}\sqrt{2\pi
n_{\textrm{h}}}}{2m_{\textrm{eff}}\Delta}$ is the superconducting coherence
length in the quantum well with $m_{\textrm{eff}}$ being the effective mass.
Using $n_{\textrm{h}}=10^{12}\unit{cm^{-2}}$, $\Delta=\qty{200}{\mu eV}$ and
$m_{\textrm{eff}}$ to be around $10\%$ of the electron mass [35], we estimated
$\xi_{\textrm{N}}\approx 500\ $\mathrm{n}\mathrm{m}$$. On the contrary,
samples D5 have $L\approx l_{\textrm{h}}$, which implies a smaller
$I_{\textrm{c}}R_{\textrm{N}}$ [34], making the interpretation challenging.
Moreover, the variations of both D5 and D8 as a function of the JoFET
dimensions is not fully understood. Therefore, such JoFETs devices are not
sufficient to characterize the quality of the proximity effect, especially
because the switching current probability distribution is rather broad at low
temperatures [36].

Figure 3: Gate and flux tunable SDE employed as a generator of complex CPRs. a
Schematic of a typical SQUID. b $V_{\textrm{sq}}$ as a function of magnetic
flux $\Phi$ and $I_{\textrm{sq}}$. $I_{\textrm{sq}}$ was swept from positive
values to zero and, subsequently, from negative values to zero. The SQUID
lobes for positive and negative currents are asymmetric with respect to half
flux and the critical current at $\frac{\Phi}{\Phi_{0}}=0.5$ is larger than
$\left|I_{\textrm{ret2}}-I_{\textrm{ret1}}\right|$, see the brown line. c
Positive ($I_{\textrm{sq,+}}$) retrapping current extracted from b. d Diode
efficiency obtained from b. $\eta=0$ for integer and half-integer fluxes,
whereas it reaches its maxima and minima around $\frac{\Phi}{\Phi_{0}}=0.5$.
The orange traces represent the expected outcome if the SQUID would be
composed of standard tunnel junctions and it would have a total inductance of
$L=\qty{110}{pH}$. The mere addition of $L$ is not enough to match the
experimental data. The red traces represent the result of the numerical fit
($K_{1}=0.611,K_{2}=0.142,K_{3}=0.102,K_{4}=0.053,K_{5}=0.035,K_{6}=0.024,K_{7}=0.019,K_{8}=0.014$),
imposing $L=\qty{110}{pH}$. e [f] $\eta$ [$I_{\textrm{sq,+}}$] as a function
of $\Phi$ and $I_{\textrm{ret2}}$ with $I_{\textrm{ret1}}=\qty{40}{nA}$. The
behavior for the switching current is shown in the Extended Fig. ED9. $\eta$
is always zero at $\frac{\Phi}{\Phi_{0}}=0.5$, independently on
$I_{\textrm{ret2}}$, and the polarity of the diode is inverted at the balanced
point (pink cross). g [h] Theoretical calculation of e [f], showing
qualitatively similar behavior like the measurements. i-j First harmonic
contribution extracted from g. The green traces highlight the points where
their contribution vanishes. Just at the balanced point and at
$\frac{\Phi}{\Phi_{0}}=0.5$ both terms vanish. k Second harmonic contribution;
it never vanishes at $\frac{\Phi}{\Phi_{0}}=0.5$. The cosinusoidal
contribution is shown in Fig. ED5 along with higher order terms. l Ratio
between second and first harmonic, $K=\frac{|b_{2}|+|a_{2}|}{|b_{1}|+|a_{1}|}$
as a function of $I_{\textrm{ret2}}$ and $\Phi$. The red trace indicates the
points where $K=1$. In c, d and g-l, the same parameters have been used.

## Tunability of the induced superconducting gap

The above reported results demonstrate that proximity induced
superconductivity can be achieved in Ge without a direct contact to the
superconductor, therefore avoiding metallization issues [37]. However, it is
not clear to what extent the ${\textrm{Si}}_{0.3}{\textrm{Ge}}_{0.7}$ spacer
of thickness D is influencing the value of the induced superconducting gap
$\Delta^{*}$ and the subgap density of states. We expect that $\Delta^{*}$
depends on the coupling $t$ between the Al and the QW, see the sketch of Fig.
2a. Since ${\textrm{Si}}_{0.3}{\textrm{Ge}}_{0.7}$ acts as a tunnel barrier,
$t$ should be strongly dependent on D. In other words, if D is very thin, we
expect $\Delta^{*}$ to be similar to the parent gap of Al $\Delta$; whereas if
D is very thick the two layers will be very decoupled (small $t$) and
$\Delta^{*}$ will be quenched.

In order to investigate the dependence of $\Delta^{*}$ on D, tunneling
spectroscopy experiments were performed to estimate the local density of
states of the hybridized Ge QW, see Figs. 2b and c for the experimental
layout.

The differential conductance $dI/dV$, plotted in logarithmic scale, as a
function of source-drain bias $V$ and fingergates voltage $V_{\textrm{dg}}$
for D8 is shown in Fig. 2d. $dI/dV$ is suppressed symmetrically around $V=0$
independently on $V_{\textrm{dg}}$ (Fig. 2e); this signals the presence of a
superconducting gap in the hybridized Ge hole gas. Interestingly, a double
peak structure is revealed. The first peak appears at $V\approx\qty{240}{\mu
eV}$ and the second at $V\approx\qty{80}{\mu eV}$. We interpret them as the
parent gap of Al, $\Delta$, and the induced gap in the Ge hole gas
$\Delta^{*}$. Their presence is more evident if $dI/dV$ is plotted in a linear
scale, like in Fig. 2f where the line-cuts have been shifted vertically for
the sake of clarity. The sub-gap conductance is suppressed by one order of
magnitude compared to the above-gap value.

We now turn our attention to the $dI/dV$ of sample D5 (Fig. 2g). The
difference between Fig. 2g and 2d is striking. The region of suppressed
conductance around $V=0$ is larger in Fig. 2g, demonstrating that the induced
gap is bigger. The line-cuts, see Fig. 2h, showcase a difference between the
normal-state conductance and the conductance at $V=0$ of about two orders of
magnitude, indicative of a hard gap [1]. Also for this device the double peak
structure is observed (Fig. 2h and Fig. 2i). The parent gap appears at a
similar value, namely $\Delta\approx\qty{230}{\mu eV}$, while
$\Delta^{*}\approx\qty{150}{\mu eV}$.

## Superconducting diode effect

Having demonstrated a hard superconducting gap, we use the hybrid Al/Ge
platform, to build a SQUID device which acts as a gate/flux tunable
superconducting diode [12] and as a generator of non-sinusoidal current-phase
relations (CPRs). The superconducting diode effect (SDE) can appear in a
simple SQUID either if its inductance $L$ is significant [38, 39] or if the
CPRs of the single junctions have higher order contributions, arising from
Andreev bound states in a semiconducting junction [17, 40] or from junctions
in the dirty limit [41, 42].

Fig. 3a shows the schematics of a SQUID, with the underlying 4-probe current-
biased electrical circuit. $I_{\textrm{sq}}$ is the current passing through
the SQUID, $J$ is the current circulating in the SQUID and $V_{\textrm{sq}}$
is the measured voltage drop across the device. In the following, we always
sweep $I_{\textrm{sq}}$ from positive/negative values to $I_{\textrm{sq}}=0$,
such that the retrapping current ($I_{\textrm{sq,+(-)}}$) is recorded for both
branches. Top gate voltages $V_{\textrm{g1}}$ and $V_{\textrm{g2}}$ are used
to tune the retrapping current $I_{\textrm{ret1}}$ of JJ1 and
$I_{\textrm{ret2}}$ of JJ2. We first tune the device to be slightly
unbalanced, i.e. $I_{\textrm{ret1}}=\qty{60}{nA}\neq
I_{\textrm{ret2}}=\qty{69}{nA}$. Fig. 3b represents a SQUID measurement for
such configuration, where $V_{\textrm{sq}}$ is recorded as a function of
$I_{\textrm{sq}}$ and $\Phi$. $I_{\textrm{sq,+ (-)}}$ is periodically
modulated by $\Phi$, as can be clearly seen in Fig. 3c. However, two features
are observed, which are not expected for a negligible inductance SQUID
composed by tunnel junctions. First, the retrapping current at
$\Phi=\frac{\Phi_{0}}{2}$ is expected to be
$|I_{\textrm{ret1}}-I_{\textrm{ret2}}|=\qty{9}{nA}$ (see brown horizontal line
in Fig. 3b), instead the measured value is around $\qty{40}{nA}$. Moreover,
the SQUID pattern is not symmetric with respect to $\Phi=\frac{\Phi_{0}}{2}$,
see green arrows in Fig. 3b. This asymmetry gives rise to a finite SDE. The
diode efficiency, defined as
$\eta=\frac{I_{\textrm{sq,+}}-|I_{\textrm{sq,-}}|}{I_{\textrm{sq,+}}+|I_{\textrm{sq,-}}|}$,
is shown in Fig. 3d. In particular, $\eta=0$ at integer ($\Phi=n\Phi_{0}$) and
it changes its sign around $\Phi=\frac{n}{2}\Phi_{0}$. The maximum value
observed for this device is around $15\%$.

In order to understand these results, we solve the static equation of the
system:

$\begin{split}\frac{I_{\textrm{sq}}}{2}+J=I_{\textrm{JJ1}}\left(\varphi_{1}\right),\\\
\frac{I_{\textrm{sq}}}{2}-J=I_{\textrm{JJ2}}\left(\varphi_{2}\right).\end{split}$
(1)

$I_{\textrm{JJ1}}$ [$I_{\textrm{JJ2}}$] is the current flowing through JJ1
[JJ2] which depends on the phase difference across the junction $\varphi_{1}$
[$\varphi_{2}$]. The phase drops are related to the fluxoid quantization:

$\varphi_{2}-\varphi_{1}=2\pi\frac{\Phi}{\Phi_{0}}+2\pi\frac{LJ}{\Phi_{0}}$
(2)

For a given $\Phi$, $I_{\textrm{sq,+}}$ [$I_{\textrm{sq,-}}$] is obtained by
finding the maximum [minimum] $I_{\textrm{sq}}$ with respect to $\varphi_{1}$.

