Cavity spin optodynamics

0.43 MB·

by  N. Brahms

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: January 5, 2011)
Thedynamicsofalargequantumspincoupledparametricallytoanopticalresonatoristreatedin
analogywiththemotionofacantileverincavityoptomechanics. Newspinoptodynamicphenomena
arepredicted,suchascavity-spinbistability,optodynamicspin-precessionfrequencyshifts,coherent
amplification and damping of spin, and the spin optodynamic squeezing of light.
Cavityoptomechanicalsystemsarecurrentlybeingex- position zˆand momentum pˆevolve as dzˆ/dt=pˆ/m and
plored with the goal of measuring and controlling me- dpˆ/dt=−mω2zˆ+fnˆ.
z
chanicalobjectsatthequantumlimit, usinginteractions To construct a spin analogue of this system, we con-
1
1 withlight[1]. Insuchsystems,thepositionofamechan- sider a Fabry-Perot cavity with its axis along k (Fig.
0 ical oscillator is coupled parametrically to the frequency 1). For the collective spin, we first consider a gas of
2 of cavity photons. A wealth of phenomena result, in- N hydrogenlike atoms in a single hyperfine manifold of

n cluding quantum-limited measurements [2], mechanical their electronic ground state, each with dimensionless
a response to photon shot noise [3], cavity cooling [4], and spin s and gyromagnetic ratio γ. The atoms are op-
J
ponderomotive optical squeezing [5]. tically confined at an antinode of the cavity field. An

3
Concurrently, spins and psuedospins coupled to elec- external magnetic field B=Bb is applied to the atoms.

tromagnetic cavities are being researched in atomic [6], The detuning ∆ between the cavity resonance is cho-
] ca
h ionic [7], and nanofabricated systems [8, 9], with appli- sen to be large compared to both the natural linewidth
p cations including magnetometry [10], atomic clocks [11], and the hyperfine splitting of the atoms’ excited state.
-
t and quantum information processing [6, 9, 12]. In con- In this limit, spontaneous emission may be ignored and
n
trast to mechanical objects, spin systems are more eas- the single-atom cavity-field interaction energy, H =
a Stark
u ily disconnected from their environment and prepared in ((cid:126)g02/∆ca)nˆ(1±υk·ˆs), comprises the scalar and vector
q quantum states, including squeezed states [11]. ac Stark shifts [19], where g quantifies the atom-cavity
0
[ In this Rapid Communication, we seek to link these coupling and υ the vector shift.

2 two fields by exploiting the similarities between large- Summing over all atoms q, we obtain the system
v spin systems andharmonic oscillators [13]to construct a Hamiltonian:
3 cavityspinoptodynamicssysteminanalogytocavityop-
(cid:18)
5 (cid:88)
tomechanics. Optomechanical phenomena map directly H=(cid:126)ω (nˆ +nˆ )+H + −(cid:126)γB·ˆs
8 c + − in/out q
to our proposed system, resulting in spin cooling and
3 q
. amplification[14,15],nonlinearspinsensitivityandspin- (cid:126)g2 (cid:19)
05 cavitybistability[16,17],andspinopto-dynamicsqueez- + ∆0 [(nˆ++nˆ−)+υ(nˆ+−nˆ−)k·ˆsq] , (2)
ing of light [14, 18]. Such a system may find application ca
0
1 as a quantum-limited spin amplifier or as a latching spin with number operators nˆ for the σ± polarized optical
±
: detector. We detail these phenomena using currently ac- modes.
