PHYSICAL REVIEW A
97
, 063854 (2018)
Editors’ Suggestion
Controlling rare-earth ions in a nanophotonic resonator using the ac Stark shift
John G. Bartholomew,
1
,
2
Tian Zhong,
1
,
2
,
3
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
3
Institute of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA
(Received 14 February 2018; published 28 June 2018)
On-chipnanophotoniccavitieswilladvancequantuminformationscienceandmeasurementbecausetheyenable
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 opportunities 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
.
3MHzper
intracavity photon, which is large compared to both the homogeneous linewidth (
h
=
84 kHz) and characteristic
width of isolated spectral features created through optical pumping (
f
≈
3 MHz). The photon-ion interaction
strength in the device is sufficiently large to control the frequency and phase of the ions for quantum 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.
DOI:
10.1103/PhysRevA.97.063854
I. INTRODUCTION
Efficient interfaces between photons and spins in solids are
one foundation for building integrable and scalable quantum
technologies for computing, communication, and metrology.
One promising system for realizing photon-spin interfaces
to create, control, and store quantum states is crystals con-
taining rare-earth ions (REIs). Experiments in REI crystals
have demonstrated entangled 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 compensate
for the weakly allowed 4
f
↔
4
f
optical transitions [
12
].
Although this approach has proved effective, the use of large
ensembles in doped crystals sets a macroscopic lower bound on
the device size. This is because increasing the spectral-spatial
density of REI dopants increases ion-ion interactions that cause
added inhomogeneity and decoherence. The size restriction
imposed by the use of large ensembles places limits on the
integration 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
*
faraon@caltech.edu
mode confinement [
14
,
23
–
26
]. Thus, these photonic crystal
resonators are suited to harnessing phenomena that are more
commonly associated with systems with strong optical transi-
tions, such as the ac Stark shift (ACSS). Although the ACSS
has been investigated in REI crystals [
27
–
29
], the interaction
resulted from
≈
10
8
photons interacting with
≈
10
8
ions.
In this work we characterize the ACSS in an on-chip
nanophotonic resonator, where the strength of the ACSS is
sufficiently large to measure the interaction between a single
photon and approximately 4
×
10
3
Nd
3
+
ions. The ACSS was
probed using a photon echo technique, which allowed the
measurement of the maximum ACSS and the inhomogeneity
of the interaction in the cavity. These measurements are then
used to assess the ACSS as a tool for REI quantum optics.
In particular, we discuss using the large ACSS to realize
all-optical quantum memories based on the hybrid photon echo
rephasing (HYPER) protocol [
30
] and the atomic frequency
comb (AFC) [
31
], and cross phase modulation using the
protocol suggested in Ref. [
29
]. Our study demonstrates that
the enhanced photon-ion interactions resulting from coupling
REIs to photonic crystal resonators offer new avenues for
quantum technologies in these materials.
II. AC STARK SHIFT CHARACTERIZATION
A. Summary of device, apparatus, and calibration
The material chosen for this work was YVO
4
doped with
aNd
3
+
-ion impurity at a nominal level of 50 ppm (Gamdan
Optics). We use the 879.9 nm transition between 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
2469-9926/2018/97(6)/063854(13)
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©2018 American Physical Society
JOHN G. BARTHOLOMEW
et al.
PHYSICAL REVIEW A
97
, 063854 (2018)
FIG. 1. (a) Schematic of the experimental apparatus. A cw Ti:sapphire laser is gated by two acousto-optic modulators (AOMs) and an
electro-optic modulator (EOM) to create pulses that are coupled into the nanophotonic device, which is maintained at approximately 500 mK in
a
3
He refrigerator. The photonic crystal cavity was fabricated on the surface of a Nd
3
+
:YVO
4
substrate and is one-sided to allow measurements
to be performed in the reflection mode. (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 contributes 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. (c) Cross sections of
|
E
z
|
in the cavity, which illustrate the spatial inhomogeneity
of the cavity field. The inset shows the variation of
|
E
z
|
along lines
y
1
and
z
1
, indicating the smaller variation of the field along the
y
axis.
(d) An example pulse sequence for the photon echo measurements used in this work.
which is lifted in an applied magnetic field resulting in
four optical transitions [see Fig.
1(b)
]. This transition has
been characterized previously [
31
–
33
] and shown to possess
narrow inhomogeneous linewidths and the largest documented
optical dipole moment for REI transitions suitable for quantum
memory applications [
34
]. Furthermore, optical pumping of
Nd
3
+
:YVO
4
allows the electron spin to be highly polarized
[
21
,
31
,
33
],becauseoflong-livedspinstates.Notably,theNd
3
+
site has D
2d
symmetry and hence has a vanishing dc Stark shift.
Asaresult,despitethehighabsorptionpossibleinthismaterial,
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 into the
Nd
3
+
:YVO
4
surface perpendicular to the crystal
c
axis using
a focused ion beam [
15
,
18
,
21
]. To couple to the transverse
magnetic (TM) mode illustrated in Fig.
1(c)
to the Nd
3
+
optical
transition, 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 broadening 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 Appendix
A
.
The ACSS is characterized through the study of two-
pulse photon echoes [
27
,
28
], which were detected by photon
counting with a silicon avalanche photodiode (APD). The two-
pulse echo sequence was augmented by additional off-resonant
ACSS pulses (AC1 and AC2) before and after the inverting
π
pulse [see Fig.
1(d)
, and the insets of Fig.
3
]. To reduce any
resonant interactions between the ions and the ACSS pulses,
spectral trenches were prepared at the ACSS frequencies prior
to the sequence by optically pumping to the other electron spin
level [see Fig.
