arXiv:quant-ph/0606033v2 4 Sep 2006
Observation of Strong Coupling between One Atom and a Monolithic Microresonator
Takao Aoki
a
, Barak Dayan, E. Wilcut, W. P. Bowen
b
, A. S. Parkins
c
, and H. J. Kimble
Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, California 91125, USA
T. J. Kippenberg
d
and K. J. Vahala
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
(Dated: February 1, 2008)
Over the past decade, strong interactions of light and matter at the single-photon level have enabled
a wi de set of scientific advances in quantum optics and quantum information science. This work
has been performed principally wi thin the setting of cavity quantum electrodynamics [1, 2, 3, 4]
with diverse physical system s [5], including single atoms in Fabry-Perot resonators [1, 6], quantum
dots coupled to micropillars and photo nic bandgap cavities [7, 8], and Cooper-pairs interacting
with superconducting resonators [9, 10]. Experiments with single, localized atoms have been at the
forefront of these advances [ 11, 12, 13, 14, 15] with the use of optical resonators in high-finesse
Fabry-Perot configurations [16]. As a result of the extreme technical challenges involved in further
improving the multilayer dielectric mirror coatings [17] of these resonators and in scaling to large
numbers of devices, there has been i ncreased interest in the development of alternative microcavity
systems [5]. Here we show strong coupling between individual Ces ium atoms and the fields of a
high-quality toroidal microresonator. From observations of transit events for single atoms falling
through the resonator’s evanescent field, we dete rm ine the coherent coupling rate for interactions
near the surface of the resonator. We develop a theoretical model to quantify our observations,
demonstrating that strong coupling is achieved, w ith the rate of coherent coupling exceeding the
dissipative rates of the atom and the cavity. Our work opens the way for investigations of optical
processes with single atoms and photons in lithographically fabricated microresonators. Applications
include the implementation o f quantum networks [18, 19], scalable quantum lo gic with photons [20],
and quantum information processing on atom chips [21].
The realization of large-s c ale quantum networks [18, 19] require s the capability to inter-connect many ‘quantum
nodes’, each of which could consist of a microresonator containing a set of trapped atoms. The ‘quantum channels’
to connect these nodes would be optical fibres, with strong interactions in cavity quantum electrodynamics (QED)
providing an efficient interface between light and matter. Here we provide a critical step towards a feasible quantum
network by demonstrating strong coupling of single atoms to microresonators fabricated on Silicon wafers in large
numbers by standard lithographic techniques followed by a laser-reflow process [22]. Combined with the capability to
couple light efficiently to and from such cavities directly via a tapere d optical fibre [23], toroidal microcavities offer
promising capabilities for new nonlinear interactions of single atoms a nd photons across distributed networks.
Our efforts follow the pioneering work of V. Braginsk y et al. [24] and later studies [25] by employing the whispering-
gallery modes of fused silica microtoro idal resonators [26]. As depicted in Fig. 1, a Silicon chip containing a collection
of 35 microtoroidal resonators is located inside a vacuum chamber at 10
−9
Torr and is positioned to couple a particular
resonator to a tapered fibre. The toroids have major diameter D ≃ 44 µm and minor diameter d ≃ 6 µm [2 6]. By
judicious choice of the point of contact b e tween the surface of the resonator and the tapered fibre, we attain critica l
coupling, in w hich the forward propagating power P
F
in the fibre drops to near zero for the pro be frequency ω
p
equal to the cavity r e sonance frequency ω
C
[23]. Measurements of the cavity transmission in the absence of atoms
are presented in Fig. 2. Note that the forward flux P
F
and associated transmission spectrum T
F
are analogous to
the reflected flux and reflection spectrum from a Fabry-Perot cavity [23]. B y varying the temperature of the Silicon
chip, the detuning ∆
AC
≡ ω
C
− ω
A
between ω
C
and the atomic resonance at ω
A
(6S
1/2
, F = 4 −→ 6P
3/2
, F
′
= 5
′
transition in Cesium) can be controlled with uncertainty ≃ ±2 MHz (see Appendix A).
