Table Of ContentPrecision frequency measurements with interferometric weak values
David J. Starling(cid:63), P. Ben Dixon, Andrew N. Jordan and John C. Howell
Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
We demonstrate an experimen√t which utilizes a Sagnac interferometer to measure a change in
opticalfrequencyof129±7kHz/ Hzwithonly2mWofcontinuouswave,singlemodeinputpower.
Wedescribethemeasurementofaweakvalueandshowhowevenhigherfrequencysensitivitiesmay
beobtainedoverabandwidthofseveralnanometers. Thistechniquehasmanypossibleapplications,
suchasprecisionrelativefrequencymeasurementsandlaserlockingwithouttheuseofatomiclines.
PACSnumbers: 42.50.Xa,03.65.Ta,06.30.Ka
I. INTRODUCTION
Fused Silica Prism
1
1 Precisionfrequencymeasurements[1–3]ofastabilized
0 lasersourceareofgreatimportanceinthefieldofmetrol-
2 Split Detector
ogy[4]aswellasatomic,molecular[5]andopticalphysics
n [6]. Here we show that weak values [7–9] in an opti-
a cal deflection measurement experiment [10] can produce
J √
frequency shift resolutions down to 129 ± 7 kHz/ Hz
7
with only 2 mW of continuous wave optical power. By
BS
] performing a weak measurement of the deflection of an 780 nm Diode Laser
h infrared laser source that has passed through a weakly
p dispersive prism, we are able to measure a change in op-
-
tical frequency comparable to precision Fabry-Perot in-
t
n terferometers[11–13]. Thistechniqueisrelativelysimple,
a requiringonlyafewcommonopticalcomponentsandop- Power Meter Isolator To External
u Experiment
erating at atmospheric pressure. Additionally, we show
q
[ that this technique has low noise over a large range of
response frequencies, making it desirable for many ap- FIG.1. (Coloronline)AGaussianlaserbeampassesthrough
1
plications such as Doppler anemometry [14], tests of the a Sagnac interferometer consisting of a 50:50 beam splitter
v
isotropy of light propagation [6] or laser locking without (BS), a mirror and a prism. The prism weakly perturbs the
4
6 the use of high finesse Fabry-Perot interferometers [15] direction of the beam as the frequency of the laser source is
modulated, denoted by the red and blue beam paths. We
4 or atomic lines.
monitorthepositionofthelightenteringthedarkportofthe
1 Firstdevelopedasawaytounderstandpreselectedand
interferometer. Welocktheinputpowertotheinterferometer
.
1 postselectedquantummeasurementsandhowtheyrelate using a power measurement before the BS. The majority of
0 to time-reversal symmetry in quantum mechanics, the the light exits the interferometer via the bright port and is
1 weakvalue AwofanoperatorAwasintroducedinasemi- collected with an isolator for use in an experiment.
1
nal1988paperbyAharonov,AlbertandVaidman(AAV)
:
v [7]. TheweakvalueisgivenbyAw =(cid:104)ψf|A|ψi(cid:105)/(cid:104)ψf|ψi(cid:105),
i where{|ψ (cid:105)}arethepreselectedandpostselectedstates ables [23]. Furthermore, due to the denominator of A ,
X i,f w
of the system, respectively. This quantity, which is therecanbealargeamplificationoftheweakvaluewhen
r
likenedtotheexpectationvalueofA,canhaveseemingly thepreselectedandpostselectedstatesarenearlyorthog-
a
strangebehavior, particularlyinthelimitwherethepre- onal; as a result, there have been a number of experi-
selected and postselected states are nearly orthogonal. mental results published in the field of optical metrology
While numerous experiments have validated the initial [10,18,24,25]. Thereisalsoavastarrayofresults,both
claims of the AAV paper [10, 16–18], there are new de- theoretical [9, 26–28] and experimental [16, 17], which
velopments concerning the interpretation of preselected havegonealongwaytofurtherourunderstandingofthe
and postselected weak measurements [8, 9, 19, 20]. weak measurement process.
