- Email: [email protected]
Quantum interference in atomic vapor controlled by a magnetic field
15 June 2002
Optics Communications 207 (2002) 227–231 www.elsevier.com/locate/optcom
Quantum interference in atomic vapor controlled by a magnetic field C.Y. Ye a,*, Y.V. Rostovtsev a, A.S. Zibrov b,c, Yu.M. Golubev d a
Department of Physics and Institute for Quantum Studies, Texas A&M University, College Station, Texas 77843, USA b Department of Physics and ITAMP, Harvard University, Cambridge, MA, 02138, USA c P.N. Lebedev Institute of Physics, RAS, Leninsky pr. 53, Moscow 117924, Russia d Physics Institute, St. Petersburg University, ul. Ulyanovskaya 1, St. Petersburg 198904, Russia Received 8 March 2002; accepted 24 April 2002
Abstract We have shown that quantum interference in a driven quasi-degenerate two-level atomic system can be controlled by an externally applied magnetic field. We demonstrate that the mechanism of optical control is based on quantum interference, which allows one to implement both electromagnetically induced transparency and electromagnetically induced absorption (EIA) in one atomic system. The experimental realization is suggested. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.50.Gy; 42.50.Hz
Quantum coherence and interference play an important role in the interaction of coherent laser fields with atomic systems. As has been shown that electromagnetically induced transparency (EIT) arises from destructive interference of atomic transitions [1], while electromagnetically induced absorption (EIA) is connected with constructive interference [2]. EIT has found a wide variety of applications in quantum optics and nonlinear optical processes [3]. EIA could have potential ap-
*
Corresponding author. Tel.: +979-4581136; fax: +9798452590. E-mail address: [email protected] (C.Y. Ye).
plications to high-speed optical modulation and quantum switching [4,5]. N-type scheme has recently attracted much attention and opened a new approach to manipulate the nonlinear phenomena in optical process [6]. It has been shown that atomic coherence among Zeeman sublevels can be spontaneously transferred from upper level to lower one, which gives rise not only to EIT, but also to EIA [2,7,8]. Doppler-free resonance absorption observed in the N scheme displays another interesting features in the Doppler-broadened medium. Ye et al. [9] have demonstrated that the restriction on Doppler-free geometry can be alleviated by atomic multicoherence in the N scheme. Zibrov et al. [10] observed three-photon resonance Doppler-free in hot Rb
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 5 2 3 - 7
228
C.Y. Ye et al. / Optics Communications 207 (2002) 227–231
vapor driven with one of two coherent fields far detuning from its resonance in the N scheme. The other type of schemes have been analysed in [11,12]. Usually, the atomic system has one type of interference, either constructive or destructive, that depends on the configuration of atomic levels and laser fields. In this paper, we show a way to coherently control the type of interference in one quasi-degenerate two-level atomic system by applying an external magnetic field. The experimental implementation is suggested as well. We consider a N-type four-level scheme, as shown in Fig. 1. Levels jbi and jci are ground states with small decay rate cbc , levels jai and jdi are the excited states with the population decay rate c to ground states (c cbc ). The drive laser couples the transition jbi ! jai, as well as jdi ! jci simultaneously, the probe monitors the changes of the transition jai and jci. The magnetic field is applied to control the transmission of the probe laser via coherent atomic vapor. The Hamiltonian of the system can be written as H ¼ H0 þ HI , where hdjcihcj hðd þ DÞjdihdj þ H :c:; H0 ¼ hDjbihbj ð1Þ and HI ¼ hXðjaihbj þ jdihcjÞ hejaihcj þ H :c:
ð2Þ
Then, the density matrix equations for this system can be written as
q_ aa ¼ ð2c þ c0 Þqaa ieðqac qca Þ iXðqab qba Þ; ð3Þ q_ bb ¼ rb þ cqaa c0 qbb þ iXðqab qba Þ;
ð4Þ
q_ cc ¼ rc þ cqaa c0 qcc þ 2cqdd þ ieðqac qca Þ iXðqcd qdc Þ;
ð5Þ
q_ dd ¼ 2cqdd c0 qdd þ iXðqcd qdc Þ;
ð6Þ
q_ ab ¼ Cab qab iXðqaa qbb Þ þ ieqcb ;
ð7Þ
q_ cb ¼ Ccb qcb iXqca þ ieqab þ iXqdb ;
ð8Þ
q_ db ¼ Cdb qdb iXqda þ iXqcb ;
ð9Þ
q_ ca ¼ Cca qca þ ieðqaa qcc Þ þ iXqda iXqcb ;
ð10Þ
q_ da ¼ Cda qda ieqdc þ iXqca iXqdb ;
ð11Þ
q_ dc ¼ Cdc qdc ieqda þ iXðqcc qdd Þ;
ð12Þ