Cold atoms are delivered to the vicinity of the toroidal resonator from a small clo ud of Cesium atoms cooled to
T ≃ 10 µK and located 10 mm above the Silicon chip. Every 5 seconds, the cloud is dropped, resulting in about
2 × 10
6
atoms in a 3 mm ball at the height of the chip, with then a few dozen atoms passing through the external
evanescent field of the toroidal resonator. By way of two single-photon detectors (D
F 1
, D
F 2
) (see Appendix A), we
continuously monitor the forward propagating signal P
F
from a frequency-stabilized probe beam P
in
coupled to the
toroidal resonator. The interaction of each individual atom with the evanescent field destroys the condition of critical
coupling, leading to an increase in P
F
. The measurement cycle then repeats itself for 2.5 seconds for a reference
2
FIG. 1: Simple diagram of the experiment. (a) A cloud of cold Cesium atoms and associated trapping lasers above an
array of microtoroidal resonators. Light from the probe beam P
in
is coupled into a resonator by way of the fibre taper, with
the forward propagating output P
F
coupled into t he taper. (b) Illustration of an SiO
2
microtoroidal resonator, fibre taper,
and atom cloud above. The calculated field distribution for the lowest order resonator mode is shown by the color contour plot
on the right. Cold Cesium atoms fall through the external evanescent field of this mode and are thereby strongly coupled to
the resonator’s field.
measurement, this time with no atomic cloud formed above the microtoroid.
Figure 3 displays typical records C(t) for the number of single-photon detection events within time bins of δt = 2
µs as functions of time t for the forward signal P
F
(t). Measurements are displayed with (Fig. 3(a)) and without (Fig.
3(b)) atoms fo r the case of equal probe and cavity frequencies, ω
p
= ω
C
, for ∆
AC
≈ 0, and with mean intracavity
photon number ¯n
0
≈ 0.3 for the forward circulating mode of the toroidal r e sonator a (see Appendix A). The traces in
both Fig. 3(a) and Fig. 3(b) exhibit background levels that result fr om the nonzero ratio P
F
/P
in
∼ 0.005 at critical
coupling in the absence of atoms . However, Fig 3(a) clearly evidences sharp peaks of duration ∆t ≈ 2 µs for the
forward propagating light P
F
(t), with a n individual peak shown more clearly in the inset. Each event arises from the
transit of a single atom through the resonant mo de of the microtoroid, with about 30 events per cycle observed. Fig.
3(c) examines the temporal profile of transit events in more detail by way of the cross correlation Γ(τ) of photoelectric
counts C
1
(t
1
), C
2
(t
1
+ τ) from the detectors (D
F 1
, D
F 2
) for P
F
(see Appendix A). This result agrees reasonably well
with the theoretical prediction for atom tr ansits through the calculated field distr ibution shown in Fig. 1(b).
By applying a thre shold requiring C(t) ≥ 6 counts for C(t) as in Fig. 3 (a, b), we find the average time dependence
¯
C
≥6
(t) over about 100 measurement cycles. Fig. 3(d) displays the results both with and without atoms, with the
average counts Σ
6
(t) derived from
¯
C
≥6
(t) by summing over successive time bins δt = 2 µs for 1 ms intervals. The
peak in transit events is consistent with the expected distribution of arrival times for atoms dropped from our atom
cloud. By c ontrast, negligible excess events (i.e., C(t) ≥ 6) are reco rded for the cases witho ut atoms.
Foc us ing attention to the central region indicated by the das hed lines in Fig. 3(d), we examine in Fig . 3(e)
the probability P (C) to recor d C counts within δt = 2 µs. Evidently, when the atom cloud is present, there is a
statistically significant incr e ase (of at least fifteen sigma) in the number of events with C ≥ 4. These are precis ely
the events illustrated by the inset in Fig. 3(a) and the cross correlation in Fig. 3(c), and are associated with single
atom tra ns its near the surface of the toroidal resonator. By varying the value of ¯n
0
, we have confirmed that the larg e
transit events evident in Fig. 3 are markedly decreased for ¯n
0
& 1 photon, which indicates the saturation of the
atom-cavity system.
