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 ampliﬁcation 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 field 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 ˆ and momentump ˆ evolve as dz/dtˆ =p/m ˆ and 2 with light [1]. In such systems, the position of a mechan- dp/dtˆ = −mωz zˆ + fnˆ. 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 first consider a gas of N hydro- response to photon shot noise [3], cavity cooling [4], and genlike atoms in a single hyperfine 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 µ. The atoms are optically confined electromagnetic cavities are being researched in atomic at an antinode of the cavity field. An external magnetic [Haroche], ionic [6], and nanofabricated systems [7, 8], field 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 hyperfine 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 fields, 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 = nˆ (1 ± υk · ˆs) , (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 amplification, 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 amplifier or as a latching spin detector. We detail the effects of these phenomena using H = ~ωc (ˆn+ +n ˆ−) + Hin/out+ currently accessible parameters, and we propose experi- µ ¶ X ~g2 mental realizations constructed either using cold atoms −µB · ˆs + 0 [(ˆn +n ˆ ) + υ (ˆn − nˆ ) k · ˆs ] , 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 ˆ± for the σ modes. field. Its Hamiltonian is The above Hamiltonian can be rewrittenP as the inter- ˆ action of the collective spin operator S ≡ q ˆsq with an † ¡ † ¢ H = ~ωcnˆ + ~ωzaˆ aˆ − fzHO aˆ +a ˆ nˆ + Hin/out. (1) effective total magnetic field Beff ≡ (~/µ)Ωeff, giving [12]

Herea ˆ is the oscillator’s phonon annihilation operator, Ωeﬀ = ΩLb + Ωc (ˆn+ − nˆ−) k. (4) nˆ is the photon number operator, ωz is the natural fre- 2 Here ΩL = µB/~ and Ωc = −υg0/∆ca. Altogether, the quency of the oscillator in the dark, and ωc 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 = ~/2mωz Ng0 ˆ H = ~ ωc + (ˆn+ +n ˆ−)+Hin/out −~Ωeﬀ ·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

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 field along i and a light-induced effective magnetic field along the cavity axis k. The evolution of the collective spin Sˆ 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 field to be static consider an ensemble of N = 5000 spin-2 Rb, Ωc/κ = −3 −2 and oriented along i, orthogonal to the cavity axis. In 1.25 × 10 ,ΩL/κ = 3.3 × 10 , andn ¯max,± = 15. (a) As ∆p the limit hSî ' Si, the spin dynamics become is varied, several stable (black) and unstable (gray) static spin configurations are found. Configurations for ∆p/κ = −4.8 are depicted. (b) The cavity exhibits hysteresis as the probe is dSˆj dSˆk = ΩLSˆk − ΩcS (ˆn+ − nˆ−) , = −ΩLSˆj. (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/κ is swept upward from -2.8 or downward from 0 involve optodynamics is established by assigningz ˆ → symmetry breaking as the cavity becomes birefringent; we dis- playn ¯+ andn ¯− assuming the stable branch closer to θ0 = 0 −zHOSˆk/∆SSQL andp ˆ → pHOSˆj/∆SSQL, where zHO is selected. Here, ∆ca/2π = 20 GHz from the D2 transition, and p = ~/(2z ) are defined with ω → Ω HO HOp z L g0/2π = 15 MHz, κ/2π = 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 Ω . Writing S = S(i sin θ + k cos θ ), this the optomechanical coupling defined through fz nˆ → eff 0 0 HO condition requiresn ¯ − n¯ = (Ω /Ω ) cot θ . The intra- −~Ω ∆S (ˆn −nˆ ). + − 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 κ, tomechanical phenomena are manifest also in cavity spin i.e.n ¯ =n ¯ [1 + (∆ ± Ω S cos θ )2 /κ2]−1 with optodynamical systems, is immediately established. Dy- ± max,± p,± c 0 ω = (ω + 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- n¯ characterizing its power. These two expressions tomechanical frequency shift [13, 14], and also to coher- max,± forn ¯ − n¯ may admit several solutions (Fig. 2). ent amplification and damping of spin precession similar + − to the cavity optical amplification 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- configurations 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 effect of cavity nonlinearity, implies that similar inhomo- torque N · j with the sign α = 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 αΩeff. Let us now elaborate on these phenomena. To obtain Quantifying the linear response of the intracavity effec- general results, we will proceed without assuming Sˆ ' Si, tive magnetic field to variations of the collective spin via except in certain cases, noted in the text, where some λ = Ωcd(¯n+ − n¯−)/dSk, the static spin orientations are 3 physical insight is gained. We begin with effects for which found to be unstable when αλ > ΩL| csc θ0|/S. both the light field and the ensemble spin may be treated Opto-dynamical Larmor frequency shift: We classically, i.e. by letting S = hSî andn ¯± = hnˆ±i. now consider the dynamics of the system precessing Cavity-spin bistability: We start with the static about one of the stable configurations. These dynam- behavior of the system by finding the fixed 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 ΩL by two effects: First, there is 3 an upward frequency shift from the static modification light drives spins to the high- or low-energy spin state of the effective magnetic field, leading to precession at according to the detuning of probe light from the cavity 0 the frequency ΩL = ΩL| csc θ0| when λ = 0. A sec- resonance, independent of the polarization. Second, this ond frequency shift occurs when the spin dynamics are amplification or damping of the intracavity spin is coher- slow compared to the response time of the cavity field ent, preserving the phase of Larmor precession, at least 0 (ΩL ¿ κ). Here the precessing spin modulates the cav- within the limits of a quantum amplifier. ity field; this modulation acts back upon the spin to mod- Spin optodynamical squeezing of light: We move ify the precession frequency, analogously to the “optical now beyond the classical treatment of cavity spin opto- spring” effect of optomechanics []. When the precession dynamics to account for quantum optical effects. One amplitude is small, a solution of the spin equations of such effect is the disturbance of the collective spin due motion derived from the Hamiltonian in Eq. (5) yields to quantum optical fluctuations of the cavity fields. In 00 an overall precession frequency ΩL , where cavity optomechanics, quantum fluctuations of the intra-

