Quantum lock-in detection of a vector light shift
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Quantum lock-in detection of a vector light shift
Kosuke Shibata, ∗ Naota Sekiguchi, and Takuya Hirano Department of Physics, Gakushuin University, Tokyo, Japan (Dated: January 25, 2021)We demonstrate detection of a vector light shift (VLS) using the quantum lock-in method. Themethod offers precise and accurate VLS measurement without being affected by real magnetic fieldfluctuations. We detect a VLS on a Bose–Einstein condensate (BEC) of Rb atoms caused by anoptical trap beam with a resolution less than 1 Hz. We also demonstrate elimination of a VLS bycontrolling the beam polarization to realize a long coherence time of a transversally polarized F =2 BEC. Quantum lock-in VLS detection should find wide application, including the study of spinorBECs, electric-dipole moment searches, and precise magnetometry. PACS numbers:
I. INTRODUCTION
The a.c. Stark shift or light shift plays significant rolesin atomic physics. One example is the optical trap [1],which has been extensively used in cold atom experi-ments and has been the subject of intriguing and impor-tant research, including low-dimensional [2] and uniformgases [3], and atoms in an optical lattice with applicationsto quantum simulation [4] and atomic clocks [5, 6]. It hasalso enabled the study of multi-component gases and, inparticular, spinor Bose–Einstein condensates (BECs) [7].The light shift has vector and tensor components andhence is state-dependent in general [8–10]. The state de-pendence has been exploited for realizing state-selectivetransport [11, 12] and confinement [13]. However, astate-dependent shift is often undesirable for situations inwhich well-controlled spin evolution is required. Escap-ing from a vector light shift (VLS), which is equivalent toa fictitious magnetic field, has been an important issue inprecise measurements, such as the search for an atomicelectric-dipole moment [14] and exotic spin-dependent in-teractions [15]. Reducing the VLS is also important inatomic magnetometers, in which the VLS introduces sys-tematic errors. The quantum noise associated with thelight shift due to the probe field ultimately limits thesensitivity [16].The VLS restricts the potential use of optically trappedatoms for magnetically sensitive experiments. While itseffect can be diminished by applying a bias magnetic fieldin a direction orthogonal to the wavevector, the VLS canstill be a significant noise source in precise measurements[14]. It is necessary to reduce the VLS when an ultralowmagnetic field is required. In addition, the relative direc-tion cannot be chosen satisfactorily in some situations,such as in 3D optical lattice experiments.In order to eliminate the VLS caused by optical trap-ping beams, the light polarization should be preciselycontrolled, because the VLS is proportional to the inten-sity of a circularly polarized component [1, 8–10]. How- ∗ Electronic address: [email protected] ever, it is a formidable task to precisely extinguish thecircular component at the atom position located in a vac-uum cell. Polarization measurements and control outsidethe cell do not assure the degree of linear polarizationdue to the stress-induced birefringence of the vacuumwindows [17].Therefore, a sensitive and robust polarization mea-surement method using atoms themselves as a probeis important. Most effective polarization measurementsare accomplished by using atoms themselves as a probe.Polarization measurements with an atomic gas havebeen performed with various methods including Lar-mor precession measurement [18], precise microwavespectroscopy [19], and frequency modulation nonlinearmagneto-optical rotation [20]. Differential Ramsey in-terferometry has been developed for spinor condensates[21]. Polarization measurements by fluorescence detec-tion have been recently demonstrated for ions [22].In this paper, we demonstrate VLS detection by apply-ing the quantum lock-in method [23, 24]. The measure-ment is immune to environmental magnetic field noise,and thus achieves excellent precision and accuracy. Wedetect a VLS induced by an optical trap beam on a BECof Rb atoms with a resolution less than 1 Hz. This de-tection method is feasible to implement and should havewide applications in various research areas involved withoptical fields.The paper is organized as follows. In Sec. II, ourexperimental method and setup are presented. The ex-perimental results are described in Sec. III. We discussthe applications and potential performance of the quan-tum lock-in VLS detection in Sec. IV. We conclude thepaper in Sec. V.
