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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 23 — Nov. 7, 2011
  • pp: 22535–22549
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Bunching-induced optical nonlinearity and instability in cold atoms [Invited]

Joel A. Greenberg, Bonnie L. Schmittberger, and Daniel J. Gauthier  »View Author Affiliations


Optics Express, Vol. 19, Issue 23, pp. 22535-22549 (2011)
http://dx.doi.org/10.1364/OE.19.022535


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Abstract

We report a new nonlinear optical process that occurs in a cloud of cold atoms at low-light-levels when the incident optical fields simultaneously polarize, cool, and spatially-organize the atoms. We observe an extremely large effective fifth-order nonlinear susceptibility of χ(5) = 7.6 × 10−15 (m/V)4, which results in efficient Bragg scattering via six-wave mixing, slow group velocities (∼ c/105), and enhanced atomic coherence times (> 100 μs). In addition, this process is particularly sensitive to the atomic temperatures, and provides a new tool for in-situ monitoring of the atomic momentum distribution in an optical lattice. For sufficiently large light-matter couplings, we observe an optical instability for intensities as low as ∼ 1 mW/cm2 in which new, intense beams of light are generated and result in the formation of controllable transverse optical patterns.

© 2011 OSA

1. Introduction

One area of sustained activity in the field of nonlinear optics is the development of schemes that enable single photons to induce nonlinear optical (NLO) interactions. Single-photon nonlinearities will enhance the efficiency of NLO devices, and are crucial components in quantum information systems. One technique for improving NLO interaction strengths is to increase the electric field strength per photon, such as by placing the NLO material in an optical waveguide [1

1. P. Kolchin, R. F. Oulton, and X. Zhang, “Nonlinear quantum optics in a waveguide: Distinct single photons strongly interacting at the single atom level,” Phys. Rev. Lett. 106, 113601 (2011). [CrossRef] [PubMed]

] or cavity [2

2. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995). [CrossRef] [PubMed]

]. Alternatively, one can enhance a material’s NLO susceptibility by taking advantage of material resonances or carefully designing and preparing the NLO medium [3

3. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008), Chap. 3.

]. One example of a material that is amenable to both approaches is a gas of atoms, which has the additional advantages that it is generally well-understood (i.e., can be modeled via first principles), can be well-controlled at the quantum level, and has been used to demonstrate extremely large NLO susceptibilities at low light intensities.

Materials with an extremely large, controllable fifth-order response have many important applications. They can be used in quantum information networks as new sources of correlated pulse pairs and quantum memories [9

9. D. Felinto, D. Moretti, R. de Oliveira, and J. Tabosa, “Delayed four- and six-wave mixing in a coherently prepared atomic ensemble,” Opt. Lett. 35, 3937–3939 (2010). [CrossRef] [PubMed]

] and for performing 3-bit quantum processing [10

10. M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A 65, 043804 (2002). [CrossRef]

,11

11. C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006). [CrossRef]

], for example. By controlling the relative contributions of the χ(3) and χ(5) NLO response, one can create novel states of light such as liquid light condensates [6

6. H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett. 96, 023903 (2006). [CrossRef] [PubMed]

] as well as use the interference of the resulting four- and six-wave mixing signals for high-precision measurements and nonlinear spectroscopy [12

12. Y. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102, 013601 (2009). [CrossRef] [PubMed]

]. In addition, quintic media can give rise to multidimensional solitons and transverse optical patterns [13

13. G. Fibich, N. Gavish, and X. P. Wang, “Singular ring solutions of critical and supercritical nonlinear Schrödinger equations,” Physica D 231, 55–86 (2007). [CrossRef]

], which might be used for low-light-level all-optical switching [14

14. A. M. C. Dawes, D. J. Gauthier, S. Schumacher, N. H. Kwong, R. Binder, and A. Smirl, “Transverse optical patterns for ultra-low-light-level-all-optical switching,” Laser Photon. Rev. 4, 221–243 (2010). [CrossRef]

, 15

15. A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Phys. Rev. A 308, 672–674 (2005).

].

At large light-by-light scattering efficiencies, we observe a transition to a collective scattering regime where the back-action of the scattered optical fields significantly affects the center-of-mass motion of the atoms and leads to longer atomic coherence times. Beyond a critical NLO interaction strength, we observe an instability that gives rise to new beams of light. Saffman et al. have described theoretically such instabilities in a thermal gas of cold atoms [16

16. M. Saffman and Y. Wang, “Collective focusing and modulational instability of light and cold atoms,” in Dissipative Solitons: From Optics to Biology and Medicine, Vol. 751 of Lecture Notes in Physics (Springer, 2008). [CrossRef]

, 17

17. G. A. Muradyan, Y. Wang, W. Williams, and M. Saffman, “Absolute instability and pattern formation in cold atomic vapors,” in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), p. ThB29.

] and have shown that the necessary nonlinear interaction strength is smaller than in other media. To the best of our knowledge, this paper represents the first experimental report of mirrorless parametric optical instabilities and pattern formation in cold atoms. Depending on the geometry of the incident beams, the generated light consists of either a sequence of randomly-oriented Gaussian spots or stable multi-petal patterns in the plane transverse to the incident beams’ wave vectors.

The organization of this paper is as follows. Section 2 develops the theoretical description of the NLO process, Sec. 3 describes the experimental setup that we use, Secs. 4 and 5 describe the steady-state and transient behavior of the nonlinearity below the instability threshold, respectively, and Sec. 6 discusses the multi-mode fields generated by the optical instability. We summarize our conclusions in Sec. 7.

