Pound-locking for characterization of superconducting microresonators
Tobias Lindström, Jonathan Burnett, Mark Oxborrow, Alexander Ya. Tzalenchuk
aa r X i v : . [ phy s i c s . i n s - d e t ] J u l Pound-locking for characterization of superconducting microresonators
T. Lindstr¨om,
1, 2
J. Burnett,
1, 2
M. Oxborrow, and A Ya. Tzalenchuk National Physical Laboratory, Hampton Road, Teddington, TW11 0LW, UK Royal Holloway, University of London, Egham Hill, Egham, TW20 0EX, UK (Dated: 26 October 2018)
We present a new application and implementation of the so-called Pound locking technique for the interro-gation of superconducting microresonators. We discuss how by comparing against stable frequency sourcesthis technique can be used to characterize properties of resonators that can not be accessed using traditionalmethods. Specifically, by analyzing the noise spectra and the Allan deviation we obtain valuable informationabout the nature of the noise in superconducting planar resonators. This technique also greatly improves theread-out accuracy and measurement throughput compared to conventional methods.
I. INTRODUCTION
Resonators are ubiquitous devices used in one form oranother in most areas of physics and in many importantapplications, from filters in mobile phones to experimentsin cavity Quantum Electrodynamics (QED) . Over thepast few years advances in on-chip quantum informationprocessing have meant that planar microwave resonatorshave taken on new roles: as quantum buses , for gen-erating Fock states and as read-out elements for solid-state qubits . Superconducing resonators are also usedas kinetic-inductance detectors . These applications in-volve measurements that need to be done quickly and ac-curately. It is natural to look at techniques already in usein the fields of optics and frequency metrology for inspira-tion on how to meet these new measurement challenges.A commonly used technique for reading-out resonators inprecision frequency metrology is Pound-locking , rou-tinely used in quantum optics and precision frequencymetrology but surprisingly little known outside thosefields.The intrinsic properties of a resonator are describedby its centre frequency ν and unloaded quality fac-tor Q u = ν / ∆ ν , where ∆ ν is the width of the reso-nance. This in turn is related to the attenuation rate α = πν / Q . All these parameters are routinely mea-sured using a vector network analyzer (VNA). Resonatormeasurements using a VNA are in principle accurate aslong as the resonance shape is known and fitting tech-niques can be used to obtain Q and ν , but are also quiteslow, only measures the average behaviour and do notprovide any direct information about the level of noisein the system. Another common approach is to use ho-modyne detection techniques where the carrier is mixeddown to DC (using a mixer) . Pound-locking has a fewimportant advantages over both these methods. Themain improvement comes from the fact that the mod-ulation frequency f m (see below) can be chosen to bemuch higher than the 1/f corner frequency of amplifiersetc; this is important since microwave amplifiers are verynoisy at low offset frequencies, meaning the noise in ahomodyne measurement scheme can be easily domi-nated by system noise. We typically use f m of 1-5 MHz,well above the corner frequency of our amplifiers. It is also a ”real time” technique (within the bandwidth of theloop) in that it always ”tracks” the centre frequency ofthe resonator irrespective of the magnitude of the fluctua-tion; leading to a significant speedup in the measurementthroughput. Finally, it also allows for accurate measure-ments of very small shifts in the centre frequency of aresonator which is important in many experiments.The purpose of this paper is to describe a new ap-plication and appropriate implementation of this well-established technique. We have adapted Pound-lockingto the read-out and characterization of superconductingmicroresonators. We will also describe how one can ob-tain information about the nature of the noise in samplesby using analysis techniques from precision frequencymetrology. It is our hope that this work will expandthe applications of the Pound locking technique. II. POUND-LOCKING
Figure 1(a) show the basic schematic of a Pound-loop.Here we will describe how this is implemented in the caseof microwave oscillators; for a very intuitive discussionof how Pound locking is used in optics see the reviewby Black . The main building blocks are a voltage-controlled oscillator (VCO) that generates a carrier offrequency f c , a phase modulator used to add sidebands ± f m to the signal, the resonator, a square-law power de-tector, a down-converting mixer and finally a regulatorcircuit which controls the VCO. The principle of opera-tion is as follows: assuming the carrier is near resonanceand the modulation frequency has been chosen so that f m ≫ ∆ ν the sidebands do not enter the resonator (notethat this means that no extra power is injected into theresonator; nor is the bandwidth limited by α ). The phasemodulated signal going into the sample will be of theform V in = V [ J ( β ) + 2 jJ ( β ) sin 2 πf m t ] e πjf c t (1)where J , are Bessel functions and β is the modulationdepth, it can be shown that the latter should be chosenso that the power in the sidebands is -3 dB smaller thanthe carrier. The resonator can be anything that has a”notch” type frequency response, most Pound loops are f (a)The basic Pound loop LO Phase ModulatorRF Lock‐in
Controller
Data aquisition f M R E F CryostatAnalysis ‐ d B
30 dB50 dB
Power detector Tunable AttenuatorMod. generator (b)Practical implementation of the Pound loop. Various auxiliarycomponents (filters etc.) have been left out for clarity.
