Using Atomic Clocks to Detect Gravitational Waves

0.11 MB·English

by  Abraham Loeb

Using Atomic Clocks to Detect Gravitational Waves
Abraham Loeb1,2 and Dan Maoz2
1Astronomy Department, Harvard University, 60 Garden St., Cambridge, MA 02138, USA
2School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
(Dated: January 29, 2015)
Atomicclockshaverecentlyreachedafractionaltimingprecisionof.10−18. Wepointoutthatan
arrayof atomicclocks, distributedalong theEarth’s orbit aroundtheSun,will havethesensitivity
needed to detect the time dilation effect of mHz gravitational waves (GWs), such as those emitted
bysupermassiveblackholebinariesatcosmological distances. Simultaneousmeasurementofclock-
rates at different phases of a passing GW provides an attractive alternative to the interferometric
5
1 detection of temporal variations in distance between test masses separated by less than a GW
0 wavelength, currently envisioned for theeLISA mission.
2
PACSnumbers: 04.80.Nn,95.55.Ym,95.85.Sz
n
a
J
Introduction. Overthe pastyear,the precisionofop- inthe frequencyrangeof0.1-100mHzandis plannedfor

8 tical lattice clocks has advanced dramatically, to a frac- launchintwodecades. eLISAisdesignedasalaserinter-
2 tionaltimingprecisionof(∆t/t) 10−18,withprospects ferometer that will record the phase shift introduced by
∼
for a future improvement by two additional orders of apassingGWasthewaveinducesachangeinthespace-
]
M magnitude through the use of other atoms [1–3]. Here, time curvature, and hence a change in light travel time
we point out that the new regime of timing precision between its three test masses, which are separated from
I
. made accessible by atomic clocks overlaps with the ex- each other by 106 km (less than the GW wavelength).
h
pected amplitudes of time dilation and compression due Here, we consider the alternative approach of detecting
p
to the passage of gravitational waves (GWs). The stan- the differential time dilation experienced by clocks lo-
-
o dardtime-dilationeffectforaclockatsomedistancefrom cated at different phases of a passing GW. For this pur-
r
ablackhole,wouldbemodulatedbytheperiodicchange pose, we propose using an array of atomic clocks.
t
s in this distance due to the orbital motion in a binary Atomic clocks in space. We envision a set of small
a
[ black hole system. Quantitatively, compact binaries of orbiting units (at a minimum two of them) equipped
supermassive black holes [4, 5] at cosmologicaldistances with atomic clocks, and a primary spacecraft between
2
produce a periodic modulation of the time-time compo- them, as illustrated in Fig. 1. For simplicity, the clocks
v
nent of the metric (in a Newtonian gauge) at a level of are assumed to be distributed along the circular orbit
6
of the Earth around the Sun at an orbital radius of
9
9 h 9 10−18 DL −1 Mz 5/3 f 2/3, 8.3 light-minutes (1AU≡ 1.5× 108km), since this con-
0 00 ≈ × (cid:18)Gpc(cid:19) (cid:18)106M⊙(cid:19) (cid:18)mHz(cid:19) figuration minimizes the kinetic energy requirements for
0 (1) launch. The typical distances between units will thus be
1. where D (z) is the luminosity distance for a source at 108 km. Additional units will naturally improve the
L ∼
0 redshift z, (1+z)(M M )3/5/(M +M )1/5 with sensitivity and directional angular resolution.
z 1 2 1 2
5 M and MMbe≡ing the masses of the binary members A passing GW characterizedby a metric perturbation
1 wh1ich are a2ssumed here to be on a circular orbit, and hµνei(kr−ωt), will induce periodic variations (over a pe-
:
v f = (ω/2π) is the observed (redshifted) GW frequency riod 2π/ω = 2π/kc) in the ticking rate of each clock
i (obtaining its maximum value based on the orbital fre- [10]. The amplitude of the timing variation would be
X quency at the binary’s innermost stable circular orbit).1 12h00 (since time is dilated by (1+h00)−1/2). For each
ar Such black hole binaries are a natural consequence of pair of clocks, the relative phase of the periodic varia-
galaxy mergers [7]. Around the Milky Way’s super- tion ∆φ = k∆r will be dictated by the projection of
massive black hole, GWs with a similar amplitude and the difference in clock position vectors ∆r on the GW
frequency can be emitted by orbiting close-in stars [8]. wavevector k (which defines the GW propagation di-
Suchsourcesconstitutetheprimarytargetsforthefuture rection). The largest timing phase difference would be
eLISA space observatory [9], which aims to detect GWs realized between clocks separated by half a wavelength,
1λ=1AU(f/1mHz)−1, correspondingto a timing phase
2
difference∆φ=π. MeasurementsoftheN(N 1)phase
−
differences ∆φ forasetofN clockswould
{ i,j}i,j=(1,2,...N)
1 In this paper, we adopt for pedagogical reasons a Newtonian allow to localize the GW source position on the sky as
gauge which is commonly used to describe the time-dilation ef- k.
