# Revealing charge-tunneling processes between a quantum dot and a
superconducting island through gate sensing

Jasper van Veen  QuTech and Kavli Institute of Nanoscience, Delft University
of Technology, 2600 GA Delft, The Netherlands  Damaz de Jong  QuTech and Kavli
Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The
Netherlands  Lin Han  QuTech and Kavli Institute of Nanoscience, Delft
University of Technology, 2600 GA Delft, The Netherlands  Christian Prosko
QuTech and Kavli Institute of Nanoscience, Delft University of Technology,
2600 GA Delft, The Netherlands  Peter Krogstrup  Center for Quantum Devices,
Niels Bohr Institute, University of Copenhagen & Microsoft Quantum Materials
Lab Copenhagen, Denmark  John D. Watson  Microsoft Quantum Lab Delft, Delft
University of Technology, 2600 GA Delft, The Netherlands  Leo P. Kouwenhoven
QuTech and Kavli Institute of Nanoscience, Delft University of Technology,
2600 GA Delft, The Netherlands  Microsoft Quantum Lab Delft, Delft University
of Technology, 2600 GA Delft, The Netherlands  Wolfgang Pfaff
<EMAIL_ADDRESS>Microsoft Quantum Lab Delft, Delft University of
Technology, 2600 GA Delft, The Netherlands

###### Abstract

We report direct detection of charge-tunneling between a quantum dot and a
superconducting island through radio-frequency gate sensing. We are able to
resolve spin-dependent quasiparticle tunneling as well as two-particle
tunneling involving Cooper pairs. The quantum dot can act as an RF-only sensor
to characterize the superconductor addition spectrum, enabling us to access
subgap states without transport. Our results provide guidance for future
dispersive parity measurements of Majorana modes, which can be realized by
detecting the parity-dependent tunneling between dots and islands.

Quantum dots coupled to superconductors can give rise to novel physical
phenomena such as $\pi$ and $\phi_{0}$-junctions De Franceschi  _et al._
(2010); Van Dam  _et al._ (2006); Szombati  _et al._ (2016), Cooper pair
splitting Hofstetter  _et al._ (2009); Herrmann  _et al._ (2010); Deacon  _et
al._ (2015), and Yu-Shiba-Rusinov (YSR) states Ralph  _et al._ (1995); Lee
_et al._ (2014). These phenomena arise because the single-electron states of
the dot hybridize with the more complicated many-particle states of the
superconductor. Recently, such hybrid systems have gained interest in the
context of Majorana zero modes (MZMs) where the quantum dot (QD) can, for
example, be used as a spectrometer Deng  _et al._ (2016). Moreover, projective
parity measurements can be achieved by coupling a QD to a pair of MZMs, which
are located on a superconducting island (SC) Higginbotham  _et al._ (2015);
Albrecht  _et al._ (2016), enabling topologically protected quantum
computation. These projective measurements rely on the parity-dependent
hybridization between a single dot level and the MZMs Plugge  _et al._ (2017);
Karzig  _et al._ (2017). Therefore, unambiguous detection of coherent
tunneling between a QD and a superconducting island is needed to implement
this readout.

Dispersive gate sensing (DGS) provides direct access to charge hybridization
between weakly coupled dots or islands. More precisely, tunneling within these
structures can impart a frequency shift on a resonant circuit that can be
observed on short time scales with high accuracy. In this way, experiments
have revealed coherent charge hybridization between superconductors Wallraff
_et al._ (2004); Duty  _et al._ (2005); Esmail  _et al._ (2017) and
semiconductor double quantum dots Petersson  _et al._ (2012); Frey  _et al._
(2012); Colless  _et al._ (2013); Landig  _et al._ (2019). Moreover,
capacitive RF sensing has been used to study charging of QDs connected to
normal and superconducting reservoirs Gonzalez-Zalba  _et al._ (2015); Bruhat
_et al._ (2016, 2018). As such, DGS presents an excellent opportunity for
studying charge-tunneling in hybrid structures containing QDs.

In this paper, we report direct detection of different types of charge-
tunneling processes between a QD and a SC through DGS on the QD. From
observations of the resonator response, supported by numerical simulations, we
find that the nature of the tunneling depends crucially on the ordering of the
relevant energy scales of the SC. When the smallest scale is the energy of the
lowest single-particle state, the QD and SC can exchange quasiparticles,
giving rise to the characteristic ‘even-odd’ effect. Conversely, when the
charging energy of the SC is lowest, we detect signatures of Cooper pairs
tunneling out of the SC. Depending on the tunneling amplitude, this results in
either 1$e$-charging of the QD, with the other electron leaving into a
reservoir, or 2$e$-charging of the QD. Tunneling to the single-particle states
can, however, be re-enabled by operating the device in a floating regime where
the total number of charges in the two systems is conserved. These results
show that DGS allows us to effectively perform RF-only tunneling spectroscopy
on the SC. To this end, we use the QD and capacitively-coupled resonator as a
probe to characterize a subgap state in the SC without need for transport via
leads. Our method is complementary to recent experiments that employed the
dispersive response of inductively-coupled resonators to probe the Andreev
bound state occupation in galvanically-isolated nanowire Josephson junctions
Hays  _et al._ (2018); Tosi  _et al._ (2019).

Figure 1:  Experimental setup and sample characterization. a) A gate sensor
detects the tunneling of charges on/off the QD. Charge hybridization between
QD and SC results in an anti-crossing in the energy spectrum at zero detuning
$\epsilon=0$, and shifts the $LC$ resonator, causing in a change in amplitude,
$A$, and phase, $\phi$, response of a probe signal at the bare resonator
frequency, $f_{0}$. b) False-colored electron micrograph of a nominally
equivalent hybrid double dot. An additional dot, shown on the right of the
device, is unused in this experiment. c) Coulomb blockade measurement of the
SC. Left: device A, measured using RF reflectometry off the source (circuit
not shown). The even-odd pattern indicates
$E_{0}<E_{\mathrm{C}}^{\mathrm{S}}$. Right: device B, measured using standard
lock-in techniques. The doubling of the period at low bias $V_{\text{b}}$
illustrates that $E_{0}>E_{\mathrm{C}}^{\mathrm{S}}$. d) Energy dispersion of
the superconducting island for device A (left) and device B (right). The even
(odd) energy levels are shown in blue (green). The odd parity sector consists
of a discrete subgap state at $E_{0}$ and a continuum of states above $\Delta$
(shaded).