FIG. 1.FIG. 2. Micrograph and equivalent circuit for one of ourlumped element resonators. The electromagnetic radiation isprimarily inductively coupled to the inductive part of the res-onator. The coupling strength is set by the mutual inductancebetween the coplanar transmission line and the resonator. implemented using a circulator (meaning the sample isactually probed in reflection) but our planar resonator(see below and ref. ) have this response in transmission,see fig. 2. The scattering parameter S for a piece oftransmission line shunted by a resonator can be written S = 2(1 + 2 jQ u y ) g + 2(1 + 2 jQ u y ) (2)Where g is a parameter determining the couplingstrength and y is the normalized centre frequency ( f − ν ) /ν . After going through the sample the signal isdetected by a square-law diode after which it is de-modulated using a mixer. The circuit therefore createsan error signal that is due to the interference betweenthe transmitted signal and the energy leaking out fromthe resonator back into the transmission line. The signalfrom the diode can be written ǫ ∝ V kJ ( β ) J ( β ) { Im[ S ( f c )](Re[ S ( f c + f m )] + Re[ S ( f c − f m )]) − Re[ S ( f c )](Im[ S ( f c + f m )] + Im[ S ( f c − f m )]) } (3)where ǫ is the error signal; and k is a constant that de-pends on the conversion efficiencies of the diode and thede-modulator. The net effect of the Pound loop is thatwhenever the carrier is off resonance, the transmitted sig-nal is phase shifted and will have an AM component, i.e.there is partial PM to AM conversion. The AM signal isthen detected and used to correct the error. Hence, ide-ally this circuit will ”lock” the frequency of the carrier f c to the resonance frequency ν . Figure 3 shows what the error signal ǫ will look like for a loop with a typicalresonator. Once the system has locked to the resonatorit will only undergo small excursions from ν and ǫ isnearly linear, using eq.2 we can simplify eq.3 ǫ lin ∝ kV gQ u ( g + 2) y (4)where we have assumed optimum modulation depth ( β =1 .
08) and coupling ( g = 2). Hence, the slope of the error −5 −4 −3 −2 −1 0 1 2 3 4 5x 10 −4 −0.5−0.4−0.3−0.2−0.100.10.20.30.40.5 y ε (−f m − ν )/ ν (f m − ν )/ ν FIG. 3. Simulated typical error signal. The parameters are f m = ν / β = 1 . signal depends on the uncoupled quality factor and thecoupling strength of the resonator. III. EXPERIMENTAL
The actual set up used in our experiment can be seenin 1(b). Our resonators are thin-film devices microfabri-cated on 5x10 mm chips (see ref for a full descrip-tion). The chip is glued to a copper cold-finger andbonded to a connectorized chip carrier which is enclosedin an aluminium box; the box is then mounted on themixing chamber of a dilution fridge which has a basetemperature of 30 mK. The in-signal to the resonatorgoes via a filtered and heavily attenuated (50 dB) line.The out-signal passes through an isolator and then intoa low-noise cryogenic amplifier with a noise temperatureof about 4 K.The microwave signal (hereon referred to as the carrier)is generated by a voltage-controlled oscillator, in practicethis is usually a microwave generator where the outputfrequency can be modulated using an external voltage(at a rate of for example 10 kHz/V). Another option isto use a local oscillator (LO) held at a fixed frequencyand mix it with an intermediate frequency (IF) from alow-frequency VCO (in practice e.g. a function genera-tor that can be frequency modulated) using for examplea single sideband modulator (SSB). The performance ofthe latter setup is in principle better since a low phase-noise LO can be used (the phase noise of the IF source isnegligible) but makes the circuit more complicated (andcare has to be taken to minimize the amplitude of theside-bands). Another advantage of this latter arrange-ment is that the frequency shift can be read out directlywith high accuracy using a standard counter. The car-rier then passes through a phase modulator which addssidebands ± f m . The shape of the error signal dependson the phase of the modulating signal, so this needs tobe adjusted to compensate for lead/lag caused by the ca-bles, filters etc. The signal from the diode then goes toan RF lock-in amplifier (referenced to f m ) which is usedwith a very short time constant (as to not limit the band-witdh of the loop), essentially being operated as a mixerwith gain. The error signal in the loop is the in-phase signal from