Minimally-destructive detection of magnetically-trapped atoms using frequency-synthesised light

We present a technique for atomic density measurements by the off-resonant phase-shift induced on a two-frequency, coherently-synthesised light beam. We have used this scheme to measure the column density of a magnetically trapped atom cloud and to monitor oscillations of the cloud in real time by making over a hundred non-destructive local density measurments. For measurements using pulses of 10,000-100,000 photons lasting ~10 microsecond, the precision is limited by statistics of the photons and the photodiode avalanche. We explore the relationship between measurement precision and the unwanted loss of atoms from the trap and introduce a figure of merit that characterises it. This method can be used to probe the density of a BEC with minimal disturbance of its phase.

3 three frequency components above (or below) the atomic transition; then the sideband closest to resonance has the main phase shift and the strong carrier acts as the local oscillator. This has the drawback that it induces a light shift in the energy of the atoms [12]. The shift can be eliminated by placing the sidebands symmetrically around the atomic resonance, but then the cloud is heated by the resonant carrier. In order to detect atoms in an optical lattice, Lodewyck et al [13], using an EOM, have suppressed the carrier by choosing a specific, high modulation index (2.4), but this has the effect of putting power into higher-order sidebands that contribute inefficiently to the signal. All the methods described above are ultimately limited by the photon shot noise [14,15], leading to the standard quantum limit. We note that there are also methods to go below that limit [16][17][18][19][20].
In this paper, we show how an acousto-optical modulator (AOM) can synthesize the required two frequencies without any other sidebands. We couple this dual-frequency light into an optical fibre that allows easy delivery to a remote site where cold atoms are to be measured. We find that the relative phase of the two beams is exceedingly robust when transported in this way. Using a phase-sensitive detector (PSD) to read out both quadratures of the observed beat note, we study the phase noise and compare this with the expected noise floor due to Poisson statistics of the coherent laser light. We then apply this two-frequency interferometer to the dispersive detection of magnetically trapped 87 Rb atoms. We measure the spectrum of the phase shift induced by the atoms and investigate the extent to which the measurement is non-destructive. To demonstrate the utility of the method, we non-destructively measure the centre-of-mass oscillations of a magnetically trapped atom cloud. Finally, we describe how one can optimize the detection scheme, using a figure of merit (FoM) based on sensitivity and destructiveness.

Experimental set-up
The light for our experiments is generated by a grating-stabilized diode laser, whose output frequency, f L , is fine-tuned by an acousto-optical modulator (AOM). Using this light we synthesize the two optical frequencies as shown in figure 1(A). A collimated, linearly polarized (LP) laser beam enters the input and passes through a half-wave plate (λ/2) that adjusts the polarization to be vertical. This light is directed by the polarizing beam splitter (PBS) through lens L 1 , AOM and lens L 2 (which re-collimates it). Retro-reflection through the quarter-wave plate rotates the linear polarization to horizontal. After passing again through L 2 , AOM and L 1 , the light is transmitted by the PBS. A final λ/2 rotates the polarization to any desired angle before the light is coupled into a single-mode fibre for transmission to the atom cloud.
The AOM is driven at two rf frequencies, rf 1 and rf 2 , produced by a four-channel DDS (Novatech DDS 409B) and passively summed. Incoming light at f L produces two beams at f L + rf 1 and f L + rf 2 after the first pass. The unshifted zero-order beam is blocked. After the second pass, the central beam emerging from the PBS contains the two desired frequencies, f L + 2rf 1 and f L + 2rf 2 . On either side of this are beams at unwanted frequency f L + rf 1 + rf 2 , produced by a shift of rf 1 in one pass and rf 2 in the other. The use of long focal length (400 mm) lenses ensures that these side beams are well resolved from the main beam, so that less than 1% of their power is coupled into the fibre when |rf 1 − rf 2 | = δ f > 10 MHz. The two rf amplitudes are adjusted to balance the power of the two desired frequency components in the light. The unwanted diffraction orders are blocked after the first pass. The quarter-wave plate (λ/4) rotates the linear polarization so that the light is coupled out by a polarizing beam cube (PBS). This light is coupled into a single-mode fibre. (B) Demodulation: light is detected by an analogue avalanche photo-diode (AAPD). A PSD determines the in-phase and quadrature part of the beat note signal. Phase-stable modulation and reference signals are produced by the same multi-channel direct digital synthesizer (DDS). (C) Experimental chamber: the beam leaving the fibre is collimated, passed through a linear polarizer (LP) and focused (L 3 ) onto the magnetic trap before re-collimation (L 4 ) and coupling back into a multi-mode fibre.
The visibility of the detected beat signal is better than 90%. By tuning the laser frequency f L , we control the mean output frequency f 0 so that the output contains two components at The single-mode fibre guiding the light to the experimental chamber, figure 1(C), enforces the best possible spatial mode overlap of the two frequency components. The light leaving this fibre is collimated, LP and then focused to a waist of w 0 = 55 µm by lens L 3 of focal length 250 mm. After passing through part of the magnetically trapped cloud of atoms, the beam is re-collimated and coupled into a multi-mode fibre, which transports up to 90% of the light collected by L 4 to the detection electronics shown in figure 1 The light is detected with 77% quantum efficiency by an AAPD 2 which produces a voltage proportional to the rate of detected photons. With equal powers at frequencies f 0 ± δ f , this rate may be written as R + R cos( t + φ), where = 4πδ f is the beat angular frequency and φ is the relative phase between the two frequency components. We set to 2π × 60 MHz in the experiments presented here. We detect the in-phase (V i ) and quadrature (V q ) components of the beat using a PSD whose local oscillator at angular frequency is generated by the same DDS that drives the AOM. The outputs of the PSD are low-pass filtered at 650 kHz and then digitized using a 14-bit analogue-to-digital converter card (NI PCI-6133) with 1.3 MHz analogue bandwidth. Typically, we work with 1-4 nW optical power, which is within the linear response range of the photodetector. The peak intensity of light is no more than 1 W m −2 , far below the atomic saturation intensity which is greater than 16 W m −2 . The frequency of the source laser is stable to ±700 kHz over tens of milliseconds, with drifts up to 3 MHz from day to day.