A schematic of our experiment is shown in Fig. 1a,b. The QD-SC hybrid double
dot is formed in an InAs nanowire with an epitaxially grown Al-shell on two of
its facets. The superconducting island consists of a proximitized wire
segment, which is defined by removing the Al outside a
$1.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ window using wet etching. The low-
carrier density in the wire allows for gate-tunable subgap states in the SC
Antipov  _et al._ (2018). Tunneling barriers are implemented with gates,
insulated from the wire by 10 nm AlOx. They are used to define the QD and to
control the various tunneling rates. Large-lever arm top gates (‘plungers’) on
both QD and SC can be used to tune the chemical potentials. We have measured
two nominally-identical devices, labelled A and B. For both devices, we
connected the QD plunger to an off-chip, superconducting resonator with a
resonance frequency of 449.5 (443.2) MHz for device A (B) (Supplemental
Material) Hornibrook  _et al._ (2014). We use its response near the resonance
frequency to probe charge tunneling on and off the dot. All measurements are
done at temperatures of $T\approx$ 20 mK and at zero magnetic field unless
otherwise indicated.

The relevant energy scales in our devices can be obtained from Coulomb
blockade measurements: Figure 1c shows Coulomb diamonds of the superconducting
island alone, measured through conductance. The diamonds of device A display a
clear even-odd pattern, indicating that the energy of the lowest odd-parity
state, $E_{0}$, is smaller than the charging energy of the superconducting
island, $E_{\mathrm{C}}^{\mathrm{S}}=e^{2}/2C_{\Sigma}^{\text{S}}$, where
$C_{\Sigma}^{\text{S}}$ is the total capacitance of the SC (Fig. 1d) Averin
and Nazarov (1992); Van Heck  _et al._ (2016). For this device, we estimate
$E_{0}=$72\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ and
$E_{\mathrm{C}}^{\mathrm{S}}=$112\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$
from the extent of the diamonds. Conversely, the charging of the
superconducting island of device B is 2$e$-periodic, indicating that
$E_{0}>E_{\mathrm{C}}^{\mathrm{S}}$ Hekking  _et al._ (1993); Van Heck  _et
al._ (2016); here, we estimate
$E_{0}\approx$90\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ and
$E_{\mathrm{C}}^{\mathrm{S}}\approx$70\text{\,}\mathrm{\SIUnitSymbolMicro
eV}$$. While in an ideal BCS superconductor $E_{0}$ is equal to the
superconducting gap $\Delta$, current measurements on device A (Supplemental
Material) and the negative differential conductance observed in device B
indicate the presence of subgap states Higginbotham  _et al._ (2015). We
attribute the difference in charging energies and $E_{0}$ between the devices
to a combination of a slightly smaller wire diameter for device A and typical
sample-to-sample variations arising from fabrication. In both devices, the
charging energy of the dot, $E_{\mathrm{C}}^{\mathrm{QD}}\approx
200-$300\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$, is the largest energy scale
in the system, and the typical QD level spacing exceeds the thermal energy
(Supplemental Material).

Figure 2:  Spin-dependent tunneling between a QD and a SC. a) and b) Charge
stability diagram of device A measured in phase a) and amplitude b). The
charge states are labeled $\left(n^{\text{SC}},n^{\text{QD}}\right)$ with
respect to the state $(N,M)$ with $N$ and $M$ even. Dashed pink lines:
expected locations of the lead-island transitions, in accordance with data
obtained from a resonator connected to the SC gate, and simulations of the
charge ground state of the system (Supplemental Material). c) Linecuts of the
phase (green dots) and amplitude (blue dots) along two interdot transitions.
Dashed line and left-pointing triangle marker, left panel: transition between
(0,2) and (1,1), representative of transitions between states of total even
parity. Continuous line and right-pointing triangle marker, right panel:
transition between (1,2) and (2,1), representative of transitions between
states of total odd parity. The lines are fits to a circuit QED model with
$t^{\text{odd}}_{\text{C}}=t^{\text{even}}_{\text{C}}/\sqrt{2}$ (Supplemental
Material). We find a good fit with
$t^{\text{even}}_{\text{C}}=$20\text{\,}\mathrm{GHz}$$ and
$g_{0}=$100\text{\,}\mathrm{MHz}$$. d) Full-frequency response of the
resonator (symbols) together with fits (lines) obtained from a pair of
interdot transitions that show the parity effect (outside the gate space shown
in a,b).  Figure 3:  Cooper pair tunneling in a hybrid double dot. a) Charge
stability diagram measured in phase (left) and amplitude (middle) along with a
simulation of the charge ground state (right) in the weakly coupled regime.
The charge states are labeled $\left(n^{\text{SC}},n^{\text{QD}}\right)$ with
respect to the state $(N,M)$ with $N$ even. Dashed pink lines: locations of
the transitions from the (0,0) state as a guide to the eye. The gray scale in
the simulation indicates the sum of the charge in the combined system. b)
Linecuts of the phase (green) and amplitude (blue) along the (-2,0) to (0,-1)
interdot transition. This transition involves a reservoir with a continuous
spectrum, indicated by the shaded region above the lowest available energy
state. The schematic shows how these states couple via crossed Andreev
reflection. c) Same as in a) for the strongly coupled regime. Dashed pink
lines: locations of the lead transitions from the (-2,-1) and (0,-1) states as
a guide to the eye. d) Linecuts of the phase (green) and amplitude (blue)
along the (2,-2) to (0,0) interdot transition. These states couple via Cooper
pair tunneling.

In the following, we investigate the change in resonator response when charges
are able to tunnel between the QD and SC at zero bias, beginning with device
A. To this end, we form a hybrid double dot by tuning the gates T1 and T2
close to pinch-off, and T3 into pinch-off. Figures 2a,b show the resonator
response as a function of the two plunger gates in the weakly coupled regime.
Both amplitude and phase response display the charge stability diagram (CSD)
of the hybrid double dot, which shows a clear 1$e$ pattern along the QD gate,
and an even-odd pattern along the SC gate; this is again a manifestation of
$E_{0}<E_{\mathrm{C}}^{\mathrm{S}}$, and the CSD shape can be readily
reproduced by computing the charge ground states of the system (Supplemental
Material).

