Electromagnetically induced transparency in a Bose–Einstein condensate

1 October 2002

Optics Communications 211 (2002) 159–165 www.elsevier.com/locate/optcom

Short Communications

Electromagnetically induced transparency in a Bose–Einstein condensate V. Ahufinger a, R. Corbal
an a,*, F. Cataliotti b, S. Burger b, F. Minardi b, C. Fort b b

a Departament de F
ısica, Universitat Aut onoma de Barcelona, E-08193 Bellaterra, Spain INFM-European Laboratory for Non-Linear Spectroscopy (LENS) and Dipartimento di F
ısica, Universit a di Firenze, Via Nello Carrara 1, I-50019 Sesto F.no, Italy

Received 18 December 2001; received in revised form 1 July 2002; accepted 18 July 2002

Abstract We report on the direct observation of the electromagnetically induced transparency (EIT) lineshape of cold 87 Rb atoms above and below the transition temperature for Bose–Einstein condensation (BEC). Similar results are observed in both temperature regimes, with an absorption reduction of about 60%. Good agreement with a theoretical model is discussed. Ó 2002 Elsevier Science B.V. All rights reserved.

Bose–Einstein condensates (BEC) of weakly interacting atomic gases are macroscopic quantum systems which exhibit a wide variety of fascinating phenomena at the interface between major areas of physics, such as statistical and condensed matter physics on the one hand, and atomic, molecular and optical physics on the other hand [1]. The study of the optical properties of coherently prepared atoms is also a very active research field. Such a preparation is usually performed by applying a strong resonant coupling laser field that induces atomic coherence and dramatically modifies the optical properties for a weak probe laser field coupled to an adjacent transition in a so called three-level scheme. Atomic coherence man-

*

Corresponding author. Fax: +34935812155. E-mail address: [email protected] (R. Corbal
an).

ifests itself in a rich variety of phenomena, namely, electromagnetically induced transparency (EIT) [2], and related phenomena such as amplification and lasing without inversion, high-refractive index without absorption, steep dispersion and either ultraslow group velocity or gain-assisted superluminal light propagation, and efficient non-linear optical processes [3,4]. EIT has many useful applications, as for instance in laser cooling for obtaining cooling to the ground state either in free space [5] or in traps [6]. Moreover, recent reports [7] have shown that using EIT a light pulse can be effectively decelerated and trapped in an atomic medium, stored for a controlled period of time, and then released on demand. This phenomenon could have important applications in the field of quantum information. Since the storage time is limited by the atomic coherence lifetime responsible for EIT, it is interesting to evaluate such effects

0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 8 2 6 - 6

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for a BEC, where one expects longer coherence relaxation times [7]. We have studied the EIT absorption spectrum with samples of cold 87 Rb atoms above and below the transition temperature Tc for Bose–Einstein condensation. In our experimental setup [8] we produce BECs of 87 Rb atoms in the (F ¼ 1, mF ¼ 1) state by means of laser cooling and forced evaporation in a magnetostatic trap. The condensates are cigar-shaped with the long axis (the z-axis) oriented horizontally. With a typical number of atoms N ¼ 4  105 the dimensions (Thomas–Fermi radii) are Rz ¼ 55 lm and R? ¼ 5:5 lm. In order to observe EIT we use the lambda-scheme shown in Fig. 1. The coupling laser beam originates from an external cavity diode laser and is locked on the F ¼ 2 $ F 0 ¼ 2 transition, within 1 MHz of the line center, via saturation spectroscopy on a separate Rb cell. From the same laser beam, with an electro-optic modulator (EOM) we produce a sideband (the probe) containing 12% of the power quasi resonant with the F ¼ 1 $ F 0 ¼ 2 transition. Both, coupling and probe laser fields, with r polarization, are copropagating through the sample along the axial z direction and we observe the absorption of both beams with the same detector. The spectra are taken with the following procedure. The magnetic trap is switched off and the atoms ballistically expand 10 ms before being illuminated with a 1 ms pulse of the bichromatic light. During the pulse the frequency of the probe beam is swept linearly