Noise
The digitized PSD outputs are integrated over a measurement time T which contains a large number of beat cycles. During this time, the mean number of detected photons is N = RT and the standard deviation of this number due to shot noise is √ N . When φ = 0, the beat note is in phase with the PSD local oscillator and the signal V i corresponds (through the various amplifier gains) to a count rate R cos 2 ( t) which integrates to a time average of 1 2 N . More generally, the time average V i (or V q ) corresponds to a count of 1 2 N cos φ (or 1 2 N sin φ). The noise power is equally distributed between the two outputs of the PSD, corresponding to standard deviations in the count of √ N /2 each. In our experiments, the standard deviation of the phase due to this shot noise is small because N 1, leading to a value σ φ =  Our instrument is not expected to reach the shot noise level because the photodiode current acquires additional noise as a result of the avalanche process [21], giving a larger phase uncertainty We have measured this extra noise factor X and find that it has the value 3.3 ± 0.3 (somewhat higher than the 2.2 indicated in the manufacturer's specifications). The correspondingly higher noise level is shown by the dashed line in figure 2. In addition, the photodiode amplifier and the PSD contribute electronic noise, which is independent of the number of photons detected and therefore appears as a further phase uncertainty proportional to 1/N . When this is added in quadrature to the other noise, we obtain the total anticipated noise, indicated by the solid line in figure 2.
The points in figure 2 show the phase noise that we have measured. In a single phase measurement, we detect a light pulse lasting T = 10 µs and we integrate the PSD outputs to obtain mean values V i and V q , from which we determine a phase arctan(V i /V q ). From the standard deviation of 50 such phase measurements, we determine one point on the graph in figure 2. The number of detected photons is determined directly from the mean AAPD signal, V 0 . This procedure is repeated over a wide range of light intensities to produce the set of data points. These measurements show that the noise in our instrument is well understood and that there are no other significant noise sources.
The electronic noise becomes increasingly important as the count rate is reduced, and is equal to the avalanche-degraded shot noise at 580 detected photons µs −1 . There is scope for suppressing the electronic noise of the PSD, in which case the photon rate could be reduced to ∼200 photons µs −1 before the noise reaches that of the AAPD electronics. If the analogue detection were replaced by pulse counting, our instrument could enjoy the noise indicated by the dash-dotted line in figure 2. However, currently available pulse-counting APDs are limited by detection dead time to less than about 10 photons µs −1 , which would slow down the precise measurement of phase.
There can also be systematic noise in the phase of the beat note due to fluctuations in the difference of optical path lengths for the two frequency components. Most important in this regard is the region between the AOM and the retro-reflection mirror (see part A of figure 1), where the two beams take different paths. After enclosing this part of the apparatus to shield it from air currents, the optical path fluctuations were too small to see as excess noise in figure 2. On increasing the number of detected photons to a million, by increasing the integration time to T = 100 µs, we found an excess σ φ of 2 mrad. Once in the optical fibre, no additional optical path length noise is observed; neither shaking the fibre nor changing its length between 2 and 20 m increases the noise. The mean phase drifts by a radian over tens of minutes, presumably because of mirror movement in the same sensitive region of the apparatus. In principle, this technical noise can be removed by using active feedback to stabilize the phase, but we have not done so here because the noise is small and the drifts are slow.

