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Home , Casimir effect, Interaction picture, Quantum mechanics

Electro-mechanical Casimir effect Mikel Sanz1, Witlef Wieczorek2, Simon Groblacher¨ 3, and Enrique Solano1,4,5

1Department of Physical , University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain 2Department of Microtechnology and Nanoscience, Chalmers University of Technology, Kemiv¨agen 9, SE-41296 G¨oteborg, Sweden 3Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628CJ Delft, The 4IKERBASQUE, Basque Foundation for , Maria Diaz de Haro 3, E-48013 Bilbao, Spain 5Department of , Shanghai University, 200444 Shanghai, China August 30, 2018

The dynamical Casimir effect is an intrigu- of modes in . Indeed, if the velocity of ing phenomenon in which are generated a mirror is much smaller than the speed of , no from vacuum due to a non-adiabatic change in excitations emerge out of the vacuum, since the elec- some boundary conditions. In particular, it con- tromagnetic modes adiabatically adapt to the changes. nects the of an accelerated mechanical However, if the mirror experiences relativistic motion, mirror to the generation of photons. While - these changes occur non-adiabatically. Hence, the vac- oneering experiments demonstrating this effect uum state of the electromagnetic field can be excited by exist, a conclusive involving a me- taking from the moving mirror, resulting in the chanical generation is still missing. We show generation of real photons. It has been suggested that, that a hybrid consisting of a piezoelec- due to the large generated on any macroscopic tric mechanical resonator coupled to a supercon- material moving with relativistic speed, the observa- ducting cavity may allow to electro-mechanically tion of the DCE might actually be unrealistic [9, 10]. generate measurable photons from vacuum, in- This challenge inspired various alternative proposals to trinsically associated to the dynamical Casimir observe the DCE [10–22], e.g., by employing nanome- effect. Such an experiment may be achieved chanical resonators, modulating two-level in- with current technology, based on film bulk side a cavity or modulating the electric boundary con- acoustic resonators directly coupled to a super- ditions of a superconducting cavity; for an overview see, conducting cavity. Our results predict a measur- e.g., [9, 23, 24]. able generation rate, which can be fur- Only recently, DCE radiation has been experimen- ther increased through additional improvements tally observed [25] in superconducting circuits by mov- such as using superconducting metamaterials. ing the electric boundary condition of a superconduct- ing cavity [20, 21] instead of the center of of a mirror. In a similar experiment, DCE photons were 1 Introduction created by modulating the effective [26] making use of a superconducting metamaterial. Ad- predicts that virtual can ditionally, relevant applications of these experiments, emerge from vacuum. This phenomenon, known as such as a robust generation of multi-partite quantum quantum fluctuations, is a cornerstone to explaining key correlations [20, 21, 27–30], were proposed, and recently effects in , ranging from the anomalous magnetic arXiv:1712.08060v3 [quant-ph] 29 Aug 2018 observed [31], thus, highlighting the potential of this ef- moment of the [1], to the enhancement in quan- fect beyond its fundamental interest. However, no ex- tum transport phenomena [2], and the of periment to date has succeeded in generating photons atomic spectra [3]. Another paramount example is the out of vacuum using a moving mechanical object in the Casimir effect, which results from a between two spirit of the original DCE proposal. separated conducting plates [4–7]. G. T. Moore [8] suggested the existence of a kinetic analogue to this In this letter, we propose an experiment consisting phenomenon, known as the dynamical Casimir effect of a nanomechanical resonator that is directly cou- (DCE). This effect, based on the fact that a moving pled to a high impedance superconducting cavity (see mirror modifies the mode structure of the electromag- Fig.1) in order to create a measurable rate of electro- netic vacuum dynamically, is the result of a mismatch mechanically generated DCE photons. Our proposal de- velops the idea to employ film bulk acoustic resonators Mikel Sanz: [email protected] (FBAR) for DCE photon creation from Ref. [17] further

