Abstract
The Casimir force, originating from vacuum zero-point energy, is one of the most intriguing purely quantum effects. It has attracted renewed interests in current field of nanomechanics, due to the rapid size decrease of on-chip devices. Here we study the optomechanically-induced transparency (OMIT) with a tunable Casimir force. We find that the optical output rate can be significantly altered by the vacuum force, even terminated and then restored, indicating a highly-controlled optical switch. Our result addresses the possibility of designing exotic optical nano-devices by harnessing the power of vacuum.
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Introduction
Cavity optomechanics1, which explores the interaction between electromagnetic waves and mechanical motion, has witnessed rapid advances in recent years, leading to a variety of applications3, such as high-bandwidth accelerometer4, quantum-limited displacement sensing6, optical self-focusing7, quantum transducer8, and most recently, achieving quantum squeezing of mechanical motion9. Another notable example, closely related to the present study, is the experimental demonstration of optomechanically-induced transparency (OMIT)10,11,12, which provides a new approach for coherent control of light with a solid device, such as delay or advance of light13,14, quantum memory15,16,17, and precision measurement of tiny objects18. The basic mechanism of OMIT is the destructive interference of two absorption channels of the signal photons (i.e. absorbed by the cavity field or the mechanical mode), thereby leading to a transparency window for the signal light in the otherwise strongly absorbed region. This is formally equivalent to that of electromagnetically-induced transparency (EIT) well-known in atomic physics19. Further interesting studies on the OMIT include, e.g., nonlinear OMIT20,21,22,23, two-color OMIT24, cascaded OMIT25, and reversed OMIT in parity-time resonators26.
On the other hand, with the unprecedented ability of fabricating and characterizing materials on the nanometer scale, research in exploring and harnessing the exotic quantum effect like the Casimir force (CF)27,28 has become active in recent years. The CF can become increasingly important in nano-devices as the space separation between the component surfaces is drastically decreased. The high-precision CF measurement was first performed in 1997 by Lamoreaux29, and then also by several other groups30,31,32. The CF-induced novel effects have been revealed, such as vacuum friction of motion33,34,35, non-touching bound of nano-particles36, nonlinear mechanical oscillations37, and giant vacuum force near a transmission line38. Practical applications of the CF in e.g. quantum sensing of motion was also presented39, highlighting its impacts on future quantum technologies.
In the present work, by combining these two research fields, we study the CF effect in a cavity optomechanical system. We note that in very recent works, the interplay of the external and zero-point radiation pressures was already investigated for the dynamics of a levitated nanosphere trapped in a cavity40,41. Here we focus on the OMIT process in the presence of a tunable vacuum force. We find an interesting CF-controlled optical switch effect, i.e. the optical transparency window can be completely shut down and also re-opened again by sorely tuning the strength of the Casimir force. To be more specific, the presence of the CF leads to the modifications of not only the steady-state values of the dynamical variables, but also the field fluctuations and subsequently, the optical output rate of the probe light. In particular, for a fixed sphere spatially separated from the moveable mirror, by reducing the air-gap distance, the conventional OMIT spectrum tends to be shifted to the red-detuning side (with some distortions as well). As a result, by tuning the CF, the output of the probe light at the probe-cavity resonance can be attenuated, or even totally shut down and then restarted again, for a fixed pump power. In addition, we find that even for the non-OMIT case, i.e. without any pump light, the CF-aided optical transparency can still be achieved in our proposed situation. A reversed pump-dependence was also revealed for the CF-aided OMIT, for the low-power cases, in comparison with the conventional OMIT. These results indicate the possibility of designing unconventional optical nano-devices by exploiting vacuum zero-point energy.
