## Bunching-induced optical nonlinearity and instability in cold atoms [Invited] |

Optics Express, Vol. 19, Issue 23, pp. 22535-22549 (2011)

http://dx.doi.org/10.1364/OE.19.022535

Acrobat PDF (1133 KB)

### 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*/10^{5}), and enhanced atomic coherence times (> 100 *μ*s). 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/cm^{2} in which new, intense beams of light are generated and result in the formation of controllable transverse optical patterns.

© 2011 OSA

## 1. Introduction

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]

*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.

*I*discovered in the context of Sisyphus cooling in a photonic lattice. For intensities below

_{d}*I*, a gas transforms into a non-equilibrium system with two distinct temperature components: a localized cold fraction (

_{d}*f*, mneumonic

_{c}*c*) and a hot fraction (

*f*, mneumonic

_{h}*h*) undergoing anomalous diffusion [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]

*χ*

^{(5)}) susceptibility. Surprisingly, this fifth-order NLO response is as large as the nonlinearities reported in previous studies that typically involve third-order

*χ*

^{(3)}processes [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]

*χ*

^{(5)}= 7.6 × 10

^{−15}(m/V)

^{4}is the largest ever reported, exceeding that obtained via electromagnetically-induced transparency (EIT) in a Bose-Einstein condensate by ∼ 10

^{5}[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]

^{22}[7

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]

_{60}by ∼ 10

^{25}[8].

*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]

## 2. Theory

*L*and diameter

*W*. We assume that the cloud is highly anisotropic (

*L/W*∼ 100), has its long axis aligned along the

*ẑ*direction, and has a large on-resonance optical thickness. In order to develop a minimal model that contains the relevant physics, we consider a

*J*= 1/2 →

_{g}*J*= 3/2 transition, as shown in Fig. 1b. A pair of balanced counterpropagating pump fields (intensity

_{e}*I*, wave vectors ±

_{p}*k⃗*) with orthogonal linear polarizations (lin⊥lin configuration) are incident on the cloud at an angle

_{p}*θ*= 10° relative to the

*z*-axis. Weak signal and idler fields (intensities

*I*, wave vectors ±

_{s,i}*k⃗*) counterpropagate along

_{s,i}*ẑ*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 =

*ω*–

*ω*from the atomic transition frequency

_{a}*ω*. We note that, while the non-zero value of

_{a}*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

*x*–

*y*plane.

*E⃗*= Σ

*exp(*

_{n}E⃗_{n}*ik⃗*·

_{n}*r⃗*) for

*n*= {

*p*1,

*p*2,

*s,i*}, which corresponds to the two counterpropagating pump, signal, and idler fields, respectively. For our beam geometry,

*k⃗*

_{p}_{1}= −

*k⃗*

_{p}_{2}=

*k⃗*and

_{p}*k⃗*= −

_{s}*k⃗*for |

_{i}*k⃗*| =

_{n}*k*= 2

*π*/

*λ*, where

*λ*= 2

*πc*/

*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

*E⃗*=

_{n}*ɛ̂*for {

_{n}E_{n}*ɛ̂*

_{p1},

*ɛ̂*

_{p2},

*ɛ̂*,

_{s}*ɛ̂*} = {

_{i}*x̂,ŷ,ŷ,x̂*}. We take the pump fields

*E*

_{p1}=

*E*

_{p2}=

*E*to be real and constant, the signal and idler fields

_{p}*E*=

_{s,i}*E*(

_{s,i}*z,t*) to have slowly-varying, complex amplitudes, and |

*E*| ≪ |

_{s,i}*E*|. We define the beam intensities as

_{p}*I*= 2

_{n}*ɛ*

_{0}

*c|E*|

_{n}^{2}and assume

*ɛ*

_{0}is the permittivity of free space,

*I*is the resonant saturation intensity, and Γ is the full width at half maximum natural linewidth of the electronic transition.

