Tunable slow and fast light in an atom-assisted optomechanical system

Optics Communications 338 (2015) 569–573

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Tunable slow and fast light in an atom-assisted optomechanical system Kai-Hui Gu a,b, Xiao-Bo Yan a, Yan Zhang c, Chang-Bao Fu a, Yi-Mou Liu a, Xin Wang a, Jin-Hui Wu a,d,n a

College of Physics, Jilin University, Changchun 130012, PR China College of Optical and Electronical Information, Changchun University of Science and Technology, Changchun 130023, PR China c School of Physics, Northeast Normal University, Changchun 130024, PR China d Center for Quantum Sciences, Northeast Normal University, Changchun 130024, PR China b

art ic l e i nf o

a b s t r a c t

Article history: Received 16 September 2014 Received in revised form 4 November 2014 Accepted 9 November 2014 Available online 15 November 2014

We theoretically investigate the slow and fast light effects of a probe laser in an optomechanical doubleended cavity filled with a bunch of identical two-level cold 87Rb atoms. It is shown that the presence of atoms can transfer the optomechanically induced transparency (OMIT) window into the steeper Fano shapes and the broader normal-mode splitting shapes. We find that the transmission group delay related to the Fano resonance is much greater than that of the OMIT and normal-mode splitting at the resonances, meanwhile, with the growing of the atomic number, the slow light becomes increasingly significant. The value of reflection group delay related to the OMIT and normal-mode splitting is positive, while the value related to the Fano resonance is significantly negative. This indicates that the reflection parts of output field contain both slow light and fast light. Therefore, via changing the atomic number and detuning, we can realize not only the manipulation of the magnitudes of the group delay and advance, but also a tunable switch from slow light to fast light. & 2014 Published by Elsevier B.V.

Keywords: Atom-assisted optomechanics Optomechanically induced transparency Fano resonances Normal-mode splitting Slow and fast light

1. Introduction Researches on slow and fast light have attracted a lot of interests from both theoretical and experimental sides [1–4] in recent years, for the reason that there could be many potential applications of this in fields like all-optical routing [5,6], all-optical switching [7], optical memories, optical telecommunication and interferometry [8–10]. Till this time, there have been different methods presented to realize the slow and fast light, e.g., electromagnetically induced transparency (EIT) [11–13], coherent population oscillation, parametric process, stimulated Brillouin scattering (SBS), and stimulated Raman scattering (SRS). The method based on EIT is a quantum interference effect which can bring dramatic change to the absorption and dispersion properties of the medium and consequently modify the nonlinear susceptibility as well as generating the slow and fast light [14–17]. EIT has been first observed in atomic vapors [18], and recently the study has extended to many hybrid systems, where some EIT-like effects have successively been discovered, including coupled resonance induced transparency [19,20], phonon induced transparency [21] and optomechanically induced transparency (OMIT) [22–27]. The n Corresponding author at: College of Physics, Jilin University, Changchun 130012, PR China. E-mail address: [email protected] (J.-H. Wu).

http://dx.doi.org/10.1016/j.optcom.2014.11.036 0030-4018/& 2014 Published by Elsevier B.V.

OMIT effect in linearly coupled optomechanical systems has been demonstrated theoretically by Agarwal and Huang [22] and experimentally by Weis et al. [23]. More recently, the Fano-like resonances in OMIT under certain conditions have been shown in cavity optomechanics as well [26]. Because of the many unique features of an optomechanical system, the realization of slow and fast light within such a system has drawn particular interest. For example, slow light with an optomechanical crystal array has been discovered both theoretically and experimentally [28,29], where the slow light with maximum transmission group delay of about 50 ns has been obtained. Chen et al. [30] also carried out the theoretical study on slow light in a cavity optomechanical system with a Bose–Einstein condensate (BEC) and demonstrated that the delay time can achieve the value of as much as0.8 ms. Experimentally, slow light and fast light have also been realized in microwave regime using a circuit nanoelectromechanics system [31]. On the other hand, some other interesting phenomena have been found in atom-assisted optomechanical systems. For instance, Chang et al. [32] carried out research on the steady-state position of bistable behaviors in an optomechanical cavity, whose standing-wave mode pattern could be modified by the collective atomic back-action, with two-level cold atoms. It has also been found that both mirror displacement and atomic susceptibility may have multiple steady-state solutions, when the cold atoms trapped in a typical optomechanical cavity were driven into the three-level Λ configuration of EIT [32,33]. In addition, the