where X is the Rabi frequency of drive laser and e is that of probe; rb and rc are the incoherent pumping rate into respective state jbi and jci; c0 is the decay rate out of all states; The off-diagonal decay rates are Cij ¼ cij þ iDij , cij are decay rates of level jii and jji. The detunings of laser from their resonance frequencies Dij with Doppler shift kv are given by Dab ¼ xab m kv ¼ D kv, Dac ¼ xac m0 k0 v ¼ d kv, Ddc ¼ xdc m kv ¼ D kv; where D ¼ xab m and d ¼ xac m, xij is the atomic transition frequency from level jii and jji, m is the frequency of the drive laser, and m0 the probe laser, k and k0 are the wavenumbers of the drive and probe lasers under the assumption of m0 ¼ m. The analytical solution for coherence term qac , which gives the absorption of the probe field, can be found with some assumptions. Here, we present the numerical calculations. To take the inhomogeneous broadening into account, one should find the velocity-averaged probe absorption coefficient, which is given by Z 1 q v¼ ð13Þ dðkvÞg ac f ðkvÞ; e 1 where g ¼ 3=8pN ck3 , f ðkvÞ is the Gaussian distribution function of atomic velocities
Fig. 1. Schematic of simplified N -type four-level system. Drive laser couples to the jai-jbi and jci-jdi transitions simultaneously, probe laser to jai-jci. Radiative decay rate from jai and jdi to lower levels is 2c, coherent decay rate for jbi-jci is cbc .
2 2 1 pffiffiffi eðkvÞ =ðkuÞ ; ð14Þ ku p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where u ¼ 2kB T =M is the most probable speed of atoms at given temperature T, M is the atomic
f ðkvÞ ¼
C.Y. Ye et al. / Optics Communications 207 (2002) 227–231
mass, kB is the Boltzmann constant. The full width at half maximum of the Doppler pffiffiffiffiffiffiffiffi broadened linewidth is given by 2WD ¼ 2 ln 2ku. The calculated spectra of probe absorption as a function of the probe detuning under different conditions are shown in Fig. 2. Let us start with the study of Doppler broadened absorption spectrum of the probe laser coupled transition jci ! jai with no the driving field (X ¼ 0), the plot is shown in Fig. 2(a). In our calculations we can decouple the driving field from the transition jci ! jdi to demonstrate the behavior of absorption spectrum. In this case, as shown in Fig. 2(b), there is a regular EIT resonance in the usual Ktype three-level system with one drive laser coupling transition jbi ! jai. Let us now turn to the situation as shown in Fig. 1. The same driving field is coupling both transitions jbi ! jai and jci ! jdi, the absorption spectrum of probe is shown in Fig. 2(c). The driving field on the transition jci ! jdi masks transparency completely, because it destroys the coherence between states jbi and jci via optically depleting the population of state jci. Moreover, the interference effect seems to be constructive, instead of transparency, that results from de-
Transmission (a.u.)
Transmission (a.u.)
structive interference. We obtain the increase of absorption at the frequency of the two-photon Raman resonance [2,7]. Thus, the presence or absence of the level jdi turns the absorption spectrum upside down (the EIT dip transforms into an enhanced absorption peak). At this point, let us note that an external magnetic field can control the coupling with level jdi via the Zeeman shifts of the energy levels. We perform numerical simulations for the case of drive detuning D ¼ 4c in corroborating this conclusion, and the results are shown in Fig. 2(d), where the restored transparency is apparently seen. To gain better understanding of the physics in this system, we examine the probe absorption for a certain velocity groups. The absorption spectra shown in Fig. 3 is plotted as a function of probe laser detuning with various atomic velocity groups. The spectra in the left column is for the case of no magnetic field, and right one with magnetic field. Fig. 3(a) shows the absorption spectra corresponding to the atoms with zero velocity kv ¼ 0, we observe the Mollow triplet [13], the maximum absorption occurs at zero detuning and small absorption at the sides. Figs. 3(b) and
(b)
(a)
(d)
(c)
20
5
0
5
5
Probe Detuning
0
(δ/γ)
5
Fig. 2. Numerically calculated the probe absorption spectra with Doppler broadening. c0 ¼ 0:01c, 2WD ¼ 100c, X ¼ c, e ¼ 0:01c, D ¼ 0: (a) two-level atom; (b) three-level K-type system; (c) four-level N-type system; (d) four-level N-type system with magnetic field on, D ¼ 4c.