v
i cessible parameters, and we propose realizations either The above Hamiltonian can be rewritten as the inter-
X usingcoldatomsandvisiblelightorusingcryogenicsolid action of the collective spin operator Sˆ ≡(cid:80) ˆs with an
q q
r state systems and microwaves. effective total magnetic field B ≡Ω /γ, giving [20]
a eff eff
An ideal cavity optomechanics system, consisting of
a harmonic oscillator coupled linearly to a single-mode Ωeff =ΩLb+Ωc(nˆ+−nˆ−)k. (3)
cavity field, obeys the Hamiltonian
Here Ω = γB and Ω = −υg2/∆ . Altogether, the
L c 0 ca
H=(cid:126)ω nˆ+(cid:126)ω aˆ†aˆ−fz (cid:0)aˆ†+aˆ(cid:1)nˆ+H . (1) cavity spin optodynamical Hamiltonian is
c z HO in/out
(cid:18) Ng2(cid:19)
Here aˆ is the oscillator’s phonon annihilation operator, H=(cid:126) ωc+ ∆ 0 (nˆ++nˆ−)+Hin/out−(cid:126)Ωeff·Sˆ. (4)
nˆ is the photon number operator, ω is the natural fre- ca
z
quency of the oscillator in the dark, and ω is the bare Now consider the external magnetic field to be static
c
cavity resonance frequency. f is the radiation-pressure and oriented along i, orthogonal to the cavity axis. In
force applied by a single photon, while z =(cid:112)(cid:126)/2mω the limit (cid:104)Sˆ(cid:105)(cid:39)Si, the spin dynamics become
HO z
is the harmonic oscillator length for oscillator mass m.
Hin/out describesthecouplingofthecavityfieldtoexter- dSˆj =Ω Sˆ −Ω S(nˆ −nˆ ), dSˆk =−Ω Sˆ . (5)
nal light modes. Under this Hamiltonian, the cantilever dt L k c + − dt L j
2
S
(b)
Ω (n -n)k 15
c + -
S n
Ω i +
L (a)
π
0
15
θ
0 n
σ+ σ - -
-π 0
FIG.1. (Color)Anensembleofatomstrappedwithinadriven -10 -5 0 5 10 -10 -5 0 5 10
Δ/κ Δ/κ
opticalresonatorexperiencesanexternallyimposedmagnetic p p
fieldalongiandalight-inducedeffectivemagneticfieldalong
the cavity axis k. The evolution of the collective spin Sˆ re- FIG. 2. (Color) Cavity-spin bistability in a cavity driven
sembles that of a cantilever in cavity optomechanics. with linearly polarized light. We consider N = 5000 spin-
2 87Rb atoms, Ω /κ=1.25×10−3, Ω /κ=3.3×10−2, and
c L
n¯ =15(similartoRef.[21]). (a)As∆ isvaried,several
max,± p
The analogy between cavity optomechanics and stable (black) and unstable (gray) static spin configurations
are found. Configurations for ∆ /κ=−4.8 are depicted. (b)
spin optodynamics is established by assigning p
zˆ → −z Sˆ /∆S and pˆ → p Sˆ /∆S , Thecavityexhibitshysteresisastheprobeissweptwithposi-
HO k SQL HO j SQL tive(dashedblue)ornegative(red)frequencychirps,withthe
where z and p = (cid:126)/(2z ) are defined with
HO HO HO spin initially along i. Rapid transitions as ∆ /κ is swept up-
(cid:112) p
ωz → ΩL [13] and ∆SSQL = S/2 is the standard wardfrom-2.8ordownwardfrom0involvesymmetrybreak-
quantum limit for transverse spin fluctuations. Eqs. (5) ingasthecavitybecomesbirefringent;wedisplayn¯ andn¯
+ −
now match the optomechanical equations of motion assumingthestablebranchclosertoθ0 =0isselected. Here,
with the optomechanical coupling defined through ∆ca/2π =20 GHz from the D2 transition, g0/2π =15 MHz,
fz nˆ→−(cid:126)Ω ∆S (nˆ −nˆ ). The main result of this κ/2π=1.5 MHz.
HO c SQL + −
work, that various cavity optomechanical phenomena
are manifest also in cavity spin optodynamical systems,
tive magnetic field to variations of the collective spin via
is immediately established.
