Gain and transmission properties of a probe field in a four-level atomic system

Optik 123 (2012) 1240–1244

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Optik journal homepage: www.elsevier.de/ijleo

Gain and transmission properties of a probe field in a four-level atomic system Zhiping Wang a,∗ , Benli Yu a , Xuqiang Wu a , Zhigang Cao a , Jun Zhu a , Shenglai Zhen a , Jun Peng b a b

Key Laboratory of Opto-Electronic Information Acquisition and Manipulation of Ministry of Education, Anhui University, Hefei 230039, China Beijing Micro-Technology Research Institute, Bejing 100200, China

a r t i c l e

i n f o

Article history: Received 22 February 2011 Accepted 16 July 2011

a b s t r a c t We theoretically investigate the absorption–amplification response of the probe field in a four-level cold 87 Rb atomic system. It is found that the incoherent pumping fields play very important roles in realizing the amplification of the probe field. The transmission property of the probe field is also studied. © 2011 Elsevier GmbH. All rights reserved.

Keywords: Optical properties Incoherent pumping fields Four-level atomic system

1. Introduction In the past few decades, a great deal of quantum optical phenomena based on coherence and quantum interference have attracted a lot of attention of many researchers in quantum optics [1–30]. Meanwhile, the absorption and amplification of light based on the quantum interference and coherence have also been extensively studied in recent years [21–28]. These studies show that one can control the absorption–amplification response of light via different approaches such as the amplitude of the driving field [23], the phases of the applied fields [24], the spontaneously generated coherence [25], the use of a low-frequency driven field [26], the incoherent pumping fields [28], and so on. In this work, we theoretically investigate the absorption– amplification response of the probe field in a four-level cold 87 Rb atomic system. It is found that the incoherent pumping fields play very important roles in realizing the amplification of the probe field. The transmission property of the probe field is also studied. Our study and the four-level system are mainly based on Refs. [21–30], however, our results are different from those papers. Firstly, we are mainly interested in studying the controllability of the absorption–amplification response of the probe field via the incoherent pumping fields. Secondly, a very important advantage of our scheme is that we provide a realistic cold 87 Rb atomic system for realizing the amplification of the probe field, which may make our scheme much more convenient in experimental realization. Thirdly, we also show the effects of the incoherent pumping fields on the transmission spectra of the probe field. To the best of

∗ Corresponding author. E-mail address: [email protected] (Z. Wang). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.08.008

our knowledge, so far no related theoretical or experimental work has been carried out to study dynamic control of light propagation through such an four-level atomic system, which motivate the current work. Our paper is organized as follows: Section 2 establishes the model, i.e., the Hamiltonian of the system and the evolution equation of the atomic operators assuming the rotating wave approximation. Section 3 is devoted to present a possible experimental realization of our scheme with cold 87 Rb atoms. In Section 4 we provide the numerical results. Finally, Section 5 provides the conclusions.

2. The model and the dynamic equations We consider the four-level 87 Rb atomic system as shown in Fig. 1(a). This system has one stable ground state |0, an intermediate level |1 and two nearly degenerate excited states |2 and |3. The ground level |0 and the intermediate level |1 are coupled by a strong coupling field ωc (amplitude Ec ) with the Rabi frequency ˝c = 10 Ec /2¯h, and 10 is the relevant dipole moment, while a weak probe field ωp (amplitude Ep ) is applied to the transitions |2 ↔ | 1 and |3 ↔ | 1 simultaneously with the respective Rabi frequencies ˝p1 = 21 Ep /2¯h and ˝p2 = 31 Ep /2¯h, and 21 , 31 are the relevant dipole moments. For the purpose of incoherent pumping fields, two broadband polarized fields 1 and 2 (can be provided by the diode laser that had a broad variable linewidth) that serve as the incoherent pumping fields and apply to the transitions |1 ↔ | 2 and |1 ↔ | 3, respectively. ω21 , ω31 , and ω10 are resonant frequencies which associates with the corresponding transitions |2 ↔ | 1, |3 ↔ | 1, and |1 ↔ | 0, and spontaneous emission rates from the levels, |1, |2 and |3 are represented by  1 ,  2 and  3 respectively. Here, we have assumed that a direct transition between the excited