_{sat}*χ⃡*is the tensor susceptibility. Substituting this polarization into Maxwell’s equation yields where we have made the slowly-varying amplitude approximation and ignored terms proportional to exp(±2

*iωt*). 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

*E*.

_{s,i}*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 where are slowly-varying polarization amplitudes such that

*μ⃗*= 〈

_{l,j}*j*|

*er⃗*|

*l*〉 is the dipole matrix element. Thus, phase-matching requires a spatial modulation of

*η*

^{±}(

*r⃗*) with wave vectors

*k⃗*−

_{s,i}*k⃗*.

_{n}*x*–

*z*-plane. We first note that the one-dimensional (1D) photonic lattice along

*k̂*formed by the pump beams dominates the atomic motion for

_{p}*I*≪

_{s,i}*I*. 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

_{p}*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]

*ρ*(

*p*) along

*k̂*in steady-state by using a Bloch-state approach [18

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

*f*= 1 –

_{c}*f*for different values of

_{h}*I*by considering the fractional area under the narrow Gaussian curve (see Fig. 2b). For the smallest

_{p}*I*, the momentum distribution is non-normalizable, and

_{p}*f*is very small and increases slowly with

_{c}*I*. For larger

_{p}*I*,

_{p}*f*∝

_{c}*I*before saturating to 1 around

_{p}*I*=

_{p}*I*, where

_{d}*I*is independent of Δ [19

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

*ρ*(

*p⃗*) where

*g*(

*p,T*) = (2

*πmk*)

_{B}T^{−1/2}exp(−

*p*

^{2}/2

*mk*),

_{B}T*m*is the atomic mass,

*k*is Boltzmann’s constant, and

_{B}*T*(

_{y}*p*),

_{y}*T*′

*(*

_{c,h}*p*

_{||}), and

*T*

_{⊥}(

*p*

_{⊥}) correspond to the temperature (momentum) along

*ŷ*,

*k̂*, and orthogonal to

_{p}*k̂*, respectively. Typical temperatures in our experiment are

_{p}*T*∼

_{y}*T*

_{⊥}∼

*T*= 30

_{eq}*μ*K,

*T*′

*∼ 3*

_{c}*μ*K, and

*T*′

*> 20*

_{h}*μ*K.

*k̂*, 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

_{p}*U*

^{±}that perturbs the spatial atomic distribution and produces a density grating along

*G⃗*=

*k⃗*+

_{p}*k⃗*that is phase-matched (see Fig. 1c). Ignoring a constant offset, For a single-component gas with temperature

_{s}*T*along

*Ĝ*, the modulation of the steady-state atomic density distribution produced by

*U*

^{±}is

*I*(

_{j}*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 to first order in

*E*, where corresponds to the degree of atomic bunching in the density grating along

_{s,i}*Ĝ*. Extending this result to the case of our two-component gas, we find that where

*T*=

_{c,h}*T*′

*cos(*

_{c,h}*θ*/2)+

*T*′

*sin(*

_{h,c}*θ*/2) ∼

*T*′

*for*

_{c,h}*θ*/2 ≪ 1. Thus, the fully-modulated atomic density is where

*η*is the average atomic density and

*g*is the degeneracy of the ground states.

*μ*is the reduced dipole matrix element for the transition.

*z*=0 and no input idler field, we find that the normalized output signal and idler intensities are 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*= 10

^{10}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

^{87}Rb, we find that

*βL*=

*I*[100

_{p}*f*(

_{c}*I*) + 13

_{p}*f*(

_{h}*I*)] for

_{p}*I*in mW/cm

_{p}^{2}. Thus, we predict a NLO phase shift of ∼ 70 rad for

*I*= 1 mW/cm

_{p}^{2}. We note that the nonlinear dependence of

*I*on

_{s,i}*η*is one type of collective behavior of this NLO system.

