1> |2 |3 |2> |3 - Temple University

VOLUME 88, NUMBER 17

29 APRIL 2002

PHYSICAL REVIEW LETTERS

Measurement of Transition Dipole Moments in Lithium Dimers Using Electromagnetically Induced Transparency J. Qi,1, * F. C. Spano,1 T. Kirova,1 A. Lazoudis,1 J. Magnes,1 L. Li,2 L. M. Narducci,3 R. W. Field,2,4 and A. M. Lyyra 1, † 1

Departments of Physics and Chemistry, Temple University, Philadelphia, Pennsylvania 19122 2 Department of Physics, Tsinghua University, Beijing, China 100084 3 Department of Physics, Drexel University, Philadelphia, Pennsylvania 19104 4 Chemistry Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received 27 September 2001; published 15 April 2002) We have observed electromagnetically induced transparency in a Doppler broadened molecular cascade system using fluorescence detection. We demonstrate that the power-dependent splitting of lines in the upper-level fluorescence excitation spectrum can be used as a new spectroscopic tool for the measurement of molecular transition dipole moment functions. DOI: 10.1103/PhysRevLett.88.173003

PACS numbers: 33.40. +f, 42.50.Hz

Coherence phenomena in laser-atom interactions have been a focus of interest for decades, beginning with Fano’s pioneering studies [1]. Coherent population trapping [2], electromagnetically induced transparency (EIT) [3], lasing without inversion [4], and ultraslow propagation of light [5], among others, have been predicted and observed in atomic systems. Fewer experimental studies have addressed coherence phenomena in molecular systems [6] and, in particular, the possible occurrence of EIT [7]. This is perhaps due to the small size of typical molecular transition dipole moments. In addition, unlike atoms, even the simplest molecules are open systems in that every excited molecular rovibrational level is radiatively coupled to many more energy levels than any atomic excited state. Therefore, coherence effects in molecular systems are more challenging in terms of both experimental observation and development of theoretical analyses. In a previous paper [8] we emphasized that the AutlerTownes (AT) splitting can be used in a four-level system as a way to facilitate all-optical control of molecular angular momentum alignment. We also demonstrated that molecular transition dipole moments can be measured through AT splitting, as done, for example, by Quesada et al. in pulsed laser experiments on the H2 molecule [9]. Thus, coherence effects may allow measurement of important molecular parameters. In this Letter we show that EIT can be observed even without sub-Doppler resolution using two frequency stabilized tunable lasers in a three-level system. We also demonstrate the use of this coherent effect to measure the transition dipole moment matrix element between two of the excited molecular levels using a much less demanding experimental arrangement than in [8]. A characteristic signature of EIT for the system shown in Fig. 1 is the enhanced transmission of a weak probe nearly resonant with the j1典 ! j2典 transition, in the presence of a strong coupling field resonant with the j2典 ! j3典 transition. EIT, however, can also be recognized by the

appearance of a sharp dip in the fluorescence excitation spectrum of the intermediate level [10], under resonance conditions for the probe. The connection between this feature and EIT can be best understood if we consider that a cascade system becomes formally equivalent to a lambda system after moving its topmost state to a position below the middle level [3(c)]. It is well known that, under EIT conditions, the population of the highest energy level of a lambda system displays a dip, which signals the emergence of a dark state. The same holds true for the population of the middle level of a cascade system. The fluorescence from the highest level (3) is also affected by the coupling field, especially if this field is sufficiently strong. The upper-level excitation spectrum, obtained by scanning the probe laser while holding the coupling laser on resonance and monitoring (filtered) side fluorescence from the

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© 2002 The American Physical Society

0031-9007兾02兾 88(17)兾173003(4)$20.00

|3 >

G 1 Π g (11,14)

L2

|2 >

A1 Σ +u (13,14)

A1 Σ +u (12,14)

L1 X 1 Σ+g (4,15)

|1 >

X 1 Σ +g (4,13)

FIG. 1. 7 Li2 three-level cascade scheme: The weak probe laser, L1 (15642.636 cm21 ), was used to excite molecules from the ground state level X 1 Sg1 共y1 苷 4, J1 苷 15兲 to an excited intermediate level A 1 S1 The laser, L2 u 共y2 苷 13, J2 苷 14兲. (17053.954 cm21 ), resonantly coupled the intermediate level to a higher electronic state level G 1 Pg 共y3 苷 11, J3 苷 14, f兲.