through resonance using the EOM while the frequency of the pump beam is kept fixed. The bandwidth of our photodiode was well above 1 MHz allowing for a resolution of 30 kHz with our sweeping time. A more critical issue is the heating of the sample due to the probe, however we estimate a maximum heating rate of 50 lK/ms causing a negligible density decrease during the scanning. Indeed we have observed two consecutive scans in opposite frequency direction without significant signal degradation. We interpret these data by considering a Bose– Einstein condensate with three internal states j1i, j2i and j3i in an K configuration (see Fig. 1): j1i and j2i are hyperfine ground states while j3i is an excited state. In our case j1i ¼ jS1=2 ðF ¼ 1; mF ¼ 1Þi, j2i ¼ jS1=2 ðF ¼ 2; mF ¼ 1Þi and j3i ¼ jP1=2 ðF 0 ¼ 2; mF 0 ¼ 2Þi. Transitions j1i $ j3i and j2i $ j3i are driven by two light beams ~ Ei ¼ 1 ~ A ðexp iðk z  x tÞ þ ccÞ ði ¼ 1; 2Þ, respectively. e i i 2 i i The single particle state, jWi, can be written as: X jWi ¼ /1 ð~ r; tÞjii; ð1Þ i¼1;2;3

r; tÞ is the centre of mass mean-field where /i ð~ wavefunction for a particle in state jii [9]. The /i ð~ r; tÞ evolution is determined by the internalenergy atomic Hamiltonian, HA , the electric dipole interaction HI ¼ ~ d ð~ E1 þ ~ E2 Þ, the interaction with the magnetic trap and the atom–atom interaction, ruled by the Gross Pitaevskii equation (GPE). Within the rotating wave approximation, and defining: /3 ð~ r; tÞ w3 ; /1 ð~ r; tÞ w1 expðix31 tÞ; /2 ð~ r; tÞ w2 expðix32 tÞ;

ð2Þ

where x31 and x32 atomic resonance frequencies [10], one obtains: ih

Fig. 1. Energy levels of 87 Rb involved in the experiment, Xi and dj are Rabi frequencies and detunings, respectively. The condensate is produced in level j1i.

o 1 w ¼  hX1 w3 expðik1 zÞ expðid1 tÞ ot 1 2 " h2 r2 þ V1 þ  2m # X 4ph2 a1j 2 jwj j w1 ; þ m j¼1;2;3

ð3aÞ

V. Ahufinger et al. / Optics Communications 211 (2002) 159–165

ih

i h

o 1 w2 ¼   hX2 w3 expðik2 zÞ expðid2 tÞ ot 2 " h2 r 2  þ V2 þ  2m # X 4p h2 a2j 2 jwj j w2 ; þ m j¼1;2;3 o 1 w ¼  hX1 w1 expðik1 zÞ expðid1 tÞ ot 3 2 1  hX2 w2 expðik2 zÞ expðid2 tÞ 2 " h2 r 2  þ  þ V3 2m # X 4p h2 a3j 2 jwj j w3 ; þ m j¼1;2;3

161

k Þ ¼ cl ð~ k Þ expðidl tÞ ðl ¼ 1; 2Þ, to eliminate ing Cl ð~ fast oscillations, and assuming that the cj ð~ k Þ are ~ r-independent, one obtains: # " 2 2 X 4ph2 a1j o h  k 2 jCj j C1 ih C1 ð~ kÞ ¼ þ ot 2m j¼1;2;3 m ð3bÞ

