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Transcript

Electromagnetically Induced Transparency with Broadband Laser Pulses D. D. Yavuz Department of Physics, University of Wisconsin, Madison, WI, 53706 Time-delay-bandwidth Product Over the last decade, counterintuitive optical effects using Electromagnetically Induced Transparency (EIT) have gained considerable attention [1-3]. The essence of EIT is to create a narrow transparency window in an otherwise opaque medium using quantum interference. Noting Fig. 1, in a three state atomic medium the quantum interference is achieved with a coupling laser beam that couples states |2 and |3. In the perturbative limit where the probe laser beam is much weaker than the coupling laser beam, the susceptibility for the probe wave is: 13 : dipole matrix N 13 ( p ) 4 0 22( ) j c 2 2 1 c p c |2 2 |1 / 0 j 2 p N 13 c1c3* z c j 2 c N 23 c2c3* z 1 In Fig. 1, the imaginary part of the susceptibility is plotted for Ωc=Г/3, and Δ=- Γ, Δ=0, and Δ=Γ respectively. In all cases perfect transparency is obtained at exact two-photon resonance. Furthermore, even though the lineshape becomes asymmetric for Δ≠0, the steep dispersion is maintained at the point of vanishing absorption. frequency The scheme of Fig. 2 is identical to the EIT with matched pulses as suggested by Harris et. al. [6]. In this work, we extend the suggestion of Harris, and show that using matched frequencies for the two transitions, one can obtain a large group delay for a broadband probe pulse. One can therefore obtain a large time-delay-bandwidth product. (a) We proceed with a perturbative analytical solution of Eqs. (1) and (2) to get an insight into the results of Figs. 3 and 4. We follow closely the formalism of Eberly and colleagues [7]. We proceed perturbatively and take the probe beam to be much weaker when compared with the coupling beam. With counter-intuitive pulse sequence the medium can be prepared in the dark state when the following two-field adiabatic condition is satisfied: p c 3 p c c z=0 z=2.5 mm p 0 1 c 0 400 time (s) 0 1 time (s) 2 (b) z=0 z=2.5 mm 2 f ( ) (5) We note that the adiabatic criteria of Eq. (5) is independent of the bandwidth of the time function f ( ) . This is the key reason why the dynamics of the medium is largely independent of the time variation common to both fields, f ( ) , as long as Eq. (5) is satisfied. With the medium prepared adiabatically, the solution of the Schrodinger equation [Eq. (1)] for the probability amplitudes, including the first non-adiabatic correction to c3 is: * 2 p c1 1, c2 * , c3 j c *c c *p 100 200 N : atomic density / 0 (2) 300 p 0 , c 0 : long envelopes (3) f ( ) q f q exp( jq ) 0 1 time (s) In Eq. (3), p 0 , c0 are long envelopes that allow adiabatic preparation of the medium and the function f ( ) defines the broad set of frequencies that are considered, f ( ) q f q exp( jq ) . The function, f ( ), is dimensionless and is 2 normalized such that q | f q | 1 . As we will show, the dynamics of the EIT medium will be determined by the long envelopes p 0 , c0 and will largely be independent of the time variation that is common to both fields f ( ). 2 0 1 time (s) 2 z=0 z=5 mm 0 50 100 150 200 time (s) Figure 3: Slowing of broadband light pulses in 87Rb vapor with an atomic density of N=1012 /cm3. At the beginning of the cell the probe beam is assumed to be p ( z 0, ) p 0 ( ) f ( ) where p 0 ( ) is a long envelope with a Gaussian width of 12 µs and f ( ) is a rapidly varying time waveform. The bottom plot shows the envelopes of the probe beam at z=0 and z=5 mm respectively. The probe beam is delayed by 84 µs. The insets zoom in on the central portion of the waveform to display the rapidly varying square wave. At the output, the structure of the square waveform is almost perfectly preserved. The time-delay-bandwidth product that is achieved in this simulation is ≈103. Figure 3 shows the envelope of the broadband probe pulse at z=0 and z=5 mm respectively. The probe pulse propagates with a group velocity of vg=58 m/s and is delayed by 84 µs at the end of the medium. The shape of the square-wave temporal waveform is almost perfectly preserved demonstrating that all Fourier components of the input pulse propagate without significant loss and phase shift. The time-delay-bandwidth product that is achieved in this simulation is ≈103. 