229
10
0
(a)
(d)
(b)
(e)
(c)
(f)
10
20 20
10
Probe Detuning
0
(δ/γ)
10
20
Fig. 3. Numerical calculation of probe absorption spectra for atoms moving along the propagation direction of laser with three certain velocity groups: (a) and (d), kv ¼ 0; (b) and (e), kv ¼ 5c, (c) and (f), kv ¼ 5c. Left and right column correspond to the drive detuning D ¼ 0 and 5c, respectively, other parameters are: c0 ¼ 0:01c, X ¼ c, e ¼ 0:01c.
230
C.Y. Ye et al. / Optics Communications 207 (2002) 227–231
(c) show the absorption spectra of atoms with group velocities kv ¼ 5c and kv ¼ 5c, respectively. The Mollow splitting disappears, the maximum absorption locate at detuning 5c, but small absorption persists at zero detuning for both groups. If one applies an external magnetic field, which results in the drive detuning, say, D ¼ 5c, the absorption spectra corresponding to Figs. 3(a)–(c) are shown in Figs. 3(d)–(f), respsectively. Compared the left column with right one, one can see that for the zero velocity group, the maximum absorption in Fig. 3(a) turns into 100% EIT in Fig. 3(d) in the external magnetic field. The central absorption disappear and the absorption linewidth at D ¼ 5c is broadened as shown in Fig. 3(e). But Fig. 3(f) slightly changes compared with Fig. 3(c). When doing Doppler average for all velocity groups, different velocity groups make different contributions to the overall spectra, which show quite a variety in the case of with and without magnetic field, as presented in Fig. 2. The experimental implementation of the calculated results can be done in Rb vapor. The 87 Rb D2 line hyperfine energy level and relevant Zeeman sublevels involved in the coherent process are shown in Fig. 4. A single-mode drive laser couples the 5S1=2 ; ðF ¼ 2Þ to 5P3=2 ; ðF 0 ¼ 3Þ, and a weak probe is tuned around the vicinity of drive transition, as shown in Fig. 4(a). The drive and probe lasers have right circular polarization and left one. Their Rabi frequency are X and e, respectively. Fig. 4(b) displays the Zeeman sublevels and their interaction with the lasers. We consider this system as a N-type four-level
Fig. 4. (a) Schematic of energy level in 87 Rb, probe and drive laser couple the F ¼ 2 and F 0 ¼ 3 hyperfine levels of D2 line, F ¼ 1 not coupled. (b) Zeeman sublevels coupled by two lasers, probe is left circular polarized light, the drive is right circular light.
scheme, as shown in Fig. 1. jbi and jci are Zeeman sublevels of F ¼ 2 in the ground state 5S1=2 , and jai and jdi are that of F ¼ 3 in the excited state 5P3=2 with the population decay rate c. The drive laser couples the transition jbi ! jai, as well as jdi ! jci simultaneously, the probe monitors the changes of the transition jai and jci. The magnetic field is applied by solenoid coil around the Rb cell and then control the transmission of the probe laser via coherently prepared atomic vapor. The effect can find broad applications for photonic switching [14] and can be used in optical modulators. In conclusion, we have proposed a coherent control on optical properties of medium. Use of an externally applied magnetic field allows one to achieve coherent control of EIT and EIA feasibly, which suggests the potential applications in the high-speed optical switching and modulator.
Acknowledgements This work was supported by the Office of Naval Research, National Science Foundation, the Texas Advanced Research and Technology Program, and the US Air Force.