ð3cÞ

di3 ~ ei Ai and di ¼ xi  x3i ði ¼ 1; 2Þ where Xi ¼ ~ are Rabi frequencies and detunings, respectively. Vi are the trapping potentials, and aij are the scattering lengths. Since we observed atoms released from the trap we take Vi ¼ 0 hereafter. Let us introduce also translational quantum numbers and denote by ji;~ pi an atom in level i with centre of mass momentum ~ p. If an atom trapped initially in state j1i has a momentum ~ p ¼ h~ k , then w1 c1 ð~ k Þ expði~ k ~ rÞ. To get excited to state j3i the atom absorbs a photon from the field ~ E1 and therefore a momentum pz ¼  hk1 , thus, w3 c3 ð~ k Þ exp½ið~ k ~ r þ k1 zÞ. As in [11], to simplify the analysis, we consider light scattering only into the j1i and j2i BEC modes. The scattering to these modes can be stimulated by a large number of atoms in condensates j1i and j2i [11]. In our case, this bosonic enhancement occurs mainly for the scattering to mode j1i. In fact, we produce the BEC in level j1i and under EIT conditions neither the coupling nor the probe laser beam is absorbed, therefore levels j2i and 3i are almost unpopulated levels. The emissions from level j3i are described by the decay rates Ci ði ¼ 1; 2Þ. Therefore, w2 c2 ð~ k Þ exp½ið~ k ~ r þ k1 z  k2 zÞ, and, for each value of ~ k , one has only three states ji;~ pi which are coupled: j1; h~ k i, j2;  hð~ k þ k1  k2 Þi, and j3;  hð~ k þ k1 Þi. Equations of motion for the cj ð~ k Þ ðj ¼ 1; 2; 3Þ are easily obtained from Eqs. (3a)–(3c). By defin-

1  hX1 C3 þ hd1 C1 ; 2 " o h2 2 2 ½k þ ky2 þ ðkz þ k1  k2 Þ  kÞ ¼ ih C2 ð~ ot 2m x # X 4ph2 a2j 2 jCj j C2 þ m j¼1;2;3 1  hX2 C3 þ hd2 C2 ; 2 " o h2 2 ½k þ ky2 þ ðkz þ k1 Þ2  kÞ ¼ ih C3 ð~ ot 2m x # X 4ph2 a3j 2 jCj j C3 þ m j¼1;2;3 1 1  hX1 C1  hX2 C2 : 2 2

ð4aÞ

ð4bÞ

ð4cÞ

Note that these equations take into account coherent (reversible) couplings only. The incoherent (decay) processes will be treated phenomenologically, and this is more conveniently done in the usual density matrix formalism. The equations of motion for the density matrix elements, qij Ci Cj , are derived from Eqs. (4a)–(4c). After phenomenologically introducing the various decay terms one finally obtains: h i i q_ 13 ¼  i d1  ðC1 þ C2 þ 2ccol Þ q13 2 i i þ X1 ðq33  q11 Þ  X2 q12 2 2 X 4ph ½ða1j  a3j Þqjj q13 ; i ð5aÞ m j¼1;2;3 h i i q_ 23 ¼  i d2  ðC1 þ C2 þ 2ccol Þ q23 2 i i þ X2 ðq33  q22 Þ  X1 q21 2 2 X 4ph ½ða2j  a3j Þqjj q23 ; i ð5bÞ m j¼1;2;3

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q_ 12 ¼ i½d1  d2  iðccol þ cÞq12 i i þ X1 q32  X2 q13 2 2 X 4p h ½ða1j  a2j Þqjj q12 ; i m j¼1;2;3

ð5cÞ

q_ 11 ¼ C1 q33 ¼ X1 Imðq31 Þ;

ð5dÞ

q_ 22 ¼ C2 q33  X2 Imðq32 Þ;