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upper level, shows a symmetrically split line shape, which is substantially narrower than the full Doppler width of the transition, and displays a separation between the split components that varies linearly with the Rabi frequency of the coupling laser (Autler-Townes splitting). In contrast, the sharp hole in the excitation spectrum from the intermediate level is largely independent of the strength of the coupling field. We use the power-dependent splitting of the level-3 excitation spectrum to measure directly the transition dipole moment matrix element between levels 2 and 3. Traditionally, transition dipole moments are measured from resolved fluorescence spectral intensities and lifetimes [11]. Such methods are time consuming and suffer from potential systematic errors. Among these, the wavelength and polarization dependence of the light detection system’s quantum efficiencies is the predominant source of error. Relative transition moments can also be calculated using ab initio methods. The experimental method described here permits determination of the absolute value of the transition dipole moment matrix element, which, in turn, requires an accurate measurement of the coupling field Rabi frequency, i.e., the coupling field spot size and power. The experimental setup was similar to that of Ref. [8] in terms of the molecular sample conditions, optical detection, and laser systems, except that it requires only two, instead of three, lasers (as shown in Fig. 1). The weak probe laser, L1 , excited molecules from the ground state level, X 1 S1 g 共y1 苷 4, J1 苷 15兲, to an in共y termediate level, A 1 S1 2 苷 13, J2 苷 14兲. The laser, u L2 , resonantly coupled the intermediate level to a higher electronic state level, G 1 Pg 共y3 苷 11, J3 苷 14, f兲. The two laser beams were counterpropagating coaxially, and linearly polarized in a common direction. By using a monochromator as a narrow-band filter, the population of A 1 S1 u 共y2 苷 13, J2 苷 14兲 was monitored by detecting the fluorescence from this level to the ground rovibrational level, X 1 S1 Similarly, the g 共y1 苷 4, J1 苷 13兲. 1 population of G Pg 共y3 苷 11, J3 苷 14, f兲 was monitored by detecting fluorescence at an auxiliary level A 1 S1 u 共y2 苷 12, J2 苷 14兲. Excitation spectra were obtained by scanning the probe laser frequency, which was calibrated to 60.002 cm21 using iodine calibration [12]. When L1 was scanned in the absence of L2 , while simultaneously monitoring the A 1 S1 u 共y2 苷 13, J2 苷 14兲 fluorescence, the usual Doppler broadened excitation spectrum was observed. When the power of the coupling laser was increased, a sharp dip emerged (as shown in Fig. 2). This dip is a signature of EIT [10].

Experiment

S 2( ∆1 )(arbitrary units)


Theory

2000

1000

0 -4000

-2000

0

2000

4000

∆1/2π (MHz)

FIG. 2. Measured excitation spectra from level 2 along with the corresponding simulations using Eqs. (1a) and (4). Dephasc c 兾2p 苷 6.78 MHz, g13 兾2p 苷 0.85 MHz, ing parameters are g12 c g23 兾2p 苷 1.69 MHz. The coupling field power is 470 mW.