Accepted in Quantum 2018-08-28, click title to verify 1 by combining it with the technology of superconduct- ity (e.g. made from Al) with length d = 33 mm, which ing circuits [20, 21, 25, 32]. Our calculations make use is capacitively coupled to a semi-infinite transmission of recent advances in mechanical oscillators based on line for the read-out, as shown in Fig.1, all assumed FBAR technology using thin films of Al-AlN-Al [32]. to be operating at a of 10 mK. To match The presence of a superconducting cavity allows for a the condition in an actual experimental im- resonant enhancement of the photon rate which sur- plementation, we note that the superconducting cav- passes the radiation from a single mirror by several or- ity can be made tunable in by incorporating ders of magnitude [12–14]. We study the minimally a flux-biased SQUID [25, 33]. We have chosen values required conditions to observe a stable flux of photons that closely follow the parameters from Ref. [32], for resulting from the electro-mechanically amplified quan- details see Appendix B.4. An AC voltage is applied to tum vacuum, concluding that such a measurement is the superconducting plates of the mechanical resonator. feasible with current technology. Finally, we propose This voltage drive has two effects: (i) the piezoelectric technical improvements to further enhance the mechan- material of the FBAR converts the AC voltage into me- ical photon production. Our also paves the way chanical contraction and expansion, which essentially to experimentally test other fundamental relativistic ef- leads to a change of the length of the superconducting fects with mechanical resonators, such as the Unruh ef- cavity; (ii) the voltage drive changes the potential of the fect or the [24] cavity’s boundary. Both effects result in the production of DCE photons. a) 2.1 FBAR modeling We make use of the modified van Dyke-Butterworth model to construct the equivalent lumped-element cir- cuit representation of the FBAR [34, 35] (for details see AppendixB). This circuit contains two parallel branches. The first one consists of a mechanical ca- pacitance Cm, mechanical inductance Lm and a resis- tor Rm modeling the mechanical dissipation connected b) x x x x Cc x in series, while the second one contains a resistor R0 L L L L L C(t) taking into account the loss and, crucially, a x x x x x x V (t) C C C C C C geometrical capacitance C(t), which slightly changes in time when an AC voltage is applied to the plates of the FBAR. Composing the impedances for both branches, Figure 1: Scheme of the coupling between an FBAR and a su- Zm and Z0, it is straightforward to prove that the total perconducting cavity. (a) Artistic view representing an FBAR impedance Z ≈ Z0, which, to first order, can be reduced (top left), i.e., a resonator composed of two superconducting to C(t) (see AppendixB). layers sandwiching a piezoelectric material. The FBAR ter- Let us estimate the change in the capacitance C(t) minates one side of a superconducting cavity that is capaci- when an AC voltage V (t) = Vpp cos(ωt + φ) is applied tively coupled to a semi-infinite transmission line (lower right). to the electrodes. An AC voltage with an amplitude Vpp (b) Equivalent circuit of the proposed implementation, which at the resonance frequency of the mechanical resonator is composed of the mechanical resonator (time-dependent ca- ω = Ω results in a change of the inter-plate pacitance C(t) and voltage drive V (t)) and the supercon- ∆x(Vpp) ≈ 1.7 · Vpp nm/V (see AppendixA). Crucial ducting cavity (with C and L capacitance and inductance per here is that the piezoelectric effect is enhanced by the unit length ∆x). We have employed the modified van Dyke- Butterworth model to provide the equivalent lumped-element mechanical factor of the FBAR, assumed to be circuit representation of the FBAR and model the cavity as an Q = 300 [32]. Applying a voltage of Vpp = 500µV thus −13 infinite set of LC circuits which will be afterwards capacitively results in ∆x = 8.5 · 10 m, which is more than five coupled via Cc to a semi-infinite . orders of magnitude smaller than the thickness tAlN = 3.5 · 10−7m of the piezoelectric layer. Thus, we model the resulting mechanical contraction and expansion as harmonic. The time-varying capacitance is then given 2 Proposed implementation as C(t) = εAlNA/[tAlN + ∆x cos(Ωt)], with A as of the FBAR and εAlN as dielectric constant of AlN. We The proposed setup consists of a piezoelectric FBAR expand C(t) for ∆x  tAlN and finally obtain resonator with resonance frequency Ω/2π = 4.20 GHz (e.g. made from Al-AlN-Al) and a superconducting cav- C(t) ≈ C0 + ∆C cos(Ωt) (1)