400 time (s) (6) With the solution of Eq. (6), the propagation equation for the probe beam becomes: Figure 4: Stopping of broadband light pulses using EIT. The probe pulse is stored for 100 µs in part (a) and 250 µs in part (b) and then released. Stopping of the probe pulse is achieved by smoothly turning down the intensity of the coupling laser beam. In both plots, the normalized intensity of the probe pulse at z=0 and z=2.5 mm is plotted. The dotted line is the intensity envelope of the coupling laser beam. The inset in (b) is a zoom in on the central portion of the released pulse. p p 0 z Following Eberly and colleagues [7], we make the following change of variable in Eq. (7), ( ) f ( ) d . With this transformation, the analytical solution of Eq. (7) is: 2 2 2 1 p p N 13 * z c c 2 1 p 0 2 2 f ( ) p N 13 2 c 0 [7] R. Grobe, F. T. Hioe, and J. H. Eberly, Phys. Rev. Lett. 73, 3183 (1994). (7) Conclusions 0 ~ p 0 ( z , ) p 0 ( z / v~g ), 1 2 p N 13 ~ , p 0 ( ) p 0 (0, ) 2 v~g c 0 2 (8) (4) With p ( z, ) p0 ( z, ) f ( ) and c ( z, ) c0 f ( ) ( c 0 independent of space and time), the two field adiabatic condition of Eq. (4) reduces to: 300 c ( z 0, ) c 0 ( ) f ( ) Below, we present a numerical simulation that demonstrate slowing of broadband light pulses in 87Rb vapor. In this simulation, we assume an atomic density of N=1012 /cm3 and take the two Raman states to be |1|F=1,mF=0 and |2|F=2,mF=0 hyperfine states of the ground electronic state 5S1/2. The excited state is chosen to be |3|F΄=2,mF΄=1 of 5P3/2. We take the two laser beams to have the same circular polarization. We take f ( ) to consist of 31 equally spaced frequencies and choose the amplitudes and the phases of the Fourier components such that the synthesized time function is a square wave with a period of 0.66 µs. The total spectral content of the probe pulse is therefore 45 MHz. We assume a Gaussian envelope, p 0 , for the probe laser beam with a Gaussian width of 12 µs. The envelope for the coupling laser beam, c 0 , smoothly turns on to its peak value of c0, peak / 3 and stays constant (not shown in Fig. 3). To assure adiabatic preparation of the medium, the coupling laser beam is turned on before the probe laser beam (counter-intuitive pulse sequence). Perturbative Analytical Solution Figure 4 is a numerical simulation that demonstrates stopping of broadband light pulses. Here, we take the coupling beam envelope to smoothly turn-off for a duration of 100 µs in (a) and 250 µs in (b) and then turn back on again. The probe beam is therefore stored in the medium for a controllable amount of time and then released. For both cases, the released pulse contains all the Fourier components of the input pulse with relative phases and amplitudes preserved. The inset in (b) is a zoom in on the central portion of the probe envelope that shows the square-wave temporal structure. 0 (1) p ( z 0, ) p 0 ( ) f ( ) Figure 2: The suggestion of our scheme. We consider the propagation of a broad set of frequencies for the probe laser beam where each frequency has a matching component in the coupling laser beam such that the two-photon resonance condition is maintained. EIT is achieved in parallel channels for each of the frequency components. Numerical Simulation 200 : excited state decay rate [4] Z. Dutton, M. Bashkansy, M. Steiner, and J. Reintjes, Proc. SPIE 5735, 115 (2005). [5] Q. Sun, Y. V. Rostovtsev, J. P. Dowling, M. O. Scully, and M. S. Zubairy, Phys. Rev. A 72, 031802(R) (2005). [6] S. E. Harris, Phys. Rev. Lett. 70, 552 (1993). [1] M. O . Scully and M. S. Zubairy, “Quantum Optics” (Cambridge University Press, 1997). [2] S. E. Harris, Phys. Today 50, No. 7, 36 (1997). [3] O. Kocharovskaya and P. Mandel, Phys. Rev. A 42, 523 (1990). 100 c : Rabi frequency of the coupling laser beam |1 Figure 1: The susceptibility for the probe laser beam as a function of two-photon detuning, for Δ=-Γ,Δ=0, and Δ=Γ respectively. 