First, we attempt to understand our results assuming standard sinusoidal CPRs,
i.e.
$I_{\textrm{JJ1}}\left(\varphi_{1}\right)=I_{\textrm{ret1}}\sin\left(\varphi_{1}\right)$
and
$I_{\textrm{JJ2}}\left(\varphi_{1}\right)=I_{\textrm{ret2}}\sin\left(\varphi_{2}\right)$,
and adding the inductive contribution. $L$ is composed of two terms, a
geometric one $L_{\textrm{geo}}$ and a kinetic one $L_{\textrm{kin}}$; we
estimated $L=\qty{110}{pH}$, see methods for details. The orange traces in
Figs. 3c-d represent the theoretical prediction. It is clear that the mere
addition of a realistic $L$ does not capture the full picture, especially
around $\frac{\Phi}{\Phi_{0}}=0.5$, where $I_{\textrm{sq,+}}$ and
$|I_{\textrm{sq,-}}|$ are greatly underestimated, see Fig. ED2.

Therefore, it is necessary to consider higher order harmonics for explaining
our results, namely we assume that our single junction CPRs are given by:

$\begin{split}I_{\textrm{JJ1}}\left(\varphi_{1}\right)=\alpha_{1}I_{\textrm{ret1}}\sum_{n}{\left(-1\right)}^{n+1}K_{n}\sin\left(n\varphi_{1}\right)\\\
I_{\textrm{JJ2}}\left(\varphi_{2}\right)=\alpha_{2}I_{\textrm{ret2}}\sum_{n}{\left(-1\right)}^{n+1}K_{n}\sin\left(n\varphi_{2}\right)\\\
,\end{split}$ (3)

where $K_{n}$ is the relative contribution of the n-th harmonic and we assumed
that the harmonics’ contribution is the same for both junctions. $\alpha_{1}$
[$\alpha_{2}$] is a dimensionless parameter which is adjusted such that
$\max{I_{\textrm{JJ1}}}=I_{\textrm{ret1}}$
[$\max{I_{\textrm{JJ2}}}=I_{\textrm{ret2}}$].

The red traces in Figs. 3c-d are the outcome of a numerical fit using up to
eight harmonic contributions. It is found that $K_{1}=0.611$ and
$K_{2}=0.142$, while higher order terms are smaller than $10\%$, see Fig. ED3
to understand the effect of higher order terms. We point out that also
asymmetric cases would give qualitatively similar results (Fig. ED4), but the
amount of free parameters of the fit would increase considerably.

We now turn our attention to the gate dependence of the SDE. Our measurements
show that the SDE can be tuned by the gate voltages $V_{\textrm{g1}}$ and
$V_{\textrm{g2}}$. In Fig. 3e, we fix $I_{\textrm{ret1}}=\qty{40}{nA}$ and we
study the behavior of $\eta$ while varying $\Phi$ and $I_{\textrm{ret2}}$.
When $I_{\textrm{ret2}}>I_{\textrm{ret1}}$, $\eta>0$ [$\eta<0$] for
$\frac{\Phi}{\Phi_{0}}>0.5$ [$\frac{\Phi}{\Phi_{0}}<0.5$], while the trend is
opposite if $I_{\textrm{ret2}}<I_{\textrm{ret1}}$. In other words, we have an
inversion of the diode polarity going from one regime to the other and most
importantly, the SDE completely vanishes independently of $\Phi$ when the two
junctions are fully balanced
($I_{\textrm{ret1}}=I_{\textrm{ret2}}=I_{\textrm{ret}}$).

$I_{\textrm{sq,+}}$, plotted in Fig. 3f, does not vanish even at half flux
quantum and for balanced junctions, see the pink cross in Fig. 3f; we refer to
this condition as the sweet spot. This is a crucial aspect, because at the
sweet spot, the first harmonic of $I_{\textrm{sq}}$
($\propto\sin\left(\varphi\right)$) is completely suppressed but not the
higher-order terms. We can understand this from considering Eqs. 1 at the
sweet spot,

$\begin{split}\frac{I_{\textrm{sq}}}{2}+J=\alpha
K_{1}I_{\textrm{ret}}\sin\left(\varphi_{1}\right)-\alpha
K_{2}I_{\textrm{ret}}\sin\left(2\varphi_{1}\right)\\\
\frac{I_{\textrm{sq}}}{2}-J=\alpha
K_{1}I_{\textrm{ret}}\sin\left(\varphi_{1}+\pi\right)-\alpha
K_{2}I_{\textrm{ret}}\sin\left(2\varphi_{1}+2\pi\right),\end{split}$ (4)

where for simplicity the inductance and higher order terms are neglected and
$\alpha_{1}=\alpha_{2}=\alpha$. Therefore the CPR of the SQUID would be
$I_{\textrm{sq}}\left(\varphi_{1}\right)=-2\alpha
K_{2}I_{\textrm{ret}}\sin\left(2\varphi_{1}\right)$. This CPR corresponds to
transport through the SQUID being governed by pairs of Cooper pairs, while the
exchange of single pairs is quenched. This is the condition needed for
creating a certain type protected qubit [20].

This behaviour can be further understood by solving Eqs.1, 2 and assuming to
have JJs with higher order contributions. Fig. 3g [h] represents the
theoretical calculation of $\eta$ [$I_{\textrm{sq,+}}$] for a SQUID with the
parameters extracted from the fit of Figs. 3c-d. The simulation results agree
well on a qualitative level with the measurements. From the theoretical
calculation it is possible to calculate the CPR of the SQUID and express it as
a Fourier expansion:

$\begin{split}I_{\textrm{sq}}\left(\varphi_{1}\right)\approx
b_{1}\sin\left(\varphi_{1}\right)+b_{2}\sin\left(2\varphi_{1}\right)+\dots{\\\
}+a_{1}\cos\left(\varphi_{1}\right)+a_{2}\cos\left(2\varphi_{1}\right)+\dots{}\end{split}$
(5)

where $b_{\textrm{n}}$ and $a_{\textrm{n}}$ represents the n-th harmonic
contribution. The values of the first harmonic terms $b_{1}$, $a_{1}$ and
second harmonic term $b_{2}$ obtained from numerical simulations are shown in
Figs. 3i-k. Finally, we show the theoretical prediction of the ratio between
the second and first harmonic, i.e.
$K=\frac{|b_{2}|+|a_{2}|}{|b_{1}|+|a_{1}|}$, see Fig. 3l. The red trace
depicts the points where the first and second harmonics equally contribute to
the SQUID CPR, whereas the ratio diverges close to the sweet spot.

We note that, different CPRs of the single Josephson junctions would give
slightly different outcomes. However, it would not change the main conclusion
that $b_{1}$ and $a_{1}$ can be completely suppressed and, importantly,
without a non-sinusoidal CPR of single JJs, the amount of $b_{2}$ in the sweet
spot would be much smaller. Even if the SDE observed in this work would not be
given by a CPR with higher order harmonics but by a high inductance value, the
main message would not be altered. In fact, also in this case, due to the gate
tunability of the semiconductor JJ, it is possible to find a sweet spot where
$I_{\textrm{sq}}\left(\varphi_{1}\right)=b_{2}\sin\left(2\varphi_{1}\right)+b_{4}\sin\left(4\varphi_{1}\right)+\dots$,
suppressing completely the first harmonic and having a non-negligible second
harmonic contribution, see Figs. ED6, ED7 for details. Importantly, the first
harmonic contribution can be suppressed over a broad range of gate space,
which also allows to tune the second harmonic contribution (Fig. ED8).

Figure 4: Half-integer Shapiro steps and ideal superconducting diode.  a
Schematics of a typical SQUID used for Shapiro experiments with a microwave
drive $I_{\textrm{ac}}\sin\left(2\pi f_{\textrm{ac}}t\right)$. b [c] Shapiro
pattern for a $\qty{30}{nm}$ thick room temperature deposited Al sample with
$R_{\textrm{shunt}}=\qty{20}{\Omega}$ in the balanced regime
($I_{\textrm{ret1}}=I_{\textrm{ret2}}=\qty{500}{nA}$) with
$f_{\textrm{ac}}=\qty{500}{MHz}$ and at $\frac{\Phi}{\Phi_{0}}=0.58$
[$\frac{\Phi}{\Phi_{0}}=0.42$]. The differential resistance
$dV_{\textrm{sq}}/dI_{\textrm{sq}}$ is plotted as a function of the RF power
$P$ and $I_{\textrm{sq}}$. Dips in $dV_{\textrm{sq}}/dI_{\textrm{sq}}$
correspond to integer Shapiro steps. d same measurement as b and c but at
$\frac{\Phi}{\Phi_{0}}=0.5$. Importantly at half flux quantum, the first half-
integer steps appear for low $P$. e Shapiro map as a function of
$I_{\textrm{sq}}$ and $\Phi$ in the balanced regime (like for the previous
plots) for $P=\qty{9}{dBm}$. The half-integer steps appear only close to
$\frac{\Phi}{\Phi_{0}}=0.5$. f Shapiro map as a function of $I_{\textrm{sq}}$
and $V_{\textrm{g2}}$ for $P=\qty{9}{dBm}$ and at $\frac{\Phi}{\Phi_{0}}=0.5$.
The half-integer steps appear when the SQUID is close to the balanced
condition, i.e. if $I_{\textrm{ret2}}\approx 500\pm 40\unit{nA}$. g and h show
data from another $\qty{30}{nm}$ thick Al sample but at
$f_{\textrm{ac}}=\qty{2}{GHz}$ and with $R_{\textrm{shunt}}=\qty{50}{\Omega}$
for achieving a better visibility. g Current-voltage characteristic for
$\Phi=0.39\Phi_{0}$ and for $f_{\textrm{ac}}=\qty{2}{GHz}$ with
$I_{\textrm{ret1}}=\qty{670}{nA}$ and $I_{\textrm{ret2}}=\qty{450}{nA}$ for
different power $P$. At $P=\qty{12}{dBm}$, $I_{\textrm{sq,+}}$ is
$\qty{120}{nA}$, whereas $I_{\textrm{sq,-}}\approx 0$, i.e. $\eta\approx 1$. h
The non-volatility of SDE at $P=\qty{12}{dBm}$ is demonstrated by switching
between the normal and superconducting behaviour alternating $I_{\textrm{sq}}$
from $\qty{90}{nA}$ to $\qty{-90}{nA}$ [upper panel]. In the lower panel, the
measured voltage $V_{\textrm{sq}}$ is reported. A time dependent offset of
$V_{\textrm{sq}}$, due to drift, was subtracted.

