## Abstract

The nonlinear spectroscopy of cold atoms in the diffuse laser cooling system is studied in this paper. We present the theoretical models of the recoil-induced resonances (RIR) and the electromagnetically-induced absorption (EIA) of cold atoms in diffuse laser light, and show their signals in an experiment of cooling ^{87}Rb atomic vapor in an integrating sphere. The theoretical results are in good agreement with the experimental ones when the light intensity distribution in the integrating sphere is considered. The differences between nonlinear spectra of cold atoms in the diffuse laser light and in the optical molasses are also discussed.

©2009 Optical Society of America

## 1. Introduction

Cooling of atoms in the diffuse laser light is an all optical laser cooling technique. In the diffuse laser light, an atom with velocity *v⃗* resonates with the photons whose propagating directions are at an angle *θ* with *v⃗*, and

where Δ is the laser detuning. The diffuse laser light can cool more atoms than optical molasses does due to the large resonant velocity-range (*kv* ≥ Δ/*k*). Because of its all optical configuration and wide cooling velocity-range, diffuse laser light is not only a laser cooling method besides the optical molasses and the magnetic-optical trap (MOT), but also a good choice in magnetic-field-free cases such as the development of a compact cold-atom clock [1].

Slowing and cooling atomic beams in diffuse light was first experimentally realized by Ketterle *et al*. in 1992 [2] and was succeeded by Batelaan *et al*. in 1994 [3]. An idea that cooling atomic beam in an integrating sphere in which the diffuse light field can be formed was first proposed by Y. Z. Wang in 1979 [4] and was realized in 1994 [5]. In this paper, we produce the diffuse laser light in a ceramic integrating sphere to have the three-dimensional cooling of ^{87}Rb atomic vapor [6]. It can be compared with the 3D cooling experiment of cesium atomic vapor in a copper integrating sphere [7].

Nonlinear spectroscopy of cold atoms in an optical-molasses as well as in a MOT has been widely studied [8, 9, 10, 11, 12], but in the diffuse laser light case it has not been reported before. The diffuse laser light is monochromatic and is generated by reflected laser beams from Lambertian-reflectance surface. It acts as both the cooling light and the pump light to the cold atoms in the pump-probe configuration. As we know, diffuse reflectance can not change the temporal coherence of monochromatic light, but the Lambertian reflectance can disorder the wave-front of the light and break its spatial coherence [13]. For the pump-probe configured nonlinear spectroscopy, the phase of pump and probe lights need to be correlated well, then their frequency difference (*ω*
_{1} - *ω*
_{0}) can oscillate with the atoms, and two-photon process can happen. Here *ω*
_{1} is the frequency of probe laser light and *ω*
_{0} is the frequency of diffuse pump
laser light. Usually pump light and probe light are from the same laser source, so they have same time-depended phase shift *ϕ*(*t*) and they are well correlated. Fortunately, the diffuse pump light has a stable *ϕ*(*t*) to the laser source because the temporal coherence is not broken by Lambertian reflection. The reason is the phase shift caused by the mechanical vibration of the Lambertian surface is tiny compared with *ϕ*(*t*) so it can be neglected, and thus the diffuse pump light is
well correlated to the laser source, as well as the probe light. It is necessary for the nonlinear spectroscopy of cold atoms in diffuse light.

The main differences between nonlinear spectroscopy of cold atoms in the diffuse light and in the optical molasses are the light shift and the steady-state population of ground state sub-levels. Because the diffuse laser light is depolarized by the Lambertian reflection [14], its polarization distribution is totally random. In this paper we assume an optimum condition that the probabilities of σ^{+}, σ^{-}, and *π* transition of atoms in the diffuse laser light are equal, and the differences among the steady-state population of all ground state sub-levels are quite small. Figure 1 shows the steady-state population and the transition of such optimum condition.

We focus on two kinds of two-photon process which are most possible to lead to pump-probe configured nonlinear spectra of cold atoms in diffuse laser light. One is recoil-induced resonance, whose introduction and theoretical model in diffuse laser light pumped case are showed in Sec. 2. The other is electromagnetically induced absorption (EIA), whose theoretical model in diffuse laser light pumped case is showed in Sec. 3. Sec. 4 is our experimental setup and the suitable condition for nonlinear spectroscopy study of cold ^{87}Rb atoms in diffuse laser light. The experimental results of RIR and EIA are compared with the theoretical ones in Sec. 5, where the influence from the intensity distribution of diffuse laser light is considered. Finally, the possible stimulated Raman process, and the differences between nonlinear spectra of cold atoms in diffuse laser light and in an optical molasses as well as in a MOT are discussed.

