arXiv:physics/0511239v3 [physics.atom-ph] 6 Feb 2006 hssse h tm,lgtfil n irritrc with interact In mirror and 1. field light Fig. atoms, see the the cavity, by system the this generated inside potential are field dipole light atoms standing-wave optical where the case in the trapped to interferometers driven sure for principle in question [18]. nontrivial conditions a boundary quantiz- is time-dependent Furthermore field 17]. light [8, the delay ing the- with for systems grounds of testing ories opportunities. excellent provide and systems challenges These many offer cavities driven cases. certain in field classical retardation electromagnetic self- the the the to instabilities, due of In chaos exhibit and can [4] oscillations addressed. motion sustained mirror be the can field regime limit light near quantum measurements the position the with to related Also, entangled questions and 15]. become [16], 14, motion could [13, way mirror of macro- this studied the states a be of can non-classical squeezing, motion including and quantized mir- the will object, state that the is scopic ground and of It mechanical possible quantum cooling be its 12]. methods to 11, these motion [10, ror with implemented that and have expected [9] pressure mir- radiation schemes proposed with Recently, end mirrors been the of [2]. cooling of background active seismic for motion be the mechanical can large from force the as ror pressure isolate systems radiation microma- to for to The used down relevant [4]. 8] is 7, cavities and [6, chined antennas 5] wave cavity. gravitation [4, optical systems as those rich an in very inside found radiation is field dynamics the nonlinear light by coupled the The affected to strongly due is damp- pressure small motion with en- whose mirrors moveable properties ing of optical creation and the mechanical ables defined structures to well mi- mechanical with due to sized attention capabilities sub-micrometer renewed end improved cromachine received the The has of developments. 3] several motion 2, mechanical [1, the mirrors to resonator optical nti ae eetn h td frdainpres- radiation of study the extend we paper this In pressure radiation view, of point theoretical a From an inside field light the of coupling the years recent In ope yaiso tm n aito rsuedie int driven pressure radiation and atoms of dynamics Coupled eue odtc h ope oino h tm n h mirro the and atoms the of motion coupled the detect to used be h iebn pcrmo h ih rnmte hog h c the through transmitted light the much of move ad spectrum atoms field band the light side where the neglec the and where be instantaneously limit field cannot light opti adiabatic field the light an the the in in on system trapped atoms coupled are results the atoms These of where backa backaction situation bistable. this the any become to to can due fai applied atoms modified be be the e is can of move to that position atoms couple reflectivity the the strongly a which field with in light mirror potential the Bragg and distributed atoms a The trapped. are ecnie h oino h n irro aiyi hs st whose in cavity a of mirror end the of motion the consider We eateto hsc,TeUiest fAioa 18E 4t E. 1118 Arizona, of University The Physics, of Department .INTRODUCTION I. .Mie n .Meystre P. and Meiser D. inbcmsbsal.Tecaatro h bifurcation the of character The posi- atomic bistable. local the two becomes nar- i.e. wavelength, exhibit tion become optical positions half sit mirror per atoms minima certain the for which and in rower wells potential pressure the motion. radiation mirror the the influence of also modifications they Furthermore, force, the the themselves. of change move qualitatively because they atoms which affects the in strongly Thus potential mirror”, “atom field. the light call the may we what by single mirror. a fictitous into this collapse of atoms position the the Thus, describing of coordinate [19]. freedom atoms of of degrees number all maxi- given of a mirror for Bragg reflectivity a mum in to themselves corresponding arrange configuration naturally a atoms As the discuss self- distribution. will the we density from atomic can calculated determined field be consistently light the can and reflection on coefficient whose atoms mirror transmission effective the an of by represented effect simple be observa- the fairly key that a The is to tion assumptions. amenable moderate is the under and of possible one analysis many system is captures the if it that of surprisingly theory aspects problem, effective that, the an is develop of to result in complexity key situations the described Our those considers be to also paper. ourselves can this of restrict atoms cases We the many of in classically. as that motion well show as the we mirror interest classically, the field of light motion first the case. from the general system Treating most this the principles. governing in equations footing the derive equal We an on other each with cavity a in atoms of mirror. moveable Schematic a online) (Color 1: Figure sacneuneo h akato ntelgtfield light the on back-action the of consequence a As described as atoms, the of back-action collective The ut otepsto fteaosand atoms the of position the to justs vt n hwta hs pcr can spectra these that show and avity l ih eaayehwtedipole the how analyze We high. rly r famr eea aueadcan and nature general more a of are e.W nlz h yaiso the of dynamics the analyze We ted. a atc nieacvt n where and cavity a inside lattice cal atrta h irr ecalculate We mirror. the than faster c te eas h tm form atoms the because other ach to fteaos eso that show We atoms. the of ction r. nigwv oeptenatoms pattern mode wave anding tet usn Z85721 AZ Tucson, Street, h P R in 1 0 T , 1 atoms erferometers L R L 2 + T , ξ 2 2 points where the single potential well branches off into A. Atoms two valleys can be modified by changing the cavity pa- rameters, such as e.g. the reflectivities of the two end Using ~ΩP as the unit of energy and ~k0 = ~2π/λ0 as mirrors. In that way it is possible to choose which of the the unit of momentum the free hamiltonian of the atoms two valleys is the deeper one and accordingly in which takes the form direction the atoms will move. The two side valleys can E U also be made equally deep, in which case the branching Hˆ = d3xψˆ†(x) R 2 + 0 ψˆ†(x)ψˆ(x) ψˆ(x), (1) point acts like an atomic beam splitter. We also show Z  − 4 ∇ 2  that the atomic motion can exhibit hysteresis, where the atoms move along different paths depending on the di- where ψˆ and ψˆ† are the atomic field annihilation and rection of motion of the mirror, giving rise to additional creation operators. Our theory can easily be adapted to instabilities of the mirror motion. describe fermions or bosons although at the level of our present theory there are no significant effects due to the In this paper we consider only the adiabatic limit where statistics of the atoms. For the sake of definiteness we the mirror moves slowly compared with the atoms, which nonetheless assume that the field operators in Eq. (1) in turn have a much slower response time than the intra- satisfy bosonic commutation relations. The energy cavity light field. Even in that limit the coupled system exhibits a remarkably rich phenomenology that we illus- 2 (2~k0) trate by analyzing and discussing the modified dipole po- ER = , 2M~Ω tential felt by the atoms as well as the modified motion P of the end mirror. with M the mass of the atoms is the two-photon recoil We propose to measure the sideband spectrum of the energy and light transmitted through the cavity as a quantitative ERa indicator of the coupled atom-mirror motion, as the cou- U0 = , λ π pling of the mirror motion to the atoms gives rise to 0 unambiguous spectral signatures that should be within with a the s-wave scattering length is the interaction easy experimental reach. strength between the atoms describing s-wave collisions. This article is organized as follows: In Section II we We assume that the atoms can be approximated as describe the model for the coupled atom-mirror-field sys- two-level atoms with transition frequency ωa and we treat tem and derive the basic dynamical equations. In section their interactions with the light field in the dipole and ro- III we consider the dynamics of the mirror with field and tating wave approximations. We consider only one state atoms following its motion adiabatically, and discuss sev- of polarization of the light field and describe this com- eral aspects of the atomic motion. In section IV we cal- ponent by a scalar field. Furthermore we assume that culate the spectrum of the light transmitted through the the intracavity light field of frequency ωL is far detuned from the atomic transition, ∆ ω ω Ω ,γ. cavity and show that it can serve as a convenient probe | | ≡ | L − a| ≫ R of the coupled mirror-atom motion. Finally, section V is Here ΩR is the Rabi frequency of the transition and γ a conclusion and outlook. is its linewidth. Then we can adiabatically eliminate the upper state and, choosing the z axis along the direction of propagation of the light field, the interaction between light field and atoms is given by the dipole potential