1(b)
]. During the off-resonant ACSS pulses
of length
τ
ac
, the optical transition of each ion is frequency
shifted by
δ
ac
(
r
)
≈
ac
(
r
)
2
/
(2
ac
)
,
(1)
where
ac
(
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 accumulated by each ion is
φ
(
r
)
=
δ
ac
(
r
)
τ
ac
,
(2)
which 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 application of an ACSS pulse results in an inhomo-
geneous phase shift across the ensemble. The inhomogeneity
resulting from the ACSS pulse cannot be rephased by the
optical
π
pulse, which leads to a decrease of the photon echo
intensity. As
φ
(
r
) increases, the echo intensity can oscillate
because subensembles with phase shifts that differ by multiples
of 2
π
(
π
) sum to increase (decrease) the coherent emission,
but this modulation is strongly damped due to the increas-
ing inhomogeneity [
30
]. It is possible, however, to rephase
this ACSS-induced inhomogeneity through the application of
additional ACSS pulses.
The average cavity photon population
n
was calculated
from the pulse input power using the known losses in the
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FIG. 2. Two-pulse photon echo intensity as a function of the
length of the second pulse for a given pulse intensity
n
. Each data
set is offset by 2
×
10
4
counts for clarity. The dashed gray lines are a
guide to the eye for the first and second maxima of the Rabi flopping.
apparatusandthemeasuredcavityparameters.Forallmeasure-
ments the cw power of the laser was recorded at (i) before fiber
coupling to the circulator [see Fig.
1(a)
]. The relative intensity
of pulses at (ii), in the device waveguide [see Fig.
1(a)
],
compared to the cw light at (i) was 0.011, which included loss
due to the finite rise time of the AOMs for 20 ns pulses (23.5%
transmission), the circulator loss (49.2% transmission), fiber
coupling and splicing losses (35% transmission), and the fiber
to waveguide coupling (27% transmission).
The average steady-state photon number in the cavity
n
given an average photon number rate in the waveguide
|
a
in
|
2
is given by
n
=
κ
in
(
κ/
2)
2
+
2
cav
|
a
in
|
2
,
(3)
where
κ
in
and
κ
are the energy decay rates through the input
coupler and entire cavity, respectively, and
cav
is the detuning
between the cavity and the ion ensemble. For the ACSS
measurements
κ
in
=
2
π
×
45 GHz,
κ
=
2
π
×
118 GHz, and
cav
=−
2
π
×
27 GHz.
To calibrate the average Rabi frequency
̄
for a given
n
,
two-pulse photon echoes were used to observe Rabi flopping
of the ensemble. Figure
2
shows indicative data where the echo
intensity was monitored as a function of pulse intensity
n
and
rephasing pulse duration.
Sustained Rabi flopping is not observed for two reasons.
First, the inhomogeneity in the Rabi frequency dampens out
the coherent oscillations. Second, it is likely that at the higher
power levels, longer pulses lead to local heating in the cavity
leading to an increased rate of decoherence. This explains the
more definitive Rabi flopping curves observed for intensities
n
<
0
.
34.
For the ACSS measurements a Rabi frequency of
̄
=
2
π
×
25 MHz was chosen for the input and rephasing pulses,
FIG. 3. ACSS control of Nd
3
+
photon echo emission. The echo
intensity is plotted against the average ACSS Rabi frequency
̄
ac
in
(a), the ACSS duration
τ
ac
in (b), and the ACSS detuning
ac
in (c).
The insets in each subfigure 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 derived from
a two-dimensional (2D) Gaussian distribution of the cavity field is
shown by the dashed red curve.
which corresponded to
n
=
0
.
53 [a cw laser power at (i) of
6.5
μ
W].
B. ac Stark shift measurement
Figure
3
shows the normalized intensity of the photon echo
for sequences that vary (a) the average Rabi frequency of the
ACSS pulses
̄
ac
, (b) their duration, and (c) their relative
frequency detuning. The data highlight 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 (circles in
Fig.
3
), the ACSS-induced inhomogeneity cannot be rephased
and the echo is attenuated. The phase accumulated due to
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JOHN G. BARTHOLOMEW
et al.
PHYSICAL REVIEW A
97
, 063854 (2018)
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
3(a)
(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 because
of limitations in the timing resolution and intensity control
of the applied ACSS pulses in our experimental setup (see
Sec.
IIC
). Importantly, the restoration of the echo is evidence
that the attenuation is caused by the ACSS interaction rather
than by other dephasing processes such as instantaneous
spectral diffusion or device heating.
The echo intensity was simulated using the cavity Maxwell-
Bloch equations under the assumption of a uniform distribution
of ions within the mode profile of the cavity (see Appendices
B
and
C
). Figure
3
shows normalized 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 smaller
variation of the field envelope along the
y
axis in comparison
to the variation along the
x
and
z
axes [see Fig.
1(c)
]. 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-difference time
domain calculation (COMSOL), and the ions’ frequencies are
chosen from a Gaussian distribution with a FWHM equal to the
input pulse average Rabi frequency (
̄
=
2
π
×
25 MHz) [
36
].
The model was fitted to the data using one free parameter
R
:the
ratio of the maximum ACSS Rabi frequency
ac
(
r
)
max
to the
average ACSS Rabi frequency
̄
ac
. 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
. From the analysis in Sec.
IIA
, pulses with an intensity in the
cavity of
n
=
0
.
53 yielded
̄
=
2
π
×
25 MHz. Given that
g
max
=
̄
R
2
n
1
/
2
,
(4)
the values of
R
and
n
specified above result in a
g
max
=
2
π
×
31
.
4 MHz, which is consistent with previous observations
in this device [
21
]. The value of
g
max
determined from our
experiments also agrees well with the calculated value of
g
calc
=
μ
n
√
ω
0
2 ̄
hε
0
V
=
2
π
×
27
.