Results
The model of the system
We consider a cavity optomechanical system with a tunable CF. The optical cavity mode, characterized by the resonance frequency ωc and the decay rate γ, is coupled to the moveable mirror via the optomechanical coupling rate g = ωc/L (L is the cavity length); also the moveable mirror interacts with a nearby gold-coated nanosphere via the CF. For a fixed sphere-plate separation d of perfect conductors, the zero-point CF is given by42,43
where the first term is the perfect reflector formula in the proximity force approximation (PFA), the second term accounts for the leading correction to the PFA43, and c is the light speed at vacuum, R is the radius of the sphere. The condition d/R ≪ 1 is the standard condition that determines the validity of the PFA; for d/R ≪ 1, the second term is safely neglected. The thermal CF, , dominates for large separations d ≳ 3 μm, but is much smaller than for d ≳ 1 μm42. Here we focus on the latter regime . In current experiments, the CF has been accurately measured for d ~ 100 nm, still showing excellent agreement with theoretical predictions44,45. Complicated calculations of various non-ideal CF corrections have also been developed42,43, leading to e.g. an increase of about 1% in the CF due to the surface roughness, for a torsion balance experiment42; for numerical calculations of the CF with finite conductivity, confirming the validity of the plasma model for the gold, see ref. 27. We stress that in this work we focus on the vaccum-assisted steering of OMIT spectrum, instead of various non-ideal CF corrections (for these efforts, see e.g. ref. 27).
The cavity is driven by a strong control laser with the frequency ωL and a weak probe laser with the frequency ωp. The field amplitudes of these two lasers are given by, respectively,
where PL and Pin denote the powers of the pump and the probe lasers. In the frame rotating at the frequency ωL, the Hamiltonian of this CF-aided optomechanical system can be written at the simplest level as1,2,3,12
Here a and a† are the creation and annihilation operators of the cavity mode respectively, mi or ωm,i (i = 1, 2) denotes the mass or resonance frequency of the oscillator respectively, the optical detuning terms are
and pi or denotes the momentum or position operator of the mechanical oscillator respectively, with and the phonon-mode operators bi, (see Fig. 1). We first focus on the single oscillator case, and then discuss the case with coupled two oscillators.
The fixed-sphere case
For x1 → x, x2 → 0, the Heisenberg equations of motion are
where , Γm is the mechanical decaying rate of the vibrating mirror, and we have neglected the noise terms and taken m1 → m, ωm,1 → ωm, for simplicity. The steady-state values of the dynamical variables are
For experimentally-accessible values of the system parameters46,47, i.e. ωc = 2πc/λ, λ = 1064 nm, L = 25 mm, R = 150 nm, m = 145 ng, γ = 2π × 80 KHz, Γm = 2π × 141 Hz, and ΔL = ωm = 2π × 947 KHz, we find d/R ≪ 1 (for d < 5 nm) and xs/d ≲ 10−2 (for d > 1.5 nm). Hence it is good enough to take the Casimir term up to the second order of xs/d, within our interested regime, as numerically confirmed later. Under this approximation, we have the balance equation of the moveable mirror
where the first (second) term in the left-hand side results from the restoring (Casimir) force. Clearly, the left-hand side should be positive, which is fulfilled for d > 0.7 nm, with the above parameters. Less values of d lead to a CF stronger than the restoring force, and thus adhesion of the mirror. We also find that the CF term is much weaker than the restoring force for d > 10 nm; in contrast, it becomes comparable with the latter for 0.7 nm < d ≲ 2 nm. We note that in current experiments, the CF measurements for d = 2 nm are challenging; nevertheless, even for a larger d, it is still possible to achieve the required strong CF by altering optical properties or geometric structures of the interacting materials31,38,48,49,50,51,52,53,54. For examples, for parallel graphene layers with d < 10 nm, it was found that the CF ~ d−5; with specific nonostructures, further enhancement as CF ~ d−7 can be achieved54. This indicates that the required CF, corresponding to d ~ 1 nm for ideal metals, can be achieved for larger values of d, e.g. d ~ 10 nm or even 50 nm, by proper designs of material properties. In fact, there is a huge list of materials whose electromagnetic response can be widely tuned, hence allowing for significant CF enhancement at fixed separations, e.g. optical crystals, semiconductors, topological insulators, or plasmonic nanostructures (see ref. 54 for a very recent review).