## Half-integer Shapiro steps and ideal SDE

The good qualitative match between the experiment and the theoretical
prediction of Fig. 3 makes us confident in our interpretation of the diode
data, but the CPR of the SQUID was not directly probed. The AC Josephson
effect would help to further elucidate the CPR periodicity. In fact, for a
standard sinusoidal CPR under microwave irradiation, the current-voltage
characteristics develop voltage steps when $V=s\frac{hf_{\textrm{ac}}}{2e}$,
the so-called Shapiro steps, where $s=0,1,2,\dots$ and $f_{\textrm{ac}}$ is
the external applied frequency. On the contrary, if the CPR becomes
$\propto\sin\left(2\varphi\right)$, signalling tunneling of pairs of Cooper
pairs, steps at half-integer values also appear, i.e. $s=0,0.5,1,\dots$ [43,
44, 45]. In our case, we expect the ratio between the second and first
harmonic to be maximized when the SQUID is balanced and
$\Phi\approx\frac{\Phi_{0}}{2}$, see Fig. 3l as an example. Therefore, we
would expect to observe half-integer Shapiro steps when approaching the sweet
spot [17].

In order to avoid flux generated by inductive effects which might lead to
similar results [46], we present results of a 30 nm-thick aluminum SQUID,
which has a much smaller inductance ($L<\qty{15}{pH}$). Furthermore, a shunt
resistor $R_{\textrm{shunt}}$ of 10-50 $\unit{\Omega}$, see Fig. 4a, was added
in order to create overdamped junctions, allowing therefore to measure Shapiro
steps at small external frequencies, avoiding issues related to Landau-Zener
transitions [47].

In the following, we study a SQUID in a balanced configuration
($I_{\textrm{ret1}}=I_{\textrm{ret2}}$) subjected to an external drive at
$f_{\textrm{ac}}=\qty{500}{MHz}$. Fig. 4b shows the differential resistance of
the SQUID $dV_{\textrm{sq}}/dI_{\textrm{sq}}$ as a function of the microwave
drive power $P$ and $I_{\textrm{sq}}$ at $\frac{\Phi}{\Phi_{0}}=0.58$. If $P$
is high enough, dips corresponding to the integer Shapiro steps $s=1,2,3$
appear. Similar results are obtained at $\frac{\Phi}{\Phi_{0}}=0.42$, see Fig.
4c. However, the situation is different if $\frac{\Phi}{\Phi_{0}}=0.5$ (Fig.
4d) a condition for which the first harmonic term should vanish. For this
situation, the first half-integer steps appears as theoretically expected (see
Fig. ED10 for the identification of the Shapiro steps).

This behavior is summarized in Fig. 4e where we fix $P=\qty{9}{dBm}$ and we
display $dV_{\textrm{sq}}/dI_{\textrm{sq}}$ as a function of $I_{\textrm{sq}}$
and $\Phi$. Far from half flux quantum, dips corresponding to integer Shapiro
steps are observed; while close to $\frac{\Phi}{\Phi_{0}}=0.5$ the integer
steps fade and the $s=0.5$ step becomes pronounced, see white numbers. In
order to further investigate the range over which the half-integer Shapiro
step is visible, we fix $\frac{\Phi}{\Phi_{0}}=0.5$ and we vary
$I_{\textrm{ret2}}$ with the gate voltage $V_{\textrm{g2}}$. When the SQUID is
close to the balanced regime ($V_{\textrm{g2}}\approx\qty{0.1}{V}$) the first
half-integer step is evident (white numbers); however it fades away if the
SQUID becomes unbalanced, i.e. $V_{\textrm{g2}}>\qty{0.2}{V}$. As expected
from the previous analysis, the half-integer step appears only if the device
is close to the balanced position and close to half flux quantum when
$I_{\textrm{sq}}\left(\varphi_{1}\right)=b_{2}\sin\left(2\varphi_{1}\right)+b_{4}\sin\left(4\varphi_{1}\right)+\dots$.
Importantly, also a second device investigated under microwave irradiation
showed the same behavior, see Fig. ED10.

The SDE indicates that the symmetry $I_{\textrm{sq,+}}=-I_{\textrm{sq,-}}$ is
broken in our system, which also implies that the widths $\Delta I_{\pm 1}$ of
the two first Shapiro steps, which eventually define $I_{\textrm{sq,+}}$ and
$I_{\textrm{sq,-}}$, can be different [17]. As the position and the width of
the plateaus depend on the microwave drive, one can envision tuning to a
situation in which the first negative plateau would start at zero current
($I_{\textrm{sq,-}}=0$) while the first positive one at a finite current
($I_{\textrm{sq,+}}\neq 0$). At this particular strength of the ac driving,
the SQUID is expected to become an ideal SDE, i.e. $\eta\approx 1$.

In order to investigate this possibility a similar SQUID, with
$R_{\textrm{shunt}}=\qty{50}{\Omega}$, at $\Phi=0.39\Phi_{0}$ for different
drive powers $P$ was investigated (Fig. 4g). For small $P$,
$I_{\textrm{sq,+}}>|I_{\textrm{sq,-}}|$ and $\eta\approx 0.18$, see blue
trace. When $P$ increases both $I_{\textrm{sq,+}}$ and $|I_{\textrm{sq,-}}|$
decrease, see orange trace. Eventually when $P$ is high enough (green trace),
$I_{\textrm{sq,-}}$ drops to zero, whereas $I_{\textrm{sq,+}}\neq 0$, yielding
a diode efficiency equal to 1. Moreover, we show that our device is non-
volatile, namely we can switch several times from the normal-state to the
superconducting branch by changing the current direction, see Fig. 4h.

Figure 5: Hybrid coplanar waveguide resonators.  a Scanning electron
microscopy image of a typical notch-type CPW resonator. The top insets show
sketches of the CPW cross-section, at the position of the cyan trace, when the
QW has been etched away (left) and when the resonator is fabricated on top of
the QW (right). In the latter case, the feedline, the resonator and the ground
plane are formed on top of the Ge QW. The scale bar corresponds to $500\
$\mu\mathrm{m}$$. b $\left|S_{21}\right|$ signal as a function of the
frequency $f$ around the resonance frequency of a device without QW. The black
line is an algebraic fit [48]. c Internal quality factor $Q_{\textrm{i}}$,
averaged for multiple $\qty{4}{GHz}<f_{\textrm{r}}<\qty{5.5}{GHz}$ resonators
(Table ED1), as a function of the average photon number $\langle
n_{\textrm{ph}}\rangle$ for samples without QW, D5 and D60. The shaded areas
correspond to the standard deviation of the data. We notice that the
proximitized Ge QW in the samples D5 leads to small additional losses,
compared to the without QW resonator samples. However, if the Ge QW gets
decoupled from the Al (D60), we observe a lower $Q_{\textrm{i}}$, suggesting
that $Q_{\textrm{i}}$ is limited by losses in Ge.

## Discussion/Outlook

In the past few years, planar germanium has established itself as a promising
platform for spin-qubit arrays [26]. Here, we demonstrate its potential also
for hybrid semiconductor-superconductor quantum devices. Inspired by more
mature technologies [33], we introduced a reliable way to induce
superconductivity by using shallow QWs. Our method does not rely on the
precise etching of the QW and/or surface treatments [31] and does not require
in-situ deposition of the superconductor. While the shallow QWs reported in
this work are of limited mobility, possible mitigation strategies of this
problem could include a careful engineering of the semiconductor/dielectric
interface [49], including the use of Ge caps [50], or growing the QWs on Ge
instead of Si wafers [51].

The reported large superconducting hard gap on a group IV material will pave
the way towards spin qubit coupling via superconductors [52, 53]. In addition,
the realized gate and flux-tunable superconducting diode can suppress the
first harmonic term, making it therefore an interesting building block for
creating protected superconducting qubits with semiconductor materials [21,
20, 24, 23, 22]. In order to realize such qubits, superconducting resonators
are key elements. A $\lambda/4$ notch-type resonator is shown in (Fig. 5a)
(see methods for details). The upper-left inset of Fig. 5a depicts the cross-
section of the resonator, pointing out that the Ge QW has been completely
etched away prior to the resonator fabrication. Fig. 5b shows the transmission
amplitude $\left|S_{21}\right|$ as a function of the probe frequency $f$. The
internal quality factors $Q_{\textrm{i}}$ were extracted [48] and found to be
around 7000 [20000] for $\langle n_{\textrm{ph}}\rangle\approx 1$ [100];
demonstrating the microwave compatibility of the used Ge/SiGe
heterostructures. Interestingly, just slightly smaller $Q_{\textrm{i}}$ values
were extracted also for superconducting resonators fabricated on Ge/SiGe
heterostructures where the Ge QW has been removed just in the gap between the
central conductor and the ground plane, showing that the proximitized Ge does
not lead to significant losses. The above demonstrated microwave compatibility
of the used Ge/SiGe heterostructures opens a path towards spin-photon
experiments [54], gate tunable transmon qubits [2, 3, 4] and superconducting
spin qubits in group IV materials [6] and allow us to envision the transfer of
quantum information between different types of qubits, all realizable on
planar Ge.

