Atomic Bloch-Zener oscillations for sensitive force measurements in a cavity

B. Prasanna Venkatesh,1 M. Trupke,2 E. A. Hinds,2 and D. H. J. O’Dell11Department of Physics and Astronomy, McMaster University,

1280 Main St. W., Hamilton, ON, L8S 4M1, Canada2Centre for Cold Matter, Imperial College, Prince Consort Road, London, SW7 2AZ, United Kingdom

Cold atoms in an optical lattice execute Bloch-Zener oscillations when they are accelerated. Wehave performed a theoretical investigation into the case when the optical lattice is the intra-cavityfield of a driven Fabry-Perot resonator. When the atoms oscillate inside the resonator, we find thattheir back-action modulates the phase and intensity of the light transmitted through the cavity. Wesolve the coupled atom-light equations self-consistently and show that, remarkably, the Bloch periodis unaffected by this back-action. The transmitted light provides a way to observe the oscillationcontinuously, allowing high precision measurements to be made with a small cloud of atoms.

PACS numbers: 37.10.Jk, 37.10.Vz, 37.30.+i, 06.20.-f

When quantum particles in a potential lattice are sub-jected to a constant force F , they execute Bloch-Zeneroscillations (BZOs) [1] with a frequency

ωB = Fd/~ , (1)

where d is the period of the lattice. This behavior wasfirst demonstrated [2] with electrons in semiconductor su-perlattices, where a DC electric field provided the force.However, rapid dephasing due to impurities [3] has pre-vented BZOs from becoming useful in solid state devices.

Cold atoms in optical lattices have recently providedan alternative realization of BZOs [4, 5, 6, 7, 8, 9, 10]in which long coherence times are possible. Initially,acceleration of the lattice induced the oscillations, butin subsequent experiments [7, 9, 10], gravity providedthe required force. In [10], the BZO damping time was12s, allowing some 4000 cycles to be measured over 7s.With such long coherence times, cold atom BZOs be-come suitable for high precision measurements, for ex-ample to determine the fine structure constant α [4], tomeasure gravity [10], or to explore Casimir-Polder forces[11]. In the experiments to date, it has been necessary toreconstruct the oscillations by making destructive mea-surements at a large number of different times, each mea-surement requiring a new cloud of atoms to be trapped,cooled, and loaded into the lattice. The process is la-borious and suffers from shot-to-shot variations in theinitial cloud conditions. In this letter, we discuss howthe measurement could be substantially improved by us-ing an optical cavity to enhance the interaction of theatoms with the light. We show how the light transmit-ted through the cavity can provide an in vivo observationof the BZOs and we assess the extent to which this per-turbs the motion of the atoms. Finally we consider thestatistical sensitivity of the method and show that it canyield high precision in a single shot.

Let us take the cavity to be a vertical Fabry-Perot res-onator illuminated by a laser, which makes a standing-wave light field inside, shown in Fig. 1(a). Cold atoms inthis lattice execute BZOs under the influence of gravity,

a+

(a)

(b)

s

d=/2

(c)

(d)

FIG. 1: Schematic of the proposed experiment. (a) A cloudof cold atoms is held in a standing-wave optical trap insidea vertical Fabry-Perot cavity. (b) The atoms execute Bloch-Zener oscillations, leading to a periodic modification of theirwave function. (c) This modulates the intra-cavity power andhence the lattice depth s. (d) The power modulation is seenin the light transmitted by the cavity.

as illustrated in Fig. 1(b). The optical dipole interactionbetween one cavity photon and an atom placed at anantinode of the field is given by ~g0 = µ

√~ωc/(ε0V ),

where ωc and V are the frequency and volume of the rel-evant cavity mode and µ is the atomic transition dipolemoment. The effect of one atom on the cavity field ischaracterised by the cooperativity C = g2

0/(2κγ), where2γ is the atomic spontaneous emission rate in free spaceand 2κ is the cavity energy damping rate. When C & 1,the cavity field is strongly perturbed by the atom, asillustrated in Fig. 1(c). Thus the light transmitted or re-flected by such a cavity can detect the presence of a singleatom [12], and can be sensitive to the motion of atomstrapped within the cavity [13], as in Fig. 1(d). Althoughwe shall not discuss Bose-Einstein condensates (BECs) inthis paper, we note in passing that several experimentshave succeeded in placing BECs inside optical cavities[14].

