Nonlinear spectroscopy of cold atoms in diffuse laser ... - OSA Publishing

Nonlinear spectroscopy of cold atoms in diffuse laser light Wen-Zhuo Zhang1,2 , Hua-Dong Cheng1 , Ling Xiao1 , Liang Liu1† , and Yu-Zhu Wang1 1 Key

Laboratory of Quantum Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China.

2 Graduate

University of the Chinese Academy of Sciences, Beijing 100039, China. † Corresponding

author: [email protected]

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 OCIS codes: (300.6210) Spectroscopy, atomic; (190.4180) Multiphoton processes; (140.3320) Laser cooling.

References and links 1. S. Tremine, S. Guerandel, D. Holleville, A. Clairon, and N. Dimarcq, “Development of a compact cold atom clock,” 2004 IEEE International Ultrasonics, Ferroelectrics, and Frequency Control Joint 50th Anniversary Conference, 65-70 (2004). 2. W. Ketterle, A. Martin, M. A. Joffe, and P. E. Pritchard, “Slowing and cooling of atoms in isotropic laser light,” Phys. Rev. Lett. 69, 2483-2486 (1992). 3. H. Batelaan, S. Padua, D. H. Yang, C. Xie, R. Gupta, and H. Metcalf, “Slowing of 85Rb atoms with isotropic light,” Phys. Rev. A 49, 2780-2784 (1994). 4. Y. Z. Wang,“Atomic beam slowing by diffuse light in an integrating sphere,” in the Proceedings of the National Symposium on Frequency Standards, Chengdu, China, 1979. 5. H. X. Chen, W. Q. Cai, L. Liu, W. Shu, F. S. Li, and Y. Z. Wang, “Laser Deceleration of an Atomic Beam by Red Shifted Diffuse Light,” Chin. J. Lasers 21, 280-283 (1994). 6. H. D. Cheng, W. Z. Zhang, H. Y. Ma, L. Liu, and Y. Z. Wang, “Laser cooling of rubidium atoms from vapor backgroud in diffuse light,” Phys. Rev. A, to be published. 7. E. Guillot, P.-E. Pottie, and N. Dimarcq, “Three-dimensional cooling of cesium atoms,” Opt. Lett. 26, 1639-1641 (2001). 8. J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, and H. J, Kimble, “Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245-3247 (1991). 9. D. Grison, B. Lounis, C. Salomon, J. Courtois, and G. Grynberg, “Raman spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149-154 (1991). 10. J.-Y. Courtois and G. Grynberg, “Probe transmission in a one-dimensional optical molasses Theory for circularlycross-polarized cooling beams,” Phys. Rev. A 48, 1378-1399 (1993). 11. G. Grynberg, B. Lounis, P. Verkerk, J. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in two- and three-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249-2252 (1993). 12. T. van der Veldt, J. F. Roth, P. Grelu, and P. Grangier, “Nonlinear absorption and dispersion of cold 87 Rb atoms,” Opt. Commun. 137, 420-426 (1997).

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Received 9 Jan 2009; revised 2 Feb 2009; accepted 7 Feb 2009; published 11 Feb 2009