## Methods

#### Growth and Al deposition

Strained Ge QW structures were grown by low-energy plasma-enhanced chemical
vapor deposition on forward-graded buffers [55] with
${\textrm{Si}}_{0.3}{\textrm{Ge}}_{0.7}$ caps of 5 and 8
$\mathrm{n}\mathrm{m}$ above the 18 $\mathrm{n}\mathrm{m}$ Ge QW. These
nominal QW and cap thicknesses vary across the wafer due to the intensity
profile of the focused plasma. Thicknesses were verified by comparing high-
resolution x-ray diffraction $\omega$–2$\theta$ scans with dynamical
simulations based on a smoothed QW profile [55]. This same composition profile
was constructed within the NextNano 1-d Poisson-Schrödinger solver, along with
a dieletric layer and top Schottky contact, in order to generate the band
profile and wavefunction density shown in Fig. 1a. The Ge/SiGe
heterostructures are cut in pieces of 6x6 $\unit{mm^{2}}$. A $\qty{3}{min}$
buffered oxide etch removes the native surface oxide of the diced samples,
after which they are transferred to a molecular beam epitaxy chamber. The
samples are cooled down to $\qty{110}{K}$ by active liquid-nitrogen cooling.
Subsequently, Al is deposited at a growth rate of $\qty{5.5}{\AA\per min}$.
Immediately after growth, samples are transferred in-situ to a chamber
equipped with an ultrahigh-purity $\textrm{O}_{2}$ source where they are
exposed to $10^{-4}\unit{mbar}$ of $\textrm{O}_{2}$ for $\qty{15}{min}$. The
formed oxide layer prevents subsequent retraction of the metal film as the
sample warms up to room temperature under ultrahigh vacuum conditions.

#### Sample fabrication

10 nm-thick, cold deposited aluminum samples:
A mesa of around $\qty{60}{nm}$ depth is obtained by first removing Al with
Transene D and then by etching the heterostructure with a
${\textrm{SF}}_{6}$-${\textrm{O}}_{2}$-${\textrm{CHF}}_{3}$ reactive ion
etching process. In a second step, Al is selectively etched away using
Transene D in order to create the Josephson junction or tunneling spectroscopy
devices. Then, for tunneling spectroscopy devices, normal metal ohmic contacts
are created by argon milling the SiGe spacer followed by a deposition of
$\qty{60}{nm}$ platinum at an angle of 5°. Finally, aluminum oxide is
deposited on top of all the sample and then Ti/Pd gates are evaporated. For
some devices, two layers of top-gates were needed.

30 nm-thick, room-temperature deposited aluminum samples:
A mesa of around $\qty{60}{nm}$ depth is obtained by etching the
heterostructure with a
${\textrm{SF}}_{6}$-${\textrm{O}}_{2}$-${\textrm{CHF}}_{3}$ reactive ion
etching process. The sample is then submerged for $\qty{15}{s}$ in buffered HF
and, subsequently, the $\qty{30}{nm}$ Al film is deposited. Gates are
patterned like for the $\qty{10}{nm}$-thick sample. Importantly, this
technique allows to fabricate devices without the need of wet etching for
removing the superconductor.

CPW resonator, without QW:
The QW is removed by a
${\textrm{SF}}_{6}$-${\textrm{O}}_{2}$-${\textrm{CHF}}_{3}$ reactive ion
etching process. Subsequently, the CPW resonator, the feed-line and the ground
plane are written by electron beam lithography followed by a 25 nm-thick Al
deposition at room temperature.

CPW resonator, with QW:
Electron beam lithography is performed on a sample with low temperature
deposited Aluminum. The area between the ground plane and the signal line is
exposed and after development the Al is removed by transene D etching.
Finally, and before removing the resist the Ge QW is etched away by a
${\textrm{SF}}_{6}$-${\textrm{O}}_{2}$-${\textrm{CHF}}_{3}$ reactive ion
etching process.

|  $\qty{10}{nm}$ Al |  $\qty{30}{nm}$ Al
---|---|---
$R_{\square}$ | $\qty{12.2}{\Omega}$ | $\qty{0.75}{\Omega}$
$T_{\textrm{C}}$ | $\qty{1.9}{K}$ | $\qty{1.4}{K}$
$L_{\textrm{kin},\square}$ | $\qty{8.9}{pH}$ | $\qty{0.75}{pH}$
$L_{\textrm{kin}}+L_{\textrm{geo}}$ | $\qty{110}{pH}$ | $\qty{13}{pH}$
Table 1: Estimation of SQUID inductance.  $R_{\square}$ and $T_{\textrm{C}}$
were estimated from 4-probe current biased measurements. The SQUIDs are
consisting of approximately 12 squares.

#### Inductance estimation

The total inductance $L$ of the measured SQUIDs has two contributions: the
geometric one ($L_{\textrm{geo}}$) and the kinetic one ($L_{\textrm{kin}}$).
For the $L_{\textrm{geo}}$ we approximated the device to a loop with a radius
$R$ and with the wire diameter $d$

$L_{\textrm{geo}}=\mu_{0}R\left[\ln\left(\frac{16R}{d}\right)-2\right]$ (6)

where $\mu_{0}$ is the magnetic vacuum permeability and we assumed the
relative magnetic permeability $\mu_{\textrm{r}}=1$. Typical values of our
SQUID geometry are $R\approx\qty{1.25}{\mu m}$ and $d\approx\qty{0.7}{\mu m}$,
which gives $L_{\textrm{geo}}\approx\qty{2}{pH}$. However, in order not to
underestimate this contribution, $L_{\textrm{geo}}$ is assumed to be as large
as $\approx\qty{5}{pH}$. As regards the kinetic inductance per square
$L_{\textrm{kin},\square}$, it was estimated from the values of the
superconducting gap and the square normal-state resistance $R_{\square}$ [56]:

$L_{\textrm{kin},\square}=\frac{h}{2\pi^{2}}\frac{R_{\square}}{\Delta}$ (7)

where $\Delta$ is estimated from the critical temperature $T_{\textrm{C}}$,
i.e. $\Delta=1.76k_{\textrm{B}}T_{\textrm{C}}$ with $k_{\textrm{B}}$ being the
Boltzmann constant. The results are summarized in Table 1.

### Acknowledgment

The authors acknowledge Alexander Brinkmann, Alessandro Crippa, Andrew
Higginbotham, Andrea Iorio, Giordano Scappucci and Christian Schonenberger for
helpful discussions. We thank Marcel Verheijen for the support in the TEM
analysis. This research and related results were made possible with the
support of the NOMIS Foundation. It was supported by the Scientific Service
Units of ISTA through resources provided by the MIBA Machine Shop and the
nanofabrication facility, the European Union’s Horizon 2020 research and
innovation programme under Grant Agreement No 862046, the HORIZON-RIA
101069515 project and the FWF Projects #P-32235, #P-36507 and #F-8606. R.S.S.
acknowledges Spanish CM “Talento Program” Project No. 2022-T1/IND-24070.

Author contribution. M.V. and T.G. fabricated the transport devices and
performed the measurements under the supervision of G.K.. M.V. did the data
analysis under the supervision of G.K.. L.B. and O.S. fabricated and measured
the CPW. M.J. and O.S. developed the microwave technology for the Ge/SiGe
heterostructures. K.A. contributed to the transport measurements and to the
device fabrication. J.A.S. fabricated the Hall bars. T. A. made the Shapiro
measurements possible. M.V. performed the simulations with input from C.S. and
R.S.S.. S.C., A.B., D.C. and G.I. were responsible for the Ge QW growth, Hall
bar measurements and NextNano simulations. J.J. and E.B. were responsible for
the growth of low temperature Al and TEM data. M. L. and J. D. contributed to
the interpretation of the experimental results. M.V. and G.K. wrote the
manuscript with input from all the coauthors.

Competing interests: The authors declare no competing interests.

Data and Materials Availability: All experimental data included in this work
and related to this work but not explicitly shown in the paper will be
available via the IST Austria repository.

