Minimally-destructive detection of magnetically-trapped atoms using frequency-synthesised light
MMinimally-destructive detection of magnetically-trapped atoms usingfrequency-synthesised light
M. Kohnen, P. G. Petrov, R. A. Nyman, ∗ and E. A. Hinds Centre for Cold Matter, Blackett Laboratory, Imperial College London,Prince Consort Road, London SW7 2AZ, United Kingdom (Dated: December 1, 2018)We present a technique for atomic density measurements by the off-resonant phase-shift inducedon a two-frequency, coherently-synthesised light beam. We have used this scheme to measure thecolumn density of a magnetically trapped atom cloud and to monitor oscillations of the cloud inreal time by making over a hundred non-destructive local density measurments. For measurementsusing pulses of 10 − photons lasting ∼ µ s, the precision is limited by statistics of the photonsand the photodiode avalanche. We explore the relationship between measurement precision and theunwanted loss of atoms from the trap and introduce a figure of merit that characterises it. Thismethod can be used to probe the density of a BEC with minimal disturbance of its phase. Ultracold atoms play a central role in modern metrol-ogy, matter-wave interferometry and many-body quan-tum physics. In all these applications one aims to detectthe atoms with high efficiency and low noise. This can beachieved by resonant excitation, detecting either the ab-sorption of the probe light or the corresponding inducedfluorescence. Often, however, the cloud is optically thick.For example in a typical trapped Bose-Einstein conden-sate (BEC) the resonant optical absorption length is only30 nm or less. In addition each spontaneous emission de-stroys quantum coherence and promotes the loss of atomsfrom the trap. These problems can both be mitigated bydetuning the light to make the sample optically thin andthe scattering rate low. Detection is then accomplishedby measuring the phase shift that the atoms impose onthe light. This dispersive detection has the advantagethat it can be repeated many times on the same cloud.It is therefore sometimes called “non-destructive”.Non-destructive optical detection permits multiplemeasurements of the same trapped atomic sample withrepetition rates in excess of 100 kHz; faster than the natu-ral timescales for density evolution in a trapped ultracoldcloud set by the trapping frequencies [1, 2]. Consequentlythe method can be used to explore dynamics in an atomiccloud or to monitor the time evolution of matter waveinterference, e.g. in a BEC. With rapid readout, activefeedback on the cloud becomes feasible, opening possi-bilities for cooling the motion [3, 4] or preparing novelquantum states.The optical phase shift induced by the atoms is de-tected by interference with a reference beam preferablyderived from the same laser in order to reject phase drift.The beams could be separated spatially, with one pathgoing through the atom cloud, however, it is technicallydemanding to maintain adequate mechanical stability [2].Alternatively, separation in frequency allows them totravel on a common path while still accumulating a dif- ∗ To whom correspondence should be addressed:[email protected] ferential phase shift through the frequency-dependenceof the atomic polarisability [5].Previously, frequency modulation (fm) sidebands havebeen used to provide additional frequencies. Atomsdropped from a MOT have been seen using fm sidebanddetection with a current-modulated diode laser [6], andwithin a MOT, atoms have been probed using electro-optic modulation (EOM) to produce fm sidebands [7].These fm methods produce (at least) three frequencies —a carrier and two sidebands — rather than two. One de-tection strategy is to tune all three frequency componentsabove (or below) the atomic transition, then the side-band closest to resonance has the main phase shift andthe strong carrier acts as the local oscillator. This has thedrawback that it induces a light shift in the energy of theatoms. The shift can be eliminated by placing the side-bands symmetrically around the atomic resonance, butthen the cloud is heated by the resonant carrier. In orderto detect atoms in an optical lattice, Lodewyck et al. [8],using an EOM, have suppressed the carrier by choosinga specific, high modulation index (2.4), but this has theeffect of putting power into higher-order sidebands thatcontribute inefficiently to the signal.In this paper, we show how an acousto-optical mod-ulator (AOM) can synthesise the required two frequen-cies without any other sidebands. We couple this dual-frequency light into an