λ = Ω d(n¯ −n¯ )/dS , the static spin orientations are
c + − k
Let us now elaborate on these phenomena. To obtain
found to be unstable when αλ>Ω |csc3θ |/S.
generalresults,wewillproceedwithoutassumingSˆ(cid:39)Si, L 0
Opto-dynamical Larmor frequency shift: The
except in certain cases, noted in the text, where some
dynamics of the spin precessing about one of the stable
physicalinsightisgained. Webeginwitheffectsforwhich
configurations can be parameterized by the precession
boththelightfieldandtheensemblespinmaybetreated
frequency,whichisshiftedfromΩ bytwoeffects. First,
classically, i.e. by letting S=(cid:104)Sˆ(cid:105) and n¯ =(cid:104)nˆ (cid:105). L
± ± there is an upward frequency shift from the static modi-
Cavity-spin bistability: We start with the static
fication of the effective magnetic field, leading to preces-
behavior of the system by finding the fixed points of the sion at the frequency Ω (cid:48) = Ω |cscθ | when λ = 0. A
L L 0
system. The collective spin vector is static when S is
secondshiftoccurswhenthespindynamicsareslowcom-
parallel to Ωeff. Writing S = S(isinθ0 +kcosθ0), this pared to the response time of the cavity field (Ω (cid:48) (cid:28)κ).
L
condition requires n¯ −n¯ =(Ω /Ω )cotθ . The intra-
+ − L c 0 Here, the precessing spin modulates the cavity field,
cavity photon numbers are determined also by the stan-
which, in turn, acts back upon the spin to modify its
dard expression for a driven cavity of half line-width κ,
precession frequency, When the precession amplitude is
i.e. n¯ = n¯ [1 + (∆ ±Ω Scosθ )2/κ2]−1 with
± max,± p,± c 0 small, a solution of the spin equations of motion derived
ω = (ω + Ng2/∆ ) + ∆ being the frequency of
± c 0 ca p,± from the Hamiltonian in Eq. (4) yields an overall preces-
laser light of polarization σ± driving the cavity and sion frequency Ω (cid:48)(cid:48), where
L
n¯ characterizing its power. These two expressions
max,±
for n¯+−n¯− may admit several solutions (Fig. 2). Ω (cid:48)(cid:48)2 =Ω (cid:48)2−λΩ Ssinθ . (6)
L L L 0
As typical in instances of cavity bistability [22], sev-
eral of the static solutions for the intracavity intensities The quantity k ≡ −λΩ Ssinθ serves as the analogue
S L 0
may be unstable. To identify such instabilities, we con- of the optical spring constant [23], and leads to shifts of
sider the torque on the collective spin when it is dis- the Larmor precession frequency with a sign and mag-
placed slightly toward +k from its static orientation. nitude that depend on the spin orientation, λ and the
Stable dynamics result when such displacement yields a frequency, intensity, and polarization of the cavity probe
torque N·j with the sign α = sgn(sinθ ). Geometri- fields. When the precession amplitude is large, the dy-
0
cally, this stability requires that the spin vector be dis- namicsbecomeessentiallynonlinear. Inthiscasethedy-
placedfurtherinthe+kdirectionthanthevectorαΩ . namicscanbedescribedbynumericalsimulation(Fig.3).
eff
Quantifying the linear response of the intracavity effec- Coherent amplification and damping of spin:
3
Now we consider the effects of the finite cavity response (a) 5000
time κ−1 on the spin dynamics. To develop an intu-
SS
itivepicture,weconsidertheunresolvedsidebandregime kk
k
Ω <κ, in a frame (indicated by the index “r”) corotat- S
L d 0
ipnoginwt.ithWteheascsoulmlecetitvheessppinin, wtoithbeirparleicgensesdintgoattheafinxeeadr S ani
constant rate, and the cavity field response to this pre- SSii
-5000
cession to be simply delayed by κ−1. Employing the 0 0.1 2 4 6 8.2 8.3 8.4
rotating-wave approximation, the delay causes the ef- t(ms)
(b) π/2 20
fective field Ω to point out of the i -k plane, with
eff,r r r
Ω ·j = −(αλS sin2θ sinφ)/2, where φ = Ω (cid:48)(cid:48)/κ.