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two-level atoms and three-level Λ atoms can induce entanglement among the atoms, cavity field and the optomechanical oscillators [34–36]. In the meantime, resonant coupling of the mirror vibrations to atomic media can cool the micromechanical oscillators to the ground-state [37–39]. Fano resonances have been recently observed in the hybrid system composed of a two-level atom and a mechanical resonator both coupled to the quantized cavity field [40]. Based on these works, it can be seen that as the media, atoms play an important role in optomechanical systems. However, the case of atom-assisted optomechanical system has rarely been studied on slow light and fast light. In this paper, we consider an ensemble of N identical two-level cold atoms, instead of a single two-level cold atom [40], embedded into the double-ended cavity with a moving nanomechanical mirror. This choice allows us to modulate the atom–cavity coupling constant in a much wider range by changing the atomic number as in Ref. [41]. Then we go beyond Ref. [40] to study not only steady-state EIT spectra and Fano resonances but also normal-mode splitting intrinsic to a sufficiently large atom–cavity coupling constant. It is more important that we also study the dynamic effects of slow light and fast light, i.e. show how the group delay and advance of a probe field can be modulated by adjusting relevant atomic parameters. The organization of this paper is as follows. In Section 2 we introduce the theoretical model and method of realizing slow light and fast light. In Section 3 we show the results of the numerical calculations and discuss the phenomena of slow light and fast light. Then the conclusion is given in Section 4.

2. The theoretical model The system we consider is shown schematically in Fig. 1, consisting of an ensemble of N identical two-level cold atoms confined in a magneto-optical trap (MOT) located at the optomechanical double-ended Fabry–Perot cavity [42]. Levels |1〉 and |2〉 may correspond, respectively, to states |5S1/2 〉 and |5P1/2 〉 on the D1 line of 87 Rb atoms. The Fabry–Perot cavity with a partially transparent movable nanomechanical mirror in the middle of this cavity is

bounded by two equal reflectivity fixed mirrors. The movable mirror is free to move along the cavity axis and is treated as a quantum mechanical harmonic oscillator with the effective mass m, the frequency ωm, and the decay rate γm. The cavity field with frequency ωc is driven by an input laser field with frequency ωd and a probe field with frequency ωp. When photons in the cavity are reflected off by the surface of a movable mirror, the movable mirror will receive the action of the radiation pressure, which is proportional to the instantaneous photon number inside the cavity. So the mirror can oscillate under the effect of the radiation pressure. Here we assume that the motion of the mirror is so slow that the scattering of photons to other cavity modes can be ignored, thus we can consider only one cavity mode [43,44], say, ωc. Under this condition, the free Hamiltonian of the system in a frame rotating at the laser frequency ωd can be written as

⎛ 2 ⎞ p 1 H0 = = (ωc − ω d ) c †c + ⎜⎜ + mωm2 q2⎟⎟ 2 ⎝ 2m ⎠ N (i) . + = ∑ (ωa − ω d ) σ22 i=1

(1)

The first term on the right-hand side of Eq. (1) is the energy of the cavity field, where c and c † are the annihilation and creation operators of the cavity mode satisfying the commutation relation [c , c †] = 1, respectively. The second term represents the energy of the mechanical oscillator with momentum p and displacement q. (i) The last term is the energy of the atoms, where σab = |a〉ii 〈b| is the projection (a = b) or transition (a ≠ b) operator of the ith atom; and ωa is the transition frequency between level |2〉 and level |1〉. The interaction Hamiltonian can be written as

N (i) (i) HI = − =χqc †c − =g ∑ (cσ21 ) + c †σ12 i=1

+ i=ε d (c † − c) + i= (c †ε p e−iδt − cε p⁎ eiδt ).

(2)

The first term on the right-hand side of Eq. (2) is the interaction between the nanomechanical oscillator and the cavity field, with χ = ωc /l being the coupling constant between the cavity field and the movable mirror. The second term gives the interaction between the atoms and the cavity field, where g = μ ωc /2=V ϵ0 is the coupling constant of the cavity field with the dipole moment μ, the cavity volume V, and the vacuum permittivity ϵ0. The last two terms describe the cavity driven by a strong control field with frequency ωd and a weak probe field with frequency ωp, where δ = ω p − ωd is the detuning between the probe and driving fields.

ϵd =

2κPd/=ωd (ϵ p =

2κPp/=ω p ) with the decay rate of the cavity

field κ is the amplitude of the driving (probe) field, which is related to its corresponding power Pd (Pp ). If most of the atoms are initially prepared in the ground state, these atoms in the cavity can be seen as a whole for the large atomic number N. We can define a collective transition operator

ρa = lim N →∞Σ iN= 1 (gi⁎/g N ) |1〉ii 〈2|, where g N =

Fig. 1. Schematic diagram of an optomechanical cavity containing N identical twolevel cold 87Rb atoms with two fixed end mirrors of equal reflectivity, which is driven by a strong field ωd and probed by a weak field ωp.

