Nanophotonic coherent light-matter interfaces based on rare-earth-doped crystals
Tian Zhong,
1
Jonathan M. Kindem,
1
Evan Miyazono,
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
(Dated: July 6, 2015)
Quantum light-matter interfaces (QLMIs) con-
necting stationary qubits to photons will enable
optical networks for quantum communications,
precise global time keeping, photon switching,
and studies of fundamental physics. Rare-earth-
ion (REI) doped crystals are state-of-the-art ma-
terials for optical quantum memories and quan-
tum transducers between optical photons, mi-
crowave photons and spin waves. Here we demon-
strate coupling of an ensemble of neodymium
REIs to photonic nano-cavities fabricated in the
yttrium orthosilicate host crystal. Cavity quan-
tum electrodynamics effects including Purcell
enhancement (F=42) and dipole-induced trans-
parency are observed on the highly coherent
4
I
9
/
2
−
4
F
3
/
2
optical transition. Fluctuations in the
cavity transmission due to statistical fine struc-
ture of the atomic density are measured, indi-
cating operation at the quantum level. Coher-
ent optical control of cavity-coupled REIs is per-
formed via photon echoes. Long optical coher-
ence times (
T
2
∼
100
μs
) and small inhomogeneous
broadening are measured for the cavity-coupled
REIs, thus demonstrating their potential for on-
chip scalable QLMIs.
QLMIs are quantum devices composed of light emitters
with quantum states that can be controlled via optical
fields and entangled to photons [1, 2]. They enable dis-
tribution of quantum entanglement over long distances
in optical quantum networks for quantum communica-
tions [1]. Quantum networks of atomic clocks have also
been proposed for precise global time-keeping and stud-
ies of fundamental physics [2]. Realizing QLMIs requires
control of light and matter at the single atom and sin-
gle photon level, which enable optoelectronic devices like
optical modulators and nonlinear optical devices operat-
ing at the most fundamental level [3]. QLMIs are also
expected to play a leading role in realizing optical to
microwave quantum transducers for interconnecting fu-
ture superconducting quantum machines via optical fi-
bres [4, 5].
Scalable and robust QLMIs require emitters to have
long spin coherence times and coherent optical transi-
tions. For integrated optical quantum networks, these
emitters need to be coupled to on-chip optical resonators
that capture the photons in a single mode and further
couple them into optical fibres or waveguides. The solid-
state emitters most investigated so far for on-chip QLMIs
are semiconductor quantum dots (QDs) [6] and nitrogen
vacancy centers in diamond (NVs) [7]. To date, com-
plete quantum control of single QD and NV spins, spin-
photon entanglement, and entanglement of remote NVs
via photons have been realized [8–10]. Both QDs [11]
and NVs [1] have been coupled to optical nano-cavities.
However, the challenge in growing optically identical QDs
limits their prospects for a scalable architecture [6]. NVs
embedded in nanostructures have long electronic spin co-
herence times [13], but suffer from optical spectral insta-
bilities such as blinking and spectral diffusion [14]. These
spectral instabilities have so far impeded the coherent
coupling between optical fields and NV centers in nano-
resonators that are essential for further developments of
QLMIs.
Rare-earth ions (REIs) embedded in host crystals at
cryogenic temperatures exhibit highly coherent quantum
states in the 4f orbital [15]. The Zeeman or hyper-
fine states of REIs can have coherence times as long
as six hours [16], the longest ever demonstrated in a
solid. These states are connected via optical transitions
with the narrowest linewidth in the solid state (sub-kHz)
and small inhomogeneous broadening (MHz to GHz) [17].
This outstanding optical and spin coherence makes REI-
doped crystals the state-of-the-art material for macro-
scopic solid-state optical quantum memories [18, 19]. In-
tegrated REI-doped waveguide quantum memories have
also been developed [20, 21]. Detection and control of sin-
gle REI spins has been recently demonstrated in bulk ma-
terial, but not using the transitions employed in optical
quantum memories at cryogenic temperatures [22, 23].