optical fibre that allows easy de-livery to a remote site where cold atoms are to be mea-sured. We find that the relative phase of the two beams isexceedingly robust when transported in this way. Usinga phase sensitive detector to read out both quadraturesof the observed beat note, we study the phase noise andcompare this with the expected noise floor due to Poissonstatistics of the coherent laser light. We then apply thistwo-frequency interferometer to the dispersive detectionof magnetically trapped Rb atoms. We measure thespectrum of the phase shift induced by the atoms andinvestigate the extent to which the measurement is non-destructive. To demonstrate the utility of the method,we non-destructively measure the centre-of-mass oscilla-tions of a magnetically trapped atom cloud. Finally, wedescribe how to optimise the detection scheme, using a a r X i v : . [ qu a n t - ph ] A p r PBSAOM V q V i MagneticTrapAAPD V LP � /2 � /2 � /4 rf A L L L L BC f ± � f T o C o m pu t e r B PS D f beat DDSIncoming light
FIG. 1. Experimental setup. (A) Synthesis: The incom-ing beam passes twice through an acousto-optic modulator(AOM) driven at two frequencies rf and rf , The quarter-wave plate ( λ/
4) rotates the linear polarisation so that thelight is coupled out by a polarising beam cube (PBS). Thislight is coupled into a single-mode fibre. (C) Experimentalchamber: The beam leaving the fibre is collimated, passedthrough a linear polariser (LP) and focussed ( L ) onto themagnetic trap before re-collimation ( L ) and coupling backinto a multi-mode fibre. (B) Demodulation: Light is detectedby an analogue avalanche photo-diode (AAPD) A phase sen-sitive detector (PSD) determines the in-phase and quadraturepart of the beat note signal. Phase-stable modulation and ref-erence signals are produced by the same multi-channel directdigital synthesizer (DDS). figure of merit based on sensitivity and destructiveness.All the methods described above are ultimately limitedby the photon shot noise [9, 10], leading to the standardquantum limit. We note that there are also methods togo below that limit [11–15]. I. THE DETECTORA. Experimental setup
The setup for synthesising the two optical frequen-cies is shown in figure 1(A). A collimated, linearly po-larised laser beam enters the input and passes througha half wave plate ( λ/
2) that adjusts the polarisationto be vertical. This light is directed by the polaris-ing beam splitter (PBS) through lens L , acousto-opticmodulator (AOM), and lens L which re-collimates it.Retro-reflection through the quarter-wave plate rotatesthe linear polarisation to horizontal. After passing againthrough L , AOM and L , the light is transmitted by thePBS. A final λ/ rf and rf ,produced by a 4-channel direct digital synthesizer DDS(Novatech DDS 409B) and passively summed. The cen-tral beam emerging from the PBS contains the two de-sired frequencies, let us call them f ± δf , produced by thedouble-pass AOM shifts 2 rf and 2 rf . On either sideof this are beams at unwanted frequency f , produced bya shift of rf in one pass and rf in the other. The useof long focal length (400 mm) lenses ensures that theseside beams are well resolved from the main beam, so thatless than 1% of their power is coupled into the fibre when | δf | >
20 MHz. The two rf amplitudes are adjusted tobalance the power of the two desired frequency compo-nents in the light.The single-mode fibre guiding the light to the experi-mental chamber, figure 1(C), enforces the best possiblespatial mode overlap of the two frequency components.The light leaving this fibre is collimated, linearly po-larised (LP) and then focused to a waist of w = 55 µ mby lens L of focal length 250 mm. After passing throughpart of the magnetically trapped cloud of atoms, thebeam is re-collimated and coupled into a multi-mode fi-bre, which transports up to 90% of the light collected by L to the detection electronics shown in figure 1(B).The light is detected with 77% quantum efficiency byan analogue avalanche photodetector (AAPD)[16] whichproduces a voltage proportional to the rate of detectedphotons. With equal power at frequencies f ± δf , thisrate may be written as R + R cos(Ω t + φ ), where Ω =4 πδf is the beat angular frequency and φ is the relativephase between the two frequency components. We set Ωto 2 π ×
60 MHz in the experiments presented here. Wedetect the in-phase ( V i ) and quadrature ( V q ) componentsof the beat using a phase sensitive detector (PSD) whoselocal oscillator at angular frequency Ω is generated bythe same DDS that drives the AOM. The outputs of thePSD are low-pass filtered at 650 kHz, then digitised usinga 14-bit analogue-to-digital converter card NI PCI-6133with 1.3 MHz analogue bandwidth. B. Noise