the lock-in amplifier, and is fed to the PIDcontroller. The output voltage from the controller thencontrols the VCO (which in our case outputs the IF) inorder to null the error; we generally use a deviation of5-100 kHz/V. The PID controller is configured so that itattempts to keep the error signal as close to zero as possi-ble while still maintaining a loop bandwidth much largerthan the frequency range of any noise we want to study.Care has to be taken to avoid dependence of the observednoise level on the loop parameters, this can happen e.g. ifthe gain is too high (resulting in oscillations that distortthe noise spectrum) or the bandwidth too low(resultingin a roll-off of the noise at higher frequencies) .Most of our measurement data is acquired using a stan-dard 16-bit data acquisition system connected to the out-put of the PID controller. Sampling rates from 10 Hz toabout 50 kHz are used, depending on the measurementtime which varies from a few seconds to tens of hours. Wealso monitor the output signal using an FFT analyzer. Inorder to get an independent measure of the frequency wemonitor the frequency of the IF source using a separatecounter. All microwave equipment, counters etc in ourexperiment use an external 10 MHz frequency referencederived from a hydrogen maser. Our current setup allowsus to track any change to the resonance frequency of theresonator with a bandwidth of a few kHz (limited by thePID controller) and a ”real-time” frequency resolution ofa few parts in 10 (limited by the intrinsic noise of theresonator).Our experiment is essentially dual to that of the con-ventional setup used to improve the stability of an oscil-lator. In a conventional experiment a ”bad” oscillator islocked to a ”good” resonator, we are here doing the ex-act opposite. The frequency noise level of our resonatoris much higher than that of the carrier; by observinghow the circuit strives to null the error signal we ob-tain direct information about the resonator. Moreover,since we are operating at offset frequencies smaller than f L = ν / Q ( ∼
75 kHz), the Leeson effect will not affectthe shape of the frequency spectrum .Our superconducting planar lumped element (LE) res-onators are about 300x300 µ m in size, fabricated fromniobium on a sapphire substrate. Several resonators(usually 5) are fabricated on the same chip and are cou-pled to the same transmission line allowing for frequencymultiplexing. The data shown here comes from a res-onator with ν of 7.5 GHz and a loaded Q=50 000, it issimilar to the ones discussed in ref . Here we only showa subset of our results, for a full discussion see .In order to verify that our data really represents prop-erties of the sample and are not measurement artifacts;we also employ a number of reference samples. Here wewill therefore also show data from a common dielectricresonator (DRO) in the shape of a ’puck’ about 7 mmin diameter mounted inside a metal box and measuredat room temperature. In this experiment we locked toa mode with a resonance frequency of 5.12 GHz and aloaded Q=5500. −2 −1 −1 f(Hz) P S D ( H z / √ H z ) −1 −0.5 FIG. 4. Root power spectral density (in units of
Hz/ √ Hz )for a 7.5 GHz lumped element resonator measured at 30mK.the solid lines have a slope of -1 and -0.5, respectivly and aremeant as guides for the eye. IV. DATA ANALYSIS
The most straightforward way to measure noise is touse a spectrum analyzer or -equivalently- to calculate thepower spectral density (PSD) from sampled time-domaindata. Figure 4 shows a typical PSD from one of our LEresonators. Note that what is plotted is the frequencynoise (in units of
Hz/ √ Hz ) meaning the PSD representspectral information of how the centre frequency of theresonator changes. The spectrum shows that randomwalk frequency noise (slope -1) dominates at low fre-quencies but that processes with a white (slope 0) spec-tra gradually takes over as the frequency increases. Thesolid line with slope -0.5 can be identified with flickerfrequency noise.One limitation of PSD is that it is often very difficultto analyze ”slow” processes. Another little known, buthighly useful tool from the arsenal of frequency metrologyis the Allan deviation (ADEV) . Here we have usedthe overlapping Allan deviation σ : σ = 12 m ( M − m + 1) M − m +1 X j =1 j + m − X i = j ( y i + m − y i ) (5)where M is the number of samples, τ is the measurementtimebase and y is the time-averaged fractional