00 2 02 cavity photon number disturb the motion of a cantilever, ΩL = ΩL − λΩLS sin θ0. (7) providing the necessary backaction of a quantum measurement of position [22]. The analogous disturbance The quantity kS ≡ −λΩLS sin θ0 serves as the analogue of the optomechanical spring constant, and leads to shifts of optically probed atomic spins (or pseudo-spins) has of the Larmor precession frequency with a sign and mag- been studied both in free-space [23] and intracavity [24] nitude that depend on the spin orientation, λ and the configurations. In an optomechanics-like configuration, frequency, intensity, and polarization of the cavity probe e.g. with B ∝ i, backaction heating of the atomic spin fields. When the precession amplitude is large, the dy- enforces quantum limits to measurement of the precess- namics become essentially nonlinear, and the picture of ing ensemble and also set limits on optodynamical cool- harmonic precession breaks down. In this case the dy- ing. In contrast with optomechanical systems, optically namics can be described by numerical simulation, as probed spin ensembles readily present the opportunity to shown in Fig. 3. perform quantum-non-demolition (QND) measurements; Coherent amplification and damping of spin: with B ∝ k, the detected spin component Sk is a QND Now we consider the effects of the finite cavity response variable representing the energy of the spin system. time τ on the spin dynamics. To develop an intuitive pic- The optically perturbed, precessing spin acts back upon the cavity optical field. This self-interaction of the ture, we consider the unresolved sideband regime ΩL < κ, in a frame (indicated by the index “r”) corotating with light field, mediated by the dynamics of the spin ensemble, can result in optical squeezing similar to that pre- the collective spin, with ir aligned to the fixed point. We assume the spin to be precessing at a near constant dicted in cavity optomechanics [15, 19]. To exhibit this + rate with, and we assume the cavity field response to the effect, we consider a cavity illumined with σ circular precessing spin to be simply delayed by τ. Employing polarized probe light with detuning ∆p,+. The dynamics the rotating-wave approximation, the delay causes the of the cavity field are given by effective field Ωeff,r to point out of the ir-kr plane, with dcˆ ³ ´ 2 00 + ˆ ˆ Ω · j = −(αλS sin θ sin φ)/2, where φ = Ω τ. = (i∆p,+ − κ + iΩcSk)ˆc+ + κ η + ξ+ (9) eff,r r k,r 0 L dt The collective spin now experiences a torque in the kr direction, giving Here, η gives the coherent-state amplitude of the drive ˆ 2 field and the noise operator ξ+ represents its fluctua- dSk,r −αλ sin θ0 sin φSi,r = Sk,r (8) tions. When evaluating the dynamics numerically, we dt 2 consider a semiclassical Langevin equation, converting ˆ For αλ > 0, the Larmor precession frequency is down- ξ+ into a Gaussian stochastic variable with statistics re- shifted and the spin is damped toward its stable point, lated to those of the noise operator, and replacing the while for αλ < 0, the Larmor precession frequency is up- operatorsc ˆ+ and Sˆ with c-numbers. This substitution is shifted and the spin is amplified away from the stable appropriate for moderately large values ofn ¯ and S. point. Similar relations apply to the case of cavity op- Fig. 3 portrays the simulated evolution of a spin pre- tomechanics [21]. The deflection of the spin toward or pared initially in a low-energy spin orientation (close away from the stable points persists even for large pre- to i), driven by a blue-detuned cavity probe. Coher- cession amplitudes, as seen in Fig. 3. ent spin amplification directs the spin toward the stable This cavity-induced spin amplification or damping dif- high-energy configuration (near −i), yielding a dynami- fers from conventional optical pumping in two important cal steady state characterized by a negative temperature. respects. First, while the spin polarization generated by To obtain analytical expressions for the evolution dy- optical pumping relies on the polarization of the pump namics, we follow the example of cavity optomechanics light, the target state for cavity-induced spin damping [25], by linearizing the Langevin equations for spin and is selected energetically. Similar to cavity optomechani- optical fluctuations about their steady-state value. The cal cooling [4], cavity enhancement of Raman scattered spin projection Sk responds to amplitude-quadrature 4

