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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 6 — Mar. 12, 2012
  • pp: 5840–5848
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Tunable slow and fast light device based on a carbon nanotube resonator

Jin-Jin Li and Ka-Di Zhu  »View Author Affiliations


Optics Express, Vol. 20, Issue 6, pp. 5840-5848 (2012)
http://dx.doi.org/10.1364/OE.20.005840


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Abstract

We report a tunable slow and fast light device based on a carbon nanotube resonator, in the presence of a strong pump laser and a weak signal laser. Detailed analysis shows that the signal laser displays the superluminal and ultraslow light characteristics via passing through a suspended carbon nanotube resonator, while the incident pump laser is on- and off-resonant with the exciton frequency, respectively. In particular, the fast and slow light correspond to the negative and positive dispersion, respectively, associating with the vanished absorption. The bandwidth of the signal spectrum is determined by the vibration decay rate of carbon nanotube.

© 2012 OSA

1. Introduction

Over the past several years, researchers have been intrigued by the possibility of using nonlinear optical methods to realize unprecedented control of light pulse propagation through different kinds of material systems [1

1. R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074 (2009). [CrossRef] [PubMed]

,2

2. B. Gu, N. H. Kwong, R. Binder, and A. L. Smirl, “Slow and fast light associated with polariton interference,” Phys. Rev. B 82, 035313 (2010). [CrossRef]

]. The interaction of light with matter can lead to extreme changes of light velocity [3

3. R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Applications of slow light in telecommunications,” Opt. Photon. News 19, 18 (2006). [CrossRef]

] i.e., fast light, slow light and even stored light [4

4. A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69 (2011). [CrossRef] [PubMed]

,5

5. V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011). [CrossRef] [PubMed]

], which have been investigated in several systems, mainly using electromagnetically induced transparency (EIT) [6

6. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594(1999). [CrossRef]

], coherent population oscillation (CPO) [7

7. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003). [CrossRef] [PubMed]

], and stimulated Brillouin scattering (SBS) [8

8. V. I. Kovalev, N. E. Kotova, and R. G. Harrison, “Slow light in stimulated Brillouin scattering: on the influence of the spectral width of pump radiation on the group index: reply,” Opt. Express 18, 8055 (2010). [CrossRef] [PubMed]

, 9

9. V. P. Kalosha, L. Chen, and X. Bao, “Slow and fast light via SBS in optical fibers for short pulses and broadband pump,” Opt. Express 14, 12693 (2006). [CrossRef] [PubMed]

]. These phenomena have been investigated in different kinds of media ranging from atomic vapors [10

10. B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776 (2010). [CrossRef]

], solid materials [11

11. S. Stepanov and M. P. Sánchez, “Slow and fast light via two-wave mixing in erbium-doped fibers with saturable absorption,” Phys. Rev. A 80, 053830 (2009). [CrossRef]

] to cavity optomechanical system [4

4. A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69 (2011). [CrossRef] [PubMed]

]. The slow and stopped light achieved in several solid state systems have thrust EIT to the forefront of detailed study during the past two decades. EIT has been proved to be a powerful technique that can be used to eliminate the effect of a medium on a propagating beam of electromagnetic radiation, while retaining the large and desirable nonlinear optical properties associated with the resonant response of a medium [12

12. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36 (1997). [CrossRef]

, 13

13. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge Iniversity Press, 1997).

]. Recently, at low temperature, Painter et al. [4

4. A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69 (2011). [CrossRef] [PubMed]

] have reported an optically tunable delay of 50 nanoseconds with near-unity optical transparency, and superluminal light with a 1.4 microsecond signal advance in cavity optomechanical system, using electromagnetically induce transparency.

Here, we show for the first time that possibilities to obtain slow and fast light are offered by a suspended carbon nanotube (CNT) resonator, whose potentialities for this purpose remain unexplored to date. Carbon nanotube resonator are particularly attractive for their unique features of strong nonlinear response, long vibration lifetime, easy fabrication and high quality factor [14

14. P. Avouris, M. Freitag, and V. Perebeinos, “Carbon-nanotube photonics and optoelectronics,” Nat. Photonics 2, 341 (2008). [CrossRef]

, 15

15. M. Muoth, T. Helbling, L. Durrer, S.-W. Lee, C. Roman, and C. Hierold, “Hysteresis-free operation of suspended carbon nanotube transistors,” Nat. Nanotechnol. 5, 589 (2010). [CrossRef] [PubMed]

]. Consequently, CNT resonators enable ultrasensitive photonic sensors such as photovoltaics, nanotherapeutics, bioimaging, and superconducting device [16

