arX

iv:1

501.

0679

7v1

[phy

sics.a

tom-p

h] 2

7 Jan

2015

Atom interferometric gravitational wave detection using heterodyne laser links

Jason M. Hogan and Mark A. KasevichDepartment of Physics, Stanford University, Stanford, California 94305

(Dated: January 28, 2015)

We propose a scheme based on a heterodyne laser link that allows for long baseline gravitationalwave detection using atom interferometry. While the baseline length in previous atom-based propos-als is constrained by the need for a reference laser to remain collimated as it propagates between twosatellites, here we circumvent this requirement by employing a strong local oscillator laser near eachatom ensemble that is phase locked to the reference laser beam. Longer baselines offer a numberof potential advantages, including enhanced sensitivity, simplified atom optics, and reduced atomicsource flux requirements.

Gravitational wave (GW) detection with atom interfer-ometry offers a promising alternative to traditional op-tical interferometry [1, 2]. Advantages of this approachinclude phase multiplication through multiple pulse se-quences, proof mass resilience [3], laser frequency noiseimmunity and quantum back-action noise immunity [4].Insensitivity to laser frequency noise also allows for thepossibility of a detector design based on a single linearbaseline, requiring only two satellites instead of the con-ventional three.

In previous proposals, technical considerations havelimited the possible baseline length of the detector. Forsuitably low frequency GWs, the sensitivity of the an-tenna scales linearly with the antenna baseline. In thisLetter, we describe a method which enables antenna op-eration with substantially longer baselines. This resultsin designs whose sensitivities exceed those of existing pro-posals (e.g. LISA), but which do not require significantadvances in the state-of-the-art for atom interferometry.For example, we describe an antenna with 10 times thesensitivity of the LISA antenna that invokes 12 photonrecoil atom optics.

The concept for an atom-based GW antenna is to com-pare two light-pulse atom interferometers, one at eachend of a long baseline. To implement the atom inter-ferometers, pulses of laser light are used to realize beamsplitters and mirrors for the atom de Broglie waves [5].In a single-baseline detector, the light pulses are sentback and forth across the baseline from alternating direc-tions, interacting with the atoms on both ends [4]. In thisscheme, the phase difference between the atom interfer-ometers is sensitive to variations of the light travel timeacross the baseline, so by monitoring the phase differ-ence it is possible to detect fluctuations in the light traveltime induced by GWs. Importantly, since the same laserpulses interact with atoms on both sides of the baseline,the common laser phase noise is substantially suppressedin the phase difference between the interferometers [1, 4].

The envisioned detector consists of two independentspacecraft, each with its own source of ultracold atoms.The atom interferometers remain inside (or nearby) thelocal satellite, while telescopes mounted on each satelliteare used to send the atom optics laser pulses across the

baseline to interact with the atoms on both ends. Previ-ous GW detection proposals using atom interferometryhave assumed that the atom optics laser beam is colli-mated, constraining the allowed baseline length L to nolarger than the Rayleigh range zR of the laser: L 2zR =2piw2/, where is the laser wavelength and w is the ra-dial beam waist [1, 4]. Assuming (/2pi) kHz Rabifrequencies with 1 m telescopes and 10 W laser powersets a practical limit for baseline length of L 103 km[1, 4]. By comparison, the traditional LISA design callsfor a baseline of at least 106 km.

Although atom interferometric detectors operating at1000 km have the potential to reach comparable sensitiv-ity to LISA [1], the advantages of increasing the baselineare tantalizing. Increased detection sensitivity could al-low for science reach beyond LISAs targets, potentiallyeven giving access to signals of cosmological origin such asthe predicted primordial gravitation waves generated byinflation [1]. In addition to substantially enhanced signalstrength, the size of many background noise sources aresuppressed. Generally, a local acceleration noise sourcea results in an effective strain response a/L, solonger baselines can reduce the technical requirementsneed to control a wide class of backgrounds. In addi-tion, at the same target GW signal strength, increasingthe baseline can reduce the need to use large momentumtransfer (LMT) and other phase enhancement techniques[4, 6], simplifying the interferometer operation.

