Feedback control of an atom laser

We report the first real-time feedback control of an atom laser. The unique feature of metastable helium atoms, the production of ions in the atom laser outcoupling process, is exploited to actively control the spatial location inside the Bose-Einstein condensate where outcoupling occurs. Unlike alkali atom lasers, this provides almost instantaneous feedback which reduces frequency, amplitude and spatial mode fluctuations in the atom laser beam. © 2008 Optical Society of America OCIS codes: (020.0020) Atomic and Molecular Physics; (020.1475) Bose-Einstein Condensates References and links 1. M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Output Coupler for Bose-Einstein Condensed Atoms,” Phys. Rev. Lett. 78, 582-585 (1997). 2. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97-105 (1983). 3. K. G. H. Baldwin, “Metastable helium: Atom optics with nano-grenades,” Cont. Phys. 46, 105-120 (2005). 4. G. V. Shlyapnikov, J. T. M. Walraven, U. M. Rahmanov, M. W. Reynolds, “Decay Kinetics and Bose Condensation in a Gas of Spin-Polarized Triplet Helium,” Phys. Rev. Lett. 73, 3247-3250 (1994). 5. C. J. Dedman, R. G. Dall, L. J. Byron and A. G. Truscott, “Active cancellation of stray magnetic fields in a Bose-Einstein condensation experiment,” Rev. Sci. Instrum. 78, 024703 (2007). 6. A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, “Hybrid apparatus for Bose-Einstein condensation and cavity quantum electrodynamics: Single atom detection in quantum degenerate gases,” Rev. Sci. Instrum. 77, 063118 (2006). 7. J.-F. Riou, W. Guerin, Y. Le Coq, M. Fauquembergue, V. Josse, P. Bouyer, and A. Aspect, “Beam Quality of a Nonideal Atom Laser,” Phys. Rev. Lett. 96, 070404 (2006). 8. M. Köhl, Th. Busch, K. Mølmer, T. W. Hänsch, and T. Esslinger, “Observing the profile of an atom laser beam,” Phys. Rev. A 72, 063618 (2005). 9. R. G. Dall, L. J. Byron, A. G. Truscott, G. R. Dennis, M. T. Johnsson, M. Jeppesen, and J. J. Hope, “Observation of transverse interference fringes on an atom laser beam,” Opt. Express 15, 17673-17680 (2007). 10. R. G. Dall, and A. G. Truscott, “Bose-Einstein condensation of metastable helium in a biplanar quadrupole Ioffe configuration trap,” Opt. Commun. 270, 255-261 (2007). 11. V. Venturi and I. B. Whittingham, “Close-coupled calculation of field-free collisions of cold metastable helium atoms,” Phys. Rev. A 61, 060703(R) (2000). 12. M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Collective Excitations of a Bose-Einstein Condensate in a Magnetic Trap,” Phys. Rev. Lett. 77, 988-991 (1996). 13. S. A. Haine, A. J. Ferris, J. D. Close, and J. J. Hope, “Control of an atom laser using feedback,” Phys. Rev. A 69, 013605 (2004).