16. R. Singhal, Z. Orynbayeva, R. V. K. Sundaram, J. J. Niu, S. Bhattacharyya, E. A. Vitol, M. G. Schrlau, E. S. Papazoglou, G. Friedman, and Y. Gogotsi, “Multifunctional carbon-nanotube cellular endoscopes,” Nat. Nanotechnol. 6, 57 (2010) [CrossRef] [PubMed]

20

20. H. Farhat, S. Berciaud, M. Kalbac, R. Saito, T. F. Heinz, M. S. Dresselhaus, and J. Kong, “Observation of electronic Raman scattering in metallic carbon nanotubes,” Phys. Rev. Lett. 107, 157401 (2011). [CrossRef] [PubMed]

]. Furthermore, carbon nanotube is not very sensitive to external temperature and perturbation, which will enhance the transmission spectrum and reduce the noise. In the present work, we design a tunable slow and fast light device based on a suspended carbon nanotube resonator, in the presence of a strong pump laser and a weak signal laser. The signal light group velocity can be accelerated and decreased significantly by turning the pump laser on- and off-resonance with the exciton frequency in CNT, which correspond to the negative and positive dispersion, respectively, associating with vanished absorption. Detailed analysis also shows that the linewidth of signal light spectrum is determined by the vibration decay rate of carbon nanotube. The smaller the decay rate of carbon nanotube is, the narrower the signal spectral bandwidth is.

2. Theory

As schematically represented in Fig. 1, the suspended carbon nanotube resonator is under the radiation of a strong pump laser (with frequency ωp) and a weak signal laser (with frequency ωs). Recently, such two-laser technique has been experimentally investigated by several groups to study the cavity optomechanical system and demonstrate the achievement of slow light and on-chip storage of light pulses [4

4. A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69 (2011). [CrossRef] [PubMed]

, 5

5. V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011). [CrossRef] [PubMed]

, 21

21. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010). [CrossRef] [PubMed]

, 22

22. J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204 (2011). [CrossRef] [PubMed]

].

Fig. 1 Schematic of a suspended carbon nanotube resonator with a localized exciton. This system is driven by a strong pump laser and probed by a weak signal laser. The inset describes the energy level of exciton while dressing with the vibration modes of carbon nanotube resonator.

The structure of a semiconducting carbon nanotube can be viewed as a graphene rolled into a cylinder, where the center of mass of the exciton is localized via the spatial modulation of the Stark-shift induced by a static inhomogeneous electric field. The quantum confinement is induced by the inhomogeneity in the field component along the CNT axis E//. In turn the normal component E can be used to induce a tunable parametric coupling between the exciton and the flexural motion of the CNT [23

23. I. Wilson-Rae, C. Galland, W. Zwerger, and A. Imamoğlu, “Nano-optomechanics with localized carbon nanotube excitons,” arXiv:0911.1330 (2009).

]. This localized exciton is formed in the segment of nanotube between the doubly clamped suspensions, leading to a quantized energy spectrum in the longitudinal direction. The inset of Fig. 1 shows the energy levels of localized exciton in suspended nanotube resonator when dressing with the vibration modes of CNT, which can be modeled as a two-level structure consisting of the ground state |g〉 and the first excited state (single exciton) |ex〉. Such an exciton can be characterized by the pseudospin −1/2 operators S± and Sz. Then the Hamiltonian of this localized two-level exciton can be described as Hex = h̄ωexSz, where ωex is the frequency of exciton. Besides, here we consider the flexural branch which is expected to present resonance with a free spectral range larger than the optical linewidth of the zero phonon line of the excitonic transition and focus on laser excitations near resonant with the lowest-frequency flexural phonon mode [23

23. I. Wilson-Rae, C. Galland, W. Zwerger, and A. Imamoğlu, “Nano-optomechanics with localized carbon nanotube excitons,” arXiv:0911.1330 (2009).

, 24

24. I. Wilson-Rae, “Intrinsic dissipation in nanomechanical resonators due to phonon tunneling,” Phys. Rev. B 77, 245418 (2008). [CrossRef]

]. In this case, this lowest-energy resonance corresponds to the fundamental in-plane flexural mode with the frequency ωn and the resonator is assumed to be characterized by sufficiently high quality factors. The eigenmode of CNT can be described by a quantum harmonic oscillator with b and b+ (the bosonic annihilation and creation operators with a quantum energy h̄ωn). The vibration Hamiltonian of nanotube resonator is given by Hn = h̄ωnb+b, where the vibration modes of CNT can be treated as phonon modes.

In the simultaneous presence of a strong pump field and a weak signal field, the Hamiltonian of the total system can be written as [23

23. I. Wilson-Rae, C. Galland, W. Zwerger, and A. Imamoğlu, “Nano-optomechanics with localized carbon nanotube excitons,” arXiv:0911.1330 (2009).

