Home
Search
Collections
Journals
About
Contact us
My IOPscience
Bunching and Antibunching in Cavity QED
This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 Commun. Theor. Phys. 56 134 (http://iopscience.iop.org/0253-6102/56/1/23) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 193.174.246.168 The article was downloaded on 31/07/2011 at 14:07
Please note that terms and conditions apply.
Commun. Theor. Phys. 56 (2011) 134–138
Vol. 56, No. 1, July 15, 2011
Bunching and Antibunching in Cavity QED H. Jabri1 and H. Eleuch2,3,∗ 1
D´epartement de Physique, Facult´e des Sciences de Tunis, Tunis, Tunisia
2
Institute for Quantum Studies, Texas A&M University, College Station Texas, U.S.A.
3
University of Carthage, Tunis, Tunisia
(Received October 28, 2010; revised manuscript received March 7, 2011)
Abstract We study the statistics of the emitted filed from Rydberg atom confined inside a microcavity and interacting with a pump laser in the strong coupling regime. We explore the manifestation of the antibunching in connection with the internal system parameters. PACS numbers: 42.50.Lc, 42.50.Ar, 42.50.Pq
Key words: quantum fluctuations, quantum noise, photon statistics, cavity quantum electrodynamics
1 Introduction Study of interaction between atoms, or similar systems,[1−8] and light in a cavity is of great interest in quantum optics.[9−36] Indeed, a prototype in physics like Rydberg atom is much investigated in such field especially when it interacts with a laser inside a good microcavity.[24−25] The autocorrelation function formalism plays a very important role to understand this coupling and to accede to the internal dynamical behavior.[17−18] Furthermore, autocorrelation function allowed us to explore the non classical effect in such systems.[19−22] In this paper, we study the autocorrelation function (2) g (0) of a Rydberg atom confined in a microcavity interacting with a pump laser in the strong coupling regime as a function of the system’s parameters. In particular we explore the domain of the internal parameters where our system shows a classical or non-classical results, in other words, when the emitted light is bunched, coherent or antibunched. This paper is organized as follows. In Sec. 2, we define the Hamiltonian of the system. Section 3 is dedicated to the study of the weak excitation regime: we give the expression of the autocorrelation function and particularly g (2) (0) then we use it to show the existence of classical or non-classical effects in the light emitted by the cavity. In Sec. 4, we study some interesting particular cases. The last section is devoted to the strong excitation regime.
2 Model A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number. These atoms have a number of peculiar properties
including an exaggerated response to external perturbation, long decay periods and electron wavefunctions that approximate under some conditions classical orbits of electrons about the nuclei. Another property is the existence of the huge dipole moment associated to a large orbit. Rydberg atoms can be created in laboratory by a number of techniques.[37] It is important to note that most experiments have been done with the atoms of the alkali metals and the alkaline earths. The physical situation we consider is an atom prepared in a Rydberg state, modelled by two level atomic states: The fundamental |0i and an excited state |1i. The frequency of transition between these states is ωat . The hamiltonian of this free atom is given by: Hat = hωat Dz , (1) Dz is the atomic inversion operator. This atom is now placed inside a cavity of volume V with high finesse, pumped by a laser of frequency ωL . The total hamiltonian of this system, a two-state atom interacting with a single cavity mode, involves other than the free atomic and free cavity field Hamiltonians two interactions. First, the electric dipole coupling between atom and cavity field mode and the second interaction between the laser and the cavity. The expression of the total hamiltonian is explicitly given in Refs. [24–25].