ˆ 3 † 2 2 II. MODEL Hint = g d xψ (x)ψ(x) E(z) u⊥(x, y; z) (2) Z | | | | where The system under consideration is shown schematically in Fig. 1. Atoms are trapped in the standing-wave light 2 ℘2 g = , (3) field of a Fabry-P´erot cavity with one of its end mirrors ~2 2 cǫ0λ0ΩP ∆ allowed to move under the effect of radiation pressure and subject to a harmonic restoring force, ξ being the where ℘ is the electric dipole moment along the polariza- displacement of the mirror from its rest position. The tion direction of the laser field and the electrical field has cavity length offset L only gives rise to a shift of the been split into a part varying along the cavity axis z and resonances and we will set it to zero in all that follows. a transverse profile u⊥(x, y; z) assumed to be normalized to unity, We measure all lengths in units of the vacuum wave- length of the light and all times in units of Ω−1, where P 2 dxdy u⊥(x, y; z) =1, (4) ΩP is the oscillator frequency of the unperturbed mov- Z | | ing mirror. The electrical fields are described in terms of their intensity integrated over the beam profile, i.e. in for every z. For simplicity we assume in the following terms of power. that the z-dependence of u⊥ along the atomic cloud can 3 be neglected. We further assume that the light field is R2,T2 red detuned, ∆ < 0, so that the atoms are attracted to E1 E4 the intensity maxima. The standing wave that forms in the cavity results in an optical lattice potential. We assume that its wells are 2 very deep, i.e. g E0 ER where E0 is the amplitude of E3 E2 the standing wave.| | In≫ the language of condensed matter physics we assume that the atoms are deeply in the Mott Figure 2: Field geometry at movable mirror. insulator regime [20]. It is then a good approximation to assume that every site has a specific number of atoms in the ground state of the harmonic oscillator potential the atomic motion is then reduced to the motion of a representing the lattice at that site. Due to the back- Gaussian wavepacket in a slowly changing harmonic po- action of the atoms on the light field the distance between tential, a situation that is well described by the classical the lattice sites is smaller than λ0/2, a point to which we dynamics of the atomic center of mass. return shortly. Because atoms at different sites see the exact same The interaction between atoms only gives rise to a con- potential at all times they will be at the same displace- stant energy shift. Its influence on the localized wave ment from their respective local minima given that they functions is irrelevant since all that will be required in started at the bottom of their respective potential wells. the following is that the atomic density at every site is Therefore it is possible to describe the atomic motion in localized within much better than an optical wavelength. terms of a single coordinate za, the position of each atom The order of magnitude of the localization is given by the modulo the lattice period. oscillator length at a single lattice site