, 25

25. R. W. Boyd, Nonlinear Optics (Academic Press, 2008), p. 313.

, 26

26. J. J. Li and K. D. Zhu, “All-optical Kerr modulator based on a carbon nanotube resonator,” Phys. Rev. B 83, 115445 (2011). [CrossRef]

]
H=Hex+Hn+Hexn+Hexo=h¯ωexSz+h¯ωnb+b+h¯ωnηSz(b++b)μ(S+Epeiωpt+SEp*eiωpt)μ(S+Eseiωst+SEs*eiωst),
(1)
where Hexn = h̄ωnηSz(b+ + b) represents the interaction between the nanotube resonator and the exciton [23

23. I. Wilson-Rae, C. Galland, W. Zwerger, and A. Imamoğlu, “Nano-optomechanics with localized carbon nanotube excitons,” arXiv:0911.1330 (2009).

, 27

27. K. F. Graff, Wave Motion in Elastic Solids (Dover, 1991) pp. 539–564.

], and η is the coupling strength. Hexo=μ(S+Epeiωpt+SEp*eiωpt)μ(S+Eseiωst+SEs*eiωst) describes the exciton coupling to the two optical fields, where Ep and Es are slowly varying envelopes of the pump field and signal field, respectively, and μ is the electric dipole moment of the exciton. In a frame rotating at the pump field frequency ωp, the total Hamiltonian of the coupled system reads as follows
H=h¯ΔpSz+h¯ωnb+b+h¯ωnηSz(b++b)h¯(ΩS++Ω*S)μ(S+Eseiδt+SEs*eiδt),
(2)
where Δp = ωexωp is the pump field-exciton detuning, δ = ωsωp is the signal-pump field detuning, and Ω = μEp/ is the Rabi frequency of the pump laser.

Applying the Heisenberg equations of motion for operators Sz, S and Q and introducing the corresponding damping and noise terms, we derive the quantum Langevin equations as follows [28

28. C. W. Gardiner and P. Zoller, Quantum Noise, 2nd ed. (Springer, 2000) pp. 425–433.

, 29

29. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994) pp. 245–265.

]:
ddtSz=Γ1(Sz+12)+iΩS+iΩ*S+iμEseiδth¯S+iμEs*eiδth¯S,
(3)
ddtS=(iΔp+Γ2)SiωnηQS2iΩSz2iμEseiδth¯Sz+F^e,
(4)
d2dt2Q+1τnddtQ+ωn2Q=2ωn2ηSz+ξ^,
(5)
where Q = b+ + b, Γ1 and Γ2 are the exciton spontaneous emission rate and dephasing rate, respectively. τn is the vibration lifetime of carbon nanotube [23

23. I. Wilson-Rae, C. Galland, W. Zwerger, and A. Imamoğlu, “Nano-optomechanics with localized carbon nanotube excitons,” arXiv:0911.1330 (2009).

, 30

30. K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. 76, 061101 (2005). [CrossRef]

], e is the δ-correlated Langevin noise operator, which has zero mean 〈e〉 = 0 and obeys the correlation function F^e(t)F^e+(t)δ(tt).

The motion of carbon nanotube resonator is affected by thermal bath of Brownian and non-Morkovian process [28

28. C. W. Gardiner and P. Zoller, Quantum Noise, 2nd ed. (Springer, 2000) pp. 425–433.

,31

31. V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001). [CrossRef]

]. The quantum effects on the resonator are only observed in the limit of high quality factor, that obeys Q = ωn/γn ≫ 1. The Brownian noise operator can be modeled as Markovian with the decay rate γn (γn = 1/τn) of the resonator mode. Therefore, the Brownian stochastic force has zero mean value 〈ξ̂〉 = 0 that can be characterized as [31

31. V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001). [CrossRef]

]
ξ^+(t)ξ^(t)=γnωndω2πωeiω(tt)[1+coth(h¯ω2kBT)].
(6)
Following standard methods from quantum optics, we derive the steady-state solution to Eqs. (3)(5) by setting all the time derivatives to zero. They are given by
S0=2ΩS0z(Δp+ωnηQ0)iΓ2,Q0=2ηS0z,
(7)
where S0z is determined by Eq. (16) (see below). To go beyond weak coupling, we can always rewrite each Heisenberg operator as the sum of its steady-state mean value and a small fluctuation with zero mean value as follows:
S=S0+δS,Sz=S0z+δSz,Q=Q0+δQ.
(8)
Inserting these equations into the Langevin equations Eqs. (3)(5), one can safely neglect the nonlinear term δQδS. Since the optical drives are weak, but classical coherent fields, we will identify all operators with their expectation values, and drop the quantum and thermal noise terms [21

21. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010). [CrossRef] [PubMed]

]. Then the linearized Langevin equations can be written as:
δS˙z=Γ1δSz+iΩδ(S)*iΩ*δS+iμEseiδth¯δ(S)*iμEs*eiδth¯δS,
(9)
δS˙=(iΔp+Γ2)δSiωnη(δSQ0+S0δQ)2iΩδSz2iμEseiδth¯δSz,
(10)
δ¨Q+1τnδ˙Q+ωn2δQ=2ωn2ηδSz.
(11)

In order to solve Eqs. (9)(11), we make the ansatz [21

21. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010). [CrossRef] [PubMed]

, 25

25. R. W. Boyd, Nonlinear Optics (Academic Press, 2008), p. 313.

] 〈δS〉 = S+e−iδt + Seiδt, δSz=S+zeiδt+S_zeiδt and 〈δQ〉 = Q+e−iδt + Qeiδt. Upon substituting the approximation to Eqs. (9)(11) and working to the lowest order in Es, but to all orders in Ep, we finally obtain the linear optical susceptibility S+ in the steady state as the following solution
χ(1)(ωs)eff=ρμS+ɛ0Es=ρμ2ɛ0h¯Γ2χ(1)(ωs)=Σ1χ(1)(ωs),
(12)
where Σ1 = ρμ2/ε0Γ2, ρ is the number density of CNT, which during the measurement the identical resonator arrays are needed because of the weak measurable signal and weak interaction with light produced by a single nanotube resonator. For a better experimental observation, ρ should be about 109m−3 [32

32. S. Yasukochi, T. Murai, S. Moritsubo, T. Shimada, S. Chiashi, S. Maruyama, and Y. K. Kato, “Gate-induced blueshift and quenching of photoluminescence in suspended single-walled carbon nanotubes,” Phys. Rev. B 84, 121409 (2011). [CrossRef]

]. ε0 is the dielectric constant of vacuum. In Eq. (12), the imaginary part and real part of χ(1) (ωs) correspond to the absorption and dispersion of the signal field, respectively [25

25. R. W. Boyd, Nonlinear Optics (Academic Press, 2008), p. 313.

]. In this case, all the interacting elements are considered during the physical treatment, except for the light-phonons coupling which is induced indirectly through the light-exciton interaction via deformation potential coupling. The dimensionless linear optical susceptibility is given by
χ(1)(ωs)=w0f(δ0)×{2e1ΩR2(e1+δ0)(e2+ωn0η2w0ζ)e1e2[2ΩR2(e1+ωn0η2w0ζ)e1(2i+δ0)(e1+δ0)]},
(13)
where e1 = Δp0ωn0η2w0 + i, e2 = Δp0ωn0η2w0i, and w0=2S0z, Γ1 = 2Γ2, ωn0 = ωn2, γn0 = γn2, ΩR = Ω/Γ2, δ0 = δ2, and Δp0 = Δp2.

The auxiliary function ζ(δ0) and function f (δ0) are given by
ζ(δ0)=ωn02ωn02iδ0γn0δ02,
(14)
f(δ0)=e1e2(e1δ0)[2ΩR2(e1+ωn0η2w0ζ)e1(2i+δ0)(e1+δ0)]2e12ΩR2(e1+δ0)(e2+ωn0η2w0ζ).
(15)
The population inversion w0 of the exciton is determined by the following equation
(w0+1)[(Δp0ωn0η2w0)2+1]+2ΩR2w0=0.
(16)

In terms of this model, we can determine the light group velocity as [33

33. R. S. Bennink, R. W. Boyd, C. R. Stroud, and V. Wong, “Enhanced self-action effects by electromagnetically induced transparency in the two-level atom,” Phys. Rev. A 63, 033804 (2001). [CrossRef]

, 34

34. S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A 46, R29 (1992). [CrossRef] [PubMed]

]
vg=cn+ωs(dn/dωs),
(17)
where n1+2πχeff(1), and then
cvg=1+2πReχeff(1)(ωs)ωs=ωex+2πωsRe(dχeff(1)dωs)ωs=ωex.
(18)
It is clear from this expression for vg that when Reχ (ωs)ωsex is zero and the dispersion is steeply positive or negative, the group velocity is significantly reduced or increased, and then we define the group velocity index ng as
ng=cvg1=cvgvg=2πωexρμ2h¯Γ2Re(dχ(1)(ωs)dωs)ωs=ωex=Γ2ΣRe(dχ(1)(ωs)dωs)ωs=ωex,
(19)
where Σ=2πωexρμ2/ɛ0h¯Γ22. One can observe the slow light when ng > 0, and the superluminal light when ng < 0 [35

35. R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074 (2009). [CrossRef] [PubMed]

].

