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Controlling the propagation of broadband light pulses by Electromagnetically Induced Transparency
Optics Communications 285 (2012) 1185–1189
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Controlling the propagation of broadband light pulses by Electromagnetically Induced Transparency Emilio Ignesti a, Roberto Buffa b, Lorenzo Fini a, c, Emiliano Sali a,⁎, Marco V. Tognetti a, Stefano Cavalieri a, c a b c
Dipartimento di Fisica e Astronomia, Università di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy Dipartimento di Fisica, Università di Siena, Via Roma 56, I-53100 Siena, Italy European Laboratory for Nonlinear Spectroscopy, Università di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy
a r t i c l e
i n f o
Article history: Received 18 March 2011 Received in revised form 29 July 2011 Accepted 4 November 2011 Available online 24 November 2011
a b s t r a c t We present an experimental and theoretical investigation, performed on hot sodium atoms in a ladder scheme, showing the control of the absorption and of the propagation velocity of a probe light pulse with a spectral bandwidth as large as 1.8 GHz. The predictions of the theoretical model compare favorably with the experimental results. © 2011 Elsevier B.V. All rights reserved.
Keywords: Electromagnetically-induced transparency Coherent control Broadband light pulses Light pulse propagation
1. Introduction One of the most interesting and studied examples of coherent control process is provided by electromagnetically-induced transparency (EIT), in which the optical properties of a material medium at wavelength λp can be deeply modified and controlled by an intense laser field at a very different wavelength λc [1]. Since the first experimental demonstrations at Stanford [2], EIT has been observed in various material media (atomic and molecular gases and solid crystals) and its possible applications have been widely discussed in many contexts: from the control of the absorption and of the group velocity of probe radiation [3,4] to the enhancement of optical nonlinearity [5–7], from light storage [8–10] to quantum memory [11–13]. In all these studies, cw or monomode radiation for the probe pulse has been used or assumed. However, existing theoretical models hint that, in order to obtain EIT, monocromaticity is not a fundamental requirement for the probe radiation, and very recent experiments have shown that some degree of control on temporal pulse shaping [14–17], delay [18] and frequency shift [19] can indeed be obtained also for broadband probe pulses. This is not a minor point in the road to a deep understanding of EIT applications. In effect, the control of the propagation velocity of light pulses may lead to the design of all-optical delay lines, in which the amount of information carried by the radiation pulse is intrinsically tied to its spectral bandwidth.
⁎ Corresponding author. E-mail address: sali@fi.infn.it (E. Sali). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.11.018
In this paper we report the results of an experimental investigation, performed on hot sodium atoms in a ladder scheme, showing the control of the absorption and of the propagation velocity of a probe light pulse with a spectral bandwidth as large as 1.8 GHz. Compared to Ref. [18], the main goal of this work was to extend our study to the characterization of the spectral transmission. Also, we provide more accurate experimental results in order to allow a more precise comparison with the theory. In Section 2 we review the theory of propagation of broadband probe pulses in an EIT-modified-medium of hot sodium atoms, and we introduce the pertinent definitions of the quantities that have been experimentally investigated. In Section 3 we describe the experimental set-up (Section 3.1), we report the measurement results of probe absorption (Section 3.2) and temporal delay (Section 3.3), and we compare the experimental data with the theoretical results. Section 4 summarizes the results of this work. 2. Theoretical framework Fig. 1 shows a schematic diagram of the interaction process. The system under consideration is a four-level scheme in sodium. The couple of states |1〉 and |2〉 are two hyperfine sublevels (F = 1, 2 F = 2) of the level S1/2 (2p 63s), which were taken into account due to their energy separation Δν = 1.772 GHz, which is comparable to 2 the laser bandwidth. States |3〉 and |4〉 correspond to P1/2 (2p 63p) 2 6 and D3/2 (2p 3d). A weak probe laser pulse connects the two close states |1〉 and |2〉 with the same excited state |3〉, which in turns is coupled to the higher state |4〉 by a much stronger coupling pulse. In this hypothesis the propagation equations for the electric-field
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E. Ignesti et al. / Optics Communications 285 (2012) 1185–1189
the explicit dependence of the coherences ρnm(v) on the velocity v of the atom has been dropped. If, upon its propagation, the probe field Ep is overlapped by a flat region of the coupling field of value Ec0, then Eqs. (2.2)–(2.3a) admit analytical solutions and the electric field envelope Ep(z, t) is given by þ∞
Ep ðz; t Þ ¼ ∫ Sp ðz; ωÞ expðiωt Þdω
ð2:4aÞ
−∞
Sp ðz; ωÞ ¼ Sp ð0; ωÞ exp½−ikðωÞz
ð2:4bÞ
with kðωÞ ¼
"
ωp ω þN c 4ħε0 c
γ 14 þ i ω þ Δp þ Δc d13 b >v γ 13 þ i ω þ Δp þ Ω2c0 γ 14 þ i ω þ Δp þ Δc # γ24 þ i ω þ Δ0p þ Δc þd23 b >v : γ23 þ i ω þ Δ0p þ Ω2c0 γ24 þ i ω þ Δ0p þ Δc ð2:5Þ
Then, through the expression of the probe field at the cell output Ep(L, t), one can define straightforwardly the probe pulse transmission Tr
Fig. 1. Schematic diagram of the interaction process.
