Fast cavity-enhanced atom detection with low noise and high fidelity

Abstract

Cavity quantum electrodynamics describes the fundamental interactions between light and matter, and how they can be controlled by shaping the local environment. For example, optical microcavities allow high-efficiency detection and manipulation of single atoms. In this regime, fluctuations of atom number are on the order of the mean number, which can lead to signal fluctuations in excess of the noise on the incident probe field. Here we demonstrate, however, that nonlinearities and multi-atom statistics can together serve to suppress the effects of atomic fluctuations when making local density measurements on clouds of cold atoms. We measure atom densities below 1 per cavity mode volume near the photon shot-noise limit. This is in direct contrast to previous experiments where fluctuations in atom number contribute significantly to the noise. Atom detection is shown to be fast and efficient, reaching fidelities in excess of 97% after 10 μs and 99.9% after 30 μs.

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Received 14 Feb 2011 | Accepted 8 Jul 2011 | Published 9 Aug 2011 DOI: 10.1038/ncomms1428

Fast cavity-enhanced atom detection with low noise and high delity

J. Goldwin1,2, M. Trupke1,, J. Kenner1, A. Ratnapala1 & E.A. Hinds1

Cavity quantum electrodynamics describes the fundamental interactions between light and matter, and how they can be controlled by shaping the local environment. For example, optical microcavities allow high-efciency detection and manipulation of single atoms. In this regime, uctuations of atom number are on the order of the mean number, which can lead to signal uctuations in excess of the noise on the incident probe eld. Here we demonstrate, however, that nonlinearities and multi-atom statistics can together serve to suppress the effects of atomic uctuations when making local density measurements on clouds of cold atoms. We measure atom densities below 1 per cavity mode volume near the photon shot-noise limit. This is in direct contrast to previous experiments where uctuations in atom number contribute signicantly to the noise. Atom detection is shown to be fast and efcient, reaching delities in excess of 97% after 10 s and 99.9% after 30 s.

1 Centre for Cold Matter, Imperial College, Prince Consort Road, London SW7 2BW, UK. 2 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK. Present address: Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria. Correspondence and requests for materials should be addressed to J.G. (email: [email protected]).

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High-nesse optical resonators can improve the sensitivity of atom detection by increasing the lifetime of photons and conning them to a small volume1. Long photon lifetime,

controlled by cavity length and mirror reectivity, increases the eective optical thickness of an intracavity sample by a factor on the order of the nesse F1. Small mode volume, which depends only on the geometry of the resonator, increases the energy density per photon and, therefore, the Einstein coefficients describing transition rates. Thus, the spontaneous emission rate of an atom is increased by coupling it to a resonant cavity2. Importantly, all the extra photons are emitted into the cavity mode, making it possible to detect uorescence even at very low atom density. For sufficiently small mode volumes, a single cavity photon becomes intense enough to saturate the atomic transition. In this regime, vacuum uctuations modify the spectral properties of the coupled atom-cavity system3

in such a way as to allow detection at the single-atom level47.

Recently, there has been growing interest in cold-atom experiments with atomic density distributions extending throughout or beyond the range of the cavity eld812. For multiple atoms, the radiative behaviour can be coherent13,14. Although the gas may be dilute, the common coupling to the electromagnetic eld produces eective long-range interactions between the atoms that can lead to self-organization15,16 and collective motion17, as well as super-radiant Rayleigh scattering and collective atomic recoil lasing18. Recently experimenters have exploited these eects to realize a quantum phase transition from a BoseEinstein condensate to a supersolid19.

A central parameter in describing cavity-enhanced detection is the dimensionless single-atom cooperativity20, C1 = g2/(2), where 2g is the single-photon Rabi frequency at the peak of the cavity intensity distribution, 2 is the cavity linewidth (full width at half maximum), and 2 is the natural atomic linewidth. The cooperativity determines both the eect of a single atom on the cavity spectrum, and the rate of uorescence into the cavity.