References [1] E. Arimondo, in: E. Wolf (Ed.), Progress in Optics, vol. XXXV, Elsevier, Amsterdam, 1996, p. 257; S.E. Harris, Phys. Today 50 (7) (1997) 36. [2] A.M. Akulshin, S. Barreiro, A. Lezama, Phys. Rev. A. 57 (1998) 2996; A. Lezama, S. Barreiro, A.M. Akulshin, Phys. Rev. A 59 (1999) 4732; A. Lezama, S. Barreiro, A. Lipsich, A.M. Akulshin, Phys. Rev. A 61 (1999) 013801; A. Lipsich, S. Barreiro, A.M. Akulshin, A. Lezama, Phys. Rev. A 61 (2000) 053803. [3] S.E. Harris, J.E. Field, A. Imamoglu, Phys. Rev. Lett. 64 (1990) 1107; B.S. Ham, M.S. Shahriar, P.R. Hemmer, Opt. Lett. 22 (1997) 1138; K. Hakuta, M. Suzuki, M. Katsuragawa, J.Z. Li, Phys. Rev. Lett. 79 (1997) 209; S.E. Harris, A. Sokolov, Phys. Rev. Lett. 81 (1998) 2894; A.B. Matsko, Y.V. Rostovtsev, M. Fleischhauer, M.O. Scully, Phys. Rev. Lett. 86 (2001) 2006;
C.Y. Ye et al. / Optics Communications 207 (2002) 227–231
[4] [5] [6] [7]
[8]
A.S. Zibrov, M. Lukin, M.O. Scully, Phys. Rev. Lett. 83 (1999) 4049. S.E. Harris, Y. Yamamoto, Phys. Rev. Lett. 81 (1998) 3611; B.S. Ham, P.R. Hemmer, Phys. Rev. Lett. 84 (2000) 4080. P. Valente, H. Failache, A. Lezama, Phys. Rev. A. 65 (2002) 023814. H. Schmidt, A. Imamoglu, Opt. Lett. 21 (1996) 1936; H. Schmidt, A. Imamoglu, Opt. Lett. 23 (1998) 1007. A.V. Taichenachev, A.M. Tumaikin, V.I. Yudin, Pis’ma Zh. Eksp. Teor. Fiz. 69 (1999) 776 (Sov. Phys. JETP. Lett. 69 (1999) 819). C.Y. Ye, A.S. Zibrov, Y.V. Rostovtsev, M.O. Scully, unpublished.
231
[9] C.Y. Ye, A.S. Zibrov, Y.V. Rostovtsev, M.O. Scully, Phys. Rev. A 65 (2002) 043805. [10] A.S. Zibrov, C.Y. Ye, Y.V. Rostovtsev, A.B. Matsko, M.O. Scully, Phys. Rev. A 65 (2002) 043817. [11] F. Renzoni, W. Maichen, L. Windholz, E. Arimondo, Phys. Rev. A 55 (1997) 3710. [12] F. Renzoni, A. Lindner, E. Arimondo, Phys. Rev. A 60 (1999) 450. [13] B.R. Mollow, Phys. Rev. A 188 (1969) 1969; B.R. Mollow, Phys. Rev. A 5 (1972) 2217. [14] G.I. Stegeman, A. Miller, in: J.E. Midwinter (Ed.), Photonics in Switching, Academic, San Diego, CA, 1993.
Optics Communications 207 (2002) 227–231 www.elsevier.com/locate/optcom
Quantum interference in atomic vapor controlled by a magnetic field C.Y. Ye a,*, Y.V. Rostovtsev a, A.S. Zibrov b,c, Yu.M. Golubev d a
Department of Physics and Institute for Quantum Studies, Texas A&M University, College Station, Texas 77843, USA b Department of Physics and ITAMP, Harvard University, Cambridge, MA, 02138, USA c P.N. Lebedev Institute of Physics, RAS, Leninsky pr. 53, Moscow 117924, Russia d Physics Institute, St. Petersburg University, ul. Ulyanovskaya 1, St. Petersburg 198904, Russia Received 8 March 2002; accepted 24 April 2002
Abstract We have shown that quantum interference in a driven quasi-degenerate two-level atomic system can be controlled by an externally applied magnetic field. We demonstrate that the mechanism of optical control is based on quantum interference, which allows one to implement both electromagnetically induced transparency and electromagnetically induced absorption (EIA) in one atomic system. The experimental realization is suggested. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.50.Gy; 42.50.Hz
Quantum coherence and interference play an important role in the interaction of coherent laser fields with atomic systems. As has been shown that electromagnetically induced transparency (EIT) arises from destructive interference of atomic transitions [1], while electromagnetically induced absorption (EIA) is connected with constructive interference [2]. EIT has found a wide variety of applications in quantum optics and nonlinear optical processes [3]. EIA could have potential ap-
*
Corresponding author. Tel.: +979-4581136; fax: +9798452590. E-mail address: [email protected] (C.Y. Ye).
