High Bandwidth Atomic Magnetometery with Continuous Quantum Non-demolition Measurements
aa r X i v : . [ qu a n t - ph ] J a n High Bandwidth Atomic Magnetometery withContinuous Quantum Non-demolition Measurements
V. Shah, G. Vasilakis and M. V. Romalis
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
We describe an experimental study of spin-projection noise in a high sensitivity alkali-metalmagnetometer. We demonstrate a four-fold improvement in the measurement bandwidth of themagnetometer using continuous quantum non-demolition (QND) measurements. Operating in thescalar mode with a measurement volume of 2 cm we achieve magnetic field sensitivity of 22 fT/Hz / and a bandwidth of 1.9 kHz with a spin polarization of only 1%. Our experimental arrangement isnaturally back-action evading and can be used to realize sub-fT sensitivity with a highly polarizedspin-squeezed atomic vapor. The limits imposed by quantum mechanics on precisionmeasurements have been the subject of long-standing in-terest. They are particularly important in atomic sys-tems that form the basis of leading frequency standards,magnetometers and inertial sensors. The Heisenberg un-certainty principle imposes a limit on measurement sen-sitivity with uncorrelated atoms known as the standardquantum limit (SQL). One can improve upon this limitusing spin squeezing techniques [1]. However, it has beenshown theoretically that in the presence of a constantrate of decoherence spin squeezing does not lead to asignificant improvement of the long term measurementsensitivity, but can be used to reduce the measurementtime [2–4]. Such an increase in the measurement band-width without loss of sensitivity is particularly impor-tant for systems with long spin coherence times, whichcan achieve the highest measurement resolution but oftenrequire impractically-long interrogation times.Here we study the limits imposed by quantum spinfluctuations in a dense hot alkali-metal vapor used in sen-sitive atomic magnetometers [5]. While the sensitivity ofmost previous atomic magnetometers has been limited byphoton shot noise or technical noise, we investigate theregime limited by spin projection noise. We show that inthis regime the measurement bandwidth can be increasedusing quantum-non-demolition paramagnetic Faraday ro-tation measurements [6] without loss of sensitivity. Wepoint out that the bandwidth increase can be realizedwith only a weak squeezing condition (∆ J x ) < J/ J , and demonstrate exper-imentally an increase in the magnetometer bandwidthfrom 420 Hz to 1.9 kHz. Increasing the measurementbandwidth is important for many applications of atomicmagnetometery, such as detection of biological magneticfields from the heart and the brain [7, 8] and nuclearmagnetic resonance [9–11], where the bandwidth of mag-netic signals often exceeds the natural bandwidth of theatomic magnetometers. We operate the magnetometerin a scalar measurement mode and obtain magnetic fieldsensitivity of 22 fT/Hz / , in agreement with theoreticalprediction for the size of spin projection noise and only afactor of 2 away from the best sensitivity previously ob- FIG. 1: Apparatus for high-bandwidth QND scalar magne-tometry. tained in a scalar magnetometer [12]. Our measurementsare performed with spin polarization of only 1%, whereback-action evasion is not necessary. However, our mea-surement configuration is naturally back-action evadingand thus much higher sensitivity can be expected in ahighly-polarized atomic vapor.The experimental geometry is shown in Fig. 1. A Pyrexglass cell 1 cm in diameter and 11.4 cm long containsenriched Rb and 60 torr of Nitrogen buffer gas and isheated to about 110 ◦ C. The cell is placed inside multi-layer magnetic shields and a magnetic field of 4.4 µ Tgenerated by an ultra-stable current source is directedperpendicular to the long axis of the cell, giving a Larmorfrequency of 31 kHz. A linearly polarized probe beamdetuned from the D1 line is directed along the length ofthe cell. Paramagnetic Faraday rotation induced by theatoms is measured with a balanced polarimeter.First we measure the spin-projection noise in an un-polarized vapor. The power spectrum of the polarimeteroutput is plotted in Fig. 2, showing a large noise peakcentered at the Larmor frequency. Detection of alkali-metal spin fluctuations by paramagnetic Faraday rota-tion was first demonstrated by Alexandrov [13] and laterstudied in [14–16]. We obtain a ratio of the peak spinnoise power to the flat photon shot noise backgroundequal to 22, substantially higher than in previous exper-iments. For our conditions the probe laser is detuned faraway relative to the excited state hyperfine structure andthe Lorentzian and Doppler linewidths, but comparableto the ground-state hyperfine splitting. The optical rota-tion angle in this case is proportional to the vector spinpolarization, with negligible contribution from the tensorpolarization [17, 18]. For the D1 line the rotation angleis given by φ = cr e f osc nl (2 I + 1) [ D ( ν − ν a ) h F ax i − D ( ν − ν b ) h F bx i ] , (1)where r e = 2 . × − cm is the classical electron ra-dius, f osc = 0 .
