Non-reciprocal energy transfer through the Casimir effect
Zhujing Xu, Xingyu Gao, Jaehoon Bang, Zubin Jacob, Tongcang Li
AA Casimir diode
Zhujing Xu, Xingyu Gao, Jaehoon Bang, Zubin Jacob,
2, 3 and Tongcang Li
1, 2, 3, 4, * Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA (Dated: February 26, 2021)A fundamental prediction of quantum mechanics is that there are random fluctuations everywhere in a vacuumbecause of the zero-point energy. Remarkably, quantum electromagnetic fluctuations can induce a measurableforce between neutral objects, known as the Casimir effect [1], which has attracted many demonstrations [2–9]. The Casimir effect can dominate the interaction between microstructures at small separations and has beenutilized to realize nonlinear oscillation [10], quantum trapping [11], phonon transfer [12], and dissipation di-lution [13]. However, a non-reciprocal device based on quantum vacuum fluctuations remains an unexploredfrontier. Here we report a Casimir diode that achieves quantum vacuum mediated non-reciprocal topologicalenergy transfer between two micromechanical oscillators. We modulate the Casimir interaction parametricallyto realize strong coupling between two oscillators with different resonant frequencies. We engineer the sys-tem’s spectrum to include an exceptional point [14–17] in the parameter space and observe the topologicalstructure near it. By dynamically changing the parameters and encircling the exceptional point, we achievenon-reciprocal topological energy transfer with high contrast. Our work represents an important development inutilizing quantum vacuum fluctuations to regulate energy transfer at the nanoscale and build functional Casimirdevices.
In 1948, Casimir speculated that two uncharged metalplates separated by a vacuum gap would experience an attrac-tive force due to quantum vacuum fluctuations [1]. Besidesits fascinating origin and importance in fundamental physics[18], the Casimir effect can dominate at sub-micrometerdistances and is essential for micro and nano technologies[19, 20]. Several studies have demonstrated Casimir effect-based devices with various functions such as nonlinear oscil-lation [10], quantum trapping [11], phonon transfer [12, 21],and dissipation dilution [13]. However, a non-reciprocal de-vice based on the Casimir effect has been elusive. We notethat many essential devices such as diodes, isolators, and cir-culators require non-reciprocity.Similar to the control of electric current with diodes, wedevelop an efficient “Casimir diode” that can rectify energytransfer coupled by Casimir interaction. The non-reciprocityis realized by robust topological operations near an excep-tional point. An exceptional point is a branch singularity of anon-Hermitian system such that the eigenvalues of the systemHamiltonian collapse with each other for both real and imag-inary parts in the parameter space [15, 22]. The exceptionalpoint has attracted broad interests in optics, optomechanics,and acoustics [23–28] since it exhibits a unique topologicalstructure. But it has not been realized with Casimir interac-tions before. We utilize the strong nonlinearity of the Casimirinteraction, and achieve the topological operations near theexceptional point by modulating the separation between twomicromechanical resonators at the desired frequency and am-plitude (Fig. 1). In this way, we realize non-reciprocal topo-logical energy transfer by Casimir force. The direction of en-ergy transfer depends on the sequence of operations. There-fore, the system provides the flexibility for future applicationsin Casimir-based devices.Our device consists of two cantilevers with resonant fre- quencies ω and ω as shown in Fig.1(a). A microsphere is at-tached to the left cantilever. The oscillating amplitudes of twocantilevers are denoted as A and A . Two cantilevers expe-rience quantum vacuum fluctuations and attract each other bythe Casimir force. We apply an additional modulation of theseparation to couple two cantilevers with different frequen-cies. For the separation considered in our experiment, quan-tum fluctuations (instead of thermal fluctuations) dominate theCasimir interaction (see Supplementary Fig. S2). In an inter-action picture, the simplified Hamiltonian (see Methods andSupplementary Information) is H int = (cid:18) − iγ g g − iγ − δ (cid:19) , (1)where γ , is the damping rates of two cantilevers and g isthe coupling strength between two cantilevers. δ is the detun-ing of the system. The coupling strength and detuning are di-rectly related to the modulation amplitude δ d and modulationfrequency f mod = ω mod / π as g = d F C dx δ d / √ m m ω ω and δ = ω + ω mod − ω . Here d F C dx is the second deriva-tive of Casimir force F C and m , is the mass of two can-tilevers. For this open system, the exceptional point is locatedat g = | γ − γ | and δ = 0 . At this specific point, the two eigen-values λ ± of the Hamiltonian are degenerate for both real andimaginary parts as shown in Fig.1(b) and 1(c). The eigenval-ues (two surfaces) intersect each other and exhibit a nontrivialtopological structure near the exceptional point. Dynamicallychanging the parameters that enclose the exceptional pointwill transfer the energy to the specific eigenstate [29, 30]. Ex-perimentally we can control the modulation frequency f mod and the modulation amplitude δ d independently as a functionof time. In this way, we can realize topological energy trans-fer by designing a control loop that encloses the exceptional a r X i v : . [ qu a n t - ph ] F e b Quantum vacuum fluctuations Resonator 1 ( 𝜔 , 𝐴 ) Resonator 2 ( 𝜔 , 𝐴 ) Modulation frequency 𝑓 𝑚𝑜𝑑 Modulation amplitude 𝛿 𝑑 (a) (b) (c) EP EP
FIG. 1.