πw2E 1/4 a = R (5) B. Moving mirror osc,z 4g E 2k2  | 0| 0 along the cavity axis and The mirror motion is described by the Newtonian equation of motion 4 1/4 πw ER FRP aosc,⊥ = 2 (6) ¨ ˙ 4g E0  ξ +Γξ + ξ = 2 αcFRP, (9) | | mP ΩP λ0 ≡ in the transverse direction. For typical laser powers and −1 detunings the oscillator length is roughly ten to a hun- where c is the speed of light, Γ is the mechanical quality dred times smaller than the optical wavelength. The os- factor of the mirror, FRP is the radiation pressure force, cillator frequencies are and

−1 2 g E 2k2E α = mP ΩP λ0c ω = | 0| 0 R (7) osc,z r πw2 is a power characteristic of the mirror. The electromag- for oscillations along the cavity axis and netic field is treated classically and from Fig. 2 we read off that 2 g E0 ER ωosc,⊥ = | | (8) 1 r πw4 F = ( E 2 + E 2 E 2 E 2) RP c | 1| | 3| − | 4| − | 2| for oscillations in the transverse direction. The param- 2 2 2 R2 E1 eters that we are interested in are such that the oscilla- = | | | | (10) c tor length in the longitudinal direction is much smaller than an optical wavelength and the oscillator length in where we have used that in our situation E2 0. the transverse direction is much smaller than the beam ≡ waist. Since the beam is typically much wider than an optical wavelength the atoms can be thought of as form- C. Light field ing a stack of pancake shaped discs. We consider an optical lattice with N sites occupied We now turn to the back-action of the atoms on the by n atoms each. Due to the phase shift suffered by the light field. The influence of the cavity mirrors will be light upon propagation through each atomic sheet the considered in the next section. lattice sites move closer together, the self-consistent pe- It is well known [21] that for large detuning, optically riod of the resulting lattice being determined later on. driving the atoms results in the polarization If the temperature of the atoms is far below the oscil- lator energy, we are justified in neglecting thermal exci- 2℘2 P = − ψˆ†ψEuˆ (11) tations. If the atoms start in the lattice ground state, ~∆ ⊥ 4

R,T leading to the inhomogeneous optical field propagation z }| { equation 2 |ψ(z)| Rs,Ts 2 2 2 (Eu⊥)+ k (Eu⊥)= χ ψ (Eu⊥) (12) ∇ 0 | | where in our units the polarizability is given by · · · 2µ Ω ℘2ω2 χ = 0 P L . (13) ~ ∆ z za za + d za + Nd In principle, the shape of the transverse profile u⊥ is the result of the complicated dynamics resulting from the Figure 3: Schematic of the Bragg mirror formed by the atoms. repeated propagation of the light field through the cav- Each atomic sheet has a reflection and transmission coefficient ity and its interaction with the complicatedly structured Rs and Ts resulting in a total reflection and transmission co- dielectric constituted by the atoms. A complete descrip- efficient R and T . tion of these processes is beyond the scope of this paper. We instead adopt the point of view that u⊥ has been es- into a z-dependent part and a transverse part. As dis- tablished in some way and is assumed to be known. We cussed above the atomic density distribution is much nar- therefore describe it an effective manner and eliminate it rower than the beam waist w. Hence we find, assuming from the theory. a Gaussian beam profile, Specifically, we assume that u⊥ varies slowly with z, 1 2 ∂u⊥/∂z k0u⊥ so that we can neglect its derivatives ψ(z) 2 = ψ (z) , (18) | | ≪ | | πw2 z with respect to z. Then, after dividing by Eu⊥ to sepa- | | | | rate the variables, we have from the wave equation where ψz(z) is the z-dependent factor of the oscillator wave function of the atoms. ∂2E(z) + k2E(z)= β2(z)E(z) (14) Equation (17) allows us to determine the effect of the ∂z2 0 atoms on the light field. The basic idea is to replace the entire configuration of atoms by a fictitious “atom mir- where β(z) is determined for every z by the eigenvalue ror” at some position za and hence to describe them in problem terms of a pair of effective reflection and transmission 2 2 2 coefficients. The period of the self-consistent lattice au- χ ψ(x,y,z) u⊥(x, y; z) u⊥(x, y; z)= β (z)u⊥(x, y; z). | | −∇⊥ tomatically fulfills the Bragg condition. Due to the self (15) consistency requirement this is even true for a lattice with In principle this formula solves our problem since it ex- nonuniform filling, see Ref. [19] for more details. (Par- presses the propagation equation for E in terms of ψ and enthetically, this means that we can take into account u⊥ independently of how the latter was obtained. In an external trapping potential through an appropriate order to obtain a more transparent formula we multiply ∗ choice of effective N and n. The thermal motion of the Eq. (15) by u⊥, integrate over x and y and after a partial atoms will lower the reflectivity of the atom mirror and integration we find could be taken into account through a Debye-Waller fac- tor.) All that remains to do to describe the effect of the 2 2 2 β (z) = χ dxdy u⊥(x,y,z) ψ(x,y,z) + atoms on the light field is to determine the transmission | | | | Z and reflection coefficient T and R of the atom mirror. A ∗ schematic of the situation is shown in Fig. 3. + dxdy ( ⊥u⊥(x, y; z)) ( ⊥u⊥(x, y; z)) Z ∇ · ∇ Consider first a single atomic sheet. Making use of 2 2 χ ψ(z) + k⊥(z), (16) the fact that the atoms are localized in the z-direction to ≡ | | a much narrower region than an optical wavelength, we ending up with the intuitively appealing equation readily find the transmission and reflection coefficient