Accepted in Quantum 2018-08-28, click title to verify 2 C0 with C0 = εAlNA/tAlN and ∆C ≈ C0∆x/tAlN. behaves as a mirror placed at x = −Leff ≈ C , such that the effective length of the resonator shown in Fig.1, which will be introduced below, is deff = d + Leff. 2.2 Lumped-element circuit model emerge due to the inelastic interaction of the photons with the oscillating mirror, and the in- The Lagrangian describing the circuit of the FBAR con- coming modes ain(ω) with frequency ω are scattered nected to an open transmission line (without the cou- elastically as aout(ω) and inelastically as aout(ω + Ω), pling capacitor Cc shown in Fig.1) can be written as aout(ω + 2Ω), ... . Keeping only the first inelastic reso- nances, as higher resonances have higher orders in ∆C, ∞ and assuming that 0 < ω ≤ Ω, which is natural since X δxC 1  L = Φ˙ 2 − (Φ − Φ )2 the condition for the rotating approximation holds 2 i+1 2δxL i+1 i 0 i=0 around Ω ≈ 2ω , the output mode is given by 1 1 + C(t)(Φ˙ − Φ˙ )2 + C Φ˙ 2. (2) 2 0 v 2 g 0 aout(ω, Leff) = h(ω, Ω) + ain(ω, Leff)

Here, C and L are the densities of capacitance and in- + S(ω, Ω + ω)ain(Ω + ω, Leff) ductance of a transmission line per unit length δx, re- † + S(ω, Ω − ω)ain(Ω − ω, Leff). (4) spectively, Cg is the capacitive coupling of the FBAR to ground, which we will discard in our analysis as it Here, the operators ain and aout are defined in the is much smaller than any other quantity involved, Φi is displaced position x = L , h(ω, Ω) is the identity in ˙ eff the i-th flux and Φv = V is the voltage of the AC , as we have treated the source classically, source. In the continuous limit, the of motion and corresponding to i = 0 is given by

S(ω0, ω00) = −i∆CZ p|ω0||ω00|θ(ω0)θ(ω00), (5) 1 ∂Φ(x, t) 0 ¨ ˙ ˙ r C(t)Φ(0, t) + C(t)Φ(0, t) − = F (t), (3) 4πZ L ∂x x=0 h(ω, Ω) = −i 0 F(ω, Ω), (6) ω d ~ with F (t) = dt (θ(t)C(t)V (t)) being the electro- −1/2 R ∞ iωt mechanical source term, and θ(t) is the Heaviside step where F(ω, Ω) = (2π) −∞ dtF (t)e is the Fourier function. In order to solve it, we follow a similar ap- transform of F (t). proach to Ref. [20, 21] and expand the field in the The aforementioned result describes an oscillatory Fourier components mirror coupled to a one-dimensional open transmission line. The photon production due to the change in the r boundary conditions can be dramatically increased by Z Z ∞ 1 h ~ 0 −i(ωt−kω x) introducing a cavity with a well chosen resonance con- Φ(x, t) = dω √ ain(ω)e 4π 0 ω dition, as shown for example in Ref. [12–14]. In order to i realize this, we will introduce a capacitor at a distance +a (ω)e−i(ωt+kω x) + H.c. , out d of the FBAR, as depicted in Fig.1. We redefine the origin of the coordinates in the coupling capacitance be- Z ≈ 55 Ω (3) with 0 . Equation can now be written as tween the cavity and the line and hence the harmonic mirror is placed at x = deff = d+Leff, with d the length Z ∞ s " of the cavity for a static mirror. Therefore, the natu- ~Z0 dω ral frequency of the resonator is ω0/2π = v/deff with 0 4π|ω| v the speed of light in the superconducting material.   The input-output relations connecting the fields inside 2 ˙ ikω −iωt + −ω C(t) − iωC(t) − ain(ω)e L the cavity {ain, aout} to fields in the transmission line {b , b }  ik  in out are given by + −ω2C(t) − iωC˙ (t) + ω a (ω)e−iωt L out       # ain(ω, 0) α¯ω βω bin(ω, 0) = ¯ , (7) + H.c. = F (t). aout(ω, 0) βω αω bout(ω, 0)

with αω = 1 + iωc/2ω, βω = iωc/2ω and the coupling R ∞ p 0 iω0t By integrating over −∞ dt |ω |e , we can see that rate ωc = 1/CcZ0. Notice that ωc is a measure of how in the case of a static capacitor (i.e. C(t) = C0), it large the coupling of the resonator to the line is. Indeed,