II. EXPERIMENTAL METHOD AND SETUP
We produce a BEC in a vacuum glass cell. A BECof 3 × atoms in the hyperfine spin F = 2 state istrapped in a crossed optical trap. The trap consists ofan axial beam at the wavelength of 850 nm and a radialbeam at 1064 nm. The axial and radial beam waists are ≈ µ m and 70 µ m, respectively. A magnetic bias field P(t) t ΔT = π/ω m π-pulse x Nπ/2( φ = 0)ω(t) ......... π/2( φ = π/2) (a) zxy g B (c)(b) m F =+2 +1 0 -1 -2 FIG. 1: (color online) (a) Experimental configuration. A BECis trapped in the axial trap beam along the z axis and the ra-dial trap beam along the x axis (not shown). (b) Typical TOFimage of a BEC measured after rf pulses for the detection.The spin components ( m F = − , − , , ,
2) are spatially re-solved by the Stern–Gerlach method. (c) Time sequence forthe quantum lock-in detection of a VLS. The beam power, P ( t ), is modulated with a frequency ω m . The phase of thespin vector evolves with an angular frequency of ω ( t ). Theaccumulated phase, Φ = R T ω ( t ) dt , is finally measured. B of 15 µ T is applied along the axial beam to define thequantization axis, as shown in Fig. 1(a). The atoms areinitially in the | F, m F i = | , i state, where m F denotesthe magnetic sublevel. The ellipticity of the axial beamat the atomic position is controlled with a quarter wave-plate (QWP) in the VLS measurement described below.The QWP is located between a polarization beam splitterfor polarization cleaning and the cell. The angle of theQWP is adjusted with a precise manual rotation stage.The minimum scale of the rotation stage is 0.28 mrad.The time sequence for the quantum lock-in detectionof a VLS is shown in Fig. 1(c). The lock-in technique en-ables enhanced sensitivity at the modulation frequencywhile reducing the effect of unwanted noise. We measurea VLS induced by the axial optical trap beam with mul-tiple rf pulses. The trap beam power, P ( t ), is modulatedwith a frequency ω m during the pulse application as P ( t ) = P (1 + p sin( ω m t )) ≡ P + P sin( ω m t ) , (1)where P is the mean power and p is the modulationindex. P can be negative by changing the modulationphase by π . ω m is set to be sufficiently higher than twicethe trapping frequency to avoid parametric heating ofthe atoms. The modulation generates an a.c. fictitiousmagnetic field to be measured, given by B fic = − α (1) C I sin( ω m t ) ≡ B sin( ω m t ) , (2)where α (1) is the a.c. vector polarizability, C is the degreeof the circularity and I is the beam intensity correspond-ing to P .The pulse set consists of an initial π/ t =0, an odd number ( N ) of π -pulses, and a readout π/ T . The spacingsatisfies ω m = π/ ∆ T so that the evolved phase due to the p m B ( n T ) FIG. 2: (color online) Detection of the VLS. The error barsrepresent the sample standard deviation. The red solid lineis the fitting curve by V F sin( ap ). The right axis represents B . It should be noted that the right axis scale is not linearsince B is proportional to arcsin( m V F ). fictitious field is constructively accumulated. The relativephase, ∆ ϕ , between the initial and read-out pulses is setto π/ π g F µ B B ¯ h T ≡ π ω T, (3)where g F is the Land´e g-factor, µ B is the Bohr magneton,¯ h is the reduced Planck constant and T = ( N + 1)∆ T is the phase accumulation time. ω / (2 π ) represents theVLS corresponding to B in units of frequency.The read-out pulse converts Φ into the magnetization, m , as m ≡ P i iN i N tot = V F sin Φ , (4)where N i is the atom number in the | F, m F = i i state( i = − , − , , ,
2) after the read-out pulse, N tot = P i N i is the total atom number, and V is the visibil-ity. V is ideally 1, but in practice it is less than 1 due tomagnetic field noise [23]. Imperfections in the initial statepreparation and spin manipulation also decrease V . Themagnetization is measured by standard absorption imag-ing after a time-of-flight with Stern–Gerlach spin separa-tion (see Fig. 1(b)). III. RESULTS
We first confirm the validity of the detection scheme.We perform a lock-in detection with ω m = 2 π × T = 0 .