Z. Wang et al. / Optik 123 (2012) 1240–1244

a

3

2ωs

Ωp

Λ2

By the straightforward semiclassical analysis, the above nonlinear density-matrix equations can be used to calculate the total complex susceptibility p of the probe transition, i.e.,

2

p =

Λ1

1

0

F=3

44 MHz

5D3/2

30 MHz

762 nm

15 MHz

5P1/2

816 MHz

F=2 F =1 F=0

(2)

F=2

∂˝p 1 ∂˝p + = i˛(21 + 31 ), c ∂t ∂z

F =1

where c is the permittivity of free space the light speed and ˛ = N |  | 2 ωp /2¯hε0 c ( =  2 =  3 ) is the propagation constant of the probe field, respectively. In the linear regime, for given ˝pin at the atomic medium input z = 0 we can easily arrive at the steady-state probe field ˝pout at the output z = L

795 nm F=2 6.834 GHz

5S1/2

N||2 (21 + 31 ) ∝ 21 + 31 , 2¯hε0 ˝p

here we assume 21 = 31 = , N is the atomic number density in the medium and ε0 is the permittivity of free space, respectively. It is well known that the absorption and dispersion are related to the susceptibility of the system. Therefore, the probe absorption–dispersion response coupled to the transitions |2 ↔ | 1 and |3 ↔ | 1 is proportional to the terms 21 + 31 . The imaginary parts Im(21 + 31 ) correspond to absorption–amplification response, yet the real parts Re(21 + 31 ) correspond to dispersion response. If Im(21 + 31 ) > 0, the probe field will be absorbed; on the contrary, the probe field will be amplified. In order to describe correctly the propagation effect of the probe laser field, the optical Bloch equations for atoms must be simultaneously solved with Maxwell’s wave equation in a selfconsistent manner. In the slowly-varying-envelope approximation (SVEA), the Rabi frequency ˝p of the probe laser field along the propagating direction of “z” obeys the following Maxwell’s wave equation

Ωc

b

1241



F =1

˝pout

Fig. 1. (a) Schematic diagram of a four-level atomic system. (b) Level hyperfine structure and the laser-coupling scheme for a four-level 87 Rb atomic system.

states |2 ↔ | 3 and that between the excited and ground states |2, |3 ↔ | 0 of the atom are forbidden in the dipole approximation. By adopting the standard approach [31], the density-matrix equations of motion under the electric dipole and rotating-wave approximations for this system can be written as follows

=

˝pin

exp −˛L Im



21 + 31 ˝p

(3)



,

(4)

where L denotes the length of the atomic sample. Then the normalized transmission coefficient of the probe laser field for the transitions |2 ↔ | 1 and |3 ↔ | 1 can be derived from expression (4) as Tp =

˝pout ˝pin

.

(5)

i˙ 22 = ˝p (21 − 12 ) + i1 11 − i2 22 , i˙ 33 = ˝p (31 − 13 ) + i2 11 − i3 33 ,

3. Possible experimental realization

i˙ 00 = ˝c (01 − 10 ) + i1 11 ,



i˙ 01 = ˝p 02 + ˝p 03 + ˝c (00 − 11 ) − i˙ 02 = ˝p 01 − ˝c 12 − (i

2

 23

i˙ 03 = ˝p 01 − ˝c 13 − i

2

i





i˙ 13 = ˝p (11 − 33 )−˝c 03 − ˝p 23 − i

Before showing our results, let us briefly discuss the possible experimental realization of our proposed scheme for the present study, which are given as follows

01 ,

− p − c + ωs 03 ,



i˙ 23 = ˝p 21 − ˝p 13 +



− p − c − ωs )02 ,

i˙ 12 = ˝p (11 − 22 ) − ˝c 02 − ˝p 32 −



1 + 1 + 2 − c 2

2 + 3 + 2ωs 2

i

1 + 2 + 1 +2 i − p − ωs 2 1 + 3 + 1 + 2 − p + ωs 2



(1)