16 February 2009 / Vol. 17, No. 4 / OPTICS EXPRESS 2892

13. Actually, the Lambertian-reflected monochromatic lights are not strictly spatially incherent. For details, see W. H. Carter and E. Wolf, J. Opt. Soc. Am. 65, 1067-1071 (1975). 14. R. Anderson, “Polarized light, the integrating sphere, and target calibration,” Appl. Opt. 29, 4235-4240 (1990). 15. J. Guo, P. R. Berman, B. Dubetsky, and G. Grynberg, “Recoil-induced resonances in nonlinear spectroscopy,” Phys. Rev. A 46, 1426-1437 (1992). 16. J. Guo and P. R. Berman, “Recoil-induced resonances in pump-probe spectroscopy including effects of level degeneracy,” Phys. Rev. A 47, 4128-4142 (1993). 17. D. R. Meacher, D. Boiron, H. Metcalf, C. Salomon, and G. Grynberg, “Method for velocimetry of cold atoms,” Phys. Rev. A 50, R1992-R1994 (1994). 18. J. -Y. Courtois, G. Grynberg, B. Lounis, and P. Verkerk, “Recoil-induced resonances in cesium: An atomic analog to the free-electron laser,” Phys. Rev. Lett. 72, 3017-3020 (1994). 19. A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59, 47324735 (1999). 20. A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in Rb vapor,” Phys. Rev. A 57, 2996-3002 (1998). 21. K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, “Dependence of electromagnetically induced absorption on two combinations of orthogonal polarized beams,” J. Phys. B 34, 2951-2961 (2001). 22. C. Affolderbach, S. Knappe, and R. Wynands, “Electromagnetically induced transparency and absorption in a standing wave,” Phys. Rev. A 65, 043810 (2002). 23. C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, “Electromagnetically induced absorption due to transfer of coherence and to transfer of population,” Phys. Rev. A 67, 033807 (2003). 24. C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, “Electromagnetically induced absorption due to transfer of population in degenerate two-level systems,” Phys. Rev. A 70, 043814 (2004). 25. J. Dimitrijevic, D. Arsenovic, and B. M. Jelenkovic, “Intensity dependence narrowing of electromagnetically induced absorption in a Doppler-broadened medium,” Phys. Rev. A 76, 013836 (2007). 26. J. Dimitrijevic, Z. Grujic, M. Mijailovic, D. Arsenovic, B. Panic, and B. M. Jelenkovic, “Enhancement of electromagnetically induced absorption with elliptically polarized light - laser intensity dependent coherence effect,” Opt. Ex. 16, 1343-1353 (2008). 27. A. Lipsich, S. Barreiro, P. Valente, and A. Lezama, “Inspection of a magneto-optical trap via electromagnetically induced absorption,” Opt. Commun. 190, 185-191 (2001). 28. H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping, Springer-Verlag, New York, (1999). 29. Weihan Tan, Weiping Lu, and R. G. Harrison, “Approach to the theory of radiation-matter interaction for arbitrary field strength,” Phys. Rev. A 46, 7128-7138 (1992). 30. S. C. McClain, C. L. Bartlett, J. L. Pezzaniti, and R. A. Chipman, “Depolarization measurements of an integrating sphere,” Appl. Opt. 34, 152-154 (1995).

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 velocityv resonates with the photons whose propagating directions are at an angle θ with v, and (1) Δ − kv cos θ = 0, 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 magneticfield-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 #106207 - $15.00 USD

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mF =

-3

mF =

-2

-1

-2

-1

0

1

2

0

1

2

3

52 P3/ 2 F=3

52 S1/ 2 F=2

Fig. 1. Pump transition and steady-state population of every ground states of 87 Rb in diffuse laser lights

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 (ω1 − ω0 ) can oscillate with the atoms, and two-photon process can happen. Here ω1 is the frequency of probe laser light and ω0 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 pumpprobe 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 #106207 - $15.00 USD

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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 ω0 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 recoilinduced resonances. The interaction Hamiltonian of pump-probe lights interacting with a twolevel atomic system is HI (ω0,1 ) = h¯ Ω0,1 |eg| cos(k0,1 · X − ω0,1t) + h¯ Ω∗0,1 |ge| cos(k0,1 · X − ω0,1t),

(2)

where Ω0 is the Rabi frequency of pump field with wave vector k0 and frequency ω0 . Ω1 is the Rabi frequency of probe field with wave vector k1 and frequency ω1 . |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

ρ = ∑ ρaa |a, pa , p |.

(3)

a,a

Then we obtain the time evolution equations for all atomic density matrix elements [15]: d ρ˜ ee (p, p ) = −(Γ + γ )ρ˜ ee (p, p ) dt 1

+ i ∑ Ω∗a exp[i(Δa + ωr − a=0 1

ka · p )t]ρeg (p, p − h¯ ka ) m

− i ∑ Ω∗a exp[i(−Δa − ωr + a=0

(4)

ka · p )t]ρge (p − h¯ ka , p ), m

p − p d · qt) ρ˜ gg (p, p ) = − ΓN(q)dqρ˜ ee (p + h¯ q, p + h¯ q) exp(i dt m − γ ρ˜ gg (p, p ) + γ W (p, p ) + i ∑ Ω∗a exp[i(Δa + ωr −