## 2. Recoil-induced resonances of cold atoms in diffuse pump light

Recoil-induced resonances (RIR) was first theoretically predicted by Guo *et al*. in 1992 [15, 16]. Its signal appears as a derivative line shape in pump-probe spectra, which can be used to measure the velocity distribution of cold atoms [17]. The first experimental observation of recoil-induced resonances is obtained by Courtois *et al*. in 1994 [18], where the pump laser is a 1D optical molasses and the probe laser beam has a small angle with respect to it.

Because the diffuse laser light is depolarized by the Lambertian reflection, and the atoms also have a three-dimensional distribution, pump and probe laser light can make atoms have all the three kinds of transitions (*σ*
^{+}, *σ*
^{-} and *π*). For a two-level atom, relative light shift between every two sub-levels of ground state do not infect the line shape of recoil-induced resonance signal [15], but only infect the detuning of the pump laser a little. So when the detuning of the diffuse pump laser light *ω*
_{0} is much larger than all the light shift among every ground state sub-levels, the atomic system is approximated to a two-level system for the recoil-induced resonances. The interaction Hamiltonian of pump-probe lights interacting with a two-level atomic system is

where Ω_{0} is the Rabi frequency of pump field with wave vector **k**
_{0} and frequency *ω*
_{0}. Ω_{1} is the Rabi frequency of probe field with wave vector **k**
_{1} and frequency *ω*
_{1}. |e〉 is the excited state and |*g*〉 is the ground state. Atomic density matrix can be expanded to the basis of the internal states |*a*〉 = |*e*〉, |*g*〉 and external center-of-mass momentum states |**p**〉

Then we obtain the time evolution equations for all atomic density matrix elements [15]:

$$\phantom{\rule{4.0em}{0ex}}+i\sum _{a=0}^{1}{\Omega}_{a}^{*}\mathrm{exp}\left[i\left({\Delta}_{a}+{\omega}_{r}-\frac{{\mathbf{k}}_{a}\xb7\mathbf{p}\prime}{m}\right)t\right]{\rho}_{\mathrm{eg}}(\mathbf{p},\mathbf{p}\prime -\stackrel{\u0304}{h}{\mathbf{k}}_{a})$$

$$\phantom{\rule{4.0em}{0ex}}-i\sum _{a=0}^{1}{\Omega}_{a}^{*}\mathrm{exp}\left[i\left(-{\Delta}_{a}-{\omega}_{r}+\frac{{\mathbf{k}}_{a}\xb7\mathbf{p}\prime}{m}\right)t\right]{\rho}_{\mathrm{eg}}(\mathbf{p}-\stackrel{\u0304}{h}{\mathbf{k}}_{a},\mathbf{p}\prime ),$$

$$\phantom{\rule{4.0em}{0ex}}-\gamma {\tilde{\rho}}_{\mathrm{gg}}\left(\mathbf{p},\mathbf{p}\prime \right)+\mathrm{\gamma W}\left(\mathbf{p},\mathbf{p}\prime \right)$$

$$\phantom{\rule{4.0em}{0ex}}+i\sum _{a=0}^{1}{\Omega}_{a}^{*}\mathrm{exp}\left[i\left({\Delta}_{a}+{\omega}_{r}-\frac{{\mathbf{k}}_{a}\xb7\mathbf{p}\prime}{m}\right)t\right]{\rho}_{\mathrm{gg}}(\mathbf{p},\mathbf{p}\prime -\stackrel{\u0304}{h}{\mathbf{k}}_{a})$$

$$\phantom{\rule{4.0em}{0ex}}-i\sum _{a=0}^{1}{\Omega}_{a}^{*}\mathrm{exp}\left[i\left({\Delta}_{a}-{\omega}_{r}-\frac{{\mathbf{k}}_{a}\xb7\mathbf{p}}{m}\right)t\right]{\rho}_{\mathrm{ee}}(\mathbf{p}-\stackrel{\u0304}{h}{\mathbf{k}}_{a},\mathbf{p}\prime ),$$

$$\phantom{\rule{4.0em}{0ex}}+i\sum _{a=0}^{1}{\Omega}_{a}^{*}\mathrm{exp}\left[i\left({\Delta}_{a}+{\omega}_{r}-\frac{{\mathbf{k}}_{a}\xb7\mathbf{p}\prime}{m}\right)t\right]{\rho}_{\mathrm{gg}}(\mathbf{p},\mathbf{p}\prime -\stackrel{\u0304}{h}{\mathbf{k}}_{a})$$

$$\phantom{\rule{4.0em}{0ex}}-i\sum _{a=0}^{1}{\Omega}_{a}^{*}\mathrm{exp}\left[i\left({\Delta}_{a}-{\omega}_{r}-\frac{{\mathbf{k}}_{a}\xb7\mathbf{p}}{m}\right)t\right]{\rho}_{\mathrm{ee}}(\mathbf{p}-\stackrel{\u0304}{h}{\mathbf{k}}_{a},\mathbf{p}\prime ),$$