ð5eÞ

q_ 33 ¼ ðC1 þ C2 Þq33 þ X1 Imðq31 Þ þ X2 Imðq32 Þ; ð5fÞ where we have introduced the total detunings   2 1  h ðk12 þ 2k1 kz Þ d1 ¼ d1  and h 2m    2 1  h 2 ðk  2k2 kz  2k1 k2 Þ ; d2 ¼ d2 þ h 2m 2  and the extra decay rates ccol and c. These decays take into account the elastic collisions of condensed and thermal atoms and the diffusion of the phase difference between the two light fields, respectively. Note that the total detunings involve not only the internal energy but also the recoil energies ð h2 ki2 Þ=2m and the Doppler shifts 2 ð h =2mÞ2ki kz ði ¼ 1; 2Þ. Since k1  k2 one can write for d2 an expression similar to that for d1 . In fact,   2 1  h 2 ðk þ 2k2 kz Þ : d2  d2  h 2m 2  Note that the only component of the atomic momentum that appears in the detunings is the one ðkz Þ in the direction of the light beams. In the experiment the light beams propagate horizontally. In this case kz is not influenced by gravity during the ballistic expansion of the atoms. To simulate the experimental procedure, we are only interested here in the time-dependent solution to Eqs. (5a)– (5e) and (5f) with a probe detuning d1 linearly swept across an interval of 30 MHz in 1 ms. As it is well known, the contribution of atoms with momentum kz to the probe absorption coefficient is proportional to Imq31 ðkz Þ, thus the total probe absorption coefficient is proportional to R dkz N ðkz Þ Imq31 ðkz Þ. Here dkz N ðkz Þ is the number of atoms per unit volume with linear momentum kz . For a thermal sample we take for N ðkz Þ the

usual Maxwellian velocity distribution, while for a pure BEC we take a gaussian distribution with 1=2 a most probable k z ¼ ð2mxz =hÞ , where xz ¼ 12:6  2p Hz is the axial trapping frequency. Since in the experiment we measure the absorption of both coupling and probe laser beams, we have calculated (and plotted in Figs. 2 and 4 below) the total intensity absorption coefficient, given by X 2hxi Xi Z a¼ dkz N ðkz ÞImq3i ðkz Þ; Ii i¼1;2 where Ii ði ¼ 1; 2Þ are the probe and coupling beam intensities. Note that for an optically thin sample of length L the normalized absorption is given by aL ¼ ðI0  IÞ=I0 , where I0 ¼ I1 þ I2 is the total transmitted intensity far out of resonance while I is the transmitted intensity near resonance. As one could expect, we have verified that for the very cold atomic samples considered in this paper, the momentum averaging has a negligible influence since the Doppler broadening of optical lines is smaller than their homogeneous width. Moreover, since the recoil shift is also much smaller than the homogeneous width, one has di  di ði ¼ 1; 2Þ for atomic samples released from the magnetic trap. Except for the non-linear terms, these Eqs. (5a)–(5e) and (5f) are identical to the ones for a non-condensed medium. At the densities used in our experiment, and taking into account that a3j ¼ 0, since j3i is an excited state, and that for Rb a1j  a2j , it results that the non-linear terms have not noticeable influence in the simulations.

Fig. 2. Theoretical spectrum for a pure condensate released from the trap for d2 ¼ 0; X2 ¼ 1:5  2p MHz, X1 ¼ 0:37X2 ; c ¼ ccol ¼ 0 and C1 ¼ 1:5  2p MHz, C2 ¼ 1:5  2p MHz.

V. Ahufinger et al. / Optics Communications 211 (2002) 159–165

This explains both why the spectrum shown in Fig. 2, for a condensed medium, is the typical EIT spectrum for a non-condensed medium [3] and also why the experimental spectra presented below (Fig. 3) show no appreciable variation with the temperature, above and below the transition temperature for Bose–Einstein condensation. In general, however, the non-linear terms could have observable manifestations if a1j 6¼ a2j . In such a case, while in steady-state conditions the EIT dip for samples released from the trap appears at d1 ¼ d2 for a thermal sample, it would move to X 4p h ða1j  a2j Þqjj d1 ¼ d2  m j¼1;2;3 for a pure BEC. The non-linear terms also influence, in general, the asymmetry of the absorption spectra for dense enough samples [12]. A case with a1j 6¼ a2j ¼ a3j ¼ 0 occurs when instead of the K configuration used here a cascade configuration (level j2i above common level j3i) is used. Simi-

Fig. 3. Absorption spectra for a thermal sample at T ¼ 500 nK (trace (a)), a mixed sample composed of a condensate and a thermal component (trace (b)), and a pure condensate (trace (c)).