Accepted in Quantum 2018-08-28, click title to verify 3 for ωc  ω it is completely decoupled, while ωc = 0 10-3 means that there is no cavity. Equation (7) holds for x = 0 [20, 21], but as we want to connect the fields 10-6 in the line with the reflected fields on the oscillating 101 mirror, we must displace the cavity fields to x = deff 10-9 by using the diag(eikω deff , e−ikω deff ). In order to 10-1 0.5 0.7 0.9 compute the reflection coefficient Rres(ω) of the cavity, we can consider that the inelastic scattering process is ( ω ) 10-3 out absent and that only the reflection-transmission process n a (ω, d ) = a (ω, d ) remains, so in eff out eff . Under this con- 10-5 dition, we get

-7 2iω 2ik d 10 1 + (1 + )e ω eff 0 0.2 0.4 0.6 0.8 1 ωc Rres(ω) = 2iω . (8) (1 − ) + e2ikω deff ω/Ω ωc Similarly, we also want to study the mode structure Figure 2: Electro-mechanical DCE photon production. We con- of the resonator. This is encoded in the sider the generated photons nout(ω) in Hz per unit bandwidth versus frequency ω in units of the mechanical fre- function Ares, defined as aout(ω, deff) = Ares(ω)bin(ω, 0), as it contains the response of the resonator to any input quency Ω. Main panel (using Vpp = 500µV, Q = 300): The orange (light gray) line depicts the total number of photons in signal coming from the line, the open transmission line that originate from the DCE effect and thermal radiation. The green (dark gray) line shows the ( 2iω )eikω deff ωc photons only generated via the electro-mechanical DCE effect, Ares(ω) = 2iω . (9) (1 − ) + e−2ikω deff whereas the purple (dotted) line shows the thermal contribu- ωc tion. The vertical gray lines depict the resonances of the su- The resonance can be deduced from the perconducting cavity. The inset shows an enhanced mechanical denominator of Eq. (9) and, as shown in Ref. [21], they DCE photon production assuming an FBAR with a higher me- are approximately the solutions of the transcendental chanical quality factor ( Q = 3·106 and using a driving voltage equation tan(2πω/ω0) = ωc/ω. The outgoing mode in of Vpp = 5µV). The additional blue (dashed) line shows the the open transmission line in the presence of a driven DCE photons only created through the mechanical motion of FBAR, keeping only the first order of S(ω0, ω00), reads the FBAR.

n (ω) = [exp( ω/k T ) − 1]−1 bout(ω) = hres(ω, Ω) + Rres(ω)bin(ω) where in ~ B is the thermal T + Sres(ω, Ω + ω)b (Ω + ω) photon occupation at temperature . The photon pro- 1 in duction of electro-mechanical Casimir origin is given by ¯res † res 2 2 + S2 (ω, Ω − ω)bin(Ω − ω), (10) the term |S2 (ω, Ω − ω)| + |hres(ω, Ω)| . The total mean photon number nout(ω) is plotted in Fig.2. For with ω/Ω = 1/2 we see that the Casimir photon production is of the order of 10−2 photons/s/unit bandwidth for a  2iω −1 −2ikω deff hres(ω, Ω) = h(ω, Ω) (1 − ) + e driving voltage of Vpp = 500 µV. This is comparable to ωc the photon production predicted by [20, 21] and should res 0 00 0 00 0 00 S1 (ω , ω ) = S(ω , ω )Ares(|ω |)Ares(|ω |) be easily measurable with current technology [36, 37]. res 0 00 0 00 ¯ 0 00 The DCE part of the total photon number is well above S2 (ω , ω ) = S(ω , ω )Ares(|ω |)Ares(|ω |). the thermal photon noise floor. The non-adiabatic change in the boundary conditions 3 Electro-mechanical DCE photon rate is of electro-mechanical nature, as the voltage driving the FBAR also produces a change in the potential of For an initial thermal state, the mean photon number the boundary, in to the mechanical movement † nout(ω) = hbout(ω)bout(ω)iT is given by of the plate. Even though it can not be measured, one may wonder about the ratio of the photon production 2 nout(ω) = |Rres(ω)| nin(ω) which is purely due to the mechanical movement. In or- res 2 der to calculate electrically generated photons, we con- + |S1 (ω, Ω + ω)| nin(Ω + ω) sider the situation in which the piezoelectric material is + |Sres(ω, Ω − ω)|2 [1 + n (Ω − ω)] 2 in removed, i.e., ∆C = 0, while the voltage source is still 2 + |hres(ω, Ω)| , (11) connected. By subtracting this from the total number of