25 ms) and N = 27, and hence T = 7 ms. P is fixed to 11 mW. The change in m is observed as p isvaried. The result is plotted in Fig. 2. Here, the angle -10 -5 0 5 10-202 m -2-1012 B ( n T ) -10 -5 0 5 10 (mrad) -4-2024 m -4-2024 B ( n T ) -60 -40 -20 0 20 40 60-0.500.5 m -2-1012 B ( n T ) FIG. 3: (color online) Polarization dependence of the signal.(a) Measurement result with T = 7 ms. (b) Measurementresult with T = 28.2 ms. The blue circles and red squaresrepresent m + and m − , respectively. (c) ∆ m as a function of θ . ∆ B is the difference between the fictitious magnetic fieldsfor positive and negative P . The solid lines in (a) and (c)are the fitting curves. of the QWP axis, θ , is approximately 4 ◦ apart from theoptimal angle, θ ∗ , minimizing the VLS. The experimentaldetermination of θ ∗ is described below. m is well fittedby a sinusoidal function V F sin( ap ), indicating the VLSwas successfully detected. The visibility in this detectionsetting is found to be V = 0 . p = 0) where ∆ ϕ isscanned.The detection is used to minimize the VLS. We con-trol the VLS by changing θ with p fixed to 0.32. The θ -dependence of m is shown in Fig. 3(a). Because C ≈ sin 2( θ − θ ∗ ) ≡ sin 2∆ θ when the birefringence inthe optical path is small [21] and | ∆ θ | ≪
1, we fit m by V F sin( β ( θ − θ ∗ )). The fit gives β = 6 . θ ∗ is foundto be -6.6(1.1) mrad. The VLS resolution is evaluated as δω = β δθ ∗ /T = 2 π × .
16 Hz, where δθ ∗ is the uncer-tainty in the θ ∗ estimation.We perform a fine estimation of θ ∗ by extending T to27.2 ms and applying a larger modulation. In this experi-ment, ω m and N are 2 π ×
625 Hz and 33, respectively. Wemeasure m for P = ±
13 mW, referred to as m ± , respec-tively. In finding θ ∗ , we use ∆ m = m + − m − to cancelthe offset due to the background field and the systematicerror in the spin measurement. The results are shown inFigs. 3(b) and (c). ∆ m is fitted by 4 V F sin( β ( θ − θ ∗ )), giving β = 117(16) and θ ∗ = 0 .
06 mrad with δθ ∗ = 0.40mrad. The angle resolution is improved 2.8 times.We observe a larger variance in m in the experimentsfor the fine θ ∗ estimation. The standard deviation of ∆ m is on average 0.64, while that for the reference data with-out modulation is 0 .
