Fast control of atom-light interaction in a narrow linewidth cavity
A. Bertoldi, C.-H. Feng, D. S. Naik, B. Canuel, P. Bouyer, M. Prevedelli
FFast control of atom-light interaction in a narrow linewidth cavity
A. Bertoldi, ∗ C.-H. Feng, D. S. Naik, B. Canuel, P. Bouyer, and M. Prevedelli Univ. Bordeaux, CNRS, IOGS, LP2N, UMR 5298, F-33400 Talence, France Dipartimento di Fisica e Astronomia, Universit`a di Bologna,Via Berti-Pichat 6/2, I-40126 Bologna, Italy (Dated: February 4, 2021)We propose a method to exploit high finesse optical resonators for light assisted coherent manip-ulation of atomic ensembles, overcoming the limit imposed by the finite response time of the cavity.The key element of our scheme is to rapidly switch the interaction between the atoms and the cavityfield with an auxiliary control process as, for example, the light shift induced by an optical beam.The scheme is applicable to many different atomic species, both in trapped and free fall configura-tions, and can be adopted to control the internal and/or external atomic degrees of freedom. Ourmethod will open new possibilities in cavity-aided atom interferometry and in the preparation ofhighly non-classical atomic states.
Narrow linewidth cavities are key devices in funda-mental physics [1], metrology [2], and they underpin theincessant progress in the study of light-matter interac-tion [3, 4]. In atom interferometry (AI) high finesse op-tical cavities can improve the instrument sensitivity byallowing very high momentum transfer beamsplitters [5–9]. Cavities with long length L are instead sought forgravitational wave (GW) detection [10] to increase thestrain sensitivity proportionally to L . The combinationof high finesse F and long L has the effect of reducingits linewidth ∆ ν = c/ (2 nL F ), where c is the speed oflight in vacuum, and n the index of refraction inside thecavity. Despite the promise of increased sensor perfor-mance, it has been pointed out in [11] that a limitationexists for ∆ ν , beyond which the pulses used to coherentlymanipulate the atomic wavefunction undergo importantdeformation, and where the effective optical power en-hancement worsens. In short, the inherent frequency re-sponse of the cavity sets a physical limit to the product L F , and forbids adopting narrow linewidth resonatorsfor manipulating matter waves.In this Letter, we propose a novel scheme to coher-ently manipulate the atomic wavefunction in a narrowlinewidth cavity, where the interaction is pulsed not bychanging the intensity of the intracavity standing wave,but by modulating the coupling between the intracav-ity light and the atoms, using an auxiliary process. Thecavity enhanced laser is always injected in the opticalresonator, hence its intensity is constant in time. Themain approach we analyze exploits light-shift engineer-ing of the atomic levels, a technique adopted in severalcontexts concerning cold atoms, e.g. to cancel the trap-ping light perturbation in optical lattice clocks [12], lasercool atoms to BEC [13], and precisely characterize thegeometry of an optical cavity [14].For the sake of clarity, we focus our study on the ex-ample of an AI-based gravimeter using Sr atoms drivenon the clock transition and vertically launched in free fall ∗ [email protected] to cross a horizontal cavity (see Fig. 1). Our schemecan easily be extended to other configurations relyingon cavity-enhanced light-pulses to manipulate the atomicstate. The coherent manipulations are performed at eachpassage of the atoms in the cavity, by pulses shorter thanthe transit interval in the Interferometric Beam (IB). Weconsider a 4 pulse sequence interferometer [17] based onthe double-diffraction scheme [18] and with a time sep-
DBIB M1 M2 atom mirrorA CB D | , ¯ hk i| , − ¯ hk i | , i | , ¯ hk i| , − ¯ hk i A | , i| , ¯ hk i| , − ¯ hk i z x FIG. 1. Schematic of the proposed experimental setup notto scale: the atomic ensemble, initially in the state | (cid:105) andmoving in the z direction, crosses the cavity-enhanced IB , andis split in the region A (see inset) in two paths with oppositehorizontal velocity ± v r . The two parts of the wavefunctionare horizontally reflected with a mirror pulse in the regionsB and C; in D their vertical velocity is inverted, and after asecond mirror pulse, again in C and B, they are recombined inA with a last split pulse. The two trajectories at the outputof the interferometer are shown in gray. The horizontal DB (yellow), not resonant with the cavity, is shone on the atomsand vertically follows their motion to have an optimal overlap.M1, M2: cavity mirrors; v r , DB and the atom labelling aredefined in the main text. a r X i v : . [ phy s i c s . a t o m - ph ] F e b S s S | i P s p P D S s s S s s S P s p P | i p P D s d D s d D