We focus on the interdot transitions, highlighted in Figs. 2a-c, where we
observe a strong amplitude and phase response on all charge degeneracy points.
Interestingly, we see a strong difference in the resonator response across
interdot transitions with a different parity of the total particle number,
indicating a difference between the coupling between the involved states
Esterli  _et al._ (2019).

This dependence on total parity can be understood as a spin-dependent tunnel
coupling Cottet  _et al._ (2011). To see this, we label the states according
to their pairing; for the SC states as even/odd, and for the QD states as
singlet/doublet: $\ket{e/o,S/D}$. We can differentiate couplings between two
sets of states; $\ket{e,D}$ to $\ket{o,S}$ and $\ket{e,S}$ to $\ket{o,D}$. The
coupling is different for these two sets because of spin degeneracy: From
$\ket{e,S}$ either of the two electrons forming the singlet can tunnel with
equal probability, giving rise to a two-fold degeneracy in the coupling
between $\ket{e,S}$ and $\ket{o,D}$, i.e., between the states with total even
parity. On the other hand, from $\ket{e,D}$ only one of the two electrons can
tunnel from QD to SC due to Pauli exclusion; in this case no degeneracy is
present. As a result, the transition rate between states of total even parity
is expected to be $\sqrt{2}$ times stronger than between states of total odd
parity, corresponding to a $\sqrt{2}$ times smaller frequency shift.

We model the resonator response across the interdot transition by using a
circuit QED model derived from input-output theory (Supplemental Material)
Petersson  _et al._ (2012). We find a good agreement between our data and the
model using $t^{\text{odd}}_{\text{C}}=t^{\text{even}}_{\text{C}}/\sqrt{2}$.
However, the weak parameter dependence of the model makes it unreliable to
confirm the spin-dependence of the tunnel coupling. To obtain quantitative
agreement, we measure the frequency-dependent resonator response at a pair of
interdot transitions that show this parity effect (Fig. 2d). From fits to the
resonator responses we extract frequency shifts with a ratio of
$(f_{0}-f^{\text{odd}})/(f_{0}-f^{\text{even}})={\color[rgb]{0,0,0}1.45}\approx\sqrt{2}$,
consistent with the expected spin-dependent tunneling amplitude.

For device B, the situation changes significantly. The energy ordering
$E_{0}>E_{\mathrm{C}}^{\mathrm{S}}$ implies that quasiparticle states are not
accessible (Fig. 1d). We form a hybrid double dot by tuning T1, T2, and T3
close to pinchoff. The CSD for a weak QD-SC coupling is shown in Fig. 3a. The
diagram is $2e$-periodic in the SC gate, indicating that the island is charged
via Andreev reflections from the lead. The QD is again 1$e$-periodic. To model
the measured CSDs, we compute the charge ground state by diagonalizing an
effective Hamiltonian of the system that includes charging effects, the
superconducting gap in the island, and coupling terms (Supplemental Material).
This model, with the energy scales extracted from the Coulomb blockade
measurements and an adjustable tunneling amplitude (rightmost panel in Fig.
3a), describes the observed CSD well.

The different gate charge periodicity for the QD and SC leads to interdot
transitions that change the total charge of the dot-island system. This
implies that a reservoir must be involved in the corresponding charge-transfer
process. The observed resonator signal, with a linecut shown in Fig. 3b,
results from tunneling on and off the QD, and thus should not contain
information of SC-lead coupling Esterli  _et al._ (2019). A possible candidate
for the precise underlying process that gives rise to our data is crossed
Andreev reflection (CAR) Hofstetter  _et al._ (2009); Herrmann  _et al._
(2010). There, a hole from the QD is converted to an electron in the lead,
consistent with the charge states involved in the experiment. This process is
exponentially suppressed in the length of the island
$\exp\left(-L/\pi\xi\right)$, where $\xi$ is the superconducting coherence
length Leijnse and Flensberg (2013). Still, with
$L=$1.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and assuming a coherence
length of $\xi\sim 260$ nm Albrecht  _et al._ (2016) this remains a plausible
scenario.

Interestingly, increasing the tunnel coupling allows for bringing the system
into a regime where a particle-conserving interdot transition emerges. The CSD
in a more strongly coupled regime, together with a simulation of the charge
ground states is shown in Fig. 3c. In this regime, we assume an induced gap in
the quantum dot, consistent with earlier studies on YSR states Lee  _et al._
(2014). Here, we observe that the regions with odd charge number in the QD
shrink, while the regions with an even number of QD charges connect, resulting
in an even-odd pattern in both gates. Now, the interdot transition appears
purely dispersive (Fig. 3d): we observe only a small phase shift, without any
amplitude response. Overall, our data is consistent with coherent Cooper pair
transfer between the dot and the island.

Figure 4: Spectroscopy of a subgap state in the floating regime a) Charge
stability diagram measured in phase in device B. The diagonal lines indicate
that the total charge in the system is conserved. b) Calculated positions of
the transitions in good agreement with the measured stability diagram. Inset:
energy spectrum with even states in black and odd states in green, showing
that the even-odd pattern is caused by the parity effect even though
$E_{0}>E_{\mathrm{C}}^{\mathrm{S}}$. c) Temperature dependence of the even-odd
pattern. d) The evolution of the free energy difference with temperature
(black dots), with a fit to the model described in Ref. 10 (green line). The
free energy difference is extracted from the even-odd pattern via
$F_{\text{o}}-F_{\text{e}}=\left(S_{\text{e}}-S_{\text{o}}\right)e\alpha_{\text{sc}}/4$
with $\alpha_{\text{sc}}=0.9$ the lever arm of the SC gate.

As we have seen, the main difference between the two devices is that the odd
states of the SC can not be directly accessed in the regime
$E_{0}>E_{\mathrm{C}}^{\mathrm{S}}$. This changes in absence of lead
reservoirs because quasiparticles that tunnel from the QD onto the SC are
confined to the system Esmail  _et al._ (2017). The additional energy
associated with decharging the QD makes Cooper pair tunneling energetically
unfavorable when
$E_{0}<E_{\mathrm{C}}^{\mathrm{S}}+E_{\mathrm{C}}^{\mathrm{QD}}$. We realize
this situation experimentally in device B by closing the outer tunnel
barriers, through gates T1 and T3. The resulting CSD and corresponding
calculation of the ground state transitions are shown in Figs. 4a,b. It can
readily be seen that no transitions to a reservoir take place, and the even-
odd pattern is indicative of the alternating occupation of even and odd states
of the SC.