Casimir effect in the dual-cantilever system and eigenval-ues near the exceptional point. (a): Two modified cantilevers withresonant frequencies ω and ω experience a Casimir force due toquantum vacuum fluctuations. The vibration amplitudes of two can-tilevers are denoted as A and A . An additional slow modulationwith a frequency f mod and an amplitude δ d is applied on resonator1 to realize parametric coupling. (b) and (c): The real part (Re ( λ ) )and the imaginary part (Im ( λ ) ) of the eigenvalues λ ± of the systemHamiltonian are shown as a function of the modulation frequency f mod and the modulation amplitude δ d . The two eigenvalues exhibita nontrivial topological structure near the exceptional point (EP). point.The schematic of the experimental setup is shown in Fig.2(a). The motion of two cantilevers are monitored indepen-dently by two fiber interferometers. The natural frequency anddamping of two modified cantilevers are ω = 2 π × Hz, ω = 2 π × Hz, γ = 2 π × . Hz, and γ = 2 π × . Hz. The microsphere and both cantilevers are coated bygold. When two surfaces are close to each other, they willexperience the Casimir interaction. At small separations, theCasimir force between an ideal conductive sphere and an idealconductive plate is [31] F C ( x ) = − π (cid:126) c Rx , (2)where R is the radius of the sphere, x is the separation be-tween the sphere and the plate, (cid:126) and c are the reduced Planckconstant and the speed of light, respectively. The Casimirforce F C between real materials can be calculated by the Lif- shitz theory [32] and more details can be found in the Supple-mentary Information. The measured Casimir force gradientdivided by the radius of the microsphere is shown in Fig.2(b).The experimental data agrees well with the theoretical predic-tion for the Casimir force between real gold films under theproximity-force-approximation.We now discuss how we achieve strong coupling betweentwo cantilevers by quantum vacuum fluctuations. To coupletwo cantilevers with different resonant frequencies, we mod-ulate the separation between them at a slow rate ω mod . Theeffective coupling strength is controllable by changing themodulation amplitude δ d . Different from direct coupling thatrequires identical resonant frequencies, parametric couplinggives us more freedom to couple arbitrary resonators and con-trol the coupling time, the coupling strength, and the effectivedetuning of two oscillators [33, 34]. If we resonantly exciteresonator 2 with a constant amplitude A and modulate theseparation at a rate of f mod = ( ω − ω ) / π , the excitationon resonator 2 is down-converted to the vibration of resonator1 (Fig.2(c)) [35]. Under weak-coupling approximation, the ra-tio of the oscillating amplitudes between two resonators ( A and A ) at their own resonant frequencies at the steady stateis given as A A = | d F C dx | ω δ d γ k . (3)Note this coupling is due to the second derivative of theCasimir force d F C /dx . For a spring force F s ∝ − x , thesecond derivative will be 0 and the parametric coupling cannot be achieved by simply modulating the separation. Themaximum transduction amplitude A /A as a function of sep-aration is shown in Fig.2(d). The experimental result agreeswith the prediction Eq. 3. Thus we have realized the energytransfer through quantum vacuum fluctuations. We can re-duce the separation x or increase the modulation amplitude δ d to achieve strong coupling. Fig.2.(e) shows the power spec-tral density (PSD) of cantilever 2 as a function of the PSDfrequency and modulation frequency f mod when parametricmodulation is applied on cantilever 1 (left). Near resonance,level repulsion is observed in Fig. 2(f), which shows strongcoupling between cantilevers. More details of the spectrumcan be found in Supplementary Fig. S4.In the experiment, topological energy transfer is realized bydynamically controlling the parameters to encircle the excep-tional point. Parametric coupling gives us the freedom to con-trol coupling strength and detuning of the system as a functionof time. Therefore, we can modify eigenvalues of the sys-tem dynamically by controlling modulation amplitude δ d andmodulation frequency f mod as shown in Fig.1.(b) and (c).To design and engineer the control loop, we first measurethe exceptional point in the parameter space experimentally.The exceptional point locates at the point when two eigen-value surfaces intersect each other. We apply an additionaldamping on cantilever 2 (right) such that γ = 2 π × . Hz to modify the location of the exceptional point in theparameter space and make the control loop easier to con- (a) (f) (b)