Coupling the highly coherent optical transitions of REIs
to nano-cavities will enable on-chip QLMIs where REI
ensembles act as quantum memories and single REIs act
as qubits [24].
Here we demonstrate high-cooperativity coupling of a
neodymium (Nd
3+
) ensemble to photonic nano-cavities
fabricated directly in the yttrium orthosilicate (YSO)
host crystal, and show coherent optical control of REIs
coupled to nanophotonic cavities. These results are en-
abled by the long coherence time and small inhomoge-
neous broadening of cavity-coupled REIs, which are es-
sential properties that may lead to nanophotonic QLMIs
with better prospects for scalability than those based on
NVs and QDs.
arXiv:1507.00977v1 [quant-ph] 3 Jul 2015
2
λ
(nm)
87
8
879
Cavity transmission (a. u.)
0
1
50
60
70
130
60
70
80
90
100
110
120
130
140
150
160
1
-1
0
1
0
0
1
5
0
2
μ
m
b
D
1
b
D
2
780 nm
b
D
1
D
2
α
c
a
b
B
= 500 m
T
500 nm
1
μ
m
Q
= 4,400
FIG. 1.
Photonic crystal nano-beam resonator fabricated in Nd:YSO. a.
Scanning electron microscope image of the
device. The red inset is a zoomed-in view of the 45
◦
angle-cut coupler that allows vertical coupling of light from a microscope
objective. The blue inset shows the grooves forming the photonic crystal.
b.
Schematics of the nano-beam resonator with
simulated field profiles of the fundamental TE resonance mode. The TE polarization aligns with the D
1
axis of the YSO crystal.
A magnetic field of 500 mT is applied in the D
1
-D
2
plane at an angle of
α
=135
◦
with respect to D
1
axis.
c.
Broadband cavity
transmission spectrum showing the cavity resonance with quality factor
Q
=4,400.
RESULTS
Design, fabrication and characterization of nano-
cavities.
The nano-cavities, one of which is shown
in Fig. 1(a), were fabricated in neodymium-doped yt-
trium orthosilicate (Nd
3+
:YSO) using focused ion beam
milling.
For this study, we used devices fabricated
in two types of samples with Nd doping of 0.2% and
0.003% (Scientific Materials Inc.). The photonic crystal
cavity is made of grooves milled in a triangular nano-
beam [25] (Fig. 1(b)) (see Methods). Finite-difference
time-domain (FDTD) simulations [26] indicate a TE
mode with quality factor exceeding 1
×
10
5
, mode vol-
ume
V
mode
= 1
.
65(
λ/n
YSO
)
3
= 0
.
2
μ
m
3
, and mode pro-
file shown in Fig. 1(b). Here
V
is defined as
V
mode
=
∫
V
(r)
|
E(r)
|
2
d
3
r
/
max(
(r)
|
E(r)
|
2
), where E(r) is the elec-
tric field and
(r) is the electric permittivity at position
r. Two 45
◦
angled cuts at both ends of the nano-beam
(i.e. couplers) allow for coupling light from a direction
normal to the chip (i.e.
~
b in Fig. 1(b)) using a confo-
cal microscope setup (See Methods). A broadband light
source was coupled into the resonator from one end and
the transmitted light was collected from the other coupler
with typical efficiencies ranging from 20% to 50%. The
transmitted spectrum shows a resonance with quality fac-
tor
Q
=4,400 (Fig. 1(c)) in the device used for the follow-
ing measurements. Arrays of devices were reproducibly
fabricated with similar performance (Supplementary S1).
Measurement of the coupling rate between REIs
and the nano-cavity.