 

12 , 13 ,

23 ,

∗ (m, n = 1, 2, 3, 4) and constrained by nm = mn 11 + 22 + 33 + 44 = 1. Here we assume ˝p1 = ˝p2 = ˝p and ˝c are real. p = ωp − (1/2)(ω21 + ω31 ) and c = ωc − ω10 are the detunings of probe and coupling fields, respectively. 2ωs is the energy separation between the states |3 and |2. For Eq. (1), it should be highlighted that we properly arrange the polarizations of the two incoherent pumping fields ε 2 and ε 1 in such a way  21 = 0 and ε 1 ·   31 = 0, thus one field acts on only one that ε 2 ·  transition so that the interference terms that may be induced by incoherent pumps are not included in the density matrix equations [32].

(i) We consider, for instance, the cold atoms 87 Rb (nuclear spin I = 3/2) on the 5S − 5P − 5D transitions as a possible candidate [29]. The detailed coupling diagram is shown in Fig. 1(b). The experimental system for this atomic scheme can be realized by the 87 Rb atom with |5S1/2 , F = 2, |5P1/2 , F = 2, |5D3/2 , F = 1, and |5D3/2 , F = 2 behaving the |0, |1, |2, and |3 state labels, respectively. (ii) Based on Ref. [30], we have found the decay rates from the excited state |1 to the ground state |0, from the excited state |2 to the excited state |1, and from the excited state |3 to the excited state |1, are  1  5.3 MHz,  3  0.67 MHz, and  3  0.67 MHz, respectively. It should also be pointed out that decay rates from states |2 and |3 to state |0 are zero, because these transitions are non-dipole allowed in our considered model. The relaxation rate of coherence among states |2 and |3 is negligible and thus can be safely neglected. (iii) Following Ref. [29], the probe field can be a linearly polarized optical wave, while the coupling field can be a left circularly

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Z. Wang et al. / Optik 123 (2012) 1240–1244

polarized optical wave. The incoherent pumping field can be provided by the diode laser that had a broad variable linewidth.

a

0.05 Λ1=γ, Λ 2=0 Λ1=0, Λ2=γ

4. Numerical results Now we show some results of numerical studies, as shown in Figs. 2–6. In the following numerical calculations, we choose the parameters to be units by scaling  = 0.67 MHz and the choices of the parameters are mainly based on results from Refs. [29,30]. In Fig. 2, we show the dependence of the gain-absorption property of the probe field on the intensity of the strong coupling field ˝c . For a small ˝c = 2, as shown in Fig. 2(a), there are only two peaks in the gain curve, and the value of probe gain near p = 0 is equal to zero. We then increase intensity of the coupling field ˝c to 10 (Fig. 2(b)) and find that the number of peaks becomes four. We can see that the value of probe gain near p = 0 is not equal to zero. When the intensity of the strong coupling field ˝c is

Probe Gain

a

0.06

Probe Gain

0.02

-0.01 -100 -80

b

40

60

80

100

0

20

40

60

80

100

-0.01 -0.02

Λ1=γ, Λ2=0 Λ1=Λ 2=2γ Λ1=Λ 2=5γ

-50

0

50

0.04 0.02

-50

0

50

-60

-40

100

Ω c =10γ

100

Δ p/γ

Probe Gain

20

Λ1=0, Λ2=γ

0.06

0.06

0

Δp/γ

0

-0.04 -100 -80

0.08

-20

0.01

0.02

c

-40

0.03

-0.03

0 -100

-60

0.02

0.04

Δ p/γ

b

Λ1=Λ2=5γ

0.03

0

Ω c =2γ

0 -100

Λ1=Λ2=2γ

0.01

Probe Gain

According to the above conditions, we believe that the experimental scientists have adequate wisdom to realize our scheme.

