Abstract
The coupling of distinct systems underlies nearly all physical phenomena. A basic instance is that of interacting harmonic oscillators, giving rise to, for example, the phonon eigenmodes in a lattice. Of particular importance are the interactions in hybrid quantum systems, which can combine the benefits of each part in quantum technologies. Here we investigate a hybrid optomechanical system having three degrees of freedom, consisting of a microwave cavity and two micromechanical beams with closely spaced frequencies around 32 MHz and no direct interaction. We record the first evidence of tripartite optomechanical mixing, implying that the eigenmodes are combinations of one photonic and two phononic modes. We identify an asymmetric dark mode having a long lifetime. Simultaneously, we operate the nearly macroscopic mechanical modes close to the motional quantum ground state, down to 1.8 thermal quanta, achieved by backaction cooling. These results constitute an important advance towards engineering of entangled motional states.
Introduction
One of the present goals in physics is the explanation of macroscopic phenomena as emerging from the quantummechanical laws governing nature on a microscopic scale. This understanding is also important for future quantum information applications^{1,2,3,4}, as many of the most promising platforms are based on nearly macroscopic degrees of freedom. For macroscopic mechanical objects, their potential quantum behaviour^{5} has been actively investigated with resonators interacting with an electromagnetic cavity mode^{6}. In particular, freezing of the mechanical Brownian motion^{7,8} down to below one quantum of energy has recently been observed^{9,10}.
Another result of radiation–pressure interaction is the mixing of the normal modes of vibration into linear combinations of the uncoupled phonon and photon modes. This has been verified with optomechanical Fabry–Perot cavity^{11,12}, and by its onchip microwave analogue^{13} using an aluminium membrane. All this work is paving the way towards engineering nonclassical motional^{5} and hybrid quantum states^{1,3,4,14} for basic tests of quantum theory^{15,16}, as well as applications in foreseeable future.
Corresponding phenomena in systems comprising more than two active degrees of freedom^{17,18,19} have received less attention. In the optomechanical crystal setup, coupling of mechanical vibrations via radiation–pressure interaction has been demonstrated with the zipper cavity^{20}, however, with some direct interaction between the beams. In the microwave regime, such measurements have remained outstanding. In this work, we take a step further, and examine a multimode system where two micromechanical beams, having resonant frequencies ω_{1}/(2π)=32.1 MHz and ω_{2}/(2π)=32.5 MHz, are each coupled to a microwave onchip cavity. We obtain the first evidence of hybridization of all the three degrees of freedom. This was made possible by operating, as opposed to the optical setup, in the goodcavity limit, that is, both ω_{1} and ω_{2} were much larger than the cavity decay rate γ_{c}. Simultaneously, the mechanical modes are found at low occupation numbers n_{m}, near the quantum ground state, down to n_{m}=1.8±0.5. These are the lowest occupations recorded with nanowire resonators to date, and they occur via the sideband cooling, starting from operation at dilution refrigerator temperatures at a few tens of millikelvin.
Results
Cavity with several mechanical modes
In the optical domain^{20}, a threemode optomechanical system can be pictured as a Fabry–Perot cavity with both massive mirrors mounted via springs, see Fig. 1a. In the microwave version, (Fig. 1b), the mirrors are replaced by movable capacitors formed by the mechanical resonators. The electromechanical coupling arises when each mechanical resonator (labelled with j), expressed as timevarying capacitors C_{j}, independently modulates the total capacitance, and hence, the cavity frequency ω_{c}. This is described by the coupling energies g_{j}=(ω_{c}/2C)∂C_{j}/∂x_{j}. In the actual device (see the micrograph in Fig. 1c), the mechanical resonators are spatially separated by about 100 μm, and we hence expect the direct interaction between the beams via the substrate material to be negligible. We drive the cavity strongly by a microwave pump signal applied at the frequency ω_{P} near the cavity frequency. The pump induces a large cavity field with the number of pump photons . This allows for a linearized description of the electromechanical interaction and results in substantially enhanced effective couplings , where is the zeropoint amplitude.
The general Hamiltonian for N mechanical resonators with frequencies ω_{j}, individually coupled to a common cavity mode, written in a frame rotating with the pump, becomes
Here, a is the annihilation operator for the cavity mode, while b_{j} are those of the mechanical resonators. If one assumes roughly similar mechanical resonators, viz. and , the cavity becomes nearly resonant to all of them at the redsideband detuned pump condition Δ=−ω_{m}.