Weak values are a result of a so-called weak measure-
ment,i.e.,ameasurementwhichgainsonlypartialinfor-
mationaboutthestateofasystem. UnlikevonNeumann II. THEORY
measurements, a weak measurement disturbs the mea-
sured state of the system only minimally. For this rea- We describe here the frequency amplification experi-
son, weak measurements have been useful in reconsider- ment shown in Fig. 1 by further developing the ideas of
ing Hardy’s paradox [21, 22] as well as making meaning- Ref. 10. Although the actual experiment uses a classical
ful, sequential measurements of noncommuting observ- beam,wechoosetocharacterizetheweakvalueeffectone
2
photon at a time; this is valid, owing to the fact that we
consider here a linear system with a coherent laser beam 6
modeled as a linear superposition of Fock states [27].
In this experiment, a single-mode Gaussian beam of 5
frequency ω and radius σ passes through an optical iso- m)
n
lator, resulting in linearly polarized light. We assume n ( 4
that the radius is large enough to ignore divergence due o
to propagation. Light then enters a Sagnac interferome- ecti
tercontaininga50:50beamsplitter(BS),amirroranda efl 3
D
prism. Thebeamtravelsclockwiseandcounter-clockwise d
tghivreonugbhyth{|e(cid:8)in(cid:105)t,e|r(cid:9)fe(cid:105)r}o;mweteerw,rditeenottheedpbhyotthoensmysetteemr ssttaatteess plifie 2
m
inthepositionbasisas{|x(cid:105)}, wherexdenotesthetrans- A 1
verse, horizontal direction.
Initially, the interferometer (including the prism) is 0
alignedsuchthatthesplitphotonwavefunctionspatially 0 1 2 3 4 5 6 7 8
overlaps (i.e., the photons travel the same path whether
Frequency (MHz)
by | (cid:8)(cid:105) or | (cid:9)(cid:105)). After the interferometer is aligned,
the photons traversing each path receive a small, con-
stant momentum kick in the vertical direction; this ver- FIG.2. Thepositionofthepostselectedbeamprofileismea-
tical kick is controlled by the interferometer mirror and sured as we modulate the input laser frequency of the inter-
results in a misalignment. Due to its spatial asymme- ferometer. Themodulationoscillatesasasinewaveat10Hz
andthesignalfromthesplitdetectorisfrequencyfilteredand
try about the input BS, this momentum kick creates an
amplified. Theerrorbarsaregivenbythestandarddeviation
overall phase difference φ between the two paths. By
of the mean. The minimum frequency change measured here
adjusting the interferometer mirror, we can control the
isaround743kHzwithaneffectiveintegrationtimeof30ms.
amountoflightthatexitstheinterferometerintothedark
The weak value amplification is approximately 79.
port. While the amplified signal ultimately depends on
thevalueofφ,andthereforeonthemagnitudeofthemis-
alignment, the signal to noise ratio (SNR) is unaffected
(discussed below). this to the standard deflection caused by a prism mea-
We then let k(ω) represent the small momentum kick sured at a distance l which is given by (cid:104)x(cid:105) ≈ lk(ω)/k0,
given by the prism to the beam (after alignment) in the where k0 is the wavenumber of the light.
