On-chip storage of broadband photonic qubits in a
cavity-protected rare-earth ensemble
Tian Zhong,
1
Jonathan M. Kindem,
1
Jake Rochman,
1
and Andrei Faraon
1,
∗
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology,
1200 E California Blvd, Pasadena, CA, 91125, USA
∗
faraon@caltech.edu
1
arXiv:1604.00143v1 [quant-ph] 1 Apr 2016
Ensembles of solid-state optical emitters enable broadband quantum storage [1,
2] and transduction of photonic qubits [3, 4], with applications in high-rate
optical quantum networks [5] for secure communications [6, 7], global time-
keeping [8], and interconnecting future quantum computers. To realize coher-
ent quantum information transfer using ensembles, spin rephasing techniques
are currently used to mitigate fast decoherence resulting from inhomogeneous
broadening [9, 10]. Here we use a dense ensemble of neodymium rare-earth
ions strongly coupled to a nanophotonic resonator to demonstrate that deco-
herence of a single photon excitation is near-completely suppressed via cavity
protection [1, 11] - a new techinique for accessing the decoherence-free subspace
of collective coupling. The protected Rabi oscillations between the cavity field
and the atomic superradiant state thereby enable ultra-fast transfer of photonic
frequency qubits into the ions (
∼
50 GHz bandwidth), followed by retrieval with
98.7% fidelity. By coupling the superradiant excitation to other long-lived rare-
earth spin states, this technology will enable broadband, always-ready quantum
memories and fast optical-to-microwave transducers.
Ensembles of rare-earth ions doped in crystals exhibit outstanding quantum coherence
properties and broad inhomogeneous linewidths that are suitable for quantum information
transfer with fast photons [13]. They are used in state-of-the-art quantum memories both in
optical and microwave domain [2, 9, 10, 14–16], and are promising candidates for optical to
microwave quantum transduction [3, 4]. 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 because of inhomogeneous broadening. To restore the
optical coherence, protocols based on spectral hole burning techniques like atomic frequency
comb (AFC) [10, 17, 18] and controlled reversible inhomogeneous broadening (CRIB) [9]
have been perfected. Although effective, these protocols involve long (hundreds of milisec-
onds) and complex preparation procedures that generally limit the interface bandwidth.
Recently, it was proposed [1, 11] 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 NV spin ensemble [19].
Here, we demonstrate for the first time strong cavity protection against decoherence in the
optical domain using a dense ensemble (a few millions) of neodymium (Nd) atoms coupled
2
to a nanophotonic cavity. Exploiting the protected mapping of photonic qubits to atomic
superradiant excitations, we realize a 50 GHz quantum light-matter interface that could find
applications in future quantum networks.
The dynamics of a cavity mode coupled to an atomic ensemble are described by the
Tavis-Cummings Hamiltonian [20]. We focus on the interaction term
H
int
=
i
~
Ω(
S
−
a
†
−
S
+
a
) where
a
†
and
a
are creation and annihilation operators of the cavity mode, and the
collective spin operators
S
±
=
1
√
N
∑
σ
±
k
act on
N
atoms each of frequency
ω
k
. Ω denotes
a collective coupling strength Ω
2
=
∑
N
k
g
2
k
which scales up the single atom coupling
g
k
by
√
N
. When the cavity and the ensemble are in resonance, the coupled system exhibits
two bright polariton states with equal mix of atomic and photonic components detuned by
±
Ω 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 [1,
11, 21]. 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
ρ
(
ω
) =
∑
k
g
2
k
δ
(
ω
−
ω
k
)
/
Ω
2
[1, 11]. In the limiting case of a Lorentzian
distribution, considerable damping given by the width of the inhomogeneous broadening
persists even with an infinite coupling Ω. When the spectral distribution exhibits a faster-
than-Lorentzian decay (e.g. Gaussian), the damping of the coherent Rabi oscillation is
diminished at increasing Ω - the system becomes ’cavity protected’ as conceptually illustrated
in Fig. 1a. In this case, the atomic component of the polariton is purely the symmetric
superradiant state [22].
