ARTICLE
Received 15 Sep 2016
|
Accepted 29 Nov 2016
|
Published 16 Jan 2017
Interfacing broadband photonic qubits to on-chip
cavity-protected rare-earth ensembles
Tian Zhong
1
, Jonathan M. Kindem
1
, Jake Rochman
1
& Andrei Faraon
1
Ensembles of solid-state optical emitters enable broadband quantum storage and transduc-
tion of photonic qubits, with applications in high-rate quantum networks for secure
communications and interconnecting future quantum computers. To transfer quantum states
using ensembles, rephasing techniques are used to mitigate fast decoherence resulting from
inhomogeneous broadening, but these techniques generally limit the bandwidth, efficiency
and active times of the quantum interface. Here, we use a dense ensemble of neodymium
rare-earth ions strongly coupled to a nanophotonic resonator to demonstrate a significant
cavity protection effect at the single-photon level—a technique to suppress ensemble
decoherence due to inhomogeneous broadening. The protected Rabi oscillations between the
cavity field and the atomic super-radiant state enable ultra-fast transfer of photonic frequency
qubits to the ions (
B
50 GHz bandwidth) followed by retrieval with 98.7% fidelity. With the
prospect of coupling to other long-lived rare-earth spin states, this technique opens the
possibilities for broadband, always-ready quantum memories and fast optical-to-microwave
transducers.
DOI: 10.1038/ncomms14107
OPEN
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, 1200 E California Boulevard, Pasadena, California 91125, USA. Corr
espondence
and requests for materials should be addressed to A.F. (email: faraon@caltech.edu).
NATURE COMMUNICATIONS
| 8:14107 | DOI: 10.1038/ncomms14107 | www.nature.com/naturecommunications
1
E
nsembles of rare-earth ions doped in crystals exhibit
outstanding quantum coherence properties and large
inhomogeneous linewidths
1
that are suitable for quantum
information transfer with broadband photons in high-speed
optical quantum communication networks
2–4
. They are used in
state-of-the-art optical quantum memories with potential for
microwave storage
5–11
and are promising candidates for optical-
to-microwave quantum transduction
12,13
. One major challenge
towards broadband quantum interfaces based on solid-state
emitters is that information stored in the collective excitation of
the ensemble quickly decoheres due to inhomogeneous broade-
ning. To restore the optical coherence, protocols based on spectral
hole burning techniques like atomic frequency comb (AFC)
7,8
and controlled reversible inhomogeneous broadening
9
have
been perfected. Although effective, these protocols involve
long (hundreds of milliseconds) and complex preparation
procedures that generally limit the interface bandwidth.
Recently, it was proposed
14,15
that ensemble decoherence can
be suppressed via strong coupling to a cavity. This phenomenon,
called cavity protection, has been experimentally observed,
though not in full effect, in the microwave domain with a
nitrogen vacancy spin ensemble
16
. Prior experiments reported
polariton linewidth narrowing in the quantum well structures
17
,
but it is not yet conclusive that those effects were results of cavity
protection.
Here, we demonstrate strong cavity protection against
decoherence in the optical domain using a dense ensemble
(a few millions) of neodymium (Nd) atoms coupled to a
nanophotonic cavity. Exploiting the protected mapping of
photonic qubits to atomic super-radiant excitations, we realize
an efficient quantum light-matter interface with
B
50 GHz
bandwidth that could find applications in future quantum
networks.
Results
Conditions for cavity protection
. The dynamics of
a coupled cavity-ensemble system are described by the
Tavis–Cummings Hamiltonian
18
. The interaction term reads
H
int
¼
i
‘
O
(
S
a
w
S
þ
a
) where
a
w
and
a
are creation and
annihilation operators of the cavity mode, and the collective spin
operators
S
¼
1
ffiffiffi
N
p
P
s
k
act on
N
atoms each of frequency
o
k
.