9MHz
,
(5)
where we have used the transition dipole moment
μ
=
8
.
48
×
10
−
32
Cm[
37
],therefractiveindex
n
=
2
.
1785(lightpolarized
parallel to the crystal
c
axis with wavelength of 879.9 nm)
[
38
], the simulated mode volume of the cavity
V
=
0
.
0564
μ
m
3
, the transition frequency
ω
0
, and the vacuum permittivity
ε
0
. Using the experimentally determined value of
g
max
,the
maximum possible single-photon ACSS in the cavity at a
FIG. 4. Two-pulse photon echo studies of the ACSS where the
pulses AC1 and AC2 were applied between the end of the input
(inversion) pulse and the beginning of the inversion pulse (echo).
The data is plotted using a natural logarithmic scale. For (a) AC1
and AC2 had detunings
ac
that had opposite sign, whereas in (b)
the detunings were equal. In both cases the black trace (full circles)
shows the photon echo decay in the absence of AC1 and AC2 and
the other traces represent the data for different values of
ac
.The
red dash-dotted curves are the exponential decay curves of the ACSS
measurements for delays
τ<
400 ns and
τ>
400 ns.
detuning
ac
/
2
π
=
160 MHz is
δ
ac
=
2
g
max
2
ac
=
2
π
×
12
.
3MHz
.
(6)
C. Investigation of incomplete echo recovery
To further investigate the incomplete echo recovery seen in
Fig.
3(a)
, a variation of the two-pulse photon echo sequence
was used. In this case, the perturbing ACSS pulse AC1 filled
the time between the input and inversion pulses, and AC2 filled
the time between the inversion pulse and the echo. The average
Rabi frequency used for the ACSS pulses was
̄
ac
=
2
π
×
6
.
5 MHz. The results of this measurement are shown in Fig.
4
along with insets showing the measurement pulse sequence.
Figure
4(a)
shows the data from measurements where the
sign of
ac
was reversed after the inversion pulse. The figure
shows that
δ
ac
increases as
ac
decreases, which results in
a more rapid echo decay. In Fig.
4(b)
, AC1 and AC2 both
possess the same
ac
. In an ideal implementation, the phase
perturbation due to the ACSS should be balanced and the
echo decay should match the case where no ACSS pulses
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are applied. In contrast, for delays
τ<
400 ns we observed
that the echo intensity decayed on a time scale similar to the
unbalancedcase[thedatapointsfrompart(a)aresuperimposed
in part (b)]. We attribute this rapid decay to the ACSS phase
perturbation remaining unbalanced on this time scale. This is
likely to be caused by a timing jitter due to imperfect alignment
of the two AOM double pass setups and intensity fluctuations
resulting from the sharp amplitude step functions in the radio
frequencies applied to the AOMs.
Fordelays
τ>
400ns,thedecaymorecloselyresemblesthe
unperturbed case. Fitting the echo decay for
τ>
400 ns to an
exponential decay (shown by the red dash-dotted curve) gives
an effective coherence time of 2
.
8
±
0
.
2
μ
s compared to 3
.
8
±
0
.
2
μ
s for the unperturbed photon echo measurement. The
difference between these decay rates will have contributions
due to the residual imbalance between the two ACSS pulses,
instantaneous spectral diffusion due to the excitation of the
residual ion population remaining in the spectral trenches, and
device heating.
III. DISCUSSION
A. All-optical hybrid photon echo rephasing memories
The results demonstrate that a single intracavity photon
can produce an ACSS
δ
ac
that is 100
×
larger than the Nd
3
+
homogeneous linewidth
h
=
84 kHz. Therefore, maintaining
n
0
.
01 during
τ
ac
allows all-optical control of the relative
phases of ions within an ensemble. 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 previously implemented
with electric field gradients [
30
]. 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 while the ensemble is
inverted, such as in the case when only a single inversion
pulse is used [
39
]. For HYPER to achieve high efficiency,
the intermediate echo resulting from the first inversion is
suppressed using an inhomogeneous phase perturbation, which
is later rephased to recover the HYPER echo [
40
]. An ideal
implementation of HYPER offers quantum storage using the
material’s full optical depth and natural bandwidth because no
preparation of the inhomogeneous line is necessary.
The photon echo measurements presented in Fig.
3
demon-
strate two of the important aspects for an all-optical HYPER
memory. The first is the suppression of the intermediate echo
using the controlled phase perturbation, and the second is
the balancing of that phase to allow the formation of the
HYPER echo [Fig.
3(a)
]. Both of these aspects are combined
in a proof-of-principle demonstration of the HYPER sequence
shown in Fig.
5
, where the secondary echo is enhanced when
the balanced ACSS pulses are applied. An all-optical version of
HYPER is promising for on-chip quantum memories because
additional electrodes on the REI substrate are not required,
the efficiency can theoretically approach unity in the limit
of an impedance matched cavity (see Appendix
B
), and
ac
can be increased so that the storage bandwidth can approach
FIG. 5. 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 Appendix
A
.
the inhomogeneous linewidth (GHz). A challenge for HYPER
memories is to simultaneously achieve high efficiency and
high fidelity. 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 offers a pathway to achieving
large bandwidth and efficient inversion [
41
], the resultant
instantaneous spectral diffusion may ultimately reduce the
maximum storage time of the memory [
42
].
B. Optically controlled atomic frequency comb memories
Other all-optical quantum protocols are possible in the
nanodevices because the
δ
ac
is large compared to the width
of absorption features prepared by optical pumping (
f
3
MHz [
21
,
31
]). As a result it is possible to create an AFC
memory with an ACSS-controlled readout. Importantly, the
ACSS allows control that is not possible using dc electric fields.