3. Numerical results and discussion

In order to show the tunable slow and fast light clearly, we choose the realistic suspended carbon nanotube resonator with an ambient temperature 4.2K, for (ωn/2π2,γn,η ) = (725MHz,310MHz,0.8MHz,0.17) [23

23. I. Wilson-Rae, C. Galland, W. Zwerger, and A. Imamoğlu, “Nano-optomechanics with localized carbon nanotube excitons,” arXiv:0911.1330 (2009).

, 26

26. J. J. Li and K. D. Zhu, “All-optical Kerr modulator based on a carbon nanotube resonator,” Phys. Rev. B 83, 115445 (2011). [CrossRef]

]. The quality factor of carbon nanotube is Q = 900.

Firstly, we interpret the basic principle of the slow and fast light in a carbon nanotube resonator, under the radiation of a strong pump laser and a weak signal laser. According to Eq. (12), we depict the resonance absorption spectrum of carbon nanotube as a function of signal-pump beam detuning δ with Δp0 = 0 (as shown in Fig. 2). Since the original energy levels of the localized exciton have been dressed by the vibration modes of the nanotube (here the vibration modes is the same as the phonon modes), the uncoupled energy levels (|ex〉 and |g〉) split into dressed states |ex,n〉 and |g,n〉, which is shown in Fig. 2(b) (|n〉 denotes the number state of the resonance mode). The curve shown in the Fig. 2(a) displays three prominent features - the middle part consisting of three weak peaks and two sharp peaks at the both sides. The middle part is analogy with conventional atomic two-level systems [25

25. R. W. Boyd, Nonlinear Optics (Academic Press, 2008), p. 313.

], which shows the origin of nanotube vibration induced stimulated Rayleigh resonance. Here the electrons make a transition from the lowest dressed level |g,n〉 to the dressed level |ex,n〉, which is signified by the transition 2. Besides, it’s worth notice that the two sharp peaks at the both sides in absorption spectrum are totally different from those in atomic system [25

25. R. W. Boyd, Nonlinear Optics (Academic Press, 2008), p. 313.

]. These two sharp peaks represent the resonance amplification and absorption of the carbon nanotube, which can be interpreted by the transition 1 and 3. The left peak corresponds to the amplification of the signal laser, where electrons making a transition from the lowest dressed level |g,n〉 to the highest dressed level |ex,n + 1〉 by the simultaneous absorption of two pump laser photons and emission of a photon at ωpωn. This process can amplify a wave at δ= −ωn. Otherwise, the right peak is an absorption process, which corresponds to the usual absorption resonance as modified by the ac-Stark effect. The underlying physical mechanism for this phenomenon can also be understood as follows. The simultaneous presence of the pump and signal fields generates a beat wave oscillating at the beat frequency δ= ωsωp to drive the CNT via the localized exciton. If the beat frequency δ is close to the resonance frequency ωn, the CNT starts to oscillate coherently, which will result in Stokes (ωs = ωpωn) and anti-Stokes (ωas = ωp + ωn) scattering of light from the pump field via the localized exciton. For the near-resonant signal laser, the signal field will interfere with the Stokes field and the anti-Stokes field, respectively. As a result the signal spectrum can be modified significantly. In the picture of the dressed states, the system is similar to the conventional three-level system in EIT. Here coupling to phonons seems to provide the exciton with additional energy levels to realize EIT phenomena. Therefore in our structure one can obtain the slow output light without absorption only by simply adjusting the pump-exciton detuning to the vibrational frequency of the nanotube resonator. In this case, we conclude that the coupling between exciton and the vibration of carbon nanotube plays an important role in the implement of these specific features.

Fig. 2 (a) The absorption spectrum of a signal field as a function of signal-pump detuning for the case ΩR2=2,ωn0 = 6, Δp0 = 0, γn0 = 0.003, and η = 0.17. (b) The energy levels of localized exciton while dressing with the vibrational modes of nanotube resonator. The transition 1,2,3 correspond to the characteristic parts shown in (a), respectively.

Fig. 3 A tunable device of slow and fast light using a carbon nanotube resonator. (a) The dimensionless imaginary part and real part of the linear optical susceptibility (in units of Σ1) as a function of the signal-exciton detuning for Δp = 0; (b) The group velocity index ng of superluminal light (in units of Σ) as a function of pump Rabi frequency Ω2; (c) The same plots with (a) except Δp = ωn; (d) The group velocity index ng(= c/vg) of slow light (in units of Σ) as a function of the pump Rabi frequency Ω2. The other parameters used in plot(a)–(d) are Ω2 = 0.1(GHz)2, ωn = 0.725GHz, γn = 0.8MHz, and η = 0.17.