O
denotes a collective coupling strength
O
2
¼
P
N
k
¼
1
g
2
k
,whichscales
up the single atom coupling
g
k
by
ffiffiffiffi
N
p
.Onresonance,thecoupled
system exhibits two bright polariton states with equal mix of atomic
and photonic components detuned by
±
O
from the mean
ensemble frequency. The polaritons decay via radiative emission
and decohere by coupling to dark subradiant states that overlap
spectrally with the ensemble
14,15,19
. The dark-state coupling
critically depends on the energy separation between the
polaritons and the subradiant states, and also on the
specific profile of the inhomogeneous spectral distribution
ro
ðÞ¼
P
k
g
2
k
do
o
k
ðÞ
=
O
2
(refs 14,15). In the limiting case of a
Lorentzian distribution, considerable damping given by the width
of the inhomogeneous broadening persists even with an infinite
O
.
When the spectral distribution exhibits a faster-than-Lorentzian
decay (for example, Gaussian), the damping of the coherent Rabi
oscillation is diminished at increasing
O
—the system becomes
‘cavity protected’ as conceptually illustrated in Fig. 1a. In this case,
the atomic component of the polariton is purely the symmetric
super-radiant state
20
.
Observation of cavity protection in Nd ensembles
. We probe
the cavity protection regime in optical nanocavities based on our
triangular beam design
21,22
fabricated in Nd-doped yttrium
vanadate (YVO) crystals (United crystals; Fig. 1e). The cavities
have fundamental TM mode resonances with measured
quality factor Q of 7,700 (
k
B
2
p
44 GHz is the energy decay
rate, that is, full-width at half-maximum (FWHM) is
44 GHz) and 17,000 (
k
B
2
p
20 GHz, FWHM
¼
20 GHz) in
0.1 and 1% Nd:YVO, respectively. A simulated mode volume
V
mode
¼
1(
l
/
n
)
3
¼
0.063
m
m
3
estimates
N
B
10
6
(10
7
) ions in the
0.1% (1%) cavity. The resonance wavelengths are close to the
4
I
9/2
(
Y
1
)
4
F
3/2
(
Z
1
) transition of Nd
3
þ
at 879.7 nm. The devices
were cooled down to 3.6 K (Montana Instruments Cryostation)
and a magnetic field of 500 mT was applied perpendicular to the
YVO
c
-axis. In 0.1% Nd:YVO, the optical coherence time is
T
2
¼
390 ns (measured via photon echoes), corresponding to a
single emitter homogeneous linewidth
g
h
/2
p
¼
1/
p
T
2
¼
0.82 MHz.
In 1% Nd:YVO, we measured an upper bound of
g
h
/2
p
r
40 MHz
via transient hole burning. The B field caused a Zeeman splitting
of the
Y
1
,
Z
1
states into four levels (Fig.1b). For this field
orientation, cross-transition probabilities are minimized
23
. There-
fore the system can be viewed as two independent distributions of
emitters separated by 17 GHz (shown as resolved absorption
lines in 0.1% but not in 1% device) both coupled to the
cavity with similar strengths. To capture the spectral shape
of the distribution, a
q
-Gaussian function was used to fit each
transition
16
, yielding a shape parameter
q
¼
1.01 (1 for Gaussian,
2 for Lorentzian) for the 0.1% ensemble. Each Zeeman branch has
a FWHM of
g
q
¼
2
D
ffiffiffiffiffiffiffiffiffi
2
q
2
2
q
2
q
¼
2
p
8.3 GHz with
D
/2
p
¼
5.0 GHz
(
D
/
p
represents the FWHM for a Lorentzian distribution). The
total FWHM of the ensemble including both branches is 24 GHz.
The 1% ensemble exhibits an asymmetric distribution with
76 GHz FWHM. However, it cannot be fitted well with any
common functions because at this concentration the ions exhibit
various interactions between themselves and with crystalline
defects that lead to satellite lines
24
.
