Control of Group Velocity via Spontaneous Generated Coherence and Kerr Nonlinearity

Transcription

1 Commun. Theor. Phys. 62 (2014) Vol. 62, No. 3, September 1, 2014 Control of Group Velocity via Spontaneous Generated Coherence and Kerr Nonlinearity Hazrat Ali, 1 Iftikhar Ahmad, 1 and Ziauddin 2,3, 1 Department of Physics, University of Malakand, Khyber Pakhtunkhwa, Pakistan 2 Department of Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan 3 Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 300, China (Received March 24, 2014; revised manuscript received May 19, 2014) Abstract A four-level N-type atomic medium is considered to study the effect of spontaneous generated coherence (SGC) and Kerr nonlinearity on light pulse propagation. A light pulse is propagating inside the medium where each atom follows four-level N-type atom-field configuration of rubidium ( 85 Rb) atom. The atom-field interaction leads to electromagnetically induced transparency (EIT) process. The atom-field interaction is accompanied by normal dispersion and in the presence of SGC and Kerr nonlinearity the dispersion property of the proposed atomic medium is modified, which leads to enhancement of positive group index of the medium. The enhancement of positive group index then leads to slow group velocity inside the medium. A more slow group velocity is also investigated by incorporated the collective effect of SGC and Kerr nonlinearity. The control of group velocity inside a four-level N-type atomic medium via collective effect of SGC and Kerr nonlinearity is the major part of this work. PACS numbers: An, Gy, k Key words: Kerr nonlinearity, spontaneous generated coherence and group index 1 Introduction Atomic coherence plays an important role in modification and controlling the optical properties of quantum systems in atom-field interaction. The atomic coherence can be generated in quantum systems by using the coherent laser fields. It has potential applications, these include for example, the modification of light pulse propagation through the dispersive medium i.e., sub-luminal pulse propagation. The slow light has a key role in quantum information processing and in all-optical communication system. Considerable interest has been shown in investigating the behavior of light propagation through a dispersive atomic medium in the last decade. It has been noticed that via manipulation of the group index of the medium the group velocity of the light pulses can be made faster or slower than the velocity of light in vacuum. In specific classes of atomic and solid state materials slow light has been noticed using different schemes these include, for example, a technique using EIT. [1] Similarly, in 1995 it was observed that rubidium atoms could be used to obtain slow light by decreasing the absorption using an EIT configuration. [2] The EIT configuration has further been used that light pulses could be halted and frozen in a magnetically trapped cold cloud of sodium atoms via control of the coupling laser [3] and in a coherently driven Doppler broadened atomic medium, [4] respectively. Slow or halted light has a direct application in optical memory ziauddin@comsats.edu.pk c 2014 Chinese Physical Society and IOP Publishing Ltd where the optical information could be stored in an atomic medium. [3,5 7] In addition, Kerr nonlinearity plays an important role in control of slow light propagation. The Kerr nonlinearity is the dispersive part of non-linear susceptibility in optical media and is essential for nonlinear optical processes. [8] The Kerr nonlinearity can be used to obtain quantum state teleportation, quantum nondemolition measurement, quantum logic gates, quantum communication, quantum state engineering, physical mechanism of realistic Kerr nonlinearity and nonlinear control of light. [9 24] A great interest has also been given to the phenomena of EIT in highly resonant optical medium. [25] The light pulse propagation in EIT atomic medium exhibits the reduction of group velocity and enhancement of Kerr nonlinearity. The ultra-slow optical solitons in EIT atomic media have been examined [26 29] based on these characters. Besides, Dey and Agarwal studied the observable effects of Kerr nonlinearity on slow light using a four-level N-type system under the condition of EIT in [30] In this report, it was noticed that the Kerr effect can make a very significant contribution to the group velocity. The idea was based on an earlier work [31] where it has been demonstrated that the Kerr nonlinearities could be enhanced using EIT. In an experiment using ultra cold gas of sodium atoms [5] large Kerr nonlinearity has been observed in a Bose condensate. It has been observed that ultra-slow optical solitons can be achieved by using kerr

