Cavity Optomechanics Mediated by a Quantum Two-Level System

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Received 12 Feb 2015 | Accepted 19 Mar 2015 | Published 27 Apr 2015 DOI: 10.1038/ncomms7981 OPEN Cavity optomechanics mediated by a quantum two-level system

J.-M. Pirkkalainen1, S.U. Cho2,w, F. Massel3, J. Tuorila4, T.T. Heikkila¨3, P.J. Hakonen2 & M.A. Sillanpa¨a¨1

Coupling electromagnetic waves in a cavity and mechanical vibrations via the radiation pressure of photons is a promising platform for investigations of quantum–mechanical properties of motion. A drawback is that the effect of one photon tends to be tiny, and hence one of the pressing challenges is to substantially increase the interaction strength. A novel scenario is to introduce into the setup a quantum two-level system (qubit), which, besides strengthening the coupling, allows for rich physics via strongly enhanced nonlinearities. Here we present a design of cavity optomechanics in the microwave frequency regime involving a Josephson junction qubit. We demonstrate boosting of the radiation–pressure interaction by six orders of magnitude, allowing to approach the strong coupling regime. We observe nonlinear phenomena at single-photon energies, such as an enhanced damping attributed to the qubit. This work opens up nonlinear cavity optomechanics as a plausible tool for the study of quantum properties of motion.

1 Department of Applied Physics, Aalto University, PO Box 11100, FI-00076 Aalto, Finland. 2 Department of Applied Physics, Low Temperature Laboratory, Aalto University, PO Box 15100, FI-00076 Aalto, Finland. 3 University of Jyva¨skyla¨, Department of Physics, Nanoscience Center, University of Jyva¨skyla¨,PO Box 35 (YFL) FI-40014 University of Jyva¨skyla¨, Jyva¨skyla¨, Finland. 4 Department of Physics, University of Oulu, FI-90014 Oulu, Finland. w Present address: Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea. Correspondence and requests for materials should be addressed to M.A.S. (email: Mika.Sillanpaa@aalto.ﬁ).

NATURE COMMUNICATIONS | 6:6981 | DOI: 10.1038/ncomms7981 | www.nature.com/naturecommunications 1 & 2015 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms7981