Although protocols that dynamically alter the comb profile
during storage using dc electric fields have been investigated
[
43
], 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 crys-
tals. In contrast, the spectral dependence of the ACSS makes it
possible to achieve a continuously tunable storage time (see
Appendix
D
). This was demonstrated in Ref. [
21
], where
ACSS pulses were applied frequency symmetrically at large
detunings about the AFC center. 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
Appendix
D
). To overcome this limitation requires control of
the spatial location of the ions within the cavity either through
spectroscopic selection or controlled placement [
21
].
For both the ACSS-controlled HYPER and AFC protocols,
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 desirable because the required
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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
,
44
]. For photons generated within the memory bandwidth,
a high memory efficiency ensures that these noise photons are
time separated from the signal photon through the protocol
storage (see Appendix
E
).
C. Quantum nondemolition measurements
In addition to offering opportunities for all-optical quantum
memories, a large single-photon ACSS can be harnessed for
quantum nondemolition measurements [
29
]. 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
transmittedthroughthecavityinatransparentwindowadjacent
to the memory. The phase shift of the retrieved probe pulse is
then a nondestructive measurement that heralds the presence
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 Ref. [
29
]. Ideally,
the phase shift would be increased further through longer
optical confinement in the cavity. Given the current quality
factor of the device (
Q
≈
2
.
9
×
10
3
), an order of magnitude
improvement should be possible with further optimization of
the cavity design and fabrication.
Although an accumulated phase shift can be increased by
improving the cavity
Q
, increasing the single-photon interac-
tion strength (the single ion
g
) requires a significantly lower
cavity mode volume or an optical transition with a larger dipole
moment. The photon-ion interaction strength achieved in our
nanocavities approaches the limit for what can be achieved
for REI quantum devices using a conventional photonic crys-
tal structure. This is because the optical dipole moment of
Nd
3
+
:YVO
4
is among the largest for 4
f
↔
4
f
transitions, and
the mode volume of the device studied in this work is within
a factor of 10 of the minimum mode volume for conventional
dielectric photonic crystal cavities
≈
(
λ/
2
n
)
3
[
45
].
D. Toward larger photon-REI interactions
A relevant next goal is to realize a single-photon ACSS that
is large compared to the ensemble inhomogeneous linewidth,
which is of the order of 100 MHz to 1 GHz. This would allow
a single photon to create a phase shift approaching 10
−
2
rad,
facilitating single-shot, nondestructive quantum measurement,
or to tune two ions into resonance with one another. To achieve
asingle-photonACSSofthisorderwouldrequireanincreasein
g
by a factor between 3 and 10 (the ACSS is proportional to
g
2
).
One effective strategy to pursue this goal would be to change
the cavity design to incorporate dielectric discontinuities [
46
–
48
], thereby reducing the mode volume. Previous work in the
design of such cavities indicates that a 100
×
reduction in the
mode volume is certainly feasible. A second strategy would
be to use the weaker 4
f
↔
4
f
transition for the storage of
photonic qubits and the parity allowed 4
f
↔
5
d
transitions to
perform the ACSS manipulation. The optical dipole moments
of the 4
f
↔
5
d
transitions are of the order of 50
×
larger
than the parity disallowed 4
f
↔
4
f
transitions, potentially
increasing the ac Stark interaction by a factor of over 10
3
.
Although doubly resonant photonic crystal cavity designs exist
[
49
], integrating two cavities, one of which is required to
operate close to the edge of the ultraviolet A band [
50
], would
be challenging. Currently, the most direct path toward such a
device would use the hybrid approach that confines the light
in a device layer bonded to an active REI substrate [
14
,
51
].
IV. CONCLUSION
In this paper we have demonstrated and characterized a
large single-photon ACSS of Nd
3
+
ions in a nanophotonic
resonator fabricated on a YVO
4
substrate. By combining the
relatively large optical dipole moment of the studied transition
with the high spatial mode confinement of the on-chip photonic
crystal cavity, it is possible to access a new regime for all-
optical control in REI crystals. Importantly, the ACSS due to
a single photon 160 MHz off-resonance was large compared
to both the homogeneous linewidth and spectral feature width
measured in this experiment configuration. Consequently, new
opportunities arise for using single photons to control quantum
information protocols in this class of materials. Given the re-
sults of this work and the avenues for increasing the strength of
the interaction, the ACSS is able to extend the versatility of an
already appealing physical system for photon-spin integration
at the quantum level.
ACKNOWLEDGMENTS
This work was funded by a National Science Foundation
(NSF)FacultyEarlyCareerDevelopmentProgram(CAREER)
award (1454607), the AFOSR Quantum Transduction Mul-
tidisciplinary University Research Initiative (FA9550-15-1-
002), and the Defense Advanced Research Projects Agency
Quiness program (W31P4Q-15-1-0012). Equipment funding
was also provided by the Institute of Quantum Information
and Matter, an NSF Physics Frontiers Center with support
from the Moore Foundation. We gratefully acknowledge crit-
ical support and infrastructure provided for this work by
the Kavli Nanoscience Institute at the California Institute of
Technology. J.G.B. acknowledges the support of the American
Australian Association’s Northrop Grumman Fellowship. The
authors also thank Jevon Longdell, Thierry Chanelière, Mikael
Afzelius, and Jean Etesse for useful comments and discussions
on the manuscript.