Σ iN= 1|gi |2 is the

total coupling strength between the atomic ensemble and the cavity field, which will enhance with the increasing of N. In addition, the operator of atoms satisfies the bosonic commutation relation [ρa , ρa† ] = 1. Moreover, comparing with the length l of the cavity, the change in the position of the mirror q is very small, thus the factorization assumption 〈c †c〉 = 〈c †〉〈c〉, 〈qc〉 = 〈q〉〈c〉 can be used, in which the correlation between them has been neglected. According to (Eqs. (1) and 2), the system is examined in the mean-

K.-H. Gu et al. / Optics Communications 338 (2015) 569–573

field limit [22]

〈q〉̇ =

〈p〉 , 〈ṗ〉 = − mωm2 〈q〉 + =χ 〈c †〉〈c〉 − γm 〈p〉, 〈c〉̇ m

= − [2κ + i (ωc − ω d − χ 〈q〉)] 〈c〉 + ε d + ε p e−iδt − ig N 〈ρa 〉, 〈ρȧ 〉 = − [γa + i (ωa − ω d )] 〈ρa 〉 − ig N 〈c〉. (3) Eq. (3) is nonlinear equations and represents the steady-state responses in the frequency domain composed of many frequency components. We suppose the steady-state solutions to Eq. (3) take the form of

⎛ 〈q〉 ⎞ ⎛ q0 ⎞ ⎛ q+ ⎞ ⎛q− ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 〈p〉 ⎟ ⎜ p0 ⎟ ⎜ p+ ⎟ ⎜ p− ⎟ i t − δ + ⎜ ⎟ ε p⁎ eiδt . ⎜ 〈c〉 ⎟ = ⎜ c0 ⎟ + ⎜ c+ ⎟ ε p e c ⎜⎜ − ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎝〈ρa 〉⎠ ⎝ ρ 0 ⎠ ⎝ ρ+ ⎠ ⎝ ρ− ⎠

(4)

Thus, Eq. (4) shows that the cavity field has three components, oscillating at the input Stokes frequency ωp, the driving frequency ωd, and a new anti-Stokes frequency 2ωd − ω p. By inserting Eq. (4) into the Heisenberg equation of motion we first obtain the final expectation values as

where c+ stands for the output field at the probe frequency, while c − describes the nonlinearly generated field in a four-wave mixing process for the driving field, the probe field, and the mechanical oscillator. We are only interested in the output properties of an input probe field but do not care about the newly generated field, so we get the reflection and transmission amplitudes at the probe frequency denoted by

ε T = 2κc+ = |T |eiϕT (ω p ) , ε R = 2κc+ − 1 = |R|eiϕ R (ω p ) .

ρ0 =

ϕT (ω p ) = arg[ε T ] =

⎛ εT ⎞ 1 ln ⎜ ⁎ ⎟, 2i ⎝ εT ⎠

ϕR (ω p ) = arg[ε R ] =

⎛ εR ⎞ 1 ln ⎜ ⁎ ⎟. 2i ⎝ εR ⎠

c0 =

2κ + iΔ +

g 2N γa + iΔa

τT =

,

D

,

(5)

where

A− =

g 2N (γa − iδ) − iΔa

, A+ =

g 2N (γa − iδ) + iΔa

,

Δa = ωa − ω d , Δ = ωc − ω d − χq0 = Δ0 − χq0, β=

⎡ 1 ∂ε ⎤ T ⎥, = Im ⎢ ⎢⎣ ε T ∂ω p ⎥⎦ ∂ω p ∂ϕT

⎡ 1 ∂ε ⎤ R ⎥. = Im ⎢ ⎢⎣ ε R ∂ω p ⎥⎦ ∂ω p ∂ϕR

(11)

Note that, if N ¼0, Eq. (11) may lead to the well-known results, i.e. the OMIT induced slow light which have been observed experimentally [28,29], for the single mode cavity optomechanical system. In what follows, we will investigate theoretically this phenomenon in the single mode atom-assisted optomechanics.

(2κ − iΔ − iδ + A− )(ωm2 − δ 2 − iγm δ) + iβ

c+ =

(10)

For an optomechanical system, in the region of the narrow transparency window the rapid phase dispersion ϕ (ω p ) can cause the group delay expressed as

τR =

−ig N c0, γa + iΔa εd

(9)

The phase of the output field can be found as

p0 = 0,

=χ |c0 |2 q0 = , mωm2

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=χ 2 |c0 |2 , D = (2κ − iΔ − iδ + A− )(ωm2 − δ 2 − iγm δ) m × (2κ + iΔ − iδ + A + ) + iβ (2iΔ − A− + A + ).

