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A Spin Optodynamics Analogue of Cavity Optomechanics A spin optodynamics analogue of cavity optomechanics N. Brahms1 and D.M. Stamper-Kurn1;2¤ 1Department of Physics, University of California, Berkeley CA 94720, USA 2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA (Dated: July 28, 2010) The dynamics of a large quantum spin coupled parametrically to an optical resonator is treated in analogy with the motion of a cantilever in cavity optomechanics. New spin optodynamic phenomena are predicted, such as cavity-spin bistability, optodynamic spin-precession frequency shifts, coherent ampli¯cation and damping of spin, and the spin optodynamic squeezing of light. Cavity optomechanical systems are currently being ex- Hin=out describes the coupling of the cavity ¯eld to exter- plored with the goal of measuring and controlling me- nal light modes. Under this Hamiltonian, the cantilever chanical objects at the quantum limit, using interactions positionz ^ and momentump ^ evolve as dz=dt^ =p=m ^ and 2 with light [1]. In such systems, the position of a mechan- dp=dt^ = ¡m!z z^ + fn^. ical oscillator is coupled parametrically to the frequency To construct a spin analogue of this system, we con- of cavity photons. A wealth of phenomena result, in- sider a Fabry-Perot cavity with its axis along k (Fig. 1). cluding quantum-limited measurements [2], mechanical For the collective spin, we ¯rst consider a gas of N hydro- response to photon shot noise [3], cavity cooling [4], and genlike atoms in a single hyper¯ne manifold of their elec- ponderomotive optical squeezing [5]. tronic ground state, each with dimensionless spin s and At the same time, spins and psuedospins coupled to gyromagnetic ratio ¹. The atoms are optically con¯ned electromagnetic cavities are being researched in atomic at an antinode of the cavity ¯eld. An external magnetic [Haroche], ionic [6], and nanofabricated systems [7, 8], ¯eld B = Bb is applied to the atoms. The detuning with applications including magnetometry [], precision ¢ca between the cavity resonance is chosen to be very frequency standards [], and quantum information pro- large compared to both the natural linewidth and the cessing [8][Haroche, Kimble]. In contrast to mechanical hyper¯ne splitting of the atoms' excited state. In this objects, spin systems can more easily disconnected from limit, spontaneous emission may be ignored. Instead, their environment [] and prepared in quantum states [], the atom-light interaction can be described entirely by including squeezed states [9]. the AC Stark shift, composed of a scalar and a vector In this Letter, we seek to link these two ¯elds, by ex- term, with interaction energy [11] ploiting the similarities between large-spin systems and 2 ~g0 harmonic oscillators [10]. Here we show how a cavity spin HStark = n^ (1 § υk ¢ ^s) ; (2) optodynamics system can be constructed in analogy to ¢ca cavity optomechanics. The phenomena of cavity optome- where g0 is the atom-cavity coupling parameter and υ chanics map directly to our proposed system, resulting in describes the strength of the vector Stark shift, which is spin cooling and ampli¯cation, nonlinear spin sensitivity positive for σ+-polarized light and negative for σ¡ light. and spin-cavity bistability, and spin optodynamic squeez- Summing over all atoms q, we obtain the system ing of light. We show how this system can be applied as Hamiltonian: a quantum-limited spin ampli¯er or as a latching spin detector. We detail the e®ects of these phenomena using H = ~!c (^n+ +n ^¡) + Hin=out+ currently accessible parameters, and we propose experi- µ ¶ X ~g2 mental realizations constructed either using cold atoms ¡¹B ¢ ^s + 0 [(^n +n ^ ) + υ (^n ¡ n^ ) k ¢ ^s ] ; q ¢ + ¡ + ¡ q and visible light or using cryogenic solid state systems q ca and microwaves. (3) An ideal cavity optomechanics system consists of a har- § monic oscillator, coupled linearly to a single-mode cavity with photon number operatorsn ^§ for the σ modes. ¯eld. Its Hamiltonian is The above Hamiltonian can be rewrittenP as the inter- ^ action of the collective spin operator S ´ q ^sq with an y ¡ y ¢ H = ~!cn^ + ~!za^ a^ ¡ fzHO a^ +a ^ n^ + Hin=out: (1) e®ective total magnetic ¯eld Be® ´ (~=¹)­e®, giving [12] Herea ^ is the oscillator's phonon annihilation operator, ­e® = ­Lb + ­c (^n+ ¡ n^¡) k: (4) n^ is the photon number operator, !z is the natural fre- 2 Here ­L = ¹B=~ and ­c = ¡υg0=¢ca. Altogether, the quency of the oscillator in the dark, and !c is the bare cavity spin optodynamical Hamiltonian is cavity resonance frequency. f is the radiation-pressurep µ 2 ¶ force applied by a single photon, while zHO = ~=2m!z Ng0 ^ H = ~ !c + (^n+ +n ^¡)+Hin=out ¡~­e® ¢S: (5) is the harmonic oscillator length for oscillator mass m. ¢ca 2 S (b) k 15 Ωc( n+-n-) S n+ Ω i L (a) π 0 15 θ0 n - σ+ σ - -π 0 FIG. 1: (Color) An ensemble of atoms trapped within a driven -10 -5 0 5 10 -10 -5 0 5 10 optical resonator experiences an