2 1 iφ ∂ E(z) 2 2 2 + (k k (z))E(z)= χ ψ(z) E(z), (17) Ts = 2 e , (19) ∂z2 0 − ⊥ | | r1+Λ Λ2 2 R = i eiφ, (20) where ψ(z) can be interpreted as the density distribu- s r1+Λ2 tion averaged| | over the transverse beam profile. In the following we neglect the reduction of the wave vector k⊥ by matching boundary conditions at the atom sheet. due to the finite extend of the beam in the transverse Here direction, an approximation valid if the beam is not too φ = arctanΛ (21) strongly focused. − For the situation at hand ψ(z) 2 is readily found, since is the phase shift suffered by light upon transmission the wave function of the harmonic| | oscillator factorizes through the sheet of atoms and we have introduced the 5 parameter ÈRÈ2,ÈTÈ2 nχ 1 Λ= 2 (22) 2πk0w for notational convenience. Equation (21) allows us to 0.8 determine the self-consistent period of the optical lattice. Due to the interaction between the light field and the 0.6 atoms the period of the optical lattice is reduced to 0.4 π 2φ d = − . (23) 0.2 k0 The entire lattice of atoms can be considered as a peri- N103 odically stratified dielectric medium. Its total reflection 2 4 6 8 10 and transmission coefficients are readily found using stan- dard methods of the optics of dielectric films [19, 22]. In Figure 4: (Color online) Reflection (blue dashed line) and essence, one can find the transfer matrix of a unit cell of transmission coefficient (red solid line) of an optical lattice 87 the lattice from the transmission and reflection coefficient consisting of ten Rb atoms at every site as a function of 9 −1 Eq. (20,19). The transfer matrix of the entire lattice is the number of lattice sites for ∆ = 10 s and w = 10λ0 as then the Nth power of the transfer matrix of the unit obtained from Eqs. (24,25). cell. This can be done in closed form using Tchebychev polynomials since we neglect scattering losses and there- fore the transfer matrix of the unit cell is unimodular. R1,T1 R,T R2,T2 From the total transfer matrix the total reflection and ε1 ε4 transmission coefficient are extracted as [25] iΛNe2iφ Pin ε2 ε3 R = − (24) 1 iΛN − za e2iφN 0 L+ξ T = . (25) 1 iΛN − Figure 5: Representation of the atoms as a mirror at za and An example of the transmission and reflection coefficient illustration of the steady state boundary conditions at za. of such a lattice of atoms is shown in Fig. 4 for the case of 87Rb. From the figure it is apparent that rather high re- flectivities can be obtained in principle. One of the main experimental difficulties will be to achieve fairly high fill- motion of the atoms which is in turn very slow compared ing factors over many sites. Also, the high reflectivities to the dynamics of the light field. For this to be true we in Fig. 4 were obtained by using a small beam waist must have ΩP ωosc,z and ωosc,z κ where κ is the and a fairly small detuning of around 100 linewidths. It linewidth of the≪ cavity. This is the most≪ relevant case for should be noted that already fairly moderate reflection the typical systems currently available. The light field coefficients of the atoms of around 10% give rise to inter- then follows the atoms adiabatically, so that both light esting effects as we discuss later in this paper. S. Slama field and atoms adjust instantaneously to any position et al. have recently reported Bragg reflection coefficients of the moving mirror. We can then eliminate the atoms of the order of 30% [23]. (Note that the assumption of and the light field and express the radiation pressure force a constant and small beam waist is not as bad as might acting on the mirror in terms of the mirror position alone. be expected, as the Gaussian shaped atomic wavepack- To obtain the effective radiation pressure force in pres- ets at every site act like lenses that constantly refocus ence of the atoms we observe that, given positions of the the beam. A more detailed analysis of the spatial mode mirror and of the atoms, the equilibrium light field is structure of the light field inside the cavity will be pre- determined by the boundary conditions at the fictitious sented in a future publication.) “atom mirror,” see Fig. 5. The steady-state boundary In the following we consider the three representative conditions are found to be cases of an empty cavity, a cavity containing atoms giving rise to an intermediate reflection coefficient of a few tens of a percent and atoms with a reflection coefficient close ik0z 2ik0z ǫ1 = T1e Pin + e R1ǫ2 (26) to one. ǫ2 = Rǫ1 + Tǫp3 (27) 2ik0(L+ξ−z) ǫ3 = R2e ǫ4 (28) III. ADIABATIC DYNAMICS ǫ4 = Tǫ1 + Rǫ3. (29)

We restrict ourselves to the limit where the motion of the moving mirror is extremely slow compared to the from which we obtain, e.g. for ǫ4, 6

ik0za e √PinTT1 ǫ4 = − . (30) RR e2ik0za + RR e2ik0(L+ξ−za) R R (R2 T 2)e2ik0(L+ξ) 1 1 2 − 1 2 − − From these fields we find the adiabatic dipole potential felt by the atoms for a given displacement of the moving mirror as

2 2 2 gPin T1T (1 + R2 2 R2 sin 2k0(L + ξ za)) V (za; ξ)= g ǫ4 + ǫ3 = | | | | − | | − 2 , (31) | | RR e2ik0za + RR e2ik0(L+ξ−za) R R (R2 T 2)e2ik0 (L+ξ) 1 1 2 − 1 2 − −

a result that holds for every lattice site provided that 0.6 R1 R2. Otherwise the depth of the potential wells is different≃ at each lattice site. [26] 0.5 2