APPENDIX A: METHODS
1. Nd
3
+
:YVO
4
nanobeam photonic crystal cavity
The nanophotonic device used in these experiments was
the same cavity that was used for the work in Ref. [
21
]. For
completeness we repeat some of the details of the design
and fabrication in this section. The nanobeam is based on a
cross section of an equilateral triangle that has a width of
690 nm. First, a suspended triangular prism waveguide is
formed through angled focused ion beam milling. Grooves
are then milled onto the top surface of the waveguide to
form the photonic crystal cavity; each groove has a width of
147 nm. The groove structure in this device was asymmetric
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(a)
(b)
(v)
FIG. 6. Left: Generalized pulse sequence for an all-optical variant of the HYPER quantum memory protocol. The pulses at
t
3
and
t
9
are
assumed to be perfect inversion pulses. The ensemble emission
ε
out
in region VI is the two-pulse photon echo that the HYPER sequence seeks
to suppress. (a)–(c) give a perspective of how the optical Bloch vectors evolve in the equatorial plane of the Bloch sphere for the HYPER pulse
sequence under different ACSS conditions. In (a), no ACSS pulses are applied. The different colored vectors represent ensemble subgroups
that accumulate phase according to their detuning (shown by the solid black arrows). Part (b) shows the sequence applied in the main article
for characterizing the ACSS where only AC1 is applied. The result is an inhomogeneous phase evolution determined by the ACSS (colored
segments and dashed arrows). Although the inversion pulse rephases the fixed inhomogeneity, the ACSS-induced inhomogeneity remains and
the echo intensity is reduced (the Bloch vectors do not fully rephase). In (c), all three ACSS pulses are applied, with the combined effect of
AC1 and AC2 suppressing the intermediate echo in region VI, but balanced by the effect of AC3 to recover complete rephasing in region IX.
Right: Schematic of the one-sided cavity assumed in the derivation.
to create a one-sided cavity: 13 grooves on one side of the
cavity and 20 grooves on the other. The period of the grooves
is increased quadratically for seven grooves on each side of
the cavity mode center to form the mode in the photonic band
gap. The measured quality factor of the resonator was initially
≈
3
.
7
×
10
3
[
21
] but had degraded slightly when used for this
work (
Q
≈
2
.
9
×
10
3
).
2. Optical measurements
The pulse sequences used in this work were created by
amplitude modulating the output from a continuous-wave
Ti:sapphire laser (M Squared Solstis). The modulation was
achieved using a series of two AOMs (Isomet 1250c), both in
double pass configuration, and an intensity EOM (Jenoptik).
When operating with a pulse duration of 20 ns, the extinction
ratio of the total system is approximately 150 dB. The po-
larization of the light was controlled through a combination of
fiber and free space polarization optics, which ensured the light
entering the waveguide was polarization matched to the TM
mode. The substrate with the nanodevice was mounted onto a
three-axis nanopositioning stage (Attocube), which was loaded
into a
3
He dilution refrigerator (BlueFors). Light was coupled
into and out of the waveguide by focusing onto a 45
◦
surface
at the end of the waveguide. The APD (Perkin Elmer) was
protected from the bright excitation and rephasing pulses using
a third AOM as a shutter. The shutter provided an attenuation
of
≈
23 dB.
Before each of the photon echo and HYPER measurements,
transparent features were prepared at the ACSS frequency
through optical pumping. The optical pumping was performed
by exciting the ions at
ω
p
+
ac
with
>
1000
π
pulses sep-
arated by 20
μ
s. The laser was then gated off for a further
250
μ
s(
>
50
×
the excited-state lifetime in the cavity) to ensure
that there was no population remaining in the excited state.
The measurement sequence was then repeated 400 times at a
repetition rate of
≈
100 kHz. The combined optical pumping
and measurement sequence was repeated at a rate of 20 Hz.
Signals were integrated over 100 s.
ForthedatapresentedinFig.
5
,thephotoncountsarebinned
in 10 ns time bins. For the region showing the HYPER echo,
the background due to incoherent emission has been subtracted
from the data for clarity.
For the Rabi flopping data presented in Fig.
2
,8
×
10
3
two-
pulse echo sequences were performed per second in bursts of
400 separated by a wait time of 40 ms. This was repeated over
a duration of 20 s and the echo intensity detected on the APD.
The input pulse duration was constant (20 ns) and the time
separation between the centers of the input and second pulse
was maintained at 500 ns.
APPENDIX B: CAVITY MAXWELL-BLOCH EQUATIONS
FOR GENERALIZED ALL-OPTICAL HYPER SEQUENCE
The all-optical HYPER sequence is divided into nine
sections as shown in Fig.
6
. It is assumed that the laser pulse
sequence is incident on a perfectly one-sided optical cavity that
contains an ensemble of
N
rare-earth ions. Let the cavity decay
rate (field) be
κ
and the single-photon Rabi frequency to be
g
.
Each ion has a homogeneous linewidth 2
γ
and a detuning
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from the input pulse frequency (shown in orange in Fig.
6
).
The ion ensemble has an inhomogeneous distribution
ρ
(
)
with a FWHM
γ
ih
such that
∫
ρ
(
)
d
=
N
, with
N
the total
number of ions. Initially it is assumed that the cavity field is
equal at the spatial location of each ion. We will later relax
this assumption, which will give rise to the ability to control
the coherent emission from the cavity-coupled ensemble. The
uniform ac Stark shift resulting from ACSS pulses AC1, AC2,
and AC3 is denoted
δ
1
,
δ
2
, and
δ
3
, respectively.
The equations of motion for the coupled ion-cavity system
in each of the regions I–IX are written below, where the
ε
i
(
t
)
and
σ
i
(
t
) are the cavity field and the atomic polarizations (for
agiven
), respectively.