Importantly, even though SC and QD are now galvanically isolated from the
environment, the gate sensor still allows us to study the quasiparticle states
in the SC. To show this, we measure the evolution of the even-odd spacing as a
function of temperature (Fig. 4c). This spacing is a measure for the free
energy difference of the SC. In particular, the temperature evolution of the
free energy difference can be used to identify and characterize subgap states
Lafarge  _et al._ (1993); for proximitized nanowires, this has earlier been
studied in transport Higginbotham  _et al._ (2015). The extracted free energy
difference $F_{\text{o}}-F_{\text{e}}$ as a function of temperature is shown
in Fig. 4d. A fit to the model from Ref. 10 yields a gap of
$\Delta=$220\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$, a supgap state energy
of $E_{0}=$106\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$, and an Al volume of
$V=2.9\times 10^{5}$ nm3, consistent with the dimensions of the island. We
note that the slightly larger energy of the subgap state is consistent with
the more negative plunger gate voltage for this measurement. The excellent
quality of the fit corroborates our initial assessment of the presence of a
supgap state (Fig. 1b). This result shows clearly that the resonator response
of the QD gate sensor can be used to characterize states of the SC, even when
leads for transport experiments are not available.

In summary, we have performed dispersive gate sensing on a quantum dot that
can exchange particles with a superconducting island. Analysis of the
resonator response has allowed us to directly detect the charge-tunneling
processes that take place between the dot and the superconductor. We have
observed single or multi-particle tunneling processes, depending on the
dominating energy scales of the hybrid double dot. Our results show that DGS
provides an excellent tool for studying subgap excitations. In particular,
using a QD and gate sensor allows performing spectroscopy without transport,
which is relevant in cases where particle number should to be conserved, such
as likely required for qubit devices that operate based on parity. The ability
of DGS to resolve differences in tunnel couplings — as seen for the case of
spin-dependent tunneling — provides a very simple means for precisely
characterizing hybridization while leaving the system in the ground state.
Going forward, these demonstrated abilities will be crucial for the
realization and operation of Majorana qubits based on proximitized nanowires
Karzig  _et al._ (2017); Plugge  _et al._ (2017). Our results thus set the
stage for the implementation of quantum measurements of topological qubits.

##  I Acknowledgements

We thank B. van Heck, K. Flensberg, A. Geresdi, A. Kou, and T. Karzig for
useful discussions; and J. M. Hornibrook and D. J. Reilly for providing the
frequency multiplexing chips. This work has been supported by the Netherlands
Organization for Scientific Research (NWO), Microsoft, the Danish National
Research Foundation, the European Research Council, and a Justus and Louise
van Effen excellence scholarship.

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# Integrated optical quantum memory controlled by electro-optic effect

Xia-Xia Li  Key Laboratory of advanced optoelectronic quantum architecture and
measurements of Ministry of Education, Beijing Key Laboratory of Nanophotonics
& Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of
Technology, 100081, Beijing, China  Pai Zhou  Key Laboratory of advanced
optoelectronic quantum architecture and measurements of Ministry of Education,
Beijing Key Laboratory of Nanophotonics & Ultrafine Optoelectronic Systems,
School of Physics, Beijing Institute of Technology, 100081, Beijing, China
Yu-Hui Chen<EMAIL_ADDRESS>Key Laboratory of advanced optoelectronic
quantum architecture and measurements of Ministry of Education, Beijing Key
Laboratory of Nanophotonics & Ultrafine Optoelectronic Systems, School of
Physics, Beijing Institute of Technology, 100081, Beijing, China  Xiangdong
Zhang  Key Laboratory of advanced optoelectronic quantum architecture and
measurements of Ministry of Education, Beijing Key Laboratory of Nanophotonics
& Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of
Technology, 100081, Beijing, China

###### Abstract

Integrated optical quantum memories are a scalable solution to synchronize a
large number of quantum nodes. Without compact quantum memories, some
astonishing quantum applications such as distributed quantum computing and
quantum sensor networks would not be possible. Rather than find a specific
material that meets all the requirements of an on-chip quantum memory as other
protocols usually do, we propose to assign the memory requirements on coherent
storage and controllability to rare earth ions and a lithium niobate crystal,
respectively. Specifically, optical quantum states are stored in an erbium-
doped lithium niobate micro-cavity by utilizing the electro-optic effect of
lithium niobate. The cavity frequency can be shifted by an external electric
field, thus modifying the resonance condition between the cavity and the
collective atomic excitation. This effect is further used to suppress or
enhance the emission of photon echoes. Our calculated results show that high
efficiency and low noise performance is achievable.

Recent developments in quantum information have placed practical quantum
computers just over the horizon. To fully unlock the potential of quantum
computing, it is desired to connect multiple quantum nodes together to build a
quantum internet[1, 2, 3], which in turn could enable some astonishing
applications such as distributed quantum computing[4] and quantum sensor
networks [5, 6]. Such a quantum internet needs quantum memories to store
quantum states for on-demand reading [7]. Otherwise, distant quantum qubits
can not be synchronized and any failure in any node would require the whole
system to restart from the beginning, making a large quantum internet almost
impossible. Similar to a quantum computer that needs many qubits to
demonstrate its quantum advances [8], accessing the full potential of quantum
networks requires scaling of multiple nodes in global distances[9]. In this
aspect, integrating multiple units into a single chip appears to be the most
effective approach for improving the scalability of a memory system, as
evidenced by the success of modern integrated circuits and integrated photonic
chips. Unlike the rapid advances in quantum processor chips, while quantum
memories of $\mu$m size have been fabricated [6, 10, 11, 12], an on-chip
quantum memory with good performance metrics, and at the same time showing
scalability with other photonic devices, remains an outstanding challenge.