163

larly, for a V configuration, with common level j3i below levels j1i and j2i, one has a3j 6¼ a1j ¼ a2j ¼ 0: Note that the percentage of transparency at line center is mainly determined by the rate of the incoherent processes that reduce the atomic coherence between the two ground states, i.e., ccol and c. In the case plotted in Fig. 2, the transparency dip reaches nearly 100% because ccol ¼ c ¼ 0. The full width at half maximum of the dip, dx ¼ 0:35  2p MHz, is determined by the strength of the coupling laser beam. Typical experimental recordings of the bichromatic light absorption are shown in Fig. 3 for a thermal sample at T ¼ 500 nK (trace (a)), a mixed sample composed of a condensate and a thermal component (trace (b)), and a pure condensate (trace (c)). For all three cases the Rabi frequency of the coupling field was about X2 ¼ 1:5  2p MHz and the ratio between the intensities of the probe laser and the coupling laser was I1 =I2 ¼ 0:14. We performed Lorentzian fits to the wings of the recorded absorption spectra, where probe absorption is linear. The maximum of the fitted Lorentzian occurs at d1 ¼ 0 and the nonlinear absorption dip at d1 ¼ d2 . In this way we deduced that recordings in Fig. 3 were obtained with d2 ¼ 0:9  2p MHz in (a) and (b) and d2 ¼ 1:05  2p MHz in (c). As expected from the discussion above the three spectra are similar. However, the absorption we detect for the various clouds is quite different being around 9% in the thermal cloud and 3% in the other two cases. This is due to the different size of the clouds, with the condensate being much smaller than the size of the light beam. At the EIT dip a maximum reduction of absorption is observed with values of 59  5%, 61  5%, and 65  5% for the three traces (top to bottom). In Fig. 4 we show a comparison between experimental data (squares) and predictions from the model (solid lines) for the cases of a thermal sample (a) or a pure condensate (b) also shown in Figs. 3(a) and (c), respectively. The parameters used in the simulations for trace (a) (trace (b)) are: d2 ¼ 0:9  2p MHz (d2 ¼ 1:05  2p MHz), X2 ¼ 1:5  2p MHz, X1 ¼ 0:37X2 , thus fulfiling the experimental ratio I1 =I2 ¼ 0:14; ccol ¼ 110  2p kHz ðccol ¼ 80  2p kHzÞ, and C ¼ C1 þ C2 ¼ 6 

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Fig. 4. Comparison between experimental and numerical data for (a) a thermal sample at 500 nK and (b) a pure condensate. For parameters see text.

2p MHz, with a ratio C1 =C2 ¼ 3 equal to the ratio of the tabulated spontaneous decay rates for the corresponding transitions. The width C ¼ 6  2p MHz is consistent with the expected natural decay time ðs ¼ 28:1 ns) and an extra broadening of about 0:3  2p MHz, probably due to a residual magnetic field. Note that power broadening is taken into account in the simulations. As expected the collisional decay rate of the ground-state coherence is smaller for the condensate than for the thermal sample, and the ccol used are reasonable and in agreement with the literature [11]. Note that level j3i decays not only to levels j1i and j2i (see Fig. 1) but also to level j4i F ¼ 2; mF ¼ 2i (not shown in Fig. 1); these four levels form a closed system that we have analyzed numerically for the parameters above. We have found that during the observation time population in level j4i remains negligible. Therefore, we used the three-level model discussed above in the simulations. Indeed, one can