To achieve strong coupling, the cavity resonance was tuned
towards longer wavelengths by gas condensation
21
while the
transmission from a broadband superluminescent input was
recorded in Fig. 2a,d using a spectrometer (Supplementary
Movies 1 and 2). The on-resonance spectra (Fig. 2b,e) reveal two
bright polariton peaks with a Rabi splitting of
O
R
/2
p
¼
110 GHz
and 48 GHz, for the 1 and 0.1% device, respectively. In Fig. 2e, a
middle peak is present in between the polaritons because the cavity
coupled simultaneously to two Zeeman branches with a resolved
splitting (Supplementary Notes 1 and 2). The decay rates
G
(
d
) were
determined from the FWHM linewidth of the left polariton peak
and are plotted against the cavity-ensemble detuning
d
¼
o
c
o
a
in Fig. 2c,f as black triangles. The data corresponds to the left anti-
crossing trajectory in Fig. 2a,d as the cavity shifted from shorter
wavelengths towards the atomic resonance.
The phenomenon of cavity protection can be observed in
both concentration samples, as the on-resonance
G
is
considerably narrower than the Lorentzian (no protection) limit
k
/2
þ
g
h
/2
þ
D
. In the 1% sample,
G
(0)/2
p
¼
21 GHz is 35 GHz
narrower than the Lorentzian limit and also narrower than the
FWHM of the initial inhomogeneous broadening of the
ensemble, thus indicating that the polariton decay is slower than
the decoherence of the initial ensemble. We point out that in
Fig. 2c (1% Nd:YVO),
G
/2
p
slightly increases around
d
¼
50 GHz
before decreasing to a minimum of 21 GHz on resonance. That
increase might be explained by coupling to one of the Nd–Nd
pair site that is known to be blue-detuned from the central
transition by 48 GHz (ref. 24; Supplementary Note 3). The data
for the 1% sample is not compared with a theoretical model
because, as mentioned above, the exact distribution of ions is
unknown. For the 0.1% sample shown in Fig. 2f,
G
(0)/2
p
¼
22 GHz and the data shows good agreement with the theoretical
decay (red curve) for a Gaussian-distributed ensemble with the
ARTICLE
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NATURE COMMUNICATIONS
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same FWHM as the joint distribution of the two Zeeman
branches (Supplementary Note 2). In the on resonance, strong
coupling limit, the theoretical decay is expressed as
G
¼
k
/2
þ
g
h
þ
p
O
2
r
(
O
) (ref. 15), which reaches the full protection limit of
G
¼
k
/2
þ
g
h
¼
2
p
22 GHz as indicated in Fig. 2f. In our case,
the experimental data approached this limit. The residual
broadening estimated from the
p
O
2
r
(
O
) term was
E
0.1 GHz,
more than two orders of magnitude suppressed compared with
the case of no protection (where the residual broadening would
be 14.6 GHz). Although close to fully protected, the total decay
rate was not much slower than the initial ensemble decoherence
(FWHM 24 GHz (
D
/2
p
¼
14.6 GHz) by treating the two Zeeman
branches as one joint distribution). To contrast with the case of
no protection, we also plot in green the theoretical decays of
upper and lower polaritons for a Lorentzian distribution
(Supplementary Note 1) assuming the same
D
as for our
ensembles. In this case, the atom- and cavity-like polariton
widths converge to the Lorentzian limit at zero detuning.