horizontalx-direction. Thesystem andmeter areentan- In order to predict the deflections (cid:104)x(cid:105) or (cid:104)x(cid:105) , we
W
gled via an impulsive interaction Hamiltonian [10] (re- must know the form of k(ω). For a prism oriented
sulting in a new state |ψi(cid:105) → |Ψ(cid:105)) such that a measure- such that it imparts the minimum deviation on a
ment of the horizontal position of the photon after it beam, the total angular deviation is given by θ(ω) =
exits the interferometer gives us some information about 2sin−1[n(ω)sin(γ/2)] − γ, where n(ω) is the index of
which path the photon took. refraction of the material and γ is the angle at the
Weconsiderahorizontaldeflectionthatissignificantly apex of the prism [29]. However, we are only inter-
smaller than the spread of the wave packet we are trying ested in the small, frequency-dependent angular deflec-
to measure, i.e., k(ω)σ (cid:28) 1. In this approximation, we tion δ(ω) = ∆θ = 2∆n{[sin(γ/2)]−2 − [n(ω)]2}−1/2,
findthatthepostselectedstateofthephotonsexitingthe where ∆n (∆θ) is the index change in the prism (angu-
dark port is given by lar deflection of the beam) for a given frequency change
of the laser. The small momentum kick is expressed as
(cid:90)
(cid:104)ψf|Ψ(cid:105)=(cid:104)ψf|ψi(cid:105) dxψ(x)|x(cid:105)exp[−ixAwk(ω)], (1) k(ω) = δ(ω)k0. We can then write the amplified deflec-
tion as
where the weak value, defined above, is given by A =
w
−icot(φ/2)≈−2i/φ for small φ. (cid:104)x(cid:105) ≈ 8k0σ2(∆n/φ) . (2)
There are two interesting features of Eq. (1). First, W (cid:112)[sin(γ/2)]−2−[n(ω)]2
the probability of detecting a photon has been reduced
to P = |(cid:104)ψ |ψ (cid:105)|2 = sin2(φ/2), and yet the SNR of
ps f i
an ensemble of measurements is nearly quantum limited The frequency-dependent index n(ω) of fused silica,
[25] despite not measuring the vast majority of the light. whichwasusedinthisexperiment,canbemodeledusing
Second, the weak value (which can be arbitrarily large the Sellmeier equation [30]. We can therefore calculate
in theory) appears to amplify the momentum kick k(ω) the expected (cid:104)x(cid:105) using Eq. (2). However, to compute
W
givenbytheprism;theresultingaveragepositionisgiven theultimatesensitivityofthisweakvaluefrequencymea-
by(cid:104)x(cid:105) =2k(ω)σ2|A |≈4k(ω)σ2/φ,wheretheangular surement, we must include possible noise sources. If we
W w
brackets denote an expectation value. We can compare consideronlyshot-noisefromthelaser,theSNRforsmall
3
φ is approximated by
60
(cid:114)
8N
R≈ k σδ(ω), (3) 50
π 0
40
as shown in Ref. 25, where N is the number of photons B)
used in the interferometer. Note that N is not the num- d 30
(
ber of photons striking the detector, which is given by y
NP . By setting R = 1 and using modest values for sit 20
ps n
N, σ and ω, we find that frequency sensitivities well be- e 10
D
low 1 kHz are possible. However, other sources of noise,
suchasdetectordarkcurrent, radiationpressureanden- ral 0
t
vironmental perturbations will reduce the sensitivity of ec -10
the device. p
S
-20
-30
III. EXPERIMENT
0 10 20 30 40 50 60 70 80 90 100
In our experimental setup (shown in Fig. 1), we used
Frequency (Hz)
a fiber-coupled 780 nm external cavity diode laser with
a beam radius of σ = 388µm. The frequency of the
laser was modulated with a 10 Hz sine wave using piezo FIG. 3. (Color online) We show the noise spectrum for a
controlled grating feedback. The frequency control was passive system (green, solid trace) and for a driven system
(blue, dashed trace), where the laser frequency modulation
calibrated using saturation absorption spectroscopy of
is 7.4 MHz at 10 Hz. We see that the first harmonic of the
the hyperfine excited states of the rubidium D2 line[31]:
signal is about 5 dB down from the fundamental and the
F = 3 → F(cid:48) = {2 − 4 crossover, 3 − 4 crossover, 4}
third harmonic is nearly 25 dB down. For a 7.4 MHz change
transitions of rubidium 85 and the F = 2 → F(cid:48) =
in laser frequency, we see that the noise is approximately 35
{1−3 crossover, 2−3 crossover, 3} transitions of rubid-
dB below the signal, demonstrating the low-noise nature of
ium 87. Linearly polarized light was divided before the this measurement.
interferometerusinga50:50BS(althoughanimbalanced
ratioherewouldbeidealforpracticalapplications). The
light in one port was measured with a photodiode and agrees with the measured P of 2-5% if we include the
ps
used to lock the power at 2 mW with an acousto-optic extra light present in the signal due to phase front dis-
modulator before the fiber. The interferometer was ap- tortions from imperfect optics.
proximately l = 27 cm in length; the mirror used to A characteristic noise scan was taken and plotted in
adjust φ was approximately 6 cm from the input BS Fig. 3 with and without frequency modulation. The sig-
(measured counter clockwise) and the prism, made of nal was passed directly from the split detector into the
fused silica, was approximately 5 cm from the input BS oscilloscope before performing a fast Fourier transform.