Consider a cavity mode, whose frequency in the ab-

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sence of any atoms is ωc, pumped by an external laser ofthe same frequency. A cloud of N atoms, each of massm, is placed inside the cavity and is sufficiently dilutethat the atoms do not interact directly with each other.The coupled atom-cavity hamiltonian becomes [15]

H =~2

2m

∫|∇zψ|2dz +

~g20

∆a†a

∫ψ†ψ cos2(kcz)dz

+F∫ψ†ψ z dz − i~(η∗a− ηa†) (2)

where the operator a(t) annihilates a photon in the cav-ity mode and the operator ψ(z, t) annihilates an atomat the point z. The first term gives the kinetic energyof the atom. The second describes the quadratic Starkinteraction in the rotating-wave approximation. Here,∆ = ωc − ωa is the detuning between the cavity modeand the atomic transition frequency ωa, and kc = ωc/c.This is an approximate form, that is valid when ∆ � γand η2 � (κ∆/g0)2. Under these conditions the atomhas negligible population in the excited state. The thirdterm accounts for the external force, and the last termdescribes the coherent excitation of the cavity by the ex-ternal laser. For a cavity with equal mirror reflectivities,the pumping rate is η =

√κI, where I is the rate of in-

cident photons matching the cavity mode. The cavityfield is a driven and damped quantum harmonic oscil-lator for which it is known that exact solutions of theFokker-Planck equation are coherent states [16]. Fur-thermore, since there is negligible spontaneous emissionby the atoms, the cavity field remains in a coherent statein the presence of atoms. Taking the expectation valueof the Heisenberg equations of motion for a and ψ in thecoherent state |α〉 yields the equations of motion [17]:

α = −iαNg2

0

∆

∫|Ψ|2 cos2(kcz)dz + η − κα , (3)

i~Ψ =(−~2

2m∂2z +

~g20 |α|2

∆cos2(kcz) + Fz

)Ψ (4)

where Ψ(z, t) = 〈ΨN−1|ψ(z, t)|ΨN 〉/√N is the wave

function occupied by all N atoms and |ΨN 〉 is the cor-responding state vector in Fock space. We have added adamping term proportional to κ in Eq. (3) to account forleakage of light through the mirrors [18]. These equationsneglect quantum fluctuations of the light field which canheat the atoms: we return to this effect later. Equa-tions (3) and (4) must be solved self-consistently: thecoupling

g2(t) = g20

∫|Ψ(z, t)|2 cos2(kcz)dz (5)

changes α, which changes the depth of the lattice. Thisalters the atomic wave function and therefore changes g,etc. In static equilibrium, α = 0 and then

α =η

κ

11 + iNg2(t)/(κ∆)

. (6)

Even if the atoms are in motion, (6) remains a verygood approximation since κ is generally much greaterthan the highest frequency in the atom dynamics, so thatthe field ‘instantaneously’ adapts to the atomic distribu-tion. Frequencies that feature in the atomic motion arethe BZO frequency, the band splitting, and the harmonicfrequency ωho = 2g0|α|

√ER/(~∆) at the bottom of each

potential well, ER = ~2k2c/(2m) being the atomic recoil

energy. For the experiments we consider here ωB and ωho

are much smaller than κ, so we assume in our analyticcalculations, though not in our numerical simulations,that Eq. (6) holds.