Time-domain measurement of extended Rabi oscillations
. The
cavity-protected system acts as a quantum interface where
a broadband photon can be transferred to a super-radiant atomic
excitation. We measured these coherent, ultra-fast dynamics
using pulsed excitations of the polaritons. The experimental
set-up is depicted in Fig. 1d. A mode-locked Ti:sapphire laser
with a 85 MHz repetition rate (Thorlabs Octavius) was filtered to
a pulse width of 4(1.5) ps using a monochromator, which was
sufficient to simultaneously excite both upper (denoted by |
o
þ
i
,
and referred to as |1
i
thereafter) and lower (|
o
i
, also referred to
as |0
i
thereafter) polaritions in 0.1% (1%) device. The filtered
laser was attenuated and sent through a Michelson interferometer
to produce two pulses with less-than-one mean photon number
separated by a variable delay
t
. These pulses were coupled into
the cavity (red path) and the transmitted signal was collected
(blue path) for direct detection using a silicon single-photon
counter. The integrated counts at varying delays produce optical
field autocorrelation signals revealing the temporal evolution
of the polaritons. The mirror at each Michelson arm was
interchangeable with a Gires–Tournois Interferometer (GTI)
etalon, which generates a
B
p
/2 phase chirp between the two
polaritons (‘Methods’ section). Furthermore, a narrow bandpass
filter was optionally inserted in either arm that allowed only one
polariton to be excited. This combination enabled a compre-
hensive polariton excitation scheme that covered individual
polariton |0
i
or |1
i
, and superposition states of two polaritons
that is, |
þi¼
1/
ffiffiffi
2
p
(|0
iþ
|1
i
)or
T
ji¼
1
=
ffiffiffi
2
p
0
ji
þ
i
1
ji
ðÞ
.
Figure 3 plots the theoretical interference fringe amplitudes
along with the measured results for several two-pulse excitation
schemes for the 0.1% cavity in which maximum protection was
y
z
x
z
x
y
Nd:YVO
B
B
Mirror
GTI etalon
a
b
d
4
I
9/2
(Z
1
)
4
F
3/2
(Y
1
)
+3/2
–3/2
879.7 nm
+1/2
–1/2
–20
(GHz)
–40
0 20 40
Spectral density
ρ
(a.u.)
0
1
c
TM
|
E
z
|
τ
Without protection
With protection
c
/2
Si APD
Mirror
e
–50 0 50
e
0.1% Nd:YVO
1% Nd:YVO
Polaritons
Lorentzian ensemble
cavity
Gaussian ensemble
2
Δ
2
Δ
(
+
γ
h
)/2+
Δ
(
+
γ
h
)/2
0
1
|
E
z
|
2
1
Figure 1 | Schematics of the cavity protection effect.
(
a
) Conceptual illustration of cavity protection for an ensemble coupled to a cavity mode. For a
Lorentzian ensemble (upper), the polaritons are not protected and undergo dephasing (linewidth broadening) due to inhomogeneous broadening
D
.
A Gaussian ensemble (lower) can be fully protected with the collective super-radiant excitation free of such dephasing, and the polariton linewidth
s do not
depend on
D
. Arrows represent the phasor of each atomic dipole. (
b
) Energy levels and transitions (dotted lines are forbidden) for Nd (left). Measured
absorption spectra for 0.1 and 1% Nd:YVO (right). Two Zeeman split sub-ensembles are resolved in the 0.1% sample. (
c
) Simulated TM resonance mode
profiles of the triangular nanobeam resonator. (
d
) Experimental set-up. Two pico-second pulses were transmitted through the cavity and the output signal
was integrated on a Si APD photon counter. (
e
) Scanning electron microscope image of the device and schematics of input and output optical coupling.
Scale bar, 1
m
m.
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14107
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3
observed. The results for the 1% device are presented in
Supplementary Fig. 3 and Supplementary Note 6. The mean
photon number per pulse coupled into the cavity was estimated to
be
m
¼
0.5. The case of an uncoupled cavity is plotted in Fig. 3a,
showing a fitted decay constant (4/
k
)of
B
14.5 ps (Supple-
mentary Note 5). When only one polariton was excited, the decay
was extended to 29.0 ps (Fig. 3b). For the superposition state |
þi
,
Ramsey-like fringes were obtained, revealing extended Rabi
oscillations between photonic and atomic excitations beyond
the cavity lifetime (Fig. 3c). In the case of Fig. 3d, the first pulse
excited the two polaritons with a phase chirp. The resulting fringe
showed the Rabi oscillations with the nodes shifted with respect
to 3c by about 5.5 ps (
1
=
4
t
R
), in agreement with the theoretical
model (Supplementary Note 4). Those nodes correspond to the
quantum excitation being entirely stored in the ensemble with no
energy left in the cavity mode, during which time the stored qubit
dephases at a significantly slower rate than the inhomogeneous
broadening.