Coupling of the mechanical resonators via the cavity ‘bus’ can be anticipated to be significant if the mechanical spectra begin to overlap when their width increased by the radiation pressure, γ_{eff}=γ_{m}+γ_{opt}, grows comparable to the mechanical frequency spacing. Here, γ_{opt}=4G^{2}/γ_{c}. The equations of motion following from equation (1) allow one to verify the above assumption. We will in the following focus on two resonators, N=2. The result of such a calculation, which is detailed in the Supplementary Methods, is that they experience an effective coupling with energy G_{12}=G_{1}G_{2}/γ_{c}.
In the limit of strong coupling, , the system is best described as a combination of two ‘bright’ modes with linewidths , and of a ‘dark’ mode having a long lifetime . In the latter, the two mechanical resonators oscillate outofphase and synchronize into a common frequency (ω_{1}+ω_{2})/2. In any case, the mode structure can be written with the normal coordinates for mode n, for position quadrature according to . Here we used the dimensionless amplitude of the mechanical oscillations , and similarly the dimensionless flux in the cavity. The mode expansion coefficients give the participation ratio of each uncoupled mode. They satisfy .
In the experiment, we use doubly clamped flexural beams, which are made by the use of ion beam milling of aluminium^{21}, having an ultranarrow nm vacuum slit to the opposite end of the cavity for maximizing the coupling. The cavity is a halfwavelength floating microstrip similar to the one used in Massel et al.^{22}, resonating at ω_{c}/(2π)=6.98 GHz. The total cavity linewidth MHz is a sum of the internal damping γ_{I}/(2π)=1.4 MHz, and the external damping γ_{E}/(2π)=4.8 MHz. At the highest powers discussed, however, we obtain a decreased γ_{I}/(2π)=0.9 MHz, typical of dielectric loss mechanism.
There are a total of three beams as shown in Fig. 1c. Two of the beams have a large zeropoint coupling of g_{1}x_{01}/2π=39 Hz and g_{2}x_{02}/2π=44 Hz. The frequencies of beams 1 and 2 are relatively close to each other, kHz, such that it is straightforward to obtain an effective coupling G_{12} of the same order. The third beam has an order of magnitude smaller coupling, and hence we neglect it in the full calculation, setting N=2 in equation (1) in what follows.
The measurements are conducted in a cryogenic temperature below the superconducting transition temperature of the Al film ( K) to reduce losses. We used a dilution refrigerator setup as in Massel et al.^{22} down to 25 mK, which allows for a low mechanical mode temperature in equilibrium. Apart from this, cryogenic temperatures or superconductivity are not necessary^{23}.
Experimental data
The modes can be probed by adding to the experimental setup another, the probe, tone^{11,13,22} with frequency ω_{in}. The maximum pump power we can reach is 4 μW, inducing a coherent photon occupancy in the cavity, and effective coupling G_{12}=(2π) × 150 kHz (with G_{1}=0.93 MHz, G_{2}=1.0 MHz). In Fig. 2a, we display changes of the cavity absorption by increasing the pump power. The two peaks at the bottom of the cavity dip are due to beams 1 and 2. They broaden with increasing n_{P}, finally leaving behind a narrow dip between them. The overall lineshape is clearly not a sum of two Lorentzian peaks.
In Fig. 2c, we show a zoomin view of the two peaks, together with theoretical predictions. One immediately sees that an attempt to simulate either peak by a single mechanical resonator coupled to cavity (N=1 in the analysis), thereby neglecting the cavitymediated coupling, fails to explain the lineshape at large . The full simulation with both beams 1 and 2 included, however, produces a remarkable agreement to the experiment. To create these curves, we further take into account a shift of the mechanical frequencies due to secondorder interaction^{8}, given as , where is the secondorder coupling coefficient. The third beam, visible in Fig. 2c as a narrow feature just below 6.982 GHz, was later fitted by running a singleresonator calculation. This is justified because its mixing is negligible owing to weak coupling.