he basic setup of cavity optomechanics1,2 involves an Results otherwise ordinary optical Fabry–Perot cavity resonator, The basic idea. The device consists of a tripartite system made Tbut having one of the mirrors compliant for mechanical with patterned thin films of superconducting aluminium on a vibrations. The oscillatory displacement x of the mirror couples to silicon chip, containing first of all a microwave resonator (called the length that corresponds to the frequency oc of the cavity. This cavity in the following). Another part is a micromechanical interaction can also be interpreted as that resulting from the resonator and the third is a charge quantum two-level system radiation pressure caused by the cavity photons and characterized (qubit), which consists of small Josephson tunnel junctions and by the coupling energy g0(qoc/qx)xzp. Here, xzp is the zero- mediates the interaction between the former two parts (see Fig. 1 point motion amplitude. Cavity optomechanics has recently and Supplementary Fig. 1). The bare cavity and the mechanical received a lot of attention, for it has turned out as an excellent test modes have the frequencies o0 and o0 , and their dynamical c m w w system for investigations on quantum–mechanical properties of variables are the creation and annihilation operators a , a and b , the motion of nearly macroscopic bodies3,4. b, respectively. The qubit is represented by the Hamiltonian Several different physical systems have been demonstrated, HQB ¼ÀBxsx/2 À Bysy/2 À Bzsz/2. Here, sj are Pauli matrices which realize the radiation-pressure interaction between electro- and Bj are the controllable pseudomagnetic fields, which also magnetic and mechanical modes. In particular, the frequency of depend on the cavity and mechanicalqPffiffiffiffiffiffiffiffiffiffiffiffi coordinates. The level the electromagnetic cavity need not be optical. One of the most spacing of the qubit is B ¼ B2. More discussion of the successful realizations in the recent past has been the use of j j microwave-regime superconducting cavities made with litho- Hamiltonian is found in Supplementary Methods. graphic techniques3,5,6. Here, the displacement changes the A fundamental reason to expect rich optomechanical physics effective capacitance C(x) of the cavity and hence its resonance in the scheme owes to the nonlinearity of the quantum two-level 17–25 frequency oc, and the coupling energy is g0 ¼ (oc/2C)(qC/qx)xzp. system . An important ingredient here is the large d.c. voltage In all the physical realizations, except in the case of bias Vg applied to the mechanical resonator represented as a 7,8 movable capacitance C (x). If coupled to a two-level system microscopic motional degrees of freedom , the radiation- g w pressure coupling energy is much smaller than the cavity decay (qubit), this creates a longitudinal coupling gm(b þ b)sz in the rate k. Therefore, the cavity has to be strongly irradiated up to a natural basis of the charge qubit, with a maximal effect on the c energy of the qubit by the mechanical resonator. When coupling high photon number nc 1 in order to enhance the intrinsically w small interaction. This, however, linearizes the interaction and this qubit to a cavity with the transverse interaction gc(aÀÁþ a)sx, 2 0 creates two linearly coupled harmonic oscillators, which is a the cavity frequency will experience a Stark shift gc = oc À B . somewhat limited system. Hence, one nowadays strives to This Stark shift contains a contribution from the vibrations due to enhance the magnitude of the interaction to exploit the their effect on qubit level spacing. The system then closely intrinsic nonlinearities9–14. The next goal is to reach a strong approximates the radiation-pressure interaction between the coupling threshold g0]k where some quantum phenomena effective cavity mode (frequency oc) and the effective 9,11,13,15,16 become observable . The largest radiation-pressure mechanical mode (frequency om), which both are Stark shifted C coupling values g0/2p 1 MHz have been demonstrated in the from the bare frequencies. The radiation-pressure coupling optical frequency domain (ref. 4). These systems also possess the strength between these modes can be amplified in this setting B largest ratio of the coupling versus cavity decay k, which is, by a factor of the order of CVg/(2e), which amounts to 4to6 B g0/k 0.2%. This is still two and half orders of magnitude away orders of magnitude for typical experimental parameters (ref. 26). from strong coupling. Moreover, there are intriguing nonlinear corrections to the basic Here we build on the scheme proposed in ref. 17. The scheme setup, first of all, a cross-Kerr type interaction in the next higher is motivated by several successful experiments coupling order. Most importantly, the damping of the mechanics becomes superconducting Josephson junction quantum circuits to affected by the nonlinearities. micromechanical resonators18–22, although canonical radiation– The physical device (see Fig. 1c and Supplementary Fig. 2) is pressure phenomena have not been observed in those setups. We fabricated by standard electron-beam lithography. The cavity is a demonstrate a microwave-regime optomechanical device that 5-mm long superconducting microstrip resonating at the (bare) 0 allows for high values of g0/k, and at the same time, displays cavity frequency oc =2p ¼ 4:93 GHz. The charge qubit has the phenomena not previously observed in cavity optomechanics island capacitance CS ¼ Cg þ C1 þ C2, which gives the charging 2 either in the microwave or the optical frequency range. energy of a single electron EC ¼ e /2CS. It is of the same order as

To cavity

ω P Vin Vout g Cc m E ,C g J1 1 Vg c Φ Cg(x) C L x

EJ2,C2

Figure 1 | Optomechanics with a qubit. (a) Illustration showing the idea for introducing a two-level system (qubit or artiﬁcial atom) into a cavity optomechanical setup. Inside the cavity (blue) there is a quantum two-level system (green), which is mechanically compliant (red). (b) Equivalent electrical circuit of the microwave design with the same colour codes as in a. The parameters in the equivalent circuit are CC0.33 pF, LC3.2 nH, CgC1.8 fF,

CSC4.9 fF, EJ ¼ EJ1 þ EJ2C10.5 GHz, ECC5.3 GHz. The reﬂected pump microwave Vout is detected outside the cryostat. The Josephson junctions forming the qubit are marked by crossed boxes, and the mechanical resonator is coupled to the qubit island via a movable capacitance Cg(x). (c) Micrographs of the device showing the qubit and the mechanical resonator. The Josephson junctions are marked by arrows. Scale bar, 2 mm.

