We study a double-cavity optomechanical system in which a movable mirror with perfect reflection is inserted between two fixed mirrors with partial transmission. This optomechanical system is driven from both fixed end mirrors in a symmetric scheme by two strong coupling fields and two weak probe fields. We find that three interesting phenomena: coherent perfect absorption (CPA), coherent perfect transmission (CPT), and coherent perfect synthesis (CPS) can be attained within different parameter regimes. That is, we can make two input probe fields totally absorbed by the movable mirror without yielding any energy output from either end mirror (CPA); make an input probe field transmitted from one end mirror to the other end mirror without suffering any energy loss in the two cavities (CPT); make two input probe fields synthesized into one output probe field after undergoing either a perfect transmission or a perfect reflection (CPS). These interesting phenomena originate from the efficient hybrid coupling of optical and mechanical modes and may be all-optically controlled to realize novel photonic devices in quantum information networks.
© 2014 Optical Society of America
Cavity optomechanics has quickly developed over the past few years as a crossover field of nanophysics and quantum optics dealing with the hybrid interaction between light fields and mechanical motions. Relevant studies allow the high-sensitivity detection of tiny mass, force, and displacement by virtue of the radiation-pressure coupling between optical modes of cavities and mechanical degrees of freedom [1–4]. One standard and simplest optomechanical setup is a Fabry-Perot cavity with one end mirror being a micro- or nano-mechanical vibrating object [5–7]. Other optomechanical devices were also designed and tested such as silica toroidal optical microresonators with radiation pressure coupling [8–11] and typical optomechanical cavities confining a Bose-Einstein condensate (BEC) [12–17]. Three-mode or multimode optomechanical systems have been further studied to attain hybrid quantum entanglement , optomechanically induced light transparency , single-photon nonlinearities , and ground-state cooling of mechanical modes [21, 22].
Electromagnetically induced transparency (EIT) and normal mode splitting (NMS), in particular, have been predicted and observed in a few optomechanical systems where the output probe field depends on the incident control field via the nonlinear optomechanical interaction [23–27]. EIT in multi-mode optomechanical systems is an analogue of that in multi-level atomic systems : a three-level atomic medium can be made transparent to a weak probe field as illuminated by a strong coupling field in the case of two-photon resonance. Then slowing down and even stopping light signals [29, 30] in the long-lived mechanical vibrations may be realized by the EIT technique. Recently, Agarwal and Huang studied a double-ended cavity optomechanical system with a movable in-between mirror of partial reflection inserted between two fixed end mirrors of equal transmission . They showed that a phenomenon opposed to EIT, coherent perfect absorption (CPA), could be attained with realistic parameters when this system was driven by one strong coupling field and two weak probe fields. Such a double-cavity optomechanical system was first studied by Paternostro et al. in 2007 under the viewpoint of quantum correlations for achieving macroscopic entanglement .
In this paper, we study a slightly different double-cavity optomechanical system with an movable mirror of perfect reflection and driven by two coupling fields and two probe fields. We find that it is possible to attain coherent perfect transmission (CPT) and coherent perfect synthesis (CPS) in addition to CPA  as far as the probe field propagation is concerned. Here CPT means that a probe field input from one end mirror is observed from the other end mirror without suffering any loss after a perfect transmission through the double-cavity system; CPS means that two probe fields input from different end mirrors are synthesized into one output probe field at one end mirror by undergoing either a perfect transmission or a perfect reflection. Note that the realization of CPA  needs a large decay rate of the movable mirror while both CPT and CPS can be realized only when the movable mirror has a small decay rate. Actual probe frequencies of CPA, CPT, and CPS can be easily changed by controlling the effective optomechanical coupling rate, i.e., by modulating one or both coupling fields in amplitude. We expect that the all-optically controlled realization of CPA, CPT, and CPS could be explored to build new tunable photonic devices in quantum information networks.