34 is the oscillator strength of the D1transition in Rb, n is the vapor density of alkali-metalatoms, l is the length of the cell along the probe direc-tion, h F ax i and h F bx i are the expectation values of theatomic spin F = I + S in the a = I + 1 / b = I − / ν a and ν b . Here D ( ν ) is the dispersion profile given by D ( ν ) = ( ν − ν ) / [( ν − ν ) + (∆ ν/ ], where ∆ ν is theLorentzian FWHM due to pressure broadening. For un-polarized, uncorrelated atoms the r.m.s. spin fluctationsare given by q h F ax i = s F a ( F a + 1)(2 F a + 1)6(2 I + 1) N , (2)where N is the number of atoms being probed, and sim-ilar for F bx . Most spin relaxation mechanisms in alkali-metal vapor affect only the electron spin and hence canintroduce correlations between F ax and F bx . However, inthe regime when the Larmor frequency ω is much largerthan the spin relaxation rate, the transverse spin compo-nents of F a and F b precess in opposite directions, quicklydestroying any correlations. Hence polarization rotationnoise from the two terms in Eq. (1) is not correlated.The number of atoms participating in the measurementdepends on the transverse intensity profile of the probebeam I ( y, z ) and is determined by summing noise vari-ance from different parts of the beam, N = nl (cid:2)R I ( y, z ) dydz (cid:3) R I ( y, z ) dydz . (3)The data in Fig. 2 are very well described by a sin-gle Lorentzian and a constant photon shot noise back-ground. The Lorentzian half-width of the noise peak isequal to 340 Hz, with about half of the width due tospin-exchange relaxation and the rest due to absorptionof the probe beam and diffusion. The flat noise back-ground is in agreement with expected level of photonshot noise. As discussed in [14–16], the noise spectrum ingeneral is a sum of Lorentzians, but the widths of the res-onances for h F ax i and h F bx i are similar and the strength ofthe h F bx i signal is smaller, so it is difficult to distinguishthem. There are only small hints of a deviation froma Lorentzian in the wings of the noise peak. The totalr.m.s. noise φ rms given by the area under the peak of the FIG. 2: Measured spin noise power spectrum of unpolarizedatoms (solid line) with a fit to a Lorentzian plus flat photonshot noise background (dashed line). The averaging time forthe noise spectrum is 350 s. power spectrum is not affected by the shape of the spec-trum. Diffusion of atoms in and out of the probe beamalso distorts the spectrum of the noise from a Lorentzianshape, causing a sharper peak due to Ramsey narrowing[19], but it also does not change φ rms .For the data presented in Fig. 2 the density of Rbatoms is equal n = 8 . × cm − , determined bya measurement of the transverse spin relaxation timedue to spin exchange collisions. The intensity profile ofthe probe beam was measured in both directions by thescanning edge technique and had effective dimensions of3 . × . . We find that the effective number ofatoms being probed is N = 1 . × . The probe laser isdetuned by 19 GHz from the F = 2 state and 25.8 GHzfrom F = 1 state, much larger than the Lorentzian op-tical FWHM ∆ ν =1.42 GHz. According to Eq (1), thisgives φ thrms = 1 . × − rad. Experimentally, the areaunder the spin noise peak is equal φ exp rms = 1 . × − rad.The agreement at the 10% level is quite good given theuncertainty in the number of atoms participating in themeasurement. Using a different cell with natural abun-dance of Rb isotopes we verified experimentally that theratio of spin noise for Rb and Rb isotopes, whichhave different nuclear spins, is consistent with our analy-sis within a few percent. In a previous detailed study ofatomic spin noise [15] the overall level of spin noise wasoff by a factor of 2 from predictions.A scalar atomic magnetometer measuring the absolutevalue the magnetic field is realized using a circularly po-larized pump beam propagating nearly parallel to theprobe beam but missing the photodetectors. We use Bell-Bloom excitation [20] of the spin precession around B z field by sinusoidally modulating the current in a DFBlaser used to generate the pumping light at the Larmor FIG. 3: Lock-in output amplitude noise spectrum for polar-ized atoms (solid line) and unpolarized atoms (dashed line).The signal expected in the magnetometer for a 22 fT rms mag-netic field is shown with dots. The spikes in the noise spec-trum for polarized atoms are from technical magnetic noise. frequency. The polarimeter signal is directly digitizedusing a fast, high resolution A/D card and lock-in de-modulation of the data is implemented