Magnetically-Induced Optical Transparency on a Forbidden Transition in Strontium for Cavity-Enhanced Spectroscopy
Matthew N. Winchester, Matthew A. Norcia, Julia R.K. Cline, James K. Thompson
MMagnetically-Induced Optical Transparency on a Forbidden Transition in Strontiumfor Cavity-Enhanced Spectroscopy
Matthew N. Winchester, Matthew A. Norcia, Julia R.K. Cline, and James K. Thompson
JILA, NIST, and Dept. of Physics, University of Colorado, 440 UCB, Boulder, CO 80309, USA (Dated: December 8, 2016)In this work we realize a narrow spectroscopic feature using a technique that we refer to asmagnetically-induced optical transparency. A cold ensemble of Sr atoms interacts with a singlemode of a high-finesse optical cavity via the 7.5 kHz linewidth, spin forbidden S to P transition.By applying a magnetic field that shifts two excited state Zeeman levels, we open a transmissionwindow through the cavity where the collective vacuum Rabi splitting due to a single level wouldcreate destructive interference for probe transmission. The spectroscopic feature approaches theatomic transition linewidth, which is much narrower than the cavity linewidth, and is highly immuneto the reference cavity length fluctuations that limit current state-of-the-art laser frequency stability. There has been a dedicated effort in recent years to im-prove the frequency stability of lasers [1] used to probeoptical atomic clocks [2–4]. Improvements in these preci-sion measurement technologies are essential for advanc-ing a broad range of scientific pursuits such as searchingfor variations in fundamental constants [5] , gravitationalwave detection [6, 7], and physics beyond the standardmodel [8, 9]. Associated improvements in atomic clockswould also advance recent work on relativistic geodesy[10].The frequency stability of current state-of-the-artlasers is limited by thermal fluctuations in the refer-ence cavity mirror coatings, substrates, and spacer [11].This problem can be alleviated by creating systems thatrely on an ensemble of atoms, rather than the referencecavity, to achieve stable optical coherence. Recent ap-proaches include cavity-assisted non-linear spectroscopy[12–14] and superradiant lasers [15–18]. Both approachesuse narrow forbidden transitions with linewidths rangingfrom 7.5 kHz to 1 mHz. These novel systems are absolutefrequency references and are intrinsically less sensitive toboth fundamental thermal and technical vibrations thatcreate noise on the optical cavity’s resonance frequency.Here we demonstrate a new linear spectroscopy ap-proach in which a static magnetic field can induce opticaltransparency in the transmission spectrum of an opticalcavity. The center frequency of the transparency win-dow is shown to be insensitive to changes in the cavity-resonance frequency and to first-order Zeeman shifts.The observed linewidth of the feature approaches thenatural linewidth of the 7.5 kHz optical transition andcan be insensitive to inhomogeneous broadening of theatomic transition frequency. The linewidth of the fea-ture is an important attribute for laser stabilization, as alaser stabilized to a narrow spectroscopic feature is lesssensitive to technical offsets than a laser stabilized to abroader feature. In the future, it might be possible to ex-tend this technique to even narrower optical transitionsfor enhanced spectroscopic sensitivity in atoms such ascalcium and magnesium.In partial analogy to electromagnetically induced transparency (EIT) [19–21], we refer to this effect asmagnetically induced transparency (MIT). In EIT, a con-trol laser is used to create a variable-width transparencywindow for slowing light [22], for stopping light [23], forquantum memories [24], and even for creating effectivephoton-photon interactions [25–27]. It might be possibleto utilize controlled magnetic fields and long-lived opticalstates to realize similar goals.In our experiment we create a strongly coupled atom-cavity system by loading up to N = 1 . × Sr atomsinto a 1D optical lattice supported by a high-finesse op-tical cavity. The peak trap depth is 100(10) µ K andthe atoms are laser-cooled to 10(1) µ K (see Fig. 1a andRefs. [16, 17, 28]). We tune a TEM00 resonance ofthe cavity at frequency ω c to be near resonance withthe dipole-forbidden singlet to triplet optical transition S to P at frequency ω or wavelength 689 nm (seeFig. 