plications to high-speed optical modulation and quantum switching [4,5]. N-type scheme has recently attracted much attention and opened a new approach to manipulate the nonlinear phenomena in optical process [6]. It has been shown that atomic coherence among Zeeman sublevels can be spontaneously transferred from upper level to lower one, which gives rise not only to EIT, but also to EIA [2,7,8]. Doppler-free resonance absorption observed in the N scheme displays another interesting features in the Doppler-broadened medium. Ye et al. [9] have demonstrated that the restriction on Doppler-free geometry can be alleviated by atomic multicoherence in the N scheme. Zibrov et al. [10] observed three-photon resonance Doppler-free in hot Rb
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 5 2 3 - 7
228
C.Y. Ye et al. / Optics Communications 207 (2002) 227–231
vapor driven with one of two coherent fields far detuning from its resonance in the N scheme. The other type of schemes have been analysed in [11,12]. Usually, the atomic system has one type of interference, either constructive or destructive, that depends on the configuration of atomic levels and laser fields. In this paper, we show a way to coherently control the type of interference in one quasi-degenerate two-level atomic system by applying an external magnetic field. The experimental implementation is suggested as well. We consider a N-type four-level scheme, as shown in Fig. 1. Levels jbi and jci are ground states with small decay rate cbc , levels jai and jdi are the excited states with the population decay rate c to ground states (c cbc ). The drive laser couples the transition jbi ! jai, as well as jdi ! jci simultaneously, the probe monitors the changes of the transition jai and jci. The magnetic field is applied to control the transmission of the probe laser via coherent atomic vapor. The Hamiltonian of the system can be written as H ¼ H0 þ HI , where hdjcihcj hðd þ DÞjdihdj þ H :c:; H0 ¼ hDjbihbj ð1Þ and HI ¼ hXðjaihbj þ jdihcjÞ hejaihcj þ H :c:
ð2Þ
Then, the density matrix equations for this system can be written as
q_ aa ¼ ð2c þ c0 Þqaa ieðqac qca Þ iXðqab qba Þ; ð3Þ q_ bb ¼ rb þ cqaa c0 qbb þ iXðqab qba Þ;
ð4Þ
q_ cc ¼ rc þ cqaa c0 qcc þ 2cqdd þ ieðqac qca Þ iXðqcd qdc Þ;
ð5Þ
q_ dd ¼ 2cqdd c0 qdd þ iXðqcd qdc Þ;
ð6Þ
q_ ab ¼ Cab qab iXðqaa qbb Þ þ ieqcb ;
ð7Þ
q_ cb ¼ Ccb qcb iXqca þ ieqab þ iXqdb ;
ð8Þ
q_ db ¼ Cdb qdb iXqda þ iXqcb ;
ð9Þ
q_ ca ¼ Cca qca þ ieðqaa qcc Þ þ iXqda iXqcb ;
ð10Þ
q_ da ¼ Cda qda ieqdc þ iXqca iXqdb ;
ð11Þ
q_ dc ¼ Cdc qdc ieqda þ iXðqcc qdd Þ;
ð12Þ
where X is the Rabi frequency of drive laser and e is that of probe; rb and rc are the incoherent pumping rate into respective state jbi and jci; c0 is the decay rate out of all states; The off-diagonal decay rates are Cij ¼ cij þ iDij , cij are decay rates of level jii and jji. The detunings of laser from their resonance frequencies Dij with Doppler shift kv are given by Dab ¼ xab m kv ¼ D kv, Dac ¼ xac m0 k0 v ¼ d kv, Ddc ¼ xdc m kv ¼ D kv; where D ¼ xab m and d ¼ xac m, xij is the atomic transition frequency from level jii and jji, m is the frequency of the drive laser, and m0 the probe laser, k and k0 are the wavenumbers of the drive and probe lasers under the assumption of m0 ¼ m. The analytical solution for coherence term qac , which gives the absorption of the probe field, can be found with some assumptions. Here, we present the numerical calculations. To take the inhomogeneous broadening into account, one should find the velocity-averaged probe absorption coefficient, which is given by Z 1 q v¼ ð13Þ dðkvÞg ac f ðkvÞ; e 1 where g ¼ 3=8pN ck3 , f ðkvÞ is the Gaussian distribution function of atomic velocities
Fig. 1. Schematic of simplified N -type four-level system. Drive laser couples to the jai-jbi and jci-jdi transitions simultaneously, probe laser to jai-jci. Radiative decay rate from jai and jdi to lower levels is 2c, coherent decay rate for jbi-jci is cbc .