0 p : Rabi frequency of the probe laser beam We analyze the propagation of a broad set of frequencies for the probe and coupling laser beams through an atomic system defined by the above coupled equations. We solve Eqs. (1) and (2) with the initial condition that all atoms start in the ground state |1 and the following boundary condition at the beginning of the cell (z=0) for the two laser beams: |2 = -3 -2 -1 c1 , c2 , c3 : probabilit y amplitudes The decay processes are assumed to be to states outside the system. With the probability amplitudes calculated by Eq. (1), the slowly varying envelope Maxwell’s equations for the two laser beams are: … power spectral density 0 |3 p =0 -2 -1 3 |3 … normalized absorption 2 c1 j p c3 2 c2 j c c3 2 c3 j j c3 *p c1 *c c2 2 2 2 p … 1 In this work, motivated by the results of Fig. 1, we consider a broad set of frequencies for the probe laser beam where each frequency has a matching component in the coupling laser beam such that the two-photon resonance condition is maintained. Noting Fig. 2, EIT and therefore slow light is achieved in parallel channels for each of the frequencies of the probe laser beam. … 0 We proceed with the analysis of schematic of Fig. 2. Working in local time, the Schrodinger equation for the probability amplitudes of the three states in the interaction picture are: element : excited state decay rate : one photon detuning : two - photon detuning =- -1 EIT provides a unique way to controllably delay and coherently store light pulses. However, narrow transparency window of EIT puts stringent limitations on the bandwidth of the light pulses that can be slowed and stopped inside the medium. A key figure of merit that is usually discussed in this context is the time-delay-bandwidth product which is obtained by multiplying the bandwidth of the optical pulse with the delay time of the pulse while propagating through the EIT medium. The largest time-delay-bandwidth product that has been experimentally demonstrated using EIT is ≈5. Recently Dutton and colleagues [4] and Zubairy and colleagues [5] have suggested schemes to overcome this limitation. Numerical Simulation Formalism normalized probe intensity Introduction In agreement with the numerical results of Figs 3 and 4, Eq. (8) shows that the probe pulse propagates without attenuation and with a group velocity determined by the intensity of the coupling laser beam, | c 0 |2. Remarkably, this group velocity, and therefore the time delay obtained while propagating through the EIT medium is independent of the bandwidth of f ( ) assuming infinite dephasing time of the Raman transition. As a result it becomes possible to obtain large group delays for large bandwidth optical pulses. One important practical application of EIT with large bandwidth pulses is to optical information processing. Our scheme may provide a unique way to controllably delay and coherently store large bandwidth optical pulses. However, a significant disadvantage of our approach is that it requires a time varying coupling laser beam with Fourier components exactly matched to the probe laser beam. Our approach may also find applications in achieving giant nonlinearities effective at single photon levels using EIT [8-12]. When compared with the traditional narrow bandwidth EIT schemes, it may be advantageous to use larger bandwidth, therefore higher peak power, single photon pulses. [8] H. Scmidt and A. Imamoglu, Opt. Lett. 21, 1936 (1996). [9] S. E. Harris and Y. Yamamoto, Phys. Rev. Lett. 81, 3611 (1998). [10] M. D. Lukin and A. Imamoglu, Phys. Rev. Lett. 84, 1419 (2000). [11] H. Kang and Y. Zhu, Phys. Rev. Lett. 91, 093601 (2003). [12] D. A. Braje, V. Balic, S. Goda, G. Y. Yin, and S. E. Harris, Phys. Rev. Lett. 93, 183601 (2004). We have extended the suggestion of Harris et. al. [6], and demonstrated that when matched Fourier components are used, the dynamics of an EIT system decouple from the time variation that is common to both fields. As a result, it becomes possible to obtain slow light and large group delays with large bandwidth optical pulses. Our numerical simulations show that time-delay-bandwidth products exceeding 103 is readily realizable with current experimental parameters. We expect possible applications to optical information processing using EIT and to achieving giant nonlinearities effective at the single photon levels. Acknowledgements I would like to thank Brett Unks and Nick Proite for helpful discussions. This work was supported by a start-up grant from the department of Physics at University of Wisconsin-Madison.