We probe the cavity-protection regime in an optical nanocavity based on our triangular
beam design [23, 24] fabricated in 0.1% Nd doped yttrium vanadate (YVO) crystal using
focused ion beam (Fig. 1e). The cavity has a fundamental TM mode resonance with mea-
sured quality factor of 7,700 (
κ/
2
π
∼
44 GHz energy decay rate), a simulated mode volume
V
mode
= 1(
λ/n
)
3
=0.063
μ
m
3
, and contains
N
∼
10
6
ions. The resonance wavelength is
877.1 nm, close to the
4
I
9
/
2
(
Y
1
)
−
4
F
3
/
2
(
Z
1
) transition of Nd
3+
at 879.8 nm. The device
was cooled down to 3.6 K (Montana Instruments Cryostation) and a magnetic field of 500
mT was applied perpendicular to the YVO c-axis, which enhanced the optical coherence
time to T
2
=390 ns (measured via photon echoes). The
B
field caused a Zeeman splitting of
the
Y
1
,Z
1
states into 4 levels. For the applied magnetic field orientation, cross-transitions
are minimized [25]. Therefore the system can be viewed as two independent distributions
3
of emitters (Fig. 1b) 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 [19],
yielding a shape parameter
q
=1.01 (1 for Gaussian, 2 for Lorentzian). For simplicity, subse-
quently we treat the two sub-ensembles as one joint distribution with an effective full-width
at half-maximum (FWHM) of
γ
q
= 2∆
√
2
q
−
2
2
q
−
2
=24 GHz and a parameter ∆=14.6 GHz (2∆
represents the FWHM for a Lorentzian distribution).
To achieve strong coupling, the cavity resonance was red-tuned by gas condensation [23]
while the cavity transmission spectra excited by a broadband superluminescent diode
was recorded using a spectrometer. In Fig. 2a, a strong anti-crossing is observed. The
on-resonance spectrum (Fig. 2b) reveals two polariton peaks with a Rabi splitting of
Ω
R
/
2
π
=48
±
2 GHz. The cavity tuning continuously changed the effective collective coupling
Ω according to Ω = Ω
0
/
√
1 + 4
δ
2
/κ
2
[23], where
δ
=
ω
c
−
ω
a
is the cavity-ensemble detuning
and Ω
0
denotes the maximum collective coupling obtained by fitting the spectrum in Fig. 2b
with a known model (Supplementary Information III). The decay rates Γ(Ω) were deter-
mined from the FWHM linewidth of the cavity (in the detuned regime) and the polariton
peak (in the strong coupling regime). The plot in Fig. 2c corresponds to the left anti-crossing
trajectory in Fig. 2a as the cavity shifted from lower wavelengths towards resonance, and
the Γ at large Ω was of the lower polariton. Cavity protection manifested as Γ decreased
rapidly to a minimum of Γ
∼
22 GHz at maximum coupling. We note the exact reverse trend
took place from the right trajectory as the cavity moved away from resonance. The data in
Fig. 2c shows excellent agreement with the analytical expression Γ =
κ/
2 +
γ
h
+
π
Ω
2
ρ
(Ω) [1]
as the red curve, which describes the Ω
→∞
limit. In our case, the full protection limit is
approached by having Γ
∼
κ/
2. The residual broadening estimated from the
π
Ω
2
ρ
(Ω) term
was
≈
0.1 GHz, more than two orders of magnitude suppressed than without protection.
To contrast with the absence of protection, we also plot in green the theoretical decay for
a Lorentzian distribution (Γ = ∆ +
κ/
2
±
√
(∆
−
κ/
2)
2
−
4Ω
2
) assuming the same ∆, of
which Γ(Ω) remains constant even in the strong-coupling limit.