Let us recall the standard theory of BZOs withouta cavity [19, 20, 21, 22, 23, 24, 25]. The SchrodingerEq. (4) can be written as i~Ψ = (H0 + Fz)Ψ, whereH0 = p2/(2m) + s cos2 kcz. The eigenfunctions χq,s,n(z)of H0 are Mathieu functions (Bloch waves) that in gen-eral depend on position z, lattice depth s, band index nand quasimomentum q, restricted to the first Brillouinzone −π/d ≤ q ≤ π/d [26]. We assume in our analyticcalculations that the atoms are in the lowest band, whoseenergy is Eq,s, and dispense with the band index so thatH0χq,s = Eq,sχq,s. The Bloch theorem allows us to writeχq,s(z) = Uq,s(z) exp[iqz], where Uq,s(z) = Uq,s(z + d)obeys

(p+ ~q)2

2mUq,s(z)+s cos2(kcz)Uq,s(z) = Eq,sUq,s(z). (7)

To tackle the full hamiltonian H0+Fz we make the gaugetransformation Ψ(z, t) = exp[−iFtz/~]Ψ(z, t), yielding

the Schrodinger equation i~ ˙Ψ = ˜HΨ, where ˜

H = (p −Ft)2/(2m) + s cos2 kcz. Comparing this with (7) wesee that the effect of the force is to evolve the quasi-momentum according to Bloch’s acceleration theorem [1]

q → q(t) = q0 − Ft/~ , (8)

where q0 is the quasimomentum at t = 0. When q(t)reaches the edge of the Brillouin zone at −π/d it ismapped to the identical point q = +π/d, giving rise to os-cillatory behaviour - the BZO. The corresponding Blochwave has the approximate form [27] (setting q0 = 0)

Ψ(z, t) ≈ Uq(t),s(z) exp[−i/~∫ t

dt′ Eq(t′),s] , (9)

within the adiabatic approximation that the rate ofchange U/U is too small to excite higher bands. HereUq(t),s(z) is the instantaneous solution of Eq. (7). Dur-ing a BZO the spatial distribution Uq(t),s(z) oscillateswith a breathing motion, as shown schematically in Fig.1(b).

Consider now the effect of the BZOs on the field insidethe cavity. The coupling g (Eq. (5)) depends on |Ψ(z, t)|2which equals |Ψ(z, t)|2. Its breathing motion changes g,which in turn modulates the cavity field through Eq. (6).

3

Inserting (6) into (4), and replacing η by√κI, we ob-

tain the Schrodinger equation for atoms in a periodicpotential i~Ψ = (p2/2m + s(t) cos2(kcz) + Fz)Ψ, withthe time-dependent potential depth

s(t) = ~I(g2

0

∆κ

)1

1 + (Ng2(t)/(∆κ))2 . (10)

When Ng20/(κ∆) � 1, s is approximately constant in

time and α becomes (η/κ)[1 − iNg(t)2/(∆κ)], i.e. thelight exhibits a small phase modulation and has negligi-ble intensity variation. In this case, the atoms oscillatein a lattice that is essentially static. With stronger cou-pling, where Ng2

0/(κ∆) ' 1, s(t) is changed significantlyduring an oscillation. The fundamental period is never-theless unchanged and is still given by Eq. (1). Physi-cally, this is because BZOs arise from an interference ofwaves in a lattice akin to Bragg scattering and latticedepth plays no role in determining the phase-matchingcondition. Rather, this is determined by the symmetryof the hamiltonian which is precisely maintained at alltimes. To examine the effect of lattice depth modula-tion in more detail, let us begin with the adiabatic casewhere the frequency spectrum of s(t) remains largely atlow frequencies unable to excite higher Bloch bands. Anexample is shown in Fig. 2(a). In that case, the low-est band energy Eq,s(t) is still determined by the instan-taneous value of s(t), with H0(t)χq,s(t) = Eq,s(t)χq,s(t),and q0 remains a constant of the motion generated byH0(t) despite the time-dependent potential ([19] reachesa similar conclusion for electrons in an ac field). Con-sequently, the Bloch wave in the presence of an externalforce can still be calculated using Eq. (9), provided weuse the instantaneous values of s(t) and s(t′). It only re-mains to find the self-consistent solution for s(t) by solv-ing Eq. (10) at each instant of time. Here s(t) appearsexplicitly on the left and also implicitly on the right asa parameter determining the Bloch wave function that isrequired to calculate the coupling

√Ng(t) using Eq. (5).