I:
̇
ε
I
=−
κε
I
+
√
2
κε
in
+
ig
∫
ρ
(
)
σ
I
d,
̇
σ
I
=−
iσ
I
−
γσ
I
+
igε
I
,
σ
I
(0)
=
0
,
II :
̇
σ
II
=−
i
(
+
δ
1
)
σ
II
−
γσ
II
,
σ
II
(
t
1
)
=
σ
I
(
t
1
)
,
III :
̇
σ
III
=−
iσ
III
−
γσ
III
,
σ
III
(
t
2
)
=
σ
II
(
t
2
)
,
IV :
̇
σ
IV
=−
iσ
IV
−
γσ
IV
,
σ
IV
(
t
3
)
=
σ
∗
III
(
t
3
)
,
V:
̇
σ
V
=−
i
(
+
δ
2
)
σ
V
−
γσ
V
,
σ
V
(
t
4
)
=
σ
IV
(
t
4
)
,
VI :
̇
ε
VI
=−
κε
VI
+
ig
∫
ρ
(
)
σ
VI
d,
̇
σ
VI
=−
iσ
VI
−
γσ
VI
−
igε
VI
,
σ
VI
(
t
5
)
=
σ
V
(
t
5
)
,
VII :
̇
σ
VII
=−
i
(
+
δ
3
)
σ
VII
−
γσ
VII
,
σ
VII
(
t
7
)
=
σ
VI
(
t
7
)
,
VIII :
̇
σ
VIII
=−
iσ
VIII
−
γσ
VIII
,
σ
VIII
(
t
8
)
=
σ
VII
(
t
8
)
,
IX :
̇
ε
IX
=−
κε
IX
+
ig
∫
ρ
(
)
σ
IX
d,
̇
σ
IX
=−
iσ
IX
−
γσ
IX
+
igε
IX
,
σ
IX
(
t
9
)
=
σ
∗
VIII
(
t
9
)
.
Region I
The general solution for the atomic polarization is
σ
I
(
t
)
=
e
(
−
γ
−
i
)
t
K
+
ige
(
−
γ
−
i
)
t
∫
t
−∞
e
(
γ
+
i
)
t
ε
I
(
t
)
dt
.
(B1)
The boundary condition gives
K
=
0. Therefore,
σ
I
(
t
)
=
ige
(
−
γ
−
i
)
t
∫
t
−∞
e
(
γ
+
i
)
t
ε
I
(
t
)
dt
.
(B2)
Substituting the solution into the equation of motion for the
cavity field gives
̇
ε
I
=−
κε
I
+
√
2
κε
in
+
ig
∫
∞
−∞
ρ
(
)
ige
(
−
γ
−
i
)
t
×
∫
t
−∞
e
(
γ
+
i
)
t
ε
I
(
t
)
dt
d,
=−
κε
I
+
√
2
κε
in
−
g
2
∫
t
−∞
e
−
γ
(
t
−
t
)
ε
I
(
t
)
×
∫
∞
−∞
ρ
(
)
e
−
i
(
t
−
t
)
ddt
,
=−
κε
I
+
√
2
κε
in
−
g
2
∫
t
−∞
e
−
γ
(
t
−
t
)
ε
I
(
t
) ̃
ρ
(
t
−
t
)
dt
,
≈−
κε
I
+
√
2
κε
in
−
Cκε
I
,
(B3)
wherewemaketheapproximationthat
γ
ih
ismuchlargerthan
γ
and the bandwidth of the input field. Under this approximation
the decoherence term
e
−
γ
(
t
−
t
)
is negligible on time scales less
than 1
/γ
, and ̃
ρ
(
t
−
t
)
≈
Nδ
(
t
−
t
)
/γ
ih
. We also introduce
the cooperativity
C
=
g
2
N/κγ
ih
.
Using the fact that
ε
out
=−
ε
in
+
√
2
κε
I
,
(B4)
the steady-state solutions of Eq. (
B3
) are then
ε
I
=
√
2
κ
κ
(1
+
C
)
ε
in
,
ε
out
=
(1
−
C
)
(1
+
C
)
ε
in
.
(B5)
Region II
Similar to the treatment of region I, the general solution for
the atomic polarization is
σ
II
(
t
)
=
e
[
−
γ
−
i
(
+
δ
1
)]
t
K,
(B6)
with the boundary condition defining the value of
K
.The
solution is then written
σ
II
(
t
)
=
ige
(
−
γ
−
i
)
t
e
−
iδ
1
(
t
−
t
1
)
∫
t
1
−∞
e
(
γ
+
i
)
t
ε
I
(
t
)
dt
.
(B7)
Region III
Again we have the general solution
σ
III
(
t
)
=
e
(
−
γ
−
i
)
t
K,
(B8)
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with the boundary condition fixing the solution as
σ
III
(
t
)
=
ige
(
−
γ
−
i
)
t
e
−
iδ
1
(
t
2
−
t
1
)
∫
t
1
−∞
e
(
γ
+
i
)
t
ε
I
(
t
)
dt
,
=
ige
(
−
γ
−
i
)
t
e
−
iδ
1
τ
1
∫
t
1
−∞
e
(
γ
+
i
)
t
ε
I
(
t
)
dt
,
(B9)
where
τ
1
is defined as (
t
2
−
t
1
), the length of pulse AC1.
Region IV
Using the same procedure, the solutions to regions IV and
V can be written down directly.
σ
IV
(
t
)
=−
ige
(
−
γ
−
i
)
t
e
iδ
1
τ
1
∫
t
1
−∞
e
(
γ
−
i
)
t
+
2
it
3
ε
∗
I
(
t
)
dt
.
(B10)
Region V
σ
V
(
t
)
=−
ige
(
−
γ
−
i
)
t
e
iδ
1
τ
1
e
−
iδ
2
(
t
−
t
4
)
×
∫
t
1
−∞
e
(
γ
−
i
)
t
+
2
it
3
ε
∗
I
(
t
)
dt
.