ka · p )t]ρgg (p, p − h¯ ka ) m

− i ∑ Ω∗a exp[i(Δa − ωr −

ka · p )t]ρee (p + h¯ ka , p ), m

1

a=0 1 a=0

(5)

d Γ ρ˜ ge (p, p ) = − ρ˜ ge (p, p ) dt 2 + i ∑ Ω∗a exp[i(Δa + ωr −

ka · p )t]ρgg (p, p − h¯ ka ) m

− i ∑ Ω∗a exp[i(Δa − ωr −

ka · p )t]ρee (p + h¯ ka , p ), m

1

a=0 1 a=0

d d ρ˜ eg (p, p ) = [ ρ˜ ge (p ), p]∗ . dt dt Here

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ρ˜ aa (p, p ) = ρaa (p, p’)e[(p

2 −p2 )t/2m¯h]

(6)

(7) eiωaa t ,

(8)



Δ0 is the detuning of pump laser and Δ1 is the detuning of probe laser, which are given by Δ0,1 = ω0,1 − ωeg ,

(9)

ωeg is the transition frequency from ground state |g to excited state |e, and the recoil frequency is h¯ k2 . (10) ωr = 2m 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 h¯ q, which for a two level atom is given by [15] 3 sin2 θ , (11) N(q) = 8π where θ is the angle between h¯ q and field polarization direction. The absorption signal is proportional to the imaginary part of the component of ρ˜ ge , which is ρ˜ ge = ρge (x,t) exp(ik1 · x − iω1t) (p2 − p2 ) 1 (p − p ) · x t + i − i dpdp ω t ρ˜ ge (p, p ) exp(ik1 · x − iω1t). = i (2π h¯ )3 2m¯h h¯ (12) It can be solved from the third-order perturbation solutions of Eqs. (4)-(7) in the limit that perturbation treatment is valid [15]. Ω2 ωr 0, momentum distribution W (−px ) > W (−px + h¯ k1 ) so the transform W (−px ) → W (−px + h¯ k1 ) means the strengthened absorption. However, w(px ) → w(px + h¯ k1 ) 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 px −py and W (px ) − W (py ) = W (px ) − W [−px (1 + cos θ )/ sin θ ] is always non-vanishing, then we can observe the RIR signal. We need to determine a critical angle θc , for which the condition |px | | − py sin θ | is satisfied in the region of 0 < θ < θc and the RIR signal is evident. We choose the minimum momentum limit px = 2¯hk, which is recoil √momentum of twophotons, and choose py equals to twice of the most probable momentum 2mkb T , with which #106207 - $15.00 USD

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Fig. 3. Calculated probe absorption signal of recoil induced resonance in diffuse pump field. Γ = 6.056MHz (87 Rb, Fg = 2 → Fe = 3), γ = 0.05Γ, Δ0 = −3Γ, T = 200μ K, S0 = 2 and S1 = 0.01.

√ 0 < |v| < 2 2mkb T contains the majority of the cold atoms. Then θc can be calculated from sin θc h¯ k1 = √ 1 + cos θc 2 2mkb T

(19)

So the solution of θc is small, which just meet the √ condition θc ≈ 0 of deriving Eq. (17). In the range 0 < θ < θc , for all px > 2¯hk and py < 8mkb T , the W (px ) − W (py ) is always nonvanishing. The total signal for isotropic pumped recoil-induced resonance is an integration of Eq. (17) over −θc to θc . Under the condition of Eq. (14), we calculated the average of the recoil-induced resonances over the range −θc ≤ θ ≤ θc . The result is showed in Fig. 3 with 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 Fe > Fg > 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]. 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, #106207 - $15.00 USD

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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 Fg = 2 → Fe = 3 transition of 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 HeIi g j = HeIi g j (ω0 )e−iω0 t + HeIi g j (ω1 )e−iω1 t ,

(20)

where HeIi g j (ω0,1 ) is the Rabi frequency for ei → g j transition. HeIi g j (ω0,1 ) = μei g j E0,1 = h¯ (−1)Fe −me

Fe −me

1 q

Fg mg

Ω0,1 .