Introduction
The success of the optical laser in many precision measurement applications is a testament to how well its parameters can be controlled.The LIGO project, for example, whose aim is to detect gravity waves using an interferometer stabilised with optical lasers, requires positional stability of the mirrors to orders of magnitude less than the diameter of a single atom.Such precision is only possible using highly stabilised lasers that use fast feedback control to compensate for environmental effects.
Derived from atoms coupled out from a Bose-Einstein condensate (BEC), an atom laser is a coherent beam of matter waves [1].While feedback loops to control the various parameters of optical lasers are common place [2], the same is not true for atom lasers.The main reason is that the slow speed of the atoms in the atom laser beam causes a time lag between the outputcoupling process and detection of the beam downstream.This time delay makes fast-feedback control almost impossible -an issue not present in the case of the optical laser because the speed of light virtually eliminates the delay.
However, the use of a metastable helium (He*) atom laser makes feedback control feasible.The He* atoms are created in an excited (2 3 S 1 ) state containing ∼ 20 eV of energy [3].This large internal energy gives rise to an inelastic process known as Penning ionisation: two colliding He* atoms have a combined internal energy of ∼ 40 eV which is sufficient to ionize one of the collision partners while transferring the other to the ground state.In the case of atoms in a magnetically trapped He* BEC, due to their spin polarisation, Penning processes are greatly reduced [4].However, the outcoupling process in a He* atom laser beam relies on transferring atoms from the m=1 BEC state to the m=0 laser state, unpolarising the gas and giving rise to Penning ions.These ions can be accelerated towards a detector enabling a fast, non-destructive control signal of the atom laser beam.
To understand our atom laser stabilisation scheme it is instructive to draw an analogy with similiar systems used to improve the stability of optical lasers.Consider the simplest optical laser consisting of a gain medium confined between two mirrors.One of the mirrors, the outputcoupler, is not totally reflective and thus allows a small portion of the circulating light inside the laser cavity to leak out.For the laser to operate stably, the position of the output coupler must remain fixed, otherwise the resonance condition of the cavity will change and the laser frequency will fluctuate.In reality there is always noise present that leads to variations in the position of the outcoupling mirror, and so active feedback is routinely used to stabilise the mirror position and hence the frequency of optical lasers.In most stabilisation schemes a small fraction of the main laser beam is analysed to determine the noise present and the outputcoupling mirror is actively adjusted to compensate for the noise.
The case is similiar for a radio frequency (RF) output-coupled atom laser [1], except instead of using a mirror to output-couple atoms into the laser state, a resonant RF field is used to flip the spin from the magnetically trapped BEC state to an untrapped atom laser state.Analogously, any variations in the detuning of the RF can lead to fluctuations in the frequency of the atom laser.However, unlike the optical laser case, noise on the atom laser output-coupler also results in amplitude and spatial mode fluctuations.
To reduce these fluctuations through feedback control, the monitoring point for the atom laser cannot be very far downstream as flight times from the condensate to the detector are typically the order of many milliseconds.A way around this problem is to probe the atom laser beam at the output-coupling process, and not rely on monitoring the atoms in the beam itself.
In this paper we report the first implementation of active feedback to stabilise a continuous wave (CW) RF atom laser.An error signal derived from ions produced during the formation of the atom laser, with appropriate gain and sign, is fed back to the RF output-coupler, locking the output-coupling surface inside the condensate.The scheme compensates fluctuations in the magnetic trap bias field, thereby actively reducing frequency, amplitude and spatial noise on the atom laser.
Fluctuations in the magnetic trap bias field are caused by ambient magnetic fields, e.g. from power supplies and other experimental devices.To minimise these effects in our experiment we use an active compensation scheme [5] to reduce the predominantly 50 Hz and higher harmonic background to aproximately 10µG RMS, (note that others have used passive magnetic shielding to achieve similiar performance [6]).Even after such precautions remnant magnetic background fields couple to the trapping potential, causing the effective output-coupling detuning to fluctuate.These variations result in amplitude, frequency and spatial mode fluctuations.
Interestingly, the degree with which amplitude noise is observed experimentally is heavily dependent on where in the condensate the RF is tuned, since this sets the spatial location at which outcoupling takes place.If the RF is tuned near the high density region of the condensate, then shot-noise-limited amplitude performance is routinely achieved in our experiment, since slight variations in the outcoupler spatial location are masked by the near flat central density profile of the condensate.However, if the RF is tuned near the outer edge of the condensate (approximately given by the magnetic trap minimum), the magnetic field noise becomes obvious in the atom laser amplitude due to the sharp change in BEC density.
In addition, the frequency (energy) of the outgoing atom laser beam broadens as a result of fluctuations in the trap bias field, independent of the detuning of the output-coupler.Fluctuations of the bias field also effect the spatial mode of an atom laser, since in most cases the profile is distorted by the interaction of the atom laser beam with the BEC [7,8,9].It follows then that the far field profile of the atom laser is dependent on how much of the condensate it has to traverse.