The coupling of Nd
3+
ions to
the nano-cavity was observed through enhancement in
photoluminescence (PL) and emission rates. With the
0.2% device cooled at 3.5 K, an 810 nm laser coupled
into the cavity excited PL in the
4
I
9
/
2
−
4
F
3
/
2
tran-
sition that was then collected from the output coupler
(Fig. 2(a)). The PL spectrum shows two lines at 883.05
nm and 884.06 nm, corresponding to two inequivalent
sites (Y
1
and Y
2
) of Nd
3+
in YSO. An important obser-
vation is that the inhomogeneous linewidth of the ions in
the cavity is the same as in the bulk material, for both the
0.2% (∆
inhom
=16.0 GHz) and 0.003% (5.9 GHz) devices
(Supplementary S2). A small inhomogeneous linewidth
(on the order of
∼
10 GHz) is important for scaling to
networks of multiple QLMIs (Supplementary S3). The
cavity resonance was tuned across the Nd
3+
PL line us-
ing a gas condensation technique [1]. The spectrograms
in Fig. 2(b-d) show enhancement of the Y
1
line when
the cavity is resonant with it, a signature of coupling.
The Y
2
line exhibits negligible enhancement because its
dipole moment is not aligned with the TE cavity polar-
ization (D
1
axis of the YSO crystal (Fig. 1(b)). The
spontaneous emission rate enhancement was character-
ized via lifetime measurements. A pulsed laser at 810
nm excited fluorescence of the Y
1
line, which was filtered
using a monochromator and detected with a single pho-
ton counter (Fig. 2(e)). From single exponential fits, we
calculated a reduction in lifetime from 254
μs
when the
cavity is detuned by ∆
λ
=0.3 nm, to 87
μs
on resonance.
Taking into account the branching ratio of the 883 nm
transition (
β
∼
4
.
5%, see Methods), the reduction in life-
times corresponds to an ensemble averaged Purcell fac-
tor [2] F=42, which agrees well with the estimations that
assume a uniform spatial distribution of Nd
3+
ions in the
3
λ
(nm)
883
884
885
Photoluminescence counts
0
50
100
150
0
50
100
150
Detuning (GHz)
-100
-50
05
0
100
Lifetime (
μ
s)
100
150
200
250
t (
μ
s)
0
200
400
600
Counts
10
1
10
2
∆
λ
=0 nm
∆
λ
=0.3 nm
883 nm
via cavity
via other
states
4
I
9/2
4
F
5/2
810 nm
4
F
3/2
a
b
f
e
d
c
Δ λ
Δ λ=0 nm
Y
1
Y
2
FIG. 2.
Coupling of Nd
3+
ions to the YSO cavity mode showing enhanced photoluminescence (PL) and reduced
lifetimes. a.
Schematic of energy levels for Nd
3+
in YSO. Optical excitation at 810 nm results in PL at several wavelengths
with only the 883 nm transition enhanced by the cavity.
b.
Spectrogram showing the Nd
3+
PL while the cavity is tuned
across resonance using gas condensation. The dashed line is a guide to the eye indicating the central wavelength of the cavity
resonance. The cavity resonance is not visible because there is no background luminescence to populate the cavity mode.
c.
d.
PL spectra in the uncoupled (
c
) and coupled (
d
) cases. The cavity resonance was drawn to indicate the cavity location.
e.
Lifetime measurements for coupled (
τ
c
=87
μ
s, ∆
λ
= 0) and uncoupled (
τ
0
=254
μ
s, ∆
λ
=0.3 nm) cases.
f.
Change in
lifetime as a function of the cavity detuning, which fits well with the calculation (red curve) using quality factor
Q
=4,400, 4.5%
branching ratio and field intensity averaged over the mode volume.
resonator (Supplementary S4). A single ion positioned
at the maximum cavity field would experience a Purcell
factor
∼
200. A similar result was obtained in a 0.003%
cavity with lower quality factor (Supplementary S5).
Measurement of the optical coherence time for
cavity-coupled REIs.
Coherent and stable optical
transitions are essential for QLMIs. We characterized the
optical coherence time
T
2
of the 883 nm transition using
two-pulse (
π/
2 -
π
) photon echo techniques (Fig. 3(a)),
with an applied magnetic field of B=0.5 T (see Methods).