Due to their solid-state nature and long coherence times[13], rare-earth doped
crystals are one of the best candidates for building on-chip quantum memories
[6]. Though there have been impressive attempts to build memories by
patterning nanostructures on rare-earth-doped crystals [14, 15, 16, 17],
achieving faithful performance is hampered by the difficulties imposed by the
small size of a chip. For example, it is desirable for an atomic frequency
comb (AFC) [18] memory to have large optical absorption such that it can have
a high storage efficiency [16]. However, for a memory based on a waveguide
architecture, its length on a chip limits the achievable optical depth.
Increasing the ion density seems a straightforward solution, but at the cost
of coherence times [19]. Besides, quantum memory protocols such as controlled
reversible inhomogeneous broadening (CRIB) [20, 21] rely on the assistance of
applied magnetic or electric fields. The need for a magnetic (or electric)
field with a large gradient on a chip not only complicates the architecture,
but also puts additional constraints on the magnetic or electric properties of
the working ions. As a result, current on-chip memories are much less
competitive in performance than their bulk counterparts [22, 23]. When
designing a practical on-chip quantum memory, both the requirements for
faithful performance and easy scalability should be taken into account.

Lithium-niobate (LiNbO3) is a material with the name of photonic silicon,
being famous for its large electro-, nonlinear-, and acousto-optic
coefficients. Photonic devices based on LiNbO3 have rapidly advanced with the
commercialization of lithium-niobate-on-insulators thin film. For example, the
development of LiNbO3 waveguides [24], high-quality micro-resonators [25],
high-frequency modulators [26], and on-chip telecom lasers [27] have made
thin-film LiNbO3 an unprecedented platform for integrated photonic circuits
for both classical and quantum applications.

Rather than imposing excessive requirements on the medium that functions as an
integrated memory, we propose a scheme based on erbium doped lithium niobate
(Er3+:LiNbO3) to realize optical quantum memories by using the electro-optic
effect of LiNbO3. The scheme relies on an electric-optically tuned cavity to
control the emission of photon echoes, resulting in high storage efficiency
and low noise performance. Using cavity detunings to silence the echo has been
investigated in the microwave regime [28, 29], the first demonstration of
which has recently been reported [30]. But its efficiency still needs to be
improved for practical quantum applications. Our memory scheme inherits some
obvious industrial advantages of thin-film Er3+:LiNbO3, such as the
fabrication maturity of LiNbO3 thin films and the 1.5 $\mu$m emission of
erbium ions, being easy to scale up with multiple independent memory units in
one chip.

Similar to photon-echo quantum memories[31, 47, 48, 28, 29, 49, 32], our
approach is modified from the two-pulse echo scheme. The two-pulse echo is not
suitable for optical quantum storage, because the echo is emitted from a
population-inverted medium. The gain and spontaneous emission of the medium
ruins the fidelity of echo [31, 33]. Applying a second $\pi$ pulse brings ions
back to their ground state; therefore the secondary echo can be used as a
readout of the stored quantum state. However, the echo rephased by the first
$\pi$ pulse already emits a large part of the stored energy, leading to
efficiency reduction of the second echo. To use the second echo as a memory
readout, it is thus necessary to eliminate the emission of the first one. In
the following, we detail how to use the electro-optic effect of LiNbO3 to
silence the undesired echo.

Figure 1: Schematic of the cavity-controlled echo scheme. A photonic-crystal
nano-beam cavity is fabricated on an Er3+:LiNbO3 thin film. A grating coupler
enables light input and output of the cavity. Electrodes located on two sides
of the cavity are used to shift the cavity frequency.

The setup that we consider is shown diagrammatically in Fig. 1. A photonic-
crystal nano-beam cavity is fabricated in an Er3+:LiNbO3 thin film. Light is
input into (and output from) a one-sided cavity through a grating coupler. Two
electrodes are used to supply electric field to utilize the electro-optic
effect of LiNbO3. The width and the thickness of the cavity are $w=900$ nm and
$h=390$ nm, respectively. A photonic bandgap around 1.5 $\mu$m is obtained by
milling oval holes with $h_{x}=220$ nm, $h_{y}=490$ nm, and a period of
$p=530$ nm, as illustrated in Fig. 2(a). A cavity resonance is formed by
changing the period of the five central holes to $460$ nm. Figure 2(b) shows
the simulated electric-field distribution of the cavity at the resonant
wavelength of 1.5 $\mu$m.

The effective refractive index of LiNbO3 in our simulations is modelled as
$n_{\text{eff}}=n_{0}-\frac{1}{2}n{{}_{0}}^{3}rE$, where $n_{0}=2.21$ is the
refractive index of LiNbO3 without applied voltage, $r=30.9\times 10^{-12}m/V$
is a constant related to the electro-optic coefficient of LiNbO3 [34], and $E$
is the applied DC electric field. According to our simulations, the cavity
frequency $\Delta_{c}$ shifts linearly with the change of refractive-index:
$\Delta_{c}\propto\Delta n_{\text{eff}}$, as shown in Fig.2(c). Applying a 10
V voltage to the structure can shift the frequency $\Delta_{c}$ by
approximately 60 GHz.

Figure 2: Simulation of the nano-beam cavity. (a) The structure of a periodic
unit of the nano-beam cavity. (b) Simulated electric field distribution of the
nano-beam cavity. Left, $x-y$ plane from the top view; right, $y-z$ plane from
the side view. (c) The cavity frequency shift $\Delta_{c}$ as a function of
applied voltage, where the space distance between two electrodes is 4 $\mu$m.

For a collection of $N$ two-level erbium ions interacting with such an optical
cavity, the dynamic equations for an ion are:

$\begin{split}\dot{\rho}_{21}(\mathbf{r})&=(-i\delta-\frac{1}{T_{2}})\rho_{21}+ig(\mathbf{r})a(\rho_{11}-\rho_{22}),\\\
\dot{\rho}_{22}(\mathbf{r})&=-\frac{1}{T_{1}}\rho_{22}+ig(\mathbf{r})a\rho_{12}-ig(\mathbf{r})a^{\dagger}\rho_{21},\\\
\end{split}$ (1)

where $\rho_{ij}$ are the elements of the two-level density matrix $\rho$ with
$i=1,2$, indicating the ground and excited state respectively, $a$ is the
cavity-mode operator, $\delta$ is the atomic frequencies, $T_{1}$ is the
excited-state lifetime, $T_{2}$ is the optical coherence time, and
$g(\mathbf{r})$ denotes the single-photon Rabi frequency. It is assumed in the
above equations that the cavity and atomic operators are uncorrelated, which
is a good approximation even in the regime of single excitation. The
inhomogeneous atomic detuning $\delta$ is subject to a Gaussian distribution
$G(\delta-\delta_{0})$, where $\delta_{0}$ is the center of the inhomogeneous
line, and $\int G(\delta-\delta_{0})d\delta=1$. The ensemble polarization of
erbium ions is

$P=N\int\int_{-\infty}^{+\infty}\rho_{21}(\mathbf{r},\delta)G(\delta-\delta_{0})d\delta
d\mathbf{r},$ (2)

where $N$ is the total number of erbium ions.