461 nmΓ gd =30.5 MHz 698 nmΓ ge =1.5 mHz 679 nm433 nm476 nm 2.6 µ m484 nm395 nm s d D λ (nm)400500600700 λ (nm)400500600700 DB DB IB
FIG. 2. Diagram with the relevant levels for Sr atoms. Thered arrow shows the IB resonant to the | (cid:105) → | (cid:105) transitionat 698 nm adopted for the coherent manipulation of matterwaves; the yellow arrows mark the DB used to shift the twolevels | (cid:105) and | (cid:105) . The action of the DB is considered whenvarying its wavelength over the range [380–740] nm, indicatedby the vertical bars referenced to the two levels | (cid:105) and | (cid:105) .The narrow red (blue) bands indicate the spectral intervalwhere the DB with parallel (perpendicular) polarization con-stitutes an effective switch for the coherent action of IB , bylight shifting in a differential fashion the clock levels | (cid:105) , | (cid:105) .The bands have been obtained as defined in Fig. 3. The levelstructure has been taken from [15, 16]. aration T − T − T ( T =0.25 s will be assumed later fornumerical evaluations). This geometry relies on a singlehorizontal IB thanks to a vertical reflection at the middleof the sequence, and other similar configurations couldbe considered [19]. At t =0 the atoms are in A (Fig. 1),in state | (cid:105) with a velocity ( v x , v z ) = (0 , − gT / g is the local acceleration of gravity. Here, they inter-act with a first splitting pulse. The atomic wavefunctionis divided in two components in state | (cid:105) with oppositehorizontal velocities v x = ± v r , where v r = (cid:126) k/m is therecoil velocity. At t = T /2 the atoms reach the apogeesof the trajectories at a height of − gT / IB .At t = T the atoms are again in the IB where a mirrorpulse reverses their respective horizontal velocity with-out changing their internal state. At t = 2 T the twoparts of the wavefunction cross in D with a vertical ve-locity v z = 3 gT /
2; they are vertically reflected by opticalmeans [20] and complete the second, symmetric half ofthe interferometric sequence. Note that during the freeevolution the atoms are always in the same internal state, thus cancelling many systematic effects, and that the res-onance condition for the IB is the same for all the pulses.The narrow linewidth cavity is locked to the linearlypolarized IB ; the beam intensity is thus increased and thespatial mode filtered. The intracavity enhanced intensityof IB is chosen to have a Rabi frequency Ω R of 2 π × τ s = π/ ( √ R ) and τ m = 2 τ s respectively [18] so we obtain τ s (cid:39) µ s. The pulsesdo not couple to spurious momentum states as long as τ m ω r (cid:29) ω r is the recoil frequency i.e. about 59kHz for Sr.As mentioned above, we focus on alkali–earth atoms,more specifically Sr, where IB is tuned on the narrowtransition at 698 nm defined by the levels | (cid:105) ≡ S and | (cid:105) ≡ P [21] (see Fig. 2), to implement the coher-ent manipulation scheme proposed in [22] and recentlydemonstrated in [23] for Sr. An additional
DressingBeam (DB) differentially shifts the levels | (cid:105) and | (cid:105) ,breaking the resonance condition for IB . Modulating theintensity of DB will allow to switch the resonance with IB on and off [24].For numerical application, we consider a narrowlinewidth cavity that can fit in a conventional laboratory:the cavity parameters are set to be L = 2 m and F =10 (∆ ν = 750 Hz). In this configuration, the fast amplitudemodulation of the IB to implement the interferometricpulses would generate strong pulse deformations in thecavity which is detrimental to the sought power enhance-ment [11].The Sr atoms are considered at very low tempera-ture, prepared in the | (cid:105) state and launched vertically,to reach point A. The spatial extension of the atomiccloud is assumed < µ m during all the duration ofthe interferometric sequence. This will require adoptingdelta-kick collimation techniques [25, 26] to prepare theatomic source. The cavity waist is set to be 1 mm, so as toobtain a rather homogeneous manipulation of the atomicensemble on its axis, even when taking into account thevertical displacement of the cloud during the manipula-tion. To obtain the required Ω R , the intracavity powerof the IB must be P (cid:39)