Here we propose a new concept for an atom inter-ferometric GW detector that can support substantiallylonger baselines without requiring proportionally largertelescopes or increased laser power. The idea is analo-gous to the traditional laser links used to connect thetest masses in the LISA concept [7, 8]. In our proposal,intense local lasers are used to operate the atom inter-ferometers at each end of the baseline. To connect theseotherwise independent local lasers, reference lasers beamsare transmitted between the two spacecraft, and the lo-cal lasers are kept phase locked to the incoming wave-fronts of these reference lasers. In this scheme, the ref-erence beams do not need to be collimated, since thephase locks can be done using much less intensity thanis required to drive the atomic transitions. This allows

2L

z

x

y Satellite 2Satellite 1

PD2PD1

BS BS AtomsAtomsR2

R1

M2

M1

TTMTTM

TTM

LO1TTM

LO2

FIG. 1. Schematic of the proposed design. M1 and M2 are the master lasers, with beams depicted as dotted and solid lines,respectively. The reference beams propagating between the satellites are denoted R1 (dotted) and R2 (solid). LO1 and LO2 arelocal oscillator lasers (dashed beam lines) that are phase locked to the incoming reference laser beams (R2 and R1, respectively).PD1 (PD2) is a photodetector used to measure the heterodyne beatnote between the incoming reference beam R2 (R1) andthe local oscillator laser LO1 (LO2) in order to provide feedback for the laser link. BS is a (non-polarizing) beamsplitter wherethe heterodyne beatnote is formed. Tip-tilt mirrors (TTM) allow for fine control of the pointing direction of each laser. Alladjacent parallel beams are nominally overlapped, but for clarity they are shown here with a small offset.

the baseline to be extended to LISA-like lengths withonly a modest telescope size and reference beam power.Critically, since the phase-locked local laser tracks thenoise of the incoming reference laser, this arrangementmaintains the essential common-mode laser phase noisecancellation between the two interferometers that allowsfor single baseline operation. The current proposal ef-fectively decouples the phase noise rejection requirementfrom the intensity demands, allowing the flexibility to in-dependently optimize the baseline and atomic transitionrate.

A schematic of the proposed design is shown in Fig.1. Each satellite contains an atom interferometer that isimplemented using laser pulses traveling along both thepositive and negative z direction. Both satellites containtheir own master laser (M1 and M2) that has enoughintensity to drive transitions in the local atom interfer-ometer. After interacting with the local atom cloud, eachmaster laser beam exits the satellite through a beamsplit-ter and then propagates across the baseline towards theopposite satellite. We refer to the beams propagating be-tween the satellites as reference beams: R1 and R2 arethe reference beams originating from satellite 1 and 2,respectively.

The reference beams are not assumed to be collimatedwhen they reach the opposite satellite, so for very longbaselines the received reference beam intensity is ex-pected to be too low to directly drive an atomic tran-sition. To address this, local oscillator lasers (LO1 andLO2) are phase locked to the incoming reference beams,and these lasers have sufficient intensity to drive transi-tions in the local atom interferometers. The phase lockfor laser LO1 is implemented by detecting the hetero-dyne beatnote formed by the incoming reference beamR2 with laser LO1 on the beam splitter BS in satellite 1

(and analogously for LO2 in satellite 2). In addition toa photodetector for measuring the phase difference be-tween the two beams, a quadrant detector (or camera)may be used to characterize the spatial interference pat-tern. This allows the pointing direction and spatial modeof the two lasers to be well matched using appropriatefeedback, as discussed later.

An essential consideration is the required noise perfor-mance of the phase lock between the reference beam andthe local oscillator in each satellite. Any noise added bythe phase lock is not common between the interferom-eters and so must be sufficiently small. One source ofnoise could arise from motion of the beam splitter (BS).The beam splitter is assumed to be rigidly connected tothe satellite bus, so any platform vibration noise will af-fect the beam splitter as well. Recall that in the originalatom GW design [1, 4], light propagates across the base-line between the two atom ensembles without encoun-tering any intervening optics, decoupling the differentialatom signal from satellite platform accelerations. In con-trast, the reference beams in the current scheme interactwith the beam splitters before reaching the atoms, con-ceivably adding noise. However, it turns out that theproposed scheme is insensitive to phase noise introducedby vibration of the beam splitters.