2 ∫ Ep ðL; t Þ dt
þ∞
envelopes Ep and Ec of the probe and coupling field, respectively, in a counter-propagating configuration read as:
ωp ∂ 1∂ ðd b ρ > þ d23 b ρ23 >v Þ þ E ¼ −i N ε0 c 13 13 v ∂z c ∂t p ∂ 1∂ − E ¼0 ∂z c ∂t c
ð2:1aÞ
ð2:1bÞ
where N is the density of the atomic sample, d13 and d23 are the electric-dipole moments of the transitions 1–3 and 2–3, respectively, and the coherences ρnm(v) are averaged over the Maxwell–Doppler velocity distribution fD(v): þ∞
ð2:2Þ
−∞
In the hypothesis of resonant coupling field the coherences ρnm(v) that appear in Eqs. (2.1a)–(2.2) satisfy the following Liouville equations: d ρ dt 13
Ωp − iΔp þ γ 13 ρ13 −iΩc ρ14 ¼ −i 2
h i d ρ ¼ −iΩc ρ13 − i Δp þ Δc þ γ 14 ρ14 dt 14 d ρ dt 23
Ωp 0 − iΔp þ γ 23 ρ23 −iΩc ρ24 2
ð2:3aÞ ð2:3bÞ
0
¼ −i
h 0 i d ρ ¼ −iΩc ρ23 − i Δp þ Δc þ γ 24 ρ24 dt 24
−∞ þ∞
ð2:6Þ
2 ∫ Ep ð0; t Þ dt
−∞
and the probe pulse delay Δt
b ρnm >v ¼ ∫ ρnm ðvÞf D ðvÞdv
Tr ¼
ð2:3cÞ ð2:3dÞ
where Ωp = d13Ep/2ħ, Ωp' = d23Ep/2ħ and Ωc = d34Ec/2ħ are the Rabi couplings, Δp = δp + ω21/2 − ωpv/c, Δp' = δp − ω21/2 − ωpv/c and Δc = ωcv/c are the laser detunings from resonances with ω21 the frequency separation of states |1〉 and |2〉, γnm represent all kind of dephasing rates, and
2 ∫ t Ep ðL; t Þ dt
þ∞
− Δt ¼ −∞ þ∞ 2 ∫ Ep ðL; t Þ dt −∞
2 ∫ t Ep ð0; t Þ dt
þ∞
−∞ þ∞
− 2 ∫ Ep ð0; t Þ dt
L : c
ð2:7Þ
−∞
In the limit ω32 →0; d13 ¼ d23 ; γ 1i ¼ γ2i , which implies to reduce our four-level scheme to a three-level one removing one of the ground close states, Eq. (2.7) reduces to the analytical expression reported in the literature for an EIT ladder scheme [17]. 3. Experiments 3.1. The experimental apparatus The experimental apparatus employed for these experiments is described in what follows. The probe field, with wavelength λp = 2πc/ωp, is provided by a frequency-tuneable multimode dye laser pumped by a frequency-doubled Q-switched Nd:YAG laser at a repetition rate of 10 Hz. The dye laser pulses have a multi-peaked temporal structure of few nanoseconds of duration. The probe laser bandwidth is δωp/2π = 1.8 GHz. The control field at λc = 2πc/ωc = 818.550 nm is provided by a frequency-tuneable, single-longitudinal-mode titanium-sapphire (Ti:S) laser [20] delivering pulses with temporal full-width at halfmaximum (FWHM) equal to 60 ns. The Ti:Sa laser pulse peak intensity can be varied from 25 KW/cm 2 to 720 KW/cm 2, while the probe pulse peak intensity can be varied variable from 50 W/cm2 to 250 W/cm 2 in our experimental conditions. The Ti:Sa laser pulse spatially overlaps the probe pulse into the cell in a counter-propagating geometry, in order to reduce the Doppler effect.
E. Ignesti et al. / Optics Communications 285 (2012) 1185–1189
The sodium cell is a 1.5-meters-long steel cylinder, where the area effectively occupied by the vapor sodium is approximately 1 meter. The electrical power supply is regulated by a feedback loop that maintains the temperature within 1 K in the range 300–700 K. By measuring the delay (see Eq. (2.7)) without the presence of the coupling laser it is possible to determine experimentally the quantity N× L, i.e. the product of the sodium density N times the interaction length L. The value of N obtained in this way was used for the comparison of the experimental data of pulse retardation in the presence of EIT with theoretical predictions obtained without any free parameter (Section 3.3). In the experiments the laser pulses of interest were sent to different photodiodes, and the signals coming from each photodiode were sent to a 1-GHz bandwidth Tektronik digital oscilloscope. A computer code realizes a shot-to-shot normalization in order to optimize the signal-to-noise ratio. This operation is necessary because of the irregularity of the coupling laser pulses, whose intensity presents a shotto-shot variation of a factor 5. An electronic discrimination upon the coupling pulse is also performed, by introducing a threshold on the integrated signal collected by the oscilloscope and thus selecting coupling pulses with a minimum value of their intensity and synchronized with a probe pulse.
