Optical resonances with anomalous magnetic field dependence

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PHYSICAL REVIEW A, VOLUME 64, 051802共R兲

Competition of dark states: Optical resonances with anomalous magnetic field dependence Gabriela Wa¸sik,1,* Wojciech Gawlik,1,2 Jerzy Zachorowski,1 and Zbigniew Kowal1 1

Instytut Fizyki M. Smoluchowskiego, Uniwersytet Jagiellon´ski, ulica Reymonta 4, 30–059 Krako´w, Poland JILA, National Institute for Standards and Technology and University of Colorado, Boulder, Colorado 80309 共Received 25 June 2001; published 2 October 2001兲

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A different type of optical resonance has been observed in Doppler-free saturated absorption spectroscopy when the pump beam has elliptical polarization slightly deviating from the ␴ ⫹ and the probe has ␴ ⫺ polarization. An interpretation in terms of competition between two trap states and a dressed-atom model is presented. The observed extreme sensitivity of the resonance frequency to the external magnetic field is attributed to the ‘‘scissors effect’’ of the dressed states’ crossing point. DOI: 10.1103/PhysRevA.64.051802

PACS number共s兲: 32.80.Bx, 32.80.Qk, 42.50.Gy, 42.50.Hz

Discovery of the phenomenon of coherent population trapping 共CPT兲关1兴 and its numerous spectacular applications 共see Ref. 关2兴 for recent review兲 is one of the most profound achievements of atomic and laser physics. The essence of this phenomenon is the existence of the superpositions of atomic states that are uncoupled from the light field 共the dark, trap, or CPT states兲. Recently, it has been shown that in systems where multiple CPT states are possible, interesting competition between them occurs, which can have important consequences. In particular, the effect of ‘‘doubledark’’ states has been investigated in Ref. 关3兴 and it has been shown that such states can be used to manipulate properties of the quantum system over a wide range of parameters. Here, we report on our study of a different observed resonance in saturated absorption spectroscopy in a Dopplerbroadened Na-D 1 line, which can be regarded as an experimental demonstration of such an effect. By setting different light polarizations and taking use of atomic velocity selection, we were able to observe Doppler-free resonances of unusual characteristics. Their origin is attributed to the competition of CPT states, created by the strong pump beam of nearly circular polarization and tested by the weak probe beam polarized orthogonally to the pump. In our experiment, counterpropagating pump and probe light beams were generated by the same laser, contrary to most of earlier works on CPT 关2兴. These resonances are also interpreted in terms of Doppler-free pump-probe spectroscopy of optically dressed atoms. Our observation is important for understanding the interaction of laser light with multilevel atomic gases, especially where CPT plays a role and for practical spectroscopy, where small polarization impurities could drastically change routinely recorded spectra. Particularly interesting is the observation and explanation of unusually strong dependence of the observed resonance position on the magnetic field. The resonance is observed in geometry typical for the Doppler-free saturated absorption spectroscopy 共Fig. 1兲. We measure the intensity of the weak probe beam transmitted by a cell about 5-cm long containing sodium vapor at 403 K and perturbed by a strong, counterpropagating pump beam. The

*Present address: Lehrstuhl fu¨r Optik, Physikalisches Institut, Universita¨t Erlangen–Nu¨rnberg, Staudtstrasse 7/B2, 91058 Erlangen, Germany. 1050-2947/2001/64共5兲/051802共4兲/$20.00

cell is placed within a solenoid and surrounded by a triple magnetic ␮ -metal shield. The shield reduces external magnetic fields by a factor of 10⫺3 , not only the dc fields, but also the ac fields radiated by the whole electronic equipment. Careful control of the magnetic field is a very important condition for our observations. This might be the reason why these resonances were not observed earlier on this extensively studied transition. The probe beam has exact circular polarization ( ␴ ⫺ ) and intensity 6 ␮ W/mm2 . Polarization of the pump beam is nearly ␴ ⫹ and can be changed by allowing some ellipticity through a weak ␴ ⫺ admixture. With the pump of pure ␴ ⫹ polarization and weak intensity 共about 18 ␮ W/mm2 ), one records typical Doppler-free spectra due to velocity-selective optical pumping 关4兴 consisting of the hyperfine structure 共hfs兲 resonances and crossovers. The lowest curve in Fig. 2共a兲 shows such a spectrum for the laser frequency scan around the F g ⫽2⫺F e ⫽1,2 hfs components of the Na-D 1 共589.6 nm兲 line. Different signs of the three components seen in the spectrum are due to different dynamics of VSOP for different lines 关5兴. When the pump-beam intensity increases, the spectrum components undergo strong broadening and eventually overlap. When a small admixture of a ␴ ⫺ component is allowed in the pump beam, such that the ratio of the ␴ ⫹ -polarized light to the admixed ␴ ⫺ light intensities I ⫹ /I ⫺ is 99.97:0.03, a narrow resonance appears in the spectrum 共seen at about 24 MHz in Fig. 2兲, which is absent when the pump beam has pure ␴ ⫹ polarization. Its width 共11 MHz兲 is close to the natural linewidth ⌫⫽10 MHz of the D 1 line and its position is different from the hfs transitions and from the cross-over