(measured clockwise). Although the prism was not sym- Data was taken with and without a 7.4 MHz optical fre-
metrically placed in the interferometer as described in quency modulation to show the noise floor over a large
the theory above, the results are the same aside from bandwidth. The noise at higher frequencies was simi-
a global offset in position which can be subtracted off larly flat. Second, to test the range over which this de-
during processing. The interferometer was first aligned vice could function, we optimized the interferometer at
to minimize light in the dark port and then, using the thelow-frequencyendofthelaser’stuningrangeandob-
aforementioned mirror, misaligned to allow a small per- tained a SNR of approximately 19 with the 7.4 MHz op-
centage of the light (∼2-5%) into the dark port. The tical frequency modulation. We then tuned to the high-
position of this light was measured using a split detector frequency end of the laser’s tuning range (∆f ≈ 141
(NewFocusmodel2921). Thesignalwaspassedthrough GHz), without adjusting or recalibrating the interfer-
two 6 dB/octave bandpass filters centered at 10 Hz and ometer, and obtained a SNR of 10. In fact, this range
amplified by a factor of about 104. can be much larger so long as the weak value condition
For Fig. 2, we measured the peak of the deflection in k(ω)σ (cid:28) 1 is satisfied; for our beam radius and optical
each 100 ms cycle, repeated 25 times; we computed the frequency, we could in principle measure over a range of
average and the standard deviation of this set as we var- 5 THz, or about 10 nm.
iedthechangeintheopticalfrequency. Wefindthatthe For our experimental parameters, we can measure be-
amplified deflection is a linear function of oscillating op- low 1 MHzof frequency change witha SNR around 1, as
ticalfrequencygivenbyabout720±11pm/MHz. Com- shown in Fig. 2. It should be noted that, although the
paredtotheunamplifieddeflectionofabout9.1pm/MHz timebetweenmeasurementsisafull100ms,ourfiltering
given by the expression for (cid:104)x(cid:105), this gives an amplifica- limits the laser noise to time scales of about 30 ms. For
tion factor of 79 ± 1.2 and a computed P of 1.3%; this analysis, we take this as our integration time in estimat-
ps
4
ing N for each measurement. The resulting sensitivity 1 kHz, although a higher sensitivity comes at the cost of
√
for our apparatus is 129 ± 7 kHz/ Hz; e.g., if we had maximum tuning range. Compare this to commercially
integratedfor1sinsteadof30ms,thisdevicecouldmea- available Fabry-Perot interferometers, which report typ-
sure a 129 kHz shift in frequency with a SNR of 1. The ical resolutions down to 5 MHz and free spectral ranges
error in frequency comes from the calibration described of only 1-5 GHz. More sensitive Fabry-Perot interferom-
above. Using Eq. (3), we find that the ideal ultimate eters exist, yet they require a host of custom equipment
√
sensitivity is approximately 67 kHz/ Hz. This implies to reduce environment noise, including vacuum systems
that this apparatus, operating at atmospheric pressure and vibration damping. Moreover, an important advan-
with modest frequency filtering, is less than a factor of tage of this technique is that a large percentage (∼90%)
two away from the shot-noise limit in sensitivity. This ofthelightusedintheinterferometercanthenbesentoff
is no longer surprising since we now understand the fact to another experiment (as indicated in Fig. 1), allowing
that weak value experiments amplify the signal, but not for real-time frequency information during data collec-
the technical noise [24, 25]. tion. While this device cannot compare to the absolute
frequency sensitivity of frequency combs [3], we believe
that this method is a simple solution for high-resolution,
IV. CONCLUSION relative frequency metrology and will serve as a valuable
laser-locking tool.