In the case of multiple atoms, the cooperativity is generalized by dening CN = C1Ne, where the eective atom number is (ref. 21)

N

Excitation laser

Optical fibre

Planemirror Falling 87 Rb atoms

Concave mirror array chip

Figure 1 | Schematic of experiment. (not to scale) Clouds of 87Rb atoms are laser cooled and dropped through a high-nesse optical microcavity. In the experiment, the atomic density is approximately Gaussian with a width three orders of magnitude larger than the cavity mode waist, allowingus to make a local density approximation for the atoms. The system is probed either by monitoring the cavity reection, with Jin incident photons per second, or by observing uorescence into the cavity mode inducedby a laser beam with Rabi frequency (and Jin = 0). In either case the output stream of Jout photons per second is detected using a single photon counting module.

transition, then the steady-state rate of photons travelling from the cavity to the detector is (ref. 23)

J

J b C

C

= C s

C s

2

1 2

+

+

out

in reflection

flu

N

N

2

2

,

(2)(2)

2 (1 2 )

gx oorescence

N

,

+ +

N

r r r

with (r) being the atomic density, L the cavity length, and (r) = sin(2z/)exp[ (x2 + y2)/w2] the cavity eld mode function ( is the wavelength). It is important to note that Ne is a random variable, generally distinct from its mean value Ne. When the atom cloud is much larger than the cavity mode volume Ncav = w2L/4, the

mean atom density is approximately uniform over the interaction region, and Ne (0)Vcav. At low densities, single-atom physics dominates, whereas, at higher densities, multi-atom eects become important 21.

Here we perform local density measurements on large dilute clouds of atoms in the crossover regime, paying particular attention to signal uctuations. We show that even at densities on the order of one atom per cavity mode volume, the eects of atomic shot noise are heavily suppressed. We then compare our results with state-ofthe-art experiments on single trapped atoms, demonstrating a fast detection time and high delity.

ResultsOptical noise suppression. Our apparatus, shown schematically in Figure 1, has been described in detail in refs 7,22 (Methods). We detect atoms either by measuring changes in the intensity of a probe beam reected from the cavity; or detecting uorescence when exciting the atoms uniformly with a laser beam propagating transverse to the cavity axis. We refer to these simply as reection and uorescence measurements, respectively. If atoms are present and the cavity and lasers are resonant with the free-space atomic

(1)(1)

eff d

=

| c r

2 3

L ( )| ( )

where Jin is the number of incident probe photons per second and b2 characterizes the reection fringe contrast in the absence of atoms. The cooperativities for reection (CN) and uorescence (CN) are not generally the same because they depend on the polarization of the probe light and the excitation light, respectively24 (see Methods). In uorescence, s = 1/2(/)2 is the free-space saturation parameter for excitation driven at a Rabi frequency , whereas is the probability for an intracavity photon to pass from the cavity into the bre. Finally, we have used the facts that (g/)4 1, and that the atomic excited state fraction is small in our reection measurements. For this work, g/(2) = 98.4(1.6) MHz, /(2) = 5200(100) MHz, and /(2) = 3 MHz, giving C1 = 0.307(11).

It is important to note that Jout as described by equation (2) is only linear in atom density for small values of the cooperativity; for reection measurements Jout saturates with increasing CN, while for uorescence Jout reaches a maximum when CN = 1/2(1 + s)1/2 and then rolls over and vanishes. We will show that this allows us to operate in a regime where we remain sensitive to variations in mean atomic density while damping out the eects of large instantaneous uctuations.

Figure 2a shows the results of repeated reection measurements of clouds falling through the cavity. A circularly polarized probe drives the atomic cycling transition, maximizing the atom-eld coupling strength. At early and late times, there are no atoms in the cavity, so the reected light is at its minimum value, determined by the incident probe power and the empty cavity fringe contrast. The reected intensity rises when there are atoms in the cavity. These experimental runs are averaged in Figure 2b, which also shows a t to

0

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10

0.6

0.8

300

200

0.4

0.6

8

Experiment no.