Quantum state transfer with protected Nd ensembles
. This
quantum interface is similar to an AFC with two teeth, one at
each polariton, that form the basis of a frequency bin qubit as
shown in Fig. 4a. Photons are stored and then released after
inverse of the teeth spacing, which is a Rabi period
t
R
. The
interface bandwidth is
B
50 GHz, spanning two polaritons, and
the qubits are of the form |0
i
,|
þi¼
1/
ffiffiffi
2
p
(|0
iþ
|1
i
)or
T
ji¼
1
=
ffiffiffi
2
p
0
jiþ
i
1
ji
ðÞ
. To characterize this process, quantum
state tomography on the released qubit after a delay
t
R
was per-
formed. As direct projection measurements were difficult given the
high-bandwidth, we adopted an interferometric scheme (Fig. 4b) to
assess the input/output fidelity
F
¼
c
in
r
out
jj
c
in
hi
, where
c
in
ji
is the
input qubit state and
r
out
is the density matrix for the retrieved
state, from a set of fringe signals including those in Fig. 3
(‘Methods’ section and Supplementary Note 7). The reconstructed
density matrices
r
out
for |0
i
,|
þi
,and
T
ji
input states along with
their respective fidelities are shown in Fig. 4c. A mean fidelity of
98.7
±
0.3% is obtained, which significantly surpasses the classical
fidelity limit (the best qubit input/output fidelity one can achieve
using a classical intersect-resend strategy
25
(Supplementary Note
8)) of 74.9
±
0.04% that takes into account the Poissonian statistics
of the coherent input photons (with
m
¼
0.5) and an imperfect but
high storage-retrieval efficiency of 25.6
±
1.2% (‘Methods’
section)
26,27
. The estimated fidelities take into account imper-
fections in the preparation and measurement of the qubit,
such as leakage of travelling waves through the cavity and
inaccurate phase shift (ideally
p
/2) by the GTI etalon. The high
fidelity indicates a robust quantum transfer with a bandwidth
that is significantly broader than existing rare-earth-based
light-matter interfaces, with the highest bandwidth demonstrated
so far of 8 GHz in a Erbium doped fibre
28
, and 5 GHz
reported in a Thulium doped LiNbO
3
waveguide
29
. To highlight
879.3
879.6
879.9
880.2
Scan time (a.u.)
60
80
100
120
140
160
180
200
220
240
Optical intensity
879.4
879.6
879.8
880
Scan time (a.u.)
40
60
80
100
120
140
160
180
200
220
0
500
1,000
1,500
2,000
abc
0
1,000
2,000
3,000
4,000
5,000
(nm)
(nm)
879.1
879.6
880.1
Transmission (a.u.)
0
1
0
GHz
100
–100
Cavity detuning (GHz)
/2
(GHz)
de
(nm)
(nm)
879.5
879.7
879.9
Transmission (a.u.)
0
1
GHz
–50
0
5
0
–100
100
1% Nd:YVO
0.1% Nd:YVO
/2+
h
/2+
Δ
/2+
h
/2+
Δ
/2+
h
/2
2
Δ
+
γ
h
f
Cavity detuning (GHz)
80
60
40
20
0
20
25
30
35
40
45
50
2
Δ
+
h
Atom-like
Cavity-like
Atom-like
Cavity-like
250
200
150
100
50
0
20
30
40
50
60
70
80
90
100
Optical intensity
/2
π
(GHz)
Figure 2 | Cavity protection of Nd ensembles against decoherence.