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C the Josephson energy EJ, which is EJ/EC 2.0. A strong qubit- cavity coupling with the coupling energy up to B500 MHz c (depending on the qubit bias) sets the system near the ultrastrong coupling27 of circuit QED, and hence gives rise to a pronounced Stark shift of the cavity. The micromechanical resonator is made as a 2-mm wide bridge suspended across a 50 nm vacuum gap m m atop the qubit island (ref. 21) and resonating at the frequency o /2pC65 MHz of the lowest ﬂexural mode. The qubit- m 0 p – m p p + m mechanics coupling varies here between gm/2pC80 MHzy 160 MHz depending on the value of Vg. We note that in spite of the large coupling exceeding the bare mechanical frequency, the Schrieffer–Wolff approach of deriving the radiation-pressure 5.0 interaction is justiﬁed (see Supplementary Methods).

4.9 Measurements. In all the measurements of the mechanical motion we use the basic idea of cavity optomechanics shown in

Fig. 2a, where the cavity is irradiated by the pump microwave at a Spectral density (arb) 4.8 frequency op, which usually resides near the red sideband oc À om. The pump detuning D(op À oc)/om is hence around 65.1 65.2 65.3 65.4 65.5 65.6 DB À 1. The motional sidebands at ±o about the pump give m f – fP (MHz) information about the mechanics. In the following, the sideband closer to the cavity is studied. The main result of the radiation- f – fP (MHz) pressure interaction is the ‘optical spring effect’ which modifies both the mechanical frequency and damping. The changes 0.5 65.2 65.4 opt opt 2 0.5 exerted are denoted as dom , and gm 4g0 nc=k, respectively. The experiments are performed in a dilution refrigerator at 20 mK temperature. We first identify the mechanical resonance V /V 9 out in by pumping the cavity very strongly up to n B10 such that the (dB) c 0.0 0.0 Josephson effect, which has its influence only in the low-photon 0

B g regime nc 1, becomes insignificant, and what is left behind is a g n 0 n –5 linear cavity with the bare frequency oc . The estimated bare optomechanical coupling between the bare cavity and the –10 pB –0.5 mechanics is only g0/2 1 Hz. Such a small coupling does not –0.5 yet allow for observing the small thermal motion of the –15 mechanical resonator at a low temperature. We hence actuate 0.5 0.6 –20 the motion via the gate line, up to about 10-pm displacement Power (arb) amplitude, obtaining the resonance curve shown in 4.8 4.85 4.9 Supplementary Fig. 3. This way we determine the intrinsic fP (GHz) mechanical frequency o /2pC65 MHz and linewidth that is m Figure 2 | Basic characterization. (a) The frequencies involved: the vertical gm/2p ¼ 15 kHz. Notice that om decreases when Vg deviates from zero due to the well-known electrostatic softening of the axis gives the spectrum of either the cavity (oc) or the mechanical mechanical spring. resonator (om). The pump (frequency op) is applied around the detuning DC À 1 below the cavity frequency. (b) Thermal motion measured at Next, we decrease the pump power by about 10 orders of C C B 20 mK. The parameters are D À 1, nc 0.05, ng ¼ 0.25, Vg ¼ 6.5 V. (c) magnitude down to Pp 1 fW. Although power levels are drastically reduced, the detection sensitivity is enhanced and we Gate charge modulation of the cavity resonance absorption. The resonance can observe the sideband peak corresponding to the thermal oc(ng) of the effective cavity appears as the periodic black-blue line. The motion, shown in Fig. 2b, at 20 mK temperature, amounting to dashed red and blue lines are theoretical fits to two of the branches, obtained by using the qubit-cavity Hamiltonian17 in numerical about six phonons at equilibrium. The peak frequency is C somewhat shifted from the calibration experiment, and the diagonalization. The flux bias was F/F0 0.39. The vertical green and grey linewidth of this peak is about twice the intrinsic linewidth. As lines represent the frequencies op and the sideband op þ om, respectively, discussed below, this results from the qubit-induced nonlinea- used to obtain the data in Fig. 3a,b. (d) Spectral density of the emission rities. We now turn into careful characterization of the novel type from the cavity around the mechanical sideband. of optomechanical interaction. We first show that the device displays the basic physics due to the radiation-pressure interaction, namely, the optical spring effect. As discussed below, the top of the large d.c. voltage, we use a fine tuning voltage to adjust latter causes the mechanical frequency and damping to differ the radiation-pressure physics by tuning the gate charge within from the intrinsic values. one period. The period of the cavity response is one electron (0.5) We now characterize the basic behaviour of the microwave instead of two as expected. This is due to a quasiparticle slowly cavity. As seen in Fig. 2c, the frequency of the effective cavity jumping on and off the island21,28. We thus see a double image, depends periodically on the gate charge ng ¼ CgVg/2e. Each shifted by one electron, of the cavity transition. These are marked period corresponds to the tunnelling of one extra Cooper pair by the red and blue dashed lines in Fig. 2c. At a given ng, there are onto the qubit island. Here we have a large dc voltage hence two different cavity frequencies. The system switches B Vg 5y10 V, which is needed to establish the electromechanical between these states while the quasiparticle randomly tunnels on 4 interaction, hence ng410 . However, because of periodicity, we and off the island at the typical rates well below a MHz henceforth refer to ng via its non-integer part for simplicity. On (refs 29,30). These rates are much slower than the mechanical