(B11)
Region VI
In region VI, there is the possibility for the ensemble
rephasing to generate emission. Therefore, the general solution
for the atomic polarization is
σ
VI
(
t
)
=
e
(
−
γ
−
i
)
t
K
−
ige
(
−
γ
−
i
)
t
∫
t
−∞
e
(
γ
+
i
)
t
ε
VI
(
t
)
dt
.
(B12)
In this region, the boundary condition gives
σ
VI
(
t
)
=−
ige
(
−
γ
−
i
)
t
e
iδ
1
τ
1
e
−
iδ
2
τ
2
∫
t
1
−∞
e
(
γ
−
i
)
t
+
2
it
3
ε
∗
I
(
t
)
dt
−
ige
(
−
γ
−
i
)
t
∫
t
t
5
e
(
γ
+
i
)
t
ε
VI
(
t
)
dt
,
(B13)
where
τ
2
is defined as (
t
5
−
t
4
), the length of pulse AC2.
Substituting the solution into the equation of motion for the
cavity field gives
̇
ε
VI
=−
κε
VI
+
ig
(
−
ig
∫
t
1
−∞
e
−
γ
(
t
−
t
)
e
iδ
1
τ
1
e
−
iδ
2
τ
2
ε
∗
I
(
t
)
×
∫
∞
−∞
ρ
(
)
e
−
i
(
t
+
t
−
2
t
3
)
ddt
−
ig
∫
t
t
5
e
−
γ
(
t
−
t
)
ε
VI
(
t
)
∫
∞
−∞
ρ
(
)
e
−
i
(
t
−
t
)
ddt
)
,
=−
κε
VI
+
g
2
(
∫
t
1
−∞
e
−
γ
(
t
−
t
)
e
iδ
1
τ
1
e
−
iδ
2
τ
2
ε
∗
I
(
t
)
×
̃
ρ
(
t
+
t
−
2
t
3
)
dt
∫
t
t
5
e
−
γ
(
t
−
t
)
ε
VI
(
t
) ̃
ρ
(
t
−
t
)
dt
)
,
≈−
κε
VI
+
2
Cκe
iδ
1
τ
1
e
−
iδ
2
τ
2
ε
∗
I
(2
t
3
−
t
)
+
Cκε
VI
(
t
)
.
(B14)
The steady state solutions of Eq. (
B14
)are
ε
VI
(
t
)
=
2
C
1
−
C
e
iδ
1
τ
1
e
−
iδ
2
τ
2
e
−
γt
ε
∗
I
(2
t
3
−
t
)
,
(B15)
ε
out
(
t
)
=
4
C
(1
−
C
2
)
e
iδ
1
τ
1
e
−
iδ
2
τ
2
e
−
γt
ε
∗
in
(2
t
3
−
t
)
.
Region VII
The same procedure as detailed in the preceding regions is
used to write down the solutions for regions VII–IX.
σ
VII
(
t
)
=−
ige
(
−
γ
−
i
)
t
e
−
iδ
3
(
t
−
t
7
)
e
iδ
1
τ
1
e
−
iδ
2
τ
2
×
∫
t
1
−∞
e
(
γ
−
i
)
t
+
2
it
3
ε
∗
I
(
t
)
dt
−
ige
(
−
γ
−
i
)
t
×
e
−
iδ
3
(
t
−
t
7
)
∫
t
7
t
5
e
(
γ
+
i
)
t
ε
VI
(
t
)
dt
.
(B16)
Region VIII
σ
VIII
(
t
)
=−
ige
(
−
γ
−
i
)
t
e
−
iδ
3
τ
3
e
iδ
1
τ
1
e
−
iδ
2
τ
2
×
∫
t
1
−∞
e
(
γ
−
i
)
t
+
2
it
3
ε
∗
I
(
t
)
dt
−
ige
(
−
γ
−
i
)
t
e
−
iδ
3
τ
3
∫
t
7
t
5
e
(
γ
+
i
)
t
ε
VI
(
t
)
dt
,
(B17)
where
τ
3
is defined as (
t
8
−
t
7
), the length of pulse AC3.
Region IX
σ
IX
(
t
)
=
ige
−
iδ
1
τ
1
e
iδ
2
τ
2
e
−
iδ
3
τ
3
×
∫
t
1
−∞
e
−
γ
(
t
−
t
)
e
−
i
(
t
+
2
t
3
−
2
t
9
−
t
)
ε
I
(
t
)
dt
+
ige
−
iδ
3
τ
3
∫
t
7
t
5
e
−
γ
(
t
−
t
)
e
−
i
(
t
+
t
−
2
t
9
)
ε
∗
VI
(
t
)
dt
+
ig
∫
t
t
9
e
−
γ
(
t
−
t
)
e
−
i
(
t
−
t
)
ε
IX
(
t
)
dt
.
(B18)
Substituting the solution into the equation of motion for the
cavity field gives
̇
ε
IX
(
t
)
=−
κε
IX
(
t
)
−
2
Cκe
−
γt
e
−
i
(
δ
1
τ
1
−
δ
2
τ
2
+
δ
3
τ
3
)
×
ε
I
(
t
+
2
t
3
−
2
t
9
)
−
2
Cκe
−
γt
×
e
−
iδ
3
τ
3
ε
∗
VI
(
t
−
2
t
9
)
−
Cκε
IX
(
t
)
.
(B19)
Under the assumption that all ions are driven by the same
field (there is no inhomogeneity in
δ
1
,
δ
2
,or
δ
3
), the term
e
−
i
(
δ
1
τ
1
−
δ
2
τ
2
+
δ
3
τ
3
)
is simply a phase shift of the emitted HYPER
echo. In the next section we will discuss how the inhomogene-
ity of the cavity field impacts this behavior.