In the picture of the dressed states shown in the Fig. 2(b), the condition Δp = ωn just corresponds to that the pump field couples to the optical transition via the Stokes process and the system becomes fully transparent to the signal beam. In this case, the system is similar to the conventional three-level systems in EIT [36

36. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005). [CrossRef]

]. Here coupling to a CNT resonator seems to provide the exciton with additional energy levels to realize EIT phenomena. Therefore in our structure one can obtain the fast or slow output light without absorption only by simply adjusting the pump laser on- or off-resonant with exciton frequency.

Figure 4 displays the the imaginary part of χ(1) as a function of Δs for three different decay rates of γn. The inset of Fig. 4 shows the amplification of the most remarkable region of transparency. From this figure, we can demonstrate that the width of the signal spectrum increases as the decay rate γn increases. Therefore the shorter the CNT resonator decay is, the narrower of the signal spectrum width is. When the decay rate of the resonator is 0.1GHz, the hole width in the spectrum becomes flat as shown in the inset. As a result, the CNT resonator with small decay rate is beneficial to the transparency window. Due to the hight quality factor and short decay rate of CNT resonator, the slow light and fast light effect performed in CNT is obviously better than that in other quantum systems such as quantum wells and quantum dots.

Fig. 4 The absorption spectrum of a signal field as a function of the detuning Δs with three different decay rates of CNT resonator. The other parameters used are Ω2 = 0.1(GHz)2, ωn = 0.725GHz, Δp = 0.725GHz, and η = 0.17. The inset shows the amplification of the most remarkable region of transparency.

Furthermore, there is an interesting feature we should notice, that the abscissa of the two sharp peak shown in Fig. 3(a) just corresponds to the vibrational frequency of CNT resonator (Δs = ωn = 725MHz). That is, fixing the pump laser on the exciton frequency Δp = 0 and scanning the exciton frequency using another signal laser, one can obtain the vibrational frequency of CNT resonator at signal absorption or dispersion spectrum efficiently. This is a precise and easy method to get the vibrational frequency of CNT resonator in all optical domain.

4. Conclusion

In conclusion, we have proposed a theoretical model for a tunable slow and fast light device based on a suspended carbon nanotube resonator, in the presence of a strong pump laser and a weak signal laser. Pump laser on- or off-resonant with exciton frequency in CNT resonator results in the fast or slow output signal light simultaneously, together with zero absorption. Absorption spectrum displays that the width of such slow and fast light is decided by the decay rate of CNT resonator. Furthermore, our model can also be used to measure the vibrational frequency of CNT resonator either in absorption spectrum or dispersion spectrum. We believe that such a tunable slow and fast light device may be have the potential applications in optical networks and engineering in nanometer scale.

Acknowledgments

The authors thanks for National Natural Science Foundation of China (No. 10774101 and No. 10974133), the National Ministry of Education Program for Ph.D.

References and links

1.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074 (2009). [CrossRef] [PubMed]

2.

B. Gu, N. H. Kwong, R. Binder, and A. L. Smirl, “Slow and fast light associated with polariton interference,” Phys. Rev. B 82, 035313 (2010). [CrossRef]

3.

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Applications of slow light in telecommunications,” Opt. Photon. News 19, 18 (2006). [CrossRef]

4.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69 (2011). [CrossRef] [PubMed]

5.

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011). [CrossRef] [PubMed]

6.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594(1999). [CrossRef]

7.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003). [CrossRef] [PubMed]

8.

V. I. Kovalev, N. E. Kotova, and R. G. Harrison, “Slow light in stimulated Brillouin scattering: on the influence of the spectral width of pump radiation on the group index: reply,” Opt. Express 18, 8055 (2010). [CrossRef] [PubMed]

9.

V. P. Kalosha, L. Chen, and X. Bao, “Slow and fast light via SBS in optical fibers for short pulses and broadband pump,” Opt. Express 14, 12693 (2006). [CrossRef] [PubMed]

10.

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776 (2010). [CrossRef]

11.

S. Stepanov and M. P. Sánchez, “Slow and fast light via two-wave mixing in erbium-doped fibers with saturable absorption,” Phys. Rev. A 80, 053830 (2009). [CrossRef]

12.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36 (1997). [CrossRef]

13.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge Iniversity Press, 1997).

14.

P. Avouris, M. Freitag, and V. Perebeinos, “Carbon-nanotube photonics and optoelectronics,” Nat. Photonics 2, 341 (2008). [CrossRef]

15.