Piezo 2
Piezo 1
Fiber 1 Fiber 2 R1 R2 (d) (e) (c) 𝑓 mod , 𝛿 𝑑 PSD x 𝑨 𝑨 𝑨 𝑨 FIG. 2.
Force measurement and energy transfer by the Casimir effect. (a). The schematic of the dual-cantilever fiber interferometer setup.A microsphere is attached to one cantilever and both cantilevers are coated with gold for good conductivity and reflectivity. Their motions aremonitored by two fiber interferometers independently. Two piezo chips at the end of cantilevers are used to drive the cantilevers and changethe separation between two cantilevers. (b). The measured force gradient divided by the radius of the microsphere is shown as a function ofthe separation. The red solid line is the theoretical prediction for real gold material. The green dashed curve is the theoretical prediction forideal conductor. (c).The ratio of the displacement of two cantilevers is shown as a function of the modulation frequency f mod at a separationof 139 nm and 161 nm. (d). Parametric modulation is applied to couple two cantilevers. The ratio of the displacement of two cantileversis shown as a function of the separation when cantilever 2 is first excited and the modulation frequency equals to the frequency differencebetween two cantilevers. The red solid curve is the theoretical prediction. (e). Power spectral density (PSD) of the cantilever 2 as a functionof the modulation frequency f mod and PSD frequency. (f) is the refined scan of the white box shown in (e). duct. Fig.3.(a) and (b) are the measured PSD of cantilever2 and they show that the exceptional point in this double-cantilever system is located approximately at δ d = 11 nm and f mod = 725 Hz. Based on the measurement of exceptionalpoint, we design a dynamical clockwise control loop as shownin Fig. 3(c). The modulation amplitude is continuously con-trolled from 6.65 nm to 13.3 nm and the modulation frequencyis tuned from 680 Hz to 785 Hz. When the control loop en-closes the exceptional point, the eigenstates can interchangeat the branch and one state dominates at the end of the controlloop. An important striking feature is that the preferred trans-fer direction of energy transfer depends on the direction of thetopological control loop. Our simulations (see SupplementaryInformation) of the energy transfer process take into accountthe nonlinearities inherent in this Casimir force coupled sys-tem.We now realize a Casimir diode by engineering the systemdynamically. Figure 3(d)-(g) show the experimental resultsof the topological energy transfer process for the clockwise (CW) and the anti-clockwise (ACW) loops with different can-tilevers being excited. We use normalized energy, which isdefined as E / ( E + E ) and E / ( E + E ) , to quantify theportion of energy for two cantilevers in the transfer process.Here E and E are the energy of two cantilevers by mea-suring their oscillating amplitudes. One of the cantilevers isfirst driven resonantly to the excited state from 0 to 80 ms.The dynamical control loop is then applied starting at 80 msas shown in the gray shaded area. We measure the energy E and E at the end of the loop ( t = 160 ms) and see whetherenergy is transferred to a different state. Fig. 3.(d) and (f)show that a clockwise loop allows energy transferring from 1to 2 while avoids the opposite direction. Cantilever 2 domi-nates the total energy in the system after the clockwise loopno matter what the initial state is. On the contrary, cantilever1 dominates the energy after the anti-clockwise loop as shownin Fig. 3(e) and (g). Two energy transfer processes with differ-ent directions for the same control loop have a high contrast.The transfer process is nonreciprocal and exhibit a topological T12 T21T12 (e) (g)
T21Clockwise (CW) 1 to 2 (f)(d) (a) (b)(c)
680 Hz785 Hz 13.3 nm6.65 nm 𝒇 𝒎𝒐𝒅 𝜹 𝒅 EP 𝒇 𝒎𝒐𝒅 ≈ 𝟕𝟐𝟓 Hz 𝜹 𝒅 ≈ 𝟏𝟏 nm 𝑓 mod , 𝛿 𝑑 Clockwise (CW) 2 to 1Anti-clockwise (ACW) 2 to 1Anti-clockwise (ACW) 1 to 2 CW FIG. 3.