Atom-light interaction
The light used in our experiment is tuned near the |F = 2, m → |F = 3, m hyperfine transition of the D 2 line of 87 Rb and propagates perpendicular to the magnetic field axis (see figure 1, part C). Almost all the atoms in the magnetic trap are prepared in the |F, m = |2, +2 7 state, and we will assume for the moment that they remain there (negligible optical pumping). If the light travels through an atomic vapour of column number density ρ a , the phase shift θ and fractional attenuation α of the optical field are, in the limit of negligible saturation, where p m denotes the fraction of light power at frequency f polarized to drive the atomic transition |m = 2 → |m , and is the square of the Wigner-3 j symbol. The detuning of light frequency f from resonance, δ m , f , includes the Zeeman shift of the transition frequency due to the magnetic field. The damping rate γ is half the spontaneous decay rate of the upper level, and λ is the wavelength.
Our apparatus measures the relative phase shift φ = θ( f 0 + δ f ) − θ( f 0 − δ f ) between the two frequency components and the fractional change in their detected beat amplitude The latter equation holds because α 1. The fraction of light power scattered is 2 .

Detection of magnetically trapped atoms by phase shift and absorption
The experimental chamber is supplied with a stream of cold 87 Rb atoms from a low-velocity intense source [22]. Over a few seconds, these are captured in the main vacuum chamber by a U-MOT [23] and then transferred into a cigar-shaped magnetic trap produced by currentcarrying wires. The magnetic field has a minimum of B 0 = 0.6 mT at a distance of 2.5 mm from the surface of the central wire, and the trapped atoms have oscillation frequencies of 75 Hz radially and 21 Hz axially. Typically, we load 2.4 million atoms at a temperature of 60 µK. Figure 3(a) shows the phase shift φ for light polarized perpendicular to the magnetic field B 0 . The abscissa is the central frequency f 0 of the light relative to an arbitrary zero. Each phase shift is determined from the time-average V i and V q over a 10 µs detection window, corresponding to ∼3 × 10 5 detected photons. The average phase φ = arctan(V q /V i ) is determined from 16 such measurements, spaced by 1 ms and taken with atoms in the trap. The atoms are then released and the measurement is repeated to determine the background phase, which we subtract. The same data are used to determine the fractional change in beat amplitude by comparing the averages V i 2 + V q 2 with and without atoms. These values are plotted in figure 3(b). The solid curve in figure 3(a) shows a least-squares fit of equation (2) to the phase data, with column density and a central frequency offset as variable parameters. We see a dispersion feature when either of the two frequency components tunes through the |m = 2 → |m = 3 resonance. This is due to the σ + component of the light. The σ − transition |m = 2 → |m = 1 , being 15 times weaker, is not seen in the data. The fit gives a column density of ρ a = 2.2 × 10 12 atoms m −2 , which is consistent with the density measured by destructive absorption imaging on a camera. Using the same fit parameters in equation (3), we obtain the line plotted in figure 3(b), in good agreement with the observed variation in the amplitude of the beat note.
In figure 4, we show how the optical phase shift can be used to monitor the density evolution of a cold atom cloud. Here we have made measurements at 1 ms intervals over a period of 120 ms to record the centre-of-mass oscillation of atoms held in the magnetic trap. The optical phase shift is made sensitive to the motion by placing the probe beam approximately one cloud radius (∼600 µm in this case) from the centre of the trap. For the purpose of this demonstration, we set the cloud oscillating by making an intentional misalignment between cloud and trap when we load it. This method allows us to determine the frequency and amplitude of the motion in a very short time, leaving the cloud largely undisturbed at the end of the measurement. This nondestructive, local monitoring with rapid analogue readout will make it possible to implement recent ideas of fast feedback and control [24,25]. It can be extended to give spatial information from many channels, e.g. using the array of atom-photon junctions [26] that we have developed.  figure 3. Dots: 120 phase shift measurements spaced 1 ms apart, each using a light pulse comprising 4 × 10 5 photons and lasting 50 µs. Line: a fit, with trap frequency measured to be 21 ± 1 Hz. After the measurement, less than 3% of the atoms have been lost due to the probe light. The damping of the oscillations is due to anharmonicity of the trap.
As shown by equations (2) and (3), the atoms cannot induce an optical phase shift without also scattering the light. Each scatter imparts a recoil momentum to the atom, which heats the cloud, and also makes it possible for the atom to be optically pumped out of the |m = 2 state. Both of these effects reduce the density of the cloud. However, the scattering during this measurement is so weak that it cannot be responsible for damping the oscillation in figure 4. We have confirmed this by allowing the cloud to oscillate in the dark and measuring at later times. We believe that the damping is in fact due to the anharmonicity of the trap. It is nonetheless relevant to consider at what level atoms are lost from the trap as a result of the measurement, and this is the subject of the next section.