The cavity-protected system acts as a quantum interface where a broadband photon can
be transferred to the superradiant atomic excitation. We measure the coherent, ultra-fast
transfer dynamics using pulsed excitations of the polariton modes. The experimental setup
is depicted in Fig. 1d. A mode-locked Ti:Sapphire laser at 85 MHz repetition rate (Thorlabs
Octavius) was filtered to a pulse width of 4 ps using a monochromator, which was sufficient
4
to simultaneously excite both upper (
|
ω
+
〉 → |
1
〉
) and lower (
|
ω
−
〉 → |
0
〉
) polaritions. The
filtered laser was attenuated and sent through a Michelson setup to produce two pulses
with less-than-one mean photon number separated by a variable delay
τ
that 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
produces optical 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
∼
π
/2 phase chirp between the two polaritons (Methods).
Furthermore, a narrow bandpass filter was optionally inserted in either arm that allowed
only one polariton to be excited. This combination enabled a comprehensive polariton
excitation scheme that covered individual polariton
|
0
〉
or
|
1
〉
, and superposition states of
two polaritons i.e.
|
+
〉
= 1
/
√
2(
|
0
〉
+
|
1
〉
) or
|
〉
= 1
/
√
2(
|
0
〉
+
i
|
1
〉
).
Figure 3 plots the theoretical interference fringe amplitudes along with the measured
results for a few representative two-pulse excitation schemes. The mean photon number
per pulse coupled in the cavity was estimated at
μ
=0.5. The case of an uncoupled cavity
is plotted in Fig. 3a, showing a fitted decay constant (4
/κ
) of
∼
14.5 ps. When only one
polariton (
|
0
〉
or
|
1
〉
) was excited, the decay was extended to 29.0 ps as a result of cavity
protection (Fig. 3b). For the superposition state
|
+
〉
, 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 system was first driven into
|
〉
state, and was subsequently excited by a
|
+
〉
pulse. The resulting fringe showed the Rabi
oscillations with the nodes shifted with respect to 3c by about 5.5 ps (
∼
1
/
4
τ
R
), in agreement
with our theoretical model (Supplementary information III). 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.
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. Similar to AFC, photons are
stored and then released after the inverse of the comb period, which in this case is a Rabi
period
τ
R
. The interface bandwidth is
∼
50 GHz, spanning two polaritons, and the qubits
are of the form
|
0
〉
,
|
+
〉
= 1
/
√
2(
|
0
〉
+
|
1
〉
) or
|
〉
= 1
/
√
2(
|
0
〉
+
i
|
1
〉
). To characterize this
process, quantum state tomography on the released qubit after a delay
τ
R
was performed.
5
As direct projection measurements were difficult given the high-bandwidth, we adopted an
interferometric scheme (Fig. 4b) to assess the input/output fidelity
F
=
〈
ψ
in
|
ρ
out
|
ψ
in
〉
, where
|
ψ
in
〉
is the input qubit state and
ρ
out
is the density matrix for the retrieved state, from a
set of fringe signals including those in Fig. 3 (Methods and Supplementary Information I).
The reconstructed density matrices
ρ
out
for
|
0
〉
,
|
+
〉
, and
|
〉
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 [3] of 74.9
±
0.04% that takes into account
the Poissonian statistics of the coherent input photons (with
μ
=0.5) and an imperfect but
high storage-retrieval efficiency of 25.6
±
1.2% (Methods) [4, 5]. The estimated fidelities take
into account imperfections in the preparation and measurement of the qubit, such as leakage
of traveling waves through the cavity and inaccurate phase shift (ideally
π
/2) by the GTI
etalon. Thus, the high fidelity indicates a robust quantum transfer with a bandwidth that
is significantly broader than existing rare-earth-based light-matter interfaces [14, 18]. To
highlight the benefit of cavity protection, we also evaluated the qubit fidelity at a delay of
2
τ
R
, which would be equivalent to the case without cavity protection where the qubit would
decohere twice as fast. The measured fidelities at 2
τ
R
dropped to 83% for
|
0
〉
, 70% for
|
+
〉
,
and 69% for
|
〉
, which no longer beats the classical fidelity.