3 Weakly Excited Atom in a Cavity We are interested in this work on the statistical behavior of the system. More precisely, we explore the domain of the parameters where the system exhibits a non classical effect. The expression of the autocorrelation function in the strong coupling regime (g 2 ≫ (κ + Γ⊥ )2 ) was derived in the previous work:[24]
∗ E-mail:
[email protected] c 2011 Chinese Physical Society and IOP Publishing Ltd
http://www.iop.org/EJ/journal/ctp http://ctp.itp.ac.cn
Communications in Theoretical Physics
No. 1
135
2 −γτ µτ /2 2 g (2) (τ ) = 1 + A e −γτ e µτ /2 cos(g+ τ ) + B e −µτ /2 cos(g− τ ) + Ae e sin(g+ τ ) + B e −µτ /2 sin(g− τ ) ,
(2)
where
κ + Γ⊥ , γ= 2
(∆b − ∆a )2 (κ − Γ⊥ ) µ= , 2g
∆a g± = g ± , 2
Also, ∆a and ∆b represent respectively the detuning between the cavity, the atom and the laser frequencies: ∆a = ωL − ωph ,
∆b = ωL − ωat ,
(3)
Γ and κ are respectively the atomic and photonic dissipation rates (Γ⊥ = Γ/2). g (2) (0) is given by, as a function of the frequency detuning between the cavity, the atom and the laser frequency ∆a and ∆b : 2κ 2 (α1 + α2 )α3 , (4) g (2) (0) = 2 (Γ⊥ + ∆2b /4)2 (β1 + β2 ) where ∆2 2 α1 = g 2 Γ⊥ + κΓ2⊥ + κ b , 4 ∆ ∆2 ∆3 2 b α2 = g 2 + Γ2⊥ ∆a + ∆a b − ∆b Γ2⊥ − b , 2 4 4 2 2 ∆ ∆ b α3 = g 2 + κΓ⊥ − b + ∆a 4 2 2 ∆ b + κ + Γ⊥ ∆a + ∆b Γ⊥ , 2 ∆2 ∆b 2 β1 = 2g 2 + 2κΓ⊥ + 2κ 2 − 2∆2a + b + ∆a , 4 2 2 ∆b ∆b β2 = 2Γ⊥ ∆a − Γ⊥ + 4κ∆a + κ . (5) 2 2 The second order correlation function g (2) (τ ) of a classical light gives values, which qualifies this light by bunched or coherent light according to a classification based on the value of g (2) (0): so, a bunched light if g (2) (0) > 1 and a coherent light when g (2) (0) = 1. Quantum theory predict values of the autocorrelation function that are not conform with the classical point of view. Such a light which exhibits this non-classical behavior is called an antibunched light (g (2) (0) < 1). A successful experiment of photon antibunchig was realized by Kimble et al. in 1977 using the light emitted by sodium atoms.[27] Figures 1 and 2 show plots of the temporarily evolution of the autocorrelation function for two cases in the strong coupling regime. The first case (Fig. 1) describes the resonance between the cavity and the laser and a quasi resonance between the Rydberg atom and the laser (g 2 ≫ ∆2b ). At delay τ = 0, g (2) has a value less than unity (g (2) (0) < 1). The second case (Fig. 2) describes the resonance between the atomic system and the laser and a quasi resonance between the cavity and the laser (g 2 ≫ ∆2a ). In
p ( g (2) (0) − 1)(µ + 2γ) A= , 2µ
B=
this case, and for some values of system parameters which are in agreement with the strong coupling regime, g (2) (0) takes values greater than unity (g (2) (0) > 1). Figure 1 shows antibunching effect. In opposite, Fig. 2 shows a bunching effect. We conclude that the detuning between the laser and the cavity destroys the non-classical effect for some particular system parameters. We will explore this effect in the following parts.
Fig. 1 The autocorrelation function is plotted as a function of the delay τ , which is given by Eq. (10) for ∆a = 0 and ∆b = 0.08 (Γ⊥ = 1/40 , κ = 1/20, g = 1).
Fig. 2 The autocorrelation function is plotted as a function of the delay τ , which is given by Eq. (10) for ∆a = 0.08 and ∆b = 0 (Γ⊥ = 1/40, κ = 1/20, g = 1).