Losses
We have measured the loss of atoms directly. The trapped atoms were illuminated by 200 pulses of probe light, each lasting for 30 µs, and then the trap was switched off. After a further delay of 3 ms, the number of atoms remaining in the cloud was measured by resonant absorption imaging on a CCD camera. The data points in figure 5 show the fraction of atoms remaining in the trap as a function of laser frequency. There are two curves. The one with weaker loss is measured with light polarized perpendicular to the magnetic field at the centre of the trap and with 6 × 10 5 photons per pulse passing through the cloud. We see resonant loss dips caused by the σ − excitation to |m = 1 . As shown in figure 6, this causes atom loss through its decays to the weakly trapped |m = 1 state and to the untrapped |m = 0 ; however, this excitation is 15 times weaker than the σ + cycling transition. The data series showing stronger loss is measured using light polarized parallel to the trap field with 9 × 10 5 photons per pulse. Here the loss is through faster excitation to the |m = 2 state, followed by its preferred decay to the weakly trapped |m = 1 state.
The curves in figure 5 are calculated using the following simple model: atoms start in the strongly trapped |m = 2 state and we use rate equations to calculate how the populations of the levels in figure 6 evolve over a single probe pulse. At the end of the pulse, atoms having m = 0, −1 or − 2 are discarded as untrapped, while those with m = 2 are trapped. Of those in state |m = +1 , we estimate that only 10% remain trapped because the gravitational force greatly lowers the barrier for escape. All trapped atoms are detected with equal efficiency in the absorption image because those having m = 1 are quickly pumped into the |m = 2 state.
After each probe pulse there is a delay of almost 1 ms before the next pulse arrives. During this time the optically pumped atoms move out of the probe region and are replaced by a new set of atoms. To a good approximation, these are all in state |m = 2 . If we write the fraction of trapped atoms in the probe beam as p and the fraction of these that are pumped to untrapped states by one pulse as q, then the survival probability after one pulse is 1 − qp. Hence the fraction of trapped atoms surviving k pulses is (1 − qp) k .
The solid lines in figure 5 plot the results of this model applied to our experiment, where the probe beam size as it passes through the cloud is w = 100 µm and p = 1.2%. The good agreement with experiment shows that, despite its simplicity, this model reliably relates the loss of atoms to the scattering rates.
With perpendicular polarization, the optical pumping probability after one pulse is small enough that the following even simpler model suffices: taking N γ incident photons, the total This follows directly from equation (3) with p 1 = 1 2 and S 1 = 1 105 . Also, ρ a is replaced by 1/A, where the beam area A is related to the waist size w by A = 1 2 πw 2 . The excited state |m = 1 decays with 40% probability to the untrapped state |m = 0 and with 53% probability to |m = 1 , which is lost 90% of the time. Thus, the probability that an atom in the probe volume will be lost as a result of the probe pulse is For perpendicular polarization, the loss spectrum given by this approximation is almost indistinguishable from the theoretical curve in figure 5. For parallel polarization, however, the optical pumping is too strong for this approximation to suffice.