1b). The excited state P spontaneously decaysback to the ground state at a rate γ = 2 π × . κ = 2 π × . Sr, the absence of nuclear spin means that the S ground state is unique, while the P excited state hasthree Zeeman sublevels.In the limit of zero applied magnetic field, our sys-tem responds to an applied probe as though each atomwere a simple two-level system. The ensemble can col-lectively absorb light from and then collectively reemitlight into the cavity mode at the so-called collective vac-uum Rabi frequency Ω = √ N g . Here 2 g = 15 kHz isthe rms value of the single-atom vacuum Rabi frequency,which accounts for averaging over the standing-wave cav-ity mode. This exchange creates two new transmissionmodes that are shifted away from the empty cavity’stransmission peak by ± Ω /
2, as shown by the central redtrace in Fig. 2a.Because two orthogonal components of the probe lightcan couple to two distinct Zeeman sublevels, the coupledatom-cavity system has three normal modes of excitation,not two. In addition to the two modes at ± Ω / a r X i v : . [ phy s i c s . a t o m - ph ] D ec frequency is equal to that of the atomic transition. Thedark mode is composed of an equal superposition of thetwo atomic excited states and a photonic component thatvanishes as the magnetic field approaches zero. FIG. 1. (a) Simplified experimental diagram. The quantiza-tion axis ˆ q is along the vertical magnetic field (cid:126)B . Horizontallypolarized probe light with amplitude c i is a linear combinationof left and right circularly polarized light. The light is trans-mitted through the input and output mirrors with character-istic rates κ and κ giving a cavity linewidth κ = κ + κ . Thelight can be coherently absorbed by the atoms (brown ovals)and reemitted into the cavity (with field amplitude c ) at thevacuum Rabi frequency Ω. The atomic transition decays intofree space at rate γ/ π = 7 . c t is detected on a single photon counting module (SPCM). (b)Corresponding energy level diagram. The applied magneticfield creates a Zeeman splitting ∆ between the two states P , m j = ± ω , the empty cavity resonance frequency ω c ,and the detunings of the probe and cavity frequencies δ p and δ c relative to the atomic frequency ω . The m j = 0 state isshown, but does not interact with the horizontally polarizedcavity-field. By applying a magnetic field B , we can mix photoniccharacter into the dark mode, inducing transmission (alsorefered to as transparency) for probe light near ω . Theprobe light is horizontally polarized and is perpendicularto the vertically oriented magnetic field. Each trace inFig. 2a corresponds to a different applied magnetic fieldwith strength parameterized by the induced Zeeman fre-quency splitting ∆ / π = (2 . B .To describe the system, we extend the linearized input- FIG. 2. (a) The transmitted power through the cavity versusthe probe detuning δ p , with δ c = 0. Each trace was takenfor different applied magnetic fields, creating different Zee-man splittings ∆ labeled on the vertical for each trace. Thecentral red trace is taken for ∆ = 0 and displays a collectivevacuum Rabi splitting Ω / π = 5 MHz. When a magnetic fieldis applied perpendicular to the probe polarization, inducinga Zeeman splitting ∆, a new transmission feature appears inbetween the two original resonances of the vacuum Rabi split-ting. (b) Linearized theory showing the power P T and phase ψ of the transmitted light, plotted here for Ω / π = 5 MHzand ∆ / π = 1 MHz. output equations of [29] to include an additional atomictransition written in a rotating frame at the averageatomic transition frequency ω as:˙ a = −