Here we show that a novel CF-controlled optical switch can be achieved in an OMIT system, even in the low-power linear regime. In order to see this, we expand each operator as the sum of its steady-state value and a small fluctuation around that value, i.e., a = as + δa, x = xs + δx. After eliminating the steady-state values, we obtain the linearized equations
By applying the ansatz10,11,12,14,26,
the linearized equations are transformed into
with an effective mechanical frequency
These equations can then be solved as
where , and is the intracavity photon number. The expectation value of the output optical field can be obtained by using the standard input-output relation, i.e. , where ain(t) and aout(t) are the input and output field operators. This leads to the optical reflection rate for the probe field, i.e. the amplitude square of the ratio of the output field amplitude to the input field amplitude at the probe field frequency,
We calculate this rate to better understand the CF-aided OMIT process under the above parameters as well as ΔL = ωm and . As Fig. 2 shows, for the conventional OMIT without any CF, at the resonance Δp = 0, the probe light is absorbed for PL = 0, while it becomes transparent by applying a pump light [see Fig. 2(a)]; in contrast, for the CF-aided OMIT, we find that by reducing d, ηp(Δp = 0) is firstly decreased until zero (at d ~ 1.8 nm) and then increased again [see Fig. 2(b,c), for a fixed value of PL = 1 mW]. This indicates a CF-controlled light switch, even shutting down and re-starting the signal [see Fig. 2(c)]. One mechanism underlying this effect is the following: the CF-induced frequency shift ωm → Ωm modifies the resonance condition as , or correspondingly, Δp = v − ωm = Ωm − ωm < 0, i.e. the OMIT spectrum tends to be shifted to the left. This effect is also reminiscent of that using the electrostatic force to tune the OMIT55 or that with an external mechanical driving25. Also Fig. 2(d) shows the result about ηp with a linearized CF, indicating that the CF-controlled light switch works well even in the linear CF regime.
Interestingly, we find that even for PL = 0, the probe light can become transparent by steering the CF [at Δp = 0, see Fig. 3(a)]. In this situation, the vacuum field, instead of the pump field, serves as the control gate for the output of the probe light. For weak pump powers, the CF-aided OMIT shows an exotic feature of reversed pump dependence at Δp = 0, in comparison with the conventional OMIT [see Fig. 3(b–d)] and also ref. 56. These results show that (i) even without any pump field, the signal light can still be transparent with the aid of the virtual photons (i.e. the Casimir potential); (ii) combining the real-photon (e.g. the pump light) and the virtual-photon (i.e. the vacuum fluctuation) fields provide more flexible and efficient ways to manipulate the light propagation.
The moveable-sphere case
For completeness, we also consider a nanosphere attached to a vibrating cantilever. We note that this configuration was recently exploited to design a Casimir parametric amplifier56. Since the linearized CF was already confirmed to be a good approximation in our system, we can expand HC in Eq. (1) up to the quadratic term of (x1 − x2)2. The linear term x1,2 and the quadratic term can be absorbed into the re-defined equilibrium positions and the mechanical frequency, respectively, and thus are unimportant; the term of interests is the inter-mode coupling
with . This kind of coupling has been achieved in various physical systems, to facilitate e.g. quantum state transfer of two spatially separated oscillators57,58,59,60.
The equations of motion of the resulting three-mode system are
from which we obtain the steady-state values of the dynamical variables
Then by following the procedure as above, we have the linearized equations of motion and their solutions. The final result about ηp is plotted in Fig. 4.
For d → ∞, as Fig. 2(a) shows, we have the conventional OMIT spectrum, i.e. a single-peak transparency at Δp = 0. In contrast, for the CF-aided OMIT with two coupled mechanical oscillators, a dip emerges at Δp = 0 [for d = 4 nm, see Fig. 4(a)] and the OMIT window is then split into a double-peak structure [for d = 2 nm, see Fig. 4(b)]. Hence, by tuning the CF, the probe light can be varied from the transparency regime to the absorption regime, or vice versa. The shape of the OMIT spectrum as Fig. 4 is very similar to that in an Autler-Townes splitting (ATS) situation61,62, and the relation between these two kinds of physical processes, or even the controllable transition of them, would be an interesting problem to be explored in our future works. We also remark that the coupling as in Eq. (16) can also be realized by using e.g. coupled charged objects, with which a similar OMIT spectrum was observed55. As a comparison, our proposal here focuses on the single-oscillator case, instead of the double-oscillator case as studied in ref. 55; more importantly, it does not require any charged or magnetic object. The Casimir force comes from the vacuum itself and plays a crucial role in chip-scale nano-devices with decreasing vacuum distances between different elements28,39,45.
Methods
Derivation of the optical output rate for the moveable sphere
Taking the expectation of each operator given in Eqs.(17)-(19), we find the linearized Heisenbrg equations as
by applying the following ansatz,
equations (21)–(23), can be transformed into the following form,
Solving these algebraic equations leads to
where we have used and
the expectation value 〈aout(t)〉 of the output field aout(t) can be calculated using the standard input-output relation , where ain(t) and aout(t) are the input and output field operators, and
Hence, the output rate of the probe field can be written as , where t(ωp) is the ratio of the output field amplitude to the input field amplitude at the probe frequency.