2 No. 3 Communications in Theoretical Physics 411 nonlinearity. [32 33] Further, the enhancing Kerr nonlinearity via SGC is reported in [34] SGC depends on quantum interference of spontaneous emissions between two channels. The quantum interference effect has been noticed earlier in three-level Λ-type system, where the spontaneous emissions interfere from a single excited state to two lower closely spaced levels. [35] Similarly, in V-type atomic system the spontaneous emissions interfere from two closely spaced upper levels to a common ground state. [36] Actually, this coherence based on nonorthogonality of the two transition dipole moments. More recently, The control of group velocity has been noticed in different atomic media via a Kerr field. [37 39] In their work, it has been noticed that a Kerr nonlinearity enhanced the group index of the medium, which leads to slow group velocity inside the medium. Now it will be more constructive to study the control of group velocity using the collective effect of SGC and Ker field. In this article, we study the light pulse propagation inside a medium via Kerr nonlinearity and SGC. Each atom of a medium consist of N-type atomic configuration of 85 Rb. A Kerr field enhance the group index which leads to slow group velocity as noticed earlier. [30,37 39] We also investigate the individual effect of SGC on group velocity. The important and major part of this article is to study the control of group velocity via the collective effect of SGC and Kerr field. 2 Model The energy-level configuration of the atom-field interaction are presented in Fig. 1. We consider a realistic four-level N-type atomic system of rubidium atoms ( 85 Rb) each having energy level a, b, c, and d. An intense driving laser field E 1 is applied between level a and b with corresponding Rabi frequency Ω 1. Similarly, a weak probe field E p and a strong Kerr field E k are applied between b c and c d with corresponding Rabi frequency Ω p and Ω k, respectively. Here, γ 1 and γ 2 are the spontaneous decay rates of the excited level b to the ground levels a and c. For generation of SGC the two lower levels a and c must be closely spaced, it is due to the fact that the two transitions of the excited state interact with the same vacuum mode. The interaction picture Hamiltonian for this simplified atom-field system, under the rotating-wave and dipole approximation, can therefore be written as V = [( 1 p ) c c + 1 b b + ( 1 p + k ) d d + Ω 1 a b + Ω p c b + Ω k c d ], (1) where p, 1, and k are the corresponding probe and driving field detunings, respectively. We consider that the driving laser field E 1 and E k are strong fields while the probe field E p is a weak field which means Ω 1 and Ω k Ω p. Now the corresponding rate equations can be written as ρ bc = [i p γ 1 γ 2 ]ρ bc + iω 1 ρ ac + iω p (ρ cc ρ bb ) iω k ρ bd, ρ ac = [ i( 1 p ) Γ 1 ]ρ ac + iω 1 ρ bc iω p ρ ab iω k ρ ad + 2q γ 1 γ 2 ρ bb, ρ bd = i( p k )ρ bd + iω 1 ρ ad + iω p ρ cd iω k ρ bc, ρ ad = [ i( 1 p + k ) γ 2 ]ρ ad + iω 1 ρ bd iω 2 ρ ac, ρ bb = ( γ 1 γ 2 )ρ bb + iω 1 ρ ab iω 1 ρ ba, ρ ab = ( i 1 γ 1 )ρ ab + iω 1 ρ bb, ρ ba = (i 1 γ 1 )ρ ba iω 1 ρ bb, (2) where γ 1, γ 2, and γ 3 are the decay rates as shown in Fig. 1 whereas Γ is the forbidden decay rate between level a and c. Fig. 1 (Color online) (a) Schematics of the four-level N-type rubidium atomic system (b) dipole moments of driving and probe fields. Here, the parameter q denotes the alignment of two dipole moments µ ba and µ bc. For the orientations of the atomic dipole moments µ ba and µ bc the effect of SGC is very sensitive. The parameter q may further be defined as q = µ ba. µ bc / µ ba. µ bc = cosθ arises due to the quantum interference between two decay channels b a and b c, whereas θ is the angle between the two dipole moments. The term q γ 1 γ 2 represents the quantum in-