FIG. 1. Simplified experimental setup: Na is the sodium cell in the solenoid producing longitudinal magnetic field B. The cell is placed in a three-fold ␮ -metal magnetic shield. PMT denotes photomultiplier.

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PHYSICAL REVIEW A 64 051802共R兲

FIG. 2. Saturated absorption spectrum of the D 1 sodium line near F g ⫽2→F e ⫽1,2 hfs components 共at 0 and 189 MHz, respectively兲 for I ⫹ /I ⫺ ⫽99.97:0.03. The Doppler background is subtracted by a lock-in. Spectra correspond to different Rabi frequencies ⍀ 13 of the 兩 1 典 → 兩 3 典 transition, curves are off-set vertically for better visualization. 共a兲 experimental results, 共b兲 calculated signals.

frequency. The position of the resonance does not noticeably depend on the intensity nor the ellipticity (I ⫹ /I ⫺ ) of the pump beam. The increase of the ellipticity causes instead a strong increase of the amplitude and width of the resonance. For still higher ellipticity 共up to 99.2:0.8兲 or for higher pump intensity, other narrow resonances appear, as seen at about 118 MHz on top curves in Fig. 2. For comparison, Fig. 2共b兲 shows signals calculated as explained below. The position of the resonances, though immune to the pump field intensity, appears to be extremely sensitive to the intensity of the longitudinal magnetic field B. Figure 3 shows the dependence of the resonance frequency on B, taken for several values of the pump beam intensity. As seen in this figure, the slope of this dependence is one order of magnitude stronger than the Zeeman shift of the nearby atomic lines. Hence, the resonance cannot be simply related to any transition between Zeeman-shifted magnetic sublevels of the unperturbed ground and excited states. Below we interpret the resonance in terms of the CPT and electromagnetically induced transparency 共EIT兲关2兴, and by using the dressed-atom model 关6兴. Due to optical pumping of the strong, circularly polarized pump, not all sublevels of the

FIG. 3. Positions of the resonances ␻ Lt 关␻ Lt (0)⬇24 MHz, triangles and lower line of black dots兴 and ␻ Ls 关␻ Ls (0)⬇118 MHz, upper black dots兴 versus the magnetic field. I ⫹ :I ⫺ ⫽99.97:0.03. The dashed line is the Zeeman shifted atomic transition F g ⫽2,m g ⫽2→F e ⫽1,m e ⫽1. 共a兲 experimental results, 共b兲 calculated positions for different values of ⍀ 13 .

FIG. 4. Double-⌳ model in the 共a兲 bare- and 共b兲 dressed-state representation. States 兩 3 典 , 兩 2 典 , and 兩 1 典 are dressed by the ␴ ⫹ pump field 共thick solid arrow兲 that yields states 兩 s(N) 典 , 兩 r(N) 典 , and 兩 t(N) 典 . The probe field 共dashed arrow兲 couples states 兩 1 典 and 兩 2 典 with 兩 4 典 . The ␴ ⫺ component of the pump field 共thin solid line兲 interacts with the same transitions as the probe.