see that the simulations have good agreement with the experimental data. The small difference between simulations and experiment may be due to experimental details not taken into account in simulations, such as the changing atomic density due both to light heating and to ballistic expansion during the observation time. We have chosen an observation time of 1 ms as a compromise: it is short enough to keep atomic density changes small and it is long enough to have a reasonable spectral resolution of the EIT dip. In conclusion, we have studied the EIT spectrum with samples of cold 87 Rb atoms above and below the transition temperature for Bose–Einstein condensation. We have compared the experimentally observed lineshapes to a theoretical modeling and obtained a good agreement. Similar results are observed in both temperature regimes, with an absorption reduction of about 60% at the EIT window. This imperfect transparency is due to the decay of the Zeeman coherence produced by atomic collisions, thus evidencing the role of atomic collisions in cold clouds. The theoretical model suggests that qualitatively different EIT spectra could be obtained above and below the transition temperature for Bose–Einstein condensation by using dense enough samples with either cascade or V configurations. Acknowledgements Support from the DGESIC (Spanish Government), from the DGR (Catalan Government), and from the EC (contracts HPRI-CT-1999-00111 and HPMT-CT-2000-00123) is acknowledged. We acknowledge funding from the INFM (project ‘‘Photon Matter’’) and the MURST (Italian Government). We wish to thank M. Artoni and M. Inguscio for many useful discussions. R.C. dedicates this work to Prof. Domingo Gonzalez, from the University of Zaragoza, on the occasion of his retirement. References [1] For reviews, see, for example: M. Inguscio, S. Stringari, C.E. Wieman (Eds.), Proceedings of the International

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School of Physics ‘‘Enrico Fermi’’, Course CXL, IOS Press, Amsterdam, 1999. In some cases termed ‘‘coherent population trapping’’ or ‘‘dark resonance’’. For reviews, see, for example E. Arimondo, in: E. Wolf (Ed.), Progress in Optics XXXV, Elsevier, Amsterdam, 1996, Chapter V; S.E. Harris, Phys. Today 50 (7) (1997) 36; J.P. Marangos, J. Mod. Opt. 45 (1998) 471; J. Mompart, R. Corbal
an, J. Opt. B: Quantum Semiclass Opt. 2 (2000) R7. F. Silva, J. Mompart, V. Ahufinger, R. Corbal
an, Phys. Rev. A 64 (2001) 033802, and references therein. A. Aspect et al., Phys. Rev. Lett. 61 (1988) 826. G. Morigi, J. Eschner, C.H. Keitel, Phys. Rev. Lett. 85 (2000) 4458; C.F. Roos et al., Phys. Rev. Lett. 85 (2000) 5547. L.V. Hau, S.E. Harris, Z. Duton, C.H. Behroozi, Nature 397 (1999) 594;

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Ch. Liu, Z. Dutton, C.H. Behroozi, L.V. Hau, Nature 409 (2001) 490; D.F. Phillips, A. Fleischhauer, A. Mair, R.L. Walsworth, M.D. Lukin, Phys. Rev. Lett. 86 (2001) 783. C. Fort et al., Europhys. Lett. 49 (2000) 8. P.B. Blakie, R.J. Ballagh, and C.W. Gardiner, Dressed states of a two component Bose–Einstein condensate, arXiv:cond-mat/9902110, 9 Feb. 1999. Note that we take as a reference origin the energy of state j3i. The atomic energies we consider now contain neither the kinetic center of mass energy nor the magnetic dipole energy. The GPE takes care of these energies. Therefore, x31 and x32 are space and time independent. J. Ruostekoski, D.F. Walls, Eur. Phys. J.D 5 (1999) 335. Notice that the spectrum in Fig. 2 is symmetric because it was computed for d2 ¼ 0 and for atomic densities such that 4p h ða21 q11 þ a22 q22 Þ  0Þ. m