12 ( γ + ı ∆) a − ı √ c (1)˙ b = −
12 ( γ − ı ∆) b − ı √ c (2)˙ c = −
12 ( κ + ı δ c ) c − ı √ a + b ) + √ κ c i e ıδ p t . (3)Here, δ c = ω c − ω is the detuning of the cavity reso-nance frequency ω c from atomic resonance, Ω is the ob-served collective vacuum Rabi splitting when ∆ = 0, γ is the decay rate of the excited atomic states, κ is thecavity power decay rate, and κ is the coupling of theinput cavity mirror that is driven by an externally in-cident probe field with complex amplitude c i and at aprobe frequency ω p and detuning from atomic resonance δ p = ω p − ω . The complex variables a = (cid:104) ˆ a (cid:105) , b = (cid:104) ˆ b (cid:105) , c = (cid:104) ˆ c (cid:105) are expectation values of bosonic lowering oper-ators describing the cavity c , and collective excitationsof the two atomic transitions a and b . The requiredHolstein-Primakoff approximation assumes weak excita-tion such that the number of atoms in the excited states M a = | a | , M b = | b | (cid:28) N is a small fraction of thetotal atom number N . The average number of photonsin the cavity is given by M c = | c | , and the complex fieldtransmitted through the cavity is c t = √ κ c with κ thecoupling of the output mirror. The transmitted probepower relative to incident probe power is P T = | c t /c i | and the relative phase is ψ = arg ( c t /c i ).Figure 2b shows the calculated steady-state transmit-ted power and phase for a single Zeeman splitting. Thephase response changes rapidly near zero probe detun-ing, which results in a narrow MIT resonance comparedto the broad vacuum Rabi splitting or bright modes forwhich the phase changes more slowly.In order to describe the linewidth of the dark stateresonance, we introduce a mixing angle θ defined bysin θ = ¯∆ / (Ω + ¯∆ ). Here, the effective detuning is¯∆ = ∆ + γ . The character of the dark state excitationis given by the ratio of the probability that the excitationis photonic-like P c = M c / ( M c + M a + M b ) = sin θ versusatomic-like P ab = ( M a + M b ) / ( M c + M a + M b ) = cos θ .The dark state excitation can decay into free space atrate R ab or by emission through the cavity mirrors R c ,with the ratio of the rates given simply by R ab /R c = γ/ ( κ tan θ ) = N C ( γ/ ¯∆) , where the single particle co-operativity parameter is C = 4 g /κγ .The linewidth of the dark state resonance can be writ-ten as: κ (cid:48) = ( γ cos θ + κ sin θ ) /b. (4)The term in parentheses is a weighted average of atomand cavity linewidths that reflects the character of themode. The correction factor is b = d cos θ +sin θ , where d = (∆ − γ ) / ¯∆ . When ∆ (cid:29) γ , both b and d ap-proach unity. At small detunings ∆ ∼ γ , the responsesof the dark and bright modes to the applied drive be-come comparable, causing a modification of the correc-tion factor. In the regime experimentally explored here( b ≈ κ (cid:48) is simply the full width at half maximumlinewidth of the power transmission feature. To definea linewidth valid in general, we define the linewidth via κ (cid:48) = 2 (d ψ/ d δ p ) − | δ c = δ p =0 . For Ω (cid:29) ∆ (cid:29) γ , the mixingangle is small and the linewidth approaches the atomiclinewidth κ (cid:48) ≈ γ , which can be much narrower than thecavity linewidth κ .We measure the linewidth for the central dark reso-nance by linearly sweeping the probe laser’s frequencyover the cavity resonance and recording a time-trace of power transmitted on a single photon counting module.A Lorentzian is fit to the central feature to extract thefull width at half maximum. This measurement is takenfor a range of different Zeeman splittings and vacuumRabi splittings by varying the applied