2. Theory

We consider the geometry shown in Fig. 1a, where optical fields interact with a pencil-shaped cloud of cold atoms with length L and diameter W. We assume that the cloud is highly anisotropic (L/W ∼ 100), has its long axis aligned along the direction, and has a large on-resonance optical thickness. In order to develop a minimal model that contains the relevant physics, we consider a Jg = 1/2 → Je = 3/2 transition, as shown in Fig. 1b. A pair of balanced counterpropagating pump fields (intensity Ip, wave vectors ±k⃗p) with orthogonal linear polarizations (lin⊥lin configuration) are incident on the cloud at an angle θ = 10° relative to the z-axis. Weak signal and idler fields (intensities Is,i, wave vectors ±k⃗s,i) counterpropagate along and are orthogonally-polarized with respect to the nearly-copropagating pump beams. We refer to the signal and idler fields collectively as probe fields. All fields have frequency ω, which is detuned by Δ = ωωa from the atomic transition frequency ωa. We note that, while the non-zero value of θ has important implications for phase-matching considerations in the wave-mixing process, it is small enough for us to justify the approximation that the field polarizations all lie in the xy plane.

Fig. 1 a) Schematic of the experimental setup. b) Atomic level scheme used in the theoretical model along with the square of the Clebsh-Gordan coefficients (defined as a fraction of the reduced dipole moment μ) for the J = 1/2 → J = 3/2 transition. c) Optical lattices that form due to the interference of the pump beams (dotted line, p) and a probe and nearly-counterpropagating pump beam (solid line, Ĝ). d) Timing scheme used in the experiment.

We write the total electric field
(r,t)=E(r,t)eiωt+c.c.,
(1)
where E⃗ = ΣnE⃗nexp(ik⃗n ·r⃗) for n = {p1, p2, s,i}, which corresponds to the two counterpropagating pump, signal, and idler fields, respectively. For our beam geometry, k⃗p1 = −k⃗p2 =k⃗p and k⃗s = −k⃗i for |k⃗n| = k = 2π/λ, where λ = 2πc/ω and c is the speed of light in vacuum. Motivated by our experimental observations of the polarization configuration that leads to the largest NLO interaction strengths, we choose E⃗n = ɛ̂nEn for {ɛ̂p1,ɛ̂p2,ɛ̂s,ɛ̂i} = {x̂,ŷ,ŷ,x̂}. We take the pump fields Ep1 = Ep2 = Ep to be real and constant, the signal and idler fields Es,i = Es,i(z,t) to have slowly-varying, complex amplitudes, and |Es,i| ≪ |Ep|. We define the beam intensities as In = 2ɛ0c|En|2 and assume Is,i,pIsatΔ, where ɛ0 is the permittivity of free space, IsatΔ=Isat(1+4Δ2/Γ2) is the off-resonance saturation intensity, Isat is the resonant saturation intensity, and Γ is the full width at half maximum natural linewidth of the electronic transition.

These fields produce an atomic polarization
(r,t)=χ(r,t)=P(r,t)eiωt+c.c.,
(2)
where χ⃡ is the tensor susceptibility. Substituting this polarization into Maxwell’s equation yields
(Es,it±cEs,iz)=iω2ɛ0Peiksr,
(3)
where we have made the slowly-varying amplitude approximation and ignored terms proportional to exp(±2iωt). The angular brackets denote a spatial average over a grating period and act to select only the contributions to the polarization that are phase-matched to efficiently drive Es,i.

We can consider the full multilevel atomic structure to be equivalent to two V-type transitions where the dynamics of the g±1/2 states (mneumonic ±) are identical (up to a spatial offset of λ/4 discussed later) because the applied fields do not drive π transitions and therefore cannot establish coherences between the g±1/2 states. Thus, we write P⃗ = P⃗+ + P⃗ and consider the single-atom polarizations p⃗± such that P⃗± = η±(r⃗)p⃗±, where η±(r⃗) is the atomic density at position r⃗. By adiabatically eliminating the excited state and solving to first-order in the field strengths for Δ ≫ Γ, we find that
p±=npn±(r)eiknr,
(4)
where
pn±(r)=μeff±h¯ΔEn(r)
(5)
are slowly-varying polarization amplitudes such that p±exp(iknr)pn±δnm. Here
μeff±=j=e1/2,e±3/2μj,g±1/2μg±1/2,j,
(6)
where μ⃗l,j = 〈j|er⃗|l〉 is the dipole matrix element. Thus, phase-matching requires a spatial modulation of η±(r⃗) with wave vectors k⃗s,ik⃗n.

We use a perturbative approach to determine the spatially-varying atomic density distribution in the xz-plane. We first note that the one-dimensional (1D) photonic lattice along p formed by the pump beams dominates the atomic motion for Is,iIp. For red detunings (Δ < 0), the atoms load into the anti-nodes of the lattice. This leads to an enhanced atomic polarization because the atoms see a higher average field strength than a homogeneously-distributed sample. In addition, the atoms are optically pumped into the stretched states such that the density modulation due to the pump beam lattice ηp+(r)=ηp+(r+qλ/2k^p)=ηp(r+qλ/4k^p) for integer q, which gives rise to long-range anti-ferromagnetic order. As the atoms move in this bright lattice, Sisyphus (or polarization gradient) cooling changes the atomic momentum distribution from a Maxwell-Boltzmann distribution to one that is well-described by a double-Gaussian [4

4. J. Jersblad, H. Ellmann, K. Stochkel, A. Kastberg, L. Sanchez-Palencia, and R. Kaiser, “Non-Gaussian velocity distributions in optical lattices,” Phys. Rev. A 69, 013410 (2004). [CrossRef]

]. The narrow (broad) Gaussian distribution component corresponds to a cold (hot), bound (unbound) fraction of atoms, which leads to an interpretation of the cooling mechanism as a transfer of population from the hot to the cold fraction. This further allows us to interpret the décrochage intensity as the lattice beam intensity at which nearly all of the atoms have been transferred to the cold fraction, which is equivalent to its original definition as the intensity at which the atomic temperature attains a minimum value (assuming a Maxwell-Boltzmann momentum distribution).