A quantitative description of our observations in Fig. 3 of individual atom transits requires the development of
a new theoretical model in cavity QED. In the appendix, we present such a model and show that the underlying
description of the interaction of an atom with the fields of the toro idal resonator is in ter ms of normal modes (A, B)
(see Fig. 5 in Appendix B), which have mode functions ψ
A,B
(ρ, x, z) that are standing waves (cos kx, sin kx) ar ound the
circumference x of the toroid, with ρ the radial distance from the surface and z the vertical coordinate. ψ
A,B
(ρ, x, z)
3
FIG. 2: Cavity transmission function T
F
= P
F
/P
in
as a function of probe frequency ω
p
. The lower trace is taken
for critical coupling, and the upper trace for conditions of under coupling [23]. From fits to such traces for critical coupling
(red dashed curve), we find (κ, h)/2π = (17.9 ± 2.8, 4.9 ± 1.3) MHz, with κ, h being the overall cavity field decay rate and the
scattering-induced coupling between two counter-propagating modes of the microtoroid, respectively (see Appendix for more
details). Inset – Photograph of a microtoroid and coupling fibre.
have a calculated peak coherent coupling g
0
/2π of 70 MHz for the lowest order modes of our resonator (such as that
illustrated in Fig. 1(b)). The normal modes A,B result from the coupling o f two oppositely directed travelling waves by
scattering at rate h, with the re sulting mode splitting manifest in Fig. 2. Note that the presence of two normal modes
leads to a
√
2 increase in the coupling constant in our case as compared to the one predicted by the Jaynes-Cummings
model for an atom interacting with a single traveling-wave mo de (see Appendix B for further deta ils ).
Guided by this theory, we have performed a series of measurements similar to those presented in Fig. 3 to determine
the coherent coupling rate g
0
for interactions of single a toms with our toroidal resonator, but now with various
values of the atom-cavity detuning ∆
AC
, keeping the probe resonant w ith the cavity: ω
p
≈ ω
C
= ω
A
+ ∆
AC
. The
qualitative idea is that large single-atom transit events will occur only over a range of detunings ∆
AC
determined by
g
0
. Specifically, the decrease in the fo rward transmission T
F
induced by atom transits as a function of ∆
AC
is described
by a Lorentzian with width β set by g
0
(see Appendix B). In our case, g
0
= g
0
(ρ, x, z) ≈ g
0
(ρ, x, V t), where V is the
velocity of the dropped atoms in the z direction. Thus, a numerical integration was perfo rmed over ρ, x, and t to
derive the theoretical expectation for T
F
(∆
AC
), pres e nted in Fig. 4(a) for three values of g
m
0
, where g
m
0
is the maximal
coupling that an atom can experience in its interaction with the cavity modes . Indeed, we see that the width β grows
monotonically with g
m
0
. However, the average value of T
F
is not a parameter that is readily measured in our current
exp eriment, in which we expect many short individual transits, some of which are too weak to be distinguished from
the background noise (Fig. 3(e)). A parameter that describes our actual exp e rimental measurements more closely
is the probability to obtain a transit which results in transmission above a certain threshold. The two meas ures are
closely related, such that this probability decreases with detuning ∆
AC
in the same fashion as does T
F
.
Figure 4(b– d) presents the results of our measur e ments for the average number of trans it events per atom drop,
N
av
drop
(C ≥ C
0
), which have photoelectric counts greater than or equal to a threshold value C
0
for a set of seven
detunings ∆
AC
. In ac c ord with the expectation set by Fig. 4(a), there is a decreas e in the occurrence of large
transit events for increasing ∆
AC
in correspondence to the decrease in the effective atom-cavity coupling coefficient
for large atom-cavity detunings. The full curves shown in Fig. 4(b– d) are the results of theoretical calculation for
these measurements, with the relevant cavity parameters (κ, h) determined from fits as in Fig. 2.