The potential V (za; ξ) is periodic with period λ0/2 in 0.4 both za and ξ direction. In the numerator we have, for R close to one, the well known cos2 k (L + ξ z)- | 2| 0 − 0.3 type potential. Its minima are moving as the cavity end- ξ mirror is moving. 0.2 The effect of the denominator, which gives rise to a series of resonances, is somewhat less intuitive. If the re- 0.1 flection coefficient of the atoms is small their back-action 0 on the cavity field is weak and we see essentially the res- onances of an empty Fabry-P´erot cavity: Whenever the −0.1 1 1 0 0.2 0.4 0.6 0.8 1 distance between the two cavity mirrors is (m + 2 )λ0/2 z with m an integer the field inside the cavity is very high a and accordingly the dipole potential very deep [27]. Fig. 6 shows how a small atomic reflectivity slightly distorts Figure 6: (Color online) Dipole potential felt by the atoms for T1 = 0.1, T2 = 0.2 and relatively high transmissivity of the this type of potential. The zeros of the numerator are atomic mirror, T = 0.90. Here, and in the potential plots to visible as dark diagonal lines and the nonzero regions are follow, lighter regions are deeper in potential and the contour structured by the back-action of the atoms on the light- lines are drawn at logarithmic intervals for clarity and, as in field. the rest of this paper, all lengths are given in units of the The situation is qualitatively different if the reflection wavelength of the injected light field. The green dashed line coefficient of the atoms is high, see Fig 7. In this case illustrates the path the atoms take when the mirror is moving the dipole potential is deep whenever the atoms are in starting at the green dot with the label 1 while the blue solid line shows the path the atoms take when the mirror moves a position where they form a resonant cavity with one back down starting at the blue dot labeled 2. of the end mirrors. Interestingly, this typically gives rise to two potential minima per λ0/2, resulting in two dis- tinct modes of motion for the atoms: They can follow the moving mirror in the local minimum along diagonals through such a resonance, the atoms have to decide which in Fig. 7, or alternatively they can remain locked with branch of the potential valley to follow on the other side the fixed mirror in the local minimum following vertical of the resonance [28]. Since we can decide which branch lines in Fig. 7. These two minima are actually already of the potential well is deeper by choosing the reflectiv- present for small atomic reflectivity, but only for a very ities of the end mirrors appropriately we can select the narrow range of ξ values near ξ 0.25. mode of motion that the atoms will perform. ≈ The relative depth of the two local minima is deter- The adiabatic motion of the atoms is particularly inter- mined by the reflectivities of the two end mirrors. The esting if the fixed mirror has the higher reflectivity. We potential well closer to the mirror of higher reflectivity is discuss it in detail for the case of small atomic reflectiv- deeper. As illustrated in Fig. 8 the system shows behav- ity. The trajectories of the local minima corresponding ior reminiscent of an “avoided crossing” where the two to the mirror moving in the positive, resp. negative ξ di- resonances corresponding to the two modes of motion rection are drawn as a green dashed line and a blue solid cross: In positions where the atoms would be on reso- line, respectively, in Fig. 6. If the moving mirror starts nance with both mirrors the potential depth has a fairly in a region far off resonance, e.g. at the green dot labeled narrow local maximum. The width of the potential bar- “1” in Fig. 6, there is only one potential minimum corre- rier arising this way becomes smaller as the reflectivity of sponding to the maximum of the numerator in the dipole the atomic mirror increases. If the moving mirror sweeps potential. Moving in the direction of growing ξ the po- 7

0.6 5 negative ξ−direction 4.5 ξ 0.5 positive −direction 4

0.4 3.5

0.3 3 ) ξ ( ξ 2.5 RP

0.2 F 2

0.1 1.5

1 0 0.5

−0.1 0 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 z a ξ

Figure 7: Dipole potential for higher atomic reflectivity corre- Figure 9: (Color Online) Radiation pressure force for the two sponding to T = 0.05 and T1 = 0.2, T2 = 0.1. For this choice directions of motion of the mirror. The atoms take the paths of reflectivities of the cavity end mirrors the potential well shown in Fig. 6 for the mirror moving in the positive and closer to the moving mirror is always deeper and, in contrast negative ξ-direction, respectively. The difference in area of to Fig. 6 the atoms move on diagonals following the bottom the two curves is a measure of the work performed on the of the potential well for both directions of the mirror motion. moving mirror. All parameters as in Fig. 6.

0.05 resonance, where this story repeats itself. 0.04 The atomic motion is totally different if the moving 0.03 mirror moves in the negative ξ direction. Then the 0.02 branch of the potential closer to the moving mirror pro- vides a local minimum all the way and the atoms never 0.01 move to the left branch, see the blue line in Fig. 6.