We conclude that despite the lattice depth modulation,the Bloch acceleration theorem (8) still holds for atomsin the lowest band and therefore the fundamental oscilla-tion frequency remains identical to the ωB of an atom ina static lattice. Furthermore, because ωB is the same forall bands, there is no frequency shift even when higherbands are excited, as we have verified numerically for awide range of Ng2

0/(∆κ).Fig. 2(a) compares the lattice depths obtained by nu-

merical solution of Eqs. (3) and (4) (lines) and by the adi-abatic approximation of Eqs. (9) and (10) (dots). Thetwo are in good agreement. Figure 2(b), shows a caseof stronger coupling, where one can see fast oscillationssuperimposed on the main motion. The Fourier trans-form of this reveals two effects, illustrated in Fig. 2(c).(i) There are higher harmonics of ωB because the os-cillations at the Bloch periodicity are not exactly sinu-

0 1 2 3 4 52.92

2.94

2.96

2.98

3

0 1 22.67

2.74

2.81

2.88

2.95

3.02

Bω t / 2π

Bω t / 2π

s / E

Rs /

ER

(a)

(b)

(c)

0 5 10 15 20 25 300

500

1000

1500

2000

2500

Bω /ω

s (ω

)~

6 8 10 12 14 160

1

2

3

4

5

FIG. 2: Calculated evolution of lattice depth s(t) normal-ized to the recoil energy for 87Rb atoms undergoing 840 HzBloch-Zener oscillations in a 780 nm lattice with s ≈ 3ER.Lines: numerical solution to Eqs. (3) and (4). Dots: self-consistent adiabatic approximation of Eqs. (9) and (10). (a)Ng2

0/(κ∆) = 0.4. (b) Close-up of lattice depth oscillationswith stronger coupling, Ng2

0/(κ∆) = 1. Non-adiabatic ef-fects are seen in the line. (c) Fourier transform s(ω) of resultin (b), showing harmonics of fundamental frequency ωB. In-set: Close-up of s(ω) at higher frequencies. Harmonics of ωB

appear as sharp vertical lines. In addition, one sees muchweaker, broad, Fourier components due to band excitation,corresponding to the rapid oscillations in (b). The actual in-dividual values of the parameters used (or assumed in the casewhere only ratios enter) in the calculations were (see text):N = 5 × 104, g0 = 2π × 2.8 MHz, κ = 2π × 1.0 MHz. In(a) ∆ = 2π × 1.0 THz, η = 2π × 39 MHz; in (b) and (c)∆ = 2π × 0.39 THz, η = 2π × 28 MHz.

soidal. In the adiabatic solution, the first four are ac-curately reproduced and the higher harmonics are verysmall. (ii) The exact solution of Eqs. (3) and (4) predictsnon-adiabatic components at higher frequencies, showninset in Fig. 2(c). These are predominantly harmonicsof ωB (the sharp lines), but in addition, there are other

4

frequency components that can be seen as broad lines.These are only found in the full numerical solution andare due to a small amount of excitation to higher Blochbands. These non-adiabatic effects become stronger asthe parameter Ng2

0/(κ∆) is increased.

The atoms are driven not only by the mean-field po-tential s(t) cos2(kcz), but also by random forces due to i)spontaneous emission, and ii) fluctuations in the photonnumber, that are associated with the decay rates γ and κ,respectively. In particular, in the strong coupling regimephoton number fluctuations can significantly heat atomsinside an optical cavity [13]. We quantify the heating ef-fect via the increase in the width σp of the atom’s momen-tum distribution according to dσ2

p/ d t = 2D. The diffu-sion constant D =

∫∞0dt′[〈Fdip(t)Fdip(t + t′)〉 − 〈Fdip〉2]