We numerically solve for the momentum distribution ρ(p) along p in steady-state by using a Bloch-state approach [18

18. Y. Castin and J. Dalibard, “Quantization of atomic motion in optical molasses,” Europhys. Lett. 14, 761–766 (1991). [CrossRef]

] (see Fig. 2a). By fitting the full distribution by a double-Gaussian function, we calculate fc = 1 – fh for different values of Ip by considering the fractional area under the narrow Gaussian curve (see Fig. 2b). For the smallest Ip, the momentum distribution is non-normalizable, and fc is very small and increases slowly with Ip. For larger Ip, fcIp before saturating to 1 around Ip = Id, where Id is independent of Δ [19

19. J. Jersblad, H. Ellmann, and A. Kastberg, “Experimental investigation of the limit of Sisyphus cooling,” Phys. Rev. A 62, 051401 (2000). [CrossRef]

].

Fig. 2 a) Momentum distribution obtained via numerical simulation (blue, solid line) and a double-Gaussian fit (dashed, red line). pr = h̄k is the recoil momentum. b) Fraction of atoms in the hot and cold component as a function of pump intensity.

This leads to an anisotropic momentum distribution ρ(p⃗)
ρ(p)=g(py,Ty)g(p,T)[fcg(p||,Tc)+fhg(p||,Th)],
(7)
where g(p,T) = (2πmkBT)−1/2exp(−p2/2mkBT), m is the atomic mass, kB is Boltzmann’s constant, and Ty (py), Tc,h(p||), and T (p) correspond to the temperature (momentum) along ŷ, p, and orthogonal to p, respectively. Typical temperatures in our experiment are TyTTeq = 30 μK, Tc ∼ 3 μK, and Th > 20 μK.

Despite the fact that the atoms can become well-localized along p, this density grating is not phase-matched for scattering pump light into the probe modes and therefore does not directly contribute to the amplification of the signal and idler beams. In contrast, the interference of a probe and nearly-counterpropagating pump field gives rise to a weak additional dipole potential U± that perturbs the spatial atomic distribution and produces a density grating along G⃗ = k⃗p + k⃗s that is phase-matched (see Fig. 1c). Ignoring a constant offset,
U±(r)=U0±cos(Gr)=(pp1±Ei*+(pp2±)*Es)eiGr+c.c..
(8)
For a single-component gas with temperature T along Ĝ, the modulation of the steady-state atomic density distribution produced by U± is ηU±(r)=eU±/kBT/eU±/kBT. We expand ηU± in terms of its spatial Fourier modes and find
ηU±(r)=12I1(ψ±)I0(ψ±)cos(Gr)+,
(9)
where ψ±=U0±/kBT and Ij(ψ) is the modified Bessel function of the first kind. In the regime where ψ± ≪ 1, we Taylor expand Eq. (9) about ψ± = 0 and find
ηU±(r)1+b±(T)eiGr+(b±(T))*eiGr
(10)
to first order in Es,i, where
b±(T)=(μeff±Ep1)Ei*+(μeff±Ep2*)EskBTh¯Δ
(11)
corresponds to the degree of atomic bunching in the density grating along Ĝ. Extending this result to the case of our two-component gas, we find that
b±(T)b±(Tc,Th)=fcb±(Tc)+fhb±(Th),
(12)
where Tc,h = Tc,hcos(θ/2)+Th,csin(θ/2) ∼ Tc,h for θ/2 ≪ 1. Thus, the fully-modulated atomic density is
η±(r)=ηηp±ηU±/g,
(13)
where η is the average atomic density and g is the degeneracy of the ground states.

Substituting Eqs. (4) and (13) into Eq. (3) and keeping only phase-matched terms to first order in the probe fields yields
(Est+cEsz)=iωη2ɛ0j=±μeff±h¯Δ[Es+b±(Tc,Th)Ep],
(14)
(EitcEiz)=iωη2ɛ0j=±μeff±h¯Δ[Ei+b±(Tc,Th)*Ep].
(15)
Because the time scale associated with the atomic motion is typically much slower than that of the optical fields, we assume that the optical fields follow adiabatically the evolution of the atomic density grating. We therefore take Eq. (14) in steady-state and find
dEsdz=iκEs+iβEi*,
(16)
dEi*dz=iκEi*+iβEs
(17)
where
β=βc+βh=2ωημ4Ip9(ɛ0ch¯Δ)2(fckBTc+fhkBTh)
(18)
κ=2ωημ23ɛ0ch¯Δ+β,
(19)
and μ is the reduced dipole matrix element for the transition.

For an input signal field at z=0 and no input idler field, we find that the normalized output signal and idler intensities are
Is(L)Is(0)=sec2(βL),
(20)
Ii(0)Is(0)=tan2(βL),
(21)
respectively. The quantity βL therefore corresponds to the nonlinear phase shift imposed upon the probe beams via the NLO interaction. For typical experimental parameters η = 1010 cm−3, μ = 2.53 × 10−29 C·m, λ = 780 nm, g = 5, Δ = −5Γ, and L = 3 cm for the D2 transition in 87Rb, we find that βL = Ip[100fc(Ip) + 13fh(Ip)] for Ip in mW/cm2. Thus, we predict a NLO phase shift of ∼ 70 rad for Ip = 1 mW/cm2. We note that the nonlinear dependence of Is,i on η is one type of collective behavior of this NLO system.

The dependence of fc on Ip (shown in Fig. 2b) gives rise to different NLO regimes. For the smallest pump intensities, fc ∼ 0 and ββh ≪ 1. For larger Ip, where fcIp,
βcIp2/TcId
(22)
and
βhIp(1Ip/Id)/Th.
(23)
Because Tc depends only weakly on Ip and Th > 5Tc for all Ip, we find that ββcIp2/Tc. We can therefore identify an effective χ(5) NLO response due to β’s locally quadratic dependence on Ip, despite the fact that a true NLO susceptibility does not exist because the Taylor series expansion about Ip = 0 is not valid in this region. This effective χ(5) response can be understood as six-wave mixing (SWM) involving four optical and two matter wave fields (i.e., the density grating along Ĝ), wherein the pump fields Bragg scatter off the density grating formed via optically-induced atomic cooling and localization. When Ip > Id, the χ(5) response saturates because fc ∼ 1 and we realize an effective χ(3) response where ββcIp/Tc.