(
a
,
d
) Cavity transmission spectra while tuning its resonance across the
inhomogeneous Nd transition. (
b
,
e
) On-resonance transmission spectra showing two bright polaritons. Red curves are the theoretical fit assuming
Gaussian ensembles. (
c
,
f
) Experimental decay rates extracted from the left anti-crossing trajectory (dotted arrows) in
a
and
d
, respectively, as a function of
detuning. In
c
(1% Nd:YVO), the polariton (21 GHz) is significantly narrower than the initial inhomogneous broadening (76 GHz), but it is not reaching the
full protection limit likely due to an asymmetric, non-Gaussian ensemble shape. In
f
(0.1%), the polariton linewidth decreases rapidly towards resonance to
the full protection limit (
k
/2
þ
g
h
/2). Red curve plots the theoretial decay for a Gaussian distribution. Green curves show the theoretical decays assuming a
Lorentzian ensemble of the same
D
.
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the benefit of cavity protection, we also evaluated the qubit
fidelity at a delay of 2
t
R
, which would be equivalent to the case
without cavity protection where the qubit would decohere
twice as fast. The measured fidelities at 2
t
R
dropped to 83%
for |0
i
,70%for|
þi
and 69% for
T
ji
, which no longer beats the
classical limit.
Count (a.u.)
0
1
Count (a.u.)
0
1
(ps)
(ps)
Counts (a.u.)
0
1
Counts (a.u.)
0
1
20
40
60
80
20
40
60
80
a
b
c
d
Theory
Experiment
R
2
R
A
0
/2
Single polariton
Two polaritons
Two polaritons
one phase chirped
Detuned cavity
=14.5 ps
=29.0 ps
12
Figure 3 | Time-domain interferometric transmission measurements.
Left panels show the cavity transmission spectra (black outlines) of the probed
system. Coloured areas show the spectral range addressed by the probe pulses (left is first pulse, right is second pulse). Grey is for uncoupled cavity;
red
for excited polariton states: yellow for the polarition with a shifted phase. (
a
) Simulated and measured cavity decay (that is, lifetime) when uncoupled from
the atoms. (
b
) Decay of singly excited lower (upper) polarition (that is, |0
i
(|1
i
)). (
c
) Decayed oscillations when both polaritons were excited with
transform-limited pulses. (
d
) Decayed and time-shifted oscillations when two polaritons were initially excited with a
p
/2 phase difference (that is,
|0
iþ
i
|1
i
) by the first pulse and in phase by the second pulse. (
b
–
d
) show extended decays times that are about twice that in
a
, confirming a nearly
full protection against ensemble-induced decoherence. The dotted lines mark multiples of Rabi periods
t
R
. The inset shows a few fine fringes scanned
around
t
R
.
ac
|1
〉
|0
〉
|0
〉
|0
〉
|1
〉
0
1
–1
Re
|1
〉
|0
〉
|0
〉
|1
〉
1
0
–1
Im
|1
〉
|0
〉
|+
〉
|0
〉
|1
〉
1
0
–1
|1
〉
|0
〉
|0
〉
|1
〉
–1
0
1
|1
〉
|0
〉
|
〉
|0
〉
|1
〉
1
0
–1
|1
〉
|0
〉
|0
〉
|1
〉
1
0
–1
F
=97.9%
F
=99.7%
F
=98.6%
|0
〉
|1
〉
b
t
|
E
|
2
Figure 4 | Broadband qubit transfer with protected Nd ensembles.
(
a
) Two polaritons serve as eigenbasis for a frequency bin qubit. A phase chirp can be
added to construct a generic qubit 0
ji
þ
e
i
f
1
ji
.(
b
) Simulated time-domain evolution of the cavity field intensities of two qubit-encoded photons, showing
temporal overlap between the retrieved photon
c
1
ji
ðÞ
and the second photon (
c
2
ji
ðÞ
) directly transmitted through the cavity at time
t
R
. The fields
dominantly overlap within a temporal window (grey area) when interference occurs yielding integrated photon counts proportional to the overlap
c
2
c
1
j
hi
jj
.
The overlap after the grey window was negligibly small due to fast cavity decays. (
c
) Reconstructed density matrices for each input qubit (top row) delayed
by
t
R
showing high fidelities.
