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Controlling rare-earth ions in a nanophotonic resonator using the ac Stark shift
John G. Bartholomew,
1, 2
Tian Zhong,
1, 2
Jonathan M. Kindem,
1, 2
Raymond Lopez-Rios,
1, 2
Jake Rochman,
1, 2
Ioana Craiciu,
1, 2
Evan Miyazono,
1, 2
and Andrei Faraon
1, 2,
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA.
2
Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125, USA.
(Dated: February 20, 2018)
On-chip nanophotonic cavities will advance quantum information science and measurement be-
cause they enable efficient interaction between photons and long-lived solid-state spins, such as those
associated with rare-earth ions in crystals. The enhanced photon-ion interaction creates new oppor-
tunities for all-optical control using the ac Stark shift. Toward this end, we characterize the ac Stark
interaction between off-resonant optical fields and Nd
3+
-ion dopants in a photonic crystal resonator
fabricated from yttrium orthovanadate (YVO
4
). Using photon echo techniques, at a detuning of
160 MHz we measure a maximum ac Stark shift of 2
π
×
12
.
3 MHz per intra-cavity photon, which is
large compared to both the homogeneous linewidth (Γ
h
= 100 kHz) and characteristic width of iso-
lated spectral features created through optical pumping (Γ
f
3 MHz). The photon-ion interaction
strength in the device is sufficiently large to control the frequency and phase of the ions for quan-
tum information processing applications. In particular, we discuss and assess the use of the cavity
enhanced ac Stark shift to realize all-optical quantum memory and detection protocols. Our results
establish the ac Stark shift as a powerful added control in rare-earth ion quantum technologies.
Efficient interfaces between photons and spins in solids
are one foundation on which to build integrable and scal-
able quantum technologies for computing, communica-
tion, and metrology. One promising system for realizing
photon-spin interfaces to create, control, and store quan-
tum states is crystals containing rare-earth ions (REIs).
Experiments in REI crystals have demonstrated entan-
gled photon-pair generation
1–3
, quantum memories for
light
4–6
, and qubit operations
7
. These results, combined
with some of the longest optical and spin coherence times
in the solid state
8–11
establish the future potential of REI
quantum technologies.
In most cases, quantum optical protocols performed in
REI materials rely on large ensembles (10
9
ions) to com-
pensate for the weakly allowed 4
f
4
f
optical transi-
tions
12
. Although this approach has proved effective, the
use of large ensembles in doped crystals sets a macro-
scopic lower bound on the device size. This is because
increasing the spectral-spatial density of REI dopants
increases ion-ion interactions that cause added inhomo-
geneity and decoherence. The size restriction imposed
by the use of large ensembles places limits on the in-
tegration and scalability of the REI platform. Thus,
there is significant impetus to develop other methods
to increase photon-ion interactions
13–17
. One solution is
to fabricate photonic crystal resonators directly in REI
crystals
15,18–21
. A large increase in photon-ion coupling
is achieved through cavity enhancement of the optical
transition
22
and strong mode confinement
14,23–26
. Given
that previous work has shown that the optical properties
of the ion ensembles are preserved within such nanopho-
tonic devices
15,21
, the platform is a significant opportu-
nity for ensemble and single REI technologies. Further-
more, these photonic crystal resonators are suited to har-
nessing phenomena that are more commonly associated
with systems with strong optical transitions, such as the
ac Stark shift (ACSS).
Strong photon-ion interactions, including a strong sin-
gle photon ACSS, offer an important additional degree of
control for REIs. In this work we characterize the ACSS
in an on-chip nanophotonic resonator containing Nd
3+
ions. The ACSS has been investigated previously in bulk
REI crystals
27,28
, where the interaction resulted from
10
12
photons interacting with
10
9
ions. We build
on this work in a different regime: where the strength
of the ACSS is sufficiently large to allow the study of
the interaction between a single photon with approxi-
mately 4
×
10
3
ions. The ACSS was probed using a pho-
ton echo technique, which allowed the measurement of
the maximum ACSS in the cavity and the inhomogene-
ity of the interaction across the ensemble of ions. These
measurements are then used to analyze the usefulness of
the ACSS as a tool in the REI quantum optics tool box.
In particular, we discuss using the large ACSS to realize
all-optical quantum memories based on the hybrid pho-
ton echo rephasing (HYPER) protocol
29
and the atomic
frequency comb
30
, and cross phase modulation using the
protocol suggested in Reference 31. Our study demon-
strates that the enhanced photon-ion interactions result-
ing from coupling of REIs to photonic crystal resonators
offer new avenues for quantum technologies in these ma-
terials.