The digitised PSD outputs are integrated over a mea-surement time T which contains an integer number ofbeat cycles. During this time, the mean number of de-tected photons is N = RT and the standard deviationof this number due to shot noise is √ N . When φ = 0the beat note is in phase with the PSD local oscilla-tor and the signal V i corresponds (through the variousamplifier gains) to a count rate R cos (Ω t ) which inte-grates to a time average of N . More generally, the timeaverage V i (or V q ) corresponds to a count of N cos φ (or N sin φ ). The noise power is equally distributedbetween the two outputs of the PSD, corresponding tostandard deviations in the count of (cid:112) N/ number of detected photons10 -2 -1 s t d . d e v . o f p h a s e [ r a d ] measuredfull modelincluding avalanche noiseideal detector FIG. 2. Standard deviation of the beat-note phase versusthe number of photons detected in an integration time of10 µ s. Dash-dot line: ideal detector having no avalanchenoise. Dashed line: analogue detector with avalanche. Solidline: full noise including that of the electronics. Dots: mea-sured noise levels. this shot noise is small because N (cid:29)
1, leading to a value σ φ = √ N/ N/ = (cid:112) /N . This level of noise is shown by thedash-dotted line in Fig. 2 versus the number of detectedphotons.Our instrument is not expected to reach the shot noiselevel because the photodiode current acquires additionalnoise as a result of the avalanche process [17], giving alarger phase uncertainty σ φ = X (cid:112) /N (1)We have measured this extra noise factor X and find thatit has the value 3 . ± . /N . When this is added in quadrature to the othernoise we obtain the total anticipated noise, indicated bythe solid line in Fig. 2.The points in Fig. 2 show the phase noise that we havemeasured. In a single phase measurement we detect alight pulse lasting T = 10 µ s and we integrate the PSDoutputs to obtain mean values V i and V q , from which wedetermine a phase arctan( V i /V q ). From the standard de-viation of 50 such phase measurements we determine onepoint on the graph in Fig. 2. The number of detectedphotons is determined directly from the mean AAPD sig-nal, V . This procedure is repeated over a wide range oflight intensities to produce the set of data points. Thesemeasurements show that the noise in our instrument iswell understood and that there are no other significant noise sources.The electronic noise becomes increasingly important asthe count rate is reduced, and is equal to the avalanche-degraded shot noise at 580 detected photons/ µ s. Thereis scope for suppressing the electronic noise of the PSD,in which case the photon rate could be reduced to ∼ µ s before the noise reaches that of the AAPDelectronics. If the analogue detection were replaced bypulse counting, our instrument could enjoy the noise in-dicated by the dash-dotted line in Fig. 2. However, cur-rently available pulse-counting APDs are limited by de-tection dead time to less about 10 photons/ µ s, whichwould slow down the precise measurement of phase.There can also be systematic noise in phase of the beatnote due to fluctuations in the difference of optical pathlengths for the two frequency components. Most impor-tant in this regard is the region between the AOM andthe retro-reflection mirror (see part A of Fig. 1), wherethe two beams take different paths. After enclosing thispart of the apparatus to shield it from air currents, theoptical path fluctuations were too small to see as excessnoise in Fig. 2. On increasing the number of detectedphotons to a million, by increasing the integration timeto T = 100 µ s, we found an excess σ φ of 2 mrad. Oncein the optical fibre, no more optical path length noiseis observed; neither shaking the fibre nor changing itslength between 2 and 20 m induces noise. The meanphase drifts by a radian over tens of minutes, presum-ably because of mirror movement in the same sensitiveregion of the apparatus. In principle, this technical noisecan be removed by using active feedback to stabilise thephase, but we have not done so here because the noise issmall and the drifts are slow. II. ATOM-LIGHT INTERACTION