Conclusion
In summary, we have demonstrated the effects of the vacuum force on the OMIT, indicating the possibility of controlling light with the vacuum. We note that the measurements of the CF for a very short distance are still missing in current experiment; however, even for a constant distance, it is still possible to significantly enhance the CF by, e.g. calibrating an unconventional surface structure38 or engineering optical properties of novel materials49,50,51,52,53,54,63. With rapid advances of nano-calibration techniques and very active efforts on controlling or enhancing the CFs, our proposal holds the promise to be realized, at least in principle. In comparison with a recent work on tuning the OMIT with a voltage-controlled electrostatic force18, our proposal here does not need any charged or magnetic object, since the CF comes from the vacuum itself, which can be of increasingly important in chip-scale nano-devices with decreasing vacuum spaces between the elements. We also note that recently in an optomechanical system, a new kind of motion-induced few percentage correction to the CF was revealed64,65, indicating that more interesting works could be performed by combining optomechanics and the CF. In the future, we plan to also study the CF-controlled slow light, the cascaded OMIT with coupled Casimir oscillators25,45, and the CF-mediated quantum mechanical squeezing.
Additional Information
How to cite this article: Liu, X.-F. et al. Casimir switch: steering optical transparency with vacuum forces. Sci. Rep. 6, 27102; doi: 10.1038/srep27102 (2016).
References
Aspelmeyer, M., Meystre, P. & Schwab, K. Quantum optomechanics. Phys. Today 65, 29–35 (2012).
Meystre, P. A short walk through quantum optomechanics. Ann. Phys. 525, 215–233 (2013).
Metcalfe, M. Applications of cavity optomechanics. App. Phys. Rev. 1, 031105 (2014).
Krause, A. G., Winger, M., Blasius, T. D., Lin, Q. & Painter, O. A high-resolution microchip optomechanical accelerometer. Nat. Photonics 6, 768–772 (2012).
Cervantes, F. G., Kumanchik, L., Pratt, J. & Taylor, J. Self-calibrating ultra-low noise, wide-bandwidth optomechanical accelerometer. Appl. Phys. Lett. 104, 221111 (2014).
Hoff, U. B. et al. Quantum-enhanced micromechanical displacement sensitivity. Opt. Lett. 38, 1413–1415 (2013).
Butsch, A., Conti, C., Biancalana, F. & Russell, P. S. J. Optomechanical self-channeling of light in a suspended planar dual-nanoweb waveguide. Phys. Rev. Lett. 108, 093903 (2012).
Stannigel, K., Rabl, P., Sørensen, A. S. & Lukin, M. D. Optomechanical transducers for long-distance quantum communication. Phys.Rev.Lett. 105, 220510 (2010).
Wollman, E. E. et al. Quantum squeezing of motion in a mechanical resonator. Science 349, 952 (2015).
Weis, S. et al. Optomechanically induced transparency. Science 330, 1520–1523 (2010).
Safavi-Naeini, A. H. et al. Electromagnetically induced transparency and slow light with optomechanics. Nature 472, 69–73 (2011).
Agarwal, G. S. & Huang, S. Electromagnetically induced transparency in mechanical effects of light. Phys. Rev. A 81, 041803(R) (2010).
Zhou, X. et al. Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics. Nat. Phys. 9, 179–184 (2013).
Jiang, C., Chen, B. & Zhu, K. D. Tunable pulse delay and advancement device based on a cavity electromechanical system. EPL 94, 38002 (2011).
Fiore, V. et al. Storing optical information as a mechanical excitation in a silica optomechanical resonator. Phys. Rev. Lett. 107, 133601 (2011).
Hill, J. T., Safavi-Naeini, A. H., Chan, J. & Painter, O. Coherent optical wavelength conversion via cavity optomechanics. Nature Commun. 3, 542–555 (2012).
Dong, C., Fiore, V., Kuzyk, M. C. & Wang, H. Optomechanical dark mode. Science 338, 1609–1613 (2012).
Zhang, J. Q., Li, Y., Feng, M. & Xu, Y. Precision measurement of electrical charge with optomechanically induced transparency. Phys. Rev. A 86, 053806 (2012).