The exceptional point and topological energy transfer by the Casimir effect . (a). PSD intensity of cantilever 2 as a function ofmodulation frequency f mod and PSD frequency when δ d = 11 . nm. (b). PSD intensity of cantilever 2 as a function of modulation amplitude δ d and PSD frequency when f mod = 725 Hz. (c). A clockwise (CW) control loop in the parameter space for the topological energy transferand it encloses the exceptional point (green star). (d)-(g). Measurement of the normalized energy in the transfer process. One cantilever is firstdriven to the excited state from 0 ms to 80 ms. The dynamical control starts at 80 ms (shaded in gray) for clockwise (CW) and anti-clockwise(ACW) loop and lasts for 80 ms. A clockwise loop allows the energy transferred from 1 to 2 and avoids the reverse direction. On the otherhand, an anti-clockwise loop allows the energy transferred from 2 to 1 and avoids the reverse direction. feature. The simulated transfer process is shown in Supple-mentary Fig. S5, which agrees with the experimental results.Our system is flexible to manipulate the preferable transferdirection by designing the control loop.We demonstrate that the topological nature is robust andleads to highly efficient directionality. To quantify the ef-ficiency of energy transfer after the dynamical control loop,we calculate the transfer efficiency which is defined as η = E / ( E + E ) at the end of the control loop for the case thatcantilever 1 is first excited. In this way, we can measure howmuch energy is transferred from cantilever 1 to cantilever 2.Similar definition applies to the reverse direction. Fig.4 showsthe dependence of transfer efficiency on the size of the controlloop when cantilever 1 is first excited. We change the max-imum modulation frequency f maxmod while keeping the mini-mum modulation frequency at 680 Hz for a clockwise controlloop as shown in the inset of Fig.4. The control on the mod-ulation amplitude δ d remains the same as mentioned above. When the control loop encloses the exceptional point (greenstar), energy is transferred to cantilever 2 when we first excitecantilever 1. The transfer process is topologically robust dueto the special topological structure near the exceptional point.It can work for a wide range of parameters. On the other hand,when the loop does not enclose the exceptional point, the pro-cess is non-topological and little energy is transferred fromcantilever 1 to cantilever 2. The experimental results agreewith the theoretical prediction.In conclusion, we report a Casimir diode that achieves non-reciprocal energy transfer with quantum vacuum fluctuations.Strong phonon coupling and energy transfer between two me-chanical resonators are realized by parametric modulation.Under the careful design of the control loop, topological en-ergy transfer with high contrast is observed. Our work devel-ops a flexible and robust method to regulate vacuum fluctua-tions and build functional Casimir devices. Maximum modulation frequency 𝒇 𝒎𝒐𝒅 𝒎𝒂𝒙 (Hz) 𝒇 𝒎𝒐𝒅 𝜹𝒅 𝜹𝒅 EP No EP 𝒇 𝒎𝒐𝒅 Non-topological Topological 𝒇 𝒎𝒐𝒅𝒎𝒂𝒙 𝒇 𝒎𝒐𝒅𝒎𝒂𝒙 FIG. 4.