3. Experimental setup

To create our NLO medium, we use a magneto-optical trap (MOT) tuned to the D2 transition to confine 87Rb atoms in the F = 2, mf = 2 state within a cylindrical region with L = 3 cm and W = 300 μm [21

21. J. A. Greenberg, M. Oriá, A. M. C. Dawes, and D. J. Gauthier, “Absorption-induced trapping in an anisotropic magneto-optical trap,” Opt. Express 15, 17699–17708 (2007). [CrossRef] [PubMed]

]. The MOT beams consist of two pairs of counterpropagating laser beams: one pair (the radial beams) cools and traps atoms in the xy-plane, and the other (the longitudinal beams) cool the atoms along the -direction. The radial beams have counter-rotating circular polarizations and an elliptical profile with a semimajor and semiminor axis of 1.5 and 1 cm, respectively. The longitudinal beams have a diameter of 1 cm and arbitrary polarizations (although we typically use a lin⊥lin configuration). All MOT beams have a detuning Δ = −3Γ (for Γ/2π = 6 MHz) and intensity IMOT =7 mW/cm2. Using this setup, we achieve atomic densities of up to ∼ 1011 atoms/cm3 for atoms isotropically cooled to Teq = 30 μK.

The pump beams used for wave mixing are detuned from the F = 2 → F′ = 3 transition by |Δ| = 3 − 25Γ and have diameters of 3 mm. The incident signal beam is detuned from the pump beams by δ = ωsω, has a diameter of 200 μm, and an intensity of 1 − 100 μW/cm2 (although Is ∼ 3 mW/cm2 unless otherwise specified). The frequency of the generated idler beam is ωi = ωδ.

Figure 1c shows the timing sequence for our experiment. First, we cool and trap the atoms for 99 ms with only the MOT beams on. We then turn off the MOT beams and conduct the wave-mixing experiment during the remaining 1 ms, which consists of turning on the pump beams at time tp for a duration Δtp. If we want to measure the NLO response below the instability threshold, then we also turn on the incident signal beam at time ts for duration Δts. This cycle is then repeated. We note that, while we leave the MOT magnetic fields on at all times, they do not influence the wave mixing experiment because the atoms are trapped in the region in which the magnetic field is essentially zero. Unlike in a spherical MOT (where magnetic field variations of > 10 G occur for large traps), the longitudinal (∼0 G/cm) and radial (10 G/cm) magnetic field gradients of our anisotropic MOT produce a total magnetic field variation of only 0.3 G across the entire trap. In addition, we have confirmed the negligible role of the MOT magnetic fields by applying additional external fields without adversely affecting the NLO response.

We use independent acousto-optic modulators (AOMs) with response times of < 100 ns to turn the beams on and off as well as scan δ. For experiments where we require δ = 0, we split off a small fraction of the pump beam for use as the signal beam and double-pass it through a separate AOM such that it incurs no net frequency shift. In this way, we can independently modulate the signal beam amplitude without deleterious effects due to the frequency noise in the AOMs (which is on the order of the spectral linewidth of the NLO resonance).

We record the transmitted signal and generated idler beam intensities via fast photomultiplier tubes (PMTs; Hamamatsu H6780-20) as well as with a fast CCD camera (Dalsa) that images the beams in the far field. We also use separate CCD cameras that directly image the trap fluorescence from the top and side when the MOT beams are present. By calibrating the recorded brightness, we continuously monitor the effective atomic density. In addition, we measure the atomic temperature components along different directions in-situ via recoil-induced resonance (RIR) velocimetry using separate, far-detuned beams [22

22. M. Brzozowska, T. M. Brzozowski, J. Zachorowski, and W. Gawlik, “Nondestructive study of nonequilibrium states of cold trapped atoms,” Phys. Rev. A 72, 061401(R) (2005). [CrossRef]

].

To measure the slow light delay td of the atomic sample, we modulate the amplitude of the incident signal beam by 10% at a frequency of 15 kHz and record the intensity of the transmitted beam. We determine the time delay due solely to nonlinear dispersion by comparing the phase of the transmitted signal when Ip > 0 to that observed when Ip = 0.

4. Steady state response

Figure 3 shows the normalized spectrum of the signal and idler beams in the presence of the pump beams. The features centered around δ = ±100 kHz correspond to Raman transitions between vibrational levels of atoms localized in the pump beam lattice. The resonance centered on δ = 0, which is the focus of this paper, arises from Bragg scattering of the pump beams off the density grating along Ĝ and has a spectral width of Δω ∼ 30 kHz. For the remainder of the paper, we consider only the case where δ = 0.

Fig. 3 Normalized intensity spectrum of the a) transmitted signal and b) generated idler beams.

In order to verify the predictions of Eq. (18), we measure the steady-state dependence of β = β(Ip, Δ, η,Is). By choosing tp,s = 0 and Δtp,s=1 ms, we monitor Is,i and observe that the system reaches steady-state after ∼ 200 μs. We find that the measured steady-state Is,i agree well with Eqs. 20 and 21, which allows us to directly determine βL. Figure 4a shows the measured and predicted (via Eq. (18)) dependence of βL on Ip, where each curve corresponds to a different Δ. In obtaining the fit, we treat the décrochage intensity as a free parameter and find that Id ∼ 1.5 mW/cm2, which is in qualitative agreement with that found in Ref. [19

19. J. Jersblad, H. Ellmann, and A. Kastberg, “Experimental investigation of the limit of Sisyphus cooling,” Phys. Rev. A 62, 051401 (2000). [CrossRef]

] and is comparable to the resonant electronic-transition saturation intensity Isat = 1.6 mW/cm2.