Here

Δ_{0} is the detuning of pump laser and Δ_{1} is the detuning of probe laser, which are given by

*ω _{eg}* is the transition frequency from ground state |

*g*〉 to excited state |

*e*〉, and the recoil frequency is

Γ is the decay rate of excited state and *γ* is the decay rate due to time of flight which is much smaller than Γ. *W*(**p**, **p**′) is the momentum distribution of atoms at ground state, which is usually considered as a Maxwell-Boltzman distribution. Another momentum distribution *N*(**q**) is the normalized probability density for emitting a photon with momentum *h̄*
**q**, which for a two level atom is given by [15]

where *θ* is the angle between *h̄*
**q** and field polarization direction.

The absorption signal is proportional to the imaginary part of the component of *$\tilde{\rho}$ _{ge}*, which is

$$\phantom{\rule{1.0em}{0ex}}=\frac{1}{{\left(2\pi \stackrel{\u0304}{h}\right)}^{3}}\int \int d\mathbf{p}\phantom{\rule{.2em}{0ex}}d\mathbf{p}\prime \left[i\frac{\left(\mathbf{p}-\mathbf{p}\prime \right)\xb7\mathbf{x}}{\stackrel{\u0304}{h}}-i\frac{\left({p}^{2}-{p\prime}^{2}\right)}{2m\stackrel{\u0304}{h}}i+\mathrm{i\omega t}\right]{\tilde{\rho}}_{\mathrm{ge}}\left(\mathbf{p},\mathbf{p}\prime \right)\mathrm{exp}\left(i{\mathbf{k}}_{1}\xb7\mathbf{x}-i{\omega}_{1}t\right).$$

It can be solved from the third-order perturbation solutions of Eqs. (4)–(7) in the limit that perturbation treatment is valid [15].

In diffuse laser cooling, the pump laser light is isotropic so the interaction range only depends on the spot size of probe beam. Condition Eq. (13) needs that the spot size of probe laser is small. It is because only if the interaction range is small enough does the decay rate due to time of flight *γ* can be large enough to fit limit Eq. (13).

Under the limits

$$\mathrm{kv}\gg \gamma $$

$$\stackrel{\u0304}{h}\mid \Delta \mid \gg {k}_{b}T$$

$$\theta \approx 0,$$

where *k _{b}* is Boltzman constant, the simplified general expression of

*$\tilde{\rho}$*for arbitrary wave vector of pump laser

_{ge}*k*

_{0}= |

**k**

_{0}| and probe laser

*k*

_{1}= |

**k**

_{1}| is expressed by

where *N*
_{0} is the atomic number, *θ* is the angle between the pump and the probe laser beams, *v* is the atomic velocity, and

Here *W*′(*y*) is the first-order derivative of one-dimensional momentum distribution *W*(*y*).

We are interested in the case of diffuse laser light pumped system, where the propagating direction of pump light is isotropic. The imaginary part of Eq. (15) is approximate to Eq. (17)under the condition of Eq. (13) and Eq. (14), where *m* is the atomic mass.

Because of the isotropic distribution of the pump laser, we need to find a critical angle *θ _{c}* that the pump beams within the range of

*θ*<

*θ*can have recoil-induced resonance with the majority of the cold atoms.

_{c}*θ*can be calculated by the energy conservation of initial and final states of the recoil-induced resonance. Recoil-induced resonance is a two-photon process with one photon being absorbed and the other being stimulatingly emitted. Its scheme is showed in Fig. 2. Probe laser beam (

_{c}*k*

_{1},

*ω*

_{1}) propagates through the direction

*ê*, and one beam of the isotropic pump laser light (

_{x}*k*

_{0},

*ω*

_{0}) has the angle

*θ*to it. For an atom with initial momentum

*p*in direction

_{x}*ê*and momentum

_{x}*p*in direction

_{y}*ê*and

_{y}, p_{x}*p*are all positive. The momentum change of the two-photon process is (

_{x}*h̄k*

_{1}+

*h̄k*

_{0}cos

*θ*)