*f*on

_{c}*I*(shown in Fig. 2b) gives rise to different NLO regimes. For the smallest pump intensities,

_{p}*f*∼ 0 and

_{c}*β*∼

*β*≪ 1. For larger

_{h}*I*, where

_{p}*f*∝

_{c}*I*, and Because

_{p}*T*depends only weakly on

_{c}*I*and

_{p}*T*> 5

_{h}*T*for all

_{c}*I*, we find that

_{p}*χ*

^{(5)}NLO response due to

*β*’s locally quadratic dependence on

*I*, despite the fact that a true NLO susceptibility does not exist because the Taylor series expansion about

_{p}*I*= 0 is not valid in this region. This effective

_{p}*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

*I*>

_{p}*I*, the

_{d}*χ*

^{(5)}response saturates because

*f*∼ 1 and we realize an effective

_{c}*χ*

^{(3)}response where

*β*∼

*β*∝

_{c}*I*/

_{p}*T*.

_{c}*βL*≪ 1 (

*i.e.*, when the back-action of the generated probe fields on the atoms is negligible). If we turn the signal beam off after the system has reached a steady state, the atom’s momentum spread gives rise to motion that washes out the density grating and therefore leads to a decay of the pump power scattered into the probe field modes. A 1D Gaussian momentum distribution with a spread

*p̄*=

*mu*= (2

*k*)

_{B}Tm^{1/2}leads to a Gaussian decay

*β*(

*t*) ∝ exp(−

*t*

^{2}/

*τ*

^{2}) with

*τ*= Λ/

*πu*, where Λ is the grating period along the direction of the atomic motion [20

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]

*β*to the total NLO response.

_{c,h}## 3. Experimental setup

^{87}Rb atoms in the

*F*= 2,

*m*= 2 state within a cylindrical region with

_{f}*L*= 3 cm and

*W*= 300

*μ*m [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]

*x*–

*y*-plane, and the other (the longitudinal beams) cool the atoms along the

*ẑ*-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/2

*π*= 6 MHz) and intensity

*I*=7 mW/cm

_{MOT}^{2}. Using this setup, we achieve atomic densities of up to ∼ 10

^{11}atoms/cm

^{3}for atoms isotropically cooled to

*T*= 30

_{eq}*μ*K.

*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*=

*ω*−

_{s}*ω*, has a diameter of 200

*μ*m, and an intensity of 1 − 100

*μ*W/cm

^{2}(although

*I*∼ 3 mW/cm

_{s}^{2}unless otherwise specified). The frequency of the generated idler beam is

*ω*=

_{i}*ω*−

*δ*.

*t*for a duration Δ

_{p}*t*. If we want to measure the NLO response below the instability threshold, then we also turn on the incident signal beam at time

_{p}*t*for duration Δ

_{s}*t*. 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.

_{s}*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).

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]

*t*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

_{d}*I*> 0 to that observed when

_{p}*I*= 0.

_{p}## 4. Steady state response

*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 Δ

*ω*∼ 30 kHz. For the remainder of the paper, we consider only the case where

*δ*= 0.

*β*=

*β*(

*I*, Δ,

_{p}*η*,

*I*). By choosing

_{s}*t*= 0 and Δ

_{p,s}*t*=1 ms, we monitor

_{p,s}*I*and observe that the system reaches steady-state after ∼ 200

_{s,i}*μ*s. We find that the measured steady-state

*I*agree well with Eqs. 20 and 21, which allows us to directly determine

_{s,i}*βL*. Figure 4a shows the measured and predicted (via Eq. (18)) dependence of

*βL*on

*I*, where each curve corresponds to a different Δ. In obtaining the fit, we treat the décrochage intensity as a free parameter and find that

_{p}*I*∼ 1.5 mW/cm

_{d}^{2}, 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]

*I*= 1.6 mW/cm

_{sat}^{2}.