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5
Discussion
While this interface efficiently maps the photonic qubit to the
ensemble, the qubit dissipates at a rate of
k
/2. Improvements in
the cavity quality factor to state of the art values of
Q
B
10
6
would
achieve storage for 1 ns (enough for perform 50 Rabi flips). The
interface with very narrow polariton lindwidths could still map a
broadband photon to the ensemble though at the expense of
efficiency. To enable long-term storage like in an AFC-spin-wave
memory
7
, the qubit can be transferred from the super-radiant
state excitation to a long-lived spin level by applying a
p
pulse
within
t
R
time. Upon recall, another
p
pulse can transfer the
qubit back to the polariton states and then a cavity photon. For
faithful spin-wave storage, further spectroscopic studies are
needed to verify the spin coherence time in the rare-earth
ensembles. Also, the Rabi frequency of the driving pulses should
exceed the polariton linewidths, which is attainable given the
strong light confinement in current nanobeam devices. Compared
with existing AFC-spin-wave memories, this interface would not
require any preparation steps or time-multiplexing to achieve
always-ready operation. Taking advantage of on-chip platforms
also allow spatially and temporally multiplexed storage by routing
photons to an array of nanocavities with different delays. Most
notably, the cavity-protected mapping of a photonic qubit to a
collective super-radiant state could compliment the reported
coupling of rare-earths to a superconducting resonator
30
to fulfil
efficient quantum transduction between optical and microwave
photons via Zeeman or hyperfine transitions in rare-earth
ensembles
12,13
.
Methods
Nanocavity design and characterization
.
The triangular nanobeam has a width
of 770 nm and length of 15
m
m. Forty periodic subwavelength grooves of 185 nm
along the beam axis were milled on top of the nanobeam. The period of the grooves
were modulated at the center of the beam to form defect modes in the photonic
bandgap. The fundamental TM mode, with side, top and cross-section views shown
in Fig. 1c, is chosen because it aligns with the strongest dipole of the 879.7 nm
transition in Nd:YVO. The theoretical quality factor is 300,000 with a mode
volume of 1(
l
/n)
3
(ref. 22). Transmission of the nanocavity was measured by
vertically coupling free-space input into the nanobeam via a 50
objective lens
and a 45
°
-angled reflector milled into the sample surface, and the cavity output was
collected via the other reflector which sent the light back vertically to free space.
The output signal was effectively isolated from the input reflections or other
spurious light by spatial filtering using a pin hole. When the cavity is on resonance,
we measured a total transmission (from free-space input to output) of 20%, which
was primarily limited by the imperfect coupling into the nanobeam. The output
signal also contained leakage travelling waves (5%) due to finite extinction of the
photonic bandgap and other spurious reflections in the system.
Polariton excitation and frequency qubit preparation
.
The GTI etalon was made
of a 250
m
m thick quartz slide with backside coated with a gold film. The front side
was uncoated, which had a reflectivity of 4%. This etalon produces a nearly linear
dispersion of 4
p
/nm over a free spectral range of 0.5 nm near 880 nm. After the
GTI etalon, the transform-limited laser pulse acquired a phase chirp, which excited
a mixed polariton state approximated by 0
ji
þ
e
i
f
1
ji
, where
f
is the phase shift
over the Rabi splitting. For our custom made etalon,
f
E
0.52
p
and the corre-
sponding polarition state was close to (|0
iþ
i
|1
i
).
Quantum state tomography based on two-pulse interferometry
.