*ê*+

_{x}*h̄k*

_{0}sin

*θê*. Because of the energy conservation, its corresponding energy change needs to be equal to the energy change of the light field 2

_{y}*πh̄*(

*ω*

_{1}-

*ω*

_{0}). Then we obtain

$$\phantom{\rule{5.5em}{0ex}}+\stackrel{\u0304}{h}\left({k}_{1}^{2}+{k}_{0}^{2}+2{k}_{1}{k}_{0}\mathrm{cos}\theta \right).$$

For atoms with temperature being above Doppler cooling limit, *h̄*(*k*
^{2}
_{1} + *k*
^{2}
_{0} + 2*k*
_{1}
*k*
_{0} cos *θ*) is quite small compared with *k*
_{0,1}
*p _{x,y}* and can be neglected. Because

*ω*=

*ck*/2

*π*, where c is the speed of light, |

*k*

_{0}| and |

*k*

_{1}| need to be very close to each other, then 4

*πm*(

*ω*

_{1}-

*ω*

_{0}) may have the same order with 2

*k*

_{(0,1)p(x,y)}.

The importance of angle *θ* can be analyzed with Eq. (18). In the recoil-induced resonance range, *k*
_{0} ≈ *k*
_{1}. If we make the transform *p _{x}* → -

*p*sin

_{y}*θ*/(1 + cos

*θ*) and

*p*→ -

_{y}*p*(1 + cos

_{x}*θ*)/sin

*θ*, Eq. (18) is approximately unchanged. When

*θ*=

*π*/2, the transform means

*p*= -

_{x}*p*and

_{y}*W*(

*p*) -

_{x}*W*(

*p*) =

_{y}*W*(

*p*) -

_{x}*W*(-

*p*) = 0. If we assume

_{x}*p*> 0, momentum distribution

_{x}*W*(-

*p*) >

_{x}*W*(-

*p*+

_{x}*h̄*

*k*

_{1}) so the transform

*W*(-

*p*) →

_{x}*W*(-

*p*+

_{x}*h̄*

*k*

_{1}) means the strengthened absorption. However,

*w*(

*p*) →

_{x}*w*(

*p*+

_{x}*h̄k*

_{1}) means the stimulated amplification of probe beam. The two signals are just canceled by each other so the total signal of recoil-induced resonance can not be observed. When

*θ*≈ 0, the result is

*p*≪ -

_{x}*p*and

_{y}*W*(

*p*) -

_{x}*W*(

*p*) =

_{y}*W*(

*p*) -

_{x}*W*[-

*p*(1 + cos

_{x}*θ*)/sin

*θ*] is always non-vanishing, then we can observe the RIR signal. We need to determine a critical angle

*θ*, for which the condition |

_{c}*p*| ≪ | -

_{x}*p*sin

_{y}*θ*| is satisfied in the region of 0 <

*θ*<

*θ*and the RIR signal is evident. We choose the minimum momentum limit

_{c}*p*= 2

_{x}*h̄k*, which is recoil momentum of two-photons, and choose

*p*equals to twice of the most probable momentum $\sqrt{2m{k}_{b}T}$, with which $0<\mid \overrightarrow{v}\mid <2\sqrt{2m{k}_{b}T}$ contains the majority of the cold atoms. Then

_{y}*θ*can be calculated from

_{c}So the solution of *θ _{c}* is small, which just meet the condition

*θ*≈ 0 of deriving Eq. (17). In the range $0<\theta <{\theta}_{c},\phantom{\rule{.2em}{0ex}}\mathrm{for}\phantom{\rule{.2em}{0ex}}\mathrm{all}\phantom{\rule{.2em}{0ex}}{p}_{x}>2\stackrel{\u0304}{h}k\phantom{\rule{.2em}{0ex}}\mathrm{and}\phantom{\rule{.2em}{0ex}}{p}_{y}<\sqrt{8m{k}_{b}T},\phantom{\rule{.2em}{0ex}}\mathrm{the}\phantom{\rule{.2em}{0ex}}W\left({p}_{x}\right)-W\left({p}_{y}\right)$ is always non-vanishing. The total signal for isotropic pumped recoil-induced resonance is an integration of Eq. (17) over -

_{c}*θ*to

_{c}*θ*. Under the condition of Eq. (14), we calculated the average of the recoil-induced resonances over the range -

_{c}*θ*≥

_{c}*θ*≥

*θ*. The result is showed in Fig. 3 with

_{c}*m*being chosen as the mass of a

^{87}Rb atom.