2. Model and equations
We consider a double-cavity hybrid system with one movable mirror (membrane oscillator) of perfect reflection inserted between two fixed mirrors of partial transmission (see Fig. 1). The membrane oscillator has an eigen frequency ωm and a decay rate γm and thus exhibits a mechanical quality factor Q = ωm/γm. Two identical optical cavities of lengths L and frequencies ω0 are got when the membrane oscillator is at its equilibrium position in the absence of external excitation. We describe the two optical modes, respectively, by annihilation (creation) operators c1 ( ) and c2 ( ) while the only mechanical mode by b (b†). These annihilation and creation operators are restricted by the commutation relation (i = 1, 2), [c1, c2] = 0, and [b, b†] = 1. Two probe (coupling) fields are used to drive the double-cavity system from either left or right fixed mirrors with their amplitudes denoted by and and ). Here ℘L, ℘R, ℘cL, and ℘cR are relevant field powers, κ is the common decay rate of both cavity modes, and ωp (ωc) is the probe (coupling) field frequency. Then the total Hamiltonian in the rotating-wave frame of coupling frequency ωc can be written as
With Eq. (1), it is easy to obtain the following quantum Langevin equations for relevant annihilation operators of mechanical and optical modesEq.s (2) can be solved with the factorization assumption 〈bci〉 = 〈b〉 〈ci〉 to generate the steady-state mean values
In the presence of both probe fields, however, we can write each operator as the sum of its mean value and its small fluctuation (b = bs + δb, c1 = c1s + δc1, c2 = c2s + δc2) to solve Eq. (2) when both coupling fields are sufficiently strong. Then keeping only the linear terms of fluctuation operators and moving into an interaction picture by introducing δb → δbe−iωmt, δbin → δbine−iωmt, δc1 → δc1e−iΔ1t, , δc2 → δc2e−iΔ2t, , we obtain the linearized quantum Langevin equations
If each coupling field drives one cavity mode at the mechanical red sideband (Δ1 ≈ Δ2 ≈ ωm), the hybrid system is operating in the resolved sideband regime (ωm >> κ), the membrane oscillator has a high mechanical quality factor (ωm >> γm), and the mechanical frequency ωm is much larger than g0 |c1s| and g0 |c2s|, Eq.s (4) will be simplified to
Noting that the mean values of quantum and thermal noise terms are vanishing, we can examine the expectation values of small fluctuations arising from both weak probe fields by the following three coupled dynamic equationsEq.s (6) are assumed to be of the form: 〈δs〉 = δs+e−ixt +δs−eixt with s = b, c1, c2. Then it is straightforward to obtain with relevant substitutions the following results Eq.s (7), we have also assumed that c1s,2s is real-valued without loss of generality.Eq.s (8) we can obtain
3. Results and discussion
Now we will implement numerical calculations based on Eq.s (7) and Eq.s (9) to show that a few interesting phenomena such as CPA, CPT, and CPS could be attained in this double-cavity optomechanical system. Each of these phenomena corresponds to a distinct parameter regime accessible by devising the optical and mechanical devices or modulating the external driving laser fields in a proper way. In the following calculations, we will adopt realistic parameters as in ref.  with L = 25 mm, m = 145 ng, κ = 2π × 215 kHz, ωm = 2π × 947 kHz, and λ = 2πc/ωc = 1064 nm for a relevant experimental setup.
3.1. Coherent perfect absorption
First we consider how to achieve CPA by pursuing a parameter regime where we have εoutL+ = εoutR+ = 0 with εL ≠ 0 and εR ≠ 0. That is, both input probe fields are perfectly absorbed by this double-cavity system without yielding any output from either fixed cavity mirror. We find by combining Eq.s (7) and Eq.s (9) that CPA can be realized only when conditions are simultaneously satisfied. These conditions31] because n could be different from 1.0 in our system.