Boller, K. J., Imamogˇlu, A. & Harris, S. E. Observation of electromagnetically induced transparency. Phys. Rev. Lett. 66, 2593 (1991).
Lemonde, M.-A., Didier, N. & Clerk, A. A. Nonlinear interaction effects in a strongly driven optomechanical cavity. Phys. Rev. Lett. 111, 053602 (2013).
Andreas, K. & Florian, M. Optomechanically induced transparency in the nonlinear quantum regime. Phys. Rev. Lett. 111, 133601 (2013).
Børkje, K., Nunnenkamp, A., Teufel, J. D. & Girvin, S. M. Signatures of nonlinear cavity optomechanics in the weak coupling regime. Phys. Rev. Lett. 111, 053603 (2013).
Xiong, H., Si, L. G., Zheng, A. S., Yang, X. & Wu, Y. Higher-order sidebands in optomechanically induced transparency. Phys. Rev. A 86, 013815 (2012).
Wang, H., Gu, X., Liu, Y.-X., Miranowicz, A. & Nori, F. Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system. Phys. Rev. A 90, 023817 (2014).
Fan, L., Fong, K. Y., Poot, M. & Tang, H. X. Cascaded optical transparency in multimode-cavity optomechanical systems. Nature Commun. 6, 5850 (2015).
Jing, H. et al. Optomechanically induced transparency in parity-time-symmetric microresonators. Sci. Rep. 5, 9663 (2015).
Lamoreaux, S. K. The casimir force: background, experiments, and applications. Rep. Prog. Phys. 68, 201–236 (2005).
Lamoreaux, S. K. Casimir forces: Still surprising after 60 years. Phys. Today 60, 40–45 (2007).
Lamoreaux, S. K. Demonstration of the casimir force in the 0.6 to 6 μm range. Phys. Rev. Lett. 78, 5 (2007).
Bressi, G., Carugno, G., Onofrio, R. & Ruoso, G. Measurement of the casimir force between parallel metallic surfaces. Phys. Rev. Lett. 88, 041804 (2002).
Chan, H. B. et al. Measurement of the casimir force between a gold sphere and a silicon surface with nanoscale trench arrays. Phys. Rev. Lett. 101, 030401 (2008).
Krause, D. E., Decca, R. S., López, D. & Fischbach, E. Experimental investigation of the casimir force beyond the proximity-force approximation. Phys. Rev. Lett. 98, 050403 (2007).
Volokitin, A. I. & Persson, B. N. J. Quantum friction. Phys. Rev. Lett. 106, 094502 (2011).
Volokitin, A. I. & Persson, B. N. J. Near-field radiative heat transfer and noncontact friction. Rev. Mod. Phys. 79, 1291–1329 (2007).
Pendry, J. B. Quantum friction-fact or fiction? New J. Phys. 12, 033028 (2010).
Rodriguez, A. W. et al. Nontouching nanoparticle diclusters bound by repulsive and attractive casimir forces. Phys. Rev. Lett. 104, 160402 (2010).
Chan, H. B., Aksyuk, V. A., Kleiman, R. N., Bishop, D. J. & Capasso, F. Nonlinear micromechanical casimir oscillator. Phys. Rev. Lett. 87, 211801 (2001).
Shahmoon, E., Mazets, I. & Kurizki, G. Giant vacuum forces via transmission lines. Proc. Natl. Acad. Sci. USA 111, 10485 (2014).
Muschik, C. A. et al. Harnessing vacuum forces for quantum sensing of graphene motion. Phys. Rev. Lett. 112, 223601 (2014).
Nie, W., Lan, Y., Li, Y. & Zhu, S. Dynamics of a levitated nanosphere by optomechanical coupling and casimir interaction. Phys. Rev. A 88, 063849 (2013).
Nie, W., Lan, Y., Li, Y. & Zhu, S. Effect of the casimir force on the entanglement between a levitated nanosphere and cavity modes. Phys. Rev. A 86, 063809 (2012).
Sushkov, A. O., Kim, W. J., Dalvit, D. A. R. & Lamoreaux, S. K. Observation of the thermal casimir force. Nat. Phys. 7, 230–233 (2011).
Bimonte, G., Emig, T., Jaffe, R. L. & Kardar, M. Casimir forces beyond the proximity approximation. EPL 97, 50001 (2012).