The light used in our experiment is tuned near the | F = 2 , m (cid:105) → | F (cid:48) = 3 , m (cid:48) (cid:105) hyperfine transition of the D line of Rb and propagates perpendicular to the mag-netic field axis (see figure 1, part C). Almost all the atomsin the magnetic trap are prepared in the | F, m (cid:105) = | , +2 (cid:105) state, and we will assume for the moment that they re-main there (negligible optical pumping). If the light trav-els through an atomic vapour of column number density ρ a the phase shift θ and fractional attenuation α of theoptical field are, in the limit of negligible saturation, θ ( f ) = 72 (cid:32) (cid:88) m (cid:48) =1 p m (cid:48) S m (cid:48) γδ m (cid:48) ,f δ m (cid:48) ,f + γ (cid:33) λ ρ a π (2) α ( f ) = 72 (cid:32) (cid:88) m (cid:48) =1 p m (cid:48) S m (cid:48) γ δ m (cid:48) ,f + γ (cid:33) λ ρ a π , (3)where p m (cid:48) denotes the fraction of light power at frequency f polarised to drive the atomic transition | m = 2 (cid:105) →
40 20 0 20 40 60 8010050050100 φ [ m r a d ] a) 40 20 0 20 40 60 80detuning [MHz]0.000.050.100.15 l o ss † b) FIG. 3. Phase shift and attenuation of 60 MHz optical beatnote due to interaction with atoms in a magnetic trap. Thelight is polarised perpendicular to the magnetic field. Theabscissa is the detuning of the central laser frequency f froman arbitrary zero. Dots: data measured in 32 shots, half withatoms and half without, each integrated over 10 µ s. (a) Phaseshifts φ . Solid lines: least squares fit to Eq.(2). The fit gives acolumn number density of ρ a = (2 . ± . × atoms m − .(b) Fractional change (cid:15) in beat amplitude. The line uses thefit parameters from (a). | m (cid:48) (cid:105) , and S m (cid:48) = (cid:18) − m (cid:48) m (cid:48) − (cid:19) is the square of the Wigner-3 j symbol. The detuning ofthe light frequency f from resonance, δ m (cid:48) ,f , includes theZeeman shift of the transition frequency due to the mag-netic field. The damping rate γ is half the spontaneousdecay rate of the upper level, and λ is the wavelength.Our apparatus measures the relative phase shift φ = θ ( f + δf ) − θ ( f − δf ) between the two frequencycomponents and the fractional change in their detectedbeat amplitude (cid:15) ≈ [ α ( f + δf ) + α ( f − δf )]. The lat-ter equation holds because α (cid:28)
1. The fraction of lightpower scattered is 2 (cid:15) . A. Detection of magnetically trapped atoms byphase-shift and absorption
The experimental chamber is supplied with a streamof cold Rb atoms from a low velocity intense source(LVIS) [18]. Over a few seconds, these are captured inthe main vacuum chamber by a U-MOT [19], then trans-ferred into a cigar-shaped magnetic trap produced bycurrent-carrying wires. The magnetic field has a mini-mum of B = 0 . . ×
21 Hz. Typically we load 2 . µ K. Figure 3(a) shows the phase shift φ for light polarisedperpendicular to the magnetic field B . The abscissa isthe central frequency f of the light relative to an ar-bitrary zero. Each phase shift is determined from thetime-average V i and V q over a 10 µ s detection window,corresponding to ∼ × detected photons. The av-erage phase φ = (cid:104) arctan( V q /V i ) (cid:105) is determined from 16such measurements, spaced by 1 ms and taken with atomsin the trap. The atoms are then released and the mea-surement is repeated to determine the background phase,which we subtract. The same data are used to determinethe fractional change in beat amplitude (cid:15) by compar-ing the averages (cid:28)(cid:113) V i + V q (cid:29) with and without atoms.These values are plotted in Fig. 3(b).The solid curve in Fig. 3(a) shows a least-squares fitof Eq. 2 to the phase data, with column density and acentral frequency offset as variable parameters. We seea dispersion feature when either of the two frequencycomponents tunes through the | m = 2 (cid:105) → | m (cid:48) = 3 (cid:105) reso-nance. This is due to the σ + component of the light. The σ − transition | m = 2 (cid:105) → | m (cid:48) = 1 (cid:105) , being fifteen timesweaker, is not seen in the data. The fit gives a columndensity of ρ a = 2 . × atoms m − , which is consis-tent with the density measured by destructive absorptionimaging on a camera. Using the same fit parameters inEq. 3, we obtain the line plotted in Fig. 3(b), in goodagreement with the observed variation in the amplitudeof the beat note.In Fig. 4 we show how the optical phase shift can beused to monitor the density evolution of a cold atomcloud. Here we have made measurements at 1 ms in-tervals over a period of 120 ms to record the centre-of-mass oscillation of atoms held in the magnetic trap. Theoptical phase shift is made sensitive to the motion byplacing the probe beam approximately one cloud radius( ∼ µ m in this case) from the centre of the trap. Forthe purpose of this demonstration we set the cloud oscil-lating by making an intentional