Fig. 4 Dependence of NLO phase shift on experimental parameters. a) Experimental data (points) and theoretical predictions from Eq. (18) (solid curves) for Δ/Γ = −3, −5, −7.3, −12.7 (from left to right, respectively). b) For Ip < Id, the experimental data (points, Δ = −3Γ) are well-fit by a quadratic function (solid curve). c) Measured (points) dependence on atomic density and linear fit (solid line) and d) scaling with the incident signal intensity for Δ = −5Γ and Ip = 1.5 mW/cm2.

We first consider the dependence of β on Ip for fixed Δ. For the shallowest lattices, where fc ∼ 0, the signal is too small to measure. For slightly larger Ip, where fcIp, we observe a nearly quadratic dependence of β on pump intensity. Beyond Id, fc ∼ 1 and the SWM process saturates but β continues to increase linearly with Ip. A collective instability occurs for βL > 1, which gives rise to new beams of light in the absence of incident probe beams (this is discussed further in Sec. 6).

One metric for quantitatively comparing the NLO response due to this χ(5) process to previously-measured χ(3) processes is obtained by considering the achievable nonlinear phase shift for a fixed Ip. For Ip = 1 mW/cm2, we find a phase shift of βL = 1.17 for Δ = −3Γ, which is only 100 times smaller than that reported in Ref. [5

5. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999). [CrossRef]

] using EIT in a BEC at larger η than used in our work. Another commonly-used metric in cross-phase-modulation (XPM) experiments is the so-called XPM figure of merit ζ, which is defined as the ratio of the nonlinear phase shift to the power loss [23

23. H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, I. A. Yu, and Y.-F. Chen, “Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Phys. Rev. A 83, 041804(R) (2011). [CrossRef]

]. For Δ = −3Γ and Ip = 1 mW/cm2, we measure a power loss of ln(0.9) = 0.1, which leads to ζ ∼ 12 and exceeds that typically obtained via EIT [23

23. H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, I. A. Yu, and Y.-F. Chen, “Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Phys. Rev. A 83, 041804(R) (2011). [CrossRef]

].

We further verify Eq. (18) by measuring how β depends on η and Is for a fixed Ip and Δ. Figure 4c shows that β scales linearly with the average atomic density, which indicates that the pump light Bragg scatters coherently off the atomic density grating. In addition, Fig. 4d shows that β is independent of Is(0) below ∼ 15 μW/cm2. Beyond this the intensity, the nonlinearity begins to saturate and β decreases (see Fig. 4). Thus, Eq. (18) accurately predicts the observed scaling of β over a broad range of all measured parameters.

In addition to a large NLO phase shift, we also observe substantial dispersion at δ = 0. Figure 5 shows the slow-light delay td and group velocity vg = td/L as a function of βL for Δ = −5Γ. For βL ∼ 1, we observe vg/c ∼ 10−5 which acts like a high-finesse cavity and increases the photon lifetime in the gas to several μs, which is comparable to the time scale associated with atomic motion over a distance d. This gives rise to additional effective long-range atom-atom interactions beyond those considered in Sec. 2.

Fig. 5 Dependence of the slow-light delay and group velocity on the NLO coupling strength for Δ = −5Γ.

We note that this NLO mechanism is not specific to the particular choice of beam polarizations described up to this point. We observe a similar quadratic scaling of β with Ip for other polarizations that allow for both atomic cooling and bunching (such as linθlin and σ+σ+), although a larger Ip is required to achieve the same NLO phase shifts. For dark lattices, where Δ > 0, we still observe an amplification of the signal beam, but β ∼ 100× smaller than in the case of the bright lattice for a given Ip. We attribute this red-blue asymmetry to the fact that, while the atoms still become spatially organized for Δ > 0, they load into the intensity nodes (thereby minimizing their polarization) and undergo heating at low beam intensities. Thus, these results further validate our physical interpretation of the NLO process as being dependent on both atomic bunching and cooling in the photonic lattice.

5. Transient response

In this section, we investigate the NLO process in the transient domain. Figure 6 shows the growth of β in time, where we turn the signal beam on at t = 0. We turn the pump beams on at t = 0 for the lower (red) curve and find that the process of cooling and loading of atoms into the grating along Ĝ occurs over ∼ 200 μs. We observe that the grating formation time is reduced to ∼ 25 μs (upper blue curve) when we separately turn the pump beams on at t = −400 μs and then turn the signal beam on at t = 0. The density grating along Ĝ forms more quickly in this case because the atoms have already been cooled and localized along p by the pump beam lattice during the 400 μs prior to t = 0. This ∼ 25 μs time scale is consistent with the time it takes atoms confined in the pump beam lattice with T = Teq to load into the grating along Ĝ by moving a distance d′ = λ/2sin(θ/2) in the direction orthogonal to p (see Fig. 1c) under the influence of U±.

Fig. 6 (Color online) Growth of βL as a function of time when the signal beam turns on at t = 0. The lower red (upper blue) curve corresponds to the case where we turn the pump beams on at t = 0 (t = −400 μs) for Δ = −5Γ.

We can measure βc,h directly by studying the SWM signal decay after we turn off the incident signal beam (while the pump beams remain on). As discussed in Sec. 2, a finite atomic momentum spread leads to motion that washes out the density grating. Thus, we can separate out the contributions to βc,h according to the observed decay time because each has a distinct temperature associated with it. In addition, when td and Is,i are small enough for us to ignore the back-action of the fields on the atomic motion (i.e., βL ≪ 1), we can directly extract the effective temperatures.

Figure 7 shows β(t) after the incident signal field is extinguished at t = 0. For small Ip, where almost all of the atoms are in the hot, unbound component, βc ∼ 0 and βh decays after a time τh = 1.8 μs (see Fig. 7a), which corresponds to Th = 25 μK for Λ = d. Increasing Ip increases fc and gives rise to a second Gaussian decay component due to the cold atomic fraction. We independently measure Tc ∼ 3 μK, which implies that the atoms are bound in the pump beam lattice and grating decay along Ĝ is highly suppressed. Therefore, atomic motion in the direction perpendicular to p determines the decay time for βc. By fitting the decay in Fig. 7b with a double-Gaussian, we find that τc ∼ 20 μs, which corresponds to T = Teq for Λ = d′ = 11.5d.