286 mW [27], which means aninput power P in (cid:39) µ W, if two lossless mirrors withequal reflectivity are considered for the cavity.The atomic interaction with the IB is controlled withthe DB (in yellow in Fig. 1 and Fig. 2), whose role is toinduce an additional energy shift ∆ ω on the | (cid:105) → | (cid:105) transition, so as to remove the resonance condition for thecavity enhanced IB . This solution has also the technicaladvantage of avoiding to have to re-lock to the cavitythe laser generating the IB at each pulse. To calculate∆ ω when varying the DB wavelength over the range380 nm < λ <
740 nm we considered the relevant levelsshown in Fig. 2, and the transition parameters reportedin [15, 16].A single DB along the cavity axis (see Fig. 1) candress the atoms along both interferometric trajectoriesduring their passages in the IB ; the cavity mirrors mustbe transparent at the DB wavelength, to maintain a highbandwidth for the variation of the beam intensity, and toallow its vertical translation to track the atomic motionas described below. A bias magnetic field B is added inthe vertical direction to define the quantization axis. The DB is linearly polarized either parallel or perpendicularto B .The main unwanted side effect of the DB on the atomicsystem is the scattering of photons at a rate Γ sc on thetwo levels of interest, which represents a decoherencechannel. Other effects, like the DB wavefront aberra-tions, are not considered here; their impact, however, ishighly reduced in the differential configuration providedby a gravity-gradiometer. By dividing ∆ ω by Γ sc weobtain a normalized light shift Ξ( λ ), plotted in Fig. 3for a DB polarized along the magnetic field (continuouscurve) and orthogonal to it (dashed curve).To optimize the DB parameters, we start by arbitrarilyfixing the maximum probability to scatter a photon fromthe DB during the whole interferometric sequence to 3%,which means a subsequent reduction of the interferome-ter contrast of the same order. Considering the atomicvertical speed at each passage in the cavity ( − gT / ∼ T ) it means a maximum nominalscattering rate Γ sc ∼ DB [28].The second parameter to set is the minimum differen-tial light shift required to effectively suppress the Rabioscillation between states | (cid:105) and | (cid:105) . To this aim, thegeneralized Rabi frequency ˜Ω R = (cid:0) Ω R + ∆ ω (cid:1) / whenthe DB is on must be (cid:29) Ω R , and the rms uncertainty ofthe interferometric phase due to the residual Rabi oscil-lation is equal to: δφ = Ω R √ ω , if ∆ ω (cid:29) Ω R [29]. We set a threshold of 3 × − - i.e.the QPN of 10 atoms - for the overall phase uncertaintydue to the residual Rabi oscillation during the 4 atomicpassages in the cavity. Any coherent evolution other thanbetween states | (cid:105) and | (cid:105) (see Fig. 2) has been neglectedin this calculation. This assumption is valid whenever the DB is far from the specific transition frequencies.To simultaneously satisfy the requirements on the scat-tering rate and residual Rabi oscillation, one must have | Ξ( λ ) | > . × . In the visible this condition is sat-isfied for a linearly polarized DB along (perpendicularto) the bias magnetic field B for 633 nm < λ <
672 nm( λ >
679 nm), as shown by the colored bands in Fig.3. At λ = 672 nm, for example, a DB with a waist of100 µ m and power ∼
10 W determines a residual oscilla-tion amplitude below the threshold mentioned above fora scattering probability < -10123×10
400 450 500 550 600 650 700 P - s S S - P P - s S P - d D P - d D P - p P Ξ ( λ ) λ [nm] FIG. 3. Ratio Ξ( λ ) between the light shift induced on the | (cid:105)−| (cid:105) transition and the overall scattering rate by a laser ata wavelength λ . The solid (dashed) curve refers to the DB po-larization parallel (perpendicular) to the magnetic field. Theregions where Ξ( λ ) > . × are indicated with a red (blue)vertical band for parallel (perpendicular) polarization of the DB . The wavelengths of the relevant transitions contribut-ing to the atomic polarizability in the visible spectrum areindicated with vertical lines and labelled. the interval [1350 nm–2.5 µ m] the ratio | Ξ( λ ) | is com-patible with an instrument sensitivity below the QPN of10 atoms with a 5% contrast reduction, and even betterparameters are obtained at CO laser wavelengths. Nev-ertheless, the required laser power for these wavelengthsis in the kW range.In the configuration studied previously, the interactionbetween the atoms and the IB is turned off when the DB is on. A trade-off must be found between having