09. Therefore, a further improvementby a factor of at least 7 is possible, because ∆ m shouldideally be independent of T and the modulation strength.We ascribe the increased variance to the actual variationof the vector shift over the experimental runs, causedby beam polarization fluctuation. The result of the sen-sitive detection implies that the beam circularity varieswith the standard deviation of approximately 3 × − .On the other hand, from an independent experiment, weexpect that the retardance of the QWP should vary byseveral mrad due to the temperature change in our ex-perimental room (within ≈ µ T/m. The gradient displaces the trap potentialfor each spin state other than the m F = 0 state, therebydriving the spin dependent motion. We observe an ac-tual motion in a transversally-spin-polarized BEC in thehyperfine spin F = 2 state, prepared after the initial π/ θ and the motion be-comes small at ∆ θ ≈
0. These observations indicate thatthe motion is induced by the fictitious magnetic field.The fictitious magnetic field gradient also causes non-linear spin evolution and thus a population change, asdoes the real magnetic field gradient [25]. The initialpolarized atomic spin state breaks due to the spin mix-ing seeded by the nonlinear spin evolution. We show p = N /N tot , p = ( N − + N +1 ) / (2 N tot ), and p =( N − + N +2 ) / (2 N tot ) in Figs. 4(e)–(h). The populationchanges are observed at an earlier time ( t <
100 ms)except for the case ∆ θ ≈
0. These changes can be at-tributed to the fictitious magnetic field gradient. Thefaster population change for ∆ θ = 4 ◦ is consistent witha qualitative estimation of the characteristic time for thechange of t ∗ ∝ b − / , where b is the magnetic field gra-dient [25]. A slow population change, which occurs re-gardless of ∆ θ , is caused by a residual axial magneticfield gradient, ∂B z /∂z . The existence of the axial gradi-ent in these data is confirmed by the fact that the spincomponents separate in the axial direction at later times.We next observed the change in the atom loss rate.Figure 5 shows the evolution of the atom numbers, cor-responding to the data in Figs. 4(b) and (d). The decay isfaster when ∆ = 4 ◦ than for ∆ ≈ ◦ . In the latter case,the decay rate starts to increase from around t = 150ms, where the population changes occur (see Fig. 4(f)). t (ms) -20020 D i s p l a c e m en t [ m ] t (ms) P opu l a t i on ° t (ms) -20020 +2-1 +1-2 t (ms) ° t (ms) -200200 100 200 t (ms) ° t (ms) -200200 100 200 t (ms) ° FIG. 4: (color online) Effects of the fictitious magnetic field on a transversally polarized BEC. (a)–(d) Vertical displacement ofthe center of mass in the TOF image. The panels show the data for ∆ θ = (-1, +0.02, +1, +4) degrees, respectively. The solidand dashed lines are guides for the eyes. (e)–(h) Population evolution corresponding to (a)–(d). t (ms) A t o m N u m be r = 4 ° = 0.02 ° optimized t (ms) P opu l a t i on FIG. 5: (color online) Atom number losses. The solid line isan exponential fit to the data with the optimized field gra-dient. The inset shows the population evolution for the op-timized case. The dotted lines are the prediction curves ofthe mean-field driven evolution without including the inelas-tic losses [26].
No increase in the loss rate is observed at the later timewhen we optimize θ and reduce the axial real magneticfield gradient. The 1 /e time for the optimized conditionis found to be 742 (31) ms. The change of the loss ratecan be understood from the property of the inelastic col-lisions in the F = 2 state [27]. Note that the inelasticcollisional loss in the polarized state is inhibited due tothe restriction of the angular momentum conservation.The break of the polarized state due to the field gradient results in rapid atom losses.However, the loss still occurs for the optimized con-dition. Although the remaining loss may be due to theresidual field