Probe Gain

0.04

Ω c =22γ

-20

Δp/γ

Fig. 3. The probe gain versus the frequency detuning of probe field p for different values of 1 and 2 , (a) ˝c = 2 and (b) ˝c = 22. The other parameters are ωs = 22.4, c = 0,  1 = 8 2 = 8 3 = 8, and ˝p = 0.5.

increased more, for example ˝c = 22 (Fig. 2(c)), the value of probe gain at p = 0 becomes very large and appears a three-peak spectral feature (shown in Fig. 2(c)). Then when the intensity of the strong coupling field ˝c is varied very strong, for the case that ˝c = 40 (Fig. 2(d)), there appear four peaks in the gain curve again, and the value of probe gain at p = 0 is suppressed by comparing with the case in Fig. 2(c). According to the above discussion, one can realize that the intensity of the coupling field can affect gain curves dramatically, however, the value of probe gain Im(21 + 31 ) is always positive, so the probe field is absorbed. This suggests that we can

0.04 0.02

0.1 Λ1=Λ2=0

Probe Gain

d

-50

0 Δ p/γ

0.06

50

100

Ω c =30γ

0.04

Λ1=Λ2=2γ

0.08

Λ1=Λ2=5γ

Probe Gain

0 -100

0.06 0.04 0.02 0

0.02

-0.02

0 -100

-50

0 Δ p/γ

50

100

Fig. 2. The probe response versus the frequency detuning of probe field p for four different intensities of coupling field ˝c , (a) ˝c = 2, (b) ˝c = 10, (c) ˝c = 22, and (d) ˝c = 40. The other parameters are ωs = 22.4, 1 = 2 = 0, c = 0,  1 = 8 2 = 8 3 = 8, and ˝p = 0.5.

-0.04 0

10

20

30

Ω c/γ

40

50

60

Fig. 4. The probe gain as a function of the intensity of coupling field ˝c with different values of 1 and 2 . The other parameters are ωs = 22.4, p = 0, c = 0,  1 = 8 2 = 8 3 = 8, and ˝p = 0.5.

Z. Wang et al. / Optik 123 (2012) 1240–1244 1.4

Transmission spectra Tp

1.3 1.2 1.1 1 0.9 0.8 0.7 Ω c =2γ

0.6

Ω c =22γ

0.5 0.4 0

Ω c =30γ

5

10

15

20

25

30

35

40

45

50

Λ 1/γ (Λ 1=Λ 2) Fig. 5. Transmission spectra of the probe field as a function of the  with different values of ˝c . The other parameters are ωs = 22.4, p = 0, c = 0, ˛L = 5,  1 = 8 2 = 8 3 = 8, and ˝p = 0.5.

not realize the amplification of the probe field just by tuning the intensity of the coupling field. The influences of the two incoherent pumping fields 1 and 2 on probe gain are given in Fig. 3. We first consider the situation that ˝c is small (shown in Fig. 3(a)). Obviously, when we apply the incoherent pumping field 1 to the transition |1 ↔ | 2 (solid red line in Fig. 3(a)) or apply the incoherent pumping field 2 to the transition |1 ↔ | 3 (dashed blue line in Fig. 3(a)), the probe gain appears negative value even if the intensity of coupling field ˝c is weak. (For interpretation of the references to color in this figure citation, the reader is referred to the web version of this article.) When both 1 and 2 are switched on, the increasing rates of the two incoherent pumping fields lead to the increasing value of probe gain (compare the dotted green line and dashed-dotted black line shown in Fig. 3(a)). In the following, we will analyze the situation that ˝c is strong (shown in Fig. 3(b)). The amplification of the probe field can easily be achieved. Two remarkable differences are that the profiles of gain curves change and the value of probe gain at p = 0 becomes very large. In order to test the validity of the analysis described above, we carry out the probe gain as a function of the intensity of coupling field ˝c with different values of 1 and 2 in Fig. 4, as already verified the above result. Based on the above analysis, we can conclude that the two incoherent pumping fields