Different from plane wave propagating along a crystal, when light is input
into a cavity [35, 36], the time that the optical field needs to form a stable
distribution inside the cavity is $\sim 1/\kappa$, where $\kappa$ is the
coupling loss of the cavity. This means that the optical field can form a
stable cavity mode within a time scale ($\sim$ns, as shown below) much less
than the coherence time and the spontaneous lifetime ($\sim$ms) of the erbium
ions. Therefore, one can use the creation and annihilation operators to
describe the dynamics of the cavity modes. The equation of motion for the
cavity mode is

$\dot{a}=(-i\Delta_{c}-\frac{\kappa}{2})a+\sqrt{\kappa}A_{in}+igP,$ (3)

where $A_{in}$ is the input driving field ( signal pulse or control $\pi$
pulses in our case).

Since the cavity frequency $\Delta_{c}$ can be electrically shifted, according
to the Purcell effect [37], emission of the atomic ensemble can either be
suppressed if $\Delta_{c}$ is off-resonance with the atomic frequency, or
enhanced if it is on-resonance. Note also that the DC-stark shift coefficient
of erbium ions in LiNbO3 is typically 10 kHz/(Vcm-1) [38], which means that
the ion frequency can also be shifted by $\sim$100 MHz under the investigated
conditions. On one hand, such a shift of the erbium ions is negligible
compared to the $\Delta_{c}$ shift. On the other hand, applying another DC
field with opposite polarity can easily cancel the effect of the erbium Stark
shift on the memory. Moreover, because the Stark interaction of rare-earth-
doped crystals tends to be anisotropic, it is possible to detune the ion
frequency in the opposite direction to the cavity frequency, being an
advantage due to the enhanced ion-cavity detuning.

Based on the effect of $\Delta_{c}$, we can both silence the first echo and
enhance the second by applying a DC electric field. To get an intuitive
picture of the requirements of our scheme, we first consider an input optical
pulse with a linewidth much narrower than $\kappa$. In such a case the field
inside the cavity can instantaneously follow the variation of the signal
field, i.e., the adiabatic approximation $\dot{a}\approx 0$ applies. Note also
that the input-output relation of a cavity is

$A_{\text{out}}+A_{\text{in}}=\sqrt{\kappa}\,a,$ (4)

At the echo-emission stage when the input $A_{\text{in}}$ is absent, one can
write the output of the cavity as

$A_{\text{out}}=\frac{\sqrt{\kappa}}{i\Delta_{c}+\frac{\kappa}{2}}igP.$ (5)

The polarization term $P$ in Eq.(5) originates from the stored collective
excitation of the erbium ions. It is clear now if $\Delta_{c}$ can be detuned
to a value much larger than $\kappa$, the output echo can be effectively
silenced.

Figure 3: Pulse sequence. (a) the input pulse at $t_{1}$ (upper, red), $\pi$
pulses at $t_{2}$ and $t_{3}$ (upper, blue), and the external electric field
at $t_{e_{1}}$ (lower, gold). The first echo of the first $\pi$-pulses is
silenced by an external electric field at $t_{e_{1}}$. The second echo (upper,
purple) is emitted at $t_{e_{2}}$. (b) The output field of the pulse sequence
with (red solid) or without (blue dashed) external electric field to shift the
cavity frequency.

The sequences of control optical pulses and applied voltage of our scheme are
illustrated in Fig. 3(a):

  * I.

At $t=t_{1}$, an optical pulse, which is the signal to be stored, enters the
cavity. To obtain high-efficiency quantum storage, perfect absorption of the
input pulse is needed. For a weak input pulse with few photons (then
$\rho_{11}-\rho_{22}\approx 1$ applies), the polarization term $P$ is a linear
function of the cavity field: $igP=-\Gamma a/2$, with $\Gamma$ being a
constant. If $\dot{a}\approx 0$ (the cavity mode $a$ can instantaneously
follow the change of input pulse), we can obtain from Eq. (3) and (4) that

$A_{\text{out}}(t)=\frac{-i\Delta_{c}+\frac{\kappa}{2}-\frac{\Gamma}{2}}{i\Delta_{c}+\frac{\kappa}{2}+\frac{\Gamma}{2}}\,A_{\text{in}}(t).$
(6)

When $\Delta_{c}=0$, there is impedance-matching $\kappa=\Gamma$ leading to
ideal absorption of the input pulse [39, 40]. By designing the coupling loss
$\kappa$ of the input grating coupler and choosing a proper density of erbium
ions $n_{\text{er}}$, the impedance matching condition can be reached so that
the input pulse are perfectly absorbed, resulting in $A_{\text{out}}(t)\approx
0$.

  * II.

At $t=t_{2}$, a rephasing $\pi$-pulse is applied to the erbium ensemble, which
reverses the phase of $\rho_{21}(\delta)$.

  * III.

At $t=t_{e1}$, where $t_{e1}-t_{2}=t_{2}-t_{1}$, the coherence $\rho_{21}$
with respect to different detuning $\delta$ restores in phase and stimulates
an echo. If no voltage is applied, the echo would be emitted into the LiNbO3
cavity. As aforementioned, this echo is noisy and not suitable for quantum
memories. To silence this echo, an external electric field is applied between
$t_{2}$ and $t_{3}$, as shown by the DC field sequence in Fig. 3(a). The DC
field can shift the cavity frequency $\Delta_{c}$ out of resonance with the
rephased collective polarization $P$, suppressing the collective emission as
an echo.

  * IV.