To see this, assume that the incoming reference laserhas phase R at the nominal position r = 0 of the beamsplitter. Due to vibration of the satellite, the beam split-ter may be displaced by some amount r, so upon re-flection from the beam splitter the reference beam willinstead have phase R = R + k r, where k is thewavevector of the incoming reference beam. On the otherhand, the LO beam that transmits through the beamsplitter is not impacted by the displacement, so the het-erodyne signal between the lasers encodes the vibration

3noise r. When the phase lock is engaged, the phase ofthe LO laser LO at r = 0 is locked to the phase of thereflected reference beam, resulting in LO = R+k r.Finally, consider the phase LO of the LO beam thatreflects off the beam splitter and that is subsequently in-cident on the atoms. Since the LO beam reflects off theopposite side of the beam splitter compared to the ref-erence beam, the reflected LO beam at r = 0 has phaseLO = LO k r. This implies that LO = R asdesired, so the vibration noise does not affect the lightreaching the atoms.The phase lock is ultimately limited by photon shot

noise of the received reference beam light. This is a newconstraint that has not been present in past designs basedon collimated beams. The optical phase noise powerspectral density (PSD) of the shot noise is approximatelySsh = h/Pr, where Pr is the power of the received ref-erence beam and = c/ is the light frequency. As-suming a Gaussian beam with power Pt and radial waistwt at the transmitting satellite, the received power col-lected by a telescope with diameter d is approximatelyPr 12Pt(d/wr)2 for d wr, where wr L/piwt isthe reference beam waist after propagating a distanceL zR to the receiving satellite location.To avoid limiting the detector strain resolution, the

photon shot noise contribution to the atom interferome-ter phase must be less than the contribution from atomshot noise. The RMS phase response of the atom in-terferometer to a laser pulse with optical phase noisePSD S() is

2rms =

S()|H()|2d, where |H()|

is the atom interferometer optical phase noise suscepti-bility [9]. For a single pulse, the susceptibility is a lowpass filter with |H()| 1 out to roughly the Rabi fre-quency = [1, 9]. Optical shot noise has a whitepower spectrum S() = Ssh, so the RMS phase noise foran interferometer with np (assumed uncorrelated) pulsesis approximately 2rms npSsh. The Rabi frequency =

It/2Isat is set by the intensity It =

Ptpiw2t /2

at the

transmitting satellite and assumes a two-level transitionwith saturation intensity Isat = 2pi

2~c/33 and natural

linewidth . Assuming an atom interferometer repetition

rate of fR, the associated noise PSD is 2

= 2rms/fR.

The final noise amplitude spectral density is then

=4

1536 ~c

pi5n1/2p 1/45/4L

f1/2R P

1/4t d

5/2(1)

where the telescope diameter is taken to be d = 2wt. Bycomparison, the phase noise PSD of atom shot noise is

2

a = 1/Na, where Na is the mean number of detectedatoms per unit time that participate in the atom inter-ferometer signal. In designing the laser link phase lockwe require that a. Assuming the Sr clock tran-sition, the telescope diameter is constrained to be

d = 28 cm(

L2109 m

)25(1 WPt

) 110(0.2 Hz/7fR/np

)15(

a103/

Hz

)25

(2)

FIG. 2. Strain sensitivity for a 2~k Sr interferometer withbaseline L = 2 109 m (green) as well as a 12~k inter-ferometer with baseline L = 6 108 m (blue). The strainresponses have been averaged over gravitational wave propa-gation direction and polarization. The 2~k curve representsan average of three alternating interferometer interrogationtimes: T = 160 s, 100 s, 40 s. The 12~k curve is an averageof T = 75 s, 69 s, 59 s, 53 s, limited at the low end by thelight travel time. The interrogation time average regularizesthe detector response by suppressing well known notches insingle interferometer transfer functions [1]. The LISA straincurve is shown for reference.