Mean counts per s

6

0.5

0.6

0.4

Photon counts per s

0.3

73.00 73.01 73.02 73.03

Count variance

4

0.4

0.4

100

1

2

Atom density

(108 cm3)

0.2

0.2

0.2

50

60

70

80

90

0

Time (ms)

0 0 0.2 0.4 0.6 0.8

Time (ms)

Mean counts

1.006

Count variance/mean

0.1

1.041.02 1

0.980.96

Figure 3 | Fluorescence measurements. (a) Fluorescence pulse, averaged over 250 drops. The exciting laser is pulsed on after the peak atomic density has passed through the cavity. At the start of the pulse, shown in detail in the inset, Neff = 1.24(5). This is slightly larger than in Figure 2 due to a higher atom number in the initial MOT. (b) Variance of uorescence counts as a function of mean. Red circles are from the data used in (a), and blue triangles are from a set where the drive beam is pulsed on at a later time in the drop, with initial Neff = 0.50(2). The green line is the photon shot-noise limit. The grey box corresponds to the inset in (a).

1.004

g(2)( )

1.002

1.000

0 10 20 30 40 50 60

(s)

50

60

70

80

90

Time (ms)

Figure 2 | Reection measurements. (a) Detected photon counts for 300 identical experiments. The atoms are released at 39.5 ms. Counts increase from blue to red. Data were taken with 2 s resolution, and the image was then re-binned to 1 ms. (b) The data in red are averages over the 300 drops shown in (a), whereas the white curve is a t to equation (2) assuming a Gaussian dependence of CN on time; the dashed blue line gives the value expected from a single atom maximally coupled to the cavity mode.

(c) Second-order correlation g(2)() as a function of time delay . Red points are from data with Neff = 0.225(17), and blue points are taken without atoms; error bars show the standard error of the mean at each delay time from 50 trials. The solid curves are the theoretical expectations. (d) Ratio of ensemble variance to mean versus time. The red curve is calculated from the raw data in (a) for each 2 s time bin, and then a 100 s running average is applied to smooth the result; the white line is the photon shot-noise level.

0 73.0 73.5 74.0 74.5

the evolution expected from equation (2) with a peak Ne = 1.06(4),

corresponding to only 4.9(2)108 atoms cm 3. For reference, the dashed line shows the expected reection with a single atom maximally coupled to the cavity mode. Note that the 10 ms width of the curve reects the size of the cloud, which is determined by its temperature. By contrast, the typical transit time for a single atom passing through the width of the cavity mode is ~14 s, so the cloud is very large compared with the extent of the cavity eld. Individual transits are revealed in Figure 2c, where we show the measured second-order (intensity) correlation, given by

g k t k t k t

( )( ) ( ) ( )( )

Figure 3a shows the uorescence signal. As the cloud falls through the cavity we switch on a resonant excitation beam whose (downward) propagation direction and polarization are both perpendicular to the cavity axis. The photon-count rate immediately jumps to a high level as a result of the laser-induced uorescence. Independent reection measurements determine that the initial Ne = 1.24(5). Although the atom number is nearly constant over several ms during the reection measurements, the signal here decays roughly exponentially with a time constant of order 100 s. This is because the atoms are heated and pushed out of the cavity by the excitation light24, which is much more intense than the probe light used in reection measurements. In Figure 3b, we plot how the variance of the uorescence count over 250 repetitions varies with the mean number of counts. Once again, we see that the uctuations are very near the photon shot-noise limit, which is indicated by the solid line.

We reiterate that these results are in direct contrast to similar experiments operating in the linear regime, that is, when the photon counts are strictly proportional to atom number or density. For example, we can compare our results in Figure 3b with Figure 2b of ref. 25, which used micro-optics to detect atoms without a cavity. They observed that the ratio of variance to mean doubled in the presence of atoms. To calculate the noise level for uorescence detection in the linear regime, one can apply Mandels theory as described in ref. 26. Atomic motion is negligible over a single 1 s time bin, so we consider each bin to have a xed number of atoms N, producing a Poissonian photon count k with a mean of photons per atom (the background count is negligible compared with ). As N uctuates over repeated experiments, the photon counts obey Var(k)/k = 1 + Var(N)/N. If atoms are positioned randomly with a uniform probability distribution, the number of atoms in a given volume follows a Poisson distribution and Var(k)/k = 1 + , independent of N. In our experiment, the yield of uorescence photons for one (hypothetical, maximally coupled) atom is 0.42(2) in 1 s. Clearly this theory does not describe our experiment, whose measured value of [Var(k)/k 1] = 0.09(3) is much smaller.