eff,r r k,r 0 L
The collective spin now experiences a torque in the k 10
r
direction, giving
0
dS −αλsin2θ sinφS 0 dB
k,r = 0 i,rS (7)
dt 2 k,r
For positive (negative) values of αλ , the Larmor pre- -10
cession frequency is shifted down (up) and the spin is -π/2
110000 220000 330000 440000
damped toward (amplified away from) its stable point.
f(kHz)
Similarrelationsapplytocavityoptomechanics[24]. The
deflection of the spin toward or away from the stable
points persists for large precession amplitudes (Fig. 3). FIG. 3. (Color) Simulations of spin dynamics for S = 5000,
Thiscavity-inducedspinamplificationordampingdif- Ω /2π = 200 kHz, Ω /2π = −2.3 kHz, κ/2π = 1.8 MHz,
L c
fersfromconventionalopticalpumpingintwoimportant nˆ+ =10,and∆p,+ =0.37κ. (a)TimeevolutionofSi (black)
respects. First, while the spin polarization generated by andSk (blue),followingspinpreparationneari,showsampli-
fication, reorientation, and damping toward the high-energy
optical pumping relies on the polarization of the pump
stable orientation near −i. Note the different scales on the
light, the target state for cavity-induced spin damping
horizontal axis. (b) Logarithmic optical spectral noise power
is selected energetically. Similar to cavity optomechani-
relative to that of coherent light, plotted vs. quadrature an-
cal cooling [4], cavity enhancement of Raman scattered gleφ(amplitudequadratureatφ=0),showsinhomogeneous
light drives spins to the high- or low-energy spin state opticalsqueezing. Simulationresultsshownincolor,andlin-
according to the detuning of probe light from the cavity earized theory (Eq. 10) as contour lines every 5 dB.
resonance, independent of the polarization. Second, this
amplificationordampingoftheintracavityspiniscoher-
that can result in optical squeezing. To exhibit this ef-
ent, preserving the phase of Larmor precession, at least
fect, we consider a cavity illumined with σ+ circular po-
within the limits of a quantum amplifier.
larized probe light with detuning ∆ . The dynamics of
Spin optodynamical squeezing of light: We now p
the cavity field are given by
consider quantum optical effects of cavity spin optody-
namics. One such effect is the disturbance of the collec-
dcˆ (cid:16) (cid:17)
tive spin due to quantum optical fluctuations of the cav- + =(i∆ −κ+iΩ Sˆ )cˆ +κ η+ξˆ . (8)
dt p c k + +
ity fields. In cavity optomechanics, intracavity photon
number fluctuations disturb the motion of a cantilever, Here, η gives the coherent-state amplitude of the drive
providing the necessary backaction of a quantum mea- field and the noise operator ξˆ represents its fluctua-
+
surement of position [25]. The analogous disturbance of tions. When evaluating the dynamics numerically, we
opticallyprobedatomicspins(orpseudo-spins)hasbeen consider a semiclassical Langevin equation, converting
studied both in free-space [26] and intracavity [11, 27] ξˆ into a Gaussian stochastic variable with statistics re-
+
configurations. In an optomechanics-like configuration, lated to those of the noise operator, and replacing the
e.g. with B ∝ i, backaction heating of the atomic spin operators cˆ and Sˆ with c-numbers. This substitution is
+
enforces quantum limits to measurement of the precess- appropriate for moderately large values of n¯ and S.
ing ensemble and also set limits on optodynamical cool- Fig. 3 portrays the simulated evolution of a spin pre-
ing. In contrast with optomechanical systems, optically pared initially in a low-energy spin orientation (close
probedspinensemblesreadilypresenttheopportunityto to i), driven by a blue-detuned cavity probe. Coher-
performquantum-non-demolition(QND)measurements; ent spin amplification directs the spin toward the stable
with B ∝ k, the detected spin component S is a QND high-energy configuration (near −i), yielding a dynami-
k
variable representing the energy of the spin system. calsteadystatecharacterizedbyanegativetemperature.