[28] involves two-time correlations of the dipole forceFdip, and hence of the intracavity electric field. It hasbeen calculated for atoms in cavities in [29] and has twoterms. The first occurs in any standing-wave light fieldand at low saturation is given by Dsw = ~2k2/2τsp [28],where τ−1

sp = 2γ|α|2g20/∆

2 is the spontaneous emissionrate at an antinode. The second term, specific to cav-ities, is Dcav = 2Dsw C sin2(2kcz). This diffusion lim-its the coherent measurement time, which we take to bethe time τ when the momentum distribution has a widthequal to one half of the first Brillouin zone, i.e. σp = ~kc.Then τ = τsp/(1+C), where we have replaced sin2(2kcz)by 1/2 - a good approximation in the ground band for alattice of depth s = 3ER.

The BZOs can be observed by detecting the photoncurrent |α(t)|2κ transmitted through the cavity whichis directly proportional to the depth s(t) of the lattice(s = ~g2

0 |α|2/∆, see Eq. (4)), whose evolution is shownin Fig. 2. For an estimate of the measurement preci-sion, let us write the detection rate as R[1 + ε cos(ωt)].After measuring this for a time τ with detectors hav-ing an efficiency ξ, the shot noise gives an uncertaintyin the oscillation frequency of σω ≈ 2πτ−3/2/(ε

√ξR),

in which τ−1 comes from the linewidth due to the fi-nite duration of the measurement, and τ−1/2 comes fromthe shot noise in this bandwidth. This simple estimateis close to the Cramer-Rao lower bound [30], the limitgiven by the information content of the signal. Forsmall ε, R ≈ |α|2κ, then the frequency uncertainty canbe written as σω ≈ 2π s~

1ε√ξ( g

20

κ∆ )2( 1C + 1)3/2. In order

to bring out the implicit dependence on the number ofatoms N contained in this result, we define the parame-ter x = Ng2

0/(κ∆). Referring to Eq. (10), if the numberof atoms is increased then proportional increases in thelaser detuning and intensity maintain constant values ofs, x and ε, while the measurement time τ and the intra-cavity power both increase by the factor N . With thisscaling, the measurement time, detuning and frequency

uncertainty are given by

τ =~sx

NC

1 + C(11)

∆ =2γxNC (12)

σω ≈2πsx2

√ξ~ε

1N2

(1C

+ 1)3/2

. (13)

The uncertainty σω therefore decreases rapidly with adramatic 1/N2 dependence which ultimately derives fromthe continuous observation of the oscillations (via theenhanced measurement time τ).

Small s reduces σω, but the lattice must support theatoms against gravity. We find that s = 3ER is a rea-sonable compromise. In the example of Fig. 2(a) we havechosen x = 0.4, which gives ε = 1.3% in this lattice.Taking a reasonable number of atoms, let us say 5× 104,a readily achieved cooperativity of C = 1.3, and a pho-ton detector with 60% efficiency, brings σω/ωB to 1ppm.From the definitions of x and C this requires a detun-ing of ∆ = 2π × 1 THz and from Eq. (11) a measure-ment time of only τ = 1 s. These numbers fix the ratiog2

0/κ = 2π × 7.8 MHz. If we choose κ = 2π × 1 MHz,then using s = ~g2

0 |α|2/∆, this means there are on aver-age > 1400 photons in the cavity.

We have not included direct atom-atom interactionsin the model discussed here. These can lead toquasimomentum-changing transitions (non-vertical tran-sitions in the language of [31]) which dephase the BZOs.However, as summarized in the opening paragraphs ofthis paper, long-lived BZOs have already been success-fully demonstrated in cold gases containing many atoms,and so it is a question of degree, i.e. at what atom den-sity and interaction strengths do the interactions be-come important? An experiment investigating the con-trol of interaction-induced dephasing of BZOs in a Bose-Einstein condensate has recently been reported [32]. Us-ing a Feshbach resonance they were able to increase theirdephasing time from a few to more than 20 thousandBZO periods. From the details of their measurements weestimate that, for our example given immediately aboveinvolving 5 × 104 atoms, the dephasing due to collisionsis negligible for reasonable cavity geometries. The effectbecomes significant on increasing the number of atomsto several million, but can be suppressed by tuning to aFeshbach resonance [32]. Large detuning and laser powerimpose a practical limit on the useful atom number atabout this level anyway.