(6)

The zeroth-order solutions p0, q0, ρ0 and c0 are values of the system at the steady state. Parameters A− and A+ show the influence of atoms on the system, and β represents the coupling relation between the cavity field and the movable mirror, Δ is the effective cavity detuning, Δa is the detuning of the driving field from the relevant atomic transition. Next, we need to find out the effect of the output field by expressing the output field as

ε Tout (t) = ε T 0 + ε T ε p e−iδt + ε T − ε p⁎ eiδt , ε Rout (t) = ε R0 + ε R ε p e−iδt + ε R − ε p⁎ eiδt .

(7)

Using the input–output relation εin (t) + εout (t) = 2κ〈c〉 [45], we can easily obtain

ε T 0 + ε T ε p e−iδt + ε T − ε p⁎ eiδt = 2κ (c0 + c+ε p e−iδt + c− ε p⁎ eiδt ), × ε d + ε p e−iδt + (ε R0 + ε R ε p e−iδt + ε R − ε p⁎ eiδt ) = 2κ (c0 + c+ε p e−iδt + c− ε p⁎ eiδt ).

(8)

3. Results For illustration of the numerical results, we choose the realistically reasonable parameters to demonstrate the fast and slow light effects based on the atom-assisted optomechanical system. All the parameters used here are accessible in experiment [33,36]: length of the cavity l ¼1 mm, wavelength of the laser λ = 2πc /ωd = 794.98 nm , m ¼10 ng, ωm = 2π × 10 MHz , mechanical quality factor Q = 105, κ = ωm/10, Δ = ωm , and the decay rate γa = 2π × 2.875 MHz . The real and the imaginary parts of εT = 2κc+ represent the absorptive and dispersive behaviors, respectively. In order to show the influence of the atoms on OMIT, we first calculate the real and the imaginary parts of the transmission probe field versus the probe frequency detuning without or with the atoms in Fig. 2. In the case of no atoms, where N ¼0, the narrow transparency window of the real part of εT (black solid line) for the transmission probe field exhibits the optomechanical induced destructive interference at the δ = ωm in Fig. 2(a). When atoms are resonant with the anti-Stokes sideband (Δa = − ωm ), interaction among the probe field, the driving field and the ensemble of two-level atoms change the position of the absorption peak (red dashed line). We observe that the OMIT window is switched from symmetric shapes to asymmetric Fano-like shapes [26,40]. Note that OMIT occurs when the system is in complete resonance, however the existence of atoms leads to the present nonresonant interactions, therefore the symmetric OMIT window is transformed into asymmetric Fano shapes. Concomitantly, when the atoms are resonant with the Stokes sideband (Δa = ωm ), the one absorption peak is switched to normal modes separated in the frequency region by the vacuum Rabi frequency 2g N [45,46] (blue dashed line). This suggests that the atoms can effectively enhance the strength of coupling between the moving mirror and

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Fig. 2. The real and the imaginary parts of ϵT as a function of the frequency detuning δ /ωm for P d = 3.8 mW . Here, N ¼0 (black solid line), g N = 6 × 2π MHz , Δa = − ωm (red dashed line), and g N = 6 × 2π MHz , Δa = ωm (blue dashed line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

the cavity field, to lead to normal-mode splitting. Simultaneously, due to the process of stimulated radiation which causes the photonic number to increase, the gain of the absorption can be attained at δ = ωm . Fig. 2(b) shows the dispersion spectra when the atoms are present. We find that the normal dispersive slope around δ = ωm is greatly changed when the atoms are resonant with the Stokes sideband or with the anti-Stokes sideband. As we have shown in the previous analysis, the presence of atoms modifies the system's absorptive and dispersive behaviors of the probe field. This is the consequence of the terms involving A− and A+ in denominator D, which is related to N and Δa.Thus, the absorptive and dispersive spectra of output field can be modified by adjusting N and Δa. The normal atomic EIT causes an extremely steep dispersion for the transmitted and reflected probe photons, and leads to a group delay and advance of the probe field. Fig. 3(a) and (b) shows the phase dispersion of the probe transmission ϕT and reflection ϕ R as a function of detuning δ /ωm for Pd = 3.8 mW . Clearly, in Fig. 3 (b) we find that the phase of the probe reflection ϕ R have two dips