externally imposed magnetic ∆p/κ ∆p/κ ¯eld along i and a light-induced e®ective magnetic ¯eld along the cavity axis k. The evolution of the collective spin S^ re- FIG. 2: (Color) In a cavity spin optodynamic system driven sembles that of a cantilever in cavity optomechanics. with linearly polarized light, several stable spin orientations and light intensities may be reached. Using parameters similar to those of existing experiments citePurdy2010, we 87 Now consider the external magnetic ¯eld to be static consider an ensemble of N = 5000 spin-2 Rb, ­c=· = ¡3 ¡2 and oriented along i, orthogonal to the cavity axis. In 1:25 £ 10 , ­L=· = 3:3 £ 10 , andn ¹max;§ = 15. (a) As ¢p the limit hS^i ' Si, the spin dynamics become is varied, several stable (black) and unstable (gray) static spin con¯gurations are found. Con¯gurations for ¢p=· = ¡4:8 are depicted. (b) The cavity exhibits hysteresis as the probe is dS^j dS^k = ­LS^k ¡ ­cS (^n+ ¡ n^¡) ; = ¡­LS^j: (6) swept with positive (dashed blue) or negative (red) frequency dt dt chirps, with the spin initially along i. Rapid transitions as The analogy between cavity optomechanics and spin ¢p=· is swept upward from -2.8 or downward from 0 involve optodynamics is established by assigningz ^ ! symmetry breaking as the cavity becomes birefringent; we dis- playn ¹+ andn ¹¡ assuming the stable branch closer to θ0 = 0 ¡zHOS^k=¢SSQL andp ^ ! pHOS^j=¢SSQL, where zHO is selected. Here, ¢ca=2¼ = 20 GHz from the D2 transition, and p = ~=(2z ) are de¯ned with ! ! ­ HO HOp z L g0=2¼ = 15 MHz, ·=2¼ = 1:5 MHz. [10] and ¢SSQL = S=2 is the standard quantum limit for transverse spin fluctuations. Eqs. (6) now match the optomechanical equations of motion with parallel to ­ . Writing S = S(i sin θ + k cos θ ), this the optomechanical coupling de¯ned through fz n^ ! e® 0 0 HO condition requiresn ¹ ¡ n¹ = (­ =­ ) cot θ . The intra- ¡~­ ¢S (^n ¡n^ ). + ¡ L c 0 c SQL + ¡ cavity photon numbers are determined also by the stan- The main result of this work, that various cavity op- dard expression for a driven cavity of half line-width ·, tomechanical phenomena are manifest also in cavity spin i.e.n ¹ =n ¹ [1 + (¢ § ­ S cos θ )2 =·2]¡1 with optodynamical systems, is immediately established. Dy- § max;§ p;§ c 0 ! = (! + Ng2=¢ ) + ¢ being the frequency of namical backaction in an optically driven cavity will re- § c 0 ca p;§ laser light of polarization σ§ driving the cavity and sult in Larmor precession frequency shifts akin to the op- n¹ characterizing its power. These two expressions tomechanical frequency shift [13, 14], and also to coher- max;§ forn ¹ ¡ n¹ may admit several solutions (Fig. 2). ent ampli¯cation and damping of spin precession similar + ¡ to the cavity optical ampli¯cation and cooling of can- As typical in instances of cavity bistability [20], sev- tilevers [15, 16]. Cavity nonlinearity and optomechanical eral of the static solutions for the intracavity intensities bistability [17, 18] imply static multistable collective-spin may be unstable. To identify such instabilities, we con- con¯gurations in a driven cavity. The ponderomotive sider the torque on the collective spin when it is dis- squeezing of light due to the cantilever's response to ra- placed slightly toward +k from its static orientation. diation pressure fluctuations [15, 19], a quantum optical Stable dynamics result when such displacement yields a e®ect of cavity nonlinearity, implies that similar inhomo- torque N ¢ j with the sign ® = sgn(sin θ0). Geometri- geneous squeezing may be generated by the response of cally, this stability requires that the spin vector be dis- intracavity spins to quantum noise. placed further in the +k direction than the vector ®­e®. Let us now elaborate on these phenomena. To obtain Quantifying the linear response of the intracavity e®ec- general results, we will proceed without assuming S^ ' Si, tive magnetic ¯eld to variations of the collective spin via except in certain cases, noted in the text, where some ¸ = ­cd(¹n+ ¡ n¹¡)=dSk, the static spin orientations are 3 physical insight is gained. We begin with e®ects for which found to be unstable when ®¸ > ­Lj csc θ0j=S. both the light ¯eld and the ensemble spin may be treated Opto-dynamical Larmor frequency shift: We classically, i.e. by letting S = hS^i andn ¹§ = hn^§i. now consider the dynamics of the system precessing Cavity-spin bistability: We start with the static about one of the stable con¯gurations. These dynam- behavior of the system by ¯nding the ¯xed points of the ics can be parameterized by the precession frequency, system. The collective spin vector is static when S is which is shifted from ­L by two e®ects: First, there is 3 an upward frequency shift from the static modi¯cation light drives spins to the high- or low-energy spin state of the e®ective magnetic ¯eld, leading to precession at according to the detuning of probe light from the cavity 0 the frequency ­L = ­Lj csc θ0j when ¸ = 0.
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