*β*on

*I*for fixed Δ. For the shallowest lattices, where

_{p}*f*∼ 0, the signal is too small to measure. For slightly larger

_{c}*I*, where

_{p}*f*∝

_{c}*I*, we observe a nearly quadratic dependence of

_{p}*β*on pump intensity. Beyond

*I*,

_{d}*f*∼ 1 and the SWM process saturates but

_{c}*β*continues to increase linearly with

*I*. A collective instability occurs for

_{p}*βL*> 1, which gives rise to new beams of light in the absence of incident probe beams (this is discussed further in Sec. 6).

*β*on Δ for fixed

*I*. In agreement with the predictions of Eq. (18), we find that

_{p}*β*∝ Δ

^{−2}. Because

*I*, the effective saturation intensity for the SWM process, is independent of Δ, we can work several Γ away from atomic resonance (where the transmission is high) while simultaneously realizing large NLO interaction strengths. While some degree of absorption is necessary to cool the atoms, the process occurs efficiently (

_{d}*i.e.*, requires the scattering of few photons) and the deleterious effects due to absorption are minimal. For the data shown in Fig. 4a, we measure a detuning-dependent signal beam transmission of > 90% in the absence of the pump beams; with the pump beams present, we observe a net amplification of the signal and idler fields.

*χ*

^{(5)}process to previously-measured

*χ*

^{(3)}processes is obtained by considering the achievable nonlinear phase shift for a fixed

*I*. For

_{p}*I*= 1 mW/cm

_{p}^{2}, 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]

*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]

*I*= 1 mW/cm

_{p}^{2}, 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]

*β*depends on

*η*and

*I*for a fixed

_{s}*I*and Δ. Figure 4c shows that

_{p}*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

*I*(0) below ∼ 15

_{s}*μ*W/cm

^{2}. 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.

*δ*= 0. Figure 5 shows the slow-light delay

*t*and group velocity

_{d}*v*=

_{g}*t*/

_{d}*L*as a function of

*βL*for Δ = −5Γ. For

*βL*∼ 1, we observe

*v*/

_{g}*c*∼ 10

^{−5}which acts like a high-finesse cavity and increases the photon lifetime in the gas to several

*μ*s, 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.

*β*with

*I*for other polarizations that allow for both atomic cooling and bunching (such as lin

_{p}*θ*lin and

*σ*

^{+}−

*σ*

^{+}), although a larger

*I*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

_{p}*β*∼ 100× smaller than in the case of the bright lattice for a given

*I*. 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.

_{p}## 5. Transient response

*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

*μ*s. We observe that the grating formation time is reduced to ∼ 25

*μ*s (upper blue curve) when we separately turn the pump beams on at

*t*= −400

*μ*s 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

*k̂*by the pump beam lattice during the 400

_{p}*μ*s prior to

*t*= 0. This ∼ 25

*μ*s time scale is consistent with the time it takes atoms confined in the pump beam lattice with

*T*

_{⊥}=

*T*to load into the grating along

_{eq}*Ĝ*by moving a distance

*d*′ =

*λ*/2sin(

*θ*/2) in the direction orthogonal to

*k̂*(see Fig. 1c) under the influence of

_{p}*U*

^{±}.

*β*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

_{c,h}*β*according to the observed decay time because each has a distinct temperature associated with it. In addition, when

_{c,h}*t*and

_{d}*I*are small enough for us to ignore the back-action of the fields on the atomic motion (

_{s,i}*i.e.*,

*βL*≪ 1), we can directly extract the effective temperatures.

*β*(

*t*) after the incident signal field is extinguished at

*t*= 0. For small

*I*, where almost all of the atoms are in the hot, unbound component,

_{p}*β*∼ 0 and

_{c}*β*decays after a time

_{h}*τ*= 1.8

_{h}*μ*s (see Fig. 7a), which corresponds to

*T*= 25

_{h}*μ*K for Λ =

*d*. Increasing

*I*increases

_{p}*f*and gives rise to a second Gaussian decay component due to the cold atomic fraction. We independently measure

_{c}*T*∼ 3

_{c}*μ*K, 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

*k̂*determines the decay time for

_{p}*β*. By fitting the decay in Fig. 7b with a double-Gaussian, we find that

_{c}*τ*∼ 20

_{c}*μ*s, which corresponds to

*T*

_{⊥}=

*T*for Λ =

_{eq}*d*′ = 11.5

*d*.