Accepted in Quantum 2018-08-28, click title to verify 4 generated photons with the same parameters as before, Finally, we would like to highlight that there are other we estimate the purely mechanical photon production traces of the Casimir effect related to quantum correla- to be 5 × 10−9 photons/s/unit bandwidth. Therefore, tion functions. For instance, techniques developed in re- only about one in every million generated photons can cent accurate experiments to measure two-time correla- be ascribed to mechanical motion alone. However, in tion functions in propagating quantum [41– our situation it is not possible to experimentally dis- 43] can be used to measure g(2)(τ). Indeed, the photons tinguish the photons generated by changes in the elec- which leak out of the cavity will show a decaying be- trical boundary conditions from the ones generated by havior of these auto-correlations, however the formalism changes in the mechanical boundary conditions. Con- developed in this letter is not suitable to calculate such sequently, we can only talk about electro-mechanically effects, as we treat both the capacitor and the voltage produced photons. source as classical elements. Improving the quality factor of the FBAR will dras- In summary, we have proposed an experiment consist- tically change the picture. Assuming a quality factor of ing of a mechanical resonator based on a FBAR directly 3·106, i.e. 4 orders of magnitude larger than in Ref. [32], coupled to a superconducting cavity, which generates and a smaller driving voltage of Vpp = 5 µV results in a resonance enhancement of the electro-mechanically ∆x = 8.5 · 10−11 m. This high-Q regime is in generated dynamical Casimir radiation. We calculate already experimentally accessible with other materials, the stable flux of photons proceeding from an electro- for example with GHz mechanical oscillators made of mechanically amplified quantum vacuum, demonstrat- [38]. The higher mechanical quality factor and ing that measuring an effect is within reach of cur- the lower driving voltage benefit the ratio of DCE pho- rent technology. We also propose a heuristic method tons generated via mechanical compared to electrical to estimate the fraction which can be ascribed to the origin, such that both can be of equal magnitude, see non-adiabatic change in the mechanical boundary con- inset of Fig.2. ditions. Further improvements can either be realized by using Josephson metamaterials or a larger mechan- ical quality factor to enhance both the photon produc- 4 Discussion and Conclusion tion and the ratio of the purely mechanically generated photons. Our work also paves the way towards experi- Further improvements could be obtained by using a ma- mentally tests of other fundamental relativistic effects, terial with higher piezoelectric coefficient, higher me- such as the Unruh effect or the Hawking radiation, by chanical quality factor and larger dielectric constant for properly modifying the proposals in Ref. [24] employing the FBAR. The most promising approach is the use mechanical resonators. of SQUID-based metamaterials for the resonator. In- deed, recent experiments with metamaterials based on Acknowledgments long arrays of Josephson junctions have demonstrated Z [39, 40] an increase in the effective impedance 0 by We would like to thank Adrian Parra, Simone Felicetti, Z ≈ 104 two orders of magnitude to 0 Ohm. This was Philipp Schmidt, Hans Huebl, Nicola Roch, and Gary achieved while keeping the total capacitance of the res- Steele for fruitful discussions and useful insights. M.S. onator approximately constant, which means that the and E.S. are grateful for funding through the Spanish v = (CZ )−1 speed of light in the metamaterial 0 is re- MINECO/FEDER FIS2015-69983-P and Basque Gov- duced by two orders of magnitude. Such a device would ernment IT986-16. S.G. acknowledges financial support lead to an enhancement in photon production by four from Foundation for Fundamental Research on orders of magnitude, from which we can again roughly (FOM) Projectruimte grants (15PR3210, 16PR1054), estimate the ratio of the mechanically produced photons the European Research Council (ERC StG Strong- with respect to the electrically generated ones, which Q), and the Netherlands Organisation for Scientific would improve by two orders of magnitude. This results Research (NWO/OCW), as part of the Frontiers of from the former scaling with Z0, as shown in Eq. (5), 1/2 Nanoscience program, as well as through a Vidi grant while the latter scaling as Z0 , as shown in Eq. (6). (016.159.369). W.W. acknowledges financial support by However, using superconducting metamaterials could Chalmers Excellence Initiative Nano. lead to other unwanted effects such as the of a finite-size lattice structure or the impedance mis- match between the resonator and the transmission line. A Piezo-electromechanical coupling Incorporating these properly into our proposal would require more detailed studies, which we leave as a fu- We base our proposal on an FBAR device of thickness ture direction. t and square surface area A = b2 with volume V = A · t