While this interface allows for efficient mapping of the photonic qubit to the ensemble,
the qubit dissipates at
κ/
2 rate. Improvements in the cavity quality factor to state of the
art values of Q
∼
10
6
would achieve storage for 1 ns (enough for perform 50 Rabi flips). To
enable long-term storage like in an AFC-spin-wave memory [17], the qubit can be transferred
from the superradiant state excitation to a long-lived spin level by applying a
π
pulse within
τ
R
time. Upon recall, another
π
pulse can transfer the qubit back to the polariton states
and then a cavity photon. For faithful spin-wave storage, the Rabi frequency of the
π
pulses should exceed the polariton linewidths, which is attainable given the strong light
confinement in current nanobeam devices. Compared to existing AFC-spin-wave memories,
this interface would not require any preparation steps, thus it is always ready. Taking
advantage of on-chip technologies also enable other proposals for long-lived memories in
the cavity-protected regime by fast detuning of the cavity-ensemble coupling [1]. Most
notably, the cavity-protected mapping of a photonic qubit to a collective superradiant 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
6
Methods
Nanocavity design and characterization.
The triangular nanobeam has a width of 770
nm and length of 15
μ
m. 40 periodic subwavelength grooves of 185 nm along the beam
axis were milled on top of the nanobeam. The grooves extend across the entire width of
the beam, which eases the alignment during fabrication. 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 879.8 nm transition in Nd:YVO. The
theoretical quality factor is 300,000 with a mode volume of 1(
λ
/n)
3
[24]. Transmission of
the nanocavity was measured by vertically coupling free-space input into the nanobeam via
a 50x 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
μ
m thick quartz slide with backside coated with a gold film. The front side was
not coated, which has a reflectivity of 4%. This etalon produces a nearly linear dispersion
of 4
π
/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
〉
+
e
iφ
|
1
〉
, where
φ
is the phase shift over the Rabi splitting. For our
custom made etalon,
φ
≈
0
.
52
π
and the corresponding polarition state was close to (
|
0
〉
+
i
|
1
〉
).
Quantum state tomography based on two-pulse interferometry.
The electric field
operators for the two consecutive photonic states
|
ψ
1
〉
,
|
ψ
2
〉
coupled to the cavity are written
as
ˆ
E
1
(
t
) =
α
1
e
−
iω
−
t
ˆ
a
−
+
β
1
e
−
iω
+
t
ˆ
a
+
and
ˆ
E
2
(
t
) =
α
2
e
−
iω
−
t
ˆ
a
−
+
β
2
e
−
iω
+
t
ˆ
a
+
, respectively. For
simplicity, we drop the finite linewidth of each polariton mode. The field operator at the
single photon detector is
ˆ
E
(
t
) =
ˆ
E
1
(
t
−
τ
R
) +
ˆ
E
2
(
t
) corresponding to the first photon delayed
8
by
τ
R
after storage in the ensemble, which interferes with the second photon. The count
rate on the detector is
C
=
〈
E
(
t
)
†
E
(
t
)
〉
= 2 + 2cos(
φ
)
|
α
1
α
∗
2
+
β
1
β
∗
2
|
, where
φ
is the phase
difference between the two photon wavepackets. This gives interference fringes with an
amplitude
A
=
C
0
|
α
1
α
∗
2
+
β
1
β
∗
2
|
as labelled in Fig. 3c, where
C
0
is a constant factor set by
the input photon number, system transmission and decay rates, and detection efficiencies.
By encoding a set of four basis states (Pauli tomography basis) on the second photon, i.e.
α
2
|
0
〉
+
β
2
|
1
〉
, we construct the set of experimental amplitude parameters
A
j
,j
= 0
...
3 (
A
0
for
|
ψ
2
〉
= 1
/
√
2
|
ψ
1
〉
;
A
1
for
|
ψ
2
〉
=
|
+
〉
,
A
2
for
|
ψ
2
〉
=
|
〉
;
A
3
for
ψ
2
〉
=
|
0
〉
) which are
directly analogous to the set of projection measurement outcomes for calculating the density
matrix ˆ
ρ
out
(see Supplementary Information for detailed derivations)
ˆ
ρ
out
=
1
2
[
ˆ
I
+ ((
A
1
/A
0
)
2
−
1)ˆ
σ
1
+ ((
A
2
/A
0
)
2
−
1)ˆ
σ
2
+ ((
A
3
/A
0
)
2
−
1)ˆ
σ
3
.