Our measurements require a dierent analysis, as the assumption of linearity is violated in equation (2) and the variable Ne should be considered rather than N. We proceed as follows. Our experiment operates with mean intracavity photon number n aa 1,

and 1 g 1, meaning excitations of the cavity eld result in immediate emission of photons rather than reabsorption by atoms.

2 t t

= +

where k is the number of photons counted during a time window centred on t, and is the relative delay between windows. Overbars denote an average over t using a 500 s segment of data, throughout which Ne is approximately constant. The solid red curve shows

the expectation for a single atom crossing the cavity according to equation (2). The only free parameter is the amplitude of the peak, which accounts for having more than one atom pass through the cavity during the trace, but with not all atoms optimally coupled. In Figure 2d, we plot as a function of time the variance of the photon counts divided by the mean, evaluated over the 300 repetitions of the experiment. Although one might have expected to see an increased variance with the arrival of the atom cloud, there is, in fact, no sign of such an increase. We return to this point below, when we see similarly low noise levels in our uorescence measurements.

(3)(3)

2

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Over a sufficiently short time T, the probability of emitting a photon is just 2aaT. Then we have

Var Var

( ) 1 2 (( ))

Table 1 | Comparison with other experiments.

Ref. S1 B F1max, T1max F2max

26 5.6 0.28 90.9,544 97.5

25 36 0.311 97.6,132 99.8

29 54.5 2.18 92.2,60 98.1

30 0.13 0.05 72.1,9853 80.5

27 94 0.05 99.773,80 99.99982

28 190 1.4 97.87,26 99.85 This work 420(20) 3.84(6) 97.46(13), 11.2(4) 99.79(2)

Rates are in cts/ms, delities in per cent, and T in s. Note that T =KT for p=1/2. Refs

26,29,27,28 use cavities, while refs 25,30 use optical waveguides without cavities. Refs 27,28

describe non-destructive detection.

= +

kk T

n n

k xe

where is the total collection efficiency from bre to detector, and T is assumed to be much longer than any correlation times for the uorescence (for example, 1/, /g2). Double brackets denote statistics taken over the conditional distribution of n given a xed Ne,

and the distribution of Ne itself. We obtain n through the master equation for the density matrix, and use the results of ref. 21 for the probability density for Ne (Methods). The value of k obtained in this way is almost identical to what one obtains by simply setting CN = C1Ne in equation (2). For the uctuations, we obtain

Var(k)/k = 1.095(8), in excellent agreement with our observed value. A similar treatment of the reection measurements in (Fig. 2) gives Var(k)/k = 1.005(2), consistent with the value 1.002(4) from the data. Note that for the two types of measurement, the ratio Var(Ne)/Ne = 3/8 is the same, but the nonlinearity of Jout is quite dierent. Calculations for both types of measurement, with the same Ne and Jin adjusted to have equal numbers of signal photons, show that the noise suppression is still much stronger for reection. This stems from the saturation of the reection signal at large instantaneous Ne versus the roll-over of the uorescence.

For both measurements, we conclude that the statistics of Ne and

the nonlinearity of the interactions are jointly responsible for the strong optical noise suppression that we observe.

Discrete detection. Our measurements do not involve trapping single atoms within the cavity mode. However, we have shown that our signal uctuations are near the photon shot noise limit, eectively allowing us to neglect uctuations in Ne. This allows a direct comparison between our measurements on falling clouds and experiments where noise in the atom number is inherently absent due to preparation of single trapped atoms. In this context, we discuss the discrete detection problem, for example, distinguishing between hyperne ground states. Because the detection linewidth is three orders of magnitude smaller than the level splitting, the |F = 1 ground state is eectively dark in our system24. When atom number uctuations are suppressed, discrete detection with Ne = 1 is thus equivalent to the problem of determining whether a single trapped atom uoresces or not, which is relevant for quantum information processing27,28. We therefore take detection of Ne = 1 as the benchmark for comparison with other experiments. From our uorescence measurements at Ne = 1.24(5), we extrapolate a mean photon count rate at Ne = 1 of S1 = 420(20) ms 1. Table 1 shows that this is high in comparison with other atom detection experiments. Following ref. 26, we could dene the single-atom efficiency of the detector as = 1 exp( S1T). This is the probability of counting 1 photon during the measurement time T, when an atom is present and assuming Poissonian photon counts with negligible background. This rises rapidly with our high count rate, reaching 98.5(3)% in only 10 s.