Accepted in Quantum 2018-08-28, click title to verify 5 and define the z-direction (subscript 3) along the thick- Evaluating the mechanical response x˜(ω) on resonance ness of the FBAR. A voltage is applied to the plates ω = Ω yields of the FBAR, resulting in an electric field along the z- direction that is assumed to be homogeneous. Due to Q E Vpp |x˜(Ω)| = d33 . (17) piezoelectric coupling this voltage results in mechanical Ω2 ρt t strain : V This formula can be rewritten by using the FBARs res-  = −d E = −d 3 , (12) 3j 3j 3 3j t onance frequency Ω = 2πv/2t (with velocity of sound p where dij is the piezoelectric coupling coefficient. This v = K/ρ, bulk modulus K = E/(3(1−2ν)), Poisson’s mechanical strain has a geometric and stress-related ef- ratio ν, density ρ) to fect, which are discussed in the following. Q |x˜(Ω)| = (3(1 − 2ν)) d V π2 33 pp A.1 Geometric effect Q = (3(1 − 2ν)) ∆z. (18) Strain leads to contraction and expansion of the me- π2 chanical device and, thus, its change. For This means that the driven response is by a factor of the z direction, one obtains: ∼ Q/π2 larger than the geometric response ∆z alone.

∆z = 33 · t = −d33V3. (13) This dimensional change shifts the resonance frequency B Proposed implementation and model- of the FBAR device. Using Ω = 2πv/2t (v velocity of sound) as approximate formula for the resonance fre- ing quency of an FBAR, one gets ∆Ω/Ω0 = −∆t/t ≈ −33. Using (12), one arrives at B.1 FBAR

∆Ω V3 We base our proposal on existing technology and em- = d33 . (14) Ω0 t ploy experimental parameters closely following Ref. [32]. The mechanical resonator is a piezoelectric FBAR, A.2 Stress-related effect made up of a heterostructure from Al-AlN-Al, whereby the Al is used to contact the piezoelectric AlN. A spe- Mechanical strain  results in mechanical stress σ via the cific device could have a total thickness of approxi- relation σ = E (E is Young’s modulus). Mechanical mately 1000 nm, consisting of a 300 nm SiO2 layer, two stress over an area A results in a force F acting on the 150 nm Al electrodes surrounding a 350 nm AlN film, crystal surface F = R σdA. A with an average speed of sound of vs = 9100 m/s. The This force can change the resonance frequency of the device could be fabricated on a high-resistivity silicon- device, if the crystal structure could not relax along on-insulator wafer. We drive the FBAR with an AC that direction. For example, this would be the case voltage source V (t) = Vpp cos(ωt + φ) with ω = Ω, i.e., for a doubly-clamped mechanical beam. However, the a driving frequency equal to the fundamental mechani- FBAR has no fixed/clamped boundaries along the z- π cal frequency Ω, and a difference φ = 2 . direction and, thus, can relax. Hence, we neglect any shift in resonance frequency of the FBAR. The force F can nevertheless result in driving the me- B.2 Modified Butterworth-Van Dyke circuit chanical amplitude of the resonator as it acts as an ad- The Modified Butterworth-Van Dyke circuit (MBVD) ditional source term in the dynamic equation. The solu- [34] enables to extract the necessary parameters for tion of a driven, damped in Fourier modeling a mechanical resonator as an equivalent elec- space is trical circuit. This has been, e.g., used in Ref. [32] x˜(ω) = χ(ω) · F˜(ω)/m, (15) for calculating the response of the FBAR made from 2 2 with the mechanical susceptibility χ(ω) = (Ω − ω − Al/AlN/Al. Note that the MBVD is an approximation −1 iγω) and the mass m of the resonator. A sinusoidal of the Generalized Butterworth-Van Dyke circuit [44] V (t) = V cos (ωt + φ) 2 driving voltage 3 pp results in a for low electro-mechanical coupling kt . force The simplest model for a FBAR is derived from ap- Z Z V3(t) proximating the 3-port Mason model to a 4 element F3(t) = σ3(t)dA = Ed33 dA A t A circuit consisting of a plate capacitance C0 in parallel V with a series Rm − Lm − Cm circuit. In our case, the = Ed V (t). (16) 33 t2 3 capacitance C0 ≡ C(t) actually changes in time when