The noise-perturbed spin acts back upon the cavity To obtain analytical expressions for the evolution dy-
opticalfield,mediatingaself-interactionofthelightfield namics,wefollowtheexampleofcavityoptomechanics[5]
4
by linearizing the Langevin equations for spin and opti- tronicstatesasthepseudospinandreplacingtheacStark
cal fluctuations about their steady-state value. The spin shift with an ac Zeeman shift from microwave radiation
projectionS respondstoamplitude-quadraturefluctua- near the 2.8-GHz crystal-field-split Zeeman transition.
k
tions of the cavity field ξ (ω) with susceptibility
A
√
S (ω) −Ω (cid:48)Ω n¯ We thank H. Mabuchi and K.B. Whaley for inspiring
χ(ω)≡ k = L c + , (9) discussions. This work was supported by the NSF and
ξA(ω) ΩL(cid:48)2+kSR(ω)−ω2+iωΓo(ω) the AFOSR. D.M.S.-K. acknowledges support from the
where Γo(ω) = 2κκΩ2+L∆(cid:48)22−−ωω22 is the cavity optodynamic Miller Institute for Basic Research in Science.
p
κ2+∆2
spin damping, and R(ω) = p . The susceptibil-
κ2+∆2−ω2
p
ity is largest for ω (cid:39) Ω (cid:48)(cid:48). The driven spin feeds the
L
fluctuations back onto the cavity field, yielding the in- ∗ [email protected]
tracavity field fluctuation spectrum [1] T.KippenbergandK.Vahala,Science321,1172(2008).
[2] V. Braginskii and F. Y. Khalili, Quantum Measurement
c+(ω)= ΩLΩ(cid:48)2+(cid:48)2i+κ∆+kpωkRS(Rω)(ω−)ω−2ω+2i+ωΓiω(Γωo)(ω)ξA+iξP, [3] K(C.amWb.riMdguercUhn,ivKer.sitLy.PMreososr,eC,aSm.brGidugpet,a1,9a9n5)d. D. M.
L S o Stamper-Kurn, Nature Physics 4, 561 (2008).
(10)
[4] V.Vuleti´candS.Chu,Phys.Rev.Lett.84,3787(2000).
where ξP(ω) is the input spectrum of phase fluctuations. [5] C.Fabreetal.,Phys.Rev.A49,1337(1994);S.Mancini
This fluctuation spectrum exhibits inhomogeneous opti- and P. Tombesi, Phys. Rev. A 49, 4055 (1994).
cal squeezing (Fig. 3b). [6] S.HarocheandJ.-M.Raimond,Exploring the Quantum:
Applications: The analogy of cavity optodynamics Atoms, Cavities, and Photons (Oxford University Press,
2006).
widenstherangeofphenomenaaccessedthroughthema-
[7] P. F. Herskind et al., Nature Physics 5, 494 (2009).
nipulation and detection of quantum spins within opti-
[8] P. E. Barclay, K.-M. C. Fu, C. Santori, and R. G. Beau-
cal cavities, enabling several applications. For example,
soleil, App. Phys. Lett. 95, 191115 (2009).
bistabilityincavity-coupledsingle-spinsystemsservesto [9] L. DiCarlo et al., Nature 460, 240 (2009).
increasethereadoutfidelityofcavity-coupledqubits[28]. [10] J. M. Geremia, J. K. Stockton, A. C. Doherty, and
Similarly here, cavity-spin bistability could be used as a H. Mabuchi, Phys. Rev. Lett. 91, 250801 (2003).