With only 2 mW of continuous wave input power, we
√
have measured a frequency shift of 129 ± 7 kHz/ Hz;
we have shown that the system is stable over our maxi- ACKNOWLEDGMENTS
mumtuningrangeof140GHzwithoutrecalibrationand
is nearly shot-noise-limited. With more optical power, This research was supported by US Army Research
longer integration and a more dispersive element such as Office Grant No. W911NF-09-0-01417, DARPA Expan-
a grating or a prism with sin(γ/2)n(ω)≈1, the sensitiv- sion Grant No. N00014-08-1-120 and the University of
ityofthisdevicecanmeasurefrequencyshiftslowerthan Rochester.
[1] M.-L. Junttila and B. Stahlberg, Appl. Opt. 29, 3510 [16] N. W. M. Ritchie, J. G. Story, and R. G. Hulet, Phys.
(1990). Rev. Lett. 66, 1107 (1991).
[2] J. Ishikawa and H. Watanabe, IEEE Trans. Instrum. [17] G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph,
Meas. 42, 423 (1993). andH.M.Wiseman,Phys.Rev.Lett.94,220405(2005).
[3] D.Jones,S.Diddams,J.Ranka,A.Stentz,R.Windeler, [18] D.J.Starling,P.B.Dixon,N.S.Williams,A.N.Jordan,
J. Hall, and S. Cundiff, Science 288, 635 (2000). and J. C. Howell, Phys. Rev. A 82, 011802 (2010).
[4] T.Udem,R.Holzwarth, andT.W.Hansch,Nature416, [19] Y. Aharonov and L. Vaidman, J. Phys. A: Math. Gen.
233 (2002). 24, 2315 (1991).
[5] S. Falke, H. Kno¨ckel, J. Friebe, M. Riedmann, E. Tie- [20] A. M. Steinberg, Nature 463, 890 (2010).
mann, and C. Lisdat, Phys. Rev. A 78, 012503 (2008). [21] L. Hardy, Phys. Rev. Lett. 68, 2981 (1992).
[6] C.Eisele,A.Y.Nevsky, andS.Schiller,Phys.Rev.Lett. [22] J.S.LundeenandA.M.Steinberg,Phys.Rev.Lett.102,
103, 090401 (2009). 020404 (2009).
[7] Y.Aharonov,D.Z.Albert, andL.Vaidman,Phys.Rev. [23] G. Mitchison, R. Jozsa, and S. Popescu, Phys. Rev. A
Lett. 60, 1351 (1988). 76, 062105 (2007).
[8] I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, [24] O. Hosten and P. Kwiat, Science 319, 787 (2008).
Phys. Rev. D 40, 2112 (1989). [25] D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C.
[9] J. Dressel, S. Agarwal, and A. N. Jordan, Phys. Rev. Howell, Phys. Rev. A 80, 041803 (R) (2009).
Lett. 104, 240401 (2010). [26] N. S. Williams and A. N. Jordan, Phys. Rev. Lett. 100,
[10] P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. 026804 (2008).
Howell, Phys. Rev. Lett. 102, 173601 (2009). [27] J.C.Howell,D.J.Starling,P.B.Dixon,P.K.Vudyasetu,
[11] T. Hori, T. Matsui, K. Araki, and H. Inomata, Opt. and A. N. Jordan, Phys. Rev. A 81, 033813 (2010).
Lett. 14, 302 (1989). [28] N.BrunnerandC.Simon,Phys.Rev.Lett.105,010405
[12] N. Nishimiya, Y. Yamaguchi, Y. Ohrui, and M. Suzuki, (2010).
Spectrochim. Acta, Part A 60, 493 (2004). [29] F.L.PedrottiandL.S.Pedrotti,IntroductiontoOptics,
[13] M. Notcutt, L.-S. Ma, J. Ye, and J. L. Hall, Opt. Lett. 2nd ed. (Prentice-Hall, Inc., 1993).
30, 1815 (2005). [30] I. H. Malitson, J. Opt. Soc. Am. 55, 1205 (1965).
[14] C. Tropea, Meas. Sci. Technol. 6, 605 (1995). [31] D. A. Steck, http://steck.us/alkalidata/ (2010).
[15] R. Pohl et al., Nature 466, 213 (2010).