The normalized output probe energy |εoutL+|2/|εL|2 (|εoutR+|2/|εR|2) is plotted as a function of the normalized input probe detuning x/κ for different values of the effective optomechanical coupling rate G in Fig. 2; for different values of the cavity photon number ratio n in Fig. 3. It is clear that CPA can occur at zero, one resonant, or two detuned points depending on the values of G [see Fig. 2]. Here CPA means that both input probe fields are fully absorbed by the mechanical mode without being reflected or transmitted out of this double-cavity system. This is the result of a perfect destructive interference between the left-going and right-going probe photons with their energy dissipated via the fast mechanical decay. In addition, we can tune the two detuned CPA points x± by increasing the value of n (right coupling field amplitude εcR) for a fixed value of G (left coupling field amplitude εcL) [see Fig. 3]. But the right probe field amplitude should be accordingly modified to guarantee the occurrence of CPA as required by εR = nεL in Eq.s (10). More discussions on CPA can be found in ref. .
3.2. Coherent perfect transmission
Then we consider how to achieve CPT by pursuing a parameter regime where we have |εoutL+/εL| = 0 and |εoutR+/εL| = 1 with εL ≠ 0 and εR = 0. In this case, the left probe field will be observed from the right cavity mirror after a perfect transmission through this double-cavity system in the absence of the right probe field. Setting εR = 0 and n = 1, we find from Eq.s (7) and Eq.s (9) that CPT will occur at three points
In Fig. 4 and Fig. 5, we plot the normalized output probe field energy |εoutL+/εL|2 and |εoutR+/εL|2 as a function of the normalized input probe detuning x/κ for different values of the effective optomechanical coupling rate G. It is clear that we have |εoutL+/εL|2 = 0 and |εoutR+/εL|2 = 1 at x0 = 0 with G = 0.2κ [see the black curves]; at x0 = 0 and x± = ±1.37κ with G = 1.2κ [see the red curves]; at x0 = 0 and x± = ±2.03κ with G = 1.6κ [see the blue curves]. These numerical results are exactly consistent with Eq.s (11) indicating that we can choose on demand the desired points where the left probe field is observed at the right cavity mirror after a perfect transmission. We stress here that CPT is a pure quantum optomechanical phenomenon because the 100% energy transfer from one cavity mode to the other cavity mode relies on the lossless tunnelling of a probe field through the membrane oscillator with the perfect reflection. Then a considerable energy loss can be successfully avoided during the probe field tunnelling process only when the membrane’s decay rate is sufficiently small. With the increasing of γm, the energy transfer from one cavity mode to the other cavity mode will become evidently less than 100% for all probe field detunings.
3.3. Coherent perfect synthesis
Finally we consider how to achieve CPS by pursuing a parameter regime where we have |εoutL+/εL| = 0 and |εoutR+/εL|2 = 2 or |εoutL+/εL|2 = 2 and |εoutR+/εL| = 0 with εL = εR ≠ 0. In this case, both input probe fields are expected to be observed (without suffering any energy loss) from either the right or the left cavity mirror. Setting εR = εL and n = 1, we find from Eq.s (7) and Eq.s (9) that CPS will occur at two points
In Fig. 6 and Fig. 7, we plot the normalized output probe field energy |εoutL+/εL|2 and |εoutR+/εR|2 as a function of the normalized input probe detuning x/κ for different values of the probe field relative phase θ. It is clear that we have |εoutL+/εL|2 = 0 and |εoutR+/εR|2 = 2 at x+ = +0.6κ but |εoutL+/εL|2 = 2 and |εoutR+/εR|2 = 0 at x− = −0.6κ with θ = 0.5π [see the red curves]; |εoutL+/εL|2 = 0 and |εoutR+/εR|2 = 2 at x− = −0.6κ but |εoutL+/εL|2 = 2 and |εoutR+/εR|2 = 0 at x+ = +0.6κ with θ = 1.5π [see the blue curves]. Then a simple relative phase modulation from θ = 0.5π to θ = 1.5π will result in the exchange of output probe field from left cavity mirror to right cavity mirror or vice versa. It is clear that in CPS one probe field experiences a perfect transmission while the other probe field experiences a perfect reflection. It is not difficult to find from Eq.s (7) and Eq.s (9) that CPS is once again the result of quantum interference between εL and εR, which is constructive at one cavity mirror but destructive at the other cavity mirror in our optomechanical system.