Na-D 1 line need to be considered. The relevant ones are 兩 F e ⫽2,m e ⫽1 典 , 兩 F e ⫽1,m e ⫽1 典 , and 兩 F g ⫽2,m g ⫽0 典 , 兩 F g ⫽2,m g ⫽2 典 , labeled respectively, 兩 1 典 , 兩 2 典 , 兩 3 典 , and 兩 4 典 . They form a double-⌳ scheme and are shown in Fig. 4共a兲 together with the transitions induced by all light fields. a. CPT interpretation. The elliptically polarized pump light can create superpositions in the ground state, which do not absorb pump light 共CPT states兲. If one considers only the ‘‘upper ⌳’’ 共states 兩 1 典 , 兩 3 典 , and 兩 4 典 ) and takes the probe as negligibly weak, such CPT state is 兩 ␾ nu 典 ⫽cos ␣ 兩 4 典 ⫺sin ␣ 兩 3 典 ,

where tan ␣ ⫽⍀ 14 /⍀ 13 , ⍀ ik is the Rabi frequency for the i ⫺k transition. In the case of the weak ␴ ⫺ component of the pump, ⍀ i3 Ⰷ⍀ i4 (i⫽1,2), this superposition consists mainly of state 兩 4 典 , 兩 ␾ nu 典 ⬀⍀ 13兩 4 典 . If one considers the ‘‘lower ⌳,’’ there is another nonabsorbing superposition 兩 ␾ nl 典 of the ground levels, consisting again mostly of state 兩 4 典 , 兩 ␾ nl 典 ⬀⍀ 23兩 4 典 . Admixtures of states 兩 4 典 to 兩 ␾ nu 典 and 兩 ␾ nl 典 depend on I ⫹ and I ⫺ and have opposite signs because of opposite signs of the dipole matrix elements for the 3⫺1 and 3⫺2 transitions. In each of the CPT states absorption of the ␴ ⫺ pump light is zero, despite efficient populating of state 兩 4 典 via incoherent optical pumping of the strong ␴ ⫹ pump. A specific case is created when interaction with both hyperfine transitions has the same strength. In our case of the Na-D 1 line, this happens when the pump beam detunings 共in the atomic frame兲 from the 3⫺2 transition and from the 3⫺1 one are in 1:3 proportion, since the transition probability ratio is 1:3. The pump beam is then at ␻ x ⫽ ␻ 23⫹⌬ x , where ⌬ x ⫽⌬ hfs/4, i.e., is detuned by 1/4 of the hyperfine splitting ⌬ hfs above the lower 3⫺2 transition. At ␻ x the two CPT states, 兩 ␾ nu 典 and 兩 ␾ nl 典 interfere fully destructively, which results in nonzero absorption of the ␴ ⫺ pump decreasing population of state 兩 4 典 . Competition of ␾ nu and ␾ nl has a dramatic effect on the Zeeman coherence ␴ 34 . This coherence, which is the essence of the CPT, undergoes resonant changes around ␻ x 共curves a in Fig. 5兲. The changes are strongest when dipole moments for 3⫺1 and 3⫺2 transi-

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FIG. 5. The ground level coherence ␴ 34 vs pump beam frequency ␻ s for an atom at rest, ⍀ 13⫽6.2⌫ and B⫽0. 共a兲 our case of ⍀ 13⫽⫺ 冑3 ⍀ 23 , 共b兲 the case of 兩 ⌽ nu 典 ⫽ 兩 ⌽ nl 典 , i.e., ⍀ 13 /⍀ 23 ⫽⍀ 14 /⍀ 24 . Optical pumping from any other than the four interesting levels is neglected for clarity.

tions have different signs. For the case of the same signs and for 兩 ␾ nu 典 ⫽ 兩 ␾ nl 典 , there is no competition and no variation of ␴ 34 , 关Fig. 5共b兲兴. Population of state 兩 4 典 is tested by the probe beam, whose frequency in the atomic frame can be tuned by the Doppler effect. Absorption of the ␴ ⫺ probe experiences a strong change in the vicinity of 4⫺1 and 4⫺2 transitions. We look for the particular case, for which the pump beam is detuned by ⌬ x and the probe is in resonance with either of the atomic transitions. These resonance conditions can be met only by atoms with nonzero velocity when the laser frequency 共in the laboratory frame兲 is midway between ␻ x and ␻ 23 or ␻ 13 . The resulting resonances have thus a cross-over character. For this frequency the probe experiences reduced absorption 共EIT兲. Since the probe interrogates a usual dipole transition and, on the other hand, the ␴ 34 resonance width is determined by the 10 MHz natural width of the Na-D 1 line 共see Fig. 5兲, the cross-over resonance has similar width. b. Dressed-state interpretation. We can also consider the atomic energy levels dressed with the principal ␴ ⫹ component of the pump field, which we call the dressing pump. After transformation to a dressed-state frame, the four relevant energy levels look as shown in Fig. 4共b兲. Two weaker ␴ ⫺ -polarized light beams: the admixture to the pump 共which we call the ␴ ⫺ pump兲 and the probe, act on the dressed states. Since these beams are counterpropagating, they can now be regarded as the pump and probe beams, respectively, in the Doppler-free spectroscopy of the atoms dressed by the strong ␴ ⫹ component. In Fig. 6 we present the energy structure of the dressed double-⌳ system of Fig. 4共b兲, as a function of the frequency of the ␴ ⫺ pump in the atomic frame, ␻ s . The straight line in Fig. 6 represents energy of the unperturbed level 兩 4 典 plus the energy of one pump photon. Since all beams we are considering here are produced by the same laser, changing ␻ s changes not only the ␴ ⫺ pump frequency, but also modifies the dressed states. This simultaneous modification is the main difference between standard saturated absorption spectroscopy and the saturated spectroscopy of dressed atoms. Our experimental arrangement, where all three relevant light beams stem from the same laser, also differs strongly from the standard pump-probe studies of CPT and EIT 关2兴, where usually the strong pump had constant intensity and frequency

