Cancellation of nonlinear Zeeman shifts with light shifts

0.31 MB·English

by  K. Jensen

Cancellation of nonlinear Zeeman shifts with light shifts
K. Jensen,1,2,∗ V. M. Acosta,3 J. M. Higbie,4 M. P. Ledbetter,3 S. M. Rochester,3 and D. Budker3,5,†
1Niels Bohr Institute, University of Copenhagen, DK 2100, Denmark
2QUANTOP, Danish National Research Foundation Center for Quantum Optics
3Department of Physics, University of California at Berkeley, Berkeley, California 94720-7300
4Department of Physics and Astronomy, Bucknell University, Lewisburg, Pennsylvania 17837
5Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720
Nonlinear Zeeman (NLZ) shifts arising from magnetic-field mixing of the two hyperfine ground-
9 states in alkali atoms lead to splitting of magnetic-resonance lines. This is a major source of
0 sensitivity degradation and the so-called “heading errors” of alkali-vapor atomic magnetometers
0 operating in the geophysical field range (B ≈ 0.2−0.7 G). Here, it is shown theoretically and
2 experimentally that NLZ shifts can be effectively canceled by light shifts caused by a laser field of
appropriateintensity,polarizationandfrequency,atechniquethatcanbereadilyappliedinpractical
n
situations.
a
J
PACSnumbers: PACS.07.55.Ge,32.60.+i,42.65.-k
2
1

I. INTRODUCTION theoretically as part of a quantum memory protocol [11]
]
h and experimentally for improving atomic spin-squeezing
p [12]. The similarities of light shifts and Zeeman shifts
Alkali-vapor atomic magnetometers [1] operating in
- was used in Refs. [13, 14], where magnetic fields were
m the geophysical range of magnetic fields, where the Zee-
simulated by light fields. It has also been proposed to
man effect is very close to linear, are nevertheless quite
o use light shifts to measure parity violation in Fr atoms
t sensitive to the nonlinear corrections to the Zeeman ef-
a [15]. In general, light shifts are important in precision
fect (NLZ) which cause broadening and splitting of the
. measurements in atomic physics, for instance in atomic
s magnetic-resonance lines, as well as line-shape asymme-
c clockexperiments,whereonecanusethe famous“magic
tries that depend on the orientation of the sensor with
i wavelengths” to eliminate light shifts, see for example
s respect to the field [2, 3]. Thus, NLZ is responsible for
y Refs. [16, 17] and references therein.
sensitivity degradation and systematic (heading) errors
h
in atomic magnetometers [4, 5]. Recently, collapse and
p
[ revival of ground-state quantum beats associated with
II. THEORY
NLZwasstudiedtheoretically[6]andexperimentally[3],
2
and a scheme for mitigating the effects of NLZ in an
v
atomic magnetometer based on double-modulated syn- For an alkali atom with nuclear spin I, total angular
1
2 chronous optical pumping was realized [3]. Alternative momentum F = I ±1/2 and projection MF of the to-
2 approaches explored recently include the use of multi- tal angular momentum on the direction of the magnetic
2 quantum transitions and high-order atomic polarization field, the Zeeman energies of the magnetic sublevels are
. moments [7, 8, 9, 10]. Here, we introduce an alterna- to second order in the field B given by
0
1 tivetechniquewhereNLZshiftsarecompensatedbylight
8 shifts due to an additional light field of appropriate in- ∆E ≈± 2 µ M B±(µBB)2 1− 4MF2 , (1)
0 tensity,polarizationandfrequency,whichispossibledue 2I+1 B F ∆hfs " (2I+1)2#
v: to the identical tensor structure of the splitting caused
Xi by the two effects. The present technique is free from whereµB istheBohrmagneton,∆hfs isthehyperfinein-
some shortcomings of the alternative techniques such as terval, and we neglect small corrections proportional to
r complexity of implementation and/ordegradationof the the nuclear magneton and the anomalous magnetic mo-
a
signal. Moreover, in contrast to the double-modulated mentoftheelectron. Thefirst-ordershiftleadstopreces-
synchronous pumping technique of Ref. [3], the present sionoftheatomicpolarizationaroundthemagneticfield
approachworkswellwhentheground-statepolarization- with Larmor frequency ν = 2µ B/[h(2I+1)], while
L B
decay rate approaches the NLZ frequency splitting (a the second-order shift leads to splitting of the magnetic
common situation for practically important Rb and Cs resonances. For the F = 2 hyperfine manifold of 87Rb,
magnetometers). there are three ∆M = 2 resonances with adjacent-
F
Compensation of NLZ with AC Stark shifts was re- frequency splittting
cently investigated in the field of quantum information,
µ2B2
δ ≈ B . (2)
h∆
hfs
∗Electronicaddress: [email protected] For 87Rb atoms placed in a magnetic field of 0.5 G, we
†Electronicaddress: [email protected] have νL =350 kHz and δ =72 Hz.