3 412 Communications in Theoretical Physics Vol. 62 terference resulting from the cross coupling between the spontaneous emission channels b a and b c. In fact, the parameter q represents the strength of the interference in spontaneous emission. If the two dipole moments are orthogonal to each other then q = 0, which clearly shows that there is no quantum interference due to spontaneous emission. When the two dipole moments are parallel to each other then the quantum interference is maximal and q = 1. The control of alignment of two dipole moments depends on the angle θ between them. The angle θ may be adjusted with the help of external driving (E 1 ) and probe (E p ) fields, see Fig. 1(b). It is due to the fact that the probe and control fields do not interact with each other s transitions so that one must be perpendicular to the dipole moment coupled to the other. [40] So the quantum interference effect can be adjusted by control the alignments of two dipole moments. The driving (Ω 1 ) and probe (Ω p ) fields associated to the angle θ and therefore we can write as Ω 1 = Ω 0 1sinθ = Ω1 0 1 q2. Now the response of the medium due to the electric field E p is related to the polarization P via the relation P = χɛ 0 E p, (3) where P = 2Nµ bc ρ bc and E p = Ω p /µ cb. After simplification the expression for the dielectric susceptibility χ for the given atom-field system as shown in Fig. 1 can be written as χ = 2N µ bc 2 ɛ 0 Ω p ρ bc, (4) where N is the atom number density whereas µ bc and ρ bc are the dipole matrix and off-diagonal density matrix elements, respectively, for the corresponding optical transition. It is clear form Eq. (4) that we need the density matrix element ρ bc. It can be calculated by considering certain perturbation approximations, i.e., first order perturbation in the probe field and all orders in the driving and Kerr fields. Following the perturbation approximations, we consider that the driving field E 1 and Kerr field E k are strong while the probe field E p is weak which implies that Ω 1 and Ω k are much greater than Ω p. Here, the zeroth order solution of the probe field elements are equal to zero except ρ 0 cc = 1, it is due to the fact that the atom is initially prepared in the ground state c. The above Eq. (2) can be solved easily following the recipe discussed in the Appendix to get ρ bc with resonance condition i.e., 1 = k = 0 ρ bc = [(iγ + p)(iγ 3 p + 2 p Ω 2 1) p Ω 2 k ]Ω p, (5) B where B = γ 2 γ 3 2 p iγ 2 3 p iγ 3 3 p 4 p + iγ 2 p Ω iγ 3 p Ω pω 2 1 Ω [iγ 2 p + iγ 3 p + 2( 2 p + Ω 2 1)]Ω 2 k Ω 4 k + Γ[(γ 1 + γ 2 i p ) (iγ 3 p + 2 p Ω 2 1) (γ 3 i p )Ω 2 k] + γ 1 p [γ 3 p i( 2 p Ω 2 1 Ω 2 k)]. (6) Using Eqs. (4) and (5) we can write the optical susceptibility for the atom-field interaction can be written as χ = β (iγ + p)(iγ 3 p + 2 p Ω 2 1) p Ω 2 k, (7) B where β = 2N µ bc 2 / ɛ 0. We consider Ω k is a Kerr field, then the effect of Kerr field Ω k on the susceptibility can be studied by the following expression [30,37 39] χ (k) = χ (0) + Ω 2 kχ (1), (8) where χ (0) is the susceptibility of the medium without Kerr nonlinearity and can be calculated as χ (0) (Γ i p ) = γ 1 p + γ 2 p i 2 p + Γ(iγ 1 + iγ 2 + p ) + iω 2. (9) 1 The second part in Eq. (8) shows the nonlinear part of the susceptibility via Kerr field Ω k, where χ (1) can be calculated as χ (1) = χ = Γ2 (iγ 3 + p ) p (iγ 3 p + 2 p + 3Ω 2 1) + 2Γ(γ 3 p i( 2 p + Ω 2 1)) Ωk =0 (iγ 3 p + 2 p Ω 2 1 )[Γ(γ 1 + γ 2 i p ) iγ 1 p iγ 2 p 2 p + Ω 2. (10) 1 ]2 Ω 2 k The group index of the medium can be calculated using the expression as g = c/v g where c and v g be the speed of light and the group velocity, respectively, can therefore be calculated using the expression [ χ g = 1 + 2πRe[χ (k) (k) ] ] + 2πν p Re, (11) p where ν p is the frequency of the probe field. 3 Result and Discussion The enhancement of group index of a medium has been studied earlier [30,37 39] via strength of Kerr nonlinearity. The enhancement of group index of a medium then leads to slow group velocity inside the medium. In the following we start our discussion by studying the control of group velocity via Kerr nonlinearity and SGC when a light pulse is propagating inside a medium. Initially, we study the control of group velocity inside a medium via SGC and then by Kerr nonlinearity. Further, we consider the collective effect of SGC and Kerr field on group velocity in four-level N-type system. We also consider that the atoms are initially prepared in level c and the corresponding parameters are γ = 1 MHz, γ 1 = γ 2 = γ 3 = 1γ, Γ = 0.002γ, Ω 0 1 = 4γ, and ν p = 1000γ.