The laser pulses were coupled in and the echoes were col-
lected via the couplers while the 0.2% and 0.003% cav-
ities were on resonance with the Nd transition. Since
only a small sub-ensemble (
<
100 ions) in the cavity was
excited, the weak echo signal required detection using
single photon counters. A typical echo from the 0.2%
cavity is shown in Fig. 3(d). The echo decays as a func-
tion of the (
π/
2 -
π
) time delay
τ
are plotted in Fig. 3(b)
together with photon echoes from bulk substrates. For
the 0.2% sample,
T
c
2
=4
τ
c
=2.8
±
0.4
μ
s was measured for
the cavity, which shows a good agreement with the bulk
value of
T
b
2
=3.2
±
0.4
μ
s. For the 0.003% sample, the
echo exhibited two exponential decays. The slower de-
cays give
T
b
2
=100
±
5
μ
s (bulk) and
T
c
2
=94
±
5
μ
s (cavity),
which match with values reported in [3]. The fast de-
cays are likely due to the superhyperfine interactions be-
tween Nd
3+
and its neighbouring yttrium ions, which
commonly manifests as modulated echoes decaying faster
than
T
2
[3]. No oscillations was observed in Fig. 3(a) be-
cause of the fast modulation frequency (
∼
1 MHz) due to
our strong magnetic field. By lowering the B field, we
observed oscillations of echoes for the initial 10
μ
s de-
lay. Additionally, we varied the excitation powers and
did not see changes in the
T
2
values, which indicates our
measurement was not significantly affected by instanta-
neous spectral diffusion. In sum, the good agreement on
T
2
between the cavity and bulk confirms that the optical
coherence property of Nd
3+
ions is not affected by the
nano-fabrication. For higher Purcell factors, the
T
2
in
cavities should decrease due to the
T
2
≤
2
T
1
limit and
would become smaller than the bulk value. This regime
is not reached here because the Purcell enhanced 2
T
1
is
not smaller than
T
b
2
.
The observation of photon echoes demonstrates coher-
ent optical control of the quantum state of cavity-coupled
ions. This control was further extended by varying the
π
pulse duration and observing Rabi oscillations in the
echo intensities as shown in Fig. 3(c). A Rabi frequency
of
∼
6 MHz is estimated. The same oscillation was not
observed in the bulk. For the coupled laser power, the op-
timal
π
pulse duration is 0.4
μ
s. The oscillations are not
visible for pulse duration less than 0.3
μ
s, because of the
4
τ
(
μ
s)
1
2
3
4
5
6
Counts
0
20
40
60
80
π
pulse width (
μ
s)
0.3
0.4
0.5
0.6
0.7
0.8
Echo intensity
10
20
30
40
50
60
τ
(
μ
s)
0
10
20
30
40
50
Photon echo counts (log scale)
-6
-5
-4
-3
-2
-1
0
0.2% cavity
0.2% bulk
0.003% cavity
0.003% bulk
time
π
echo field
π/2
1
2
3
-5
-4
-3
-2
a
c
d
π 3π
5π
coupled
uncoupled
τ
τ
b
FIG. 3.
Photon echo measurements showing long optical coherence time and enhanced echo intensity from an
ensemble of Nd
3+
ions in the cavity. a.
Two-pulse photon echo sequence (
π/
2
−
π
) used to measure
T
2
.
b.
Two-pulse
photon echo decays measured in both the cavity (red) and the bulk (black) samples with two different doping concentrations.
The inset plots the echo decays measured with a 0.2% doped sample.
c.
Oscillation of echo intensity with increasing width
of the
π
rephasing pulse. The periodic signal reveals the ensemble averaged Rabi frequency of the coupled ions. The ideal
π
pulse duration for the input power was 0.4
μ
s.
d.