misalignment when weload it. This method allows us to determine the fre-quency and amplitude of the motion in a very short time,leaving the cloud largely undisturbed at the end of themeasurement. By comparison, the same measurementusing the standard method of resonant absorption imag-ing would require several hundred seconds to performbecause the cloud is destroyed by a single laser shot andhas to be replaced by a fresh cloud each time. Sinceit is a routine part of any experiment on magneticallytrapped atoms to measure the trap frequency and to nullthe trap oscillations from time to time, our methods isof practical value. Moreover, it opens the possibility oftracking dynamical evolution of the cloud in real time andof doing so with high spatial resolution and bandwidth.Rapid, non-destructive, local monitoring would make itpossible to implement recent ideas of fast feedback andcontrol [20, 21]. This could be extended to many chan-nels using the array of atom-photon junctions [22] thatwe have developed. p h a s e [ r a d ] c o l u m n d e n s i t y [ m − ] FIG. 4. Centre-of-mass oscillations of a trapped atom cloud,observed through the modulated phase shift of the beat notein a single realisation of the experiment. The laser detuningcorresponds to the frequency 17 . × photons and lasting 50 µ s. Line:a fit, with trap frequency measured to be 21 ± As shown by Eqs. (2) and (3) the atoms cannot inducean optical phase shift without also scattering the light.Each scatter imparts a recoil momentum to the atom,which heats the cloud, and also opens the possibility thatthe atom may be optically pumped out of the | m = 2 (cid:105) state. Both of these effects reduce the density of thecloud. However, the scattering during this measurementis so weak that it cannot be responsible for damping theoscillation in Fig. 4. We have confirmed this by allowingthe cloud to oscillate in the dark and measuring at latertimes. We believe that the damping is in fact due to theanharmonicity of the trap. It is nonetheless relevant toconsider at what level atoms are lost from the trap as aresult of the measurement, and this is the subject of thenext section. B. Losses
We have measured the loss of atoms directly. Thetrapped atoms were illuminated by 200 pulses of probelight, each lasting for 30 µ s, then the trap was switchedoff. After a further delay of 3 ms the number of atomsremaining in the cloud was measured by resonant ab-sorption imaging on a CCD camera. The data points inFig. 5 show the fraction of atoms remaining in the trap asa function of laser frequency. There are two curves. Theone with weaker loss is measured with light polarised per-pendicular to the magnetic field at the centre of the trapand with 6 × photons per pulse passing through the
40 20 0 20 40 60detuning [MHz]0.00.20.40.60.81.0 f r a c t i o n o f a t o m s r e m a i n i n g light, theory light, theory light, experiment light, experiment FIG. 5. Fraction of atoms remaining in a magnetic trap aftera measurement with 200 pulses. Loss is mostly due to atomsbeing pumped into a weakly-trapped state, where their ther-mal energy is sufficient to take them out of the trap. Thispumping effect is much stronger for light polarised parallelto the magnetic field than perpendicular. The solid lines aretheoretical predictions from a rate equation model followingthe internal states of the atoms during the light pulses. cloud. We see resonant loss dips caused by the σ − exci-tation to | m (cid:48) = 1 (cid:105) . As shown in Fig. 6, this causes atomloss through its decays to the weakly trapped | m = 1 (cid:105) state and to the untrapped | m = 0 (cid:105) , however, this exci-tation is 15 times weaker than the σ + cycling transition.The data series showing stronger loss is measured usinglight polarised parallel to the trap field with 9 × pho-tons per pulse. Here the loss is through faster excitationto | m (cid:48) = 2 (cid:105) state, followed by its preferred decay to theweakly trapped | m = 1 (cid:105) state.The curves in Fig. 5 are calculated using the follow-ing simple model. Atoms start in the strongly-trapped | m = 2 (cid:105) state and we use rate equations to calculate howthe populations of the levels in Fig. 6 evolve over a sin-gle probe pulse. At the end of the pulse, atoms having m = 0 , − , or − m = 2 are trapped. Of those in state | m = +1 (cid:105) , weestimate that only 10% remain trapped because the grav-itational force greatly lowers the barrier for escape. Alltrapped atoms are detected with equal efficiency in theabsorption image because those having m = 1 are quicklypumped