We first ask whether the data might be explained by an effective value g
e
0
for the coherent coupling of atom and
cavity field, without taking into account the fact that individual atoms transit at radial distances ρ which vary
from atom to atom. Fig. 4(b) examines this possibility for various va lues of g
e
0
, assuming a coupling c oefficient
g
e
0
ψ
A,B
(x) = g
e
0
[cos kx, sin kx] averaged along one period in x (as in Fig. 4(a)). Apparently, an effective value
g
e
0
/2π ≈ 40 MHz provides r e asonable correspondence between theory and experiment for large events C ≥ 6.
We adapt our theory to the actual situation o f atoms arriving randomly at radial and circumferential coordinates by
introducing a mes h grid over (ρ, x), and then computing the cavity transmission function T
F
(t) from ψ
A,B
(ρ, x, z(t))
for atomic trajectories z(t) over this gr id. We account for the time resolution δt = 2 µs of our data acquisitio n by
a suitable average of T
F
(t) over such time bins (as was a lso tr ue in Fig. 4(b)). The re sults from these calculations
4
P(C)
C
e
Without atoms
With atoms
Poissonian
0 2 4 6 8 10 12
10
−6
10
−4
10
−2
10
0
Γ(τ)
Delay [µs]
c
−10 −5 0 5 10
1
1.2
1.4
1.6
1.8
Σ
6
(t)
Time [ms]
d
35 40 45 50 55
0
2
4
6
x 10
−3
C(t)
Time [ms]
b
35 40 45 50 55
0
2
4
6
8
10
C(t)
Time [ms]
a
35 40 45 50 55
0
2
4
6
8
10
Time [µs]
−10 0 10
0
5
10
FIG. 3: Measurements of the forward signal P
F
in the presence of falling atoms (blue) and without atoms
(green). (a, b) Single-photon counting events C(t) as a function of time t after the release of the cold atom cloud at t = 0,
with (a) and without (b) atoms dropped. C(t) gives the t otal number of counts recorded for time bins of δt = 2 µs duration.
The inset in (a) shows the time profile for a single-atom transit. (c) Normalized cross-correlation Γ(τ) of the forward signal
counts from two detectors (D
F 1
, D
F 2
) showing the time profile associated with atom transit events. The smooth (red) curve
is the theoretically predicted average cross correlation for a transit event with one atom, taking into account drop height of 10
mm and the spatial sh ape of t he mode, as depicted in Fig. 1(b). (d) Counts Σ
6
(t) obtained from
¯
C
≥6
(t) by summing over
1 ms intervals, compared to a Gaussian distribution which fits the rate of atom transits assuming a 3 mm (FWHM) cloud of
cold atoms d ropped from 10 mm above the microtoroid. (e) Probability P (C) to detect C counts within δt = 2 µs bins for
the central interval shown by the vertical dashed lines in (d), compared with Poissonian statistics (red) with the same mean
number of counts (∼ 0.25 per 2 µs). The excess probability above the poissonian level in the no atoms case is predominately
due to instability in the cavity temperature, which results in small fluctu ations in the forward flux. Error bars show ±1 s.d.
5
∆
AC
T
F
av
/ T
F
av
(∆
AC
= 0)
a
0 20 40 60
0
0.2
0.4
0.6
0.8
1
g
0
= 35
g
0
= 50
g
0
= 65
∆
AC
N
drop
av
(C ≥ 6)
b
g
0
e
=50
40
30
0 20 40 60
0
1
2
3
4
MOT
no MOT
∆
AC
N
drop
av
(C ≥ 5)
c
0 20 40 60
0
2
4
6
8
10
g
0
m
= 35
g
0
m
= 50
g
0
m
= 65
∆
AC
N
drop
av
(C ≥ 6)
d
0 20 40 60
0
1
2
3
4
g
0
m
= 35
g
0
m
= 50
g
0
m
= 65
FIG. 4: Measurements of transit events as a function of the atom-cavity detuning ∆
AC
. Events are shown in the
presence of atoms (filled circles) and without atoms (empty circles), compared with the theoretical calculations (lines). (a)
Theoretical calculation for the average of the transmission T
F
(ω
p
= ω
C
) as a function of (∆
AC
, g
0
). Red, g
0
=35; blue, g
0
=50;
green, g
0
=65. (b-d) Measurements for t he average number of events per drop of the atom cloud N
av
drop
(C ≥ C
0
) plotted against
the atom-cavity detuning ∆
AC
, with C
0
= 6; (b, d) and C
0
= 5 (c). Error bars show ±1 s.d. The data are taken for probe
frequency ω
p
≈ ω
C
= ω
A
+ ∆
AC
. The full curves are theoretical results as discussed in the text. The widths of the curves are
determined from the experimental uncertainties in (κ, h). (b) Theory for N
av
drop
(C ≥ 6) without radial averaging to deduce an
effective coupling g
e
0
/2π = 40 MHz. Theory for (c) N
av
drop
(C ≥ 5) and (d) N
av
drop
(C ≥ 6) with radial and azimuthal averaging
leading t o g
m
0
/2π = 50 MHz. Red, g
m
0
=35; blue, g
m
0
=50; green, g
m
0
=65.