FIG. 6. Dressed-state energies 共dashed lines兲 versus pump frequency ␻ s for atom at rest, ⍀ 13⫽3.7⌫ and B⫽0. The continuous line shows the energy of the fourth unperturbed level, 兩 4,N 典 , plus the energy of one photon of the pump beam, ប ␻ s . The intersection of this line with any of the dotted lines 共shown here in the circle兲 indicates the position of a dressed-state resonance ␻ x . This resonance appears here only for ␻ 23⫹⌬ x ,⌬ x /2␲ ⫽47.7 MHz.

and only the weak probe was tuned across the range of interest. In any saturation spectroscopy experiment, it is required that pump and probe beams are resonantly coupled for a given velocity class of atoms. In our case, the ␴ ⫺ pump beam interacts resonantly with the dressed atom when the solid line in Fig. 6 crosses either one of the dressed states. For the system of interest, this happens only when the ␴ ⫺ pump beam is in resonance with the 兩 4 典 ⫺ 兩 r(N) 典 transition, i.e., at a single frequency ␻ x ⫽ ␻ 23⫹⌬ x . This value is determined exclusively by the ratio of the transition probabilities on 3⫺2 and 3⫺1 transitions, as described above. The resonant change of transmission of the ␴ ⫺ probe beam occurs whenever the probe beam is in resonance with transitions from 兩 4,N 典 to any of the three states: 兩 r(N) 典 , 兩 s(N) 典 , or 兩 t(N) 典 . The requirement of simultaneous resonance of the pump and probe defines the relevant atomic velocity group. The first of these simultaneous conditions can be met for the zero velocity group of atoms 共regular resonance兲, the two other cases are fulfilled by the nonzero velocity atoms 共cross-over resonances兲. In the laboratory frame this occurs for laser frequencies: ␻ Lr ⫽ ␻ x , ␻ Ls ⫽( ␻ x ⫹ ␻ 4s )/2, and ␻ Lt ⫽( ␻ x ⫹ ␻ 4t )/2. At ␻ s ⫽ ␻ x the 兩 r(N) 典 state is 兩 r 共 N 兲典 ⫽ 共 cos ␸ 兩 1,N 典 ⫺sin ␸ 兩 2,N 典 )sin ␪ ⫹cos ␪ 兩 3,N⫹1 典 , 2 tan ␸ ⫽⍀ 13 /⍀ 23 , tan ␪ ⫽2 冑(⍀ 13/3) 2 ⫹⍀ 23 /⌬ hfs . For our case tan ␪ Ⰶ1, the 兩 r(N) 典 state has only a small admixture of excited states, the regular resonance transition is very weak, and therefore, we do not see it in our spectra. In the experiment we clearly observe the crossover at frequency ␻ Lt and the second crossover at ␻ Ls with much smaller amplitude. They are seen in Fig. 2共a兲 at about 24 MHz and 118 MHz, respectively. In this dressed-state interpretation, the changes of the probe absorption are directly related to the EIT processes in either of the V structures: 兩 r,N 典 ⫺ 兩 4,N 典 ⫺ 兩 t,N 典 or 兩 r,N 典 ⫺ 兩 4,N 典 ⫺ 兩 s,N 典 . Still, since the dressed states are su-