2
Considernow87Rbatomsilluminatedwithlinearlypo-
larizedlightnear-resonantwiththeD1transition. Based y
onthecalculationsinRefs.[18,19]wecanfindanalytical x
results for the AC Stark shifts ofthe energy levels in the z
F = 2 ground-state manifold. For light polarized along Intensity Pump Probe
the magnetic field, we find a change in the splitting of
Time
adjacent ∆M =2 resonance frequencies of
F
λ3ǫ E2Γ A
∆Dπ1 =−16π2h0∆0∆D−12AP1/2 , (3) B
P1/2
(cid:0) (cid:1) Magnetic Shields
where ǫ is the permittivity of vacuum, A is the
0 P1/2
52P excited-state hyperfine-structure coefficient mea-
1/2
FIG. 1: (Color online) Schematic of the experimental appa-
sured in Hz, Γ is the natural linewidth of the 52P
D1 1/2 ratus. AOM – acousto-optical modulator; LP – linear polar-
excited state, λ is the transition wavelength, E0 is the izers. A vapor cell with 87Rb is located within a multi-layer
electric field amplitude of the light and the light fre- magneticshield. Coilswithintheinnershieldisusedtoapply
quency detuning ∆ is measured relative to the F =2→ theEarth-range magnetic field and to compensate gradients.
F′ = 1 transition and assumed to be much larger than Theintensityoflight from laserL1ismodulatedasshownin
the Doppler width and the upper-state hyperfine inter- the upper inset, so the same laser beam is used to optically
val. For light polarized transverse to the magnetic field, pump and probe the atoms. Laser L2 is used to induce light
wefindthatthechangeinsplittingofthe∆M =2reso- shifts.
F
nancesareofhalfthemagnitudeandoftheoppositesign
be
compared to the longitudinal case. In general, if the L2
light is linearly polarized at an angle θ to the magnetic
µ2B2Γ
fipeolrdt,iotnhaeldtoiff3erceonst2iaθl−sh1i,ftsseeo,fftohreeMxaFmpsuleb,lePvreolsblaerme p2.r1o1- ΓDπ1 = 2hABP1/2∆Dh1fs. (5)
in Ref. [20]. Since the splitting increases in response to
lightoftransversepolarizationanddecreasesinresponse For D2 light with transverse polarization we find that
to longitudinal polarization, NLZ can only be compen- the broadening at the compensation power is around 20
sated on this transition using longitudinal polarization. times larger compared to the D1 case. It will therefore
Using Eqs. (2) and (3) we can calculate the D1 light be preferable to use D1 light to cancel NLZ.
powerneededto cancelthe nonlinearZeemaneffect, and Another source of light broadening results from the
we find nonuniform intensity profile of the light beam over the
cross-sectional area of the cell. If, as in our case, the
P = µ2BB22π3cd2∆ ∆−2AP1/2 . (4) compensating light illuminates only a small portion of
∆ λ3Γ A the cell, the atoms are subject to short periods of com-
hfs (cid:0)D1 P1/2 (cid:1) pensating light at randomintervals, so that the phase of
each atom’s evolution undergoes a random walk. This
Herewe asssumethatthe atomsarekeptinacylindrical
leads to dephasing of the atomic evolution at a rate
cellofdiameterd,andthattheaveragelightintensityI =
cǫ E2/2 = P/ πd2/4 , where c is the speed of light in Γdephase = φ2/τ, where φ is the phase shift an atom
0 0 experiences in each pulse of compensating light, and τ
vacuum,determinestheamountofcompensation. Thisis
(cid:0) (cid:1) is the average time between pulses. In order that the
thecaseforatomsinanantirelaxation-coatedcell,where
light properly compensates the nonlinear Zeeman effect,
eachatomsamplesthelightfieldintheentirecellvolume.