are displayed in Fig. 4 (c–d) as the set of full curves for three values of c oherent coupling g
0
for the cavity mode
functions ψ
A,B
(ρ, x, z), where in Fig. 4(b– d) the theory is scaled to match the measured N
av
drop
(C ≥ C
0
) a t ∆
AC
= 0.
From such comparisons, we determine the maximal accessible g
m
0
/2π = (50 ± 12) MHz. Note that this conclusion is
insensitive to the choice of cutoff C
0
over the range 4 ≤ C
0
≤ 9 for which we have significant transit events. Strong
coupling with g
m
0
> (κ, γ) is thereby achieved, where (κ, γ)/2π = (17.9 ± 2.8, 2.6) MHz are the dissipative rates for
the cavity field and the atom.
According to our calculations, g
m
0
/2π = 50 MHz corresponds to the coupling rate expected at a distance o f roughly
45 nm from the surface of the microtoroid. We estimate that due to the attractive van der Waals forces (see Ref [27]),
atoms which enter the evanescent field with a distance ρ ≤ 45 nm from the microtoroid are expected to strike its
surface in less than 1 µs, thus preventing such atoms from generating appreciable transit events in the transmission
function T
F
.
In summary, we report the first observation of strong coupling for single atoms interacting with an optical resonator
other than a conventional Fabr y-Perot cavity. The mo nolithic microtoroidal resonators [22] employed her e have the
capability of input-output coupling with small parasitic losses , with a demonstrated ideality of more than 9 9.97%
[23]. Moreover, quality factors Q = 4 × 10
8
have b e e n rea lize d at λ = 1550 nm [28] and Q ≃ 10
8
at λ = 850 nm
[26], with good prosp e c ts for impr ovements to Q ≃ 10
10
[29]. For these parameters, the efficiency for c oupling single
photons into and out of the r e sonator could approach ǫ ∼ 0.99 − 0.999 [23], while still re maining firmly in the regime
of strong coupling [26]. Such high efficiency is critical for the realization of scalable quantum networks [18, 19] and
photonic quantum computation [20]. Indeed, of the diverse possibilities for the distribution and processing of quantum
information with optical cavities [5, 7, 8], the system of single atoms coupled to microtoro idal re sonators arguably
provides one of the most promising avenues. Beyond efficient input-output coupling [23], strong coupling to a material
system with long-lived internal states has now been achieved, although here in a primitive, proof-of-principle setting.
An outstanding technical challenge is to trap single atoms near the surface of the microtoroid, with one poss ibility
![FIG. 2: Cavity transmission function TF = PF /Pin as a function of probe frequency ωp. The lower trace is taken for critical coupling, and the upper trace for conditions of under coupling [23]. From fits to such traces for critical coupling (red dashed curve), we find (κ, h)/2π = (17.9± 2.8, 4.9± 1.3) MHz, with κ, h being the overall cavity field decay rate and the scattering-induced coupling between two counter-propagating modes of the microtoroid, respectively (see Appendix for more details). Inset – Photograph of a microtoroid and coupling fibre.](/figures/fig-2-cavity-transmission-function-tf-pf-pin-as-a-function-3ujcy769.webp)