a
3.2. Absorption spectral profiles
c
The experimental apparatus described in the previous section has been used to measure the absorption spectral profiles (normalized energy versus wavelength) of the probe laser beam for various intensities of the coupling laser pulse and various temperature of the Na cell. In all these measurements a resonant coupling field has been used (i.e. δc = ω3 − ω2). A fast Hamamatsu S1722-02 photodiode was used to detect the probe laser pulse coming out of the sodium cell. A (second) photodiode was used to detect the reference probe laser pulse, i.e. the probe pulse before entering the sodium cell. A third photodiode (ElectroOptics Technology ET2000) was finally used in order to detect the coupling pulse from the Ti-Sapphire laser. As an example, Fig. 2 shows typical spectra obtained at a temperature of 250 °C, with coupling laser intensities Ic in the range 65 kW/cm2– 380 kW/cm2. For coupling laser intensities above a certain threshold, the absorption profiles show the two peaks structure typical of EIT, with a transmission at line center and a peak separation both increasing with Ic. Theoretical predictions of the same spectra, obtained using Eq. (2.6), are also shown in Fig. 2 as continuous curves. The spectrum of the input probe field Sp(ω,0) is generated as the superposition of a series of Gaussian modes with a 1.8-GHz wide Gaussian envelope. The modes are separated from each other by 0.38 GHz and each of them has a width of 0.11 GHz with a different random constant phase. With these choices the temporal shape of the obtained pulses reproduces quite well the main features of the real pulses generated by the multimode dye laser. We verified by systematic numerical calculations that the probe laser absorption is not affected by different choices of the constant random phases. The atomic parameters are d13 =d23 = 7.2× 10− 30 C m and d34 =1.66× 10− 29 C m [21]. The atomic density N=2 ×1013 at/cm3 and dephasing rates γij =108 rad/s for a cell temperature T = 250 °C were independently measured in another experiment using the same cell. We performed our calculation for a pulsed field having the characteristics of the real probe pulse of the dye laser (1.8-GHz bandwidth) and also for a monochromatic (temporally constant) field. In the first case, the integrals in Eq. (2.9) must be evaluated numerically, whereas in the second case the evaluation can be analytical. The theoretical results are in quite good agreement with the experimental data (solid blue circles). Remarkably, the results for the two cases are essentially identical, to the extent that the two curves in Fig. 2 (which are both effectively plotted) cannot be distinguished.
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b
λ
λ
Fig. 2. Typical transmission spectra obtained at a temperature of 250 °C and coupling laser intensities Ic in the range 65 kW/cm2–380 kW/cm2. The solid dots with error bars are the experimental results. The continuous curves are theoretical calculations (see text for details).
Figs. 3 and 4 show, respectively, the transmission at line center as a function of the coupling laser intensity Ic and the peak distance as a function of the square root of Ic. The measurements are performed in the range of Ic ranging from 20 kW/cm 2 to 500 kW/cm 2 and at temperatures T = 180 °C (black squares) and T = 250 °C (blue circles), the transmission results to be decreasing with the temperature T, while the peak separation results to be independent of T. In both figures the solid dots with error bars represent the experimental data, while the continuous curves are theoretical results, evaluated numerically using the “real” multimode broadband pulse. Again, the theoretical results obtained by using Eq. (2.6) reproduce quite well the experimental data. The increase of the sodium atomic density N, owing to a higher cell temperature, explains the decrease of transmission with T shown in Fig. 3 (the imaginary part of Eq. (2.5) is proportional to N). On the other hand, Eq. (2.6) in the limit Ωc0 > > δωp, δωD provides a peak separation equal to the Rabi coupling Ωc0 and thus independent from the temperature, in agreement with the experimental findings shown in Fig. 4. 3.3. Delay measurements One of the most interesting effects related to EIT is the reduction of the group velocity for the probe pulse. This effect has been investigated in several publications since the first results from Stanford's group [2,3]. However, all the results reported so far have concerned temporal delay
E. Ignesti et al. / Optics Communications 285 (2012) 1185–1189
Δ
1188
Fig. 3. Transmission vs. coupling intensity performed at temperatures T = 180 °C (black dots and curve) and T = 250 °C (blue dots and curve) with coupling laser intensities Ic in the range 20 kW/cm2–500 kW/cm2. The solid dots with error bars are the experimental results. The continuous curves are theoretical calculations (see text for details).