It is also worthwhile to write down the specific solution
that represents an efficient HYPER protocol. That is, when
δ
1
τ
1
−
δ
2
τ
2
+
δ
3
τ
3
=
0 and
ε
VI
is suppressed, the steady-state
063854-9
JOHN G. BARTHOLOMEW
et al.
PHYSICAL REVIEW A
97
, 063854 (2018)
solutions of Eq. (
B19
) around
t
=
2(
t
9
−
t
3
)are
ε
IX
(
t
)
=−
2
C
1
+
C
e
−
γt
ε
I
(0)
,
(B20)
ε
out
(
t
)
=−
4
C
(1
+
C
)
2
e
−
γt
ε
in
(0)
.
Spatial inhomogeneity
In this section we consider the impact of the field inho-
mogeneity within the optical cavity. This inhomogeneity can
be used to suppress
ε
VI
to achieve an efficient HYPER echo.
The photon-ion interaction is represented by the coupling
parameter
g
(
r
):
g
(
r
)
=
μ
n
√
ω
2 ̄
h
0
V
E
(
r
)
max[
E
(
r
)]
=
g
0
ψ
(
r
)
,
(B21)
where
μ
is the optical transition dipole moment,
n
is the
refractive index of the material,
ω
is the transition frequency
of the ion (assumed to be in resonance with the cavity),
0
is
the vacuum permittivity,
V
is the mode volume of the cavity,
and
ψ
(
r
) is the function describing the field amplitude as a
function of position. Equation (
B21
)alsoassumesthatthe
dipole moment is aligned with the electric field
E
in the cavity.
The Rabi frequency can then be expressed as
(
r
)
=
2
√
n
p
g
(
r
)
,
(B22)
where
n
p
is the average number of photons in the cavity.
The equations of motion for region I must be rewritten to
incorporate the spatial inhomogeneity of the electric field
I:
̇
ε
I
=−
κε
I
+
√
2
κε
in
+
i
V
∫
V
(
g
(
r
)
∫
ρ
(
)
σ
I
d
)
d
r
,
̇
σ
I
=−
iσ
I
−
γσ
I
+
ig
(
r
)
ε
I
,
σ
I
(0)
=
0
.
Here the
∫
V
represents integration over the cavity mode of
volume
V
.
The solution in region I has the same form as Eq. (
B5
)
ε
out
=
(1
−
C
)
(1
+
C
)
ε
∗
in
.
(B23)
When
ac
γ
ih
,
δ
1
and
δ
2
only depend on an ion position
r
and not on their frequency
. The modified solution for the
case of spatial inhomogeneity in region VI is
ε
out
(
t
)
=
4
C
(1
−
C
2
)
1
V
∫
V
ψ
2
(
r
)
×
e
iδ
1
(
r
)
τ
1
e
−
iδ
2
(
r
)
τ
2
d
r
e
−
γt
ε
∗
in
(2
t
3
−
t
)
.
(B24)
Therefore, the amplitude of the first output echo is proportional
to
η
=
∫
V
ψ
2
(
r
)
e
iδ
1
(
r
)
τ
1
e
−
iδ
2
(
r
)
τ
2
d
r
.
(B25)
Because there is no analytical expression for the cavity
mode profile
ψ
(
r
) for the nanocavity used in this work, we
calculate the dependence of the echo intensity numerically
from the simulated mode profile. In addition, below we use
an approximation of
ψ
(
r
) to form an analytically tractable
solution.
The mode envelope of the photonic nanocavity can be
approximated by a two-dimensional Gaussian
ψ
(
x
,
y
,
z
)
≈
e
−
x
2
2
x
2
−
z
2
2
z
2
.
(B26)
Transforming to polar coordinates allows
ψ
(
x
,
y
,
z
)tobe
written as
ψ
(
r
)
≈
e
−
r
2
.
(B27)
In the case where only AC1 is applied and
ac
,
Eq. (
B25
) becomes
η
≈
∫
∞
0
re
−
2
r
2
e
−
iAe
−
2
r
2
dr,
(B28)
where
A
=
2
n
g
0
2
τ
1
ac
,
(B29)
which is the maximum value of the ACSS-induced phase shift
in the cavity max(
φ
ac
).
Therefore, the output echo field will be proportional to
η
∝
[
e
−
i
max(
φ
ac
)
−
1
i
max(
φ
ac
)
]
,
(B30)
and the output echo intensity will be proportional to
|
η
|
2
∝
sinc
2
[
max(
φ
ac
)
2
]
.
(B31)
APPENDIX C: EFFECT OF ENSEMBLE AVERAGING
IN ACSS MEASUREMENTS
For the data shown in Fig.
3
, each photon echo sequence
was accumulated for 100 s and the repetition rate of the
measurement sequence was 20 Hz. Thus, the signal represents
an average of 8
×
10
5
individual photon echoes. Each photon
echo sequence will probe a different ensemble of ions within
the device. This is due to the combination of the slow laser drift
of the order of 10 MHz/h, the probabilistic excitation of ions
within the the input pulse bandwidth, the imperfect inversion
of the applied
π
pulses, and the optical pumping resulting from
optical relaxation to the other Zeeman ground state. As a result,
the experimental data shown in Fig.
3
(and repeated here in
Fig.
7
) represent the average of many ensembles, all of which
have different frequency and spatial distributions.
Also included in Fig.
7
is the predicted signal for the
pulse sequences used in this work if only a single ensemble
is probed. The most important impact of using a single
ensemble is that the echo intensity can be attenuated to
zero, which is important for applications such as the HYPER
memory protocol. In the case where the signal is the result
of averaging over many ensembles, the echo intensity only
approaches zero in the limit of very large ACSS-induced phase
shifts.
Experimentally, there are several ways in which to move
toward using only a single ensemble. First, eliminating long-
063854-10