Accepted in Quantum 2018-08-28, click title to verify 6 an AC driving is applied, due to the change in the ge- note that a similar FBAR was recently demonstrated ometric structure. Indeed, mechanical resonators con- with C0 = 1.00 pF [46]. This higher coupling capac- sisting of a piezoelectric material sandwiched between itance would allow increasing the DCE photon rate superconducting layers suffer from a geometrical change further. Taking into account that the of in their structure due to the piezoelectric effect when a AlN is εAlN ≈ 9.2ε0 = 81 pF/m and that the dis- voltage is applied. Therefore, the value of the electric el- tance between the plates is tAlN = 350 nm, we can ements describing the electric response of the resonator model a parallel plate capacitor and estimate its area −10 2 depends on the applied voltage (note that our voltage A = tAlNC0/ε ≈ 7.7 · 10 m . source is a function of time). For the usual applica- In this work, we are primarily interested in calculat- tions of FBARs and other mechanical oscillators, this ing the DCE photon production. Therefore, we sim- dependence is negligible, since it is a tiny correction of plify the full model to the elements containing the the mean value. However, this dependence has already most relevant information, as shown in Fig.3. Tak- been studied, for instance, in Ref. [45], and references ing the parameters considered in Appendix B.4, then 1 4 thereof. The aforementioned 4-element circuit has a se- Zm = Rm +j(ΩLm − ) ≈ 146+j2.8·10 Ohm, while √ ΩCm ries resonance ω (given by L ,C as ω = 1/ L C ) 1 Z0Zm s m m s m m Z0 = R0 − j ≈ 8 − j189 Ohm. As Zeq = ≈ ω C ΩC0 Z0+Zm and a parallel resonance p (set by 0 in series with Z0, since |Z0|  |Zm| and the resistor is much smaller 2 Lm,Cm as (ωp/ωs) = 1 + 1/r). The capacitance ra- than the capacitor, then our approximation follows. tio r is defined as r = C0/Cm. The electro-acoustic In order to understand the connection between the 2 kt can be derived to be mechanical properties described in the AppendixA and the modified Butterworth-Van Dyke model, let us ex- 2 2   plain how the mechanical Q-factor which appears in 2 π ωs ωp − ωs π 1 1 (kt) = = 1 − (19) Eq. (17) is described in terms of the electric elements 4 ωp ωp 8 r r of the circuit. This connection is deeply related, as one may expect, to the resistances in the circuit and the coupling of the losses of the circuit when coupled to other circuits. A detailed theoretical and experimental C(t) analysis of this dependence may be found in Ref. [34]. Cm In this reference, it is proven that C(t) V (t) Lm 1  1 1  ≈ + (20) R V (t) 0 Q Qs Qe Rm ⇡ −1 −1 with (Qs) = ωRmCm, (Qe) = ωR0Cm, and ω the resonance frequency of the circuit. Hence, the quality factor is indeed determined by the electric elements of the circuit. Figure 3: Equivalent circuit for the FBAR. We have employed the modified van Dyke-Butterworth model to provide the equiv- alent lumped-element circuit representation of the FBAR. B.3 Superconducting cavity

R The superconducting cavity has a length of Adding a series resistor s at the input to this model −2 allows to account for the electrode electrical loss. This is d = 3.3 · 10 m, with a fundamental frequency the so-called Butterworth-Van Dyke circuit, which has of ωc,0 = 2π · v/d = 2π · 3.03 GHz with the speed of light in the superconducting material of v = 108 m/s. 5 parameters: C0,Rs,Rm,Lm,Cm. One can add a sixth It is capacitively coupled to a semi-infinite transmission parameter, a resistor R0 in series with C0, accounting for material loss. In Ref. [34], it is derived a set of line with impedance Z0 ≈ 55 Ohm. The capacitor coupling the resonator to the transmission line has a to obtain the 6 parameters from measurable −1 quantities: series and shunt resonant frequencies, the frequency ωc/2π = (2πZ0Cc) ≈ 2π · 29.1 GHz. effective quality factors Qs0,Qp0, and the capacitance and resistance far away from the resonances. To model the FBAR we use experimental parame- B.4 Parameters for the proposed implementa- ters closely following Ref. [32]: Cm = 0.655 fF, Lm = tion 1.043 µH and Rm = 146 Ohm for the mechanical part, and C0 = 0.4 pF (average value in time) and The following table summarizes the experimental pa- R0 = 8 Ohm for the geometric part. We would like to rameters used in this proposal.