The material chosen for this work was YVO
4
doped
with a Nd
3+
-ion impurity at a nominal level of 50 ppm
(Gamdan Optics). We use the 879.9 nm transition be-
tween the lowest crystal field components of the
4
I
9
/
2
and
4
F
3
/
2
multiplets. Both these levels are Kramers doublets,
the degeneracy of which is lifted in an applied magnetic
field resulting in four optical transitions (see Figure 1(b)).
This transition has been characterized previously
30,32,33
arXiv:1802.06172v1 [quant-ph] 17 Feb 2018
2
FIG. 1. (a) Photonic crystal cavity fabricated on the surface of a Nd
3+
:YVO
4
substrate. The cavity is one-sided allowing
measurements to be performed in the reflection mode. Below the scanning electron microscope images are cross-sections of
|
E
z
|
in the cavity, which illustrate the spatial inhomogeneity of the cavity field. (b) The optical absorption of the
4
I
9
/
2
(Z1)
4
F
3
/
2
(Y1) transition in Nd
3+
:YVO
4
modeled from the spin Hamiltonian, which shows the spectral subset of ions that contribute
to the photon echo signal (at frequency
ω
p
), and the transmission trench prepared for the ACSS pulses (at frequency
ω
p
+ ∆
ac
).
The inset on the left shows the energy level structure of the transition, and the inset on the right shows the spectral region of
interest in greater detail.
and shown to possess narrow inhomogeneous linewidths,
and the largest documented optical dipole moment for
REI transitions suitable for quantum memory applica-
tions
34
. Furthermore, optical pumping of Nd
3+
:YVO
4
allows the electron spin to be highly polarized
21,30,33
, be-
cause of long lived spin states. Notably, the Nd
3+
site
has D
2d
symmetry and hence, has a vanishing dc Stark
shift. As a result, despite the high absorption possible
in this material, its use in quantum memory applications
has been limited because electric fields cannot be used to
control the ions.
A one-sided photonic crystal cavity was milled on the
Nd
3+
:YVO
4
surface perpendicular to the crystal c-axis
using a focused ion beam. The nanophotonic cavity is
based on a triangular nanobeam
15,18
and is described in
more detail in Reference 21. To couple the TM mode
illustrated in Figure 1(a) to the Nd
3+
optical transi-
tion, the cavity is frequency tuned through nitrogen gas
condensation. The device was cooled to approximately
500 mK in a
3
He cryostat to reduce transition broaden-
ing due to phonon interactions. In addition, a constant
magnetic field of approximately 340 mT was applied at a
small angle from the YVO
4
c-axis to reduce broadening
from Nd
3+
-Nd
3+
magnetic dipole interactions. Further
details about the cavity and the experimental setup are
provided in the Supplemental Material.
The ACSS is characterized through the study of two
pulse photon echoes
27,28
, which were detected by pho-
ton counting with a silicon avalanche photodiode (APD).
The two pulse echo sequence was augmented by addi-
tional off-resonant ACSS pulses (AC1 and AC2) before
and after the inverting
π
-pulse (see the insets of Fig-
ure 2). During the off-resonant pulses of length
τ
ac
,
the optical transition of each ion is frequency shifted by
δ
ac
(
r
)
Ω(
r
)
2
/
(2∆
ac
), where Ω(
r
) is the Rabi frequency
at spatial position
r
, and ∆
ac
is the detuning of the ACSS
pulse from the echo input pulse. The resulting phase ac-
cumulated by each ion
φ
(
r
) =
δ
ac
(
r
)
τ
ac
, is governed by
the field amplitude of the cavity mode at the ion’s spatial
location. Because there is no correlation between an ion’s
resonant frequency and its position in the cavity, the ap-
plication of an ACSS pulse results in an inhomogeneous
phase shift across the ensemble. The inhomogeneity re-
sulting from the ACSS pulse cannot be rephased by the
optical
π
pulse leading to a modulation of the photon
echo intensity. However, rephasing the ACSS-induced in-
homogeneity is possible through the application of addi-
tional ACSS pulses. To reduce any resonant interactions
between the ions and the ACSS pulses, spectral trenches
were prepared at the ACSS frequency prior to the se-
quence by optically pumping to the other electron spin
level (see Figure 1(b)).
Figure 2 shows the normalized intensity of the emit-
ted photon echo for sequences that vary (a) the average
Rabi frequency of the ACSS pulses
̄
Ω, (b) their dura-
tion, and (c) their relative frequency detuning. The data
highlights the ability to control the coherent emission
intensity by manipulating the relative phase evolution
throughout the ensemble using the ACSS. In the case
where only a single ACSS pulse is applied (solid blue cir-
cles in Figure 2), the ACSS-induced inhomogeneity can-
not be rephased and the echo is attenuated. The phase
accumulated due to AC1 can be balanced through the
application of an identical ACSS pulse after the
π
-pulse,
which in principle can restore the echo to full intensity.
Figure 2(a) (solid squares) demonstrates where this has
been partially achieved through the application of AC2.