Fig. 7 Decay of the nonlinear coupling coefficient as a function of time after the signal beam is turned off at t = 0 for Δ = −5Γ and Ip = a) 0.45, b) 0.7 and c) 1.25 mW/cm2. The solid and dashed lines correspond to experimental data and a double-Gaussian fit, respectively (note the change in scale of the horizontal axis).

For larger Ip where βL > 0.5, the back-action of the amplified probe fields strongly influences the coupled light-matter dynamics. In this regime, additional cooling along the direction perpendicular to Ĝ begins to occur via the lattice formed by the pump and nearly-copropagating probe beams and reduces T. Atoms cooled in both the x- and z-directions give rise to a third decay time, as shown in Fig. 7c. In addition, the large values of td invalidate our assumption that the field evolution is slaved to the atomic motion and prevents us from directly extracting a temperature from the observed decay time. Nevertheless, the strong coupling between the evolution of the atomic and optical fields represents a secondary form of collective behavior and leads to the observation of long coherences times of > 100 μs.

We use a multi-Gaussian function to fit β(t) for various Ip and relate the amplitudes associated with the slow and fast decay components with βc,h, respectively. Figure 8 shows the dependence of βc,h on Ip for Δ = −5Γ. We find that the predictions of Eqs. 22 and 23 agree reasonably well with the experimental results, where we use fc,h(Ip) shown in Fig. 2 and Tc/Th = 10. The main source of discrepancy between the measured and predicted values stems from the oversimplified model used to determine fc,h [24

24. S. Jonsell, C. M. Dion, A. Kastberg, S. J. H. Petra, and P. Sjolun, “A non-adiabatic semi-classical method for Sisyphus cooling,” Eur. Phys. J. D. 39, 67–74 (2006). [CrossRef]

]. Instead, we can consider this approach to represent a new technique for independently determining fc,h, as our method is particularly sensitive to the small cold fraction and works in the region IpId. Thus, this technique may be useful for building a phenomenological model of Sisyphus cooling well below décrochage.

Fig. 8 Dependence of βc,h on Ip. Points correspond to experimental data extracted from multi-Gaussian fits to the transient decay measurements, and the solid curves are obtained by a fit using Eqs. 22 and 23.

6. Transverse optical instabilities

We observe an optical instability that generates new beams of light for βL > 1, which occurs for pump intensities as low as 1 mW/cm2 and can be understood as either parametric self-oscillation [25

25. A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977). [CrossRef] [PubMed]

] or collective scattering [26

26. R. Bonifacio and L. De Salvo, “Collective atomic recoil laser (CARL) optical gain without inversion by collective atomic recoil and self-bunching of two-level atoms,” Nucl. Instrum. Methods Phys. 341, 360–362 (1994). [CrossRef]

]. We consider first the geometry shown in Fig. 1a, where we shine only the pump beams on the atoms. We find that light is emitted in the signal and idler modes (i.e., the direction of maximum gain), which implies that initial density fluctuations are amplified via the NLO process and lead to a spontaneous, long-range spatial organization along Ĝ. The profile of the generated light in the xy plane is roughly Gaussian and aligned with the center of the trap, but we find that multiple modes (within a few mrad angular width) can fire sequentially within a single shot (i.e., MOT realization). This result is similar to the predictions of Moore et al. [27

27. M. G. Moore and P. Meystre, “Theory of superradiant scattering of laser light from Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5202–5205 (1999). [CrossRef]

], where multi-mode emission occurs via an atomic-bunching-induced instability in an anisotropic BEC. In addition, we note that the instability only occurs for Δ < 0, in agreement with the physical mechanism described above but contrary to previously-observed wave mixing instabilities based on electronic nonlinearities (which typically require Δ > 0).

Figure 9 shows the measured time-dependent intensity of the generated light in the signal and idler modes (where we turn the pump beams on at t = 0). While the details of Is,i(t) vary from shot-to-shot, we observe several universal characteristics. Immediately after turning the pump beams on, there is a period of time (typically lasting 20–50 μs and dependent on Ip) during which the atoms are cooled and the density grating forms. After this time, we observe an exponential growth of the intensity before it reaches a maximum value of ∼ 0.1Ip. The generated light persists and displays complicated temporal dyanmics for several hundred μs, after which time free expansion of the MOT in the y-direction reduces η and Is,i decreases. We note that the signal and idler modes show strong temporal correlations of r > 0.9 (where r=(−)1 for perfectly (anti-)correlated signals) due to their mutual coupling.

Fig. 9 Time-dependent intensity of the instability-generated light in the + (signal, red) and − (idler, blue) directions for Ip = 5 mW/cm2 and Δ = −5Γ, where the instability threshold is Ip = 2.3 mW/cm2. The temporal correlation of the signal and idler modes is r = 0.99.

If we alter the beam geometry such that the pump beams propagate along and are focused down to a diameter on the order of W (see Fig. 10a), we observe multi-petal transverse optical patterns generated in both the ± directions. Because the focused pump beams weakly confine the atoms in the xy plane, the patterns can persist for ∼ 2 ms (where we modify our experimental timing scheme to allow for a 2-ms-long wave-mixing period). Depending on the relative powers and exact alignment of the pump beams, we find many different types of patterns. Figure 10b shows several examples of patterns, including two-, four-, and six-spot patterns as well as a nearly-continuous ring. In addition, we can choose a configuration where the orientation of the patterns rotate randomly (when there is a high degree of symmetry) or are fixed in a particular orientation (when we impose an asymmetry externally).