M. Muoth, T. Helbling, L. Durrer, S.-W. Lee, C. Roman, and C. Hierold, “Hysteresis-free operation of suspended carbon nanotube transistors,” Nat. Nanotechnol. 5, 589 (2010). [CrossRef] [PubMed]

16.

R. Singhal, Z. Orynbayeva, R. V. K. Sundaram, J. J. Niu, S. Bhattacharyya, E. A. Vitol, M. G. Schrlau, E. S. Papazoglou, G. Friedman, and Y. Gogotsi, “Multifunctional carbon-nanotube cellular endoscopes,” Nat. Nanotechnol. 6, 57 (2010) [CrossRef] [PubMed]

17.

W. Belzig, “Hybrid superconducting devices: bound in a nanotube,” Nat. Phys. 6, 940 (2010). [CrossRef]

18.

A. Pályi, P. R. Struck, M. Rudner, K. Flensberg, and G. Burkard, “Spin-orbit induced strong coupling of a single spin to a nanomechanical resonator,” ArXiv:1110.4893v1 (2011).

19.

C. Ohm, C. Stampfer, J. Splettstoesser, and M. R. Wegewijs, “Readout of carbon nanotube vibrations based on spin-phonon coupling,” ArXiv:1110.5165v1 (2011).

20.

H. Farhat, S. Berciaud, M. Kalbac, R. Saito, T. F. Heinz, M. S. Dresselhaus, and J. Kong, “Observation of electronic Raman scattering in metallic carbon nanotubes,” Phys. Rev. Lett. 107, 157401 (2011). [CrossRef] [PubMed]

21.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010). [CrossRef] [PubMed]

22.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204 (2011). [CrossRef] [PubMed]

23.

I. Wilson-Rae, C. Galland, W. Zwerger, and A. Imamoğlu, “Nano-optomechanics with localized carbon nanotube excitons,” arXiv:0911.1330 (2009).

24.

I. Wilson-Rae, “Intrinsic dissipation in nanomechanical resonators due to phonon tunneling,” Phys. Rev. B 77, 245418 (2008). [CrossRef]

25.

R. W. Boyd, Nonlinear Optics (Academic Press, 2008), p. 313.

26.

J. J. Li and K. D. Zhu, “All-optical Kerr modulator based on a carbon nanotube resonator,” Phys. Rev. B 83, 115445 (2011). [CrossRef]

27.

K. F. Graff, Wave Motion in Elastic Solids (Dover, 1991) pp. 539–564.

28.

C. W. Gardiner and P. Zoller, Quantum Noise, 2nd ed. (Springer, 2000) pp. 425–433.

29.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994) pp. 245–265.

30.

K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. 76, 061101 (2005). [CrossRef]

31.

V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001). [CrossRef]

32.

S. Yasukochi, T. Murai, S. Moritsubo, T. Shimada, S. Chiashi, S. Maruyama, and Y. K. Kato, “Gate-induced blueshift and quenching of photoluminescence in suspended single-walled carbon nanotubes,” Phys. Rev. B 84, 121409 (2011). [CrossRef]

33.

R. S. Bennink, R. W. Boyd, C. R. Stroud, and V. Wong, “Enhanced self-action effects by electromagnetically induced transparency in the two-level atom,” Phys. Rev. A 63, 033804 (2001). [CrossRef]

34.

S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A 46, R29 (1992). [CrossRef] [PubMed]

35.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074 (2009). [CrossRef] [PubMed]

36.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005). [CrossRef]

OCIS Codes
(230.1150) Optical devices : All-optical devices
(270.0270) Quantum optics : Quantum optics

ToC Category:
Optical Devices

History
Original Manuscript: November 14, 2011
Revised Manuscript: December 24, 2011
Manuscript Accepted: January 20, 2012
Published: February 27, 2012

Citation
Jin-Jin Li and Ka-Di Zhu, "Tunable slow and fast light device based on a carbon nanotube resonator," Opt. Express 20, 5840-5848 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-5840