where np = 7 corresponds to a 2~k interferometer [4].Figure 2 shows the strain sensitivity curves for two

long-baseline designs using Sr atoms. The more con-servative design (green) uses an L = 2 109 m base-line and the photon shot-noise limited laser link assumes1 W laser power, a d = 30 cm diameter telescope, aswell as = 50 concurrent interferometers to support afR = /2T 0.2 Hz sampling rate. The long base-line allows for high sensitivity even though the designassumes conservative 2~k atom optics and atom shot-noise of a = 10

3 rad/Hz. A long interrogation time

of T = 160 s is used to support low frequency sensitivity,but despite this long drift time the maximum wavepacketseparation is bounded to < 2 m, minimizing satellite size.The atom source design assumes ensembles of 7 106atoms with a 20 pK longitudinal temperature, allowingfor a /2pi = 60 Hz Rabi frequency. Such design criteriaare readily met using existing technology [10].LMT techniques allow for enhanced sensitivity, as

shown by the second strain sensitivity curve (blue) inFig. 2. This design is based on a 12~k interferome-ter sequence with an L = 6 108 m baseline and im-proved phase noise a = 10

4 rad/Hz. Photon shot

noise requirements are met using 1 W laser power and ad = 50 cm diameter telescope, giving Rabi frequency/2pi = 40 Hz. The design has a sampling rate offR = /2T 1 Hz and assumes = 120 concurrentinterferometers [11]. The increased phase sensitivity ofthis design allows for improved low frequency responseeven using a smaller interrogation time. Using T = 75 s,

4the maximum wavepacket separation is < 4 m.

Another source of noise is timing delay and jitter in thepulses emitted by the phase-locked local oscillator laser.Referring to Fig. 1, consider a pulse emitted from M1 attime t that arrives at the satellite 2 beam splitter at timeta. If the pulse emitted from LO2 is delayed by some timetd after the arrival of the reference pulse, the laser phaseLO(ta+ td) imprinted on interferometer 2 will be differ-ent from the phase M (t) written onto interferometer 1by the amount LO(ta+ td)M (t) = td+ , where is the instantaneous frequency of laser LO2 at time ta and LO(ta) M (t) quantifies any imperfection in thephase lock between R1 and LO2. In addition, the delayedtransition in interferometer 2 implies that interferometer1 spends more time in the excited state by comparison,leading to an extra differential phase Atd for excitedstate energy ~A [12]. The total gradiometer phase dueto the delayed pulse is then delay = ( A)td + .Assuming a perfect phase lock ( = 0), noise can arisefrom either timing jitter td or frequency noise , giving

a noise PSD of 2

delay = (N td )2 + (N td)

2, where = A is the pulse detuning and N is any LMTphase enhancement factor. Keeping each term below arequires noise amplitude spectral densities of

=2pi80 HzHz

(2N

)(1 std

)(a

103 rad/Hz

)(3)

td =1.3sHz

(2N

)(60 Hz/2pi

)(a

103 rad/Hz

)(4)

at frequencies in the GW detection band. In particular,this shows that the long-time frequency stability require-ments of the LO laser can be reduced by ensuring thatthe pulses from the LO are well synchronized with theincoming reference pulses, keeping td small. In practice,td 10 ns with RMS noise td 1 ns appears straight-forward, suggesting that LO pulse timing constraints aremanageable.

Satellite and laser beam pointing jitter can also in-troduce noise. Consider a beam propagating approxi-mately along the z axis in Fig. 1 from satellite 1 towardssatellite 2 that is tilted by a small angle y about they-axis. Near satellite 1, the phase of the Gaussian beamvaries with position as 1(x, z) kz + kyx, where thecenter of rotation of the beam is taken to be x = 0,z = 0. By comparison, the phase near satellite 2 is2(x, z) k(z + L) + ky(zR/L)2x for baseline lengthL much longer than the Rayleigh range zR. Here along baseline is advantageous since when zR L thebeam arriving at satellite 2 is approximately a spheri-cal wave, so the dependance of the phase on angle isgreatly suppressed. In this limit, the pointing jitter con-straint is set by the 1 coupling and has noise amplitude