## 3. Electromagnetically induced absorption of cold atoms in diffuse light

Electromagnetically induced absorption (EIA) [19] is a nonlinear optical effect due to the atomic coherence. Pump-probe configured EIA [20, 21, 22] can happen when a strong pump laser and a weak probe laser are interacting with a degenerated two-level atomic system which satisfies *F _{e}* >

*F*> 0 [19, 20]. It leads to a sharp-peak enhancement effect in the absorption spectrum at the position where probe laser resonates with the pump one. It is just opposite to the electromagnetically induced transparency (EIT) which leads to a sharp-dip attenuation at the pump-probe resonating position. Pump-probe configured EIA was studied in details and classified into two cases: EIA-TOC (transfer of coherence) [23] and EIA-TOP (transfer of population) [24]. The EIA-TOC requires the pump and probe laser have different polarizations and the EIA-TOP requires that they have the same polarization. Some interesting phenomenon in Hanle configured EIA was also studied recently [25, 26].

_{g}Most experimental researches of EIA were carried in atomic vapor cells at room temperature. In fact, the cold atoms are more suitable to study atomic coherence. The first EIA of cold atoms was observed by Lipsich *et al*. in a MOT [27]. Although the MOT is the most common method to obtain cold atoms, the small interaction region and the strong magnetic field restrict itself to be an optimum choice for pure pump-probe configured EIA studies.

In diffuse laser cooling, atoms can easily be cooled to the temperature near its Doppler limit, and no magnetic field is added. The diffuse laser light then becomes a strong pump field with random polarization. When an arbitrary polarized probe beam is added, the EIA-TOC happens easily. For theoretical study, the model of pump-probe configured EIA of stationary atoms is also more suitable for cold atoms in diffuse laser light than room-temperature atoms.

In our model, we select the *F _{g}* = 2 →

*F*= 3 transition of

_{e}^{87}Rb atoms as the cooling transition because this transition is used for the diffuse laser cooling of

^{87}Rb atomic vapor in our experiment. The cooling laser, which is also the pump one, interacts with the cold atoms together with the probe laser, then the interaction Hamiltonian for every two Zeeman sub-levels in the rotating wave approximation are

where *H ^{I}_{eigj}*(

*ω*

_{0,1}) is the Rabi frequency for

*e*→

_{i}*g*transition.

_{j}Here *ω*
_{0} is the frequency of pump laser and *ω*
_{1} is the frequency of probe laser. Ω_{0,1} are Rabi frequency caused by the two laser.

Time evolution equations of every density matrix element can be solved in two stages [23]. The first stage contains only the strong pump laser, and the equations are showed in Eqs. (22)–(24).

$$\phantom{\rule{16.0em}{0ex}}+\frac{i}{\stackrel{\u0304}{h}}\sum _{k=-2}^{2}\left[{\rho}_{{e}_{i}{g}_{k}}\left({\omega}_{0}\right){H}_{{g}_{k}{e}_{i}}^{I}\left(-{\omega}_{0}\right)-{H}_{{e}_{i}{g}_{k}}^{I}\left({\omega}_{0}\right){\rho}_{{g}_{k}{e}_{i}}\left(-{\omega}_{0}\right)\right],$$

$$\phantom{\rule{15.0em}{0ex}}+\frac{i}{\stackrel{\u0304}{h}}\sum _{k=-3}^{3}\left[{\rho}_{{e}_{i}{e}_{k}}\left({\omega}_{0}\right){H}_{{e}_{k}{g}_{j}}^{I}\left(-{\omega}_{0}\right)-i\sum _{k=-2}^{2}{H}_{{e}_{i}{g}_{k}}^{I}\left({\omega}_{0}\right){\rho}_{{g}_{k}{e}_{i}}\left(-{\omega}_{0}\right)\right],$$

$$\phantom{\rule{13.0em}{0ex}}+\frac{i}{\stackrel{\u0304}{h}}\sum _{k=-3}^{2}\left[{H}_{{e}_{i}{g}_{k}}^{I}\left({\omega}_{0}\right){\rho}_{{g}_{k}{e}_{i}}\left(-{\omega}_{0}\right)-{\rho}_{{e}_{i}{g}_{k}}\left({\omega}_{0}\right){H}_{{g}_{k}{e}_{i}}^{I}\left({-\omega}_{0}\right)\right],$$

where

and

$$\phantom{\rule{12.2em}{0ex}}\times \left(\begin{array}{ccc}{F}_{g}& 1& {F}_{e}\\ -{m}_{{g}_{i}}& q& {m}_{e}\end{array}\right){\rho}_{{e}_{k}{e}_{l}}\left({\omega}_{\mathrm{0,1}}\right)\left(\begin{array}{ccc}{F}_{e}& 1& {F}_{g}\\ -{m}_{e}& q& {m}_{{g}_{i}}\end{array}\right),$$

which is the spontaneous emission term. In Eqs. (22)–(24), Γ is the decay rate of excited state *F _{e}* = 3, and Γ