For atoms being continuously measured, an impor-tant source of dephasing is quantum measurement back-action. This effect is included in the estimate above ina quasi-classical way through the diffusive heating of theatoms by fluctuations of the cavity light field. The cavityfield suffers fluctuations because it is dissipatively cou-pled to the outside world and it is precisely the light

5

escaping from the cavity (at rate κ) that contains the in-formation about the state of the atoms. In other words,it is the cavity decay that is doing the measuring. Forour parameters, τ < 1

2τsp because C > 1, and thereforeit is the fluctuations due to cavity decay that limit themeasurement time, rather than the spontaneous emis-sion. We plan to perform a more microscopic study ofthe measurement back-action in the future.

In conclusion, we predict that the force on a small coldatom cloud can be measured very accurately by a newmethod based on BZO oscillations in an optical cavity.The BZO oscillations are measured continuously by mon-itoring the light that leaks out of the cavity. This enablesa relatively fast and high precision measurement of theoscillation frequency, the error being given by Eq. (13).Our treatment of the problem is based upon solving thecoupled equations of motion for the atoms and light. Thisgives a detailed picture of the dynamics, including theadiabatic and non-adiabatic aspects.

We note that since the submission of this paper a re-lated proposal on monitoring of Bloch oscillations usingan optical cavity has appeared [33].

We thank Pavel Abumov, Donald Sprung and Wytsevan Dijk for discussions. Funding was provided byNSERC (Canada), EPSRC, QIPIRC and the Royal Soci-ety (UK). Part of this work was done during D.O.’s stayat the Institut Henri Poincare - Centre Emile Borel. Hethanks this institution for hospitality and support.

[1] F. Bloch, Z. Phys. 52, 555 (1928); C. Zener, Proc. R.Soc. A 145, 523 (1934).

[2] J. Feldmann, K. Leo, J. Shah, D. A. B. Miller, J. E.Cunningham, T. Meier, G. von Plessen, A. Schulze, P.Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252(1992); C. Waschke, H. G. Roskos, R. Schwedler, K.Leo, and H. Kurz, K. Kohler, Phys. Rev. Lett. 70, 3319(1993).

[3] J.P. Reynolds and M. Luban, Phys. Rev. B 54, R14301(1996).

[4] R. Battesti, P. Clade, S. Guellati-Khelifa, C. Schwob,B. Gremaud, F. Nez, L. Julien, and F. Biraben, Phys.Rev. Lett. 92, 253001 (2004); P. Clade, E. de Miran-des, M. Cadoret, S. Guellati-Khelifa, C. Schwob, F. Nez,L. Julien, and F. Biraben, Phys. Rev. Lett. 96, 033001(2006).

[5] M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C.Salomon, Phys. Rev. Lett. 76, 4508 (1996); E. Peik, M.Ben Dahan, I. Bouchoule, Y. Castin, and C. Salomon,Phys. Rev. A 55, 2989 (1997).

[6] S.R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu,and M. G. Raizen, Phys. Rev. Lett. 76, 4512 (1996).

[7] B.P. Anderson and M.A. Kasevich, Science 282, 1686(1998).

[8] O. Morsch, J. H. Muller, M. Cristiani, D. Ciampini, andE. Arimondo, Phys. Rev. Lett. 87, 140402 (2001).

[9] G. Roati, E. de Mirandes, F. Ferlaino, H. Ott, G. Mod-ugno, and M. Inguscio, Phys. Rev. Lett. 92, 230402(2004).

[10] G. Ferrari, N. Poli, F. Sorrentino, and G. M. Tino, Phys.Rev. Lett. 97, 060402 (2006).