(21)

Here ω0 is the frequency of pump laser and ω1 is the frequency of probe laser. Ω0,1 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). d ρe e (ω0 ) = − (Γ + iωei e j )ρei e j (ω0 ) dt i j i 2 + ∑ [ρei gk (ω0 )HgIk ei (−ω0 ) − HeIi gk (ω0 )ρgk ei (−ω0 )], h¯ k=−2

(22)

Γ + Γgi d + i(ωei g j − ω0 )]ρei e j (ω0 ) ρei g j (ω0 ) = − [ dt 2 2 i 3 + ∑ [ρei ek (ω0 )HeIk g j (−ω0 ) − i ∑ HeIi gk (ω0 )ρgk ei (−ω0 )], h¯ k=−3 k=−2

(23)

d ρg g (ω0 ) = −iωgi g j ρgi g j (ω0 ) + 7Γρgsi g j (ω0 ) dt i j i 3 + ∑ [HeIi gk (ω0 )ρgk ei (−ω0 ) − ρei gk (ω0 )HgIk ei (−ω0 )], h¯ k=−3

(24)

ρei e j (ω0 ) = ρei e j exp(iω0t),

(25)

where and

ρgsi g j (ω0,1 ) =

1

3

∑ ∑

(−1)−mk −ml

q=−1 k,l=−3

×

Fg −mgi

1 q

Fe me

ρek el (ω0,1 )

Fe −me

1 q

Fg mgi

(26) ,

which is the spontaneous emission term. In Eqs. (22)-(24), Γ is the decay rate of excited state Fe = 3, and Γgi is the decay rate from ground state sub-level gi . Because 87 Rb atom has another ground state hyperfine level Fg = 1 whose energy is 6.8GHz lower than Fg = 2, the Zeeman #106207 - $15.00 USD

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sub-levels of Fg = 2 can decay to Fg = 1 with rate gi . The repumping laser is needed to pump the population from Fg = 1 back to Fg = 2. Another decay rate is the γ 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. In the second stage, a weak probe laser is added as a perturbation to the system. The density matrix elements of population ρei e j and ρgi g j can oscillates with frequencies ω1 − ω0 and ω0 − ω1 , while the coherence term of ground and excited state is

ρei g j = ρei g j (ω0 ) exp[−iω0t + iφ ] + ρei g j (ω1 ) exp[−iω1t].

(27)

Then we can get the optical Bloch equations of the second stage d ρe e (ω1 − ω0 ) = − [Γ + γδei e j − i(ω1 − ω0 − ωei e j )]ρei e j (ω1 − ω0 ) dt i j i 2 + ∑ [ρei gk (ω1 )HgIk ei (−ω0 ) − HeIi gk (ω1 )ρgk ei (−ω0 )], h¯ k=−2 Γ + Γg j d + i(ωei g j − ω1 )]ρei e j (ω1 ) ρe g (ω1 ) = − [ dt i j 2 2 i 3 + ∑ ρei ek (ω0 )HeIk g j (ω1 ) − i ∑ HeIi gk (ω1 )ρgk ei (−ω0 ) h¯ k=−3 k=−2 +

(28)

(29)

2 i 3 ρei ek (ω1 − ω0 )HeIk g j (ω0 ) − i ∑ HeIi gk (ω0 )ρgk ei (ω1 − ω0 ), ∑ h¯ k=−3 k=−2

Γgi + Γg j d − i(ω1 − ω0 − ωgi g j )]ρgi g j (ω1 − ω0 ) ρgi g j (ω1 − ω0 ) = − [ dt 2 + 7Γρgsi g j (ω1 − ω0 ) +

2

∑

Γgi gk δgi g j ρgk gk (ω1 − ω0 )

k=−2

+

(30)

i 3 ∑ [HeIi gk (ω1 )ρgk ei (−ω0 ) − ρei gk (ω1 )HgIk ei (−ω0 )]. h¯ k=−3

There are also another three equations that describe the time evolution of the density matrix elements at the frequency ω0 − ω1 , which can be written via replacing ω1 − ω0 by ω0 − ω1 in Eqs. (28)-(30). The probe absorption intensity is proportional to Im[∑ HeIi g j (ω1 )ρei g j (ω1 )],