we must have φ = δτ. The average time for an atom
Similar calculations for light near-resonant with the D2
to cross the cell is on the order of d/v, where v is the
transition shows that the change in splitting is of the
rms atomic velocity, and the probability that an atom
oppositesigncomparedtothe D1case,andthatroughly
passes through the light beam in one trip across the cell
8-10 times more power is needed to cancel NLZ.
is on the order of b/d, where b is the beam diameter, so
In addition to shifting the magnetic resonances, light
τ ≈d2/(bv). Thus
alsobroadensthem. Onesourceofthisbroadeningisab-
sorption of light by the ground-state atoms. The broad-
d2µ4B4
ening depends on the light power needed to compensate Γ ≈τδ2 ≈ B . (6)
NLZ.Forlargedetunings thebroadeningatthe compen- dephase bv~2∆2hfs
sation power will be independent of detuning, since the
power needed to cancel NLZ goes as ∆2 and the absorp- By inserting typical experimental values (B = 0.397 G,
tion rate goes as 1/∆2. Based on the calculations in d ≈ 2 cm and b ≈ 1.5 mm), we find ΓD1 ≈ 0.3 Hz and
π
Refs. [18, 19], we find the broadening at the compensa- Γ ≈ 11 Hz, and we see that dephasing dominates
dephase
tion power for D1 light with longitudinal polarization to the broadening.
3
20 80
18 mrad)1124 70 πy ppoollaarriizzeedd lliigghhtt
Optical rotation (mrad)11110246468 Optical rotation (10468392 394 396 tim3e9 (8µs) 400 402 404 ∆Splitting of M=2 resonances (Hz)F2345600000
10
2
00 5 10 15 20 25 30 35 40 45 00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
time (ms) light power (mW)
FIG. 2: (Color online) An example of data showing optical FIG. 3: (Color online) Splitting between adjacent magnetic
rotationoftheprobelight. Attime=0ms,thepumpingstage resonances (see Fig. 4) as a function of L2 light power for
ended. The signal undergoes several collapses and revivals polarizations along (circles) and transverse (squares) to the
during the decay. The magnetic field was set to 0.477 G. magnetic field (denoted π and y polarization in the figure).
The inset shows a zoom-in revealing the fast oscillation at Linesarelinearfittothedata. Themagneticfieldwassetto
2νL =667 kHz. 0.397 G.
the form
III. EXPERIMENTAL SETUP, PROCEDURE
AND RESULTS S(t) = {C cos(2π[2ν −δ]t−α)+C cos(4πν t)
2 L 1 L
+C cos(2π[2ν +δ]t+α)}e−t/T, (7)
2 L
Figure 1 shows the experimental setup. A paraffin-
which consists of a central component of amplitude C
1
coated cylindrical glass cell (with diameter and length
oscillating at twice the Larmor frequency and two side-
of2cm)atroomtemperaturecontaining87Rbislocated
bandswithfrequencies2ν ±δofequalamplitudeC and
L 2
withinamulti-layermagneticshieldwithasystemofcoils
oppositephase±α. Thethreefrequencycomponentsde-
in the inner volume designed to produce a homogeneous
cay with time, and for simplicity it is assumed that the
Earth-range magnetic field along the x direction. The
1/e decay time T is the same for all three components.