(a) 5000 Bistability in cavity-coupled single-spin systems has been proposed as a tool to increase the readout ﬁdelity of Sk cavity-coupled qubits [26]. Similarly, this system could k 0 be used as a Schmitt trigger for the collective spin: If

and S the probe power is turned on diabatically in the bistable i S regime, the cavity transmission will latch into either a

Si bright or dark state, depending on whether the initial -5000 0 0.1 2 4 6 8.2 8.3 8.4 spin state is below or above a parametrically chosen t (ms) threshold value. (b) π/2 20 For spin dynamics occurring near the shifted precession 00 frequency ΩL , the system behaves as a phase-preserving 10 quantum-limited amplifier, with amplification noise given by Eq. (11). This amplification could be used to amplify 0 0 dB the effects of AC fields above technical sensitivity limits. Conversely, cavity spin optodynamics may serve as a powerful simulator of cavity optomechanics, with the spin -10 system allowing for new means of control. For exam- -π/2 ple, precession frequencies may be tuned rapidly by vary- 10010 0 20020 0 30 3000 40040 0 f (kHz) ing the applied magnetic field, simulating optomechanics with a dynamically variable mechanical spring constant. Alternately, spatial control of inhomogeneous magnetic FIG. 3: (Color) Simulations of spin dynamics for S = 5000, fields may be used to divide a spin ensemble into sev- ΩL/2π = 200 kHz, Ωc/2π = −1.9 kHz, κ/2π = 1.8 MHz, eral independent subensembles, simulating optomechan- nˆ+ = 10, and ∆p,+ = 0.37κ. (a) Time evolution of Si (black) ics with several mechanical modes. and Sk (blue), following spin preparation near i, show amplification, reorientation, and damping toward the high-energy In addition to the dilute gas implementation we have stable spin orientation near −i. Note the different scales on already detailed, a similar system could be constructed the horizontal axis. (b) Logarithmic optical spectral noise using solid-state spin ensembles with microwaves. Re- power relative to that of coherent light, plotted vs. quadra- cently, experiments have been proposed to couple en- ture angle φ (amplitude quadrature at φ = 0), shows inhomo- sembles of nitrogen-vacancy defects in diamond to the geneous optical squeezing. Simulation results shown in color, evanescent radiation of a coplanar microwave resonator, and linearized theory (Eq. 11) as contour lines every 5 dB. with collective cavity coupling on the order of 2 MHz [27]. The Hamiltonian of Eq. (5) is obtained by using coherent superpositions of the ground ms = ±1 states fluctuations of the cavity field ξA(ω) with susceptibility for the spin, using linearly-polarized microwaves on the 0 √ S (ω) −Ω Ω n¯ ms = ±1 ↔ 0 transition at 2.8 GHz. ΩL is chosen to be χ(ω) ≡ k = L c + , (10) 02 large compared to the hyperfine splitting (on the order ξA(ω) Ω + k R(ω) − ω2 + iωΓ (ω) L S o of a couple megahertz [28]), and the vector stark shift is 02 2 ΩL −ω obtained by choosing ∆ca to be only a few times greater where Γo(ω) = 2κ 2 2 2 is the cavity optodynamic κ +∆p−ω 2 2 than ΩL. κ +∆p spin damping, and R(ω) = 2 2 2 . The susceptibil- We thank H. Mabuchi and K.B. Whaley for inspiring κ +∆p−ω 00 ity is largest for ω ' ΩL . The driven spin feeds these discussions. This work was supported by the NSF and fluctuations back onto the cavity field, yielding the intra- the AFOSR. D.M.S.-K. acknowledges support from the cavity field fluctuation spectrum Miller Institute for Basic Research in Science.

02 κ+ω 2 ΩL + i kSR(ω) − ω + iωΓo(ω) ∆p c(ω) = ξA + iξP , 02 2 ΩL + kSR(ω) − ω + iωΓo(ω) (11) ∗ Electronic address: [email protected] where ξP (ω) is the input spectrum of phase fluctuations. [1] T. Kippenberg and K. Vahala, Science 321, 1172 (2008). This fluctuation spectrum exhibits inhomogeneous opti- [2] V. Braginskii and F. Y. Khalili, Quantum Measurement cal squeezing (Fig. 3b). (Cambridge University Press, Cambridge, 1995). Applications: In this work, we use the analogy [3] K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, Nature Physics 4, 561 (2008). of cavity optodynamics to widen the range of phenom- [4] V. Vuletićand S. Chu, Phys. Rev. Lett. 84, 3787 (2000). ena accessed through the manipulation and detection of [5] C. Fabre et al., Phys. Rev. A 49, 1337 (1994); S. Mancini quantum spins within optical cavities. We end by detail- and P. Tombesi, Phys. Rev. A 49, 4055 (1994). ing some applications of this new system. [6] P. F. Herskind et al., Nature Physics 5, 494 (2009). 5

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