_{gi}is the decay rate from ground state sub-level

*g*. Because

_{i}^{87}Rb atom has another ground state hyperfine level

*F*= 1 whose energy is 6.8GHz lower than

_{g}*F*= 2, the Zeeman sub-levels of

_{g}*F*= 2 can decay to

_{g}*F*= 1 with rate

_{g}*g*. The repumping laser is needed to pump the population from

_{i}*F*= 1 back to

_{g}*F*= 2. Another decay rate is the y due to time of flight in the interaction range between atoms and pump-probe light. In diffuse cooling system the diffuse light and the cold atoms distribute all over the cavity, so atoms can always in the interaction range of the diffuse laser light. After the probe laser beam is added, the interaction range is only determined by the probe laser.

_{g}In the second stage, a weak probe laser is added as a perturbation to the system. The density matrix elements of population *ρ _{eiej}* and

*ρ*can oscillates with frequencies

_{gigj}*ω*

_{1}-

*ω*

_{0}and

*ω*

_{0}-

*ω*

_{1}, while the coherence term of ground and excited state is

Then we can get the optical Bloch equations of the second stage

$$\phantom{\rule{10.0em}{0ex}}+\frac{i}{\stackrel{\u0304}{h}}\sum _{k=-2}^{2}\left[{\rho}_{{e}_{i}{g}_{k}}\left({\omega}_{1}\right){H}_{{g}_{k}{e}_{i}}^{I}\left(-{\omega}_{0}\right)-{H}_{{e}_{i}{g}_{k}}^{I}\left({\omega}_{1}\right){\rho}_{{g}_{k}{e}_{i}}\left(-{\omega}_{0}\right)\right],$$

$$\phantom{\rule{13.0em}{0ex}}+\frac{i}{\stackrel{\u0304}{h}}\sum _{k=-3}^{3}\left[{\rho}_{{e}_{i}{e}_{k}}\left({\omega}_{0}\right){H}_{{e}_{k}{g}_{j}}^{I}\left({\omega}_{1}\right)-i\sum _{k=-2}^{2}{H}_{{e}_{i}{g}_{k}}^{I}\left({\omega}_{1}\right){\rho}_{{g}_{k}{e}_{i}}\left(-{\omega}_{0}\right)\right]$$

$$\phantom{\rule{17.0em}{0ex}}+\frac{i}{\stackrel{\u0304}{h}}\sum _{k=-3}^{3}\left[{\rho}_{{e}_{i}{g}_{k}}\left({\omega}_{1}-{\omega}_{0}\right){H}_{{e}_{k}{g}_{j}}^{I}\left({\omega}_{0}\right)-i\sum _{k=-2}^{2}{H}_{{e}_{i}{g}_{k}}^{I}\left({\omega}_{0}\right){\rho}_{{g}_{k}{e}_{i}}\left({\omega}_{1}-{\omega}_{0}\right)\right],$$

$$\phantom{\rule{6.0em}{0ex}}+7\Gamma {\rho}_{{g}_{i}{g}_{j}}^{s}\left({\omega}_{1}-{\omega}_{0}\right)+\sum _{k=-2}^{2}{\Gamma}_{{g}_{i}{g}_{k}}{\delta}_{{g}_{i}{g}_{j}}{\rho}_{{g}_{k}{g}_{k}}\left({\omega}_{1}-{\omega}_{0}\right)$$

$$\phantom{\rule{7.0em}{0ex}}+\frac{i}{\stackrel{\u0304}{h}}\sum _{k=-3}^{3}\left[{H}_{{e}_{i}{g}_{k}}^{I}\left({\omega}_{1}\right){\rho}_{{g}_{k}{e}_{i}}\left(-{\omega}_{0}\right)-{\rho}_{{e}_{i}{g}_{k}}\left({\omega}_{1}\right){H}_{{g}_{k}{e}_{i}}^{I}\left({-\omega}_{0}\right)\right].$$

There are also another three equations that describe the time evolution of the density matrix elements at the frequency *ω*
_{0} - *ω*
_{1}, which can be written via replacing *ω*
_{1} - *ω*
_{0} by *ω*
_{0} - *ω*
_{1} in Eqs. (28)–(30).