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## Extended data

Figure ED1: Additional data for Fig.3. $dI/dV$ as a function of $V$ and $V_{\textrm{dg}}$ plotted in logarithmic scale for sample D5. The data for less positive $V_{\textrm{dg}}$ show indications of quantum dots formation.  sample |  $f_{\textrm{r}}($GHz$)$ | $Q_{\textrm{e}}$ | $Q_{\textrm{i},\langle n_{\textrm{ph}}\rangle=0.1}$ | $Q_{\textrm{i},\langle n_{\textrm{ph}}\rangle=1}$ | $Q_{\textrm{i},\langle n_{\textrm{ph}}\rangle=10}$ | $Q_{\textrm{i},\langle n_{\textrm{ph}}\rangle=100}$
---|---|---|---|---|---|---
without QW | $4.01$ | $6150\pm 150$ | $2800\pm 200$ | $6000\pm 300$ | $13000\pm 600$ | $30000\pm 3000$
without QW | $4.04$ | $6300\pm 150$ | n.a. | $6270\pm 130$ | $10000\pm 130$ | $15700\pm 170$
without QW | $4.18$ | $3250\pm 70$ | n.a. | $9240\pm 350$ | $15350\pm 450$ | $26350\pm 635$
without QW | $4.23$ | $3380\pm 40$ | n.a. | $5670\pm 140$ | $8220\pm 125$ | $11000\pm 175$
without QW | $4.27$ | $3800\pm 150$ | $2700\pm 200$ | $7000\pm 300$ | $15000\pm 800$ | $26000\pm 3000$
without QW | $4.60$ | $2750\pm 150$ | $3800\pm 200$ | $8800\pm 300$ | $17000\pm 500$ | $24000\pm 1000$
without QW | $4.66$ | $4000\pm 300$ | $5200\pm 500$ | $10000\pm 500$ | $21000\pm 700$ | $40000\pm 2000$
without QW | $4.77$ | $2000\pm 40$ | n.a. | $7780\pm 200$ | $11900\pm 400$ | $17100\pm 1120$
without QW | $4.84$ | $1370\pm 15$ | n.a. | $7600\pm 360$ | $12220\pm 1220$ | $18350\pm 3000$
without QW | $4.91$ | $2500\pm 200$ | $5000\pm 400$ | $10000\pm 600$ | $17000\pm 1000$ | $27000\pm 2000$
without QW | $5.02$ | $2100\pm 70$ | $3100\pm 100$ | $4300\pm 100$ | $6560\pm 240$ | $9000\pm 725$
without QW | $5.24$ | $1550\pm 20$ | $4100\pm 175$ | $5340\pm 160$ | $7430\pm 300$ | $9640\pm 600$
without QW | $5.72$ | $1660\pm 25$ | $4300\pm 135$ | $5500\pm 145$ | $7660\pm 260$ | $10200\pm 930$
without QW | $5.78$ | $1365\pm 20$ | $4830\pm 180$ | $5950\pm 140$ | $8230\pm 220$ | $10270\pm 500$
without QW | $6.00$ | $1700\pm 20$ | $5250\pm 220$ | $6300\pm 150$ | $9060\pm 230$ | $12580\pm 500$
D5 | $4.04$ | $4740\pm 800$ | n.a. | $5365\pm 225$ | $8920\pm 550$ | $14360\pm 1240$
D5 | $4.13$ | $3200\pm 215$ | n.a. | $6400\pm 260$ | $10530\pm 825$ | $15400\pm 1000$
D5 | $4.22$ | $3300\pm 130$ | n.a. | $3230\pm 230$ | $11000\pm 610$ | $15750\pm 1160$
D5 | $4.35$ | $3085\pm 215$ | n.a. | $5850\pm 230$ | $10150\pm 570$ | $15200\pm 1200$
D5 | $4.39$ | $1440\pm 55$ | n.a. | $5690\pm 220$ | $10400\pm 680$ | $14200\pm 1750$
D5 | $4.43$ | $2100\pm 100$ | n.a. | $5500\pm 210$ | $8900\pm 470$ | $13040\pm 720$
D5 | $4.52$ | $1680\pm 240$ | n.a. | $5230\pm 510$ | $6880\pm 1060$ | $10000\pm 3050$
D5 | $4.56$ | $2450\pm 170$ | n.a. | $7100\pm 220$ | $12600\pm 1530$ | $18150\pm 3700$
D5 | $4.61$ | $1250\pm 50$ | n.a. | $6500\pm 400$ | $20000\pm 4000$ | $36000\pm 17000$
D5 | $4.73$ | $800\pm 20$ | $4000\pm 200$ | $7600\pm 500$ | $14000\pm 2000$ | $23000\pm 5000$
D5 | $5.23$ | $770\pm 25$ | $2730\pm 200$ | $4545\pm 340$ | $6980\pm 1230$ | $8530\pm 1400$
D5 | $5.45$ | $740\pm 25$ | $3600\pm 200$ | $5500\pm 200$ | $10000\pm 900$ | $17000\pm 1000$
D5 | $5.83$ | $970\pm 20$ | $1645\pm 70$ | $2160\pm 70$ | $2680\pm 100$ | $2910\pm 170$
D5 | $5.97$ | $630\pm 10$ | n.a. | $2900\pm 150$ | $3400\pm 100$ | $4000\pm 200$
D60 | $4.47$ | $3220\pm 50$ | n.a. | $4460\pm 75$ | $5660\pm 50$ | $6600\pm 110$
D60 | $4.70$ | $2870\pm 50$ | n.a. | $3365\pm 65$ | $4130\pm 90$ | $4830\pm 110$
D60 | $4.98$ | $2745\pm 145$ | $2700\pm 75$ | $3200\pm 35$ | $3700\pm 30$ | $4100\pm 65$
D60 | $5.09$ | $2610\pm 160$ | $2910\pm 130$ | $3175\pm 55$ | $3570\pm 65$ | $4000\pm 130$
D60 | $5.24$ | $2100\pm 30$ | $2440\pm 60$ | $2870\pm 45$ | $3260\pm 50$ | $3470\pm 60$
D60 | $5.34$ | $683\pm 5$ | $2300\pm 80$ | $2555\pm 35$ | $2840\pm 50$ | $3070\pm 60$
D60 | $5.68$ | $627\pm 3$ | $2040\pm 70$ | $2240\pm 30$ | $2530\pm 20$ | $2670\pm 20$
D60 | $6.14$ | $730\pm 10$ | $1740\pm 60$ | $1930\pm 40$ | $2100\pm 40$ | $2200\pm 50$
Table ED1: Summary of the resonator data. Resonance frequency
$f_{\textrm{r}}$, external quality factor $Q_{\textrm{e}}$ and internal
quality factor $Q_{\textrm{i}}$ for different average photon number $\langle
n_{\textrm{ph}}\rangle$ extracted using an algebraic fit of the parameter
$S_{21}$ [48]. The values reported here are average values and the error is
given by the standard deviation. The error of $f_{\textrm{r}}$ is not reported
because it is smaller than $\qty{1}{MHz}$. n.a. stands for not available. We
note that higher frequency resonators show lower $Q_{\textrm{i}}$.  Figure
ED2: Inductance effect on the SQUID behaviour. a [b] $I_{\textrm{sq,+}}$
[$\eta$] as a function of $\Phi$. The green dots represent the experimental
values extracted from Fig. 3b. Red curves represent the theoretical prediction
for different values of $L$ and ignoring higher harmonic contribution. c [d]
same as a [b] but for higher $L$ values. The theoretical prediction with
$L\approx\qty{5}{nH}$ matches qualitatively the experimental data; however
this value is much higher than the experimentally extracted value, i.e.
$L\approx\qty{110}{pH}$.  Figure ED3: Effect of higher harmonics on the SQUID
behavior Left column [right column] $I_{\textrm{sq,+}}$ [$\eta$] as a function
of $\Phi$. The green dots represent the experimental values extracted from
Fig. 3b. The blue traces in a-b represent the theoretical calculation using
$L\approx\qty{110}{pH}$, $K_{1}=1-K_{2},K_{3}=K_{4}=0$ and $K_{2}$ varies from
$0$ to $0.3$. The purple traces in c-d are for $L\approx\qty{110}{pH}$,
$K_{1}=0.83-K_{3},K_{2}=0.17,K_{4}=0$ and $K_{3}$ varies from $0$ to $0.2$.
The black traces in e-f are for $L\approx\qty{110}{pH}$,
$K_{1}=0.755-K_{4},K_{2}=0.17,K_{3}=0.075$ and $K_{4}$ varies from $0$ to
$0.1$. Figure ED4: Expected results with asymmetric harmonic contributions. a
[b] shows the same plot as Fig. 3e [Fig. 3f]. c [d] Theoretical prediction of
a [b], where, like in the main text, it was assumed that $L=\qty{110}{pH}$,
$I_{\textrm{ret1}}=\qty{40}{nA}$ and the harmonic contributions of
$I_{\textrm{JJ2}}$ are $K_{1}=0.705,K_{2}=0.17,K_{3}=0.075,K_{4}=0.05$.
However, $I_{\textrm{JJ1}}$ was assumed to have less higher harmonic
contribution, namely $I_{\textrm{JJ1}}$ has
$K_{1}=0.8,K_{2}=0.15,K_{3}=0.03,K_{4}=0.02$. e-f First-order harmonics
contribution as a function of $I_{\textrm{ret2}}$ and $\Phi$. Both terms
vanish close to the balanced point and at $\frac{\Phi}{\Phi_{0}}=0.5$. g
Sinusoidal second harmonic contribution as a function of $I_{\textrm{ret2}}$
and $\Phi$. Importantly, $b_{2}$ never vanishes at
$\frac{\Phi}{\Phi_{0}}=0.5$. h Ratio between second and first harmonic as a
function of $I_{\textrm{ret2}}$ and $\Phi$. For this situation, the second
component would dominate the CPR not only at the sweet spot. Therefore a small
asymmetry between the harmonic contributions would not change the fact that
the second harmonic contribution never vanishes at
$\frac{\Phi}{\Phi_{0}}=0.5$. In c-h the same parameters have been used.
Figure ED5: Higher harmonic contribution as a function of $\Phi$ and
$I_{\textrm{ret2}}$ estimated using the same parameters as in Fig. 3. Upper
[lower] row shows the cosinus [sinus] terms. Interestingly, at the sweet spot
($I_{\textrm{ret1}}=I_{\textrm{ret2}}$ and $\frac{\Phi}{\Phi_{0}}=0.5$) all
terms, but $b_{2}$ and $b_{4}$, vanish.  Figure ED6: Expected results with
standard sinusoidal CPRs and low inductance ($\boldsymbol{L=\qty{110}{pH}}$) a
[b] shows the same plot as Fig. 3e [Fig. 3f]. c [d] Theoretical calculation of
a [b], assuming standard sinusoidal CPR and $\qty{110}{pH}$. The theoretical
prediction does not resemble the experimental data. In c, $\left|\eta\right|$
is much smaller than what has been experimentally observed (a) and $\eta$ is
finite only slightly away from sweet spot, located at
$I_{\textrm{ret2}}=\qty{40}{nA}$ and at $\Phi=0.5\Phi_{0}$. In addition, the
expected $I_{\textrm{sq,+}}$ is much lower than the measured values,
especially at the sweet spot. In the theoretical calculation,
$I_{\textrm{sq,+}}\approx 0$ at the sweet spot (d); whereas in the
experimental data $I_{\textrm{sq,+}}\approx\qty{20}{nA}$ (b). e-f First-order
harmonics contribution as a function of $I_{\textrm{ret2}}$ and $\Phi$
extracted using the same parameters as in c-d. Like for the case described in
the main text, at the balanced point and at $\frac{\Phi}{\Phi_{0}}=0.5$ both
first harmonic terms vanish. g Second sinusoidal harmonics contribution as a
function of $I_{\textrm{ret2}}$ and $\Phi$. Importantly, $b_{2}$ never
vanishes at $\frac{\Phi}{\Phi_{0}}=0.5$. Despite the fact that it is
qualitatively similar to the situation of the main text (Fig. 3k), there is a
major difference. At the sweet spot $\left|b_{2}\right|\approx 0.1\
$\mathrm{n}\mathrm{A}$$, while in Fig. 3k $\left|b_{2}\right|\approx 20\
$\mathrm{n}\mathrm{A}$$. h Ratio between second and first harmonic. For this
situation, the second component would dominate the CPR just at the sweet spot.
Figure ED7: Expected results with standard sinusoidal CPRs and with an
inductance ($\boldsymbol{L=\qty{5}{nH}}$) more than one order of magnitude
higher than what was measured. a [b] shows the same plot as Fig. 3e [Fig. 3f].
c [d] Theoretical calculation of a [b], assuming standard sinusoidal CPRs and
$L=\qty{5}{nH}$, which is much higher than what has been experimentally
estimated. The theoretical calculation resembles qualitatively the
experimental data. e-f First-order harmonics contribution as a function of
$I_{\textrm{ret2}}$ and $\Phi$ extracted using the same parameters as in c-d.
Similar to the case described in the main text, close to the balanced point
and at $\frac{\Phi}{\Phi_{0}}=0.5$ both first harmonic terms vanish. g Second
sinusoidal harmonics contribution as a function of $I_{\textrm{ret2}}$ and
$\Phi$. Importantly, $b_{2}$ never vanishes at half flux quantum. h Ratio
between second and first harmonic. For this situation, the second component
would dominate the CPR not only at the sweet spot, but also in the area
enclosed by the red line corresponding to $K=1$.  Figure ED8: Procedure to
identify and quantify the region of the sweet spot. a $\eta$ for
$\frac{\Phi}{\Phi_{0}}=0.55$ as a function of $V_{\textrm{g1}}$ and
$V_{\textrm{g2}}$. If
$\left|V_{\textrm{g2}}\right|>\left|V_{\textrm{g1}}\right|$
[$\left|V_{\textrm{g2}}\right|<\left|V_{\textrm{g1}}\right|$] $\eta<0$
[$\eta>0$]. Importantly, $\eta$ vanishes along the diagonal line, i.e. if
$V_{\textrm{g1}}\approx V_{\textrm{g2}}$ which corresponds to
$I_{\textrm{ret1}}\approx I_{\textrm{ret2}}$. The black dots indicate the
points at which $\left|\eta\right|<0.015$. Once the $I_{\textrm{ret1}}\approx
I_{\textrm{ret2}}$ region is identified, we move to
$\frac{\Phi}{\Phi_{0}}=0.5$ in order to be able to assess the sweet spot
regime. b shows $I_{\textrm{sq,+}}$ as a function of $V_{\textrm{g1}}$ and
$V_{\textrm{g2}}$. $I_{\textrm{sq,+}}$ is finite even when the junctions are
balanced, see black spots. c $I_{\textrm{sq,+}}$ as a function of
$V_{\textrm{g2}}$ in the sweet spot configuration. The identification of the
balanced region cannot be carried out at half flux quantum because $\eta$ is
always zero.  Figure ED9: Additional SDE results. a [b] $\eta$
[$I_{\textrm{sq,+}}$] as a function of $\Phi$ and $V_{\textrm{g2}}$ for a
$\qty{30}{nm}$-thick Al device. In a $\eta$ is always zero at
$\frac{\Phi}{\Phi_{0}}=0.5$, independently on $V_{\textrm{g2}}$, and the
polarity of the diode is inverted at the balanced point. Differently from Fig.
3e, the SDE does not vanish for all values of $\Phi$ at the balanced point.
This can be due to an asymmetric harmonic contribution (see Fig. ED4c). In a
[b], $\eta$ [$I_{\textrm{sq,+}}$] was extracted by recording the retrapping
current for both branches. c [d] is the same measurement as a [b] but now the
switching current was recorded. Importantly, both approaches give similar
results. e $\eta$ as a function of $\Phi$ and $V_{\textrm{g2}}$ for another
sample D8 with thin film of Al deposited at low temperature. f same as e but
with the addition of an applied RF power, namely with $P=\qty{-40}{dBm}$ and
$f_{\textrm{ac}}=\qty{1.5}{GHz}$. Importantly $\eta$ increases with an applied
drive.  Figure ED10: Observation of half-integer steps for another device and
identification of Shapiro steps. a [b] Shapiro pattern for a $\qty{30}{nm}$
thick room temperature deposited Al sample with
$R_{\textrm{shunt}}=\qty{10}{\Omega}$ in the balanced regime
($I_{\textrm{ret1}}\approx I_{\textrm{ret2}}\approx\qty{400}{nA}$) with
$f_{\textrm{ac}}=\qty{500}{MHz}$ and at $\frac{\Phi}{\Phi_{0}}=0.45$
[$\frac{\Phi}{\Phi_{0}}=0.5$]. The differential resistance
$dV_{\textrm{sq}}/dI_{\textrm{sq}}$ is plotted as a function of the RF power
$P$ and $I_{\textrm{sq}}$. Dips in $dV_{\textrm{sq}}/dI_{\textrm{sq}}$
correspond to integer Shapiro steps. Importantly at half flux quantum (b), the
first half-integer step appears for low $P$, see white numbers. c Shapiro map
as a function of $I_{\textrm{sq}}$ and $\Phi$ in the balanced regime (like for
the previous plots) for $P=\qty{4}{dBm}$. The half-integer steps appear only
close to $\frac{\Phi}{\Phi_{0}}=0.5$. d Shapiro map as a function of
$I_{\textrm{sq}}$ and $V_{\textrm{g2}}$ for $P=\qty{4}{dBm}$ and at
$\frac{\Phi}{\Phi_{0}}=0.5$. The half-integer step appears when the SQUID is
close to the balanced condition, i.e. if $I_{\textrm{ret2}}\approx 400\pm
25\unit{nA}$. The white numbers in c and in d emphasize the appearance of the
half-integer step. The second row corresponds to the same measurements as the
first row but they are plotted as a function of $V_{\textrm{sq}}$ instead of
$I_{\textrm{sq}}$. The third row corresponds to the device shown in the main
text in Fig. 4 but the data are plotted as a function of $V_{\textrm{sq}}$ as
well. i corresponds to Fig. 4b, j corresponds to Fig. 4d, k corresponds to
Fig. 4e and l corresponds to Fig. 4f. $V_{\textrm{sq}}$ is obtained by
integrating the signal of $dV_{\textrm{sq}}/dI_{\textrm{sq}}$.