atoms in the F = 2 ground-state are optically pumped
We analyze the envelope of the signal by postprocessing
and probed using a beam from a diode laser (L1) with
the data with a digital lock-in amplifier with reference
frequency close to either the D1 or D2 line. The in-
frequency2ν . TheenvelopeR(t)canbecalculatedfrom
L
tensity of the light is modulated with an acousto-optical
Eq. (7) and we find
modulator (AOM). The same laser beam, initially lin-
earlypolarizedinthez direction,isusedtocreateatomic 1 1
alignment(therank-twoatomicpolarizationthatischar- R(t) ≈ 4C12+ 2C22+C1C2cos(2πδt−α)
acterized by a preferred “alignment axis”) and to probe (cid:20)
1/2
the alignment via optical rotation due to the polarized 1
+ C2cos(4πδt−2α) ·e−t/T. (8)
atoms [21]. A measurement consists of two consecutive 2 2
(cid:21)
stages. During the first, pumping stage, the laser in-
tensity is square-wave modulated (duty cycle 1/8) at a ItisfoundexperimentallythatthedemodulatedFIDsig-
rate equal to twice the Larmor frequency, and ground- nalsarewelldescribedbyEq.(8). Forthedatapresented
state atomic alignment is synchronously pumped for 3 inFig.2,afitoftheenvelopetoEq.(8)givesthevalueof
ms. The beam diameter is ≈3 mm, and the power dur- δ =65.37(7) Hz, consistent with the expected value due
ingthe“on”partofthecycleis≈5mW.Oncetheatoms to the NLZ effect for the magnetic field of B = 0.477 G
arepumped,theAOMissettotransmitaround4µWof usedforthedatapresentedinFig.2. Theoverall1/ede-
light, which probes the atomic alignment created during cay time extracted from the fit is 20.8(7) ms, limited by
the pumping stage. The optical rotation of the trans- magnetic-field gradients and drifts in the bias magnetic
mitted light is measured by a balanced polarimeter. An field combined with signal averaging [for comparison, at
exampleofafree-inductiondecay(FID)signal(averaged lower magnetic field of 24 mG the relaxation time was
over 1024 cycles) is shown in Fig. 2. The FID signal os- measured to be 55(2) ms].
cillates at667kHz (see inset) correspondingto twice the In order to cancel NLZ, linearly polarized light from a
Larmor frequency. The NLZ-induced beats in the signal second diode laser (L2) tuned close to the D1 resonance
areobservedasachangeintheoscillationamplitudewith wasdirectedthroughthevaporcellalongthez direction.
time. The signal has an offset of around 7 mrad due to The splitting of the ∆MF =2 resonances was measured
imperfect balancing of the polarimeter. as described above for different detunings, polarizations
and powers of the L2 light. Figure 3 shows the splitting
WemodelthemeasuredFIDsignalswithafunctionof as a function of light power inside the cell for two polar-
4
9000 W)10
8000 m
Fourier amplitude squared 234567000000000000000000 Compensation power ( 02468 −4 −2 0 2 4 6 8
1000
5505.45 555.5 555.55 555.6 555.65 Hz)30
frequency (kHz) ht (
g
FIG.4: (Coloronline)PowerspectrumofaFIDsignalshow- o L2 li20
ingthesplittingofthe∆MF =2magneticresonancesdueto ue t
d
NLZ(solidline),andofasignalwhereNLZwascompensated ng 10
ni
by light shifts induced by the L2 light (dashed line). The de
a
magnetic field was set to 0.397 G. Bro 0 −4 −2 0 2 4 6 8
detuning from 87Rb F=2−>F´=1 D1 transition (GHz)
izations, along and transverse to the magnetic field (de-
noted π and y polarization in the figure), for a detuning
FIG. 5: (Color online) Top: L2 light power needed to cancel
∆=−3.8 GHz from the F =2→F′ =1 D1 transition.