Experiment and results
The experimental setup for creating a He* BEC has been reported elsewhere [10].Briefly, we use a cryogenic beamline to produce a He* magneto-optic trap (MOT).Atoms are extracted from this low vacuum MOT into a high vacuum MOT via an LVIS + setup.Atoms are trans-ferred from the MOT into a BiQUIC magnetic trap [10], where a BEC is produced using forced RF evaporative cooling.Using this setup we are able to create almost pure BECs containing up to 5×10 6 atoms in a magnetic trap with a radial frequency of 460 Hz and an axial frequency of 55 Hz.Fig. 2. Solid line is the ion signal that results when the outcoupling frequency is stepped from just below the trap minimum to the lock point for a 5 ms period.The 5 ms timing pulse is shown as a dashed line.
Our locking scheme relies on the atom laser signal being highly correlated with the fast responding ion signal.As previously mentioned the origin of the ion signal is due to Penning processes.The rate constants for these various processes have been calculated theoretically [11] and vary depending on the total angular momentum of the colliding atoms.At the very low temperatures in our experiment the rate constants are calculated to be ∼ 10 −14 cm 3 /s, 10 −13 cm 3 /s, 10 −9 cm 3 /s, for a j=2, j=1, j=0 collison respectively (where j is the quantum number associated with the total angular momentum).It follows then that for our outcoupling rate (∼ 10 7 atoms/s) the majority of ions we detect are due to m=0 atoms colliding with each other, since the rate constant for an m=1 atom colliding with an m=0 atom is four orders of magnitude smaller than for a j=0 collision.Note that although the rate constant for an m=-1 atom colliding with an m=1 atom is high, the number of m=-1 atoms should be relatively small and thus should lead to minimal ion production.
To measure the correlation between the atom and ion signals we use a single electron multiplier (EM).This is possible since the ions arrive within 100 µs after creation while the atoms have a flight time ∼ 150 ms. Figure 1 shows the strong correlation between atoms and ions for a constant output-coupling intensity (Rabi frequency ∼ 2 kHz) as the RF detuning is varied.When the output-coupler is tuned to the minimum of the magnetic trap (zero detuning) there is a region where both signals vary linearly with the outcoupling frequency which is ideal for locking the atom laser.Moreover, with the RF outcoupler tuned to this region the noise on the atom laser becomes more obvious, due to the sharp density change of the condensate.
The step response of the feedback system is also important, since any delays will adversely affect the performance of the locking loop.To determine this we measure the ion signal that results when the detuning of the outcoupler is suddenly changed from below the edge (zero detuning) of the condensate to the lock point (see Fig. 1).The resulting ion signal is shown in Fig. 2 and interestingly it displays an asymmetric response, with a rise time of ∼ 250 µs and a fall time of ∼ 1.2 ms.The rise time is due to the flight time of the ions to the detector as well as the pumping interval required to transfer atoms into the m=0 atom laser state.The long decay time of the ion signal is indicative of the time taken for the density of the m=0 atom laser to decrease driven by mean field interactions.The atom laser stabilisation scheme is implemented as follows.After the production of the atom laser the atoms fall ∼ 14.5 cm under gravity where they strike an EM and are detected with high efficiency.Ions produced from the atom laser process are accelerated towards a mesh located in front of the same EM by a -2 KV potential.A constant fraction discriminator is used to produce pulses of constant width (500 ns) and amplitude (5 V) from the EM ion signal.The detected ion pulse rate is typically 100 kHz.After some integration and filtering the ion signal is fedback to control the RF out-coupling frequency.In operation the controller gain is advanced until instability occurs, and then slightly reduced.As with all control loops, the performance is ultimately limited by unwanted phase lags within the system.In our case the ∼ 1.2 ms decay time of the ions leads to the feedback loop oscillating at a frequency of 400 Hz as the gain is increased, indicating a phase lag of 90 degrees, limiting attainable loop gain and performance.
To demonstrate the performance of the stabilisation technique we create CW atom laser beams of 100 ms in duration.As with conventional locking schemes we tune the atom laser RF detuning close to the lock point to minimise the work the loop has to do.The locking loop is closed at the same time that the RF atom laser pulse begins.In order to make a direct comparison between the locked and unlocked performance we only apply the stabilisation for the first 50 ms of the atom laser pulse.The resulting EM signal is shown in Fig. 3 as well as the lock signal (inset).
With the stabilisation enabled the first half of the ion signal locks to a predetermined level set by the lock point.The large correcting excursions present on the controller output signal (inset) indicates that the controller is actively reducing the large 50 Hz (mains frequency) spikes that are present (see the second half of the atom laser signal).A small overshoot at the start of the ion signal is observed due to the initial turn on of the stabilisation loop.Note that the ion signal does not sit on a zero background due to the presence of ions produced by BEC atoms colliding.The unperturbed BEC background is stable on the timescale of our experiment, with the Penning limited lifetime of the BEC being of order 1.5 seconds.However, production of the atom laser leads to BEC loss and consequently a change in this background level.Ideally, the atom laser loss would be a very small percentage of the BEC and so this effect would be minimal.In our experiments we need to couple out ∼ 30% of the condensate over the 100 ms atom laser pulse to reduce the measurement shot noise to an acceptable level, consequently the ion background rate is changing.Although this rate change was not corrected for, it could be compensated by actively reducing the set point appropriately.This is a technical point, however, which a BEC containing more atoms would solve.
While the ion signal is effectively stabilised, the real proof is in the corresponding atom laser signal.As shown in Fig. 3, the stabilised part of the atom laser signal is almost completely devoid of the large 50 Hz spikes present in the unstabilised part.Interestingly, the locked atom laser does not maintain the same amplitude, rather there is a small slope to the signal.The reason for this is the changing ion background explained previously.As the atoms are coupled out, the number of atoms in the BEC decreases and the background ion rate decreases.Thus to achieve the same ion lock point more atoms need to be outcoupled.Nonetheless, the reduction of noise on the locked atom laser is apparent, and is best quantified by taking a comparison of the power spectrums of the locked and unlocked parts of the atom laser signal.Figure 4 shows a power spectrum comparison for both the ion signal and atom signal.Both comparisons show a reduction of power at the main 50 Hz noise frequency of about an order of magnitude.The ion signal also shows a large reduction in the 100 Hz and 150 Hz noise components, while the atom signal only shows a small improvement at these frequencies, possibly due to blurring of the atom laser signal caused by its long time of flight.