of narrowband Fourier-transform-limited optical pulses. Here we want to study the same effect using the large-bandwidth (1.8 GHz), multimode pulses provided by our dye laser, in order to show that controllable retardation of a light pulse can be obtained without stringent requirements as far as the laser source is concerned. The experimental apparatus that we employed is essentially the same that was described before, where the detection system in this case is constituted by a fast photodiode and an oscilloscope. A fraction of the probe laser beam is reflected from a glass plate before entering the cell in order to be used as a reference pulse to measure the delay. The reference pulse is sent to the detection system together with the main part of the probe pulse that exits the cell. The lengths of the two optical paths are adjusted in order to be able to detect the two pulses with the same detector without temporal overlapping. With
60
50
Fig. 5. Probe pulse delay vs coupling intensity measured at a temperature T = 250 °C. The solid dots with error bars are the experimental results. The continuous curve is the theoretical calculation (see text for details).
this configuration, several measurements of temporal delay of the probe pulse were realized by changing the control laser intensity. All the measurements were done with a probe pulse peak intensity inside the cell of approximately 100 W/cm 2. In order to obtain various delays, we varied the peak intensity of the control pulse from 50 to nearly 700 kW/cm 2. In all our measurements the relation Ωc > > δωp is satisfied, providing that the probe pulse propagates in EIT conditions. The experimental results are shown in Fig. 5 (solid dots), which reports the measured probe pulse delay as a function of the coupling pulse intensity. The continuous curve is the result of a theoretical calculation, performed by numerically evaluating Eq. (2.7) without free parameters and using the “real” multimode broadband pulse. Probe pulse delays as large as 15 ns were obtained and the delay can be effectively controlled by changing the intensity of the coupling pulse. One parameter of interest in this kind of experiment is the distortion suffered by the delayed pulse. This parameter is particularly difficult to characterize for multimode non-transform-limited pulses such as those that we are using. In fact, in our results we remain within a fractional broadening of approximately 2 for laser pulse delays of the order of Δt = 5. For a detailed analysis of pulse distortion with pulses like ours we refer to our previous work [18].
Peak distance (pm)
4. Summary 40
30
20
10
0
0
2
4
6
8
10
12
14
16
18
20
I1/2 (kW1/2m) c0 Fig. 4. Peak distance vs. square root of the coupling intensity performed at temperatures T = 180 °C (black squares) and T = 250 °C (blue circles) with coupling laser intensities Ic in the range 20 kW/cm2–500 kW/cm2. The solid dots with error bars are the experimental results. The continuous curve is the theoretical calculation (see text for details).
To summarize, in this paper we presented an extensive investigation, both theoretical and experimental, concerning the possibility to optically control the absorption and the propagation velocity of a probe light pulse with a very large spectral bandwidth in the presence of electromagnetically-induced transparency. We reported experimental results obtained with hot sodium atoms in a ladder scheme, using a probe light pulse with a spectral bandwidth as large as 1.8 GHz. The predictions of the theoretical model are in good agreement with the experimental results. The possibility to control the absorption and the propagation velocity of a laser pulse makes systems based on EIT promising for the design of all-optical delay lines. References [1] [2] [3] [4] [5]
M. Fleischhauer, A. Imamoglu, J.P. Marangos, Rev. Mod. Phys. 77 (2005) 633. K.-J. Boller, A. Imamoglu, S.E. Harris, Phys. Rev. Lett. 66 (1991) 2593. A. Kasapi, M. Jain, G.Y. Yin, S.E. Harris, Phys. Rev. Lett. 74 (1995) 2447. P.W. Milonni, J. Phys. B 35 (2002) R31. S.E. Harris, J.E. Field, A. Imamoglu, Phys. Rev. Lett. 64 (1990) 1107.
E. Ignesti et al. / Optics Communications 285 (2012) 1185–1189 [6] R. Buffa, S. Cavalieri, M.V. Tognetti, Phys. Rev. Lett. 93 (2004) 129401. [7] R. Buffa, S. Cavalieri, M.V. Tognetti, Phys. Rev. A 72 (2005) 037801. [8] D.F. Phillips, A. Fleischhauer, A. Mair, R.L. Walsworth, M.D. Lukin, Phys. Rev. Lett. 86 (2001) 783. [9] G. Nikoghosyan, Eur. Phys. J. D 36 (2005) 119. [10] Liu Chien, Z. Dutton, C.H. Berhoozi, M.V. Hau, Nature 409 (2001) 490. [11] M. Fleischhauer, M.D. Lukin, Phys. Rev. A 65 (2002) 022314. [12] A.S. Zibrov, A.B. Matsko, O. Kocharovskaya, Y.V. Rostovtsev, G.R. Welch, M.O. Scully, Phys. Rev. Lett. 88 (2002) 103601. [13] M.D. Lukin, Rev. Mod. Phys. 75 (2003) 457. [14] R. Buffa, S. Cavalieri, M.V. Tognetti, Phys. Rev. A 69 (2004) 033815. [15] R. Buffa, S. Cavalieri, L. Fini, E. Sali, M.V. Tognetti, Opt. Comm. 264 (2006) 471. [16] V.G. Arkhipkin, I.V. Timofeev, Phys. Rev. A 73 (2006) 025803.