In this work, we propose to explore the effect of the system parameters on the statistical behavior. Indeed, in order to distinguish the boundaries of the bunching and the antibunching effects, let us consider the expression of g (2) (0). The expression of autocorrelation function g (2) (0) is given by Eq. (12):
∆2 2 g 2 ∆b ∆a ∆2b ∆3 2 g 2 Γ⊥ + κΓ2⊥ + κ b + + Γ2⊥ ∆a + − ∆b Γ2⊥ − b 4 2 4 4 2 2 2 2 g + κΓ⊥ − ∆b /4 + ∆a (∆b /2) + κ(∆b /2) + Γ⊥ ∆a + ∆b Γ⊥ × 2 Γ2⊥ + ∆2b /4
g (2) (0) = 2κ 2
µ − 2γ . µ + 2γ
Communications in Theoretical Physics
136
×
Vol. 56
1 , (G1 + (2Γ⊥ ∆a − (∆b /2)Γ⊥ + 4κ∆a + κ(∆b /2))2 )
where
∆2 ∆b 2 G1 = 2g 2 + 2κΓ⊥ + 2κ 2 − 2∆2a + b + ∆a , 4 2
using the strong coupling approximation (g 2 ≫ (κ + Γ⊥ )2 ) and the quasi resonance (g 2 ≫ ∆2a , ∆2b ) we get: ∆2 2 ∆b ∆2 2 g (2) (0) ≈ 2κ 2 g 2 Γ⊥ + κ b + g 2 + Γ2⊥ ∆a + ∆a b 4 2 4 2 2 ((g + ∆a (∆b /2)) + (κ(∆b /2) + Γ⊥ ∆a + ∆b Γ⊥ )2 ) × 2 . (Γ⊥ + ∆2b /4)2 ((2g 2 + ∆a (∆b /2))2 + (2Γ⊥ ∆a − (∆b /2)Γ⊥ + 4κ∆a + κ(∆b /2))2 )
(6)
In order to simplify the calculation, we consider the case where ∆a = ∆b = ∆ and Γ⊥ = κ. In this case g (2) (0) becomes: ∆2 2 2 ∆ ∆3 2 ((g 2 + ∆2 /2)2 + (κ(∆/2) + 2κ∆)2 ) g (2) (0) ≈ 2κ 2 g 2 κ + κ + g + κ2 ∆ + (7) 4 2 4 (κ 2 + ∆2 /4)2 ((2g 2 + ∆2 /2)2 + (6κ∆)2 ) ≈
2κ 2 (g 4 κ 2 + g 4 (∆2 /4))(g 4 + 25(∆2 κ 2 /4)) . (κ 2 + ∆2 /4)2 (4g 4 + 36∆2 κ 2 )
After some algebraic manipulations, we obtain for ∆ four solutions for the case g (2) (0) = 1:
(8)
p 4g 8 + 625g 8κ 8 1/2 ∆3 ≈ , (11) 36κ 2 p 25g 4 κ 2 g4 4g 8 + 625g 8κ 8 1/2 ∆4 ≈ − + + . (12) 18κ 2 36 36κ 2 At these values of the parameter ∆, emitted light by the cavity is coherent. Other than these particular values we get:
g4 25g 4 κ 2 − + − 18κ 2 36
p g4 25g 4 κ 2 4g 8 + 625g 8 κ 8 1/2 ∆1 ≈ − − + − , (9) 2 18κ 36 36κ 2 p g4 25g 4 κ 2 4g 8 + 625g 8κ 8 1/2 ∆2 ≈ − − + + , (10) 18κ 2 36 36κ 2 if ∆ ∈]∆1 , ∆2 [∪]∆3 , ∆4 [=⇒ g (2) (0, g, κ, ∆) − 1 > 0 =⇒ bunching , otherwise g (2) (0, g, κ, ∆) − 1 < 0 =⇒ antibunching .
4 Some Particular Cases Let us plot some particular cases: 4.1 First Case: Resonance Between Laser and Rydberg Atom (∆b = 0) We study here the case where is a resonance between the atom and the laser frequency (∆b = 0).
Fig. 3 The autocorrelation function g (2) (0) is plotted as a function of the frequency detuning between the cavity and the laser frequency ∆a for g = 1, κ = 1/20, Γ⊥ = 1/40: antibunching effect.