*I*where

_{p}*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

*t*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

_{d}*μ*s.

*β*(

*t*) for various

*I*and relate the amplitudes associated with the slow and fast decay components with

_{p}*β*, respectively. Figure 8 shows the dependence of

_{c,h}*β*on

_{c,h}*I*for Δ = −5Γ. We find that the predictions of Eqs. 22 and 23 agree reasonably well with the experimental results, where we use

_{p}*f*(

_{c,h}*I*) shown in Fig. 2 and

_{p}*T*/

_{c}*T*= 10. The main source of discrepancy between the measured and predicted values stems from the oversimplified model used to determine

_{h}*f*[24

_{c,h}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]

*f*, as our method is particularly sensitive to the small cold fraction and works in the region

_{c,h}*I*≪

_{p}*I*. Thus, this technique may be useful for building a phenomenological model of Sisyphus cooling well below décrochage.

_{d}## 6. Transverse optical instabilities

*βL*> 1, which occurs for pump intensities as low as 1 mW/cm

^{2}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]

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]

*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

*x*–

*y*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]

*ẑ*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 ±

*ẑ*directions. Because the focused pump beams weakly confine the atoms in the

*x*–

*y*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).

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]

## 7. Conclusions

*I*and SWM occurs. This corresponds to an extremely large fifth-order nonlinearity, and leads to substantial NLO phase shifts with high transparency.

_{p}## 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. |

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. |

3. | R. W. Boyd, |

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 |

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 |

6. | H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett. |

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. |

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. |

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 |

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 |

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. |

13. | G. Fibich, N. Gavish, and X. P. Wang, “Singular ring solutions of critical and supercritical nonlinear Schrödinger equations,” Physica D |

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. |

15. | A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Phys. Rev. A |

16. | M. Saffman and Y. Wang, “Collective focusing and modulational instability of light and cold atoms,” in |

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. |

19. | J. Jersblad, H. Ellmann, and A. Kastberg, “Experimental investigation of the limit of Sisyphus cooling,” Phys. Rev. A |

20. | M. Mitsunaga, M. Yamashita, M. Koashi, and N. Imoto, “Temperature diagnostics for cold sodium atoms by transient four-wave mixing,” Opt. Lett. |

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 |

22. | M. Brzozowska, T. M. Brzozowski, J. Zachorowski, and W. Gawlik, “Nondestructive study of nonequilibrium states of cold trapped atoms,” Phys. Rev. A |

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 |

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. |

25. | A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. |

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. |

27. | M. G. Moore and P. Meystre, “Theory of superradiant scattering of laser light from Bose-Einstein condensates,” Phys. Rev. Lett. |

**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

Sort: Year | Journal | Reset

### References

- 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]
- 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]
- R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008), Chap. 3.
- 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]
- 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]
- 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]
- 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]
- 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.
- 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]
- 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]
- 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]
- 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]
- G. Fibich, N. Gavish, and X. P. Wang, “Singular ring solutions of critical and supercritical nonlinear Schrödinger equations,” Physica D231, 55–86 (2007). [CrossRef]
- 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]
- A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Phys. Rev. A308, 672–674 (2005).
- 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]
- 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.
- Y. Castin and J. Dalibard, “Quantization of atomic motion in optical molasses,” Europhys. Lett.14, 761–766 (1991). [CrossRef]
- J. Jersblad, H. Ellmann, and A. Kastberg, “Experimental investigation of the limit of Sisyphus cooling,” Phys. Rev. A62, 051401 (2000). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.