into the | m = 2 (cid:105) state.After each probe pulse there is a delay of almost 1 msbefore the next pulse arrives. During this time theoptically-pumped atoms move out of the probe regionand are replaced by a new set of atoms. To a good ap-proximation these are all in state | m = 2 (cid:105) . If we writethe fraction of trapped atoms in the probe beam as p and the fraction of these that are pumped to untrappedstates by one pulse as q , then the survival probabilityafter one pulse is 1 − qp . Hence the fraction of trappedatoms surviving k pulses is (1 − qp ) k . m'=1 m'=2 m'=3m=2m=0 m=1 100%157%40%53 % 33%67% 51 P , F'=3 S , F=2 fullytrappedweaklytrappednottrapped Coupling strength b r an c h i ng r a t i o s FIG. 6. Relative coupling strengths for exciting trappedatoms in the | F, m (cid:105) = | , (cid:105) state. Transitions driven by lightperpendicular (parallel) to the magnetic field are indicated bya solid (dashed) line. Branching ratios for decay back to theground states are also shown. Atoms pumped to state | , (cid:105) are untrapped and therefore lost. Most of the atoms pumpedto the weakly-trapped state | , (cid:105) are also lost. Those remain-ing in state | , (cid:105) are trapped. The solid lines in Figure 5 plot the results of this modelapplied to our experiment, where the probe beam size inthe cloud is w = 100 µ m and p = 1 . N γ incidentphotons, the total number of σ − excitations per atom is (cid:15) N γ , where (cid:15) = λ πA (cid:32) γ γ + δ ,f + δf + γ γ + δ ,f − δf (cid:33) . (4)This follows directly from Eq. 3 with p = and S = . Also, ρ a is replaced by 1 /A , where the beam area A is related to the waist size w by A = πw . Theexcited state | m (cid:48) = 1 (cid:105) decays with 40% probability tothe untrapped state | m = 0 (cid:105) and with 53% probabilityto | m = 1 (cid:105) , which is lost 90% of the time. Thus theprobability that an atom in the probe volume will belost as a result of the probe pulse is q = 0 . (cid:15) N γ . (5)For perpendicular polarisation, the loss spectrum givenby this approximation is almost indistinguishable fromthe theoretical curve in Fig. 5. For parallel polarisation,however, the optical pumping is too strong for this ap-proximation to suffice.
60 40 20 0 20 40 60detuning of centre frequency [MHz]01234 × F o M µ T 10 µ T FIG. 7. Figure of merit (defined in text) for detecting Rbatoms using light with perpendicular polarisation. The ab-scissa shows detuning of the mean frequency f from the F = 2 → F (cid:48) = 3 cycling transition. The two light frequenciesare separated by 60 MHz. Dashed line: 10 µ T magnetic field.Solid line: 650 µ T magnetic field.
C. Figure of merit
The detection scheme presented in this paper is devel-oped for multiple measurements of the atomic density onthe same sample, as for example shown in figure 4. Theaim is to infer the number of atoms in the probe beam N a from the measured phase φ with the smallest uncertainty σ N a . Since φ ∝ N a , the uncertainty is σ N a = σ φ | φ | . Here φ = φ/N a is the phase shift per atom in the light beam,given by Eq. (2) on replacing ρ a by 1 /A . It is also de-sirable to minimise q , the fraction of atoms lost from theprobe region as a result of the probe pulse. We thereforedefine the figure of merit for the case of perpendicularpolarisation as F oM = 1 σ N a √ q (cid:39) | φ | X √ (cid:15) (cid:115) NN γ , (6)where the last step above makes use of Eqs. (1) and (5)and we take √ . (cid:39)
1. The factor
N/N γ is the fractionof photons in the pulse that are detected. For our detec-tor this quantum efficiency is 0.77. This figure of merithas the virtue that it does not depend on the number ofatoms or the number of photons. It can be seen as thesignal-to-noise ratio per atom, when the atoms have a 1 /e probability of surviving the measurement. If the area ofthe beam is reduced, the FoM increases as A − / , whichmotivates efforts to probe atom clouds using beams ofsmall cross section [9] [22].The dashed line in Fig. 7 shows how the figure of meritvaries with the detuning of the central frequency f fromthe atomic resonance frequency. We take the beat fre-quency 2 δf = 60 MHz used in the experiment. Strongminima appear when either of the optical frequencies isresonant with the atomic transition, since then the phaseshift is close to zero and the loss is maximum. The op-timum condition, with the optical frequencies symmet-rically on either side of resonance, gives FoM (cid:39) / ∼
400 atoms in our 100 µ m beam can bedetected with a signal:noise ratio of 1. The solid lineshows FoM when we add a magnetic field of 0 .