alarge DB power to effectively switch off the IB, and min-imizing the residual scattering rate it causes. Anotherscheme, which is not analyzed in details in this publica-tion, consists in using the DB to turn on the interaction.Photon scattering is strongly reduced because the DB isonly on during the coherent manipulation pulses. As aconsequence, the DB can be set closer to a transitionbetween | (cid:105) and an excited level. This has three advan-tages: (i) lowers the DB power; (ii) adds the IB detuningas a parameter to reduce even further the residual Rabioscillation; (iii) makes CB unnecessary, with a suitablechoice of the DB ’s wavelength and intensity. The priceto pay is that the control of the coherent manipulationnow depends not only on the stability of the IB , but alsoon the stability of the DB .Two other effects of the cavity can affect the coherentmanipulation. First, the intracavity IB light intensitycan decay during the time the DB is turned off, becauseof the modified effective atomic index of refraction thatshifts the cavity resonance [30]. The cavity narrow band-width prevents, however, the intracavity field to evolvesignificantly during the duration of the light pulses, whichis much shorter than the cavity response time. Second,the atomic absorption can spoil the cavity linewidth [31];again, for the adopted parameters, namely the numberof atoms and the threshold set on the allowed scatteringrate, the effect has been evaluated to be negligible.We now focus on the specific case of a cavity-aidedgravity-gradiometer for GW detection which motivatedthis proposal [11]. An instrument with a long baselinelength L and a 4 pulse sequence gives a phase sensitivity δφ ≈ kLh + sin (2 πf T ) sin (cid:18) πf T (cid:19) , to a plus-polarized GW of frequency f [10, 22]. Consid-ering L =10 km (i.e. the design value for the EinsteinTelescope [32]), and a finesse F =100 (i.e lower than thesystem design Finesse of aLIGO, which is 450 [33]) oneobtains ∆ ν =150 Hz. Such value excludes the possibil-ity to realize interferometric pulses shorter than 1 msby varying the intensity of the IB injected in the cavity.Adding a DB to design the pulses removes the limitationon the minimum pulse length.With our parameters and for T=0.25 s, the peak strainsensitivity is h + ∼ . × − / √ Hz at 2 Hz for a shotnoise limited detection of 10 atoms per second. Im-proved strain sensitivities can be obtained by adoptinga higher atomic flux, and exploiting the cavity to im-plement sub-shot-noise sensitivity [31, 34] and large mo-mentum splitting [35–37]. The latter can be achieved byinserting several π pulses in the sequence as describedin [18], and using the amplitude of the DB to maintainthe Doppler shift compensation. At the same time back-ground noise signals, arising from the residual phase noiseinduced by the out-of-resonance IB , must be proportion-ally reduced, exploiting the common mode rejection ra-tio of the gradiometer or improving the frequency andamplitude stability of the DB . The sensitivity curve canbe shifted at lower frequency by increasing the atomicinterrogation time, which requires to adapt accordinglythe specifications of the atom mirror pulse [19].We have proposed a new coherent manipulationscheme to bypass the limitations of cavity linewidth incavity-aided AI. Our method enables fast and pulsed ma-nipulation of matter waves with the intracavity resonant light without any restrictions on cavity length and fi-nesse. The scheme described here relies on light-shift en-gineering to control the atomic coupling on a narrow opti-cal transition to the light stored in the cavity. It could beextended to manipulation schemes with freely-falling ortrapped atoms [35–37], or relying on moderately narrowtransitions with relatively higher single-photon Rabi fre-quency [38]. Other control processes could be adopted,such as magnetic field induced spectroscopy [39], threephoton resonance [40], DC Stark effect [41]; notably, theycould introduce a more homogeneous control of the coher-ent switching, and a mitigation of the related aberrationissue.This method opens perspectives to push the use atomiccavities in long baseline atom interferometers, such asproposed for GW detection, and to exploit high finesse(narrow linewidth) cavities to improve the spatial filter-ing of the coherent manipulation beams [11]. This can beused for shorter pulses, large momentum transfer atomoptics, and may even lead to universal AI [42]. 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