Fig. 10 a) Beam geometry for pattern-forming instability, where θ′ ∼ 5 mrad. b) Transverse optical patterns, including i) a nearly full ring and ii) a six- iii) four- and iv) two-spot pattern, for Ip = 15 mW/cm2, Δ = −8Γ and an instability threshold of Ip = 6 mW/cm2. The central spot in the images corresponds to residual pump light.

For the highly-symmetric case, we find that we can inject a weak control beam along the cone of the generated light and pin the pattern orientation. In the case where a particular pattern orientation is set by a broken symmetry in the system, we find that we can use the control beam to rotate the pattern to a specified orientation. Thus, we anticipate that this system may be useful for low-light-level all-optical switching applications [14

14. A. M. C. Dawes, D. J. Gauthier, S. Schumacher, N. H. Kwong, R. Binder, and A. Smirl, “Transverse optical patterns for ultra-low-light-level-all-optical switching,” Laser Photon. Rev. 4, 221–243 (2010). [CrossRef]

, 15

15. A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Phys. Rev. A 308, 672–674 (2005).

].

7. Conclusions

In this paper, we present the theory for and experimental demonstration of a new, dissipation-induced NLO process that arises when a cloud of cold atoms are simultaneously polarized, spatially-organized, and cooled by incident optical fields. By using a weak beam to probe the atomic NLO response, we observe that a regime exists for which the NLO coupling strength scales quadratically with Ip and SWM occurs. This corresponds to an extremely large fifth-order nonlinearity, and leads to substantial NLO phase shifts with high transparency.

For sufficiently large NLO coupling strengths, we enter a regime in which the back-action of the amplified probe fields strongly influences the system dynamics. This leads to additional atomic cooling, enhanced atomic coherence times, slow optical group velocities, and, eventually, optical instabilities and pattern formation.

References and links

1.

P. Kolchin, R. F. Oulton, and X. Zhang, “Nonlinear quantum optics in a waveguide: Distinct single photons strongly interacting at the single atom level,” Phys. Rev. Lett. 106, 113601 (2011). [CrossRef] [PubMed]

2.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995). [CrossRef] [PubMed]

3.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008), Chap. 3.

4.

J. Jersblad, H. Ellmann, K. Stochkel, A. Kastberg, L. Sanchez-Palencia, and R. Kaiser, “Non-Gaussian velocity distributions in optical lattices,” Phys. Rev. A 69, 013410 (2004). [CrossRef]

5.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999). [CrossRef]

6.

H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett. 96, 023903 (2006). [CrossRef] [PubMed]

7.

C. M. Cirloganu, P. D. Olszak, L. A. Padilha, S. Webster, D. J. Hagan, and E. W. V. Stryland, “Three-photon absorption spectra of zinc blende semiconductors: theory and experiment,” Opt. Lett. 33, 2626–2628 (2008). [CrossRef] [PubMed]

8.

R. W. Boyd, K. Dolgaleva, and H. Shin, “Strong, fifth-order, nonlinear optical response resulting from local-field-induced microscopic cascading in C60,” in Nonlinear Optics: Materials, Fundamentals and Applications (Optical Society of America, 2009), p. NWB2.

9.

D. Felinto, D. Moretti, R. de Oliveira, and J. Tabosa, “Delayed four- and six-wave mixing in a coherently prepared atomic ensemble,” Opt. Lett. 35, 3937–3939 (2010). [CrossRef] [PubMed]

10.

M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A 65, 043804 (2002). [CrossRef]

11.

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006). [CrossRef]

12.

Y. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102, 013601 (2009). [CrossRef] [PubMed]

13.

G. Fibich, N. Gavish, and X. P. Wang, “Singular ring solutions of critical and supercritical nonlinear Schrödinger equations,” Physica D 231, 55–86 (2007). [CrossRef]

14.

A. M. C. Dawes, D. J. Gauthier, S. Schumacher, N. H. Kwong, R. Binder, and A. Smirl, “Transverse optical patterns for ultra-low-light-level-all-optical switching,” Laser Photon. Rev. 4, 221–243 (2010). [CrossRef]

15.

A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Phys. Rev. A 308, 672–674 (2005).

16.

M. Saffman and Y. Wang, “Collective focusing and modulational instability of light and cold atoms,” in Dissipative Solitons: From Optics to Biology and Medicine, Vol. 751 of Lecture Notes in Physics (Springer, 2008). [CrossRef]

17.

G. A. Muradyan, Y. Wang, W. Williams, and M. Saffman, “Absolute instability and pattern formation in cold atomic vapors,” in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), p. ThB29.

18.

Y. Castin and J. Dalibard, “Quantization of atomic motion in optical molasses,” Europhys. Lett. 14, 761–766 (1991). [CrossRef]

19.

J. Jersblad, H. Ellmann, and A. Kastberg, “Experimental investigation of the limit of Sisyphus cooling,” Phys. Rev. A 62, 051401 (2000). [CrossRef]

20.

M. Mitsunaga, M. Yamashita, M. Koashi, and N. Imoto, “Temperature diagnostics for cold sodium atoms by transient four-wave mixing,” Opt. Lett. 23, 840–842 (1998). [CrossRef]

21.

J. A. Greenberg, M. Oriá, A. M. C. Dawes, and D. J. Gauthier, “Absorption-induced trapping in an anisotropic magneto-optical trap,” Opt. Express 15, 17699–17708 (2007). [CrossRef] [PubMed]

22.

M. Brzozowska, T. M. Brzozowski, J. Zachorowski, and W. Gawlik, “Nondestructive study of nonequilibrium states of cold trapped atoms,” Phys. Rev. A 72, 061401(R) (2005). [CrossRef]

23.

H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, I. A. Yu, and Y.-F. Chen, “Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Phys. Rev. A 83, 041804(R) (2011). [CrossRef]

24.

S. Jonsell, C. M. Dion, A. Kastberg, S. J. H. Petra, and P. Sjolun, “A non-adiabatic semi-classical method for Sisyphus cooling,” Eur. Phys. J. D. 39, 67–74 (2006). [CrossRef]

25.