inhomogeneity, it is associated with the spinmixing induced by the quadratic Zeeman energy [26, 28].The model presented in [26, 28], however, needs to bemodified to explain the observed population conserva-tion, shown in the inset of Fig.5. According to [26], thepopulation evolution in the limit of small quadratic Zee-man energy, q , is approximately given by p = 38 (cid:20) q g n (1 − cos(4 g nt/ ¯ h )) (cid:21) , (5) p = 14 , (6) p = 116 (cid:20) − q g n (1 − cos(4 g nt/ ¯ h )) (cid:21) , (7)where g = π ¯ h m a − a is the interaction strength with a F being the s -wave scattering length for the collisionalchannel of the total angular momentum F and n is themean atomic density. Following these equations, p and p would undergo oscillations, which is not in agreementwith the observed experimental result. We therefore at-tribute the population conservation to polarization pu-rification by inelastic collisional losses [29]. It shouldbe noted that the observed population conservation con-trasts with the case of the F = 1 state, in which themagnitude of the polarization modulates [30]. IV. DISCUSSION
The quantum lock-in VLS detection is of practical usein cold atom experiments. It can be used for evaluat-ing the degree of circular polarization of an optical trapbeam at the atomic position, as we have shown. As thevacuum window birefringence introduces a maximum el-lipticity of 10 − or 10 − [19], a beam with no special caretaken with respect to the in vacuo polarization may gen-erate a fictitious field of several nT or a VLS of tens ofHz, even with a shallow trap for ultracold atom experi-ments. Quantum lock-in detection is sensitive enough toensure better linear polarization at the atomic positionand therefore will greatly improve the magnetic condi-tions in cold atom experiments. The sensitivity is suf-ficient to suppress the VLS below the requirements formagnetically sensitive experiments, including studies ofspinor BECs. Although a homogeneous linear Zeemanshift does not affect the spinor physics due to spin con-servation [31], a magnetic field gradient below several µ T/m is typically required to prevent magnetic polariza-tion and observe the intrinsic magnetic ground state [32]or dynamics. The sub-Hz VLS resolution of quantumlock-in detection meets this challenging demand.Reducing the VLS is also important for precise mea-surements. In addition to a direct energy shift, an in-homogeneous fictitious field is also detrimental to mea-surement accuracy [33]. VLS reduction leads to a longcoherence time, which is a mandatory requirement forhighly sensitive measurements. We have constructed aprecise BEC magnetometer using a transversally polar-ized F = 2 BEC with a long coherence time, realizedusing VLS elimination as we have shown. The detail ofthe F = 2 BEC magnetometer will be presented else-where [34].We finally discuss the sensitivity limitations. The sen-sitivity of the quantum lock-in detection is essentiallythe same as that of a Ramsey interferometer with anequal phase accumulation time. As the atom shot noise is dominant over the photon shot noise in typical absorp-tion imaging, the standard quantum limit in the VLSmeasurement is given by [35, 36] δω = 1 T √ N tot . (8)Here we replace the factor π in Eq. (3) due to the sinu-soidal modulation with the maximal value of 1, which isrealized with a rectangular waveform modulation. Sub-stituting N tot = 3 × and T = 30 ms into Eq. (8), weobtain δω = 2 π ×
10 mHz. This is equivalent to a singleshot field sensitivity of ∼ V. CONCLUSIONS
We have demonstrated precise detection of a VLS dueto an optical trap using the quantum lock-in method.We have applied the detection to eliminating the VLS,to observe the extension of the lifetime of transversallypolarized F = 2 BEC. The attained resolution of sub Hzis sufficient to suppress the VLS below the required levelfor magnetically sensitive research, including the studyof spinor BECs. Although our demonstration was per-formed with a BEC, the scope of the detection methodis not limited to cold atom gases; the proposed methodcan be applied to spin systems such as trapped ions anddiamond NV centers, where coherent spin control is pos-sible. Acknowledgments