# Steady-state squeezing transfer in hybrid optomechanics

Hugo Molinares<EMAIL_ADDRESS>Centro de Optica e Información
Cuántica, Universidad Mayor,
camino la Piramide 5750, Huechuraba, Santiago, Chile  Vitalie Eremeev
<EMAIL_ADDRESS>Facultad de Ingeniería y Ciencias, Universidad Diego
Portales, Av. Ejercito 441, Santiago, Chile  Miguel Orszag
<EMAIL_ADDRESS>Centro de Optica e Información Cuántica, Universidad
Mayor,
camino la Piramide 5750, Huechuraba, Santiago, Chile  Instituto de Física,
Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile

###### Abstract

A hybrid scheme is presented that allows the transfer of squeezed states (TSS)
from the mechanical part to an optical cavity in the steady-state. In a
standard optomechanical scheme, a three-level atom acts as an intermediate
element for TSS. Two different procedures are developed that allow the
visualization of the TSS effect: In the first one, we apply a coherent pump of
squeezed phonons in our hybrid system, and the second method is achieved by
placing the system in contact with a phonon squeezed bath. Our model and
procedures show that in optomechanical systems TSS can be achieved with a high
fidelity.

††preprint: APS/123-QED

With the remarkable state-of-the-art of the hybrid systems [1, 2, 3] composed
by mechanical oscillators (MO), cavities, spins, etc., it is becoming more and
more feasible to control such systems in their quantum regime in search for
non-classical features of their elements. Besides MO have the ability to
easily interact with a wide range of physical systems, such as ultracold
atomic BECs [4], superconducting qubits [3], spin states in quantum dots [5,
6] and color/NV centers [7, 8, 2], cavity fields in the opto-mechanical
systems [9, 10], etc.

The hybrid setups are of paramount importance regarding the development of the
quantum technologies. Therefore, to move forward the limits of the fields such
as quantum computing, quantum networks, quantum metrology and sensing, a very
demanded tool should be the high-fidelity transfer of the quantum states
between the components of a hybrid system. For instance, in the realm of
quantum networks, the mechanical object can serve as light-matter transducer
[11, 12] or to map/encode information from a qubit [13, 3]. Particularly,
spin-mechanical systems have attained major attention, mainly because spin
systems exhibit long-coherence times and they can be easily manipulated and
read-out [14, 15]. A rapidly emerging field is the quantum metrology and
sensing [16] by using the quantum states and protocols with the aim to
overcome the precisions unreachable by the classical sensing. There are
several experiments proving the sensing near and beyond the standard quantum
limit, e.g. [17, 18, 19]. Commonly the quantum sensors are based on the
photonics setups, involving cavities, atoms, photons, trapped ions, and solid-
state systems with electrons, superconducting junctions, etc. On the other
hand, the hybrid systems like opto/spin-mechanical, opto-electromechanical
setups, etc., actually become more attractive for their effectiveness and
usefulness for a wide range of quantum applications, from gravitational wave
detectors [20, 21] to force microscopes [22, 23], hence considered as the
leader candidates for the quantum metrology and sensing.

In this context, the squeezing of the modes in a hybrid system, and
particularly the squeezing transfer between them, is of a major importance and
applicability. On the one hand, the preparation of the mechanical and light
modes in the squeezed states, has been widely investigated theoretically [24,
25, 26, 27] and is nowadays experimentally feasible in versatile hybrid setups
[28, 29, 30, 31, 32, 33, 10]. On the other hand, there are less proposals for
squeezing transfer between different elements in a hybrid system, e.g. some
theoretical work over the last two decades [34, 35, 36, 37, 38]. To advance in
this direction, in the present work we propose an optomechanical protocol of
high-fidelity squeezing transfer from the mechanical mode to the cavity field.
Our results show that such a transfer could be realized by a coherent
mechanical squeezed pump, as well as under the coupling with a squeezed
phononic bath.

Figure 1: $(a)$ Schematic diagram of a hybrid cavity-atom-mechanics system.
$(b)$ Visualization of Wigner function to explain how squeezing is transferred
from the mechanical part to the cavity field. $(I)$ Initial states of the
cavity field and mechanical part. $(II)$ States in presence of the mechanical
squeezing. The mechanical part of the system is squeezed via $i)$ Applying a
coherent squeezed pump of the form $\mathcal{H}_{q}$, and $ii)$ by coupling
the system to an incoherent pump, that is, a squeezed mechanical reservoir.
$(III)$ Final steady states. The squeezed state is transferred from the
mechanical part to the cavity field through the tripartite atom-photon-phonon
interaction coupling strength $J$.