The narrow dip between the peaks manifests the onset of the dark mode, where the cavity participation approaches zero with growing G_{j}, as occurs also in other tripartite systems^{24}. As usual for weakly decaying states, the dark modes can be useful for storage of quantum information^{20,25}. The other two dips (best discerned in the theoretical plots at higher G_{j}, see Fig. 2d) are the bright modes, and they retain a fully mixed character of all three degrees of freedom. In Fig. 2e, we display the prediction for the mode expansion coefficients with respect to coupling. Asymmetry in the role of the mechanical modes is sensitive to parameters. With the maximum coupling in the experiment (), the normal modes are well mixed: the dark mode can be expressed as .
One can also vary the pump detuning, thereby effectively detuning the cavity and the mechanical resonators^{11,13}. An anticrossing of the cavity and mechanical modes is seen in Fig. 3a up to detuning , above which the system exhibits parametric oscillations. The pertinent simulation, Fig. 3b, portrays the main features of the measurement, except bending of the cavity frequency towards lower value in an upward sweep of Δ. We expect such possibly hysteretic behaviour of the cavity frequency to be due to secondorder effects beyond the present linear model.
Sideband cooling
We now turn the discussion into showing how the tripartite system resides nearly in a pure quantum state during the modemixing experiments. The mechanical resonators can be sideband cooled with the radiation pressure backaction^{8,26,27,28} under the effect of the pump, which also creates the mode mixing. Although the ground state of one mode was recently achieved this way^{9,10}, cooling in multimode or nanowire systems is less advanced. The efficiency of cooling with two mechanical resonators is supposed to become hindered at those values of effective coupling where mode mixing starts to set in^{29}. At modest pump powers, relevant for achieving the lowest phonon numbers in the present case, mixing remains insignificant.
The incoherent output spectrum can be due to either a finite phonon number n_{m}, or nonequilibrium population number of the cavity. The former manifests as a Lorentzian centred at the upper sideband, as in Fig. 4a, with the linewidth γ_{eff}=γ_{m}+4G^{2}/γ_{c}, whereas the latter gives a small emission at broader bandwidth. The analytical approximation^{8} and experimental details are discussed in Methods.
At low input power, when the cavity backaction damping is small and , phonon number is set by thermal equipartition . This is best observed by the area of the mechanical peak in the output spectrum, as in Fig. 4a, which should be linear in cryostat temperature. The linearity holds down to about 150 mK temperatures (Fig. 4b), below which we observe intermittent heating, which is sensitive to parameters.
The cooling of beam 2 is observed in the mechanical peaks shown in Fig. 4c (see Methods), and the resulting phonon numbers are displayed in Fig. 4d. At a relatively high temperature of 245 mK, the data points follow theory well up to input powers corresponding to . At the minimum cryostat temperature of 28 mK, the mode is not thermalized in a manner that depends irregularly on power. We obtain a minimum phonon number in beam 2 of , where the mechanical mode spends onethird of the time in the quantum ground state. The other mechanical mode, beam 1, cooled simultaneously down to 2.5 quanta, possibly compromised by the slightly smaller coupling. The cooling is quite clearly bottlenecked by heating, by the pump microwave, of the bath to which the mechanical mode is coupled. For the optimum data points, the starting temperature for cooling was raised to 150 mK, and above this it grows roughly quadratically with n_{P}.
Discussion
The coupling of micromechanical resonators mediated via microwave photons in an onchip cavity is a basic demonstration of the control of a multimode mechanical system near the quantum limit. The setup provides a flexible platform for creation and studies of nonclassical motional states entangled over the chip^{30}, or over macroscopic distances^{31,32}. The setup is easily extended to embody nearly arbitrarily many mechanical resonators, hence allowing for designing an electromechanical metamaterial with microwavetunable properties. For future quantum technology applications even at elevated temperatures^{23}, data may be transferred for storage^{33} from photonic modes into the dark states, which decay very slowly at the mechanical damping rate.
Methods
Device fabrication
The lithography to define the meandering cavity and jigs for the beams consisted of a single layer of electronbeam exposure, followed by evaporation of 150 nm of aluminium on a fused silica substrate. As the cavity is of planar structure, it has no potentially problematic crossovers, which tend to have weak spots in superconducting properties and hence could heat up the cavity. We thus expect the cavity to portray a high critical current limited only by the intrinsic properties of the strip. With a 2 μm wide strip, we expect the critical current to be several milliamperes. With the maximum circulating currents in cavity (Fig. 2), about 0.5 mA of the peak value, there was no sign of nonlinearity in the cavity response.