[11] I. Carusotto, L. P. Pitaevskii, S. Stringari, G. Modugno,and M. Inguscio, Phys. Rev. Lett. 95, 093202 (2005).

[12] C.J. Hood, M. S. Chapman, T. W. Lynn, and H. J. Kim-ble, Phys. Rev. Lett. 80, 4157 (1998); P. Munstermann,T. Fischer, P. W. H. Pinkse, and G. Rempe, Opt. Comm.159, 63 (1999); M. Trupke, J. Goldwin, B. Darquie, G.Dutier, S. Eriksson, J. Ashmore, and E. A. Hinds, Phys.Rev. Lett. 99, 063601 (2007).

[13] J. Ye, D. W. Vernooy, and H. J. Kimble, Phys. Rev.Lett. 83, 4987 (1999); C. J. Hood, T. W. Lynn, A. C.Doherty, A. S. Parkins, and H. J. Kimble, Science 287,1447 (2000); P. W. H. Pinkse, T. Fischer, P. Maunz, G.Rempe, Nature 404, 365 (2000); S. Gupta, K. L. Moore,K. W. Murch, and D. M. Stamper-Kurn, Phys. Rev. Lett99, 213601 (2007).

[14] S. Slama, C. von Cube, B. Deh, A. Ludewig, C. Zim-mermann, and P. W. Courteille, Phys. Rev. Lett. 94,193901 (2005); P. Treutlein, D. Hunger, S. Camerer, T.W. Hansch, and J. Reichel, Phys. Rev. Lett. 99, 140403(2007); Y. Colombe, T. Steinmetz, G. Dubois, F. Linke,D. Hunger, and J. Reichel, Nature 450, 272 (2007); F.Brennecke, S. Ritter, T. Donner, T. Esslinger , Science322, 235 (2008).

[15] C. Maschler and H. Ritsch, Phys. Rev. Lett. 95, 260401(2005); I. B. Mekhov, C. Maschler and H. Ritsch, Nat.Phys. 3, 319 (2007).

[16] W. H. Louisell and J. H. Marburger, IEEE J. QuantumElectron. QE-3, 348 (1967).

[17] P. Horak and H. Ritsch, Phys. Rev. A 63, 023603 (2001).[18] Langevin fluctuation terms do not appear in (3) if the

bath is at thermal equilibrium. See p 343 of C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Atom-Photon Interactions (Wiley, New York, 1992).

[19] M. Holthaus and D. W. Hone, Phil. Mag. B 74, 105(1996).

[20] Q. Thommen, J. C. Garreau and V. Zehnle, Phys. Rev.A 65, 053406 (2002).

[21] M. Gluck, A. R. Kolovsky, H. J. Korsch and N. Moiseyev,Eur. Phys. J. D 4, 239 (1998).

[22] M. Holthaus, J. Opt. B: Q. Semiclass. Opt. 2, 589 (2000).[23] J. Zapata, A. M. Guzman, M. G. Moore, and P. Meystre,

Phys. Rev. A 63, 023607 (2001).[24] T. Hartmann, F. Keck, H. J. Korsch and S. Mossmann,

N. J. Phys. 6, 2 (2004).[25] J. Larson, Phys. Rev. A 73, 013823 (2006).[26] M. Abramowitz and I. Stegun, Handbook of Mathematical

Functions (National Bureau of Standards, Washington,1964).

[27] W. V. Houston, Phys. Rev. 57, 184 (1940).[28] J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606

(1980).[29] G. Hechenblaikner, M. Gangl, P. Horak, and H. Ritsch,

Phys. Rev. A 58, 3030 (1998).[30] S. M. Kay, Fundamentals of Statistical Signal Processing:

Estimation theory (Prentice Hall International Editions,London, 1993).

[31] E. J. Mueller, Phys. Rev. A 66, 063603 (2002).[32] M. Gustavsson et al., Phys. Rev. Lett. 100, 080404

(2008).[33] B. M. Peden, D. Meiser, M. L. Chiofalo and M. J. Hol-

land, Phys. Rev. A 80, 043803 (2009).