1.3

1.4

a

1.3

1.2

Transmission spectra Tp

Transmission spectra Tp

1 0.9 0.8 0.7

1.1 1 0.9 0.8 0.7

Λ 1=Λ 2=0 0.6

Λ 1=Λ 2=γ

0.6 0.5 -50

= Λ 2=5γ Λ 10.5 0 Δ p/γ

50

0.4 -100

1 and 2 play important roles in realizing the amplification of the probe field. At last, we display the transmission properties of the probe field propagating through the atomic medium in Figs. 5 and 6. In Fig. 5, we plot transmission spectra of the probe field as a function of the 1 (1 = 2 ) with different values of ˝c . It can clearly be shown from the figure that the probe transmission spectrum Tp > 1 with the increasing rates of the two incoherent pumping fields. Specifically, for the case when the ˝c = 22, the value of probe transmission spectrum Tp nearly reaches 1.4. Of course, too large ˝c will have a destructive effect on the probe transmission spectrum (compare the blue line and green line shown in Fig. 5). In order to further illustrate explicitly the dependence of the transmission spectra of the probe field on the coupling field ˝c and two incoherent pumping fields, the transmission spectra of the probe field as a function of the frequency detuning of probe field p are plotted in Fig. 6 for two different values of the coupling field intensity: ˝c = 2 (panel (a)) and ˝c = 22 (panel (b)). At the line center of the probe transmission spectrum p = 0, the probe transmission spectrum Tp > 1 when the two incoherent pumping fields are switched on, and a high transmission can take place when the coupling field ˝c and two incoherent pumping fields are tuned to reasonable values (see, for example, the blue line shown in Fig. 5). These results are good agreement with the probe gain-absorption spectra given in Fig. 3(a) and (b). 5. Conclusions To sum up, we have investigated the absorption–amplification response of the probe field in a four-level cold 87 Rb atomic system. The system interacts with a weak probe field, a strong control field and two incoherent pumping fields. By numerically solving the coupled Bloch–Maxwell equations, we have studied the propagation dynamics of a weak probe field in such a four level atomic medium, where the incoherent pumping fields play very important roles in realizing the amplification of the probe field. We anticipate that this can be used as a kind of optical modulation in the cold atomic medium. Acknowledgment This work has been supported by the Doctoral Fund of Ministry of Education of China under Grant no. 20093401110002. References

b

1.2 1.1

1243

-50

0

50

100

Δ p/γ

Fig. 6. Transmission spectra of the probe field as a function of the frequency detuning of probe field p with different values of 1 and 2 , (a) ˝c = 2 and (b) ˝c = 22. The other parameters are ωs = 22.4, c = 0, ˛L = 5,  1 = 8 2 = 8 3 = 8, and ˝p = 0.5.

[1] M.O. Scully, M. Fleischhauer, Lasers without inversion, Science 263 (1994) 337–338. [2] M. Xiao, Y.Q. Li, S.Z. Jin, J. Gea-Banacloche, Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms, Phys. Rev. Lett. 74 (1995) 666–669. [3] Y. Wu, X. Yang, Electromagnetically induced transparency in V-, -, and cascade-type schemes beyond steady-state analysis, Phys. Rev. A 71 (2005), 053806/1–7. [4] S.E. Harris, Electromagnetically induced transparency, Phys. Today 50 (1997) 36–42. [5] G.R. Welch, M.D. Lukin, Y. Rostovtsev, E.S. Fry, M.O. Scully, Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas, Phys. Rev. Lett. 82 (1999) 5229–5232. [6] M. Fleischhauer, A. Imamoglu, J.P. Marangos, Electromagnetically induced transparency: optics in coherent media, Rev. Mod. Phys. 77 (2005) 633–673. [7] D.F. Phillips, A. Fleischhauer, A. Mair, R.L. Walsworth, M.D. Lukin, Storage of light in atomic vapor, Phys. Rev. Lett. 86 (2001) 783–786. [8] A. Lezama, A.M. Akulshin, A.I. Sidorov, P. Hannaford, Storage and retrieval of light pulses in atomic media with “slow” and “fast” light, Phys. Rev. A 73 (2006), 033806/1–5. [9] Y. Wu, X. Yang, Four-wave mixing in molecular magnets via electromagnetically induced transparency, Phys. Rev. B 76 (2007), 054425/1–6. [10] Y. Zhang, U. Khadka, B. Anderson, M. Xiao, Temporal and spatial interference between four-wave mixing and six-wave mixing channels, Phys. Rev. Lett. 102 (2009), 013601/1–4.