ξ 0 The motion of the atoms is similar for high atomic reflectivity. If the mirror is moving in the positive ξ di- −0.01 rection the atoms turn left at every resonance into the −0.02 vertical potential valley. Just before reaching the zero −0.03 potential line they roll to the right and fall into the di- agonal valley. If the mirror moves back in the negative ξ −0.04 direction the atoms also turn left at every resonance and −0.05 move to the left into the diagonal potential valley before 0.2 0.22 0.24 0.26 0.28 0.3 z reaching the zero potential line. a Interestingly, the radiation pressure force is different Figure 8: Enlargement of the resonance region of Fig. 7, for the two distinct paths the atoms take as the mirror is showing that the potential well corresponding to motion with moving back and forth, see Fig. 9. Hence the light field the moving mirror is deeper than the one corresponding to performs work on the mirror in every round trip, and this the resonance with the fixed mirror, and explaining why the work can be fairly large. It should be noted however that atomic motion is locked to the mirror motion. the figure also our model is too simplistic to study this feature quantita- shows the avoided crossing of the dipole potential. tively. First, the assumption of adiabaticity of the atomic motion is likely to break down near the zeros of the nu- merator, where the atoms take a sharp turn to the right. tential valley eventually reaches a branching point and The potential well becomes extremely shallow in this re- the atoms follow it to the left towards the fixed mirror, gion and when the atoms fall into the other branch of since the potential well to the left is deeper if the fixed the potential, both situations for which the assumption mirror has the higher reflectivity. Near the diagonal zero of adiabaticity of the atomic motion may not hold. If the lines of the numerator the potential well becomes shal- atoms have some inertia, they could overcome the poten- lower and shallower until finally it ceases to provide a tial wall that prevents them from going over the potential local minimum. The atoms “roll” to the second branch hill and continue their motion on the other side. Tunnel- of the potential valley closer to the moving mirror and ing through the barrier is another possibility. For these follow this local minimum until the system hits the next reasons we concentrate in the following on the case where 8

8 20 1 with moving mirror no atoms 7 with fixed mirror small R empty cavity 6 0 large R 15 5 −1 4

) −2 ) ξ (

ξ 3 10 ( RP RP

F −0.5 0 0.5 1

V 2

1

5 0

−1

−2 0 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ξ −3 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ξ Figure 10: (Color online) Radiation pressure force for atomic motion locked to the motion of the moving mirror for in- Figure 11: (Color online) Radiation pressure potential for termediate atomic transmissivity T = 0.9 (red dashed line), atoms moving with the moving mirror and an empty cavity. small atomic transmissivity T = 0.05 (green solid line) and All parameters as in Fig. 10. an empty cavity (blue dash dotted line) for comparison. To ensure that the atoms are following the desired branch of the potential we choose T1 = 0.2, T2 = 0.1. The injected laser of the potential is near ξ 5 in this case. They are −1 power is 0.01W and α = 10W . smaller for the double peaks≈ for high atomic reflectivity and smallest for the empty cavity. the atoms move with the moving mirror, i.e. where the reflectivity of the moving mirror is higher than that of IV. SIGNATURES OF THE COUPLED MOTION the fixed mirror. For this type of motion the adiabaticity assumption holds at all times. The question of how to detect the coupled motion of Once the position of the atom mirror z (ξ) has been a the mirror and the atoms naturally arises. A possible ob- determined we can find the radiation pressure force by in- servable is the spectrum of the light transmitted through serting z (ξ) into the expression for ǫ . Fig. 10 shows the a 4 the cavity. The amplitude and phase modulation of the three different cases of low atomic reflection coefficient, optical field due to the mirror motion are detectable as high atomic reflection coefficient as well as an empty cav- sidebands in the spectrum of transmitted light. ity for comparison. A small atomic reflectivity broadens To find the spectrum of the transmitted light we inte- the resonance. Also, the peak radiation pressure force grate the Newtonian equations of motion of the mirror increases. As the atomic reflectivity increases the reso- subject to the modified radiation pressure force using the nance becomes wider and wider and develops a broad dip velocity Verlet algorithm. [29] in the middle. For very high atomic reflectivities adjacent resonances “collide” and give rise to an avoided crossing behavior that results in the double peak structure of Fig. 10. The double resonances in this regime are much wider A. Large amplitude oscillations than the empty cavity resonances and the peaks are sub- stantially higher. Also the radiation pressure force never First we analyze what happens as the mirror undergoes falls off to values as low as for the empty cavity. The low large amplitude oscillations such that in every period it intensity region between the double peaks are the limit- sweeps over several resonances. [30] We consider again ing shape of the aforementioned dips that develop in the the three cases of atoms with high reflectivity, interme- resonant lineshape. diate reflectivity and an empty cavity. The cavity and From the radiation pressure force we find the total po- atomic parameters are the same as before and as for the tential of the moving mirror by a simple integration and quality factor of the cavity we choose Γ−1 = 104. by adding the harmonic potential due to the restoring Figure 12 shows the resulting sideband spectra. For force. Figure 11 shows the total potentials, again for the clarity each data point has been obtained by averaging three different cases considered for the radiation pressure over 1000 frequency bins. The spectrum of the empty force above. Every time the mirror moves through a res- cavity falls off nearly exponentially, a consequence of the onance the potential makes a steep step down. The steps near-Lorentzian lineshape of the resonances and the fact are highest if the atoms have intermediate reflectivity due that the mirror moves through each resonance at nearly to the broad and high resonances. The global minimum constant velocity. 9

3 4

2 2 4

1 0 3 ) ))

ω −2 0 ξ 0 20 40 ( S( loc 2 10 Ω (

−1 10 log

log 1 −2

no atoms 0 −3 no atoms small R small R large R large R −4 −1 0 1000 2000 3000 4000 5000 6000 7000 8000 ω 1 2 3 4 5 6 7 ξ

Figure 12: (Color online) Sidebands of the transmitted light for atoms moving with the mirror, staying with the fixed mir- Figure 13: (Color online) Approximate harmonic frequencies ror and for an empty cavity. near the local minima for atoms with high reflectivity, inter- mediate reflectivity and an empty cavity. All parameters as in Fig 12.