at the analysisstage. The noise spectrum of the out-of-phase componentof the lock-in signal, proportional to small variations of B z field, is shown in Fig. 3 with and without the pumpbeam. The intensity of the pump beam is increased un-til the noise level just starts to increase due to technicalnoise sources and is equal to about 100 µ W. The spinpolarization of the atomic vapor, determined from theamplitude of the oscillating rotation signal, is equal to1.0%. Discrete noise peaks from magnetic interferencecan be seen for polarized atoms. The magnetic field sen-sitivity of the rotation signal is calibrated by applyinga known modulation to B z field at various frequencies.We show in Fig. 3 with filled circles the expected mag-netometer signal for a 22 fT rms oscillating magnetic fieldas a function of frequency. It can be seen that at higherfrequencies the response of the magnetometer drops, butthe noise level decreases as well, so the magnetometerretains its sensitivity up to much higher frequencies thanthe resonance linewidth.In fact, the sensitivity remains constant as long as thenoise spectrum is dominated by spin noise. This can beeasily seen in the simpler case of a spin-1/2 system whenthe magnetic resonance is described by Bloch equations.One can show that the absolute value of the magnetome-ter signal in response to an oscillating B z field decreaseswith frequency as the square root of a Lorentzian with awidth given by the inverse of the spin coherence time T , S ( f ) = S / [1 + (2 πf T ) ] / . (4)The shape of the spin noise spectrum is also describedby the square root of a Lorentzian with the same width FIG. 4: Experimentally observed magnetic field noise spec-tral density corrected for the frequency response of the mag-netometer (solid line). The spikes are due to narrow-bandmagnetic noise. Dashed line shows expected magnetometersensitivity for a demolition measurement assuming a flat noisespectrum. The shaded area represents improvement in themagnetometer sensitivity at high frequencies as a result ofQND measurements. For QND measurements the sensitivitydecreases by √ [21]. Hence the signal and the noise decrease simultane-ously, maintaining constant sensitivity. In Fig. 4 we showthe magnetic field sensitivity of the magnetometer as afunction of frequency by dividing the noise spectrum bythe response curve. The sensitivity remains nearly con-stant up to 2 kHz, while the resonance linewidth of themagnetometer is equal to 420 Hz in this case. The in-crease in the bandwidth is a direct result of the non-whitenature of the spin noise that is realized with QND mea-surements which preserve temporal correlations of thespin expectation value. In contrast, if the spin polar-ization is monitored using optical absorption instead ofFaraday rotation, or if the optical density of the vapor onresonance is less than one, the noise spectrum would bewhite. We show in Fig. 4 the sensitivity that would beobtained with such a flat noise spectrum for comparison.This demonstrates the role of quantum-non-demolitionmeasurements, they do not improve the performance atlow frequencies but increase the bandwidth of measure-ments without any penalty in sensitivity. The neces-sary condition for bandwidth increase can be written as(∆ J x ) < J/ J x ) < h J z i / / √ f BW = 1 / (2 πT ) . Using electronic feedback orself-oscillating operation it is possible to obtain a flatfrequency response, however, even in this case the sensi-tivity of the magnetometer becomes worse at frequencieshigher than 1 /T because of noise induced by the feed-back [22]. For a quantum non-demolition paramagneticFaraday rotation measurement with a far-detuned probelaser, the spectral noise density can be written as φ n ( f ) = 1 √ (cid:26) η + N ab Γ pr T β π ( f − f ) T ] (cid:27) / , (5)where Φ is the probe laser photon flux, η is the quan-tum efficiency of the photodetectos, N ab is the numberof absorption lengths (optical density) of the atomic sam-ple on resonance, Γ pr is the optical pumping rate of theprobe laser and β is a factor of order unity dependingon the nuclear spin and polarization of the atomic en-semble ( β = 1 for I = 0). Using Eq. (4) and (5)it’s easy to show that the bandwidth of the magnetome-ter with quantum-non-demolition measurements is givenby f BW = ( ηN ab Γ pr T β + 1) / / (2 πT ). In the regimewhere the spin relaxation rate is dominated by the op-tical pumping rate of the probe laser, Γ pr T ≃