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References

  1. R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science326, 1074 (2009). [CrossRef] [PubMed]
  2. B. Gu, N. H. Kwong, R. Binder, and A. L. Smirl, “Slow and fast light associated with polariton interference,” Phys. Rev. B82, 035313 (2010). [CrossRef]
  3. R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Applications of slow light in telecommunications,” Opt. Photon. News19, 18 (2006). [CrossRef]
  4. A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature472, 69 (2011). [CrossRef] [PubMed]
  5. V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett.107, 133601 (2011). [CrossRef] [PubMed]
  6. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature397, 594(1999). [CrossRef]
  7. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett.90, 113903 (2003). [CrossRef] [PubMed]
  8. V. I. Kovalev, N. E. Kotova, and R. G. Harrison, “Slow light in stimulated Brillouin scattering: on the influence of the spectral width of pump radiation on the group index: reply,” Opt. Express18, 8055 (2010). [CrossRef] [PubMed]
  9. V. P. Kalosha, L. Chen, and X. Bao, “Slow and fast light via SBS in optical fibers for short pulses and broadband pump,” Opt. Express14, 12693 (2006). [CrossRef] [PubMed]
  10. B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics4, 776 (2010). [CrossRef]
  11. S. Stepanov and M. P. Sánchez, “Slow and fast light via two-wave mixing in erbium-doped fibers with saturable absorption,” Phys. Rev. A80, 053830 (2009). [CrossRef]
  12. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today50, 36 (1997). [CrossRef]
  13. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge Iniversity Press, 1997).
  14. P. Avouris, M. Freitag, and V. Perebeinos, “Carbon-nanotube photonics and optoelectronics,” Nat. Photonics2, 341 (2008). [CrossRef]
  15. M. Muoth, T. Helbling, L. Durrer, S.-W. Lee, C. Roman, and C. Hierold, “Hysteresis-free operation of suspended carbon nanotube transistors,” Nat. Nanotechnol.5, 589 (2010). [CrossRef] [PubMed]
  16. R. Singhal, Z. Orynbayeva, R. V. K. Sundaram, J. J. Niu, S. Bhattacharyya, E. A. Vitol, M. G. Schrlau, E. S. Papazoglou, G. Friedman, and Y. Gogotsi, “Multifunctional carbon-nanotube cellular endoscopes,” Nat. Nanotechnol.6, 57 (2010) [CrossRef] [PubMed]
  17. W. Belzig, “Hybrid superconducting devices: bound in a nanotube,” Nat. Phys.6, 940 (2010). [CrossRef]
  18. A. Pályi, P. R. Struck, M. Rudner, K. Flensberg, and G. Burkard, “Spin-orbit induced strong coupling of a single spin to a nanomechanical resonator,” ArXiv:1110.4893v1 (2011).
  19. C. Ohm, C. Stampfer, J. Splettstoesser, and M. R. Wegewijs, “Readout of carbon nanotube vibrations based on spin-phonon coupling,” ArXiv:1110.5165v1 (2011).
  20. H. Farhat, S. Berciaud, M. Kalbac, R. Saito, T. F. Heinz, M. S. Dresselhaus, and J. Kong, “Observation of electronic Raman scattering in metallic carbon nanotubes,” Phys. Rev. Lett.107, 157401 (2011). [CrossRef] [PubMed]
  21. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science330, 1520 (2010). [CrossRef] [PubMed]
  22. J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature471, 204 (2011). [CrossRef] [PubMed]
  23. I. Wilson-Rae, C. Galland, W. Zwerger, and A. Imamoğlu, “Nano-optomechanics with localized carbon nanotube excitons,” arXiv:0911.1330 (2009).
  24. I. Wilson-Rae, “Intrinsic dissipation in nanomechanical resonators due to phonon tunneling,” Phys. Rev. B77, 245418 (2008). [CrossRef]
  25. R. W. Boyd, Nonlinear Optics (Academic Press, 2008), p. 313.
  26. J. J. Li and K. D. Zhu, “All-optical Kerr modulator based on a carbon nanotube resonator,” Phys. Rev. B83, 115445 (2011). [CrossRef]
  27. K. F. Graff, Wave Motion in Elastic Solids (Dover, 1991) pp. 539–564.
  28. C. W. Gardiner and P. Zoller, Quantum Noise, 2nd ed. (Springer, 2000) pp. 425–433.
  29. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994) pp. 245–265.
  30. K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum.76, 061101 (2005). [CrossRef]
  31. V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A63, 023812 (2001). [CrossRef]
  32. S. Yasukochi, T. Murai, S. Moritsubo, T. Shimada, S. Chiashi, S. Maruyama, and Y. K. Kato, “Gate-induced blueshift and quenching of photoluminescence in suspended single-walled carbon nanotubes,” Phys. Rev. B84, 121409 (2011). [CrossRef]
  33. R. S. Bennink, R. W. Boyd, C. R. Stroud, and V. Wong, “Enhanced self-action effects by electromagnetically induced transparency in the two-level atom,” Phys. Rev. A63, 033804 (2001). [CrossRef]
  34. S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A46, R29 (1992). [CrossRef] [PubMed]
  35. R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science326, 1074 (2009). [CrossRef] [PubMed]
  36. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633 (2005). [CrossRef]

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