At $t=t_{3}$, where $t_{3}-t_{2}=\tau$, a second $\pi$-pulse is applied to
cancel the phase differences accumulated between $t_{3}$ and $t_{e1}$, and, in
addition, to bring those ions excited by the first $\pi$ pulse back to their
ground states.

  * V.

At $t=t_{e2}$, where $t_{e2}-t_{3}=t_{3}-t_{e1}$, a secondary echo with low
noise is released from the cavity.

To show that efficient storage is experimentally feasible, we calculate the
output of the scheme by numerically solving Eq. (1)-(4). In the present work,
the focus is on proposing the principles and mechanisms of a storage scheme,
and the building and computation of the realistic model [41] will be addressed
in future work . The loaded quality-factor of the nano-beam cavity is
$Q=9\times 10^{4}$ (the intrinsic $Q$ in simulations can be as high as
$5\times 10^{6}$), and the mode volume of which is $V=0.19\,\mu m^{3}$. While
erbium ions in bulk LiNbO3 crystal have an inhomogeneous linewidth of
$G_{in}=180$ GHz and optical coherence time of $T_{2}=1.8$ ms and electric
dipole moment of $d_{12}=3.5\times 10^{32}$ C$\cdot$m [13], we used the
parameters when erbium ions are doped in LiNbO3 thin film, which are
G${}_{in}=166$ GHz and $T_{2}=0.18$ ms[42].

The performance with (red line) and without (blue dashed line) an applied
voltage is shown in Fig. 3(b). At $t_{1}$, a light pulse with a duration of
0.16 $\mu$s is input to the cavity. According to a given coupling loss
$\kappa$ of our cavity, we set the concentration of erbium ion as 225 ppm to
realize the impedance-matching condition. Perfect absorption is observed in
Fig. 3(b), where the output at $t_{1}$ is $A_{\text{out}}(t_{1})\sim 0$.

With the absence of an applied DC field, both a primary echo and a secondary
echo can be seen after applying $\pi$ pulses, as shown by the blue dashed line
in Fig. 3(b). However, if an electrical field is turned on to shift
$\Delta_{c}$$=$230 GHz out of resonance during the emission period of the
first echo, the first echo can be silenced to a negligible level. At the same
time the amplitude of the second echo, compared to its counterpart without
applied field, is enhanced. These results suggest that one can both silence
the unwanted echo and enhance the second by using the electro-optic effect of
LiNbO3.

Figure 4: Memory efficiency and noise. (a) Efficiencies of the first echo
(dashed blue) and second echo (solid red) as a function of $\Delta_{c}$. (b)
Efficiency as a function of the interval between the signal pulse and the
revival echo. The maximum efficiency is limited by $T_{2}$ of erbium ions,
which is 180 $\mu$s in calculations. $\eta$ depends on $\tau$ through a
exponential function $\exp(-4\tau/T_{2})$. (c) Noise analysis. The noise level
with and without applied DC electric field are shown in solid red and dashed
blue line. The noise caused by the free induction decay of $\pi$-pulses can be
significantly reduced when switching on a large $\Delta_{c}$ (grey shade). The
inset shows the noise ratio of two situations with $\Delta_{c}=0$ and
$\Delta_{c}=230$ GHz. The noise can be switched off by $\Delta_{c}$ whenever
needed.

To efficiently cancel the first echo, one needs a large $\Delta_{c}$ as
suggested by Eq (5). In Fig. 4(a), we compare the emissions of the two echoes
for increasing $\Delta_{c}$. When there is no external DC field
$\Delta_{c}=0$, both the first and the second echo are present and of similar
weights. With increasing $\Delta_{c}$, the efficiency of the second echo rises
up to 97%, accompanied by the decrease of the first one. Note that to obtain a
high-contrast ratio between the first and the second echo, $\Delta_{c}$ is
just required to be larger than the cavity linewidth $\kappa$ (2 GHz in our
case), rather than the inhomogeneous linewidth of erbium ions (166 GHz in our
case). This is evident in Fig. 4(a), where for $\Delta_{c}\approx 60$ GHz the
ratio between the first and the second echo is already 1.3%. Therefore a
high-Q cavity can reduce the needed DC voltage and facilitate the application
of the scheme.

Further increasing $\Delta_{c}$ above 60 GHz leads to little increase of the
efficiency above $97\%$. This is because the memory operation takes time,
during which some ions decohere and cannot be rephased by $\pi$-pulses. The
efficiency depends on the decoherence time according to
$\eta=\exp(-{4\tau}/{T_{2}})$, where $4\tau$ is twice the interval between the
signal pulse and the revival echo, as shown in Fig. 4(b). In this paper,
$T_{2}=180$ $\mu$s is used from a relevant experiment that measured the
coherence time of erbium ions in thin-film LiNbO3 waveguides [43]. When rare-
earth ions are doped into LiNbO3, static defects like charge compensation and
compositional disorder cause large inhomogeneous linewidths and can also lead
to decoherence [44]. In addition, the presence of non-zero nuclear spin
isotopes 93Nb, 7Li, and 6Li causes additional decoherence due to their dynamic
magnetic interactions with erbium ions. Note that the $T_{2}=180$ $\mu$s in
our scheme already enables a theoretical efficiency of 97%. If the operation
time $4\tau$ is reduced, such an efficiency can be extended to meet the
requirement of quantum-error-correction applications [45]. To further extend
the memory storage time without losing high efficiency, one can use the
hyperfine ground states of erbium ions. When an erbium isotope with nonzero
nuclear spin is used, hyperfine ground states offer the possibility of
realizing the Zero-First-Order-Zeeman effect, showing the prospects of
extending the coherence times by orders of magnitudes [53, 54, 55].

By introducing a tunable LiNbO3 cavity, not only can the emission of echoes be
controlled, but also the noise of such a memory can be suppressed to a
negligible level, as shown in Fig. 4(c). For quantum memories based on photon
echoes, typical noise originates from free induction decay or spontaneous
emission generated by $\pi$-pulses. Shown in Fig. 4(c) is the output with
$\pi$ pulses present but no input signal. Without an applied electric field,
the free-induction-decay tails of both $\pi$ pulses immediately arise once the
$\pi$ pulses are switched off, as shown by the blue dashed curve of Fig. 4(c).
If an electric field is applied after the second $\pi$ pulse to shift the
cavity frequency by 230 GHz, both the free induction decay and spontaneous
emission (red curve) are suppressed. In our calculations, a ratio of more than
104 between the noise levels with and without an applied field can be
obtained, as shown in the inset of Fig. 4(c). The low noise thus makes the
memory suitable for working at single photon levels. If the input optical
state is a one-photon state, then the noise-limited fidelity of the output,
according to the calculation method in reference [46], can be more than 99.8%
(depending on the read-out time of the memory).