Schmitttriggerforthecollectivespin: iftheprobepower [11] I.D.Leroux,M.H.Schleier-Smith,andV.Vuletic,Phys.
Rev. Lett. 104, 073602 (2010).
is turned on diabatically in the bistable regime, the cav-
[12] Q. Turchette et al., Phys. Rev. Lett. 75, 4710 (1995).
ity transmission will latch into either a bright or dark
[13] T. Holstein and H. Primakoff, Phys. Rev. 58, 1098
state, depending on whether the initial spin state is be-
(1940).
low or above a parametrically chosen threshold value. [14] V. B. Braginsky and A. B. Manukin, Sov. Phys. JETP
The cavity-spin system may also be used as a phase- 25, 653 (1967).
preserving amplifier for spin dynamics occurring near [15] T.J. Kippenberg et al., Phys. Rev. Lett. 95, 033901
the shifted precession frequency Ω (cid:48)(cid:48), with amplification (2005); S. Gigan et al., Nature 444, 67 (2006); O. Ar-
L
cizet et al., Nature 444, 71 (2006).
noise given in Eq. (10). Both applications may aid mea-
[16] A. Dorsel et al., Phys. Rev. Lett. 51, 1550 (1983).
surements of ac magnetic fields, amplifying weak signals
[17] S. Gupta, K. L. Moore, K. W. Murch, and D. M.
above technical sensitivity limits.
Stamper-Kurn, Phys. Rev. Lett. 99, 213601 (2007).
Conversely, cavity spin optodynamics may be applied [18] H. J. Kimble et al., Phys. Rev. D 65, 022002 (2001).
asapowerfulsimulatorofcavityoptomechanics,withthe [19] W. Happer and B. S. Mathur, Phys. Rev. Lett. 18, 577
spinsystemallowingfornewmeansofcontrol. Forexam- (1967).
ple,precessionfrequenciesmaybetunedrapidlybyvary- [20] C. Cohen-Tannoudji and J. Dupont-Roc, Phys. Rev. A
5, 968 (1972).
ingtheappliedmagneticfield,simulatingoptomechanics
[21] T. Purdy et al., preprint, arXiv:1005.4085.
with a dynamically variable mechanical spring constant.
[22] H. M. Gibbs, Optical bistability: Controlling light with
Alternately, spatial control of inhomogeneous magnetic
light (Academic Press, New York, 1985).
fields may be used to divide a spin ensemble into sev- [23] A. Buonanno and Y. B. Chen, Phys. Rev. D 65, 042001
eral independent subensembles, simulating optomechan- (2002).
ics with several mechanical modes. [24] T. Corbitt et al., Phys. Rev. Lett. 98, 150802 (2007).
Inadditiontothedilutegasimplementationdiscussed [25] C. M. Caves, Phys. Rev. Lett. 45, 75 (1980).
[26] C.Schori,B.Julsgaard,J.L.Sørensen,andE.S.Polzik,
so far, a similar system could be constructed using solid-
Phys. Rev. Lett. 89, 057903 (2002).
state spin ensembles and microwave resonators. For ex-
[27] M.H.Schleier-Smith,I.D.Leroux,andV.Vuletic,Phys.
ample,usingensemblesofnitrogen-vacancydefectsindi-
Rev. Lett. 104, 073604 (2010).
amond coupledto thecircular polarizedevanescentradi- [28] I. Siddiqi et al., Phys. Rev. B 73, 054510 (2006).
ationofacrossedmicrowaveresonator[29],theHamilto- [29] J.J.Henderson,C.M.Ramsey,H.M.Quddusi,andE.d.
nian of Eq. (4) is obtained by the ground m =±1 elec- Barco, Rev. Sci. Instrum. 79, 074704 (2008).
s