In Fig. 3, g (2) (0) decreases considerably and rapidly
(13)
when ∆a increases and tends towards zero when ∆a becomes infinite. It is important to mention here that at an infinite value of ∆a , g (2) (0) = 0, in this case we have a total antibunching effect. In Fig. 4, g (2) (0) changes slowly when ∆a increases and it takes always values greater than 2: near the total resonance, we have a classical bunching effect. g (2) (0) > 2 indicates that the system emits a super thermal light.[38−39]
Fig. 4 The autocorrelation function g (2) (0) is plotted as a function of the frequency detuning between the cavity and the laser frequency ∆a for g = 1, κ = 1/20, Γ⊥ = 1/40: near resonance we have a classical bunching effect.
No. 1
Communications in Theoretical Physics
137
4.2 Second Case: Resonance Between Laser and Cavity ∆a = 0
the expression of the autocorrelation function in the strong coupling regime is:[25]
Now we take the case where we have a resonance between the laser frequency and the cavity (∆a = 0). Figure 5 shows a coexistence of the bunching and antibunching effects in the chosen range of ∆b values.
g (2) (τ ) = 1 + µ e −τ /τpop − (µ + 1) cos(Ωτ ) e −τ /τcoh , (14) where µ is a parameter given by: (ωL − ω)2 /g 2 , (15) 1 + (ωL − ω)2 /g 2 and the population and the coherence lifetimes τpop and τcoh are respectively defined by: 1 2 τpop = = , (16) Γpop (µ + 1)Γ µ=
1 1 = . (17) Γcoh Γ(µ + 1)/4 + (1/2)(Γ + κ)(1 − µ) Ω represents the Rabi frequency of the system given by: q Ω = (ωph − ω)2 + g 2 . (18) τcoh =
Fig. 5 The autocorrelation function g (2) (0) is plotted as a function of the frequency detuning between the cavity and the atomic frequency ∆b for, g = 1, κ = 1/20, Γ⊥ = 1/40.
The expression shows that independently of the parameters, the system exhibits a total antibunching effect g (2) (0) = 0.
4.3 Third Case: Non Resonance ∆a 6= 0, ∆b 6= 0 In this part we see the general situation where the two frequency detuning ∆a and ∆b are non equal to zero (∆a 6= 0, ∆b 6= 0). Figure 6 shows again the coexistence of the bunching and antibunching effects in the chosen range of ∆b values. For fixed value of ∆b the statistics of the emitted light is independent of ∆a . Fig. 7 The autocorrelation function g (2) (τ ) is plotted as a function of time delay in the strong coupling regime (∆L = 0.01, g = 1, Ω = 1, Γ = 1/20, κ = 1/40).
6 Conclusion
Fig. 6 The autocorrelation function g (2) (0) is plotted as a function of the frequency detuning between the cavity and the laser frequency ∆a , and the frequency detuning between the cavity and the atomic frequency ∆b for g = 1, κ = 1/20, Γ⊥ = 1/40: coexistence of the classical bunching effect and the quantum antibunching effect.
5 Highly Excited Rydberg Atom in a Cavity In this section, we explore the case of highly excited Rydberg atom. In the previous work, we have shown that
In this paper, we study the statistics of the emitted field from a cavity of high finesse, where a two level atom is confined inside it. We show that the non-classical effect, namely the antibunching effect, depends crucially on the system parameters in the weak excitation regime. In the high excitation regime, we show a total antibunching independently of the internal parameters. The autocorrelation function experiment constitutes an alternative to the direct measurement experiments. It is based on interferometric technique known to be accurate. As the autocorrelation function explores the evolution of the system it offers the opportunity to access some characteristic parameters of the system such as atomic dipolar moment, mean radiative lifetime, cavity dissipation, atomic frequency, and Rabi frequency.
138
Communications in Theoretical Physics
Vol. 56
References
[21] H. Eleuch, Eur. Phys. J. D 48 (2008) 139.