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[17] R. Buffa, S. Cavalieri, M.V. Tognetti, Opt. Lett. 29 (2004) 2432. [18] E. Ignesti, S. Cavalieri, L. Fini, E. Sali, M.V. Tognetti, R. Eramo, R. Buffa, Phys. Rev. A 80 (2009) 011811 (R). [19] E. Ignesti, S. Cavalieri, E. Sali, L. Fini, M.V. Tognetti, R. Buffa, Phys. Rev. A 81 (2010) 023807. [20] E. Sali, E. Ignesti, S. Cavalieri, L. Fini, M.V. Tognetti, R. Buffa, Opt. Commun. 282 (2009) 3330. [21] Ralchenko Yu., A.E. Kramida, J. Reader, NIST ASD Team, NIST Atomic Spectra Database (version 4.0) [Online]. Available: http://physics.nist.gov/asd [Tuesday, 26-Jul-2011 12:47:10 EDT]., National Institute of Standards and Technology, Gaithersburg, MD, 2010.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Controlling the propagation of broadband light pulses by Electromagnetically Induced Transparency Emilio Ignesti a, Roberto Buffa b, Lorenzo Fini a, c, Emiliano Sali a,⁎, Marco V. Tognetti a, Stefano Cavalieri a, c a b c
Dipartimento di Fisica e Astronomia, Università di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy Dipartimento di Fisica, Università di Siena, Via Roma 56, I-53100 Siena, Italy European Laboratory for Nonlinear Spectroscopy, Università di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy
a r t i c l e
i n f o
Article history: Received 18 March 2011 Received in revised form 29 July 2011 Accepted 4 November 2011 Available online 24 November 2011
a b s t r a c t We present an experimental and theoretical investigation, performed on hot sodium atoms in a ladder scheme, showing the control of the absorption and of the propagation velocity of a probe light pulse with a spectral bandwidth as large as 1.8 GHz. The predictions of the theoretical model compare favorably with the experimental results. © 2011 Elsevier B.V. All rights reserved.
Keywords: Electromagnetically-induced transparency Coherent control Broadband light pulses Light pulse propagation
1. Introduction One of the most interesting and studied examples of coherent control process is provided by electromagnetically-induced transparency (EIT), in which the optical properties of a material medium at wavelength λp can be deeply modified and controlled by an intense laser field at a very different wavelength λc [1]. Since the first experimental demonstrations at Stanford [2], EIT has been observed in various material media (atomic and molecular gases and solid crystals) and its possible applications have been widely discussed in many contexts: from the control of the absorption and of the group velocity of probe radiation [3,4] to the enhancement of optical nonlinearity [5–7], from light storage [8–10] to quantum memory [11–13]. In all these studies, cw or monomode radiation for the probe pulse has been used or assumed. However, existing theoretical models hint that, in order to obtain EIT, monocromaticity is not a fundamental requirement for the probe radiation, and very recent experiments have shown that some degree of control on temporal pulse shaping [14–17], delay [18] and frequency shift [19] can indeed be obtained also for broadband probe pulses. This is not a minor point in the road to a deep understanding of EIT applications. In effect, the control of the propagation velocity of light pulses may lead to the design of all-optical delay lines, in which the amount of information carried by the radiation pulse is intrinsically tied to its spectral bandwidth.
⁎ Corresponding author. E-mail address: sali@fi.infn.it (E. Sali). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.11.018
In this paper we report the results of an experimental investigation, performed on hot sodium atoms in a ladder scheme, showing the control of the absorption and of the propagation velocity of a probe light pulse with a spectral bandwidth as large as 1.8 GHz. Compared to Ref. [18], the main goal of this work was to extend our study to the characterization of the spectral transmission. Also, we provide more accurate experimental results in order to allow a more precise comparison with the theory. In Section 2 we review the theory of propagation of broadband probe pulses in an EIT-modified-medium of hot sodium atoms, and we introduce the pertinent definitions of the quantities that have been experimentally investigated. In Section 3 we describe the experimental set-up (Section 3.1), we report the measurement results of probe absorption (Section 3.2) and temporal delay (Section 3.3), and we compare the experimental data with the theoretical results. Section 4 summarizes the results of this work. 2. Theoretical framework Fig. 1 shows a schematic diagram of the interaction process. The system under consideration is a four-level scheme in sodium. The couple of states |1〉 and |2〉 are two hyperfine sublevels (F = 1, 2 F = 2) of the level S1/2 (2p 63s), which were taken into account due to their energy separation Δν = 1.772 GHz, which is comparable to 2 the laser bandwidth. States |3〉 and |4〉 correspond to P1/2 (2p 63p) 2 6 and D3/2 (2p 3d). A weak probe laser pulse connects the two close states |1〉 and |2〉 with the same excited state |3〉, which in turns is coupled to the higher state |4〉 by a much stronger coupling pulse. In this hypothesis the propagation equations for the electric-field
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E. Ignesti et al. / Optics Communications 285 (2012) 1185–1189
the explicit dependence of the coherences ρnm(v) on the velocity v of the atom has been dropped. If, upon its propagation, the probe field Ep is overlapped by a flat region of the coupling field of value Ec0, then Eqs. (2.2)–(2.3a) admit analytical solutions and the electric field envelope Ep(z, t) is given by þ∞
Ep ðz; t Þ ¼ ∫ Sp ðz; ωÞ expðiωt Þdω
ð2:4aÞ
−∞
Sp ðz; ωÞ ¼ Sp ð0; ωÞ exp½−ikðωÞz
ð2:4bÞ
with kðωÞ ¼
"
ωp ω þN c 4ħε0 c
γ 14 þ i ω þ Δp þ Δc d13 b >v γ 13 þ i ω þ Δp þ Ω2c0 γ 14 þ i ω þ Δp þ Δc # γ24 þ i ω þ Δ0p þ Δc þd23 b >v : γ23 þ i ω þ Δ0p þ Ω2c0 γ24 þ i ω þ Δ0p þ Δc ð2:5Þ
Then, through the expression of the probe field at the cell output Ep(L, t), one can define straightforwardly the probe pulse transmission Tr
Fig. 1. Schematic diagram of the interaction process.