(31)

i, j

whose steady-state value can be solved from Eqs.(28)-(30). There are relative light shifts ωgi g j 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 sublevel δgi 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 2 2 2 2 ground-state Zeeman sub-levels are ωgi g j = h¯ ( Δ + Ωgi − Δ + Ωg j )/2 [28], where Ωgi = (Ωgi e(i−1) + Ωgi ei + Ωgi e(i+1) )/3 is the average Rabi frequency of σ + , σ − , and π transitions. Rabi #106207 - $15.00 USD

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mF =

-3

1 30

2

5 P3 / 2 F=3 1 2

2

5 S1/ 2

1 6

-2

0

-1

1 10

1 10

4 15 1 3

F=2

mF =

-1

-2

1 5

0

2 1 30 1 6

4 15

3 10

1 5

1

3 1 2

1 3

1

2

Fig. 4. Clebsch-Gordan coefficients and light shifts of every sub-levels for Fg = 2 − Fe = 3 transition.

Fig. 5. Calculation results of probe absorption. Γ = 6.056MHz (87 Rb, Fg = 2 → Fe = 3), γ = Γg = 0.05Γ, Δ0 = −3Γ, S1 = 0.1, S0 = 1, 5, 10.

frequencies Ωgi e(i,i±1) are proportional to the square of Clebsch-Gordan coefficients of corresponding transitions, so after the calculation we find that Ωg1 = Ωg−1 = Ωg2 = Ωg−2 < Ωg1 , then the terms of ωg(±1,±2) g(±1,±2) are all equal to zero. Only ωg0 g(±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 ωgi g j and an absolute light shift δ in Fg = 2 Fe = 3 transition. S0,1 = 2|Ω0,1 |2 /Γ2 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. Ωgi e(i,i±1) equals to Ω0 multiplied by Clebsch-Gordan coefficients of corresponding transitions in Fig. 4, while δ can be obtain directly from Ω0 in dressed atom picture, that is shifts can be calculated from S0 .

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Ω20 + Δ20 − Δ0 [29], so all light



Muti-mode fibers +

Laser for repumping

LF356

l/2 Laser for cooling and probe

Glass cavity l/2

Light-balance circuit

Digital oscilloscope

Integrating sphere

l/2

Fig. 6. Experimental setup of the diffuse cooling sphere.

4.

87 Rb

atomic vapor in an integrating

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 48mm and the diameter of the spherical glass cavity is 45mm. The cooling laser is locked with detuning Δ0 to the Fg = 2 → Fe = 3 transition frequency, and the repumping laser is lock to the Fg = 1 → Fe = 2 transition. 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 48mm, a photon can travel at most 116cm 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 Δ1 . The large absorption peak around Δ1 − Δ0 = 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 Δ0 . The strength of the F = 2 → F = 3 absorption signal and the nonlinear signal are highest at Δ0 = −3Γ because at this detuning the largest number of 87 Rb atoms are cooled and captured in our experiment system [6]. Figure #106207 - $15.00 USD

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Fig. 7. Experimental signal varying with the detuning of probe laser light at three different diffuse light detunings: (a) Δ0 = −2Γ, (b) Δ0 = −3Γ, (c) Δ0 = −2Γ. Power of injected cooling laser beams are 40 mW/cm2 .

Fig. 8. Experimental signal varying with power of injected cooling laser: (a) 40 mW/cm2 , (b) 32 mW/cm2 , (c) 24 mW/cm2 , (d) 16 mW/cm2 . Δ0 = −3Γ.

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Fig. 9. (a) The two dot line are the calculated signals of recoil-induced resonances when S0 = 2.5, S1 = 0.03 and EIA when S0 = 3.80, S1 = 0.03. Their sum is the solid line. (b) Experimentally observed signal under Δ0 = −3Γ when the total power of injected cooling laser beams is 40 mW/cm2 .

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 Δ1 − Δ0 = 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 #106207 - $15.00 USD

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b a

Cooling laser

Cooling laser

b

Probe laser

Fig. 10. light-field distribution in the integrating sphere, the injected laser are reflected by the inner surface of the integrating sphere to create diffuse laser light for laser cooling. Before the first-time reflection, the injected laser are two expanded beams due to the fibers have a the numerical aperture. Through the light path of the probe beam, the two expanded beams and the diffuse laser light are all pump light in region (a), while only diffuse laser light are the pump light in region (b).

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, Fg = 2, mF = 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]. 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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