A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977). [CrossRef] [PubMed]

26.

R. Bonifacio and L. De Salvo, “Collective atomic recoil laser (CARL) optical gain without inversion by collective atomic recoil and self-bunching of two-level atoms,” Nucl. Instrum. Methods Phys. 341, 360–362 (1994). [CrossRef]

27.

M. G. Moore and P. Meystre, “Theory of superradiant scattering of laser light from Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5202–5205 (1999). [CrossRef]

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(020.3320) Atomic and molecular physics : Laser cooling

ToC Category:
Nonlinear Materials and Spectroscopy

History
Original Manuscript: September 6, 2011
Revised Manuscript: October 16, 2011
Manuscript Accepted: October 20, 2011
Published: October 25, 2011

Virtual Issues
Nonlinear Optics (2011) Optical Materials Express

Citation
Joel A. Greenberg, Bonnie L. Schmittberger, and Daniel J. Gauthier, "Bunching-induced optical nonlinearity and instability in cold atoms [Invited]," Opt. Express 19, 22535-22549 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22535


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References

  1. P. Kolchin, R. F. Oulton, and X. Zhang, “Nonlinear quantum optics in a waveguide: Distinct single photons strongly interacting at the single atom level,” Phys. Rev. Lett.106, 113601 (2011). [CrossRef] [PubMed]
  2. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett.75, 4710–4713 (1995). [CrossRef] [PubMed]
  3. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008), Chap. 3.
  4. J. Jersblad, H. Ellmann, K. Stochkel, A. Kastberg, L. Sanchez-Palencia, and R. Kaiser, “Non-Gaussian velocity distributions in optical lattices,” Phys. Rev. A69, 013410 (2004). [CrossRef]
  5. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature397, 594–598 (1999). [CrossRef]
  6. H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett.96, 023903 (2006). [CrossRef] [PubMed]
  7. C. M. Cirloganu, P. D. Olszak, L. A. Padilha, S. Webster, D. J. Hagan, and E. W. V. Stryland, “Three-photon absorption spectra of zinc blende semiconductors: theory and experiment,” Opt. Lett.33, 2626–2628 (2008). [CrossRef] [PubMed]
  8. R. W. Boyd, K. Dolgaleva, and H. Shin, “Strong, fifth-order, nonlinear optical response resulting from local-field-induced microscopic cascading in C60,” in Nonlinear Optics: Materials, Fundamentals and Applications (Optical Society of America, 2009), p. NWB2.
  9. D. Felinto, D. Moretti, R. de Oliveira, and J. Tabosa, “Delayed four- and six-wave mixing in a coherently prepared atomic ensemble,” Opt. Lett.35, 3937–3939 (2010). [CrossRef] [PubMed]
  10. M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A65, 043804 (2002). [CrossRef]
  11. C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A74, 012319 (2006). [CrossRef]
  12. Y. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett.102, 013601 (2009). [CrossRef] [PubMed]
  13. G. Fibich, N. Gavish, and X. P. Wang, “Singular ring solutions of critical and supercritical nonlinear Schrödinger equations,” Physica D231, 55–86 (2007). [CrossRef]
  14. A. M. C. Dawes, D. J. Gauthier, S. Schumacher, N. H. Kwong, R. Binder, and A. Smirl, “Transverse optical patterns for ultra-low-light-level-all-optical switching,” Laser Photon. Rev.4, 221–243 (2010). [CrossRef]
  15. A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Phys. Rev. A308, 672–674 (2005).
  16. M. Saffman and Y. Wang, “Collective focusing and modulational instability of light and cold atoms,” in Dissipative Solitons: From Optics to Biology and Medicine, Vol. 751 of Lecture Notes in Physics (Springer, 2008). [CrossRef]
  17. G. A. Muradyan, Y. Wang, W. Williams, and M. Saffman, “Absolute instability and pattern formation in cold atomic vapors,” in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), p. ThB29.
  18. Y. Castin and J. Dalibard, “Quantization of atomic motion in optical molasses,” Europhys. Lett.14, 761–766 (1991). [CrossRef]
  19. J. Jersblad, H. Ellmann, and A. Kastberg, “Experimental investigation of the limit of Sisyphus cooling,” Phys. Rev. A62, 051401 (2000). [CrossRef]
  20. M. Mitsunaga, M. Yamashita, M. Koashi, and N. Imoto, “Temperature diagnostics for cold sodium atoms by transient four-wave mixing,” Opt. Lett.23, 840–842 (1998). [CrossRef]
  21. J. A. Greenberg, M. Oriá, A. M. C. Dawes, and D. J. Gauthier, “Absorption-induced trapping in an anisotropic magneto-optical trap,” Opt. Express15, 17699–17708 (2007). [CrossRef] [PubMed]
  22. M. Brzozowska, T. M. Brzozowski, J. Zachorowski, and W. Gawlik, “Nondestructive study of nonequilibrium states of cold trapped atoms,” Phys. Rev. A72, 061401(R) (2005). [CrossRef]
  23. H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, I. A. Yu, and Y.-F. Chen, “Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Phys. Rev. A83, 041804(R) (2011). [CrossRef]
  24. S. Jonsell, C. M. Dion, A. Kastberg, S. J. H. Petra, and P. Sjolun, “A non-adiabatic semi-classical method for Sisyphus cooling,” Eur. Phys. J. D.39, 67–74 (2006). [CrossRef]
  25. A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett.1, 16–18 (1977). [CrossRef] [PubMed]
  26. R. Bonifacio and L. De Salvo, “Collective atomic recoil laser (CARL) optical gain without inversion by collective atomic recoil and self-bunching of two-level atoms,” Nucl. Instrum. Methods Phys.341, 360–362 (1994). [CrossRef]
  27. M. G. Moore and P. Meystre, “Theory of superradiant scattering of laser light from Bose-Einstein condensates,” Phys. Rev. Lett.83, 5202–5205 (1999). [CrossRef]

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