NATURE COMMUNICATIONS | 6:6981 | DOI: 10.1038/ncomms7981 | www.nature.com/naturecommunications 3 & 2015 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms7981 frequency, and hence in the long-time average we see responses causes its own sideband response. These arise via the detuning from both cavity states superimposed on top of each other. and coupling shown in Fig. 3c,d. The blue curves in reality C C From the geometry we find qCg/qx 24 nF/m and xzp 6 fm. originate from ng values between 0.25...0.5, but reflect to the left We determine the coupling from the experimental data using about ng ¼ 0.25 due to the quasiparticle. The data and theory are g0 ¼ (qoc/qx)xzp, where (qoc/qx) ¼ Vg/2e(qoc/qng)(qCg/qx) can marked with the compatible red and blue colour codes in Figs 2 be related to the slope of the gate charge modulation of the cavity and 3 and Supplementary Fig. 4. We obtain a good agreement frequency seen in Fig. 2c. For ng ¼ 0.25 where the cavity is between the experiment and theory for the mentioned two C reasonably well visible, we have (qoc/qng) (2p) Á 250 MHz and branches. However, the peaks visible around ng ¼ 0...0.15, marked hence g0/2pC0.5 MHz when Vg ¼ 4.6 V. Since |g0(ng ¼ 0.25)| ¼ in black, are not directly explained by this model. We attribute |g0(ng ¼ 0.75)|, this value is the same for both cavity branches. them to that the qubit spends some part of the time in the excited state as observed in this kind of measurements21. The excited Gate charge dependence. One can characterize the radiation- state gives rise to another cavity frequency above oc, having the pressure physics as a function of, for example, the value of the highest qoc/qng around ng ¼ 0. The qubit being excited, however, is beyond the current model and we cannot make a quantitative gate charge ng. Changes in ng change the value of g0, which is comparison to theory. proportional to the slope (qoc/qng). However, a change in ng has also other consequences. With a given pump frequency, the pump detuning D changes according to the effective cavity frequency. Damping due to the qubit. In the rest of the paper, we select a Similarly, also the effective mechanical frequency om will change fixed value ng ¼ 0.25, which simplifies the analysis since there is (due to Stark shift) with ng. Finally, these effective modes exhibit only one cavity branch and a constant g0. Another way to the radiation-pressure interaction visible as further changes in examine the radiation-pressure interaction is to change the pump frequency and damping due to the optical spring effect. We detuning (Fig. 3e). The total damping is maximized close to the denote the resulting total frequency of the mechanics as sideband resonance in agreement with theory. tot opt om ¼ om þ dom . The total damping is given by The total damping should increase linearly with the pump tot opt f f opt gm ¼ gm þ gm þ gm, where the last term gm is due to the qubit photon number nc owing to the contribution gm , moreover, it and is discussed below. In Fig. 2d we show a result of this kind of should depend quadratically on g0 at a given nc. We test this in a measurement, allowing us to connect the thermal emission peak Fig. 4 by varying the photon number at a few fixed values of g0, with the cavity response. A maximum emission occurs at which are set by changing the gate voltage Vg. The data points are C ng 0.25, which here coincides with the sideband resonance and extracted from the data sets in Supplementary Fig. 5, which we hence the expected maximum signal. The two crossing branches here show as an example of thermal motion peaks. As observed in - are associated with the two cavity branches. Fig. 4, the damping towards nc 0 is clearly higher than the We next look in detail at the frequency and linewidth of the intrinsic gm. The extra damping is attributed to that the qubit thermal motion peak in the measurement as in Fig. 2d. The opens another dissipation channel due to the hybridization of the extracted parameters are displayed in Fig. 3a,b revealing three qubit and the mechanics17, analogously to the Stark shift, and is distinct sets of peaks. To understand this data, we first calculate given by the Stark-shifted om and oc, as well as g0 from the diagonaliza- g g2 B2 gf QB m x : tion of the qubit-cavity Hamiltonian (see Supplementary m ’ 3 ð1Þ Methods) and then apply the results in refs 17,31 to obtain a B om prediction for the optical spring. Each of the two cavity branches