Fig. 3. Transmission and reflection phases as a function of the frequency detuning δ /ωm for P d = 3.8 mW . Here, N ¼ 0 (black solid line), g N = 6 × 2π MHz , Δa = − ωm (red dashed line), and g N = 6 × 2π MHz , Δa = ωm (blue dashed line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 4. (a) Transmission group delay τT as a function of the frequency detuning δ /ωm for P d = 3.8 mW . Here, N ¼0 (black solid line), g N = 6 × 2π MHz , Δa = − ωm (red dashed line), and g N = 6 × 2π MHz , Δa = ωm (blue dashed line). (b) Transmission group delay τT as a function of the driving field power Pd. Here, N¼ 0 (black solid line), g N = 6 × 2π MHz , Δa = − ωm (red dashed line), and g N = 6 × 2π MHz , Δa = ωm (blue dashed line). (c) Transmission group delay τT as a function of the atomic number g N for P d = 0.1 mW . Here, Δa = − ωm (red dashed line) and Δa = ωm (blue dashed line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

around δ = ωm , which will bring in two tunable group delays of the reflected probe field. Fig. 4(a) shows that group delay τT is positive, which means that the transmission parts of the output field contain slow light. At the presence of atoms, the Fano resonance induced τT (red dashed line) is much greater than that induced by the OMIT (black solid line) and by normal-mode splitting (blue dashed line) at the resonance δ = ωm . Fig. 4(b) shows τT as a function of the pump power Pd (0.1 ∼ 10 mW) at the resonance δ = ωm . It is clear that τT decreases with the increasing of the driving field strength. As for the maximum of τT , the Fano resonance induced τT (red dashed line) is about 10 times as that of the OMIT induced τT (black solid line), and 5 times as of the normal-mode splitting induced τT (blue dashed line) in the same case. In Fig. 4(c), with the increasing of N at Pd = 0.1 mW , the Fano resonance induced τT will increase.

Fig. 5. (a) Reflection group delay τR as a function of the frequency detuning δ /ωm for P d = 3.8 mW . Here, N ¼0 (black solid line), g N = 6 × 2π MHz, Δa = − ωm (red dashed line), and g N = 6 × 2π MHz , Δa = ωm (blue dashed line). (b) Reflection group delay τR as a function of the frequency detuning δ /ωm for P d = 3.8 mW and Δa = − ωm . Here, g N = 2 × 2π MHz (black solid line), g N = 4 × 2π MHz (blue solid line), and g N = 6 × 2π MHz (red solid line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

K.-H. Gu et al. / Optics Communications 338 (2015) 569–573

However, the normal-mode splitting induced τT will increase first and then decrease after the point of inflexion at the resonance δ = ωm . Thus, τT can be tuned by adjusting Pd and N. Fig. 5(a) shows the reflection group delay τR versus the frequency detuning δ /ωm for Pd = 3.8 mW . We find that the OMIT induced τR (black solid line) is positive value, the normal-mode splitting induced τR (blue dashed line) is almost zero, and the Fano resonance induced τR (red dashed line) is negative value. It means that the reflection parts of output field contains both slow light and fast light. In Fig. 5(b), with the increasing of N, τR related to the Fano resonance will get close to the point δ = ωm from left-hand side, and gradually disappear on the right-hand side as Δa = − ωm .

4. Conclusion In this paper, we study slow light and fast light in an optomechanical system of tripartite atom–cavity-mechanics mode. The main conclusions are as follow. (i) At the absence of the atoms, the generic OMIT window can be obtained; with the presence of the atoms, we obtain the steeper Fano shapes and the broader normalmode splitting shapes by changing the detuning between the atomic transition and the driving field. (ii) The transmission parts of the output probe field contain slow light. The group delay related to the Fano resonance is much greater than those of the OMIT and normal-mode splitting at the resonance δ = ωm . Meanwhile, the slow light of the transmitted probe field can be modified by modulating the power of the driving field and atomic number. (iii) The reflection parts of the output probe field consist of both slow light and fast light. With the absence of atoms, the reflection parts of output probe field have slow light; when the atoms are resonant with the Stokes sideband, the OMIT window is transferred to the normal-mode splitting shapes, where the slow light of the reflection probe field almost disappears; when the atoms are resonant with the anti-Stokes sideband, the OMIT window transfers from symmetric shapes to asymmetric Fano shapes, where the slow light of the reflection probe field turns into fast light. It is worth noting that by adjusting the atomic number and detuning, we can not only change the magnitudes of the group delay and advance, but also turn the slow light into fast light.

Acknowledgments This work is supported by the National Natural Science Foundation of China (11174110 and 61378094), the National Basic Research Program of China (2011CB921603), and the Fundamental Research Fund of Jilin University and Northeast Normal University.

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