Casimir switch: steering optical transparency with vacuum forces

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.


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 ω c and the decay rate γ, is coupled to the moveable mirror via the optomechanical coupling rate g = ω c /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 zeropoint CF is given by 42,43 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 PFA 43 , 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, F C T ( ) , dominates for large separations d ≳ 3 μm, but is much smaller than = F C T ( 0) for d ≳ 1 μm 42 . 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 predictions 44,45 . Complicated calculations of various non-ideal CF corrections have also been developed 42,43 , leading to e.g. an increase of about 1% in the CF due to the surface roughness, for a torsion balance experiment 42 ; 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 ω L and a weak probe laser with the frequency ω p . The field amplitudes of these two lasers are given by, respectively, where P L and P in denote the powers of the pump and the probe lasers. In the frame rotating at the frequency ω L , the Hamiltonian of this CF-aided optomechanical system can be written at the simplest level as 1-3,12 Here a and a † are the creation and annihilation operators of the cavity mode respectively, m i or ω m,i (i = 1, 2) denotes the mass or resonance frequency of the oscillator respectively, the optical detuning terms are denotes the momentum or position operator of the mechanical oscillator respectively, with and the phonon-mode operators b i , † b i (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 x 1 → x, x 2 → 0, the Heisenberg equations of motion are   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 materials 31,38,[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 achieved 54 . 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 = a s + δ a , x = x s + δ x . After eliminating the steady-state values, we obtain the linearized equations By applying the ansatz 10-12,14,26 with an effective mechanical frequency These equations can then be solved as , and = n a s s 2 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. γ = − a t a at ( ) 2 ( ) out i n , where a in (t) and a out (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, p p 2 We calculate this rate to better understand the CF-aided OMIT process under the above parameters as well as As Fig. 2 shows, for the conventional OMIT without any CF, at the res- , or correspondingly, Δ p = v − ω m = Ω m − ω m < 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 OMIT 55 or that with an external mechanical driving 25 . Also Fig. 2(d) shows the result about η p 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 P L = 0, the probe light can become transparent by steering the CF [at Δ p = 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 Δ p = 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 amplifier 56 . Since the linearized CF was already confirmed to be a good approximation in our system, we can expand H C in Eq. (1) up to the quadratic term of (x 1 − x 2 ) 2 . The linear term x 1,2 and the quadratic term x 1,2 2 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 d 120 3 4 . This kind of coupling has been achieved in various physical systems, to facilitate e.g. quantum state transfer of two spatially separated oscillators [57][58][59][60] .
The equations of motion of the resulting three-mode system are  Then by following the procedure as above, we have the linearized equations of motion and their solutions. The final result about η p 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 Δ p = 0. In contrast, for the CF-aided OMIT with two coupled mechanical oscillators, a dip emerges at Δ p = 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) situation 61,62 , 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 observed 55 . 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 elements 28,39,45 .

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 i t i t 1 1 1 Figure 4. The output rate η of the probe light versus the optical detuning Δ p , with different values of d.
Here we take P L = 1 mW, and assume the same parameters for the two mechanical oscillators, just for simplicity. All the other parameters are the same as in Fig. 2.
Scientific RepoRts | 6:27102 | DOI: 10.1038/srep27102 the expectation value 〈 a out (t)〉 of the output field a out (t) can be calculated using the standard input-output relation where a in (t) and a out (t) are the input and output field operators, and

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 structure 38 or engineering optical properties of novel materials [49][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 force 18 , 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 revealed 64,65 , 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 oscillators 25,45 , and the CF-mediated quantum mechanical squeezing.