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WA ¸ SIK, GAWLIK, ZACHOROWSKI, AND KOWAL


perpositions of the unperturbed ones, they also involve the ␴ 34 coherence and it can be easily shown that the two pictures are equivalent. The dressed-state analysis allows also an easy explanation of the most striking feature of the observed resonance: its anomalously strong magnetic field dependence of the resonance position ␻ z 共B兲. In a weak magnetic field the unperturbed level 兩 4,N 典 is shifted by g␮ B m g B 共g is the Lande´ factor, ␮ B the Bohr magneton, m g is the ground-state magnetic number兲, while the dressed states remain almost unaffected 关8兴. Since the slope of level 兩 4,N 典 versus detuning 共solid line in Fig. 6兲 is almost equal to the slope of the dressed level 兩 r(N) 典 , the effect of the translating level 兩 4,N 典 by g␮ B m g B is such that the crossing of these two lines, i.e., the value of ␻ x , changes rapidly with B, the scissors effect. With the increase of the dressing field intensity, the crossing angle between the two lines increases, and the magnetic field dependence becomes less strong 共Fig. 3兲. We investigate further our four-level model. Assuming parameters 共hyperfine splitting, electric dipole moments, etc.兲 of the transitions between particular sodium sublevels, we calculate the absorption coefficient of the Doppler broadened medium in the density matrix formalism. We solve the appropriate Liouville-von Neumann equation semiclassically in steady-state approximation, taking into account all orders of the pump field amplitudes and the first order of the probe field amplitude. We assume that the absorption of the probe beam depends only on population differences introduced by the elliptically polarized pump beam. To obtain proper signs of the resonant probe beam absorption changes 关comparable to the experimental data shown in Fig. 2共a兲兴, we assume appropriate resonant optical pumping into states 兩 3,N 典 and 兩 4,N 典 , which does not influence the nonoptical coherences.

The calculated, velocity averaged absorption spectra are presented in Fig. 2共b兲. The calculated and observed spectra agree very well, in particular two resonances are well visible around 24 MHz and 118 MHz. Their positions agree with the positions of dressed-atom cross-over resonances, described above. The described resonance is due to EIT resulting from competition between two CPT states associated with two coupled ⌳ systems 共multilevel interference兲. This effect can be viewed as an interesting interplay of dressing and pumping of a multilevel system by appropriately polarized counterpropagating beams. It can also be regarded as experimental realization of the double-dark states, recently discussed in Ref. 关3兴. We show that it can be observed in a very simple setup with just one laser. The effect could be important for practical situations where laser beams are not in a pure polarization state. It should be noted that this resonance differs from the one studied by Himbert et al. 关7兴, which appears for three-level atoms of nonzero velocity and the pump beam of comparable intensity of the two circular components. In our case it is essential to consider double ⌳, i.e., a four-level system, and most importantly, the resonance appears to be an order of magnitude more sensitive to the magnetic field than Zeeman splitting. Indeed, due to the scissors effect, the crossing point of dressed states in Fig. 6, or equivalently the maximum of destructive interference of two CPT states, moves significantly with only a small variation of the magnetic field.

关1兴 G. Alzetta, A. Gozzini, L. Moi, and G. Oriols, Nuovo Cimento B 36, 5 共1976兲. 关2兴 E. Arimondo, in Progress in Optics, edited by E. Wolf 共Elsevier, Amsterdam, 1996兲, Vol. 35, pp. 257–354. 关3兴 M.D. Lukin, S.F. Yelin, M. Fleischhauer, and M.O. Scully, Phys. Rev. A 60, 3225 共1999兲. 关4兴 M. Pinard, C.-G. Aminoff, and F. Laloe¨, Phys. Rev. A 19, 2366 共1979兲; W. Gawlik and G. W. Series, in Laser Spectroscopy IV, edited by H. Walther and K.W. Rothe 共Springer-Verlag, Berlin,

1979兲, p. 210. 关5兴 O. Schmidt, K.-M. Knaak, R. Wynands, and D. Meschede, Appl. Phys. B: Lasers Opt. 59, 167 共1994兲. 关6兴 C. Cohen-Tannoudji, B. Zambon, and E. Arimondo, J. Opt. Soc. Am. B 10, 2107 共1993兲. 关7兴 M. Himbert, S. Reynaud, J. Dupont-Roc, and C. CohenTannoudji, Opt. Commun. 30, 184 共1979兲. 关8兴 The dressed states are a combination of a ground state of m g ⫽0 and excited states with m e ⫽1 and small Lande´ factor.

This work was supported by the KBN through Grant No. 2P03B01516. G.W. thanks the Alexander von Humboldt Foundation. W.G. thanks the people at JILA for their hospitality during 1998-1999.

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