Figure of merit
The detection scheme presented in this paper is developed for multiple measurements of the atomic density on the same sample, as for example shown in figure 4. The aim 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 1 | . Here φ 1 = φ/N a is the phase shift per atom in the light beam, given by equation (2) on replacing ρ a by 1/A. It is also desirable to minimize q, the fraction of atoms lost from the probe region as a result of the probe pulse. We therefore define the FoM for the case of perpendicular polarization as where the last step above makes use of equations (1) and (5) and we take √ 0.88 1. The factor N /N γ is the fraction of photons in the pulse that are detected. For our detector this quantum efficiency is 0.77. This FoM has the virtue that it does not depend on the number of atoms or the number of photons. It can be seen as the signal-to-noise ratio per atom, when the atoms have a 1/e probability of surviving the measurement. When measuring density within a cloud larger than the probe beam, the FoM is improved by reducing the beam size, as noted in [27]. It increases as A −1/2 , which motivates efforts to probe atom clouds using beams of small cross section [14,26]. When tracking the evolution of an entire ensemble, the size of the cloud sets a natural lower limit on the useful size of the light beam [13,28].
The dashed line in figure 7 shows how the FoM varies with the detuning of the central frequency f 0 from the atomic resonance frequency. We take the beat frequency 2δ f = 60 MHz used in the experiment. Strong minima appear when either of the optical frequencies is resonant with the atomic transition, since then the phase shift is close to zero and the loss is maximum. The optimum condition, with the optical frequencies symmetrically on either side of resonance, gives FoM 1/400, indicating that ∼400 atoms in our 100 µm beam can be detected with a signal : noise ratio of 1. The solid line shows FoM when we add a magnetic field of 0.65 mT, which is typical of the field experienced by atoms in our magnetic trap. This changes the frequency dependence of FoM quite significantly in the vicinity of the resonances. We see the two sharp minima associated with the dispersive character of |φ 1 | move to higher frequency because they are due primarily to the σ + transition. In addition, we see two broader minima that are down-shifted. These are due to the σ − resonances that maximize the loss. This separation of the σ + and σ − transitions makes it possible to double the highest FoM. When detecting with parallel polarization, the loss is more severe, as we have discussed in section 2.2. In addition, one cannot use the Zeeman shift to improve the FoM since the same transition produces both the phase shift and the loss.
The FoM described above expresses the compromise between determining the atom number as accurately as possible and minimizing the scattering of photons. Equation (6) illustrates this idea under the reasonable assumptions of sufficient optical power to render the technical noise negligible and weak absorption of the light. In our case, with 3 × 10 5 photons per pulse, and atomic column density of 2.2 × 10 12 atoms m −2 , both assumptions are valid. In addition, we have specialized to the case of 87 Rb. However, the idea is readily applied to any experiment where the complex amplitude of a coherent light beam provides state-selective information about an atomic density through the one-photon interaction.

Application to measurements with Bose-Einstein condensates
Our technique is intended to be used with BECs. Any spontaneous emission will remove an atom from the condensate, so all the σ + excitations also cause loss. This typically increases the value of q in equation (6) by a factor of 16 because of the 15 : 1 ratio of σ + and σ − coupling strengths shown in figure 6. A condensate can be addressed on an atom chip by a beam of small waist (∼2 µm [26]), which enhances the FoM by a factor of 50 in comparison with the 100 µm beam used here. The net effect is a FoM very similar to the dashed curve in figure 7 but with a peak at the centre of 0.03. With a typical 87 Rb condensate of 3 × 10 4 atoms in a prolate trap of trapping frequencies 20 Hz and 1 kHz, 10 3 atoms are illuminated when the probe beam is centred on the cloud. Thus, the FoM shows that with our method, a 10% precision measurement of the central density entails a loss of only 100 atoms from the condensate.
The passage of an off-resonant light pulse through a BEC normally imposes a phase shift on the condensate equal to N γ N a φ, where φ is the phase shift that the atoms impose on the light. For probe light that is homogeneous over the whole trapped cloud (e.g. in non-destructive density imaging experiments [1][2][3][4][5]), a global phase shift is of no real importance. If the light beam is smaller than the cloud, the inhomogeneous light shift [29] can excite unwanted phonons in the BEC. However, in our case, the two frequency components impose a total phase shift of zero on the atoms because they are symmetrically disposed on either side of resonance. Thus, the net force on the atoms is zero [12] 3 . As a result, the BEC will be undisturbed apart from the noise in the force due to photon statistics.

Conclusion
In conclusion, this paper presents a method of measuring the column density through a small part of an atom cloud with minimum disturbance of the atoms. The method measures the phase shift between two synthesized frequency components of a laser beam, tuned on opposite sides of an atomic resonance. We have used this scheme to measure the column density of a magnetically trapped atom cloud and to monitor oscillations of the cloud in real time. Measurement sensitivity is principally limited by photon shot noise and excess noise due to the avalanche amplification in the photodiode. We have measured how many atoms are lost from the trap as an unwanted byproduct of the measurement. We have developed a FoM for this scheme, which quantifies the relationship between the sensitivity and the destructiveness of the measurements. Using this we have anticipated the performance of the technique when applied to BECs trapped on an atom chip.