[1] M. Abdel-Aty, Commun. Theor. Phys. 37 (2002) 723; M. Abdel-Aty, T. El-Shahat, and A.S.F. Obada, Commun. Theor. Phys. 38 (2002) 715. [2] A. Baas, et al., Phys. Rev. A 69 (2004) 023809. [3] N. Boutabba, et al., Synthetic Metals 159 (2009) 1239. [4] Sh. Barzanjeh and H. Eleuch, Physica E 42 (2010) 2091. [5] G. Dridi, et al., Phys. Rev. A 80 (2009) 043408; S. AbdelKhalek, Appl. Math. Inf. Sci. 1 (2007) 53. [6] J. McKeever, et al., Nature (London) 425 (2003) 268. [7] F.N.M. Al-Showaikh, Appl. Math. Inf. Sci. 2 (2008) 21. [8] Z. Ficek, Appl. Math. Inf. Sci. 3 (2009) 375. [9] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Processus D’interaction Entre Photons Et Atomes, Edition of CNRS, Paris (1988). [10] Y.V. Rostovstev and H. Eleuch, J. Mod. Opt. 57 (2010) 1311. [11] H. Eleuch, Eur. Phys. J. D 49 (2008) 391. [12] H. Eleuch, N. Ben Nessib, and R. Bennaceur, Eur. Phys. J. D 29 (2004) 391. [13] H. Eleuch and R. Bennaceur, J. Opt. B: Quantum Semiclass. Opt. 6 (2004) 189. [14] H. Eleuch, D. Elser, and R. Bennaceur, Laser Phys. Lett. 1 (2004) 391. [15] H. Eleuch and R. Bennaceur, J. Opt. A: Pure Appl. Opt. 5 (2003) 528. [16] H. Eleuch, et al., J. Opt. B.: Quantum Semiclass. Optics 1 (1999) 1. [17] H. Eleuch, J. Phys. B 41 (2008) 055502. [18] H. Eleuch, Appl. Math. Inf. Sci. 3 (2009) 185. [19] H. Eleuch and N. Rachid, Eur. Phys. J. D 57 (2010) 259. [20] H. Eleuch, Int. J. Mod. Phys. B 24 (2010) 5653.
[22] Mark Fox, Quantum Optics: An Introduction, Oxford University Press, Oxford (2006). [23] E. Giacobino, et al., C. R. Physique 3 (2002) 41. [24] H. Jabri, et al., Laser Phys. Lett. 2 (2005) 253. [25] H. Jabri, et al., Phys. Scr. 73 (2006) 397. [26] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51 (1963) 89 [27] H.J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39 (1977) 691. [28] Li-Hui Sun, Gao-Xiang Li, and Zbigniew Ficek, Appl. Math. Inf. Sci. 4 (2010) 315. [29] G. Messin, et al., J. Phys. 11 (1999) 6069. [30] M.M. Nassar, et al., J. At. Mol. Sci. 1 (2010) 54. [31] E.A. Sete, et al., J. Mod. Opt. 57 (2010) 1311. [32] Eyob A. Sete and H. Eleuch, Phys. Rev. A 82 (2010) 043810. [33] R.J. Schoelkopf and S.M. Girvin, Nature (London) 451 (2001) 664. [34] S. Osnaghi, et al., Phys. Rev. Lett. 87 (2001) 037902. [35] Y. Rostovstev, et al., Phys. Rev. A 79 (2009) 063833. [36] S. Abdel-Khalek, Phys. Scr. 80 (2009) 045302; A.S.F. Obada, S. Abdel-Khalek, and A. Plastino, Physica A 390 (2011) 525. [37] S. Haroche, G. Vitrant, and J.M. Raimond, RydbergRydberg Interactions in a Dense System of Very Excited Atoms in “Laser spectroscopy V”, A.R.W. Mc Kellar, T. Oka, B.P. Stoiche, eds. Springer, Verlag (1980). [38] R. Loudon and P.L. Knight, J. Mod. Opt. 34 (1987) 709. [39] M. Sebawe Abdalla, E. Khalil, and A.S.F. Obada, Il Nuovo Cimento, DOI 10.1393/ncb/i2010-10942-y.