For most applications, however, it is not enough to detect the

bright state (logical 1) efficiently; the detector must also be able to identify the dark state (logical 0) correctly. A more useful gure of merit is thus the delity, which is the probability of a correct measurement result. Let us take the detection of K photons as indicating logical 1, and < K as logical 0. Then, for Poissonian distributions, the single-photon delity is FK = 1 = (1 P)e BT + P[1 e (S+B)T], where

B is the background photon counting rate and P is the probability that the state being measured is logical 1. The rst (second) term is the probability of having logical 0 (1) and identifying it correctly. The four logical possibilities are shown schematically in (Fig. 4a). The red curve in (Fig. 4b) shows the expected value of FK = 1 in our

(4)(4)

100

Atomic state 1

99

0

98

P Prob(k ~ K )

1 P Prob(k ~ K )

P Prob(k < K)

1 P Prob(k < K)

Fidelity

Identification

1

0

97

96

95 5 10 15 20 25 30

Integration time (ms)

Figure 4 | Detection delity. (a) Calculating the delity. Given two atomic states there are four possible outcomes of the experiment. Columns (rows) correspond to the actual (identied) state. In each box, the upper quantity is the probability to have the state, and the lower is the probability to make the identication; the total probability for the corresponding outcome is the product of the two. Correct identications are in green (diagonal), and their sum equals the delity as given by equation (5) in the text. (b) Detection delity FK(T) for K = 1, 2, and 3 counts (red , blue , and green ). Points are from 500 measurements with p = 1/2 and curves show equation (5), assuming Poisson distributions with mean signal and background count rates obtained from the data at the beginning of the pulse (the width of the curves reects the statistical uncertainties in these rates). The steps in the data are in units of the minimum resolution of 0.2% for 500 trials.

experiment over a data set for which Ne = 1.24 and P = 1/2. The delity rises quickly as the detection of logical 1 becomes increasingly successful but eventually falls, due to false positives from the background. Superimposed on this curve are our measured values of the delity versus detection time, which agree well with our expectations. In general, the maximum delity F1max increases with S/B, reaching its peak at a time T1max proportional to 1/S for xed S/B.

Table 1 compares our values of F1max and T1max with those for

other atom detection experiments. The highest delity by far is that of ref. 27, whereas our high signal rates result in the shortest detection time. It is worth noting that the measurements in refs 27,28 are non-destructive, whereas the rest are carried out on resonance. Lossless uorescence detection of single trapped atoms in free space has been observed with 95% (98.6%) accuracy in 0.3 ms (1.5 ms) 31,32.

Additionally, recent renements to our cavity manufacturing process have increased the nesse by two orders of magnitude 33. This suggests the possibility of single-atom strong coupling with g>(,) and C1 in the hundreds, allowing non-destructive measurements in our system as well.

A simple way to improve the delity is to increase the detection threshold K. This leads to the general result

F P PK BT

K P

PK B S T

K

K = +

+

(1 ( 1)! 1

]

) [ , ] [ ,( )

(5)(5)

( 1)!