Enhanced photon echo intensity (by
∼
12 fold) when the cavity is coupled,
compared to the uncoupled case (cavity detuned by ∆
λ
=15 nm so that the transition is outside the photonic bandgap).
limited rise/fall times (
∼
200 ns) of the pulse-generating
setup (Methods). A
∼
12 fold increase in the echo inten-
sity is observed in the cavity-coupled case compared to
the uncoupled case (i.e. detuning ∆
λ
=15 nm) as shown
in Fig. 3(d). This enhancement can be attributed to a
combination of several effects: the higher atomic absorp-
tion rate through the Purcell effect [35], stronger intra-
cavity field intensity, and high echo collection efficiency
as the ions emit dominantly into the cavity mode. The
spectral diffusion of the coupled ions using three-pulse
photon echoes was also investigated. The homogeneous
linewidths were broadened at rates of 6.1 kHz
μ
s
−
1
for
the 0.2% doped cavity and 380 Hz
μ
s
−
1
for the 0.003%
cavity. These slow spectral diffusions permit repeated op-
tical addressing of the ions for 10s of
μ
s (Supplementary
S6).
Measurement of dipole-induced transparency and
statistical fine structure.
QLMIs require efficient in-
teractions between atoms and photons, which is why
quantum memories use long atomic clouds or doped crys-
tals to achieve large optical depth. One key advantage
provided by nano-resonators is that efficient atom-photon
interaction can be achieved in a small volume with only
a handful of ions. This is readily observable in our sys-
tem, where the Nd
3+
ions coherently interact with the
intra-cavity field and control its transmission via dipole-
induced transparency [29]. With the cavity tuned to 883
nm, the cavity transmission was probed using broadband
light and a dip was observed at resonance (Fig. 4(a)).
The depth of the dip depends on the collective coupling
cooperativity
η
= 4
N
̄
g
2
/
(
κ
Γ
h
), where ̄
g
is the ensemble
averaged coupling strength,
κ
is the cavity full linewidth,
Γ
h
is the Nd
3+
homogeneous linewidth, and
N
is the
number of ions per Γ
h
. Considering an empirical collec-
tive dipole-cavity coupling model, the normalized cavity
transmission in the presence of unsaturated resonant ions
is,
T
=
∣
∣
∣
∣
κ
i
∆ +
κ
+ 4
N
̄
g
2
/γ
h
∣
∣
∣
∣
2
,
(1)
which simplifies to
T
= (1 +
η
)
−
2
for zero detuning [30].
The cavity transmission can be controlled by varying the
5
Detuning (GHz)
-30
-20
-10
0
Counts
0
100
200
300
400
500
600
700
800
900
10
20
30
Transmission
λ
(nm)
882
882.5
883
883.5
a
b
4
F
3/2
-
4
I
9/2
Counts
150
200
250
trace 1
trace 2
100 MHz
c
trace 2
trace 1
FIG. 4.
Control of cavity transmission and observation of statistical fine structure (SFS) of coherently driven
Nd
3+
ions in the cavity. a.
Broadband transmission spectra as the cavity is tuned to the 883 nm Nd transition. A dip is
observed when the two are on resonance. The negligible dip at far detunings confirms that this effect is not due to absorption,
but quantum interference between the intra-cavity field and the ions.
b.
High resolution transmission spectrum (red curve)
obtained by scanning a narrow linewidth (
∼
20 KHz) Ti:Sapphire laser over the inhomogeneous line. Green curve is the fit
using parameters: ̄
g
=2
π
×
6 MHz, Γ
h
=100 kHz, Γ
inhom
=16.0 GHz, and assuming a Gaussian ion density distribution. The
green shaded region is the estimated fluctuation in the transmitted laser intensity caused by
√
N
statistical variations of the
ion density. Large fluctuations are expected because the density
N
is low (few tens), which agrees with the measurement. The
fluctuations within the inhomogeneous linewidth is noticeably larger than that at far detunings (
>
25 GHz) and the technical
background noise (grey area), confirming that they are caused by SFS of the ion spectral density.
c.