4 No. 3 Communications in Theoretical Physics 413 Fig. 2 (Color online) Plots of (a) real (solid) and imaginary (dashed) parts of the susceptibility χ (0) (b) group index n (0) g versus probe field detuning p for q = 0, γ = 1 MHz, γ 1 = γ 2 = γ 3 = 1γ, Γ = 0.002γ, Ω 0 1 = 4γ, and ν p = 1000γ. Fig. 3 (Color online) Plots of (a) real (solid) and imaginary (dashed) parts of the susceptibility χ (0) (b) group index n (0) g versus probe field detuning p for q = 0.99, the remaining parameter remain the same as in Fig. 2. Fig. 4 (Color online) Plots of (a) real (solid) and imaginary (dashed) parts of the susceptibility χ (k) (b) group index g versus probe field detuning p for q = 0 and Ω k = 2γ, the remaining parameters remain the same as in Fig. 2. We investigate the effect of SGC on group velocity when a light pulse is propagating inside a medium. We study the group velocity in the absence of Kerr field i.e., Ω k = 0, then the system becomes a simple three-level Λ configuration. In Fig. 2(a) we plot the real and imaginary parts of susceptibility χ (0) versus probe field detuning without considering SGC i.e., q = 0. The plot shows a normal dispersion (subluminal behavior) which has been noticed earlier for EIT process. [41 42] In this plot we consider q = 0, which means that there is no quantum interference between spontaneous emission channels. We also plot the group index versus probe field detuning and at resonance condition we calculate the group index n (0) g = 393. Next, we increase the value of q from 0 to 0.99 and again we plot the susceptibility χ (0) and group index n (0) g versus probe field detuning, see Fig. 3. A strong quantum interference effect of the spontaneous emission channels occurs for the maximum value of q. In this time we get a steep dispersion along with narrow EIT window as shown in Fig. 3(a) which has been noticed earlier. [34,43] As EIT window depends on stark splitting, which is directly related to the control or driving field. When the strength of the control field increases the width of EIT window also increases and vice versa. Now it is obvious from the relation Ω 1 = Ω1 0 1 q2 that q can affect the control field Ω 1. If the value of q increases the control field decreases and the EIT window also decreases. So at high value i.e., q = 0.99 the width of the EIT window decreases. For the maximum value of q we also study the group index of the medium and at resonance condition and calculate n (0) g =

5 414 Communications in Theoretical Physics Vol. 62 Fig. 5 (Color online) Plots of (a) real (solid) and imaginary (dashed) parts of the susceptibility χ (k) (b) group index g versus probe field detuning p for q = 0.99 and Ω k = 2γ, the remaining parameters remain the same as in Fig. 2. Fig. 6 (Color online) Plots of (a) group index versus Kerr field for q = 0 and p = 0 (b) group index versus Kerr field for p = 0 and q = 0.99, the remaining parameters remain the same as in Fig. 2. Now we switch on a Kerr field Ω k and investigate the light pulse propagation inside the medium. We consider initially the value of q = 0, which shows that there is no quantum interference effect of the spontaneous emission channels. We plot the susceptibility χ (k) and group index using Eqs. (8) and (11), respectively. The dispersion as well as the group index increases with increasing the Kerr field, see Fig. 4. We consider the Kerr field Ω k = 2γ and calculate the group index at resonance condition i.e., g = 687. This clearly tells us that the Kerr field enhances the group index, which leads to slow group velocity. Similar effect of Kerr field on the dispersion and group index has been investigated earlier in four-level N-type atomic medium. [37] Further, it will be more appropriate to study the collective effect of SGC and Kerr field on control of group velocity. As discussed previously that the individual effect of SGC and Kerr field control the group velocity when a light pulse is propagating inside the medium. In our proposed atomic configuration both SGC and Kerr field are present then it is more constructive to study the collective effect of these phenomena on the slow light propagation. We are expecting a more slow group velocity inside a medium as compared to the individual effect of SGC and Kerr field. We consider that there is a strong quantum interference effect i.e., q = 0.99 and the Kerr field Ω k = 2γ. We plot the real and imaginary parts of susceptibility χ (k) and group g index g versus probe field detuning, see Fig. 5. Due to the collective effect of SGC and Kerr field the normal dispersion (solid curve) becomes more and more steep as compared to the individual effect, see Fig. 5(a). We also plot the group index as shown in Fig. 5(b), at resonance condition we calculate