As illustrated in Fig. 1$(a)$, we consider a hybrid system where a confined
photon mode is coupled both to two upper levels of the three-level atom and to
a mechanical oscillator as ($\hbar=1$)

$\displaystyle\mathcal{H}_{0}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{2}\omega_{i}\sigma_{ii}+\omega_{c}a^{\dagger}a+\omega_{m}b^{\dagger}b$
(1) $\displaystyle+\imath
g_{ac}\left(a\sigma^{+}_{21}-a^{\dagger}\sigma^{-}_{21}\right)-\imath
g_{cm}a^{\dagger}a\left(b^{\dagger}-b\right),$

where $\sigma^{\pm}_{ij}$ are the ladder operators for the atom, $a(b)$ is the
annihilation operator of the cavity (mechanical) mode of frequency
$\omega_{c}(\omega_{m})$ and $\omega_{i}$ are the energy levels of the three-
level atom. $g_{ac}$ and $g_{cm}$ are the atom-cavity and optomechanical
coupling strengths, respectively.

When this hybrid system is driven by the coherent pump $\mathcal{H}_{q}=\imath
q(b^{\dagger 2}-b^{2})$ ($q$ is proportional to the driving field strength),
the mechanical resonator is prepared in a squeezed state [39]. The full
Hamiltonian in the interaction picture (rotating at the mechanical frequency
$\omega_{m}$) takes the form

$\displaystyle\mathcal{\tilde{H}}_{1}$ $\displaystyle=$ $\displaystyle\imath
g_{ac}\sigma_{21}^{+}ae^{\imath\Delta t}e^{-\imath F^{*}(t)}+\imath
qb^{\dagger 2}+2\imath Ka^{\dagger}ab^{\dagger}\eta$ (2) $\displaystyle+H.c.$

Here we used the Hermitian operator
$F\equiv\frac{g_{cm}}{\omega_{m}}\left(b^{\dagger}\eta^{*}+b\eta\right)$, with
$\eta\equiv e^{\imath\omega_{m}t}-1$;
$\Delta\equiv\omega_{2}-\omega_{1}-\omega_{c}$ is the detuning, and the
parameter $K\equiv q\cdot g_{cm}/w_{m}$ depends on the optomechanical coupling
and the amplitude of the coherent squeezing pump. Considering the experiments
in optomechanics [1], it is expected that $K$ takes very small values.

In the following, we assume that the optomechanical coupling $g_{cm}$ is much
smaller than the mechanical frequency $\omega_{m}$, so that $e^{-\imath
F^{*}(t)}\approx
1-\imath\frac{g_{cm}}{\omega_{m}}\left(b^{\dagger}\eta^{*}+b\eta\right)$.
Using the rotating wave approximation under the blue-detuned regime
$\Delta=\omega_{m}$, and keeping only the time independent terms, the above
Hamiltonian becomes

$\mathcal{\tilde{H}}_{2}=\imath J\sigma_{21}^{+}ab^{\dagger}+\imath
qb^{\dagger 2}-2\imath Ka^{\dagger}ab^{\dagger}+H.c.,$ (3)

where $J\equiv g_{ac}\cdot g_{cm}/w_{m}$ is the tripartite atom-photon-phonon
interaction strength.

In order to produce squeezed light in the optical cavity, two lasers $E_{1}$
and $E_{2}$ (proportional to the field strengths) are introduced in the
system, which are resonant with the transitions of the levels
$|2\rangle\longleftrightarrow|0\rangle$ and
$|1\rangle\longleftrightarrow|0\rangle$, respectively. These coherent drives
are described by the Hamiltonian in the interaction picture
$\mathcal{\tilde{H}}_{E}=\imath
E_{1}\left(\sigma_{20}^{-}-\sigma_{20}^{+}\right)+\imath
E_{2}\left(\sigma_{10}^{-}-\sigma_{10}^{+}\right)$. If we now include the
dissipation caused by the system-environment coupling, the dissipative
dynamics of the hybrid quantum system is described by the Markovian master
equation

$\displaystyle\frac{d\rho}{dt}$ $\displaystyle=$
$\displaystyle-\imath[\mathcal{\tilde{H}}_{2}+\mathcal{\tilde{H}}_{E},\rho]+\frac{\gamma_{21}}{2}\mathcal{L}[\sigma_{21}]\rho$
(4)
$\displaystyle+\frac{\gamma_{10}}{2}\mathcal{L}[\sigma_{10}]\rho+\frac{\kappa_{a}}{2}\mathcal{L}[a]\rho+\frac{\kappa_{b}}{2}\mathcal{L}[b]\rho,$

where $\rho$ is the density matrix of the hybrid system and the common
Lindblad dissipative terms are defined by: $\forall\mathcal{O},$
$\mathcal{L}[\mathcal{O}]=2\mathcal{O}\rho\mathcal{O}^{\dagger}-\mathcal{O}^{\dagger}\mathcal{O}\rho-\rho\mathcal{O}^{\dagger}\mathcal{O}$,
with all the baths at $n_{th}=0$. Here $\gamma_{21}$ $(\gamma_{10})$
correspond to spontaneous emission rates from level 2 to 1 (level 1 to 0),
respectively and $\kappa_{a}$ $(\kappa_{b})$ are the decay rates of the
optical (mechanical) mode. The approach of $n_{th}=0$ is realistic in recent
experiments, for example for a hybrid system with atoms, cavity and mechanics
in the regime of microwave frequencies, as in recent experiments [3, 12] the
mechanical mode has $\omega_{m}\approx 30$ GHz, and by cooling the system to
the temperatures of $\sim 30$ mK, one gets
$n_{th}\equiv[\exp(\hbar\omega_{m}/k_{B}T)-1]^{-1}\approx 10^{-4}$.

After building the theoretical model, we discuss the degree of squeezing
present in the states of the cavity and the mechanical part. For this, we rely
on numerical methods according to [40] to solve Eq. 4 in the steady-state,
i.e. $\dot{\rho}=0$, and so calculating the quadrature fluctuations. Here, we
study the parameter space in order to minimize fluctuations and achieve an
optimal result for TSS in the stable region.

For a better understanding the TSS effect and stability of the hybrid system,
we get a set of first order differential equations from Eq. 4:

$\displaystyle\frac{d\langle a^{\dagger}a\rangle}{dt}$
$\displaystyle=-2J\langle
a^{\dagger}b\rangle\langle\sigma^{-}_{21}\rangle-\kappa_{a}\langle
a^{\dagger}a\rangle,$ (5) $\displaystyle\frac{d\langle
b^{\dagger}b\rangle}{dt}$ $\displaystyle=2J\langle
a^{\dagger}b\rangle\langle\sigma^{-}_{21}\rangle+4q\langle
b^{2}\rangle-\kappa_{b}\langle b^{\dagger}b\rangle,$ (6)
$\displaystyle\frac{d\langle a^{2}\rangle}{dt}$ $\displaystyle=-2J\langle
ab\rangle\langle\sigma^{-}_{21}\rangle-\kappa_{a}\langle a^{2}\rangle,$ (7)
$\displaystyle\frac{d\langle b^{2}\rangle}{dt}$ $\displaystyle=2J\langle
ab\rangle\langle\sigma_{21}^{+}\rangle+4q\langle
b^{\dagger}b\rangle-\kappa_{b}\langle b^{2}\rangle+2q,$ (8)
$\displaystyle\frac{d\langle a^{\dagger}b\rangle}{dt}$
$\displaystyle=-J\left(\langle b^{\dagger}b\rangle-\langle
a^{\dagger}a\rangle\right)\langle\sigma^{+}_{21}\rangle+2q\langle ab\rangle$
$\displaystyle-\frac{\kappa_{a}}{2}\langle
a^{\dagger}b\rangle-\frac{\kappa_{b}}{2}\langle a^{\dagger}b\rangle,$ (9)
$\displaystyle\frac{d\langle ab\rangle}{dt}$ $\displaystyle=-J\left(\langle
b^{2}\rangle\langle\sigma^{-}_{21}\rangle-\langle
a^{2}\rangle\langle\sigma^{+}_{21}\rangle\right)+2q\langle
a^{\dagger}b\rangle$ $\displaystyle-\frac{\kappa_{a}}{2}\langle
ab\rangle-\frac{\kappa_{b}}{2}\langle ab\rangle,$ (10)

where we have considered the factorization of the form $\langle
a^{\dagger}b\sigma^{-}_{21}\rangle=\langle
a^{\dagger}b\rangle\langle\sigma^{-}_{21}\rangle$ (adiabatic approximation)
and since $\omega_{m}\gg\left\\{g_{cm},q\right\\}$ we have considered $K=0$.

Figure 2: $(a)$ Quadrature fluctuations as functions of the parameter $q$, for
the cavity $\left(\Delta y_{a}\right)^{2}$ (red and blue curves) and for the
mechanics $\left(\Delta y_{b}\right)^{2}$ (green and magenta curves). $(b)$
Time evolution of the quadrature fluctuations. All the parameters are scaled
by $\omega_{m}$: $g_{ac}=10^{2}$, $\kappa_{a}=0.2$,
$\kappa_{b}=\kappa_{a}\cdot 10^{-2}$, $\gamma_{10}=20$, $\gamma_{21}=0$,
$E_{1}=E_{2}=25$.