Two traces of the
transmitted intensities over the same 100 MHz bandwidth near zero detuning at different times. The high degree of correlation
confirms the static and repeatable nature of SFS.
probe light power and observing the saturation of the
ions at increasing intra-cavity photon number (Supple-
mentary S7). The saturation photon number in the nano-
cavity was measured to be
〈
n
cav
〉
= 2
×
10
−
5
.
To better resolve the spectrum, a narrow (
∼
20 kHz)
Ti:Sapphire laser was scanned across the resonance (See
Methods) to give the transmitted signals shown in
Fig. 4(b, c). A 75 % decrease in transmission was mea-
sured at zero detuning, which corresponds to a collec-
tive cooperativity
η
∼
1
.
2. Fitting using a Gaussian
spectral density distribution (green line) with measured
parameters ̄
g
=2
π
×
6 MHz, Γ
h
=100 kHz, Γ
inhom
=16.0
GHz, gives a peak ion density of
N
≈
53. Due to the
statistical fine structure (SFS) of the inhomogeneously
broadened line, a variation in the transmitted intensity
(
〈
δT
〉
2
∼
d
2
T
dN
2
〈
δN
〉
2
) due to
δN
=
√
N
fluctuations in
the ion spectral density is expected [31]. This expected
variation is represented in Fig. 4(b) by the green shaded
region and shows good agreement with that of the mea-
sured signal. This variation within the inhomogeneously
broadened line, on which the statistical fluctuations of
the ion spectral density are imprinted, is significantly
larger than the spectrometer technical noise (grey area)
and laser shot noise at far detunings (
>
25GHz), thus con-
firming that the static SFS in spectral density
N
(∆
λ
) is
probed. Two traces of the laser scan over the same 100
MHz bandwidth near zero detuning at different times
are shown in Fig. 4(c). The high degree of correlation
reflects the static and repeatable nature of SFS. Notably,
the current platform would allow detection and control
of a single ion coupled to the cavity if
N <<
1 and the
laser linewidth were narrower than Γ
h
(Supplementary
S8).
DISCUSSION
The results reported in this paper (long optical coher-
ence time, small inhomogeneous broadening, enhanced
coherent optical control, and resonant probing of cavity-
coupled REIs) demonstrate REI-based nanophotonics
as a promising approach for robust and scalable quan-
tum photonic networks integrating memories and sin-
gle REI qubits. Single photon rates exceeding 1 MHz
can be achieved with single REIs in nano-cavities with
Q
/
V
∼
10
4
-10
5
(
V
is normalized to (
λ/n
)
3
), and the in-
homogeneous broadening allows for frequency multiplex-
ing of multiple REIs. To use the interface as an optical
quantum memory, efficient optical pumping into the long-
lived Zeeman level needs to be demonstrated. Bulk REI
quantum memories already boast high storage efficiency
[32] with multi-mode capacity [3]. Their implementa-
tions in our nanophotonic platform open the possibility
of multiplexed systems for on-chip quantum repeaters.
For Nd, high-fidelity storage of entanglement based on
atomic frequency comb (AFC) has been demonstrated
[33, 34]. With cavity impedance matching [35], unit stor-
age efficiency is achievable with a mesoscopic ensemble of
6
cavity-coupled ions. Meanwhile, long-lived nuclear spin
coherence of 9 ms in
145
Nd [36] bodes well for spin-wave
quantum memories using our nano-resonators. These de-
vices can be further coupled to superconducting or op-
tomechanical devices to enable hybrid quantum systems
[4]. Furthermore, the technology can be readily trans-
ferred to other wavelengths, such as 1.5
μ
m for telecom
quantum memories using Er
3+
:YSO or 590 nm for long-
haul quantum hard drives using Eu
3+
:YSO [16].
METHODS
YSO nano-resonator design and fabrication.