1244

Z. Wang et al. / Optik 123 (2012) 1240–1244

[11] L. Deng, M.G. Payne, W.R. Garrett, Effects of multi-photon interferences from internally generated fields in strongly resonant systems, Phys. Rep. 429 (2006) 123–241. [12] Y.P. Niu, S.Q. Gong, Enhancing Kerr nonlinearity via spontaneously generated coherence, Phys. Rev. A 73 (2006), 053811/1–6. [13] Y. Wu, L. Deng, Ultraslow optical solitons in a cold four-state medium, Phys. Rev. Lett. 93 (2004), 143904/1–4. [14] G. Huang, L. Deng, M.G. Payne, Dynamics of ultraslow optical solitons in a cold three-state atomic system, Phys. Rev. E 72 (2005), 016617/1–10. [15] A. Joshi, M. Xiao, Optical multistability in three-level atoms inside an optical ring cavity, Phys. Rev. Lett. 91 (2003), 143904/1–4. [16] A. Chen, Z. Wang, D. Chen, Optical bistability and multistability via atomic coherence in the quasi--type atomic system, Sci. Chi. Ser. G 52 (2009) 524–528. [17] Z. Wang, M. Xu, Control of the switch between optical multistability and bistability in three-level V-type atoms, Opt. Commun. 282 (2009) 1574–1578. [18] M.O. Scully, S.Y. Zhu, A. Gavrielides, Degenerate quantum-beat laser: lasing without inversion and inversion without lasing, Phys. Rev. Lett. 62 (1989) 2813–2816. [19] Y.F. Zhu, A.I. Rubiera, M. Xiao, Inversionless lasing and photon statistics in a V-type atomic system, Phys. Rev. A 53 (1996) 1065–1071. [20] S.E. Harris, Lasers without inversion: interference of lifetime-broadened resonances, Phys. Rev. Lett. 62 (1989) 1033–1036. [21] X.M. Hu, J.S. Peng, Dynamically irreversible channels of population transfer in coherently driven  systems, Opt. Commun. 170 (1999) 259–263.

[22] S.Q. Gong, Y. Li, S.D. Du, Z.Z. Xu, Unexpected population inversion via spontaneously generated coherence of a  system, Phys. Lett. A 259 (1999) 43–48. [23] H. Kang, L. Wen, Y. Zhu, Normal or anomalous dispersion and gain in a resonant coherent medium, Phys. Rev. A 68 (2003), 063806/1–5. [24] J.H. Wu, J.Y. Gao, Phase control of the probe gain without population inversion in a four-level V model, J. Opt. Soc. Am. B 19 (2002) 2863–2866. [25] J.H. Wu, H.F. Zhang, J.Y. Gao, Probe gain with population inversion in a fourlevel atomic system with vacuum-induced coherence, Opt. Lett. 28 (2003) 654–656. [26] B.P. Hou, S.J. Wang, W.L. Yu, W.L. Sun, Control of one- and two-photon absorption in a four-level atomic system by changing the amplitude and phase of a driving microwave field, J. Phys. B 38 (2005) 1419–1434. [27] X.L. Zhou, X.D. Yang, Effects of amplitude and phase of microwave field on single and two-photon absorption properties in a four-level -type atom, Phys. Lett. A 343 (2005) 20–26. [28] Y. Zhu, Light amplification mechanisms in a coherently coupled atomic system, Phys. Rev. A 55 (1997) 4568–4575. [29] D. Steck, 87 Rb D line data, http://steck.us/alkalidata. [30] M. Yan, E.G. Rickey, Y. Zhu, Electromagnetically induced transparency in cold rubidium atoms, J. Opt. Soc. Am. B 18 (2001) 1057–1062. [31] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, 1997. [32] W.-H. Xu, J.-H. Wu, J.-Y. Gao, Gain with and without population inversion via vacuum-induced coherence in a V-type atom without external coherent driving, J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 1461–1471.