We next monitored the upper-level fluorescence as described above. When L2 was below saturation (,10 mW), scanning the probe laser produced the narrow spectrum shown in Fig. 3(a). The effect of increasing the coupling field power to 470 mW is shown in Fig. 3(b). The line shape is significantly power broadened and symmetrically split into two non-Lorentzian lines. Figure 4 shows the splitting as a function of the coupling field amplitude. When the Rabi frequency of the coupling laser was made larger than the partially Doppler broadened line, the splitting became observable and it was found to be proportional to the Rabi frequency. In order to confirm the nature of the fluorescence line shapes in the presence of EIT, we solved the density matrix equations for the open, three-level, cascade system of Fig. 1. The probe laser (L1 ) was treated perturbatively and the coupling field (L2 ) was treated exactly. As described in Ref. [8], the linearly polarized electric fields can be approximated by Ei 共៬r , t兲苷 Ei ´៬ exp关2共r兾wi 兲2 兴 cos共ki z 2 vi t兲 ,

where wi , ki , and vi denote the spot size at the beam waist, the wave number, and the frequency of the ith laser, respectively, and ´៬ is the unit polarization vector. In steady state, the populations of the second and third levels for a molecule with orientation M, radial position r relative to the common axis of the laser beams, and velocity yz are given by 2 2 # " V2,M g23 V2,M W32 2 2 2 1 D 1 ig 兲关D 1 g 1 兴 1 共1 2 兲共D 2 ig 兲共D 2V 1 2 13 2 23 2 23 2W 4 W 1,M 3 3 M r22 共r, yz , t ! `兲苷 , (1a) Im 2 2FM 共D2 兲共D1 1 D2 1 ig13 兲共D1 1 ig12 兲 2 V2,M 兾4 ∏ ∑ 2 2 V1,M V2,M 22g23 共D1 1 D2 1 ig13 兲 1 W2 共D2 2 ig23 兲 M , (1b) Im r33 共r, yz , t ! `兲苷 2 8W3 FM 共D2 兲共D1 1 D2 1 ig13 兲共D1 1 ig12 兲 2 V2,M 兾4

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180

S 3(∆1)(arbitrary units)

(a)

Experiment

160

Theory

140

upper level splitting (MHz)

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1000

120

Experimental data

100

Set 1 Set 2 Set 3

80 60 40

0 20

-300

-200

-100

0

100

200

300

0

∆1/2π (MHz)

0

5

10

15

20

25

1/2

square root of power (mW )

FIG. 4. Dependence of the observed and calculated splitting on the square root of the laser power. Three sets of calculations using the following sets of dephasing rates apc c c 兾2p 苷 0.85 MHz, g23 兾2p 苷 6.78 MHz, g13 兾2p 苷 pear: g12 c c c 1.69 MHz for Set 1, g12 兾2p 苷 1.69 MHz, g13 兾2p 苷 g23 兾2p 苷 c c 5.08 MHz for Set 2, and g12 兾2p 苷 0.42 MHz, g13 兾2p 苷 c g23 兾2p 苷 7.62 MHz for Set 3.

4000

S 3( ∆1 )(arbitrary units)

(b)

Experiment

3000

Theory

2000

transitions induced by L1 and L2 respectively, we have [13] p 2 2 mM 1,2 苷 mk j具y1 j y2 典j 共J1 2 M 兲兾共2J1 1 1兲共2J1 2 1兲 , (3a) q M (3b) m2,3 苷 m⬜ j具y2 j y3 典Mj兾 J2 共J2 1 1兲 ,

1000

0 -1500

-1000

-500

0

500

1000

1500

∆1/2π (MHz) FIG. 3. (a) Measured and calculated weak-field excitation spectra. The dephasing parameters are the same as in Fig. 2. (b) Measured excitation spectra from level 3 along with the corresponding simulations using Eqs. (1b) and (4). The dephasing parameters and the coupling field power are the same as in Fig. 2.