fectdueto stationarygravity, as measuredinthe Pound-Rebka −Tocommunicatetheperiodictimedilationorcompres-
experiment[6]. Inthisgauge, anoscillatingperturbationinthe
sion signal associated with a GW, the clock in each unit
time-timecomponent of the metric, h00, would trigger periodic
shoulddriveanoptical-frequency(ν 1015Hz)laser,
variationinthePound-Rebkatimedilationandamismatchbe- laser
∼
tweenthetickingrateofclocksseparatedapart. pointed at the primary spacecraft. The required signal-
2
riod). The change in the light travel time between the
clocks due to the GW averages out over a GW wave-
length, and thus amounts to a higher ordercorrectionto
the beat frequency, which can be neglected.
Our proposal is to measure timing variations rather
than distance variations. Our detection scheme is not
concernedwithlaserphasevariationsthatareinducedby
distancevariationsbetweenfree-floatingunits,butrather
with the change in the rate at which clocks are ticking
relativetoeachother. Thelasersarelockedtotheclocks
so as to keep their timing stability at a high level.
The precision achieved by optical lattice clocks scales
inversely with the square root of the integration time
[3] and limits the clock-timing method to low-frequency
GWs with f .1 mHz, allowing integration for f−1 &
103s. This, in turn, requires clocks at 1 AU∼separa-
∼
FIG. 1: Schematic illustration of an array of atomic clocks tions, as considered here. Secular trends (due to slow
in a circular orbit around the Sun, as an observatory for orbitalvariationsinthegravitationaleffectofthe Sunor
low-frequency GWs. Clocks separated by half a wavelength, planets,theenergylossoftheSun,orthephaseshiftdue
12λ = π/k = 1AU(f/mHz)−1, of a passing GW, would show to the solar wind) over the short period of the measure-
the largest difference in fractional timing, h00, varying pe- ment(.1hour)canbeseparatedfromthe periodicGW
riodically over the GW period, f−1 = 2π/kc. We adopt a
signal in the frequency range of interest here. Further-
Newtonian gauge for pedagogical reasons, but the measured
more,forGWsignalslastingweeksormonths,theorbital
physical effect should be independentof gauge.
motionwillscanthe ecliptic, improvingthe sourcelocal-
izationaccuracy. SolaroscillationsatmHz frequency (p-
mode and especially g-mode) could also be detected by
to-noiseratioisnotdictatedbytheprecisionfordistance
the proposed array.
measurements through laser phase shifts, but rather by
AsindicatedbyEq. (1),thetimingprecisionofatomic
what is needed to maintain an optical phase lock be-
clocks is sufficient to detect a GW from a supermassive
tweenthedifferentunitssuchthatthetickingrateoftheir
black hole binary at cosmological distances within one
clocks can be compared at a high precision. Techniques
GW period. The signal-to-noise ratio (SNR) of GW de-
for remote optical clock comparison had been developed
tection will improve in proportion to the square-root of
over the past decade [11, 12] and are conceptually dif-
the number of wavecycles being observed. Therefore,to
ferent from the interferometric technique currently envi-
achieveSNR=1,thefractionaltimingprecisionpercycle
sioned for eLISA. In the absence of phase noise in space,
could be worse than the GW amplitude by the square-
the distance noise level of 5 10−8 cm/√Hz expected
∼ × root of the number of wave cycles observed.