2 2 1 pffiffiffi eðkvÞ =ðkuÞ ; ð14Þ ku p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where u ¼ 2kB T =M is the most probable speed of atoms at given temperature T, M is the atomic
f ðkvÞ ¼
C.Y. Ye et al. / Optics Communications 207 (2002) 227–231
mass, kB is the Boltzmann constant. The full width at half maximum of the Doppler pffiffiffiffiffiffiffiffi broadened linewidth is given by 2WD ¼ 2 ln 2ku. The calculated spectra of probe absorption as a function of the probe detuning under different conditions are shown in Fig. 2. Let us start with the study of Doppler broadened absorption spectrum of the probe laser coupled transition jci ! jai with no the driving field (X ¼ 0), the plot is shown in Fig. 2(a). In our calculations we can decouple the driving field from the transition jci ! jdi to demonstrate the behavior of absorption spectrum. In this case, as shown in Fig. 2(b), there is a regular EIT resonance in the usual Ktype three-level system with one drive laser coupling transition jbi ! jai. Let us now turn to the situation as shown in Fig. 1. The same driving field is coupling both transitions jbi ! jai and jci ! jdi, the absorption spectrum of probe is shown in Fig. 2(c). The driving field on the transition jci ! jdi masks transparency completely, because it destroys the coherence between states jbi and jci via optically depleting the population of state jci. Moreover, the interference effect seems to be constructive, instead of transparency, that results from de-
Transmission (a.u.)
Transmission (a.u.)
structive interference. We obtain the increase of absorption at the frequency of the two-photon Raman resonance [2,7]. Thus, the presence or absence of the level jdi turns the absorption spectrum upside down (the EIT dip transforms into an enhanced absorption peak). At this point, let us note that an external magnetic field can control the coupling with level jdi via the Zeeman shifts of the energy levels. We perform numerical simulations for the case of drive detuning D ¼ 4c in corroborating this conclusion, and the results are shown in Fig. 2(d), where the restored transparency is apparently seen. To gain better understanding of the physics in this system, we examine the probe absorption for a certain velocity groups. The absorption spectra shown in Fig. 3 is plotted as a function of probe laser detuning with various atomic velocity groups. The spectra in the left column is for the case of no magnetic field, and right one with magnetic field. Fig. 3(a) shows the absorption spectra corresponding to the atoms with zero velocity kv ¼ 0, we observe the Mollow triplet [13], the maximum absorption occurs at zero detuning and small absorption at the sides. Figs. 3(b) and
(b)
(a)
(d)
(c)
20
5
0
5
5
Probe Detuning
0
(δ/γ)
5
Fig. 2. Numerically calculated the probe absorption spectra with Doppler broadening. c0 ¼ 0:01c, 2WD ¼ 100c, X ¼ c, e ¼ 0:01c, D ¼ 0: (a) two-level atom; (b) three-level K-type system; (c) four-level N-type system; (d) four-level N-type system with magnetic field on, D ¼ 4c.
229
10
0
(a)
(d)
(b)
(e)
(c)
(f)
10
20 20
10
Probe Detuning
0
(δ/γ)
10
20
Fig. 3. Numerical calculation of probe absorption spectra for atoms moving along the propagation direction of laser with three certain velocity groups: (a) and (d), kv ¼ 0; (b) and (e), kv ¼ 5c, (c) and (f), kv ¼ 5c. Left and right column correspond to the drive detuning D ¼ 0 and 5c, respectively, other parameters are: c0 ¼ 0:01c, X ¼ c, e ¼ 0:01c.