The widths of the transmission spectra are compara- ble in all three cases, confirming that the widths of the B. Small amplitude oscillations resonances are of the same order. However the spectra acquire a peak near ω = 0 for nonzero atomic reflec- In our units the average thermal energy of the mir- tivities, a consequence of the larger resonance widths in ror at room temperature is T 10−5 . Comparing with these cases; in addition the light field never falls off to the energy scale in Fig. 11 we≈ see that the mirror can zero, in contrast to the empty cavity case where it does undergo many oscillations in each one of the sawtooth- so to a good approximation for high-reflectivity mirrors. shaped local potential minima before it hops to a different Especially for high atomic reflectivity the cavity always local minimum due to thermal activation or some other leaks a significant amount of light that gives rise to an perturbation. Thus it is an interesting question to deter- enhanced DC component in the transmission spectrum. mine the oscillation frequency at every local minimum for small amplitude oscillations. The fringes that appear in the case of high atomic re- flectivity can be understood in the following way: two Figure 13 shows these frequencies for the three cases light pulses exit the cavity whenever the mirror passes already discussed. For high atomic reflectivity, and in through a resonance, see Fig. 10, and they give rise to a contrast to the intermediate reflectivity and empty cav- double-slit diffraction pattern in frequency space. Their ity cases, the oscillation frequency near ξ = 0 is much spacing corresponds to the inverse of the time it takes the lower than the restoring frequency of the mirror. This is mirror to travel from the first peak to the second peak of because the resonances have wide wings in this case and, the double resonance. if added to the harmonic oscillator potential of the mir- ror, they can compensate the harmonic restoring force The inset of Fig. 12 shows the details of the low fre- almost completely over a large range of displacements ξ quency part of the spectrum for the case of high atomic and give rise to an extremely flat potential near the local reflectivity. (That low frequency part is similar in all minimum. In the other two cases the onset of the radia- cases.) It consists of a peak at the frequency correspond- tion pressure force near the resonances is very abrupt and ing to the time it takes the mirror to move from one res- the frequency of the mirror motion can never be signif- onance to the next and of a series of additional peaks at icantly smaller than the oscillator frequency. At higher each harmonic. These strong components are a result of displacements from the mirror rest positions the stronger the highly nonlinear potential. The fact that the mirror resonances found with atoms inside the cavity give rise to moves at different speeds between different resonances stronger confinement and lead to resonance frequencies gives rise to a broadening of the harmonics into bands that eventually become higher than in the empty cavity whose width increases with growing harmonic number to case. Note that for the empty cavity cases we could find finally overlap. Effects due to the damping of the mirror only a few more local minima beyond ξ = 8 and these motion cannot be resolved because of this “inhomoge- could not be reasonably approximated by a harmonic po- neous broadening”. tential. 10

V. CONCLUSION AND OUTLOOK to see how all these effects are influenced by the presence of the atoms. We have developed a model that describes the coupled A related question of practical significance is what hap- motion of a cavity end-mirror and atoms trapped in the pens if the atoms are subject to a friction force. Atoms standing-wave intracavity field. The key result is that can be efficiently cooled to very low temperatures with the resulting model is simple enough to allow for a fairly laser cooling and one could hope that by means of the straightforward analysis. The self-consistent electric field coupling to the mirror through the cavity field one should inside the cavity containing the atoms plays a central role be able to also cool the mirror. Loosely speaking, if the since from this field we find the modified dipole potential atoms are subject to a friction force, they lag slightly in which the atoms move as well as the modified radiation behind the position at which they ought to be accord- pressure force acting on the moving mirror. We find that ing to the position of the mirror. This retardation has as a result of the strong coupling between light field and been seen to lead to the damping of the mirror motion atoms due to the collective back-action of the atoms the in preliminary results, but these calculations need to be dipole potential changes qualitatively, the most notable carried out in much more detail, especially since the mo- feature being that the atomic position becomes bistable. mentum kicks in laser cooling, and hence the fluctuations The resonances of the cavity are broadened and develop in the atomic position, are substantial and cannot be ne- a double peak structure in the limit of high atomic re- glected. They can be taken into account e.g. using a flectivity. We have studied the dynamics of the mirror master equations approach. subject to the modified radiation pressure force in the Finally, one needs to address the situation where all adiabatic limit where the atomic motion is much faster constituents of our model are quantized. Quantization of than the motion of the mirror and the light field is slaved the atomic motion should allow one to study whether the to both atoms and mirror. We have shown that the side- resonances can be used as beam splitters, and the prop- bands of the light transmitted through the cavity can erties of the resulting interferometer. Also, a detailed un- serve as an indicator of the coupled motion of the mirror derstanding of the role of damping of the atomic motion and the atoms. requires a proper quantum treatment. The quantization Many open questions remain to be studied. First of of the mirror motion is called for in view of the the exper- all a more careful analysis of the spatial mode structure imental goal to cool it to its quantum mechanical ground inside the cavity in the presence of the atoms is desir- state. Finally, a quantum treatment of the light field, able. Such an analysis could start from the Maxwell- and especially of its fluctuations, is interesting because Bloch equations and should include scattering losses due of the powerful methods to measure the fluctuations of to spontaneous emission from the atomic excited state. the electromagnetic field. If they are linked to fluctu- It would yield more accurate values for the effective re- ations of the atomic position or the mirror position, as flection coefficient of the atoms and the modified dipole must be expected, one could then use the light field as a potential, especially if R1 and R2 are very different. powerful diagnostic tool for the study of these mechanical One should also consider different adiabaticity regimes, fluctuations. or abandon the assumption of adiabaticity altogether. The situation where the atomic motion is roughly as fast as the mirror motion and the atom and mirror influence VI. ACKNOWLEDGEMENTS each other on an equal footing is particularly interest- ing, and so is the case where the retardation of the light This work is supported in part by the US Office of field is taken into account. Already in an empty cavity Naval Research, by the National Science Foundation, by the retardation of the light field gives rise to many in- the US Army Research Office, by the Joint Services Op- teresting phenomena such as instabilities, self sustained tics Program, and by the National Aeronautics and Space oscillations and chaotic behavior. It will be interesting Administration.