65.5 80 80 70 65.4 60 60 50 65.3 40

40 (kHz) (kHz) (MHz) tot m tot tot m m 30 f 20 65.2 20 10 0 65.1 0 4.76 4.78 4.8 4.82 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 fP (GHz) ng ng x 105 2 8 1 6 m

(Hz) 4

0 0 Δ/ω g 2 –1 0 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 n g ng

Figure 3 | Optomechanics enhanced by a qubit. (a,b) Optical spring effect as a function of gate charge. The solid lines are a fit to theory and are plotted in the region where detuning and coupling are such that the peaks are visible. The red and blue colours refer to the two cavity branches as in Fig. 2. The black circles present data that presumably originate from higher excited levels of the qubit (see main text). The pump frequency op/2p ¼ 4.79 GHz, and the sideband (op þ om)/2p ¼ 4.855 GHz. (c) Illustration of the pump detuning for the two branches for the measurements in a and b.(d) Radiation-pressure coupling extracted from oc(ng). (e) Optical spring as a function of pump detuning, at fixed gate charge ng ¼ 0.25. The solid line is a fit to theory. In all data, the cavity photon number is in the range 0.1...1. This is estimated based on the input power and cable attenuation, and in the end fine-tuned as an adjustable parameter. In all the panels, Vg ¼ 4.6 V.

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350 optomechanical setting. This was achieved by integrating, for the ﬁrst time, a quantum two-level system into such a setup, allowing 300 boosting of the radiation-pressure coupling by six orders of magnitude. The ratio of coupling to the cavity linewidth, B4%, 250 V = 9.5 V can likely be further increased by designing the coherence g properties of the qubit. We estimate that boosting factors up to V = 6.5 V 200 g eight orders of magnitude, giving a radiation-pressure coupling (kHz) Vg= 4.6 V up to 100 MHz, are in principle possible with an optimized device tot m 150 (see Supplementary Fig. 6). The setup introduces long-sought 100 nonlinearities into the framework of cavity optomechanics and enables their utilization, for example, in Fock-state measurements 50 of the phonon states, or for tests of the foundations of quantum mechanics32–34. 0 0 1 2 3 4 5 6 n Methods c Chip layout. A photograph of the chip is shown in Supplementary Fig. 2. The ends of the meandering inductance are close to each other such that the double-junction charge qubit can be placed to connect the ends of the meander, which is required 100 for the scheme. The device is fabricated on a regular silicon substrate using a similar process as in ref. 21. The device was designed to not have a too small cavity capacitance for the system to reside in the Lamb–Dicke limit for qubit-cavity 80 coupling (see discussion in Supplementary Methods). Most of the capacitance is provided by a bonding pad-like structure visible between the coupling capacitance and the meandering inductance, marked C in Supplementary Fig. 2. 60 (kHz) Experimental setup. The measurements are done at the base temperature of a