]
(1)
where
ˆ
I
is the identity operator and ˆ
σ
1
,
2
,
3
are the Pauli spin operators. Then we perform
a maximal likelyhood estimation [2] to obtain a physical density matrix, which is used to
calculate the fidelity
F
=
〈
ψ
in
|
ˆ
ρ
out
|
ψ
in
〉
Qubit storage and retrieval efficiency.
The storage efficiency is defined as the probabil-
ity 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 superradiant 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) vesus the total transmitted pho-
tons. Based on 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 stor-
age 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.
9
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storage of a photonic polarization qubit in a solid.
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Acknowledgements
This work is funded by NSF CAREER 1454607, NSF Institute for Quantum Information
and Matter PHY-1125565 with support from Gordon and Betty Moore Foundation GBMF-
12500028, 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.
Author contributions
A.F. and T.Z. conceived the experiments. T.Z. J. R fabricated the device. T.Z., J.M.K.
performed the measurements. T.Z. and J.M.K. analyzed the data. T.Z. and A.F. wrote the
manuscript with input from all authors.
Competing financial interests
The authors declare no competing financial interests.
12
On-chip storage of broadband photonic qubits in a cavity-protected rare-earth
ensemble: Supplementary Information
S1. Construction of qubit density matrix from interference fringes of two pho-
tons
The state of an arbitrary qubit
|
ψ
〉
=
α
|
0
〉
+
β
|
1
〉
, where
|
α
|
2
+
|
β
|
2
= 1, can be determined by
taking a set of four projection measurements represented by the operators ˆ
μ
0
=
|
0
〉〈
0
|
+
|
1
〉〈
1
|
,
ˆ
μ
1
=
|
+
〉〈
+
|
, ˆ
μ
2
=
|
〉〈
|
, ˆ
μ
3
=
|
0
〉〈
0
|
[2]. The outcome of these measurements are
n
j
=
C
Tr
{
ˆ
ρ
ˆ
μ
j
}
,
(2)
where
ρ
=
|
φ
〉〈
φ
|
for a pure state and the scaling factor
C
is the number of detected photons,
which will be set to 1 in the following derivation. We explicitly write out the four
n
j
values
for an arbitrary qubit as
n
0
= 0
.
5(
|
α
|
2
+
|
β
|
2
) = 0
.
5
n
1
= 0
.
5
|
α
+
β
|
2
n
2
= 0
.
5
|
α
−
iβ
|
2
n
3
=
|
α
|
2
.
(3)
From these values, the four Stokes parameters are calculated as
S
0
= 2
n
0
S
1
= 2(
n
1
−
n
0
)
S
2
= 2(
n
2
−
n
0
)
S
3
= 2(
n
3
−
n
0
)
.
(4)
The density matrix ˆ
ρ
is then constructed from the Stokes parameters by
ˆ
ρ
=
1
2
3
∑
j
=0
S
j
S
0
ˆ
σ
j
,
(5)
where ˆ
σ
0
is the identity operator
ˆ
I
and ˆ
σ
1
,
2
,
3
are the Pauli spin operators.