This work was supported by the MEXT Quantum LeapFlagship Program (MEXT Q-LEAP) Grant Number JP-MXS0118070326 and JSPS KAKENHI Grant NumberJP19K14635. [1] R. Grimm, M. Weidem¨uller, and Y. B. Ovchinnikov, Adv.At. Mol. Opt. Phys. , 95 (2000).[2] A. G¨orlitz, J. M. Vogels, A. E. Leanhardt, C. Raman,T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur,S. Gupta, S. Inouye, T. Rosenband, et al., Phys. Rev.Lett. , 130402 (2001).[3] A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P.Smith, and Z. Hadzibabic, Phys. Rev. Lett. , 200406(2013).[4] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[5] A. Derevianko and H. Katori, Rev. Mod. Phys. , 331(2011).[6] H. Katori, Nat. Photon. , 203 (2011).[7] D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. ,1191 (2013). [8] I. H. Deutsch and P. S. Jessen, Phys. Rev. A , 1972(1998).[9] J. M. Geremia, J. K. Stockton, and H. Mabuchi, Phys.Rev. A , 042112 (2006).[10] I. H. Deutsch and P. S. Jessen, Opt. Commun. , 681(2010).[11] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W.H¨ansch, and I. Bloch, Phys. Rev. Lett. , 010407 (2003).[12] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W.H¨ansch, and I. Bloch, Nature (London) , 937 (2003).[13] A. Heinz, A. J. Park, N. ˇSanti´c, J. Trautmann, S. G.Porsev, M. S. Safronova, I. Bloch, and S. Blatt, Phys.Rev. Lett. , 203201 (2020).[14] M. V. Romalis and E. N. Fortson, Phys. Rev. A , 4547(1999).[15] D. F. Jackson Kimball, J. Dudley, Y. Li, D. Patel, and J. Valdez, Phys. Rev. D , 075004 (2017).[16] M. Fleischhauer, A. B. Matsko, and M. O. Scully, Phys.Rev. A , 013808 (2000).[17] G. E. Jellison, Appl. Opt. , 4784 (1999).[18] K. Zhu, N. Solmeyer, C. Tang, and D. S. Weiss, Phys.Rev. Lett. , 243006 (2013).[19] A. Steffen, W. Alt, M. Genske, D. Meschede, C. Robens,and A. Alberti, Rev. Sci. Instrum. , 126103 (2013).[20] D. F. Jackson Kimball, J. Dudley, Y. Li, and D. Patel,Phys. Rev. A , 033823 (2017).[21] A. A. Wood, L. D. Turner, and R. P. Anderson, Phys.Rev. A , 052503 (2016).[22] W. H. Yuan, H. L. Liu, W. Z. Wei, Z. Y. Ma, P. Hao,Z. Deng, K. Deng, J. Zhang, and Z. H. Lu, Rev. Sci.Instrum. , 113001 (2019).[23] S. Kotler, N. Akerman, Y. Glickman, A. Keselman, andR. Ozeri, Nature (London) , 61 (2011).[24] G. de Lange, D. Rist`e, V. V. Dobrovitski, and R. Hanson,Phys. Rev. Lett. , 080802 (2011).[25] Y. Eto, M. Sadgrove, S. Hasegawa, H. Saito, and T. Hi-rano, Phys. Rev. A , 013626 (2014).[26] J. Kronj¨ager, C. Becker, P. Navez, K. Bongs, and K. Sen-gstock, Phys. Rev. Lett. , 189901 (2008).[27] S. Tojo, T. Hayashi, T. Tanabe, T. Hirano,Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev.A , 042704 (2009). [28] J. Kronj¨ager, C. Becker, P. Navez, K. Bongs, and K. Sen-gstock, Phys. Rev. Lett. , 110404 (2006).[29] Y. Eto, H. Shibayama, K. Shibata, A. Torii, K. Nabeta,H. Saito, and T. Hirano, Phys. Rev. Lett. , 245301(2019).[30] M. Jasperse, M. J. Kewming, S. N. Fischer, P. Pakkiam,R. P. Anderson, and L. D. Turner, Phys. Rev. A ,063402 (2017).[31] D. M. Stamper-Kurn and W. Ketterle, in Coherentatomic matter waves , edited by R. Kaiser, C. Westbrook,and F. David (Springer, Berlin, Heidelberg, 2001), pp.139–217.[32] J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Mies-ner, A. P. Chikkatur, and W. Ketterle, Nature (London) , 345 (1998).[33] G. D. Cates, S. R. Schaefer, and W. Happer, Phys. Rev.A , 2877 (1988).[34] N. Sekiguchi, A. Torii, H. Toda, R. Kuramoto,D. Fukuda, T. Hirano, and K. Shibata (2020),arXiv:2009.07569.[35] V. Giovannetti, S. Lloyd, and L. Maccone, Science ,1330 (2004).[36] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev.Lett.96