65 mT,which is typical of the field experienced by atoms in ourmagnetic trap. This changes the frequency dependence ofFoM quite significantly in the vicinity of the resonances.We see the two sharp minima associated with the disper-sive character of | φ | move to higher frequency becausethey are due primarily to the σ + transition. In addition,we see two broader minima that are down-shifted. Theseare due to the σ − resonances that maximise the loss.This separation of the σ + and σ − transitions makes itpossible to double the highest figure of merit. When de-tecting with parallel polarisation, the loss is more severe,as we have discussed in Sec. II B. An addition, one can-not use the Zeeman shift to improve the figure of meritsince the same transition produces both the phase shiftand the loss.The figure of merit is a convenient way to explore theapplication of the technique to other circumstances. Forexample in a Bose-Einstein condensate (BEC), any spon-taneous emission will remove an atom from the conden-sate, so the σ + excitations all contribute to the loss. Thistypically increases the value of q in Eq. (6) by a factorof 16 because of the 15 : 1 ratio of coupling strengthsshown in Fig. 6. The condensate can be addressed on anatom chip by a beam of small waist ∼ µ m [22], whichenhances FoM by a factor of 50 in comparison with the100 µ m beam used here. The net effect is a FoM very sim-ilar to the dashed curve in Fig. 6 but with a peak at thecentre of 0 .
03. With a typical Rb condensate of 3 × atoms in prolate trap of trapping frequencies 20 Hz and1 kHz, 10 atoms are illuminated when the probe beamis centred on the cloud. Thus, the figure of merit showsthat with our method, a 10% measurement of the cen-tral density entails to loss of only 100 atoms from thecondensate. The passage of a light pulse through a BECnormally imposes a local phase shift on the condensate equal to N γ N a φ . The gradient of this phase correspondsto a force that can excite phonons in the BEC. How-ever, the two frequency components of the probe pulseused in this case induce opposite forces, which are equalin magnitude because they are symmetrically disposedon either side of resonance. As a result, the BEC willbe undisturbed apart from the noise in the force due tophoton statistics. III. CONCLUSION
In conclusion, this paper presents a method of mea-suring the column density through a small part of anatom cloud with minimum disturbance of the atoms. Themethod is to measure the phase shift between two syn-thesised frequency components of a laser beam, tunedon opposite sides of an atomic resonance. We have usedthis scheme to measure the column density of a magnet-ically trapped atom cloud and to monitor oscillations ofthe cloud in real time. Measurement sensitivity is princi-pally limited by photon shot noise and excess noise dueto the avalanche amplification in the photodiode. Wehave measured how many atoms are lost from the trapas an unwanted byproduct of the measurement. We havedeveloped a figure of merit for this scheme, which quan-tifies the relationship between the sensitivity and the de-structiveness of the measurements. Using this we haveanticipated the performance of the technique when ap-plied to Bose-Einstein condensates trapped on an atomchip.We acknowledge the technical expertise of ValerijusGerulis, who built the demodulation electronics and thesupport and advice of John Dyne and Stephen Maine inconstructing the apparatus. We thank Tim Cable fromAnalog Modules for making sure we could use the lastever photo detector of the 712A-4 series. This researchwas supported by EPSRC (UK), the Royal Society (UK)and European projects AQUTE and HIP (EU). [1] M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S.Durfee, D. M. Kurn, and W. Ketterle. Direct, non-destructive observation of a bose condensate.
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