On average the echo is restored to greater than 75% of the
unperturbed echo intensity
35
. The incomplete recovery
is likely to be dominated by imperfections in balancing
the phase evolution from pulses AC1 and AC2, largely be-
cause of limitations in the timing resolution and intensity
control of the applied ACSS pulses in our experimental
setup (see further analysis in the Supplemental Material).
Importantly, the restoration of the echo is evidence that
the attenuation is caused by the ACSS interaction rather
3
AC1
AC2
AC1
AC1
FIG. 2. ACSS control of Nd
3+
photon echo emission. The
echo intensity is plotted against the average ACSS Rabi fre-
quency
̄
Ω in (a), the ACSS duration
τ
ac
in (b), and the ACSS
detuning ∆
ac
in (c). The insets in each sub-figure illustrate
the pulse sequence used. The duration of the input and
π
pulses in all measurements was 20 ns, and
τ
represents the
pulse-center to pulse-center time separation. The expected
echo intensity based on the simulated cavity mode is shown
by the solid black curve, and the analytic approximation de-
rived from a 2d Gaussian distribution of the cavity field is
shown by the dashed red curve.
than by other dephasing processes such as instantaneous
spectral diffusion, or device heating.
The echo intensity can be simulated using the cavity
Maxwell-Bloch equations under the assumption of a uni-
form distribution of ions within the mode profile of the
cavity (see Supplemental Material). Figure 2 shows nor-
malized echo intensities from both a simplified analytical
model (dashed line) and a numerical simulation (solid
line). The analytical model assumed a two-dimensional
Gaussian distribution
28
for the cavity mode profile. This
is a coarse approximation that captures the small vari-
ation of the field envelope along the y-axis in compar-
ison to the variation along the x and z axes. Despite
this, the analytical solution is a useful reference point for
understanding the echo behavior. The numerical model
used the simulated cavity mode profile from a finite dif-
ference time domain calculation (COMSOL), and the
ions’ frequencies are chosen from a Gaussian distribution
with a FWHM equal to the input pulse Rabi frequency
(
̄
Ω = 2
π
×
25 MHz). The numerical simulation is fit-
ted to the data using one free parameter
R
: the ratio of
the maximum ACSS Rabi frequency Ω(
r
)
max
to the av-
erage ACSS Rabi frequency
̄
Ω. The agreement between
the experimental and simulated data for the least squares
fit value of
R
= 1
.
83
±
0
.
02 is further evidence that the
ACSS is the dominant perturbation to the system.
With the experimentally determined value for
R
it is
possible to calculate the single photon-ion interaction
strength
g
by calibrating the average cavity photon pop-
ulation
n
to
̄
Ω. From the known transmission losses in
our system we estimate that
n
= 0
.
53 for pulses with
̄
Ω = 2
π
×
25 MHz. The specified values of
R
and
n
result
in a
g
= 2
π
×
31
.
4 MHz. This is consistent with previous
observations in this device
21
and only
10% greater than
the value of
g
calculated from the published optical os-
cillator strength
32
and the simulated mode volume of the
nanophotonic cavity (see Supplemental Material). Using
the experimentally determined value of
g
, the maximum
possible single photon ACSS in the cavity at a detuning
ac
/
2
π
= 160 MHz is 2
g
2
/
ac
= 2
π
×
12
.
3 MHz.
The results demonstrate that a single intra-cavity pho-
ton can produce an ACSS
δ
ac
that is 100
×
larger than
the Nd
3+
homogeneous linewidth Γ
h
100 kHz (see Sup-
plemental Material). Therefore, maintaining
n
〉 
0
.
01
during
τ
ac
allows all-optical control of the relative phases
of ions within an ensemble using time domain techniques,
such as the photon echo. Thus, this work establishes a
path toward realizing all-optical variations of techniques
that have previously relied on applied electric fields
27,28
.
In particular, our measurements form the basis for
achieving an all-optical quantum memory based on the
hybrid photon echo rephasing (HYPER) protocol previ-
ously implemented with electric field gradients
29
. The
HYPER protocol uses two inversion pulses to recall an
input photon (the HYPER echo) when the ensemble is
almost completely in the ground state. This avoids the
stimulated emission noise that occurs when the recalled
photon is emitted whilst the ensemble is inverted, such as
in the case when only a single inversion pulse is used
36
.
For HYPER to achieve high efficiency, the intermediate
echo resulting from the first inversion is suppressed using
a controlled phase perturbation throughout the ensem-
ble, which is later balanced to recover the HYPER echo.
The largest benefit of HYPER is that, in the ideal im-
plementation, no preparation of the inhomogeneous line
is required, allowing the full optical depth and natural
bandwidth of the material to be harnessed for quantum
optical storage.