Bordag, M., Klimchitskaya, G. L. & Mostepanenko, V. M. Casimir force and in situ surface potential measurements on nanomembranes. Phys. Rev. Lett. 109, 027202 (2012).
Zou, J. et al. Casimir forces on a silicon micromechanical chip. Nat. commun. 4, 1845 (2013).
Gröblacher, S., Hammerer, K., Vanner, M. & Aspelmeyer, M. Observation of strong coupling between a micromechanical resonator and an optical cavity field. Nature 460, 724–727 (2009).
Huang, S. & Agarwal, G. S. Electromagnetically induced transparency with quantized fields in optocavity mechanics. Phys. Rev. A 83, 043826 (2011).
Intravaia, F. et al. Strong casimir force reduction through metallic surface nanostructuring. Nat. Commun. 4, 2515 (2013).
Klimchitskaya, G. L., Mohideen, U. & Mostepanenko, V. M. The casimir force between real materials: experiment and theory. Rev. Mod. Phys. 111, 1827–1885 (2009).
Dalvit, D. A. R. et al. Casimir physics (Lecture Notes in Physics) (Springer, 2011).
Rodriguez-Lopez, P. & Grushin, A. G. Repulsive casimir effect with chern insulators. Phys. Rev. Lett. 112, 056804 (2014).
Cysne, T. et al. Tuning the casimir-polder interaction via magneto-optical effects in graphene. Phys. Rev. A 90, 052511 (2014).
Klimchitskaya, G. L. & Mostepanenko, V. M. Impact of graphene coating on the atom-plate interaction. Phys. Rev. A 89, 062508 (2014).
Woods, L. M. et al. A materials perspective on casimir and van der waals interactions. arXiv: 1509.03338 (2015).
Ma, P.-C., Zhang, J.-Q., Xiao, Y., Feng, M. & Zhang, Z.-M. Tunable double optomechanically induced transparency in an optomechanical system. Phys. Rev. A 90, 043825 (2014).
Imboden, M., Morrison, J., Campbell, D. K. & Bishop, D. J. Design of a casimir-driven parametric amplifier. J. Appl. Phys. 116, 134504 (2014).
Hensinger, W. K. et al. Ion trap transducers for quantum electromechanical oscillators. Phys. Rev. A 72, 041405(R) (2005).
Tian, L. & Zoller, P. Coupled ion-nanomechanical systems. Phys. Rev. Lett. 93, 266403 (2004).
Brown, K. R. et al. Coupled quantized mechanical oscillators. Nature 471, 196–199 (2011).
Harlander, M., Lechner, R., Brownnutt, M., Blatt, R. & Hänsel, W. Trapped-ion antennae for the transmission of quantum information. Nature 471, 200–203 (2011).
Peng, B., Özdemir, S. K., Chen, W., Nori, F. & Yang, L. What is and what is not electromagnetically induced transparency in whispering-gallery microcavities. Nat. Commun. 5, 5082 (2014).
Sun, H.-C. et al. Electromagnetically induced transparency and autler-townes splitting in superconducting flux quantum circuits. Phys. Rev. A 89, 063822 (2014).
Yang, Y., Zeng, R., Chen, H., Zhu, S. & Zubairy, M. S. Controlling the casimir force via the electromagnetic properties of materials. Phys. Rev. A 81, 022114 (2010).
Armata, F. & Passante, R. Vacuum energy densities of a field in a cavity with a moblie boundary. Phys. Rev. D 91, 025012 (2015).
Butera, S. & Passante, R. Field fluctuations in a one-dimensional cavity with a mobile wall. Phys. Rev. Lett. 111, 060403 (2013).
Acknowledgements
We thank Chang-Pu Sun, Franco Nori, and Yu-xi Liu for discussions. This work is supported partially by the CAS 100-Talent program and the National Natural Science Foundation of China under Grand numbers 11274098, 11474087, and 11422437.
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H.J. conceived the idea and performed the calculations with the aid of X.F.L. and H.J. wrote the manuscript with the input of Y.L.; and all the authors discussed the content of the manuscript.
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Liu, Xf., Li, Y. & Jing, H. Casimir switch: steering optical transparency with vacuum forces. Sci Rep 6, 27102 (2016). https://doi.org/10.1038/srep27102
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DOI: https://doi.org/10.1038/srep27102
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