Efficiency of the topological energy transfer by theCasimir effect.
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Experimental set-up.
We use two AFM (atomic force mi-croscope) cantilevers to construct our Casimir interaction sys-tem. The left cantilever has a size of µ m × µ m × µ m .A 70- µ m-diameter polystyrene sphere is attached on the freeend of the left cantilever by a vacuum-compatible conductiveepoxy. The right cantilever has a size of µ m × µ m × µ m . 70-nm thick gold layers are coated on both sidesof cantilevers and the sphere to create metallic surfaces forthe Casimir interaction and a better reflectivity for detection.Films with nearly equal thicknesses are coated on both sidesof cantilevers in order to reduce strains on the cantilevers.Each cantilever is mounted on a piezoelectric stack to controlits equilibrium position and vibration. One cantilever is con-nected to the electrical ground. A biased voltage is appliedon the other cantilever to minimize the effects of the patchpotential (see Supplementary Fig. S3).The motion of two cantilevers are monitored by two fiberinterferometers. A 50- µ W laser with a wavelength of 1310nm is incident on the back side of a cantilever from a fiberand gets reflected back. The reflected light is directed intothe same optical fiber and interferes with the reflected lightfrom the fiber-air interface. The interference signal gives theinformation of the separation between the cantilever and thefiber and hence we can measure the motion of the cantilever.The jacket and the cladding of the fibers have been removedand the bare fiber has a diameter of 125 µ m. The fibers areplaced at a distance of more than 200 µ m from the cantileversto reduce the electrostatic effect. To minimize the environ-mental impact, the system is placed on top of optical tables bytwo stages of pneumatic vibration isolation (see Supplemen-tary Fig. S1). The experiment is conducted under a pressureof − torr at room temperature. To measure the Casimirforce, we monitor the frequency shift of the cantilever at eachseparation using a phase-lock loop (see Supplementary Fig.S3). Casimir force.
We use the Lifshitz theory to calculate theCasimir force between real materials [32]. The Casimir en-ergy per unit area of two parallel plates at zero temperaturewith a finite separation x is [32] E ( x ) = (cid:126) π (cid:90) ∞ k ⊥ dk ⊥ (cid:90) ∞ dξ { ln[1 − r T M ( iξ, k ⊥ ) e − xq ]+ ln[1 − r T E ( iξ, k ⊥ ) e − xq ] } , (4)where ξ is the imaginary frequency and k ⊥ is the wave vectorperpendicular to the surface. r T E ( iξ, k ⊥ ) and r T M ( iξ, k ⊥ ) are the reflection coefficients of the transverse electric andmagnetic modes [32]. At a finite temperature T , both quantumfluctuations and thermal fluctuations contribute to the Casimirenergy per unit area E ( x, T ) . In our experiment, the radiusof the microsphere and the dimensions of the bare cantileverare far larger than the separation. Therefore, we can useproximity-force approximation (PFA) to evaluate the Casimirforce between a sphere and a plate. The Casimir force for oursystem is F C ( x, T ) = − πRE ( x, T ) , where R is the radiusof the microsphere. As shown in Supplementary Fig. S2, thecontribution from thermal fluctuations at T = 300 K is lessthan 7% when the separation is less than 1000 nm, and is onlyabout 3% when the separation is 200 nm. Thus the effects ofquantum vacuum fluctuations dominate in our experiment.
Effective Hamiltonian.