with

µ

∂ 2 V2,M g23 FM 共D2 兲 ⬅ 1 1 2W3 µ ∂ 2 V2,M W32 . g23 1 2 1 2 W3 In Eqs. (1a) and (1b), the velocity-dependent detunings of the probe and coupling lasers from the molecular transition frequencies, vij , are D1 ⬅ v21 2 v1 1 k1 yz and D2 ⬅ v32 2 v2 1 k2 yz , respectively. The Rabi frequencies of the two lasers (i 苷 1, 2) depend on both the orientation and distance of the molecule from the beam axis. For the ith laser tuned near resonance with the i, i 1 1 transition, the Rabi frequency is W2 D22

2 g23

¯ exp关2共r兾wi 兲2 兴 , Vi,M 共r兲 ⬅ 共mM i,i11 Ei 兾h兲 mM i,i11

(2)

where is the M-dependent transition dipole moment between levels i and i 1 1. For the P and Q branch 173003-3

where yi is the vibrational quantum number for level i, and mk (m⬜ ) is the j1典 ! j2典 (j2典 ! j3典) electronic transition dipole moment, which is along (perpendicular to) the internuclear axis. In Eq. (1), Wi is the damping rate of the ith level, including both radiative and collisional contributions. The decay rate of the coherence between levels i and j (ﬁ i) is gij 苷共Wi 1 Wj 兲兾2 1 gijc , where gijc is the pure dephasing contribution induced by phase-changing collisions. For open systems, W32 , W3 , where W32 is the level-3 ! level-2 decay rate. In a closed system, W32 苷 W3 , and Eqs. (1a) and (1b) simplify considerably. The signal is obtained from Eqs. (1) by summing over M and averaging over the Doppler and transverse laser profiles. The final excitation spectrum Si 共D1 兲 from level i (i 苷 2, 3) is Z ` XZ ` Si 共D1 兲苷 r dr dyz N 共yz 兲riiM 共r, yz , t ! `兲 , M

0

2`

(4) where N共yz 兲 is the Maxwell velocity distribution. The parameters in Eqs. (1a) and (1b) are determined as follows: From Ref. [14], the lifetime of the intermediate level 2 is known, W221 苷 18 ns. Although W321 and the gijc have not been directly measured, a set of values which reproduce the experimental line shapes in Figs. 2 and 3 can be found. In the following we take W321 苷 0.5W221 , 173003-3



consistent with Ref. [8], and the three collisional dephasing rates reported in Fig. 2. However, as is shown below, the upper-level splitting does not critically depend on these parameters. The branching ratio was set to W32 兾W3 苷 0.1, indicating an open system, although all calculated line shapes in Figs. 2 and 3 remain essentially unchanged for W32 兾W3 , 0.5. For the spatial and Doppler averaging in Eq. (4), we used the measured value of the FWHM Doppler linewidth 2.6 GHz, and the beam spot size parameters w1 苷 208 mm and w2 苷 557 mm for L1 and L2 , respectively. Figures 2 and 3 also show the line shape simulations using Eqs. (1) and (4). In order to obtain agreement with the experimental spectra in Fig. 3(b), the Rabi frequency of the second laser, V2,M , was varied until the peak-topeak separation in S3 共D1 兲 matched the experimental value. The simulated splitting was found to be approximately ¯ for sufficiently large E2 , allowing 0.42m⬜ j具y2 j y3 典jE2 兾h, m⬜ to be determined experimentally from S3 共D1 兲, once j具y2 j y3 典j and E2 are known. The linear dependence of the upper-level splitting on E2 is confirmed experimentally in Fig. 4, which also shows simulations using several sets of collisional dephasing parameters. Collisional dephasing influences only the low power region in which the emergence of the dip depends on details of the linewidth. Once the splitting is sufficiently large, it becomes independent of homogeneous broadening and scales linearly with E2 . The best match between our theory and experiment is obtained for an electronic transition dipole moment m⬜ 苷 2.4 6 0.2 a.u. for the G 1 Pg 2 A 1 S1 u system [15]. The main sources of uncertainty are slight variations in the laser power and frequency of L2 , which are indicated by the error bars in Fig. 4 obtained by combining independent measurements. Recent ab initio calculations of these transition dipole moment matrix elements, 2.44 a.u. [16] and 2.28 a.u. [17], are in excellent agreement with our measured value. An experimental method to determine systematically the transition dipole moment matrix elements for a sequence of electronic states is of critical value. For example, in the 1 1 Rydberg series nlLg 共y兲 √ A 1 S1 u [nlLg 共y兲苷 ns Sg , 1 1 1 nd Sg , and nd Pg ], transition moments to members of the same Rydberg series are expected to follow a simple scaling rule, MnlL 共R兲 ⬅ 具nlLjerjA 1 S1 典 ~ n23兾2 M1lL 共R兲 . Since, at n . 10, successive n members of the same l, L, y series are separated by less than 220 cm21 , it should be possible to measure the transition moment matrix elements