The mechanical resonators were defined by focussed ion beam cutting, as in Sulkko et al.^{21}. We used low gallium ion currents of 1.5 pA, which gives the nominal beam width of about 7 nm. To maximize sputtering yield with minimal gallium contamination, we used a singlecutting pass mode. Otherwise, the cut beams tend to collapse to the gate. With the current recipe, fabrication of down to 10 nm gap widths over 5–10 μm distance can be done with about 50% yield.
Theory of cavitycoupled resonators
The Hamiltonian for N mechanical resonators each coupled to one cavity mode via the radiation–pressure interaction is
The pump with the Hamiltonian induces a large cavity field with the number of pump photons .
Under the strong driving, the cavity–mechanics interaction can be linearized individually for each beam, resulting in equation (1). Further details of the calculations are given in the Supplementary Methods.
Sideband cooling
For one beam, the measured output spectrum divided by system gain, S_{out}(ω), carries information on the phonon number according to^{8}
which is a Lorentzian centred at the cavity frequency. A nonzero base level is due to possible broadband emission from the cavity, due to a thermal state with occupation . At modest pump (cooling) powers of utmost interest, coupling of the mechanical resonators remains insignificant, and thus we can apply equation (3) separately for each resonator. Some deviations can, however, occur at the highest powers with .
To obtain the phonon number under strong backaction, we fit a Lorentzian to each peak as in Fig. 4c, obtaining independently the base level, amplitude and linewidth for every input power. These values are then compared with equation (3). This leaves, however, yet too many unknowns to obtain the occupations of the mechanics and cavity. This is basically because of uncertainties in the attenuations of both input and amplifier lines, which limit the accuracy of estimating n_{P}. We can obtain a calibration using the linear temperature dependence, and by fitting the dependence of γ_{eff} on input power.
As opposed to, for instance, Rocheleau et al.^{8}, the bottleneck for cooling is not heating of the cavity by strong pump, as seen in Fig. 4d, where the cavity temperature is increasing only little up to the optimum cooling powers corresponding to about 0.15 mA of peak current values in cavity. This can be due to the simplistic structure of the cavity.
The above analysis supposes the presence of nearly no nonequilibrium photons in the cavity at zero or low input powers. This situation can be investigated by observing a possible emission from about cavity linewidth under these conditions. A complication arises because the cavity linewidth is so large that a small change is easily overwhelmed by modest standing waves in cabling. We avoided this issue by using temperature dependence of the cavity frequency, due to the kinetic inductance in the long microstrip. Thereby, a broad emission peak moving according to the cavity frequency would be easily distinguishable. We observed no such signal down to the level of , which justifies the above analysis.
Authors contributions
F.M. and T.H. developed the theory. J.M.P., S.U.C. and M.A.S. contributed to design and fabrication of the samples, and cryogenic setup. P.J.H. and M.A.S. conceived the work.
Additional information
How to cite this article: Massel, F. et al. Multimode circuit optomechanics near the quantum limit. Nat. Commun. 3:987 doi: 10.1038/ncomms1993 (2012).
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Acknowledgements
We thank S. Paraoanu and Lorenz Lechner for useful discussions. This work was supported by the Academy of Finland, by the European Research Council (grants No. 240362Heattronics and 240387NEMSQED), and by the Vaisala Foundation.
Author information
Author notes
 Sung Un Cho
Present address: Korea Research Institute of Standards and Science, Daejeon 305340, Republic of Korea
 Mika A. Sillanpää
Present address: Department of Applied Physics, Aalto University School of Science, P.O. Box 11100, FI00076 Aalto, Finland
Affiliations
Low Temperature Laboratory, Aalto University School of Science, P.O. Box 15100, FI00076 Espoo, Finland
 Francesco Massel
 , Sung Un Cho
 , JuhaMatti Pirkkalainen
 , Pertti J. Hakonen
 , Tero T. Heikkilä
 & Mika A. Sillanpää
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Correspondence to Mika A. Sillanpää.
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Supplementary Figures S1S4 and Supplementary Methods
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