In summary, we have studied in theory a two-mode optical cavity with an in-between membrane oscillator driven at most by two strong coupling fields and two weak probe fields. Our analytical and numerical results show that three interesting phenomena on the probe field transmission (CPA, CPT, and CPS) could be attained in this optomechanical system. In CPA, both probe fields experience a complete absorption without yielding any energy output from the two cavity mirrors when the membrane oscillator has a very large decay rate. In CPT, one probe field travels through the optomechanical system without suffering any energy loss in the absence of the other probe field if the membrane oscillator has a very small decay rate. In CPS, one probe field is totally transmitted while the other one is totally reflected to generate a perfect synthesis at one cavity mirror when the membrane oscillator has a very small decay rate. Each phenomenon is realized at one, two, or three points of probe frequency depending mainly on the two coupling field amplitudes and the two probe field phases. A flexible control of CPA, CPT, and CPS should be important in quantum information networks, e.g., for achieving photonic switching, routing, swapping, and entanglement.
This work is supported in part by the National Natural Science Foundation of China ( 61378094) and the National Basic Research Program of China ( 2011CB921603).
References and links
2. F. Marquardt and S. M. Girvin, “Trend: Optomechanics,” Physics 2, 40 (2009). [CrossRef]
3. P. Verlot, A. Tavernarakis, T. Briant, P.-F. Cohadon, and A. Heidmann, “Backaction amplification and quantum limits in optomechanical measurements,” Phys. Rev. Lett. 104, 133602 (2010). [CrossRef] [PubMed]
4. S. Mahajan, T. Kumar, A. B. Bhattacherjee, and ManMohan, “Ground-state cooling of a mechanical oscillator and detection of a weak force using a Bose-Einstein condensate,” Phys. Rev. A 87, 013621 (2013). [CrossRef]
5. S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006). [CrossRef]
6. D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature (London) 444, 75–78 (2006). [CrossRef]
7. G. S. Agarwal and Sumei Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010). [CrossRef]
9. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421, 925–928 (2003). [CrossRef]
10. A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008). [CrossRef]
11. S. I. Schmid, K. Y. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011). [CrossRef]
13. G. De Chiara, M. Paternostro, and G. M. Palma, “Entanglement detection in hybrid optomechanical systems,” Phys. Rev. A 83, 052324 (2011). [CrossRef]
14. S. K. Steinke and P. Meystre, “Role of quantum fluctuations in the optomechanical properties of a Bose-Einstein condensate in a ring cavity,” Phys. Rev. A 84, 023834 (2011). [CrossRef]
15. S. Singh, H. Jing, E. M. Wright, and P. Meystre, “Quantum-state transfer between a Bose-Einstein condensate and an optomechanical mirror,” Phys. Rev. A 86, 021801(R) (2012). [CrossRef]
16. B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,” Phys. Rev. A 86, 042323 (2012). [CrossRef]
17. A. Dalafi, M. H. Naderi, M. Soltanolkotabi, and Sh. Barzanjeh, “Nonlinear effects of atomic collisions on the optomechanical properties of a Bose-Einstein condensate in an optical cavity,” Phys. Rev. A 87, 013417 (2013). [CrossRef]
19. M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013). [CrossRef]
20. P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013). [CrossRef]
23. S. Huang and G. S. Agarwal, “Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity,” Phys. Rev. A 80, 033807 (2009). [CrossRef]
25. 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 (London) 472, 69–73 (2011). [CrossRef]
26. Y. X. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110, 223603 (2013). [CrossRef] [PubMed]
27. A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the single-photon strong coupling regime,” arXiv:1034.5230.
28. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005) [CrossRef]
29. D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. 13, 023003 (2011). [CrossRef]
30. 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]
31. G. S. Agarwal and Sumei Huang, “Coherent perfect absorption in cavity opto-mechanics,” arXiv:1304.7323.
32. M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007). [CrossRef]
33. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).
34. S. Gröblacher, K. Hammerer, M. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature (London) 460, 724–727 (2009). [CrossRef]