Accepted in Quantum 2018-08-28, click title to verify 7 Parameter Symbol Value Material properties AlN Youngs modulus E 308 GPa density ρ 3230 kg/m3 −12 piezoelectric coupling [47, 48] d33 5.1 · 10 m/V Poissons ratio along 0001 ν 0.287 average velocity of sound{ } v 9100 m/s [32] permittivity εAlN 9.2ε0 FBAR parameters quality factor Q 300 resonance frequency Ω 2π · 4.2 GHz thickness AlN layer tAlN 350 nm drive voltage Vpp 0.5 mV driven motional amplitude ∆x(Vpp) 1.7 · Vpp nm/V Superconducting cavity parameters length d 3.3 · 10−2 m cavity frequency ωc,0 2π · 3.03 GHz impedance Z0 55 Ohm −12 DCE capacitance C0 0.4 · 10 F transmission line coupling rate ωc 2π · 29.1 GHz Further parameters Temperature T 10 mK

C DCE miscellanea in the context of the FBAR scheme, the v/c ratio in photon production still holds. C.1 DCE photon rate and its relation to v/c For the sake of completeness, let us estimate the ra- tio of v/c as follows. The maximal velocity of the me- In the following, we will relate the photon rate from chanical resonator on resonance is given as v = ∆x · Ω. Eq. (11) to the commonly used speed ratio v/c in the The speed of light in the superconducting cavity is c = DCE, where v is the velocity of the mirror (in our case (CZ )−1 = 1 · 108 m/s. We obtain with Ω = 2π · 4.2GHz the velocity of the vibrational motion of the FBAR) 0 a ratio v/c of ∼ 2 · 10−10 and ∼ 2 · 10−8 for ∆x = and c is the speed of light (in our case the speed in the 8.5·10−13 m (low mechanical Q) and ∆x = 8.5·10−11 m superconducting material). (high mechanical Q), respectively. The mechanical photon production rate from Eq. (11) depends essentially on the term hnout(ω)i = res 2 0 00 |S2 (ω, Ω − ω)| , which is a function of S(ω , ω ) = p 0 00 0 00 −i∆CZ0 |ω ||ω |θ(ω )θ(ω ). Therefore, hnout(ω)i ∝ C.2 DCE as a parametric effect 2 2 ∆C Z0 ω(Ω − ω), which in resonance ω = Ω/2 gives 2 2 2 In the following, we show by using a simplified model hnout(Ω/2)i ∝ ∆C Z0 Ω /4. Taking into account that how the parametric effect of the DCE emerges from a ∆C ≈ C0∆x/tAlN and that the maximal speed of the change in boundary conditions. To this end, we use the FBAR is given by v = ∆x · Ω, then hnout(Ω/2)i ∝ 2 2 2 2 Hamiltonian of a cavity ended by an oscillating mirror, C0 Z0 v /(4tAlN). Finally, the speed of light in the ma- −1 with CT = C + C0, the sum of the total capacitance of terial is given by c = (CZ0) , with C the density of capacitance of the superconducting cavity, so the cavity and the capacitor. Therefore,

2 2 2 2 2 2 2 2 1 1 hnout(Ω/2)i ∝ C0 C Z0 v /(4C tAlN) = Qv /c , (21) H = q2 + Ψ2, (22) 2CT 2L 2 2 2 with Q = C0 /(4C tAlN) is related with the impedance mismatch between the FBAR and the superconducting which is the Hamiltonian corresponding to an LC cir- cavity and, consequently, with the quality factor of the cuit. Let us assume that the face of the resonator vi- resonator. This expression coincides with Eq. (15) of brates due to the as a classical harmonic os- Ref. [12] and Eq. (3) of Ref. [21] for photon production cillator of frequency Ω and amplitude ∆x, so d(t) = in the presence of a cavity, and shows that, effectively, d0 + ∆x cos(Ωt) and the capacitance, that we assume

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Accepted in Quantum 2018-08-28, click title to verify 10