Conclusion
In summary we have demonstrated the first successful feedback control of an atom laser.By tuning the outcoupler of the atom laser to the edge of the condensate we become sensitive to magnetic trap noise that causes the amplitude, frequency and the spatial profile to fluctuate.Use of the ions, produced in the production of a He* atom laser, as a fast real time control signal has allowed us to actively feedback to the RF detuning and reduce these fluctuations by at least an order of magnitude within a bandwidth of ∼ 200 Hz.At present the results of our system are limited by the size of the condensate and the efficiency with which we can extract ions which at present is approximately 10%.A larger condensate and better ion extraction would allow us reduce the shot noise on the ion signal as well as minimise drift in the locked atom laser amplitude due to a varying background BEC ion rate.Besides being able to stabilise the output of an atom laser, a similar technique might be used to stabilise oscillations in a BEC [12].In many BEC experiments the trap frequency is altered and under some circumstances this can lead to excitation of unwanted modes.Since these oscillations can lead to density changes they should be detectable in an ion signal, which could then be used to feedback to the relevant magnetic trap currents [13].

Fig. 1 .
Fig.1.Atom laser signal (solid line) as a function of the outcoupling RF detuning.Also shown (dashed line) is the corresponding ion signal.The locking point for the experiments outlined in this paper is also indicated.Measurements were taken at a temperature T ∼ 0.5T c .

Fig. 3 .
Fig.3.Electron multiplier trace, averaged over four runs of the experiment, demonstrating stabilisation of the atom laser beam.The ion signal is first to arrive on the left of the trace, while the atom signal arrives ∼ 150 ms later which is the time of flight from the BEC to the EM.Shown in the inset is the output of our control circuitry.

Fig. 4 .
Fig. 4. (A) Power spectrum of the locked (dashed line) and unlocked (solid line) ion signal.(B) as in (A) except this time the power spectrums for the atom signal are displayed.