[1] A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and (1989). H. Walther, Phys. Rev. Lett. 51, 1550 (1983). [7] V. Pierro and I. M. Pinto, Phys. Lett. A 185, 14 (1994). [2] P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, [8] J. M. Aguirregabiria and L. Bel, Phys. Rev. A 36, 3768 J. Opt. Soc. Am. B 2, 1830 (1985). (1987). [3] S. Solimeno, F. Barone, C. de Lisio, L. D. Fiore, L. Mi- [9] I. Wilson-Rae, P. Zoller, and A. Imamo¯glu, Phys. Rev. lano, and G. Russo, Phys. Rev. A 43, 6227 (1991). Lett. 92, 075507 (2004). [4] F. Marquardt, J. G. E. Harris, and S. M. Girvin (2005), [10] P. F. Cohadon, A. Heidmann, and M. Pinard, Phys. Rev. cond-mat/0502561. Lett. 83, 3174 (1999). [5] C. Stambaugh and H. B. Chan (2005), cond- [11] Constanze H¨ohberger Metzger and K. Karrai, Nature, mat/0504791. London 432, 1002 (2004). [6] B. Meers and N. MacDonald, Phys. Rev. A 40, 3754 [12] D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, J. 11

Opt. Soc. Am. B 20, 1054 (2003). tial wells, i.e. they are always near the bottom of their [13] K. Jacobs, I. Tittonen, H. M. Wiseman, and S. Schiller, respective wells and never see the tops of the potential Phys. Rev. A 60, 538 (1999). barriers separating them. [14] I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. M¨uller, [27] The additional λ0/4 offset comes from our particular R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. choice of phase of the reflection and transmission coef- F. de Rooij, Phys. Rev. A 59, 1038 (1999). ficients. [15] J.-M. Courty, A. Heidmann, and M. Pinard, Eur. Phys. [28] The resonances act like beam splitters for the atoms and a J. D 17, 399 (2001). more exact analysis should treat the motion of the atoms [16] W. Marshall, C. Simon, R. Penrose, and near these branching points quantum-mechanically. Also D. Bouwmeester, Phys. Rev. Lett. 91, 130401 (2003). one can easily convince oneself that atoms originally in [17] L. Bel, J. Boulanger, and N. Deruelle, Phys. Rev. A 37, neighboring potential wells can end up in the same po- 1563 (1988). tential well if they choose to go into different branches at [18] C. K. Law, Phys. Rev. A 51, 2537 (1995). a resonance. This provides one with the essential ingredi- [19] I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. ents for an interferometer that can in principle operate in Phillips, Phys. Rev. A 52, 1394 (1995). a massively parallel fashion. This problem will be studied [20] M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ansch, and in greater detail in a future publication. I. Bloch, Nature (London) 415, 39 (2002). [29] The canonical nature of this algorithm is necessary be- [21] P. Meystre and M. Sargent III, ”Elements of Quantum cause we have to integrate the equations of motion over Optics” (Springer-Verlag, 1999), 3rd ed. very long time intervals, typically several ten thousand [22] M. Born and E. Wolf, Principles of Optics (Cambridge cycles. At the same time we have to integrate with high University Press, 1980), 6th ed. temporal resolution because the radiation pressure force [23] S. Slama, C. von Cube, M. Kohler, C. Zimmermann, and is so sharply peaked. The position of the atoms is deter- P. W. Courteille (2005), quant-ph/0511258. mined at every time step using Brent’s method [24] in an [24] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. interval centered at the atomic position of the previous Flannery, Numerical recipes in C (Cambridge University time step and with a width that depends on the velocity Press, Cambridge, UK, 1992), 2nd ed. of the mirror. This way we ensure that the atoms follow [25] Strictly speaking, Eq. 24 is the the reflection coefficient a continuous path along a potential valley without jumps for a wave propagating in the positive z direction with to different valleys. The transmitted field at every time the reflection coefficient in the negative z direction differ- is easily found from ǫ4. The side band spectrum is found ing by an additional phase e4iNφ. This phase is however by numerically fourier transforming the field. immaterial for our discussion since it can be absorbed in [30] To avoid too large velocities at the bottom of the poten- a redefinition of za. The transmission coefficients for the tial, which would make an accurate numerical integration two directions are exactly equal as they have to be for more difficult, we choose the initial conditions such that general reasons. the mirror swings through roughly eight resonances. This [26] This should bring about only minor quantitative changes number is not exact since the turning point on the left is that are irrelevant for the limit of the adiabatic motion typically at a resonance. that we are considering here: We assume from the outset that the atoms are adiabatically moving in deep poten-