from the shot noise of a 1 Watt laser detected by tele-
So far we assumed a monochromatic GW signal and
scopes of 30 cm diameter across a 1-2AU path length,
ignored the slow drift in the GW frequency (so-called
would be more than sufficient to phase lock a single op-
chirp) due to the inspiral of the supermassive black hole
tical cycle. The needed laser is available and already
binary. The lifetime of tight binaries is in fact limited
in use. The primary spacecraft will interfere the laser
by the rate of GW energy loss [14]. For a circular bi-
signals from the two daughter clocks to produce a beat
nary orbit, the fractional frequency increase during an
signal with a frequency ν h ν , i.e. with a pe-
riod of . 103 s, comparabbealet ∼to t0h0elapseerriod of the GW, infinitesimal observing time ∆tobs is given by [4],
and hence detectable. ∆f 5/3 f 8/3 ∆t
z obs
Discussion The proposed method is fundamentally =0.3 M . (2)
f (cid:18)106M⊙(cid:19) (cid:18)mHz(cid:19) (cid:18)1 hour(cid:19)
differentfromthecurrentinterferometricmethodswhich
underline the design of Advanced-LIGO [13] and eLISA. The temporal rise in frequency f (chirp) of the GW sig-
In GW interferometry, the GW wavelength is consid- nalcan also be detected throughclock timing to provide
erably longer than the interferometer arm length, and asecondconstraintonthe valuesof andD inaddi-
z L
at any given moment one is essentially measuring the tion to Eq. (1). Since there are & 1M012 galactic mergers
roughly-uniform space curvature in the region between within a Hubble time in the observable volume of the
the test masses by means of the light-traveltime. Half a universe, the duty cycle of detectable signals (event rate
GW period later, the change in light-travel time reveals times lifetime) could be high [7, 15, 16].
the passage of the GW. In our clock-timing method, the The use of a large number of uncorrelated clocks, N,
separation between clocks is of order half a GW wave- ineveryclockunit,canimprovethetimingprecisionbya
length and one measures the simultaneous difference in factor of 1/√N. This may become another advantage of
clockratesbetweenthelow-curvatureandhigh-curvature the proposed method, as atomic clocks become progres-
phases ofthe GW (whichagainflips after halfa GW pe- sivelysmall,low-weightandinexpensive. Iftheclocksare
3
quantum entangled, the timing precision could improve andJunYeforhelpfulcommentsonthemanuscript. A.L.
as 1/N and inversely with integration time [17]. acknowledges generous support from the Sackler Profes-
sorship by Special Appointment at Tel Aviv University
Acknowledgments. We thank Pete Bender, Bence (TAU), the Raymondand BeverlySacklerTAU-Harvard
Kocsis, Pawan Kumar, Misha Lukin, Ron Walsworth, programinAstronomy,andtheNSFgrantAST-1312034.
[1] N. Hinkley,J. A. Sherman, N. B. Phillips, M. Schioppo, [10] M. J. Koop, & L. S. Finn, Phys. Rev. D 90, 062002
N.D.Lemke,K.Beloy,M.Pizzocaro,C.W.Oates,&A. (2014).
D.Ludlow, Science341, 121 (2013). [11] J.Ye,J.L.Peng,R.J.Jones,K.W.Holman,J.L.Hall,
[2] B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L., D. J. Jones, S. A. Diddams, J. Kitching, S. Bize, J. C.
Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Brom- Bergquist, L. W. Hollberg, L. Robertsson, & L.-S. Ma,
ley,& J. Ye,Nature 506, 71 (2014). J. Opt.Soc. Am. B 20, 1459 (2003).
[3] K. Beloy, N. Hinkley, N. B. Phillips, J. A. Sherman, M. [12] S. M. Foreman, K. W., Holman, D. D. Hudson, D. J.
Schioppo,J.Lehman,A.Feldman,L.M.Hanssen,C.W. Jones, & J. Ye,Rev.Sci. Instrum.78, 021101 (2007).
Oates, & A. D. Ludlow, Phys. Rev. Lett. 113, 260801 [13] https://www.advancedligo.mit.edu
(2014). [14] P. C. Peters, Phys. Rev.136, 1224 (1964).
[4] A. Sesana, Classical and Quantum Gravity 30, 244009 [15] J. S. B. Wyithe, & A. Loeb, Astrophys. J. 590, 691
(2014); arXiv: 1307.4086 (2003).
[5] D.E.Holz,&S.A.Hughes,Astrophys.J.629,15(2005). [16] A. Loeb, Phys. Rev.D81, 047503 (2010).
[6] Pound, R. V., & Rebka, G. A., Phys. Rev. Lett. 3, 439 [17] P. K´oma´r, E. M. Kessler, M. Bishof, L. Jiang, A. S.
(1959). Sorensen, J. Ye,&M. D.Lukin,NaturePhysics10, 582
[7] M.Colpi,&M.Dotti,AdvancedSci.Lett.4,181(2011). (2014).
[8] M. Freitag, Astrophys.J. 583, L21 (2003).
[9] https://www.elisascience.org