1, thebandwidth is increased by a factor on the order of thesquare root of the optical density on resonance N / ab .For our conditions the back-action of the probe beam isnot significant because of low spin polarization. However,our experimental arrangement is naturally back-actionevading and can be used to generate conditional spinsqueezing. In the regime of far deturning the back-actionof the probe beam is due to the light shift created by thequantum fluctuations of the circular polarization of theprobe beam and is equivalent to a fictitious magnetic fieldparallel to the propagation direction of the probe beam[17]. Our probe beam is directed perpendicular to a largestatic magnetic field and thus the light shift fluctuationsonly contribute in second order to the absolute value ofthe magnetic field measured by the scalar magnetometer.A fluctuating magnetic field along the x direction willgenerate a small z component of the polarization that isnot directly measured. Therefore this arrangement canbe used to generate conditional spin squeezing in a highlypolarized vapor [23].The magnetic field sensitivity of 22 fT/Hz / obtainedin this experiment is the best measured sensitivity with asingle-channel scalar magnetometer and within a factorof 2 of the best measured sensitivity obtained in a gra-diometer arrangement [12]. Scalar atomic magnetome-ters have lagged in sensitivity compared to other typesof atomic magnetometers [5] because of spin-exchangebroadening. The measurement volume used in our sen-sor is about 2 cm and the spin polarization is equalto 1%. The sensitivity of the magnetometer will be opti-mized if the intensity of the pumping light is increased toobtain spin polarization of 50% with a factor of 2 broad-ening of the magnetic resonance. The signal will increaseby a factor of 50, while the spin-projection noise will notchange appreciably, resulting in a magnetic field sensitiv-ity of about 0.6 fT/Hz / , consistent with spin-exchange limited sensitivity for alkali-metal magnetometers [12].Alternatively, operating in the regime of low spin polar-ization can be advantageous if it is desired to minimizethe heading errors of the magnetometer which depend onthe degree of polarization [24].In conclusion, we investigated operation of an atomicmagnetometer in the spin-projection noise limitedregime. The magnitude of the spin noise is in good agree-ment with theory. We demonstrated an increase in themagnetometer bandwidth by a factor of 4 using quan-tum non-demolition measurements. Such increase of themeasurement bandwidth without loss of sensitivity is im-portant for many practical applications of atomic magne-tometery, such as detection of NMR and biological fields.Similar QND measurements can also be used to increasethe bandwidth of magnetometers based on nuclear spinswith very long spin coherence times and the update rateof atomic clocks based on very narrow transitions. Thiswork was supported by an ONR MURI award. [1] D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore,and D. J. Heinzen, Phys. Rev. A , R6797 (1992).[2] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert,M. B. Plenio, and J. I. Cirac, Phys. Rev. Lett. , 3865(1997).[3] A. Andr´e, A. S. Sørensen, and M. D. Lukin, Phys. Rev.Lett. , 230801 (2004).[4] M. Auzinsh, D. Budker, D. F. Kimball, S. M. Rochester,J. E. Stalnaker, A. O. Sushkov, and V. V. Yashchuk,Phys. Rev. Lett. , 173002 (2004).[5] D. Budker and M. Romalis, Nature Physics , 227 (2007).[6] Y. Takahashi, K. Honda, N. Tanaka, K. Toyoda,K. Ishikawa, and T. Yabuzaki, Phys. Rev. A , 4974(1999).[7] G. Bison, R. Wynands, and A. Weis, Optics Express ,904 (2003).[8] H. Xia, A. B.-A. Baranga, D. Hoffman, and M. V. Ro-malis, Appl. Phys. Lett. (2006).[9] I. M. Savukov and M. V. Romalis, Phys. Rev. Lett. ,123001 (2005).[10] S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M.Rochester, D. Budker, and A. Pines, Proc. Nat. Acad.Sci. , 12668 (2006).[11] M. P. Ledbetter, I. M. Savukov, D. Budker, V. Shah,S. Knappe, J. Kitching, D. J. Michalak, S. Xu, andA. Pines, Proc. Nat. Acad. Sci. , 2286 (2008).[12] S. J. Smullin, I. M. Savukov, G. Vasilakis, R. K. Ghosh,and M. V. Romalis, Phys. Rev. A , 033420 (2009).[13] E. Alexsandrov and V. Zapassky, Zh. Eksp. Teor. Fiz. , 132 (1981).[14] S. Crooker, D. Rickel, A. Balatsky, and D. Smith, Nature , 49 (2004).[15] B. Mihaila, S. A. Crooker, D. G. Rickel, K. B. Blagoev,P. B. Littlewood, and D. L. Smith, Phys. Rev. A ,043819 (2006).[16] G. E. Katsoprinakis, A. T. Dellis, and I. K. Kominis,Phys. Rev. A , 042502 (2007).[17] B. S. Mathur, H. Tang, and W. Happer, Phys. Rev. ,
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