dc magnetic fieldand atom number respectively. Figure 3a shows collecteddata plotted against the theoretical prediction at severalRabi frequencies. For very small ∆ the feature becomesincreasingly narrow, approaching the atomic transitionlinewidth γ = 2 π × . κ . (a)(b) FIG. 3. (a) The measured linewidth of the central MIT trans-mission feature versus the induced Zeeman splitting betweenexcited states. The traces are taken for three different col-lective vacuum Rabi frequencies Ω / π = 4.6(5) (red), 10(1)(blue), and 16(1) (green) MHz, with values set by changingthe total atom number N . The upper dashed line is the emptycavity’s linewidth κ , and the lower dashed line is the atomictransition’s linewidth γ . The minimum observed linewidthwas 11 kHz. The shaded regions are no-free parameter pre-dictions from the linearized model introduced in the text, in-dicating the ± ± Figure 3b shows the peak transmitted power at theMIT feature’s resonance for the same data show inFig. 3a. The linearized theory predicts that the peaktransmitted power is given by: P max = 4 κ κ κ (cid:0) γκ tan θ (cid:1) . (5)Note that the term in the denominator above is just theratio of excitation decay rates R ab /R c . For large detun-ings ∆ (cid:29) Ω , γ , the peak transmission goes to that of anempty cavity P max → P empty = 4 κ κ /κ .In the regime experimentally explored here, a changein the cavity resonance frequency ω c by ∆ ω c leads toa change in the dark state resonance frequency ω D bya much smaller amount ∆ ω D . The pulling coefficient P = ∆ ω D / ∆ ω c expresses this ratio. A small pulling co-efficient P (cid:28) P = sin θb . (6)In the typical regime of operation ( b ≈ ω D with a Lorentzian fit model.This is then repeated while toggling ω c between two val-ues separated by 100 kHz, and the pulling coefficient isdetermined from the change in ω D versus ω c . Our lowestmeasured pulling coefficients are below P = 0 . P = 1 / (1 + N/ (8 M c )) at ∆ = √ γ , where M c = ( γ/ (2 g )) is the so-called critical photon number.The critical photon number is proportional to the cavitymode volume and atomic linewidth, but does not dependon the mirror reflectivity. As a result, small pulling coef-ficients are reached by working with small cavity volumesand very narrow linewidth transitions. For spectroscopicapplications, one would want to balance the desire for alow pulling coefficient against the need to collect trans-mitted photons without inducing heating in the atomicensemble due to free-space scattering. The optimal pa-rameter regime will depend on the specific requirementsof the system. Zeeman Splitting ∆ / π [MHz] P u lli n g C o e ff i c i e n t P FIG. 4. The pulling coefficient P versus Zeeman splitting ∆for several collective vacuum Rabi frequencies Ω / π = 5(1)(red), 10(1) (blue), and 17(1) (green) MHz. The predictionfrom the linearized theory is shown with ± While the majority of this work has been done withthe atoms trapped in the Lamb-Dicke regime (i.e. con-fined to much less than the wavelength of the probe light)with respect to the cavity axis, we have also performedscans of the cavity transmission spectrum in which theatoms were unconfined along the cavity axis. In this con-figuration, the rms Doppler shift along the cavity axis isroughly 45 kHz. Despite this inhomogeneous broadening,we observe a center feature linewidth of 18.5 kHz, whichwe believe is limited by technical noise on the cavity fre-quency that arises when we turn down the lattice depthto release the atoms. We expect the linewidth of the darkfeature to be insensitive to inhomogeneous broadening solong as ∆ is much larger than the inhomogeneous broad-ening [30]. This insensitivity to Doppler broadening maymake such techniques suitable to continuously operatingatomic beam experiments, where confining the atoms tothe Lamb-Dicke regime would be challenging.To summarize, we