2 ∫ Ep ðL; t Þ dt
þ∞
envelopes Ep and Ec of the probe and coupling field, respectively, in a counter-propagating configuration read as:
ωp ∂ 1∂ ðd b ρ > þ d23 b ρ23 >v Þ þ E ¼ −i N ε0 c 13 13 v ∂z c ∂t p ∂ 1∂ − E ¼0 ∂z c ∂t c
ð2:1aÞ
ð2:1bÞ
where N is the density of the atomic sample, d13 and d23 are the electric-dipole moments of the transitions 1–3 and 2–3, respectively, and the coherences ρnm(v) are averaged over the Maxwell–Doppler velocity distribution fD(v): þ∞
ð2:2Þ
−∞
In the hypothesis of resonant coupling field the coherences ρnm(v) that appear in Eqs. (2.1a)–(2.2) satisfy the following Liouville equations: d ρ dt 13
Ωp − iΔp þ γ 13 ρ13 −iΩc ρ14 ¼ −i 2
h i d ρ ¼ −iΩc ρ13 − i Δp þ Δc þ γ 14 ρ14 dt 14 d ρ dt 23
Ωp 0 − iΔp þ γ 23 ρ23 −iΩc ρ24 2
ð2:3aÞ ð2:3bÞ
0
¼ −i
h 0 i d ρ ¼ −iΩc ρ23 − i Δp þ Δc þ γ 24 ρ24 dt 24
−∞ þ∞
ð2:6Þ
2 ∫ Ep ð0; t Þ dt
−∞
and the probe pulse delay Δt
b ρnm >v ¼ ∫ ρnm ðvÞf D ðvÞdv
Tr ¼
ð2:3cÞ ð2:3dÞ
where Ωp = d13Ep/2ħ, Ωp' = d23Ep/2ħ and Ωc = d34Ec/2ħ are the Rabi couplings, Δp = δp + ω21/2 − ωpv/c, Δp' = δp − ω21/2 − ωpv/c and Δc = ωcv/c are the laser detunings from resonances with ω21 the frequency separation of states |1〉 and |2〉, γnm represent all kind of dephasing rates, and
2 ∫ t Ep ðL; t Þ dt
þ∞
− Δt ¼ −∞ þ∞ 2 ∫ Ep ðL; t Þ dt −∞
2 ∫ t Ep ð0; t Þ dt
þ∞
−∞ þ∞
− 2 ∫ Ep ð0; t Þ dt
L : c
ð2:7Þ
−∞
In the limit ω32 →0; d13 ¼ d23 ; γ 1i ¼ γ2i , which implies to reduce our four-level scheme to a three-level one removing one of the ground close states, Eq. (2.7) reduces to the analytical expression reported in the literature for an EIT ladder scheme [17]. 3. Experiments 3.1. The experimental apparatus The experimental apparatus employed for these experiments is described in what follows. The probe field, with wavelength λp = 2πc/ωp, is provided by a frequency-tuneable multimode dye laser pumped by a frequency-doubled Q-switched Nd:YAG laser at a repetition rate of 10 Hz. The dye laser pulses have a multi-peaked temporal structure of few nanoseconds of duration. The probe laser bandwidth is δωp/2π = 1.8 GHz. The control field at λc = 2πc/ωc = 818.550 nm is provided by a frequency-tuneable, single-longitudinal-mode titanium-sapphire (Ti:S) laser [20] delivering pulses with temporal full-width at halfmaximum (FWHM) equal to 60 ns. The Ti:Sa laser pulse peak intensity can be varied from 25 KW/cm 2 to 720 KW/cm 2, while the probe pulse peak intensity can be varied variable from 50 W/cm2 to 250 W/cm 2 in our experimental conditions. The Ti:Sa laser pulse spatially overlaps the probe pulse into the cell in a counter-propagating geometry, in order to reduce the Doppler effect.