= 4kNx , where 2is the angle noise PSD and

x is the transverse position offset of the atom relative

to the baseline [2, 13]. The pointing requirement is then

= 10 nradHz

(2N

)(1 mmx

)(a

103 rad/Hz

). (5)

To avoid introducing additional pointing noise, the LOlaser beam incident on the atoms must point in the samedirection as the incoming reference laser pulse. This canbe facilitated by monitoring the relative angle betweenthe two beams at the beam splitter. In addition to mea-suring the beat note for the phase lock, a position sensi-tive detector such as as quadrant photodiode or a CCDcamera could be used to record the spatial interferencepattern between the reference and LO beams. Feedbackapplied to a tip-tilt mirror (show as TTM in Fig. 1 be-fore the BS) can then be used to control the angle of theLO laser. Similarly, the angle of the master laser itselfcan be controlled by comparing it to LO laser directionand using another tip-tilt mirror. The interference signalbetween the LO and the master can be generated usinga Michelson interferometer geometry, inserting an addi-tional beam splitter at any point along the path wherethe two beams are counter-propagating. In this configu-ration, the pointing stability of all the beams is tied to thestability of the incoming (nearly) spherical wavefronts ofthe reference beam.The performance of the angle control loop is ultimately

limited by the shot noise of the received reference wave-front. To estimate this, note that the power differenceP between the two sides of a quadrant detector due tothe spatial interference pattern caused by a small angle between the LO and the reference beam is P 42piPLOPr wt/ for a telescope diameter d = 2wt

and received powers PLO and Pr from the two beams[14]. The noise in P is dominated by the strong LO

beam, giving a noise PSD of (P )2 P 2LO = hPLO

assuming shot noise for the LO optical power noise. Theshot noise limit for a measurement of is then givenby amplitude spectral density

() =Ssh

42pi

wt 1 nrad

Hz

(10 cmwt

)(a

103 rad/Hz

)(6)

where Ssh = h/Pr is the phase noise PSD of the refer-

ence beam and we assume a design with Ssh = 2

a asbefore. Comparing this with the requirement in Eq. 5suggests that the angle can be sufficiently well measuredto control the LO pointing direction. This also suggeststhat overall satellite bus pointing requirements are mod-est ( 106 rad/Hz, limited by the dynamic range ofthe pointing servos), and substantially reduced from pre-vious proposals.Past designs have required large momentum transfer to

reach design sensitivity. The designs proposed here op-erate at lower momentum transfer and thus place muchless stringent constraints on laser phase front stability.For example, here we require a phase stability of /30for the telescope. Following references [1, 2, 15, 16],

5laser wavefront aberrations / couple to satellite trans-verse position noise x, resulting in phase noise ampli-tude = 16pi

2N(/)x/, where is the aberrationwavelength [13]. The estimated wavefront requirementfor the interferometer beam is then [17]

= 30

(2N

)(

1 cm

)(a

103 rad/Hz

)(x

1 m/Hz

)(7)

where we have assumed satellite transverse position jitterof x = 1 m/

Hz. This suggests modest satellite bus

jitter requirements.In addition to pointing errors, measuring the spatial

interference pattern can also provide information aboutthe wavefront of the LO laser. Since most abberationshave diffracted out of the beam after propagating acrossthe long baseline, the reference beam is a pristine op-tical wavefront reference. It might be possible to miti-gate wavefront requirements further [2] by combining thiswavefront measurement with appropriate feedback on theLO mode cleaning optics.Instrument constraints imposed by backgrounds that

have their origin in spurious forces or phase shifts (due to,for example, magnetic field gradients, blackbody shifts,AC Stark shifts or gravitational gradients) are signifi-cantly eased due to the longer baseline. Since previousshorter baseline designs could meet these requirements[18], in the longer baseline designs proposed here thesebackgrounds can be brought to levels where they do notimpact instrument performance.Several variations of the basic laser link concept are