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frequency of the laser, and used the fact that the cavity and elds are on resonance with the free-space |g|e transition. The system evolves in time according to the master equation for the density matrix

ddt i H A

j

N

where [K,a] is the incomplete gamma function. These delities are plotted in Figure 4 versus measurement time T for the cases of K = 2 and 3. They peak at 99.79 and 99.98% when T = 22.4 s and 33.6 s, respectively. The data points again show that our measurements are consistent with expectations. Similar methods were exploited in ref. 29, where two photons were required within a short-time window to register a logical 1 result. With a 1-s detection window they found that 99.719(6)% of observed 2-photon coincidences were due to atoms and described this percentage as the delity. In that experiment, however, there was only a 0.2% chance that the logical 1 state would produce a 2-photon count in the detection window. Thus, although the detection condence was high, the efficiency was low, resulting in a low delity in the usual sense that we adopt here.

Discussion

We have characterized a cavity-enhanced atom detector with low noise and high spatial resolution (set by the small cavity mode waist). We have shown that the nonlinear, multi-atom nature of the interactions results in a strong suppression of signal noise due to atomic uctuations. Our detector is fast and efficient, and suitable for detecting dilute samples below the level of a single atom per mode volume. Although we have focused here on measurements of low atomic densities, the dynamic range can be extended upwards simply by detuning the cavity and/or the probe eld.

We envision a variety of applications for making local density measurements on cold atom clouds and quantum gases. For example, small impurities can be detected for studies of Fermi polaron physics34 and quantum transport35. Cavity-enhanced detection also allows a greater collection efficiency for scattered photons than in conventional high-numerical-aperture (NA) optical systems20. The

maximum fraction of photons which can be captured in such systems is approximately NA2/4, which even for the best available lenses is an order of magnitude smaller than the fraction 2CN/(1 + 2CN) that

can be captured by a cavity with CN~1. This could improve the speed and efficiency of atom trap trace analysis, where laser-induced uorescence is used to detect radioactive atoms for dating environmental samples over time scales not accessible with 14C (ref. 36). Finally, the compatibility of our detector with atom chips makes it attractive for studying quantum gases in the Tonks-Girardeau regime37.

Producing one-dimensional gases requires trapping potentials of extremely high aspect ratio, as are typical with atom chips, and strong interactions require low densities which can be detected locally very quickly with our cavity.

Methods

Experiment. We work with 87Rb, near the D2 spectral lines at = 780 nm. Our optical microcavity is formed between the end of a single-mode optical bre anda spherical surface microfabricated in silicon, both being coated with multilayer dielectric mirrors. The resulting plano-concave cavity mode has a lengthof L = 139(1)m and a waist whose e 1 eld radius is w = 4.46(7)m. To our knowledge, the only FabryPerot cavity with a smaller mode waist is the all-bre design of ref. 12 (w = 3.9 m). Because C1F/w2, a small waist makes it possible to detect single atoms using a cavity of relatively modest nesse. This relaxes the usual need for very high mirror quality and reduces the sensitivity to noise in the cavity length. As stated in the main text, g/(2) = 98.4(1.6) MHz, /(2) = 5200(100) MHz, and /(2) = 3 MHz, giving C1 = 0.307(11). We begin each experimental sequence by cooling and trapping ~2107 87Rb atoms in a magneto-optical trap formed abovea mirror38, followed by sub-Doppler cooling to 16 K in optical molasses. We then release the atoms, which fall through a hole in the mirror and pass through a cavity mounted immediately below.

Master equation. The Hamiltonian describing our system is ( = 1)

H i a a i g a j

N

r r r

= =

[ , ] [ ]0

D

j

(7)(7)

(8)(8)

(9)(9)

(10)(10)

, A0(2)1/2a, and Aj>0(2)1/2j. For reection measurements, under the assumption aa 1 and with jj 1 one obtains a coherent intracavity eld with amplitude || = aa1/2

obeying

a h

k

= +

where D A A A A A A A

j j j j j j j

[ ] ( )

r r r r

= +

12

11 2CN

The total eld reected from the cavity comprises a component reected immediately from the input/ouput mirror that interferes with the fraction of the intracavity eld (8) being transmitted back out. This leads to equation (2) for reection.