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for many consecutive members of several Rydberg series. These measurements could have considerable diagnostic value and would provide new insights into the electronic structure and dynamics of Rydberg states. In summary, we have observed EIT in a Doppler broadened cascade molecular system using fluorescence detection. We have demonstrated that the power-dependent splitting in the level j3典 √ j2典 excitation spectrum can be used as a new spectroscopic diagnostic of transition dipole moment functions in molecular (and atomic) systems. We are grateful for support from NSF PHY 9983533 and the Lagerqvist Research Fund (L. L.).

*Current address: Physics Department, University of Connecticut, Storrs, Connecticut 06269. † Corresponding author. Email address: [email protected] [1] U. Fano, Phys. Rev. 124, 1866 (1961). [2] For a review of this subject, see E. Arimondo, in Progress in Optics XXXV, edited by E. Wolf (North-Holland, Amsterdam, 1996), p. 259. [3] (a) A. Imamoglu et al., Opt. Lett. 14, 1344 (1989); (b) K. J. Boller et al., Phys. Rev. Lett. 66, 2593 (1991); (c) J. Gea-Banacloche et al., Phys. Rev. A 51, 576 (1995). [4] O. A. Kocharovskaya et al., Zh. Eksp. Teor. Fiz. 48, 581 (1988) [JETP Lett. 48, 630 (1988)]; A. S. Zibrov et al., Phys. Rev. Lett. 75, 1499 (1995). [5] L. V. Hau et al., Nature (London) 397, 594 (1999); M. M. Kash et al., Phys. Rev. Lett. 82, 5229 (1999); D. Budker et al., Phys. Rev. Lett. 83, 1767 (1999). [6] R. Sussmann et al., J. Chem. Phys. 103, 3315 (1995); T. Halgmann et al., J. Chem. Phys. 104, 7068 (1996). [7] Claims to this effect have been advanced in N. N. Rubtsova, Opt. Spectrosc. 91, 53 (2001). [8] J. Qi et al., Phys. Rev. Lett. 83, 288 (1999). [9] M. A. Quesada et al., Phys. Rev. A 36, 4107 (1987), and references therein. [10] K. Ichimura et al., Phys. Rev. A 58, 4116 (1998). [11] J. B. Koffend et al., J. Chem. Phys. 70, 2366 (1979). [12] S. Gerstenkorn et al., Atlas du spectre d’absorption de la molecule d’iode (CNRS, Paris, 1978); S. Gerstenkorn et al., Rev. Phys. Appl. 14, 791 (1979). [13] F. C. Spano, J. Chem. Phys. 114, 276 (2001). [14] G. Baumgartner et al., Chem. Phys. Lett. 107, 13 (1984). [15] Reference [8] reported a value ofp3.4 a.u. However, after omitting an erroneous factor of 1兾 2 in Eq. (3) of Ref. [8] [J. Qi et al., Phys. Rev. Lett. erratum (to be published)], the value becomes 2.4 a.u., in excellent agreement with the present measurement. [16] S. Magnier (private communication). [17] G.-H. Jeung (private communication).

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