possible. For example, by employing a second laser linkon each satellite, a dedicated reference laser can be usedto generate the reference beams instead of relying on thetransmitted master laser light as shown in Fig. 1. In thismodified setup, the heterodyne signal between the mas-ter laser and the new reference laser is formed on theexisting beam splitter (BS) such that the reflected refer-ence laser beam is directed towards the opposite satellite.Additional beam splitters, samplers, and polarization op-tics may be used to overlap the new beam paths with theexisting LO lock paths.Generalizing further, if the reference beam is a sepa-

rate laser then in principle its wavelength can be differentfrom that of the atomic transition. An optical frequencycomb would then be used to implement the heterodynelock, spanning the frequency difference between the refer-ence laser and the lasers responsible for interrogating theatoms. Changing the reference wavelength could lead tolower optical shot noise ( 5/4 in Eq. 1) or could exploitexisting laser technology at particular wavelengths.

[1] S. Dimopoulos, P. Graham, J. Hogan, M. Kasevich, andS. Rajendran, Physical Review D 78, 122002 (2008).

[2] J. M. Hogan, D. M. S. Johnson, S. Dickerson, T. Kovachy,A. Sugarbaker, S.-w. Chiow, P. W. Graham, M. A. Ka-sevich, B. Saif, S. Rajendran, P. Bouyer, B. D. Seery,L. Feinberg, and R. Keski-Kuha, General Relativity andGravitation 43, 1953 (2011).

[3] The properties of the atomic proof-masses are regular (allatoms are identical) and knowable (well defined magneticsusceptibility). They are immune from charging events.

[4] P. W. Graham, J. M. Hogan, M. A. Kasevich, and S. Ra-jendran, Physical Review Letters 110, 171102 (2013).

[5] M. Kasevich and S. Chu, Physical Review Letters 67,181 (1991).

[6] S.-w. Chiow, T. Kovachy, H.-C. Chien, and M. A. Kase-vich, Physical Review Letters 107, 130403 (2011).

[7] LISA Study Team, LISA Pre-Phase A Report, (1998).[8] D. Gerardi, G. Allen, J. W. Conklin, K.-X. Sun, D. De-

Bra, S. Buchman, P. Gath, W. Fichter, R. L. Byer, andU. Johann, The Review of Scientific Instruments 85,011301 (2014).

[9] P. Cheinet, B. Canuel, F. Pereira Dos Santos, A. Gau-guet, F. Yver-Leduc, and A. Landragin, IEEE Trans-actions on Instrumentation and Measurement 57, 1141(2008).

[10] T. Kovachy, J. M. Hogan, A. Sugarbaker, S. M. Dicker-son, C. A. Donnelly, C. Overstreet, and M. A. Kasevich,(2014), arXiv:1407.6995.

[11] Concurrent interferometers can be implemented usingdifferent initial velocities so that each interferometer isaddressable with a unique Doppler shift. Strategies forefficient allocation of Doppler bandwidth for LMT inter-ferometers are discussed in [19].

[12] The description here is of a transition from the groundto excited state. For transitions from excited to ground,the signs of all the phase shifts are flipped.

[13] Assumes a conventional pi/2pipi/2 N~k LMT sequenceand a perturbation frequency = pi/T equal to the peakGW response frequency.

[14] See, for example, [20].[15] P. L. Bender, Physical Review D 84, 028101 (2011).[16] S. Dimopoulos, P. W. Graham, J. M. Hogan, M. A. Ka-

sevich, and S. Rajendran, Physical Review D 84, 028102(2011).

[17] See Eq. 6 of Ref. [2]. Here we neglect averaging overthe cloud. Averaging would further relax requirementsat short wavelengths.

[18] See, for example, the analysis in Refs. [2, 4].[19] J.M. Hogan, (to be published).[20] J. M. Hogan, J. Hammer, S.-W. Chiow, S. Dickerson,

D. M. S. Johnson, T. Kovachy, A. Sugarbaker, and M. A.Kasevich, Optics letters 36, 1698 (2011).