The probe absorption intensity is proportional to

whose steady-state value can be solved from Eqs.(28)–(30). There are relative light shifts *ω _{gigj}* between each two ground-state Zeeman sub-levels in Eq. (30), which can shift the position of EIA signal. For diffuse laser light, we have assumed that the atoms have same probabilities for σ

^{+}, σ

^{-}, and

*π*transitions, then the light shift of every ground-state Zeeman sub-level

*δ*can be calculated from the Clebsch-Gordan coefficient given in Fig. 4. The square of these coefficients give the probabilities of corresponding transitions. Relative light shifts of ground-state Zeeman sub-levels are ${\omega}_{{g}_{i}{g}_{j}}=\stackrel{\u0304}{h}\sqrt{{\Delta}^{2}+{\Omega}_{{g}_{i}}^{2}}-\sqrt{{\Delta}^{2}+{\Omega}_{{g}_{j}}^{2}}/2$ is the average Rabi frequency of σ

_{gi}^{+}, σ

^{-}, and

*π*transitions. Rabi frequencies Ω

_{gie(i,i±1)}are proportional to the square of Clebsch-Gordan coefficients of corresponding transitions, so after the calculation we find that Ω

_{g1}= Ω

_{g-1}= Ω

_{g2}= Ω

_{g-2}< Ω

_{g1}, then the terms of

*ω*

_{g(±1, ±2)g(±1, ±2)}are all equal to zero. Only

*ω*

_{g0g(±1,±2)}. is non-vanishing and contributes to the relative light shifts of ground-state Zeeman sub-levels in Eq. (30).

Figure 5 shows the calculated result of EIA signal of cold atoms (*v* ≈ 0) under the relative light shifts *ω _{gigj}* and an absolute light shift

*δ*in

*F*= 2 ⇌

_{g}*F*= 3 transition.

_{e}*S*

_{0,1}= 2|Ω

_{0,1}|

^{2}/Γ

^{2}is the saturation parameter, subscript “0” denotes to the pump light and “1” denotes to the probe light. As we know the intensity of isotropic pump laser lights is much higher than the probe laser light, so the light shift is mainly determined by the pump laser light. Ω

_{gie(i,i±1)}equals to Ω

_{0}multiplied by Clebsch-Gordan coefficients of corresponding transitions in Fig. 4, while

*δ*can be obtain directly from Ω

_{0}in dressed atom picture, that is $\sqrt{{\Omega}_{0}^{2}+{\Delta}_{0}^{2}}-{\Delta}_{0}$ [29], so all light shifts can be calculated from

*S*

_{0}.

## 4. Experimental setup

Figure 6 shows the experimental setup of diffuse laser cooling of ^{87}Rb atoms. The ^{87}Rb atomic vapor is filled in a spherical glass cavity which connects to an ion pump. The vacuum in the cavity is about 10^{-9} Torr. A ceramic integrating sphere is settled surrounding the spherical glass cavity. The cooling and probe lasers for are from one *Toptica* TA100 semiconductor laser, while the repumping beam is from a *Toptica* DL100 laser. The cooling and repumping beam enter the integrating sphere from two multi-mode fibers and generates a diffuse light field via being reflected by the integrating sphere. The inner diameter of the integrating sphere is 48*mm* and the diameter of the spherical glass cavity is 45*mm*. The cooling laser is locked with detuning Δ_{0} to the *F _{g}* = 2 →

*F*= 3 transition frequency, and the repumping laser is lock to the

_{e}*F*= 1 →

_{g}*F*= 2 transition.

_{e}The probe system is made up of a probe laser beam, a detector, and a digital oscilloscope. The probe laser is split from cooling laser so the phase of pump and probe lights are highly correlated. The detector is a light-balanced amplification circuit with two photodiodes, one receives the probe laser beam that propagates through the spherical glass cavity and the other receives the laser beam from probe laser directly. We sweep the frequency of the probe laser with AOM to cover the transmission signals. With the detector, the amplified signals can be obtained and seen in the digital oscilloscope.

The reflectance of our integrating sphere is about 98%, and the aperture area of the integrating sphere is 2.2% of its whole inner surface area. It means a photon can be averagely reflected about 24 times when its remanent probability decays to 1/*e*. Because the diameter of our integrating sphere is 48*mm*, a photon can travel at most 116*cm* for 24 time reflections, which is very small compared with the coherent length of semiconductor laser (usually several hundred meters), and the temporal coherence of diffuse pump light is then well maintained. The integrating sphere here can also randomize the polarization of diffuse cooling laser [30], so atoms have all σ^{+}, σ^{-}, and *π* transitions, which are necessary for the EIA-TOC [23].