For uorescence, we ignore correlations between atoms, which are distributed randomly, and consider a single atom with coupling gNeff

1 2

/ experiencing a Rabi frequency from the driving laser. The excited state population is

= +

s s g

1 2

22 2

/

2

/

2

tot

where 2tot is the total radiation rate, and we have assumed that the externaleld is much stronger than the cavity eld ||2|g|n1/2. From the Purcell eecttot = (1 + 2CN), with a fraction 2CN/(1 + 2CN) going into the cavity mode2. With the assumptions and denitions in the text, we recover equation (2). To determine CN, we solved the master equation for a toy model including all 12 of the Zeeman substates of the F = 2 ground and F = 3 excited states, but neglecting the cavity. The equilibrium excited-state populations were determined and the corresponding total decay rate of transitions was calculated (the quantization axis was taken along the cavity axis, so transitions did not contribute). The calculated dependence of Jout on the polarization of the drive laser agreed well with experimental results24.

The ratio CN/CN has a weak dependence on s. In this work, CN/CN = 0.53(2). The validity of all our results and conclusions were supported by direct numerical solution of equations (6) and (7), as well as quantum jump simulations3941. For our

parameters, the intracavity eld is indistinguishable from a coherent state for any xed arrangement of atoms and either type of detection.

Probability density for Neff . Carmichael and Sanders derived an expression for the probability density P(G)dG for the collective dipole G N

eff

1 2

/ in ref. 21, focusing on the case of travelling-wave cavities. The distribution depends on Ne and must generally be obtained numerically. Taking into account our

standing wave geometry, and in the limit Ne1, one can obtain an approximate distribution for G. Transforming to Ne gives

P N N N N N

N N

( ) exp ( )

eff eff eff eff

eff

2

d d

=

4 3

eff

where N is a normalization factor that approaches 2/(3Ne)1/2 as Ne. Note that in this limit, one obtains by inspection Var(Ne)/Ne = 3/8, which holds for all Ne in a Fabry-Perot cavity, highlighting the dierence between Ne and the total number of atoms in a small volume around the cavity. We have used the full numerical distribution for analysing our results with Ne = 1.24(5). However, we note that even in this regime the approximation (10) predicts Var(k)/k = 1.081(6) for our uorescence measurements, which is still in agreement with our observations.

References

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4. Thompson, R. J., Rempe, G. & Kimble, H. J. Observation of normal mode splitting for an atom in an optical cavity. Phys.Rev.Lett. 68, 11321135 (1992).

5. Childs, J. J., An, K., Otteson, M. S., Dasari, R. R. & Felds, M. S. Normal mode line shapes for atoms in standing-wave optical resonators. Phys.Rev.Lett. 77, 29012904 (1996).

6. Mnstermann, P., Fischer, T., Mauns, P., Pinske, P. W. H. & Rempe, G. Observation of cavity-mediated long-range light forces between strongly coupled atoms. Phys.Rev.Lett. 84, 40684071 (2000).

7. Trupke, M. etal. Atom detection and photon production in a scalable, open, optical microcavity. Phys.Rev.Lett. 99, 063601 (2007).

= +

1

2 H.c.

where we have used the rotating-wave approximation and neglected atomic centre-of-mass motion. The operator a annihilates a cavity photon and j = |ge|

lowers the jth atom from the excited state |e to ground state |g, and the pump strength Jin1/2 (ref. 23). We have expressed H in a frame rotating with the angular

(6)(6)

h s

( )

1

j j j

=

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Acknowledgements

We thank J. Dyne for technical expertise, and S. Barrett for an introduction to quantum jump theory and simulations. This work was funded by EPSRC and the Royal Society, and EU programmes HIP, AQUTE and CHIMONO.

Author contributions

J.G., M.T., J.K., and A.R. built the apparatus under the planning and supervision of E.A.H. J.G., M.T., and A.R. took the data and J.G., M.T. and E.A.H. analysed the data. J.G. and E.A.H. prepared the manuscript and all authors contributed to the editing.

Additional information

Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Goldwin, J. etal. Fast cavity-enhanced atom detection with low noise and high delity. Nat.Commun. 2:418 doi: 10.1038/ncomms1428 (2011).

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NATURE COMMUNICATIONS | 2:418 | DOI: 10.1038/ncomms1428 | www.nature.com/naturecommunications

2011 Macmillan Publishers Limited. All rights reserved.