NLZatamagneticfieldof0.397GasafunctionofL2laserfre-
Thesplittingswerefittostraightlinesyieldingtheslopes quency. Dotsrepresent measured values, and thesolid lineis
−9.4(13) Hz/mW and 5.6(8) Hz/mW, which within the thetheoretical calculation given by Eq. (4). Bottom: Broad-
uncertainties confirms that the light shifts due to y po- ening of the magnetic resonance due to the L2 light at the
larized light have the opposite sign and are of half the compensation power. Dashed line is the 11 Hz estimate due
magnitude compared to the light shifts due to π polar- to dephasinggiven by Eq.(6).
ized light. It should be noted that when the splitting is
within -4 to 5 GHz of the F =2→F′ =1 D1 transition
smallerthanthelinewidthoftheresonances,itisdifficult
the broadeningis approximatelyconstantandconsistent
to extract the splitting from the FID signal. Therefore,
with the dephasing estimate. For the detuning ∆ = 7.6
inordertofindthelightpowerwherethelightshiftscan-
GHz (which is on the F = 1 → F′ = 2 resonance) a
cel NLZ, an extrapolation from the linear fit was used.
slightly larger broadening was measured. We note that
In Fig. 3 the light power where the splitting is zero is
the broadening due to dephasing can be avoided if the
4.8(7) mW.
compensating light has a homegeneous intensity profile
Figure 4 (solid line) shows the power spectrum of a
over the cell volume. In the experiment the beam diam-
FID signal with the L2 light blocked. Three ∆M = 2
F
eter of the compensating light was rather small, and we
resonances are seen, split from each other in frequency
thereforeexpectthatitispossibletosignificantlyreduce
due to NLZ. When the L2 light is on (dashed line), with
the broadening by for instance expanding the beam.
the appropriate power needed to cancel NLZ, the peaks
arecombinedintoasingle,strongerresonance. However,
the L2 light also broadens the resonance. The FWHM
of the resonances are 14.0(4) Hz (solid line) and 27.8(4) IV. DISCUSSION AND CONCLUSIONS
Hz (dashed line), giving a broadening of 13.8(6) Hz due
to L2 light. By comparing the two spectra in Fig. 4, To be useful in practical magnetometers, our method
it is seen that there is a shift of ≈ 8 Hz in the central for compensating NLZ should be robust against, for in-
frequency of the resonances. The shift is not thought to stance, intensity fluctuations of the L2 laser light. At
be due to the L2 light, but instead due to drifts in the Earth-rangemagneticfields,NLZsplitsthemagneticres-
currentsupply forthe bias magnetic fieldin betweenthe onances considered in the experiment by approximately
two measurements. 70 Hz. We compensate NLZ by overlapping the reso-
TheL2lightpowerneededtocanceltheNLZeffectwas nances. Theresonanceswilloverlapiftheyarepositioned
measured for different frequencies of the L2 light close withintheirwidth,whichinourparticularcaseisaround
to the D1 resonance; the results are plotted in Fig. 5 30 Hz (see Fig. 4). Since the splitting between the reso-
(top). For these measurements, the magnetic field was nancesislinearinL2laserintensity,therelativeintensity
set to B = 0.397 G, producing a NLZ shift δ ≈ 45 Hz. noise of the laser should be (much) less than the ratio of
Also plotted in Fig. 5 (top) is the theoretical calculation the linewidth to the splitting without compensation, in
for the compensation power given by Eq. (4), showing this case 30 Hz / 70 Hz ≈ 40 %. In practical situations
a reasonable agreement between theory and experiment. one can easily stabilize lasers to have less than 1 % in-
The broadeningofthe magneticresonancedue to the L2 tensitynoise. Theintensitynoisehasthereforeverylittle
light at the compensation power was also measured and effect on the compensation. Similarly, we estimate that
isplottedinFig.5(bottom)togetherwiththeestimated the frequency drifts of the L2 laser within easily achiev-
broadening due to dephasing. In the frequency range able±1MHzleadtosub-1-%changesintheinducedlight
5
shifts, much smaller than typical resonance width. the manuscript and to E. Corsini for helpful discussions.
In conclusion, using AC Stark shifts, we compensated This work has been supported by NURI grant HM1582-
the nonlinear Zeeman effect for 87Rb atoms located in 08-1-0006,ONR MURI and STTR grants.
a magnetic field comparable to the Earth’s magnetic
field. Themethodcandirectlybe appliedtoalkali-vapor
atomic magnetometers in order to reduce heading errors
and increase sensitivity.
ACKNOWLEDGMENTS
The authors are grateful to E. S. Polzik for encour-
agement and support, to W. Gawlik for comments on
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