Now we turn to the measurement of interference between two photonic states represented
by the electric field operators
ˆ
E
1
(
t
) =
α
1
e
−
iω
−
t
a
−
+
β
1
e
−
iω
+
t
a
+
and
ˆ
E
2
(
t
) =
α
2
e
−
iω
−
t
a
−
+
β
2
e
−
iω
+
t
a
+
, where
ω
−
,
ω
+
are the optical frequencies of the lower and upper polaritons,
respectively, with frequency difference
ω
+
−
ω
−
= Ω
R
. For simplicity, we do not include
13
the finite linewidth of each polariton as it does not affect the results of the tomography
measurement. The field operator at the single photon detector is
ˆ
E
(
t
) =
ˆ
E
1
(
t
−
τ
R
) +
ˆ
E
2
(
t
)
corresponding to the first photon delayed by
τ
R
after storage in the ensemble. The count
rate on the detector is
C
∝〈
E
(
t
)
†
E
(
t
)
〉
= 2 + 2 cos
φ
|
α
1
α
∗
2
+
β
1
β
∗
2
e
−
i
Ω
R
τ
R
|
= 2 + 2 cos
φ
|
α
1
α
∗
2
+
β
1
β
∗
2
|
,
(6)
where the last equality holds for Ω
R
τ
R
= 2
π
. Here
φ
is the carrier phase difference between
the two photons, which is varied from 0 to 2
π
to produce interference fringes with a peak-to-
peak amplitude
C
max
−
C
min
= 4
|
α
1
α
∗
2
+
β
1
β
∗
2
|
. We define a set of experimentally measurable
fringe amplitude parameters
A
=
C
0
|
α
1
α
∗
2
+
β
1
β
∗
2
|
, where
C
0
is a constant factor representing
the integrated counts. These parameters closely resemble the projection measurement out-
comes in Eq. (1) (different by a power of 2) depending on the states encoded on the second
photon, i.e.
α
2
|
0
〉
+
β
2
|
1
〉
. For instance, if we encode the second photon in the same qubit as
the first photon but attenuate the intensity by a factor of 2 , i.e.
α
2
=
α
1
/
√
2,
β
2
=
β
1
/
√
2,
we get the amplitude
A
0
=
C
0
√
2
(
|
α
1
|
2
+
|
β
1
|
2
). For
α
2
=
β
2
= 1
/
√
2,
A
1
=
C
0
/
√
2
|
α
1
+
β
1
|
.
For
α
2
=
−
iβ
2
= 1
/
√
2,
A
2
=
C
0
/
√
2
|
α
1
−
iβ
1
|
. For
α
2
= 1,
β
2
= 0,
A
3
=
C
0
|
α
1
|
. Based on
the four amplitude values, we construct an equivalent set of Stokes parameter
S
A
j
S
A
0
= 2
A
2
0
S
A
1
= 2
A
2
1
−
2
A
2
0
S
A
2
= 2
A
2
2
−
2
A
2
0
S
A
3
= 2
A
2
3
−
2
A
2
0
,
(7)
from which the density matrix is calculated by
ˆ
ρ
=
1
2
3
∑
j
=0
S
A
j
S
A
0
ˆ
σ
j
=
1
2
[
ˆ
I
+ ((
A
1
/A
0
)
2
−
1)ˆ
σ
1
+ ((
A
2
/A
0
)
2
−
1)ˆ
σ
2
+ ((
A
3
/A
0
)
2
−
1)ˆ
σ
3
]
.
(8)
Then we perform a maximal likelyhood estimation [2] to obtain a physical density matrix,
which is used to calculate the fidelity
F
=
〈
ψ
in
|
ˆ
ρ
|
ψ
in
〉
14
S2. Classical storage fidelity for weak coherent photons
The classical fidelity for any storage device measures the best input/output fidelity one can
achieve using a classical method. For a given photon number of the input state
N
ph
, the
maximum classical fidelity is known to be
F
=
N
ph
+1
N
ph
+2
[3]. For an input pulse that is in a
coherent state with a mean photon number
μ
, the Poissonian statistics give a
N
-photon
probablity of
P
(
N
ph
) =
e
−
μ
μ
N
ph
N
ph
!
. Accounting for each
N
-photon component, the classical
fidelity of a coherent state then is
F
=
∞
∑
N
ph
≥
1
N
ph
+ 1
N
ph
+ 2
P
(
N
ph
)
1
−
P
(0)
(9)
In addition, for an imperfect memory with storage and retrieval efficiency
η <
1, the classical
fidelity would be higher because a classical memory can preferentially measure the higher
photon component of the input and send out a new qubit. We follow the strategy in [4, 5]
that there exists a threshold photon number
N
min
that the classical memory sends out a qubit
when the input photon number is greater than this value, which happens with a probability
1
−
p
. Otherwise the memory sends out a result for input photon
N
min
with probability
p
.