In our scheme, the primary echo is silenced by $\Delta_{c}$. This is quite
different from other protocols such as the Revival-of–Silenced-Echo (ROSE)
scheme that utilizes the spacial phase mismatching effect[47], the Hybrid-
Photon-Echo-Rephasing scheme that utilizes the electric Stark effect[48], and
the Light-Shift-Photon-Echo-Rephasing scheme that utilizes the light-induced
frequency shift effect[49]. The echo cancellation in all these schemes is the
result of a vanishing ensemble polarization due to the destructive
interference of ions with different accumulated phases (spatial or temporal).
At the moment $t_{e1}$ of our scheme, all ions are already rephased, which
means that an ensemble polarization is built up and can otherwise emit photons
if $\Delta_{c}$ is in tune. The silenced echo is based on the Purcell effect
rather than the out-of-phase superposition of ions. This difference has
brought several advantages, as listed below, to our scheme.

The most important one is that the scheme not only shifts the requirement on
controllability from erbium ions to LiNbO3, but also bring in the feature of
LiNbO3. Many memory protocols, such as CRIB and AFC, rely on preparation
steps. The preparation not only introduces technical difficulties, but also
imposes additional requirements on working ions. For example, in addition to
excellent coherence properties of working ions, CRIB also relies on extra
electric or magentic field to realize controllable linewidth broadening, which
can lead to a compromise between coherence and controllability. The AFC
protocol only works for ions with auxiliary long-life-time states. In our
scheme, all these requirements are relaxed. When building a memory, one can
just focus on utilizing the best experimental conditions to achieve the
longest coherence time, as the write-in and read-out of the memory are now
controlled by the LiNbO3. This means that the protocol can take full advantage
of the 1.5 $\mu$m telecom emission and the long coherence times of erbium
ions. Moreover, benefiting from the host LiNbO3 thin film, such a device is
easy to integrate within modern photonic circuits. The number of memory units
can be scaled up with current technology, and the operation of each unit can
be independently controlled by electrodes, ideal for integrated quantum chips.

Table 1: Theoretical speculation for single-ion quantum memory. Q | V(${\mu m}^{3}$) |  $n_{\text{er}}$$(\text{ppm})$ | $N_{h}$ | Efficiency($\%$)
---|---|---|---|---
$2\times 10^{5}$ | 0.28 | 105 | 3.1 | 97
$3\times 10^{5}$ | 0.3 | 70 | 2.2 | 97
$4\times 10^{5}$ | 0.28 | 50 | 1.5 | 97
$8\times 10^{5}$ | 0.26 | 26 | 0.7 | 97
$7\times 10^{6}$ | 0.23 | 3 | 0.07 | 97

The multimode capacity of the memory can be limited by the linewidth of the
optical cavity. In the above calculation, the linewidth of the cavity is $\sim
2$ GHz and the input pulse is $\sim 10$ MHz, thus the multimode capacity is on
the order of $10^{2}$. Using a high $Q$ cavity reduces the multimode capacity,
but on the other hand, can further enhance the light-matter interaction. For
high $Q$ cavities, fewer erbium ions are needed for high optical absorption,
which, together with the small mode volume of photonic-crystal cavities,
enables us to exploit the possibility of quantum memories based on single
ions. The numerical results of calculations on $\eta$ for several combinations
of $Q$ and erbium concentrations $n_{\text{er}}$ are shown in Table 1 (note
that in these calculations, the mode volume V and the duration of the two
$\pi$ pulses are also adjusted to maintain the efficiency $\eta>97\%$). The
number of ions that can be resolved in its homogeneous linewidth is
approximated by $N_{h}=n_{\text{er}}V\Gamma_{h}/\Gamma_{in}$, where
$\Gamma_{h}=1/(\pi T_{2})\sim$1 MHz is chosen for the homogeneous linewidth ,
and $\Gamma_{in}=166$ GHz is the inhomogeneous linewidth [42]. For $Q\approx
10^{5}$ (LiNbO3 micro cavities with $Q\approx 10^{8}$ has been reported [50]),
the number of interacting erbium ions can easily be reduced to single digits.
For erbium doped crystals with Gaussian broadening, working on the edge of the
inhomogeneous line can further reduce $N_{h}$ and facilitate single ion
detection. With decreasing erbium concentration, the scheme shows potential
for realizing solid-state single-ion quantum memories, where the property of
addressable single ions can be used to predict events of successful storage
[51] and to establish deterministic entanglement [52].

In summary, we propose an on-chip quantum memory scheme by utilizing the
electric-optic effect of LiNbO3. Applying a DC electric field can change the
refractive index of LiNbO3 and the frequency. Therefore, it can change the
frequency of a LiNbO3 photonic cavity and control the emission of echoes due
to the Purcell effect. Our numerical results show that the echo readout
efficiency approaches unity. Compared with the technical difficulties in
memory schemes, the electric-field controllability of lithium niobate on a
phtonic chip is industrially compatible. It has been demonstrated that the
erbium ions in a patterned thin-film LiNbO3 are able to preserve the coherence
properties of bulk material. At high magnetic field and low concentrations,
the coherence time can be further extended, which in turn can increase the
maximum memory efficiency. Furthermore, when an erbium isotope with nonzero
nuclear spin is used, hyperfine interactions offer the possibility of using
the ZEFOZ effect[53, 54, 55] to extend the coherence times by orders of
magnitude. We thus expect it is experimentally feasible to build an integrated
quantum memory with high efficiency and long coherence times using Er3+:LiNbO3
thin films. The scheme also offers new ideas for exploiting quantum memories,
such as noise suppression by electric fields and single-ion quantum memories.

The authors wish to acknowledge financial support from the National Natural
Science Foundation of China (No. 62105033 and No. 12174026), the Start-up Fund
of Beijing Institute of Technology, and the Science and Technology Innovation
Project of Beijing Institute of Technology.

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