E. Ignesti et al. / Optics Communications 285 (2012) 1185–1189
The sodium cell is a 1.5-meters-long steel cylinder, where the area effectively occupied by the vapor sodium is approximately 1 meter. The electrical power supply is regulated by a feedback loop that maintains the temperature within 1 K in the range 300–700 K. By measuring the delay (see Eq. (2.7)) without the presence of the coupling laser it is possible to determine experimentally the quantity N× L, i.e. the product of the sodium density N times the interaction length L. The value of N obtained in this way was used for the comparison of the experimental data of pulse retardation in the presence of EIT with theoretical predictions obtained without any free parameter (Section 3.3). In the experiments the laser pulses of interest were sent to different photodiodes, and the signals coming from each photodiode were sent to a 1-GHz bandwidth Tektronik digital oscilloscope. A computer code realizes a shot-to-shot normalization in order to optimize the signal-to-noise ratio. This operation is necessary because of the irregularity of the coupling laser pulses, whose intensity presents a shotto-shot variation of a factor 5. An electronic discrimination upon the coupling pulse is also performed, by introducing a threshold on the integrated signal collected by the oscilloscope and thus selecting coupling pulses with a minimum value of their intensity and synchronized with a probe pulse.
a
3.2. Absorption spectral profiles
c
The experimental apparatus described in the previous section has been used to measure the absorption spectral profiles (normalized energy versus wavelength) of the probe laser beam for various intensities of the coupling laser pulse and various temperature of the Na cell. In all these measurements a resonant coupling field has been used (i.e. δc = ω3 − ω2). A fast Hamamatsu S1722-02 photodiode was used to detect the probe laser pulse coming out of the sodium cell. A (second) photodiode was used to detect the reference probe laser pulse, i.e. the probe pulse before entering the sodium cell. A third photodiode (ElectroOptics Technology ET2000) was finally used in order to detect the coupling pulse from the Ti-Sapphire laser. As an example, Fig. 2 shows typical spectra obtained at a temperature of 250 °C, with coupling laser intensities Ic in the range 65 kW/cm2– 380 kW/cm2. For coupling laser intensities above a certain threshold, the absorption profiles show the two peaks structure typical of EIT, with a transmission at line center and a peak separation both increasing with Ic. Theoretical predictions of the same spectra, obtained using Eq. (2.6), are also shown in Fig. 2 as continuous curves. The spectrum of the input probe field Sp(ω,0) is generated as the superposition of a series of Gaussian modes with a 1.8-GHz wide Gaussian envelope. The modes are separated from each other by 0.38 GHz and each of them has a width of 0.11 GHz with a different random constant phase. With these choices the temporal shape of the obtained pulses reproduces quite well the main features of the real pulses generated by the multimode dye laser. We verified by systematic numerical calculations that the probe laser absorption is not affected by different choices of the constant random phases. The atomic parameters are d13 =d23 = 7.2× 10− 30 C m and d34 =1.66× 10− 29 C m [21]. The atomic density N=2 ×1013 at/cm3 and dephasing rates γij =108 rad/s for a cell temperature T = 250 °C were independently measured in another experiment using the same cell. We performed our calculation for a pulsed field having the characteristics of the real probe pulse of the dye laser (1.8-GHz bandwidth) and also for a monochromatic (temporally constant) field. In the first case, the integrals in Eq. (2.9) must be evaluated numerically, whereas in the second case the evaluation can be analytical. The theoretical results are in quite good agreement with the experimental data (solid blue circles). Remarkably, the results for the two cases are essentially identical, to the extent that the two curves in Fig. 2 (which are both effectively plotted) cannot be distinguished.
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b
λ
λ
Fig. 2. Typical transmission spectra obtained at a temperature of 250 °C and coupling laser intensities Ic in the range 65 kW/cm2–380 kW/cm2. The solid dots with error bars are the experimental results. The continuous curves are theoretical calculations (see text for details).