The photon echo measurements presented in Figure 2,
4
FIG. 3. ACSS controlled HYPER protocol sequence. The
lower trace shows the signal when no ACSS pulses are applied,
and the upper trace (offset for clarity) shows the signal when
both ACSS pulses are applied. Superimposed above the data
is the pulse sequence used, where
τ
= 700 ns,
τ
= 250 ns,
and ∆
ac
= 160 MHz. Further details are provided in the
Supplemental Material.
demonstrate two of the important aspects for an all-
optical HYPER memory. The first is the suppression
of the intermediate echo using the controlled phase per-
turbation, and the second is the balancing of that phase
to allow the formation of the HYPER echo (Figure 2(a)).
Both of these aspects are combined in a proof-of-principle
demonstration of the HYPER sequence shown in Fig-
ure 3, where the secondary echo is enhanced when the
balanced ACSS pulses are applied. An all-optical ver-
sion of HYPER is promising for on-chip quantum mem-
ories because additional electrodes on the REI substrate
are not required, the efficiency can theoretically ap-
proach unity in the limit of an impedance matched cavity
(see Supplemental Material), and ∆
ac
can be increased
so that the storage bandwidth can approach the inho-
mogeneous linewidth (GHz). Achieving the impedance
matching condition requires approximately doubling the
nanocavity quality factor, which is possible through im-
proved fabrication. A challenge for HYPER memories
is to simultaneously achieve high efficiency and high fi-
delity. To do so requires efficient inversion pulses over
the bandwidth of interest, which is not achieved by the
simple Gaussian pulses used in this work. Although the
use of more complex adiabatic pulses offer a pathway to
achieve large bandwidth and efficient inversion
37
, the re-
sultant instantaneous spectral diffusion may ultimately
reduce the maximum storage time of the memory
38
.
Whilst well suited to the current devices, the HY-
PER memory is not the only protocol that could be
achieved all-optically in the nanodevices. This is be-
cause the
δ
ac
of a single photon is large compared to the
width of absorption features prepared by optical pump-
ing (Γ
f
3 MHz
21,30
). As a result it is possible to create
an atomic frequency comb (AFC) memory with an ACSS
controlled readout delay. This was demonstrated in Ref-
erence 21, where ACSS pulses were applied frequency-
symmetrically at large detunings about the AFC center.
Importantly, the ACSS allows control that is not possi-
ble using dc electric fields. Although protocols that dy-
namically alter the comb profile during storage using dc
electric fields have been investigated
39
, they are not able
to achieve a continuously tunable delay. This is because
there is no correlation between an ions’ spatial position
and resonant frequency in stochastically doped crystals.
Because the ACSS is spectrally dependent, it is possible
to achieve a continuously tunable storage time (see Sup-
plemental Material). To realize ACSS controlled AFC
memories operating with high efficiency further steps are
required. In samples with a uniform distribution of ions
the recall efficiency of a pulse stored with a controlled
delay will be limited due to the ACSS inhomogeneity
21
(see Supplementary Material). To overcome this limita-
tion requires control of the spatial location of the inter-
acting ions within the cavity either through spectroscopic
selection or controlled placement
21
.
For both the ACSS controlled HYPER and AFC pro-
tocols, operation at the quantum level will require the
suppression of noise photons that are generated by ions
excited resonantly or off-resonantly by the ACSS control
pulses. Therefore, a large single photon ACSS is desir-
able because the required frequency shift can be achieved
with fewer photons, reducing the number of excited ions
contributing to the noise. To suppress noise photons that
are generated outside the memory bandwidth with high
extinction, spectral filters created by optical pumping in
another Nd
3+
:YVO
4
crystal can be applied
2,40
. For pho-
tons generated within the memory bandwidth, a high
memory efficiency ensures that these noise photons are
time-separated from the signal photon through the pro-
tocol storage (see Supplemental Material for quantitative
analysis using the AFC as an example).
In addition to offering opportunities for advancing
quantum memory protocols, large single photon ac
Stark interactions can be harnessed for quantum non-
demolition measurements
31
. Sinclair
et al
. discuss the
measurement of the phase shift of an optical probe pulse
stored in an AFC due to the ACSS of a single photon
transmitted through the cavity in a transparent window
adjacent to the memory. The phase shift of the retrieved
probe pulse is then a non-destructive measurement that
heralds the presence or absence of a single photon. The
maximum single photon phase shift resultant from the
experiments performed here is 3
×
10
4
rad, which is
consistent with the prediction from Reference 31. The
parameters used in this work would require the probe
pulse to be stored in a comb with a bandwidth of the
order of 10s of MHz. Ideally, the phase shift would be
increased further through longer optical confinement in
the cavity. Given the current quality factor of the de-
vice (
Q
2
.
8
×
10
3
), an order of magnitude improvement
should be possible with further optimization of the cavity
design and fabrication, and would not require significant
changes to the proposed scheme.