The
electric field operators for the two consecutive photonic states
c
1
ji
;
c
2
ji
coupled to the cavity are written as
E
1
t
ðÞ¼
a
1
e
i
o
t
a
þ
b
1
e
i
o
þ
t
a
þ
and
E
2
t
ðÞ¼
a
2
e
i
o
t
a
þ
b
2
e
i
o
þ
t
a
þ
, respectively. The field operator at the
single-photon detector is
Et
ðÞ¼
E
1
t
t
R
ðÞþ
E
2
t
ðÞ
corresponding to the
first photon delayed by
t
R
after storage in the ensemble, which interferes
with the second photon. The count rate on the detector is
C
¼
Et
ðÞ
y
Et
ðÞ
DE
¼
2
þ
2cos
f
ðÞ
a
1
a
2
þ
b
1
b
2
, where
f
is the phase difference between the two photon
wavepackets. This gives interference fringes with an amplitude
A
¼
C
0
a
1
a
2
þ
b
1
b
2
as labelled in Fig. 3c, where
C
0
is a constant factor. By encoding a set of four basis
states (Pauli tomography basis) on the second photon, that is,
a
2
|0
iþ
b
2
|1
i
,we
construct the set of experimental amplitude parameters
A
j
,
j
¼
0...3 (
A
0
for
c
2
ji¼
1
=
ffiffiffi
2
p
c
1
ji
;
A
1
for
c
2
ji¼þ
ji
,
A
2
for
c
2
ji¼
T
ji
;
A
3
for
c
2
ji¼
T
ji
) which are
analogous to the set of projection measurement outcomes for calculating the
density matrix
r
out
(see Supplementary Note 7 for detailed derivations),
r
out
¼
1
2
I
þ
A
1
=
A
0
ðÞ
2
1
s
1
þ
A
2
=
A
0
ðÞ
2
1
s
2
þ
A
3
=
A
0
ðÞ
2
1
s
3
;
ð
1
Þ
where
I
is the identity operator and
s
1
;
2
;
3
are the Pauli spin operators. Then we
perform a maximal likelyhood estimation
31
to obtain a physical density matrix,
which is used to calculate the fidelity
F
¼
c
in
r
out
jj
c
in
hi
Qubit storage and retrieval efficiency
.
The storage efficiency is defined as the
probability that a photon in the cavity mode is transferred to a collective excitation
in the ensemble. This storage efficiency is intrinsically 100% as the polariton modes
under cavity protection condition are maximally entangled states between the
cavity and the super-radiant state. The retrieval efficiency is defined by the number
of photons emitted at the cavity output during the second Rabi period (grey
window in Fig. 4b) versus the total transmitted photons. On the basis of the
temporal distribution of the transmitted photons (deconvolved from the oscillation
signal in Fig. 3c), the integrated counts during the second Rabi period (grey
window in Fig. 4b) was 25.6
±
1.2% of the total transmitted photons. Note that this
storage and retrieval efficiency does not take into account the input coupling and
scattering loss. Including the 20
±
2% transmission efficiency through the device,
the overall system efficiency was 5.1
±
0.7%. The corresponding classical bound for
qubit storage fidelity would be 78.9
±
0.05%, still significantly below the measured
fidelities in Fig. 4c.
Data availability
.
The data that support the findings of this study are available
from the corresponding author on request.
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Acknowledgements
This work is funded by NSF CAREER 1454607 and AFOSR Quantum Transduction
MURI FA9550-15-1-002. Device fabrication was performed in the Kavli Nanoscience
Institute with support from Gordon and Betty Moore Foundation. Some equipment
funding was provided by NSF Institute for Quantum Information and Matter PHY-
1125565 with support from Gordon and Betty Moore Foundation GBMF-12500028.
Author contributions
T.Z. and A.F. conceived and designed the experiments. T.Z. and J.R. fabricated the
device. T.Z. and J.M.K. performed the measurements and analysed the data. T.Z. and
A.F. wrote the manuscript with input from all authors.
Additional information
Supplementary Information
accompanies this paper at http://www.nature.com/
naturecommunications
Competing financial interests:
The authors declare no competing financial interests.
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information is available online at http://npg.nature.com/
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How to cite this article:
Zhong, T.
et al.
Interfacing broadband photonic qubits
to on-chip cavity-protected rare-earth ensembles.
Nat. Commun.
8,
14107
doi: 10.1038/ncomms14107 (2017).
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The Author(s) 2017
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14107
ARTICLE
NATURE COMMUNICATIONS
| 8:14107 | DOI: 10.1038/ncomms14107 | www.nature.com/naturecommunications
7