have demonstrated a technique torealize a narrow spectroscopic feature based on collectiveinteraction between an ensemble of atoms and a high fi-nesse optical cavity. The center frequency of the featurecan be made highly insensitive to changes in cavity reso-nance frequency. In analogy to EIT, this technique mayalso be applicable to tasks relevant for information pro-cessing, especially if applied to an optical transition witheven narrower linewidth.We acknowledge contributions to the experimental ap-paratus by Karl Mayer. All authors acknowledge finan-cial support from DARPA QuASAR, ARO, NSF PFC,and NIST. J.R.K.C. acknowledges financial support fromNSF GRFP. This work is supported by the National Sci-ence Foundation under Grant Number 1125844. [1] T. Kessler, C. Hagemann, C. Grebing, T. Legero,U. Sterr, F. Riehle, M. Martin, L. Chen, and J. Ye,Nature Photonics , 687 (2012).[2] B. Bloom, T. Nicholson, J. Williams, S. Campbell,M. Bishof, X. Zhang, W. Zhang, S. Bromley, and J. Ye,Nature , 71 (2014).[3] N. Hinkley, J. Sherman, N. Phillips, M. Schioppo,N. Lemke, K. Beloy, M. Pizzocaro, C. Oates, and A. Lud-low, Science , 1215 (2013).[4] I. Ushijima, M. Takamoto, M. Das, T. Ohkubo, andH. Katori, Nature Photonics , 185 (2015).[5] T. Rosenband, D. Hume, P. Schmidt, C. Chou, A. Br-usch, L. Lorini, W. Oskay, R. Drullinger, T. Fortier,J. Stalnaker, et al. , Science , 1808 (2008).[6] S. Kolkowitz, I. Pikovski, N. Langellier, M. D.Lukin, R. L. Walsworth, and J. Ye, arXiv preprintarXiv:1606.01859 (2016).[7] P. W. Graham, J. M. Hogan, M. A. Kasevich, and S. Ra-jendran, Physical review letters , 171102 (2013).[8] A. Derevianko and M. Pospelov, Nature Physics , 933(2014).[9] S. Blatt, A. Ludlow, G. Campbell, J. W. Thomsen,T. Zelevinsky, M. Boyd, J. Ye, X. Baillard, M. Fouch´e,R. Le Targat, et al. , Physical Review Letters , 140801(2008).[10] C.-W. Chou, D. Hume, T. Rosenband, and D. Wineland,Science , 1630 (2010).[11] K. Numata, A. Kemery, and J. Camp, Phys. Rev. Lett. , 250602 (2004).[12] M. J. Martin, D. Meiser, J. W. Thomsen, J. Ye, andM. J. Holland, Phys. Rev. A , 063813 (2011).[13] P. G. Westergaard, B. T. R. Christensen, D. Tieri,R. Matin, J. Cooper, M. Holland, J. Ye, and J. W.Thomsen, Phys. Rev. Lett. , 093002 (2015).[14] B. T. R. Christensen, M. R. Henriksen, S. A. Sch¨affer, P. G. Westergaard, D. Tieri, J. Ye, M. J. Holland, andJ. W. Thomsen, Phys. Rev. A , 053820 (2015).[15] J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J.Holland, and J. K. Thompson, Nature , 78 (2012).[16] M. A. Norcia and J. K. Thompson, Phys. Rev. X ,011025 (2016).[17] M. A. Norcia, M. N. Winchester, J. R. K. Cline, andJ. K. Thompson, Science Advances , 10 (2016).[18] D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland,Phys. Rev. Lett. , 163601 (2009).[19] K.-J. Boller, A. Imamo˘glu, and S. E. Harris, PhysicalReview Letters , 2593 (1991).[20] M. M¨ucke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr,S. Ritter, C. J. Villas-Boas, and G. Rempe, Nature ,755 (2010).[21] G. Hernandez, J. Zhang, and Y. Zhu, Physical ReviewA , 053814 (2007).[22] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi,Nature , 594 (1999).[23] C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature , 490 (2001).[24] R. Zhao, Y. Dudin, S. Jenkins, C. Campbell, D. Mat-sukevich, T. Kennedy, and A. Kuzmich, Nature Physics , 100 (2009).[25] T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth,A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuleti´c,Nature , 57 (2012).[26] Y. Dudin and A. Kuzmich, Science , 887 (2012).[27] J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine,A. Sommer, and J. Simon, Physical Review A , 041802(2016).[28] M. A. Norcia and J. K. Thompson, Phys. Rev. A ,023804 (2016).[29] Z. Chen, J. G. Bohnet, J. M. Weiner, K. C. Cox, andJ. K. Thompson, Phys. Rev. A , 043837 (2014).[30] R. Houdr´e, R. P. Stanley, and M. Ilegems, Phys. Rev. A53