Under a slow modulation andCasimir interaction, the separation between two cantileversis time-dependent such that x ( t ) = d + δ d cos( ω mod t ) + x ( t ) − x ( t ) . Here d is the equilibrium separation whenthere is no modulation applied, δ d is the modulation ampli-tude. ω mod = 2 πf mod , where f mod is the modulation fre-quency. x ( t ) and x ( t ) describe vibrations of the cantileversnear their equilibrium positions. The motions of the can-tilevers follow equations m ¨ x + m γ ˙ x + m ω x = F C ( x ( t )) ,m ¨ x + m γ ˙ x + m ω x = − F C ( x ( t )) . (5)When the modulation amplitude and the oscillation amplitudeof two cantilevers are far smaller than the separation suchthat δ d , x , x (cid:28) d , we can expand the Casimir force term F C ( d + δ d cos( ω mod t ) + x − x ) to the second order. Sincetwo cantilevers have a frequency difference over 700 Hz, thedirect coupling is neglected. The zero-order and first-orderterms shift the frequency of two cantilevers but have no con-tribution to energy transfer since they are off-resonant. Thecontribution comes from the term d F C dx | d δ d cos( ω mod t )( x − x ) . Therefore, we can rewrite the equations as ¨ x + γ ˙ x + ω x = Λ m cos( ω mod t )( x − x ) , ¨ x + γ ˙ x + ω x = Λ m cos( ω mod t )( x − x ) , (6)where we have Λ = d F C dx δ d .We now separate the fast-rotating term and the slow-varying term for two cantilevers such that x ( t ) = A ( t ) e − iω t ,x ( t ) = A ( t ) e − iω t . (7)Here A ( t ) and A ( t ) are slow-varying amplitudes and wecan neglect their second derivative terms ¨ A ( t ) and ¨ A ( t ) inthe equations of motion. Besides, we consider the conditionthat the damping rate of two cantilevers are far smaller thanthe resonant frequency such that γ (cid:28) ω and γ (cid:28) ω .Therefore, the equations of motion can be rewritten as − iω γ A ( t ) e − iω t − iω ˙ A ( t ) e − iω t = Λ2 m ( A ( t ) e − i ( ω + ω mod ) t − A ( t ) e − i ( ω − ω mod ) t ) , − iω γ A ( t ) e − iω t − iω ˙ A ( t ) e − iω t = Λ2 m ( A ( t ) e − i ( ω − ω mod ) t − A ( t ) e − i ( ω + ω mod ) t ) , (8) where we have taken the rotating frame approximation and ne-glected the fast-rotating term. Now we apply the transforma-tion such that A (cid:48) ( t ) = A ( t ) and A (cid:48) ( t ) = A ( t ) e iδt , where δ = ω + ω mod − ω . Then the equation of motion becomes i (cid:18) ˙ A (cid:48) ( t )˙ A (cid:48) ( t ) (cid:19) = (cid:18) − i γ m ω Λ4 m ω − i γ − δ (cid:19) (cid:18) A (cid:48) ( t ) A (cid:48) ( t ) (cid:19) , (9)where we have neglected the fast-rotating terms. The vibra-tions of two cantilevers can be quantized as phonons. Intro-ducing normalized amplitudes c = (cid:112) m ω (cid:126) A (cid:48) and c = (cid:112) m ω (cid:126) A (cid:48) , we obtain the equation of motion for phononmodes i (cid:18) ˙ c ˙ c (cid:19) = (cid:18) − i γ g g − i γ − δ (cid:19) (cid:18) c c (cid:19) , (10)where g = Λ2 √ m m ω ω = d F C dx δ d √ m m ω ω and δ = ω + ω mod − ω . Thus the effective Hamiltonian of the systemis Eq. (1) in the main text. The eigenvalues of the Hamiltonianare λ ± = − δ − i γ + γ ± (cid:114) − ( γ − γ ) δ + g − i ( γ − γ ) δ. (11)The eigenvalue depends on modulation amplitude δ d andmodulation frequency ω mod . We can also see that the excep-tional point locates at δ = 0 and g = | γ − γ | which meansthat ω mod = ω − ω and δ d = | γ − γ |√ m m ω ω d F C /dx . DATA AVAILABILITY
The data that support the findings of this study are availablefrom the corresponding authors upon reasonable request.
ACKNOWLEDGMENTS
We are grateful to supports from the Defense Advanced Re-search Projects Agency (DARPA) NLM program, and the Of-fice of Naval Research under Grant No. N00014-18-1-2371.
AUTHOR CONTRIBUTIONS
T.L., Z.X. and Z.J. conceived and designed the project.Z.X., T.L., X.G. and J.B. built the setup. Z.X. performed mea-surements. Z.X. and X.G. performed calculations. T.L. andZ.J. supervised the project. All authors contributed in dataanalysis and writing of the manuscript.