Combing the two cases, the memory efficiency is expressed as
η
=
p
+
∑
N
ph
≥
N
min+1
P
(
N
ph
)
1
−
P
(0)
(10)
For a given
μ
and
η
, the value of
N
min
can be readily calculated according to [5],
N
min
= min
i
:
∑
N
ph
≥
i
+1
P
(
N
ph
)
≤
(1
−
P
(0))
η,
(11)
which is used to obtain the final classical fidelity
F
class
=
N
min
+1
N
min
+2
p
+
∑
N
ph
≥
N
min+1
N
ph
+1
N
ph
+2
P
(
N
ph
)
η
(1
−
P
(0))
(12)
S3. Simulation of the dynamics of the coupled cavity-ensemble system
The model of the system follows that presented in [1] and consists of a cavity mode
a
of
frequency
ω
0
coupled with strength
g
k
to a distribution of
N
two-level emitters described by
modes
b
k
. We define the cavity frequency
ω
0
to be the origin and label the frequency of each
emitter as
ω
k
. We account for the atomic dephasing rate
γ
and label the cavity field decay
as
κ
. Using the standard input-output formalism [1] for a two sided cavity with input field
15
c
in
, reflected field
c
r
, and transmitted field
c
t
gives the Heisenberg equations for the system:
̇
a
=
−
[
κ
2
+
i
(
ω
0
−
ω
)
]
a
−
√
κ
2
c
in
+
∑
k
g
k
b
k
̇
b
k
=
−
[
γ
2
+
i
(
ω
k
−
ω
)
]
b
k
−
g
k
a
c
t
=
√
κ
2
a
c
r
=
c
in
+
√
κ
2
a.
By solving this set of equations in the steady state, we arrive at the complex transmission
of the cavity:
t
(
ω
) =
〈
c
t
〉
〈
c
in
〉
=
−
κ
2
i
ω
0
−
iκ
2
−
ω
−
∑
k
g
2
k
ω
k
−
iγ
2
−
ω
.
(13)
For a large number of emitters, we can define the distribution of emitters in terms of a
continuous spectral density
ρ
(
ω
) =
∑
k
g
2
k
δ
(
ω
−
ω
k
)
Ω
2
.
Here Ω is the collective coupling strength
defined by Ω
2
=
∑
k
g
2
k
. The continuum form of the transmission is then [1]
t
(
ω
) =
−
κ
2
i
ω
0
−
iκ
2
−
ω
−
Ω
2
∫
ρ
(
ω
′
)
dω
′
ω
′
−
ω
+
iγ
2
.
(14)
In the case of a decoupled cavity (Ω = 0), this reduces to a Lorentzian with full-width
at half maximum (FWHM) of
κ/
2
π
. From the transmission of the far-detuned cavity,
we measure
κ/
2
π
= 44 GHz. The measured distribution of atoms
ρ
(
ω
) is described by
a q-gaussian with q
≈
1 and FWHM of 24 GHz. We measure
γ
=1/
π
T
2
= 0.82 MHz using
two-pulse photon echoes. Fitting the on-resonance transmission spectrum with these known
parameters allows us to determine the collective coupling strength Ω = 25 GHz.
To simulate the dynamics of the system (Fig. 3 theoretical plots and Fig. 4b), we start
from the discrete form of the differential equations. In this case, simulating the entire
system entails solving a set of
N
∼
10
6
coupled differential equations. To make the problem
more computationally tractable, we instead solve the system of
N
sim
N
coupled emitters
with frequencies randomly assigned according to the experimentally measured frequency
distribution. The coupling strength of each emitter is set as
g
= Ω
/
√
N
sim
such that the
collective coupling strength is held constant at the experimentally measured value. This
reduced set of equations was then solved numerically in Mathematica using the built-in
differential equation solver (NDSolve). The number of simulated emitters was increased
until the solution converged (i.e. until increasing the number of simulated emitters no
16