Figs. 3 and 4 show, respectively, the transmission at line center as a function of the coupling laser intensity Ic and the peak distance as a function of the square root of Ic. The measurements are performed in the range of Ic ranging from 20 kW/cm 2 to 500 kW/cm 2 and at temperatures T = 180 °C (black squares) and T = 250 °C (blue circles), the transmission results to be decreasing with the temperature T, while the peak separation results to be independent of T. In both figures the solid dots with error bars represent the experimental data, while the continuous curves are theoretical results, evaluated numerically using the “real” multimode broadband pulse. Again, the theoretical results obtained by using Eq. (2.6) reproduce quite well the experimental data. The increase of the sodium atomic density N, owing to a higher cell temperature, explains the decrease of transmission with T shown in Fig. 3 (the imaginary part of Eq. (2.5) is proportional to N). On the other hand, Eq. (2.6) in the limit Ωc0 > > δωp, δωD provides a peak separation equal to the Rabi coupling Ωc0 and thus independent from the temperature, in agreement with the experimental findings shown in Fig. 4. 3.3. Delay measurements One of the most interesting effects related to EIT is the reduction of the group velocity for the probe pulse. This effect has been investigated in several publications since the first results from Stanford's group [2,3]. However, all the results reported so far have concerned temporal delay
E. Ignesti et al. / Optics Communications 285 (2012) 1185–1189
Δ
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Fig. 3. Transmission vs. coupling intensity performed at temperatures T = 180 °C (black dots and curve) and T = 250 °C (blue dots and curve) with coupling laser intensities Ic in the range 20 kW/cm2–500 kW/cm2. The solid dots with error bars are the experimental results. The continuous curves are theoretical calculations (see text for details).
of narrowband Fourier-transform-limited optical pulses. Here we want to study the same effect using the large-bandwidth (1.8 GHz), multimode pulses provided by our dye laser, in order to show that controllable retardation of a light pulse can be obtained without stringent requirements as far as the laser source is concerned. The experimental apparatus that we employed is essentially the same that was described before, where the detection system in this case is constituted by a fast photodiode and an oscilloscope. A fraction of the probe laser beam is reflected from a glass plate before entering the cell in order to be used as a reference pulse to measure the delay. The reference pulse is sent to the detection system together with the main part of the probe pulse that exits the cell. The lengths of the two optical paths are adjusted in order to be able to detect the two pulses with the same detector without temporal overlapping. With
60
50
Fig. 5. Probe pulse delay vs coupling intensity measured at a temperature T = 250 °C. The solid dots with error bars are the experimental results. The continuous curve is the theoretical calculation (see text for details).
this configuration, several measurements of temporal delay of the probe pulse were realized by changing the control laser intensity. All the measurements were done with a probe pulse peak intensity inside the cell of approximately 100 W/cm 2. In order to obtain various delays, we varied the peak intensity of the control pulse from 50 to nearly 700 kW/cm 2. In all our measurements the relation Ωc > > δωp is satisfied, providing that the probe pulse propagates in EIT conditions. The experimental results are shown in Fig. 5 (solid dots), which reports the measured probe pulse delay as a function of the coupling pulse intensity. The continuous curve is the result of a theoretical calculation, performed by numerically evaluating Eq. (2.7) without free parameters and using the “real” multimode broadband pulse. Probe pulse delays as large as 15 ns were obtained and the delay can be effectively controlled by changing the intensity of the coupling pulse. One parameter of interest in this kind of experiment is the distortion suffered by the delayed pulse. This parameter is particularly difficult to characterize for multimode non-transform-limited pulses such as those that we are using. In fact, in our results we remain within a fractional broadening of approximately 2 for laser pulse delays of the order of Δt = 5. For a detailed analysis of pulse distortion with pulses like ours we refer to our previous work [18].
Peak distance (pm)
4. Summary 40
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I1/2 (kW1/2m) c0 Fig. 4. Peak distance vs. square root of the coupling intensity performed at temperatures T = 180 °C (black squares) and T = 250 °C (blue circles) with coupling laser intensities Ic in the range 20 kW/cm2–500 kW/cm2. The solid dots with error bars are the experimental results. The continuous curve is the theoretical calculation (see text for details).
To summarize, in this paper we presented an extensive investigation, both theoretical and experimental, concerning the possibility to optically control the absorption and the propagation velocity of a probe light pulse with a very large spectral bandwidth in the presence of electromagnetically-induced transparency. We reported experimental results obtained with hot sodium atoms in a ladder scheme, using a probe light pulse with a spectral bandwidth as large as 1.8 GHz. The predictions of the theoretical model are in good agreement with the experimental results. The possibility to control the absorption and the propagation velocity of a laser pulse makes systems based on EIT promising for the design of all-optical delay lines. References [1] [2] [3] [4] [5]
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[17] R. Buffa, S. Cavalieri, M.V. Tognetti, Opt. Lett. 29 (2004) 2432. [18] E. Ignesti, S. Cavalieri, L. Fini, E. Sali, M.V. Tognetti, R. Eramo, R. Buffa, Phys. Rev. A 80 (2009) 011811 (R). [19] E. Ignesti, S. Cavalieri, E. Sali, L. Fini, M.V. Tognetti, R. Buffa, Phys. Rev. A 81 (2010) 023807. [20] E. Sali, E. Ignesti, S. Cavalieri, L. Fini, M.V. Tognetti, R. Buffa, Opt. Commun. 282 (2009) 3330. [21] Ralchenko Yu., A.E. Kramida, J. Reader, NIST ASD Team, NIST Atomic Spectra Database (version 4.0) [Online]. Available: http://physics.nist.gov/asd [Tuesday, 26-Jul-2011 12:47:10 EDT]., National Institute of Standards and Technology, Gaithersburg, MD, 2010.