## 5. Results and discussions

Figure 7 is the absorption signals varying with detuning of the probe laser Δ_{1}. The large absorption peak around Δ_{1} - Δ_{0} = 0 is the signal of *F* = 2 → *F* = 3, which is not exactly on the zero position due to the light shift caused by the diffuse laser light. The position of the nonlinear spectra is around the detuning of cooling laser Δ_{0}. The strength of the *F* = 2 → *F* = 3 absorption signal and the nonlinear signal are highest at Δ_{0} = - 3Γ because at this detuning the largest number of ^{87}Rb atoms are cooled and captured in our experiment system [6]. Figure 8 is the absorption signal varying with the power of cooling laser that injected into the integrating sphere. We can see two phenomena in the nonlinear spectra. First, the position of the amplification peak and the small absorption peak do not change with the power of the diffuse light, which is the feature of recoil-induced resonances signal because the width of RIR signal only depends on the velocity-distribution of cold atoms [15]. Second, the light shift leads to a deviation of the absorption peak of *F* = 2 → *F* = 3 transition, as well as the Δ_{1} - Δ_{0} = 0 position. The deviation is proportion to the power of diffuse laser light. This is just the feature of EIA that discussed in Sec. 3.

Figure 9 compares the theoretical and experimental results of nonlinear spectra in diffuse laser light. The two dot line in Fig. 9(a) are calculated from theoretical model of RIR in Sec. 2 and EIA in Sec. 3 by Beer-Lambert law. Solid line in Fig. 9(b) is our experimentally observed signal. We see an interesting result that when the power of pump laser of EIA is about 1.5 times to that of recoil-induced resonances, the theoretical result matches the experimental data well. The reason is in our experiment, the two injected cooling laser beams are not diffuse light before the first-time reflection by the inner surface of integrating sphere. Figure 10 describes the distribution of light-field intensity in the integrating sphere. Intensity of the two injected cooling laser beam is higher than that of the diffuse laser light in the center of the sphere in our experiment. The two beams are approximately vertical to the probe laser beam, so they do not participate in the RIR, but they still participate in the EIA. Then we can see the signal is observed under the condition that intensity of pump laser in EIA is about 1.5 times to it in RIR.

The two counter-propagating injected laser beams are linearly polarized. If we replace them by two σ^{+}σ^{-} configured laser beams, an one-dimensional optical molasses will be formed. The light shift caused by the σ^{+}σ^{-} one-dimensional optical molasses will be large enough to cause significant population difference among all ground-state Zeeman sub-levels of cold atoms [9]. For cold ^{87}Rb atoms in one-dimensional optical molasses, *F _{g}* = 2,

*m*= 0 has the largest weight of population and the lowest energy, so stimulated Raman process can happen and its signal may be observed [9, 10].

_{F}Another feature of pump-probed nonlinear spectra of cold atoms in diffuse laser cooling system is the much stronger signals than that in optical molasses. The reason is that their interaction ranges of the cold atoms and pump-probe laser lights are different. In our experiment, the diffuse laser light, as well as the cold atoms, distribute all over the integrating sphere, so cold atoms can be pumped and probed coherently through the whole light path of the probe laser within the integrating sphere. However, the pump-probed interaction range in an optical molasses is the overlap range of probe beam and 3D optical molasses. For same spot size probe beams, the pump-probe interaction range in diffuse laser light is much larger than it is in optical molasses, so it makes diffuse laser cooling method as an optimum technique in studying the nonlinear spectroscopy of cold atoms, as well as their application.

## 6. Conclusions

In conclusion, we have studied the recoil-induced resonances (RIR) and the electromagnetic induced absorption (EIA) of cold atoms in diffuse laser light. We present their completed theoretical models and observe their compound signal of cold ^{87}
Rb atoms which are cooled and pumped by the diffuse laser light in an integrating sphere. Theoretical result can match the experimental one well when the intensity of pump laser light of EIA needs to be 1.5 times to that of RIR, which is because the injected cooling laser before first-time diffuse reflectance only participates in the EIA, while the diffuse laser light participates in both EIA and RIR. Comparing with such two nonlinear spectra of cold atoms in optical molasses, we show the feature of diffuse light case is the much larger pump-probe interaction range. It can provide much stronger signals of the RIR, as well as EIA, which may benefit their future applications.

## Acknowledgment

This work is supported by the National Nature Science Foundation of China under Grant No. 10604057 and National High-Tech Programme under Grant No. 2006AA12Z311.

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