The
nano-beam has an equilateral triangular cross-section
with each side of 780 nm. This geometry allows a cir-
cular fundamental mode field that can be efficiently cou-
pled with a free space laser beam. The cavity is formed
by forty equally spaced grooves of lattice constant 340
nm on the nano-beam, except for a defect introduced
at the center by perturbing the lattice constant. The
depth of the grooves is 65% of the beam height. The
triangular nano-beam resonator was fabricated using fo-
cused ion beam (FIB) milling followed by wet etching of
Ga+ contaminated YSO in diluted (10%) hydrochroric
acid. An ion beam of 20 kV, 0.2 nA was used to fabri-
cate the suspended nano-beam waveguide by milling at
30
◦
angle with respect to the crystal surface normal. A
small ion beam of 23 pA was then used to accurately
pattern the grooves on top of the nano-beam. Limited
by the finite width of the FIB beam, the side-walls of the
grooves in the actual device were not vertical, but had
an angle of 6
◦
. This leads to a degraded theoretical
Q
of
5
.
0
×
10
4
. We were able to reproducibly fabricate arrays
of resonators (up to six) in a batch (Supplementary In-
formation S1), with all the devices measuring resonances
close to 883 nm and quality factors varying from 1,100
to 10,000.
Experimental setup for the photon echo measure-
ments.
A 500 mT external magnetic field was ap-
plied at
α
=135
◦
relative to the crystal D
1
axis using
a pair of permanent magnets (see Fig. 1(b)). The
π/
2
and
π
Gaussian pulses were generated by amplitude-
modulating the Ti:Sapphire laser with two acousto-optic
modulators (AOM) in series, with each in a double-pass
configuration. The two pulse widths were 250 ns and 400
ns at a repetition rate of 1 kHz.
The average (peak)
power of the excitation pulses was 210 nW (320
μ
W)
measured after the objective lens. The extinction ratio
between the on and off level of the pulses was
∼
120 dB,
ensuring sufficient signal-to-noise ratio for detecting echo
photons using a Si single-photon counter (Perkin Elmer
SPCM). A third shutter AOM in single-pass configura-
tion was inserted just before the photon counter to block
the strong excitation pulses from saturating the detector.
The extinction ratio of this shutter AOM was 30 dB.
High-resolution laser spectroscopy on cavity-
coupled Nd
3+
ions.
For the cavity transmission exper-
iments, the Ti:Sapphire laser (M Squared SolsTiS) was
continuously scanned at a rate of 10 MHz per second.
The high-sensitivity CCD camera in the spectrometer
(Princeton Instruments PIXIS) registers the transmitted
photon counts intermittently at an adjustable frame rate
(frames per second (fps)) with an exposure time 0.01 s
for each frame. Therefore one exposure corresponds to a
spectral width of 10 MHz
×
0.01=100 kHz scanned by the
laser, which is equal to the homogeneous linewidth of the
0.2% doped sample. The long term drift of the laser is 10
MHz per hour, so the drift during each exposure should
be inconsequential. Each data point in Fig. 4(b,c) rep-
resents the photon count collected in one camera expo-
sure, corresponding to the signal contributed by the ions
within one homogeneous linewidth. The frame rate was
0.1 fps for the coarser scan in Fig. 4(b), corresponding
to a spectral interval of 100 MHz between two adjacent
data points. The frame rate was 8.2 fps for the fine scan
in Fig. 4(c), with a spectral interval
∼
1 MHz. Each
data point was obtained with one scan. Several scans
at different spectral regions were performed and stitched
together to cover the entire bandwidth in Fig. 4(b).
Estimation of the branching ratio.
The measured
optical depth of a 15-
μ
m-long nano-beam resonator at
3.8 K was
d
=0.1, from which we deduce an oscillator
strength of
f
= 6
.
5
×
10
−
7
and a spontaneous emission
rate of this transition to be
γ
883
= 1
/τ
883
=1/5.6 ms [24].
With a measured bulk medium lifetime
τ
0
=250
μs
, the
branching ratio was thus estimated to be
τ
0
/τ
883
≈
4
.
5%.