230
C.Y. Ye et al. / Optics Communications 207 (2002) 227–231
(c) show the absorption spectra of atoms with group velocities kv ¼ 5c and kv ¼ 5c, respectively. The Mollow splitting disappears, the maximum absorption locate at detuning 5c, but small absorption persists at zero detuning for both groups. If one applies an external magnetic field, which results in the drive detuning, say, D ¼ 5c, the absorption spectra corresponding to Figs. 3(a)–(c) are shown in Figs. 3(d)–(f), respsectively. Compared the left column with right one, one can see that for the zero velocity group, the maximum absorption in Fig. 3(a) turns into 100% EIT in Fig. 3(d) in the external magnetic field. The central absorption disappear and the absorption linewidth at D ¼ 5c is broadened as shown in Fig. 3(e). But Fig. 3(f) slightly changes compared with Fig. 3(c). When doing Doppler average for all velocity groups, different velocity groups make different contributions to the overall spectra, which show quite a variety in the case of with and without magnetic field, as presented in Fig. 2. The experimental implementation of the calculated results can be done in Rb vapor. The 87 Rb D2 line hyperfine energy level and relevant Zeeman sublevels involved in the coherent process are shown in Fig. 4. A single-mode drive laser couples the 5S1=2 ; ðF ¼ 2Þ to 5P3=2 ; ðF 0 ¼ 3Þ, and a weak probe is tuned around the vicinity of drive transition, as shown in Fig. 4(a). The drive and probe lasers have right circular polarization and left one. Their Rabi frequency are X and e, respectively. Fig. 4(b) displays the Zeeman sublevels and their interaction with the lasers. We consider this system as a N-type four-level
Fig. 4. (a) Schematic of energy level in 87 Rb, probe and drive laser couple the F ¼ 2 and F 0 ¼ 3 hyperfine levels of D2 line, F ¼ 1 not coupled. (b) Zeeman sublevels coupled by two lasers, probe is left circular polarized light, the drive is right circular light.
scheme, as shown in Fig. 1. jbi and jci are Zeeman sublevels of F ¼ 2 in the ground state 5S1=2 , and jai and jdi are that of F ¼ 3 in the excited state 5P3=2 with the population decay rate c. The drive laser couples the transition jbi ! jai, as well as jdi ! jci simultaneously, the probe monitors the changes of the transition jai and jci. The magnetic field is applied by solenoid coil around the Rb cell and then control the transmission of the probe laser via coherently prepared atomic vapor. The effect can find broad applications for photonic switching [14] and can be used in optical modulators. In conclusion, we have proposed a coherent control on optical properties of medium. Use of an externally applied magnetic field allows one to achieve coherent control of EIT and EIA feasibly, which suggests the potential applications in the high-speed optical switching and modulator.
Acknowledgements This work was supported by the Office of Naval Research, National Science Foundation, the Texas Advanced Research and Technology Program, and the US Air Force.
References [1] E. Arimondo, in: E. Wolf (Ed.), Progress in Optics, vol. XXXV, Elsevier, Amsterdam, 1996, p. 257; S.E. Harris, Phys. Today 50 (7) (1997) 36. [2] A.M. Akulshin, S. Barreiro, A. Lezama, Phys. Rev. A. 57 (1998) 2996; A. Lezama, S. Barreiro, A.M. Akulshin, Phys. Rev. A 59 (1999) 4732; A. Lezama, S. Barreiro, A. Lipsich, A.M. Akulshin, Phys. Rev. A 61 (1999) 013801; A. Lipsich, S. Barreiro, A.M. Akulshin, A. Lezama, Phys. Rev. A 61 (2000) 053803. [3] S.E. Harris, J.E. Field, A. Imamoglu, Phys. Rev. Lett. 64 (1990) 1107; B.S. Ham, M.S. Shahriar, P.R. Hemmer, Opt. Lett. 22 (1997) 1138; K. Hakuta, M. Suzuki, M. Katsuragawa, J.Z. Li, Phys. Rev. Lett. 79 (1997) 209; S.E. Harris, A. Sokolov, Phys. Rev. Lett. 81 (1998) 2894; A.B. Matsko, Y.V. Rostovtsev, M. Fleischhauer, M.O. Scully, Phys. Rev. Lett. 86 (2001) 2006;
C.Y. Ye et al. / Optics Communications 207 (2002) 227–231
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