By considering
$\left\\{E_{1},E_{2},\gamma_{10}\right\\}\gg\left\\{J,q,\kappa_{a},\kappa_{b}\right\\}$
and $\gamma_{21}=0$, we get the following semiclassical results for the
quadrature fluctuations in steady-state:

$\displaystyle\left(\Delta y_{a}\right)^{2}$
$\displaystyle=\frac{m+n+s+1}{4(m+1)(n+1)},$ (11) $\displaystyle\left(\Delta
y_{b}\right)^{2}$ $\displaystyle=\frac{n+s}{4(m+1)(n+1)},$ (12)

where

$\displaystyle m$ $\displaystyle=\frac{4q}{\kappa_{a}+\kappa_{b}},$ (13)
$\displaystyle n$
$\displaystyle=\frac{\kappa_{a}\kappa_{b}+4J^{2}\langle\sigma_{21}^{+}\rangle^{2}}{4q\kappa_{a}},$
(14) $\displaystyle s$
$\displaystyle=\frac{\kappa_{b}}{\kappa_{a}+\kappa_{b}},$ (15)
$\displaystyle\langle\sigma_{21}^{+}\rangle$
$\displaystyle=\frac{2E_{1}E_{2}}{\gamma_{10}^{2}+4\left(E_{1}^{2}+E_{2}^{2}\right)}.$
(16)

The stability of the system is obtained by solving the system of equations of
the form $\frac{d\vec{x}}{dt}=M\vec{x}+c$, where we have defined a vector
$\vec{x}$ with the moments shown in the left-side of Eqs. 5-10, $M$ is a
matrix built with the elements on the right-side of Eqs. 5-10 and $c$ is a
constant vector arising from the constant term $2q$ in the equation for
$\langle b^{2}\rangle$. The steady-state is only achieved if the real part of
the eigenvalues of the matrix are all negative. Under the present
approximations, we get a stationary state only if $4q<\kappa_{a}+\kappa_{b}$.

Figure 3: $(a)$ Fidelity between the cavity field and mechanical state as a
function of the optomechanical coupling, $g_{cm}$, and squeezing pump, $q$. As
seen in $(b)$, an election of higher optomechanical coupling,
$g_{cm}=10^{-2}$, gives the optimal TSS.

In panel $(a)$ of the Fig. 2 we show the fluctuations as a function of the
parameter $q$ for different couplings $g_{cm}$. One sees that for
$g_{cm}=0.01$, the TSS shows an optimal performance, since the mechanical and
optical fluctuations are close as well as the semiclassical solutions. In
fact, the numerical solution could reach the analytical solution for a large
enough Hilbert space, however, such dynamics for larger $q$ values increases
dramatically the computation resources. To support the above, in panel $(b)$
of the Fig. 2, we show the time evolution of the quadrature fluctuations.
Without loss of generality, we have fixed the parameters $q=0.01$ and
$g_{cm}=0.01$. We can see how the injection of mechanical squeezing in the
system modifies the original cavity and mechanical wave functions, bringing
these to squeezed steady-states. Furthermore, one observes that both
fluctuations are quite close to each other.

According to this last result, we investigate numerically the steady-state TSS
by computing the fidelity [41], defined as
$\mathcal{F}(\rho_{c}^{ss},\rho_{m}^{ss})\equiv
Tr\sqrt{\sqrt{\rho_{c}^{ss}}\rho_{m}^{ss}\sqrt{\rho_{c}^{ss}}}$, where
$\rho_{c}^{ss}$ and $\rho_{m}^{ss}$ are the steady-state density operators of
the cavity and mechanical part. In panel $(a)$ of Fig. 3 we show the fidelity
as a function of $g_{cm}$ and $q$. Notice that the fidelity is close to unity
as the optomechanical coupling increases, even in the presence of dissipation
in the system. In panel $(b)$ of Fig. 3 we show the fidelity as a function of
$q$ for some values of $g_{cm}$. This result allows us to conclude that in the
strong coupling regime the open quantum system allows the TSS with a fidelity
close to unity.

We propose another method for performing TSS under incoherent driving of
phonon squeezing, that is the hybrid system with $q=0$ and in contact with a
squeezed reservoir. To treat such an effect, considering $q=0$ in the
Hamiltonian (3), one has the following interaction Hamiltonian:

$\mathcal{\tilde{H}}_{3}=\imath
J\left(\sigma_{21}^{+}ab^{\dagger}-\sigma_{21}^{-}a^{\dagger}b\right).$ (17)

Figure 4: $(a)$ Quadrature fluctuations as functions of the parameter $r$.
Here $\left(\Delta x_{a}\right)^{2}$ (red and blue curves) and $\left(\Delta
x_{b}\right)^{2}$ (green and magenta curves). $(b)$ Time evolution of the
fluctuations evidencing the steady-state. The parameters are: $g_{ac}=10^{2}$,
$\kappa_{a}=\kappa_{b}=0.2$, $\gamma_{10}=20$, $\gamma_{21}=0$,
$E_{1}=E_{2}=25$, $\theta=\pi$.

Now, we use the master equation with atomic and cavity losses (for
$n_{th}^{(at,cav)}=0$), as well as a phonon squeezed vacuum reservoir:

$\displaystyle\frac{d\rho}{dt}$ $\displaystyle=$
$\displaystyle-\imath[\mathcal{\tilde{H}}_{3}+\mathcal{\tilde{H}}_{E},\rho]+\frac{\gamma_{21}}{2}\mathcal{L}[\sigma_{21}]\rho$
(18)
$\displaystyle+\frac{\gamma_{10}}{2}\mathcal{L}[\sigma_{10}]\rho+\frac{\kappa_{a}}{2}\mathcal{L}[a]\rho+\frac{\kappa_{b}}{2}\mathcal{L}_{sq}[b]\rho,$

with the Lindbladian corresponding to the incoherent pump of squeezed phonons
as $\mathcal{L}_{sq}[b]=(N_{sq}+1)\left(2b\rho
b^{\dagger}-b^{\dagger}b\rho-\rho
b^{\dagger}b\right)+N_{sq}\left(2b^{\dagger}\rho b-bb^{\dagger}\rho-\rho
bb^{\dagger}\right)+M_{sq}\left(2b^{\dagger}\rho
b^{\dagger}-b^{\dagger}b^{\dagger}\rho-\rho
b^{\dagger}b^{\dagger}\right)+M_{sq}^{*}\left(2b\rho b-bb\rho-\rho bb\right)$;
here $N_{sq}=\sinh^{2}{r}$ and $M_{sq}=-\exp{(\imath\theta)}\sinh{r}\cosh{r}$,
obey the relation $\sqrt{N_{sq}(N_{sq}+1)}=|M_{sq}|$.

Using the above scheme, we explore the effects of the squeezing parameters
($r$, $\theta$) on the quantum fluctuations and fidelity, in order to achieve
the best TSS. In the semiclassical approximation we get following results for
the quadrature fluctuations in steady-state:

$\displaystyle\left(\Delta x_{a}\right)^{2}$ $\displaystyle=\frac{1}{4}p+l,$
(19) $\displaystyle\left(\Delta x_{b}\right)^{2}$
$\displaystyle=\frac{1}{4}p\left[2\left(N_{sq}-M_{sq}\right)+1\right]+l,$ (20)

where

$\displaystyle p$
$\displaystyle=\frac{\kappa_{a}\kappa_{b}}{4J^{2}\langle\sigma_{21}^{\dagger}\rangle^{2}+\kappa_{a}\kappa_{b}},$
(21) $\displaystyle l$
$\displaystyle=\frac{J^{2}\langle\sigma_{21}^{\dagger}\rangle^{2}\left[\kappa_{b}\left(1+2\left[N_{sq}-M_{sq}\right]\right)+\kappa_{a}\right]}{\left[\kappa_{a}+\kappa_{b}\right]\left[4J^{2}\langle\sigma_{21}^{\dagger}\rangle^{2}+\kappa_{a}\kappa_{b}\right]}.$
(22)

Now, we study the parameters in our hybrid system for which both quadratures
reach their maximum squeezing and that allow the optimal TSS via
optomechanical coupling. In panel $(a)$ of the Fig. 4 the quadrature
fluctuations are presented as functions of the parameter $r$. The results show
that for $g_{cm}=0.01$, the TSS performed in an optimal way, since the curves
are closer, that is, both fluctuations are similar. In panel $(b)$ of the Fig
4, we show the time evolution of the quadrature fluctuations. Without loss of
generality, we have fixed the parameters $r=0.3$ and $g_{cm}=0.01$. Analogous
to the aforementioned process of the coherent pump, we can see how the
injection of a squeezed bath leads to cavity and mechanical squeezing in
steady state.

In [36] the authors propose a setup that allows the transmission from an
atomic squeezed state to a membrane with a high fidelity in presence of
relevant decoherence rates. Motivated for this, we simulate the influence of
the Jaynes-Cummings coupling on the fidelity of the TSS in our system. In
panel $(a)$ of Fig. 5 we show the fidelity as a functions of $g_{cm}$ and
$g_{ac}$. Notice that the fidelity is close to unity as the optomechanical and
Jaynes-Cummings couplings increase, even in the presence of dissipation in the
system. In panel $(b)$ of Fig. 5 we show the fidelity as a function of
$g_{ac}$ for some values of $g_{cm}$. This result allows us to conclude that
in the strong coupling regime the open system allows the TSS with a
reliability close to 100%.

In conclusion, we have proposed a hybrid system consisting of a three-level
atom, an optical cavity, and mechanical resonator that allows the steady-state
squeezing conversion. We demonstrated that by pumping coherently and
incoherently phonon squeezing in our hybrid system and also adding two
coherent atomic drives, we can transfer steady-state squeezing to the cavity
photons with an extremely high fidelity, close the unity (see Figs. 3 and 5).

Figure 5: $(a)$ Fidelity between the cavity field and mechanical state as a
function of the couplings $g_{cm}$ and $g_{ac}$. $(b)$ Fidelity as a function
of $g_{ac}$ for some couplings $g_{cm}$. As is observed, the fidelity is close
to unity as the optomechanical and Jaynes-Cummings couplings increase. The
parameters here are the same as in Fig. 4 with $r=0.3$.

Furthermore, in the parameter range used in this work, as we increase the
optomechanical coupling, we get better results in producing closer phonon and
photon squeezed states, i.e. improving the protocol of TSS. The parameters
used in this work are compatible with recent experiments for the
optomechanical hybrid setups, e.g. [1, 3, 12].

###### Acknowledgements.

H.M. acknowledge financial support from Universidad Mayor through the Doctoral
fellowship. V.E. and M.O. acknowledge the financial support from Fondecyt
Regular No. 1180175.

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