the group index i.e., g = This enhancement of group index then leads to more slow group velocity inside the medium. To study the enhancement of group index of the medium with increasing the strength of the Kerr filed Ω k, we plot the group index g versus strength of the Kerr field. In Fig. 6(a), the plot shows the group index g versus strength of a Kerr field by considering no quantum interference effect present i.e., q = 0, here we notice that the group index of the atomic medium becomes more positive for the normal dispersion with an increase in the strength of a Kerr field. Next, we consider a strong quantum interference effect in our system i.e., q = 0.99 and plot the group index g of the medium versus the strength of Kerr field and investigate similar behavior as we examined earlier (in Fig. 6(a)), see Fig. 6(b). But at this time the group index becomes more and more positive via the collective effect of SGC and Kerr field. Now this is obvious that by increasing the strength of the Kerr field one can obtain much slower group velocity or even one can halt the light during its propagation inside the atomic medium via

6 No. 3 Communications in Theoretical Physics 415 control of the Kerr field. The slow light has potential applications, these include for example, slow light devices are considered for enhancing other optical nonlinearities. [44] Slowing or stopping light is also used to achieve the long storage times to perform quantum operations. [45 46] Slow light could be used to enhance the sensitivity of spectral interferometer, which has been noticed by Shi and co-workers in [47] Similarly, slow light has been used in laboratory settings to achieve true time delay to synchronize the radio frequency emitters of a phased-array radar system. [48 49] 4 Conclusion In conclusion, we have suggested a scheme for control of the group velocity inside an atomic medium. We theoretically investigated the control of group velocity via an SGC as well as Kerr field inside a medium. Each atom of a medium consist of four-level N-type atomic configuration. It is noticed that the individual as well as the collective effect of SGC and Kerr field enhanced the positive group index. This enhancement in the positive group index leads to slow light propagation. It is emphasized that a much slower group velocity inside the medium is investigated by incorporating the collective effect of SGC and Kerr field. A very strong control of the group velocity was examined via external control Kerr field and SGC. Appendix Here, we consider the details of solving Eq. (2). We follow the same method as has been used in [42, 50]. We can write Eq. (2) in the form as Ṙ = MR + C, (12) where R, C, and M are the column vectors and matrix, respectively, as given below R = (ρ bc ρ ac ρ bd ρ ad ρ bb ρ ab ρ ba ) T, C = (iω p ) T, i p +γ 1 +γ 2 iω 1 iω k iω 1 i( 1 p ) + Γ 0 iω k 2q γ 1 γ iω k 0 i( p k ) iω M = 0 iω k iω 1 i( 1 p + k )+γ γ 1 + γ 2 iω 1 iω iω 1 i 1 + γ iω 1 0 i 1 + γ 1 Now the formal solution of such an equation can be written as R(t) = We use Eq. (13) and get the solution for ρ bc which is given in Eq. (5). t e M(t t) C d t = M 1 C. (13) References [1] S.E. Harris, J.E. Field, and A. Imamoĝlu, Phys. Rev. Lett. 64 (1990) [2] M. Xiao, Y.Q. Li, S.Z. Jin, and J.G. Banacloche, Phys. Rev. Lett. 74 (1995) 666. [3] C. Liu, Z. Dutton, C.H. Behroozi, and L.V. Hau, Nature (London) 409 (2001) 490. [4] O. Kocharovskaya, Y. Rostovtsev, and M.O. Scully, Phys. Rev. Lett. 86 (2001) 628. [5] L.V. Hau, S. E. Harris, Z. Dutton, and C.H. Behroozi, Nature (London) 397 (1999) 594. [6] M.M. Kash, V.A. Sautenkov, A.S. Zibrov, L. Hollberg, George R. Welch, M.D. Lukin, Y. Rostovtsev, E.S. Fry, and M.O. Scully, Phys. Rev. Lett. 82 (1999) [7] M.S. Bigelow, N.N. Lepeshkin, and R.W. Boyd, Phys. Rev. Lett. 90 (2003) [8] R.W. Boyd, Nonlinear Optics, 2nd edition, Academic, San Diego (2003). [9] J.F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, Phys. Rev. Lett. 78 (1997) 634. [10] Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, and H.J. Kimble, Phys. Rev. Lett. 75 (1995) [11] C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, Phys. Rev. Lett. 90 (2003) [12] C. Hang, Y. Li, L. Ma, and G. Huang, Phys. Rev. A 74 (2006) [13] S.E. Harris and Y. Yamamoto, Phys. Rev. Lett. 81 (1998) [14] B. He, Y. Ren, and J.A. Bergou, Phys. Rev. A 79 (2009) [15] Y.B. Sheng, L. Zhou, S.M. Zhao, and B.Y. Zheng, Phys. Rev. A 85 (2012) [16] Y.B. Sheng, L. Zhou, and S.M. Zhao, Phys. Rev. A 85 (2012) [17] Q. Lin and B. He, Phys. Rev. A 80 (2009) [18] K. Nemoto and W.J. Munro, Phys. Rev. Lett. 93 (2004) [19] H. Jeong, Phys. Rev. A 72 (2005) [20] B. He, M. Nadeem, and J.A. Bergou, Phys. Rev. A 79 (2009)

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