SOM_v4.doc. Parametric amplification and noise squeezing by a qubit-coupled nanomechanical

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1 upporti foratio Paraetric aplificatio ad oise squeezi by a qubit-coupled aoechaical resoator Juho uh, Matt LaHaye, Pierre Echterach, Keith chwab, Michael Roues Fitti Δ f vs. G i Fiure c The periodicity of the qubit eery is periodic i, thus Δf also shows a periodic respose vs, or G = qe CG. The period Δ G is 9 ad we et C G = qe ΔG = 7aF which ives us the scali betwee ad G. Now we approiate Equatio as, ± λ ± h E J 3 ΔE = ± h[4e C λ E J E J ] 3 sice ΔE >> h. We fit the data Δf = π vs. to this equatio, with EJ = E J cos Φ Φ = 3GHz cos.8π = 8.3GHz tae fro the previous spectroscopic easureet. Resulti E C h =.5GHz arees with the spectroscopy data 3GHz. Also fro the fit, we fid λ h = 3.MHz, which ives

2 9 = λ 4EC zp = 3.3. This is cosistet with our estiate based upo fiite eleet siulatios of capacitace betwee the CPB ad aoresoator, where we have C G = 5pF, which ives 9 =.5. Fitti the oliear dissipatio odel ad estiate of the coiciet We siplify the equatio 4 to y = A B C ad fit it with =echaical displaceet X before oralizatio ad y =. With the calculated fit paraeters, a 3 cubic equatio AX C X B = is solved for each ad this produces the solid curves i Fi.b. Fro the fit, we et the oliear dissipatio coiciet 9 η 8 s. A estiate based o the easured quality factor of the aoresoator ives a value areei withi a order of aitude. The PLL circuit we use i the resoace shift easureet also oitors the aitude siultaeously, ivi us iforatio about the aoresoator s quality factor with respect to the ate chare. Fiure shows the width of the aoechaical resoace = Q coverted fro the easured quality factor. t is clear that there is additioal dissipatio due to coupli to the CPB, which is aiized at the deeeracy poit. The source of this additioal dissipatio is ot clear yet, but a

3 3 possible reaso is the resoace frequecy fluctuatio due to the fiite rate of quasiparticle poisoi. A aaloous lie broadei ect of atoic trasitio fro rado teleraphtype resoace fluctuatio was aalyzed elsewhere. With the CPB biased at 63., a sall fractio of ate chare is odulated by the aoechaical otio accordi to e C N G = δ, which results i odulatio of of ad. Assui that the equatio of otio taes the for, f = up to the secod order of, we ca idetify hiher order ters such as,, ad collect the followi Lifshitz et.al. 3 The oliear dissipatio coiciet is the ive by, 3 = = η Fro the easured resoace liewidth vs. ate chare i Fi., we et 5 5 s, 4 6 s. Ad fro the resoace shift data, we have 4 s, 9 3.3, yieldi 9 s η. We ote that, fro the fluctuatio-dissipatio theore, this oliear dissipatio should have bee accopaied by displaceet-depedet ters i the aoresoator s force

4 oise correlatio 4. However, we did ot attept to easure this force oise correlatio i the curret eperiet. Noise odel The aoresoator biased at dc = at its resoace ca be represeted as a N resistor R G Q C = dc 5. The ipedace atchi circuit trasfors the ipedace as N : where N = L Z 5 = L C T T T T T. The aplifier is assued to have oise sources represeted by two ucorrelated oes 6 Fi.a, spectral desities of oise voltae ad curret,, with a oiseless iput ipedace Z i = Z = 5Ω. our set-up, R. 8MΩ for G = ad R N >> Z i. The echaical displaceet is proportioal to the curret throuh Z i, thus the oise curret throuh Z i ives the displaceet oise. The spectral desity of oise curret throuh Z i is, = R N Zi, R N Z i R N Z i R N We see the secod ter does ot deped o the echaical resoator ad it is siply additive cotributio fro the aplifier. The first ter, by cotrast, icreases whe R 4

5 decreases, i.e. the coupli to the aplifier icreases. Thus we idetify the aplifier bacactio oise as, R N N C = GNQ C, BA = dc dc ad the additive oise as, N C, ADD = dc t is evidet that oly the bac-actio oise is aplified or squeezed depedi o the paraetric ai G. Ad also, sice oly, depeds o the echaical Q, the oise BA floor uder the otioal pea of the displaceet oise spectru is,. ADD For the first step, the paraetric pup is set to zero, ad the total displaceet oise spectru, ADD, BA = is recorded for each of quadraturesx ad Y of loc-i. The two oise spectru show o differece i pea heiht ad oise floor level with each other as epected, ad we choose X quadrature data to plot i Fi.3a ad to estiate,. ADD ice the oise floor has a slope due to sliht offset of LC atchi circuit resoace ad echaical resoace, a quadratic polyoial is fitted to the oise floor as a estiate dashed lie i Fi. ad the fitted oise desity at the echaical resoace is chose as, ADD. The we tur o the paraetric pup, ad the phase of the resoator ecitatio is 5

6 swept, while at the echaical resoace is oitored., BA, ADD = ives us the bac-actio oise for each phase. Now the total displaceet oise is, =, BA, ADD GNQ dc C N C dc where we do ot iclude the theroechaical oise of the aoresoator cosideri the oise teperature of the aplifier ~3K which is uch hiher tha the saple teperature. We cofir this assuptio later i the sectio. The displaceet oise at the aoechaical resoace = ives the oise i the force easureet by, f = GQ. Thus the force oise due to the aplifier is, f = N = dc f, BA C f, ADD Q dc C t is iiized whe f BA f, ADD, = or R = R N which ives the oise atchi coditio. The iiu force oise is the, i f = G Q = 4 B T G Q N 6

7 where B is the Boltza costat ad T N is the iiu oise teperature of the aplifier. Fro the easured, ADD ad BA,, we calculate ad ad et = 36 p Hz ad =. pa Hz which are close to what easured i a separate easureet o the cryoeic aplifier, = 34 p Hz ad =. pa Hz. Also, we etract the force sesitivity 83aN Hz with o paraetric ai. Copari this with the epected theroechaical oise force, 4 Hz BT Q = 5.aN, we see the preaplifier oise is doiat ad the paraetric aplificatio ideed iproves the force sesitivity ad it also cofirs the validity of our iitial assuptio electi the theroechaical oise. 7

8 Refereces LaHaye, M. D.; uh, J.; Echterach, P. M.; chwab, K. C.; Roues, M. L. Nature 9, 459, 96. Wodiewicz, K.; hore, B. W.; Eberly, J. H. Phys. Rev. A 984, 3, Lifshitz, R.; Cross, M. C. Reviews of Noliear Dyaics ad Copleity; Wiley-CH, 8; ol.. 4 Zaitsev,.; htepluc, O.; Bus, E.; Gottlieb, O. 9, arxiv: Truitt, P. A.; Hertzber, J. B.; Hua, C. C.; Eici, K. L.; chwab, K. C. Nao Lett. 7, 7,. 6 Rothe, H.; Dahle, W. Proc. RE 956, 44,

9 s Fiure. Naoechaical resoace width vs. ate chare Fiure. Noise floor estiate 9