tot m 40 dilution refrigerator using the cabling setup as shown in Supplementary Fig. 7. The pump microwave reflects from the cavity and is amplified at a 4 K stage using a low-noise amplifier. The system noise temperature is in the range of 3...4 K. 20 Filtering the room temperature noise away from reaching the sample via the m cable connected to the mechanical resonator (that is, qubit gate) is challenging 0 because the typical solution (resistive attenuators) cannot be used owing to the high 0 0.5 1 1.5 2 2.5 voltages involved. We hence use reflective filtering. From the qubit point of view as well, the filter needs to be reactive, because dissipative 50 O environment could nc cause too high spontaneous decay. The filtering task is carried out by a combined bias-T and low-pass filter, marked yellow in Supplementary Fig. 7. Without the Figure 4 | Nonlinear cavity damping. (a) The total (tot) mechanical filter, we observe substantial excitation of the qubit to the higher levels, such that the spectrum of the effective cavity changes qualitatively and cannot be modelled linewidth as a function of cavity photon number. Red circles: Vg ¼ 6.5 V, by ground-state transitions. With the filter installed, the excitation is modest, and g /2pC0.7 MHz. Black circles: V ¼ 9.5 V yielding higher g /2pC1.0 MHz. 0 g 0 cavity response well understood. The filter provides enough transmission Squares: smaller g /2pC0.5 MHz obtained with V 4.6 V. The solid lines 0 g ¼ ( À 33 dB) at frequencies around om such that we can actuate the mechanics if tot opt are expectations from the basic linear model, that is gm ¼ gm þ gm .The needed. dashed lines include the qubit-induced extra linewidth, tot opt f gm ¼ gm þ gm þ gm.(b) Zoom-in of the linear regime of a. In all the data, Cavity characterization. The cavity frequency has a strong dependence on the DC À 1, F/F0C0.39. magnetic flux F applied through the loop, as shown in Supplementary Fig. 8, which displays the microwave reflection response quite similar to Fig. 2c in the main text. The cavity resonance appears as the yellow-blue coloured dip varying between 4.99 g Here, QB is the qubit relaxation rate. We could not and 4.84 GHz, periodically in flux with the period of one flux quantum F0.Inthe independently measure it, but a good fit of the data is obtained measurement, we applied noise to the gate such that the response is a weighted C average over all the possible gate n values. In the range B0.4yn y0.6 the with a reasonable value gQB (2p) Á 60 MHz. ÀÁ g g Although the damping due to the basic cavity cooling gopt resonance is not well visible because of the strong gate dispersion. The theoretical m curves plotted on top of the data in Supplementary Fig. 8 are obtained by gives rise to back-action sideband cooling of the mechanical numerical diagonalization of Supplementary Equation 9, but disregarding the - resonator, further theoretical work is needed to understand the mechanics (that is, ng(x) ng), which here is insignificant. The overlaid curves f indicate a good match between the experiment and the modelling. effect of the qubit contribution gm as well as the nonlinearities such as the cross-Kerr effect on the final temperature. The cavity linewidth k has a strong dependence on the qubit gate charge, as displayed in Supplementary Fig. 9. While the changes are due to the qubit, we In Fig. 4 we see that the damping grows first linearly up to B cannot make a definite conclusion of whether it is caused by qubit relaxation or about nc 1 and then saturates. The saturation of the radiation- dephasing. We also checked that simulating the inhomogenous broadening by B pressure physics around nc 1 is due to reaching the limits of the establishing the theory with bare k, and selecting a ensemble of fc, and averaging linear regime of the effective cavity.pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi This is set by the requirement over the ensemble does not markedly change the results. that the total phase excursions nc þ 1=2fzp should be small as compared to 2p in order not to hit the nonlinearity of the References Josephson cosine function. This is analogous to the transition of a 1. Kippenberg, T. J. & Vahala, K. J. Cavity optomechanics: Back-action at the trapped ion out from the Lamb–Dicke regime (see Supplementary mesoscale. Science 321, 1172–1176 (2008). B 2. Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. Methods). Here, the phase zero-point fluctuation is fzp 0.2, Mod. Phys. 86, 1391–1452 (2014). which agrees with the scale we observed. 3. 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