frequency.ADEV is the most common measure of time-domain sta-bility, but other related types of measures such as themodified Allan deviation (MDEV), Hadamard deviationetc are also very useful for analyzing and identifying noise(e.g. the MDEV in particular can distinguish betweenwhite and flicker phase noise). Using the Allan deviationit is relatively easy to identify noise processes that hap-pen over a long timescale. Just as in the case of the PSDwe categorize the noise depending on the slope of thecurve when plotted in a log-log plot. Table I shows asummary. Note that we have included frequency drift in −4 −3 −2 −1 −1 τ (s) σ f ( τ ) α =−0.5 α =1 FIG. 5. (colour online) Allan deviation plot for the dielectricresonator, with(diamonds) and without(points) temperaturestabilisation. table I; although stricly speaking not a noise, drift dueto e.g. temperature fluctuations are often seen in realmeasurements.In order to demonstrate how ADEV can be used toanalyze the intrinsic noise of resonators we first show theexperimental data from the DRO in figure 5. White fre-quency noise (slope -0.5) dominates as expected for shorttimes until linear drift (slope +1) takes over and the devi-ation increases rapidly. This large drift in the resonancefrequency (about 10-20 Hz/second)is due to a changingtemperature and agrees with the specified temperaturestability of a few ppm/K. By employing some rudimen-tary temperature stabilization we can shift the onset ofthe drift further to the right on the plot. According tothis plot the best one can hope for is a short term stability(and read-out accuracy δf ) of about 0.2 Hz; which agreeswith the white noise level measured using an FFT ana-lyzer as well as the direct read-out using a counter. Thiscorresponds to a line splitting factor ∆ ν/δf of 5 · .Figure 6 shows the ADEV -calculated using the samedataset as was used for the PSD- for the lumped elementresonator measured at 30 mK. It is quite clear that thebehaviour is very different from that of the dielectric res-onator. Flicker frequency noise (slope 0) dominates atboth long and short timescales, but the dominant fea-ture is the sudden rise at timescales between 1 ms to TABLE I. Noise processes and the resulting slope in the rootPSD and Allen deviation(ADEV) plotsNoise type root PSD slope ADEV slopeWhite phase 1 -1Flicker Phase 0.5 -1White frequency 0 -0.5Flicker frequency -0.5 0Random walk -1 0.5Linear drift - 1 −4 −3 −2 −1 τ (s) σ f ( τ ) α =0.6 FIG. 6. Allan deviation plot for a 7.5 GHz lumped elementresonator measured at 30mK. The power in the resonator isapproximately -100 dBm. ; the random walk noise mightbe due to random changes in the dielectric environment(due to hopping) or the movement of trapped vortices inthe superconducting film. V. CONCLUSION
By bringing well-known measurement techniques andanalysis methods from time and frequency metrology tobear on the problem of the nature of noise in supercon-ducting resonators one can gain a lot of information notreadily obtainable using more widely known techniques.By implementing a Pound loop and analyzing the datausing the Allan deviation we are able to identify noiseprocesses that are hard to distinguish by conventionalmeans. Moreover, the fact that the loop directly mea-sures the resonance frequency makes the analysis morestraigtforward. Finally, we would also like to mentionthat the ability to ”track” the value of ν can also leadto significant improvement in measurement throughput.A good example is measurements of ν vs. temperaturewhich can e.g. be used to determine the intrinsic loss tan- gent due to two-level fluctuators in the sample. This canbe a time-consuming process if one needs to ”step” thetemperature and wait for it to stabilize before acquiringthe scattering parameters (S21 and/or S11) using a VNA.However, by using a Pound-loop one is only limited bythe thermal time-constant between the sample and thetemperature sensor, allowing for the temperature to beramped relatively quickly. ACKNOWLEDGMENTS
The authors would like to thank Phil Meeson, GregoireIthier, John Gallop, Ling Hao and Enrico Rubiola forstimulating discussions, advice and help. This work wassupported by EPSRC and the Pathfinder program of theNational Measurement Office. C. C. Gerry and P. L. Knight,
Introductory Quantum Optics (Cambridge University Press, 2005). M. A. Sillanpaa, J. I. Park, and R. W. Simmonds, “Coherentquantum state storage and transfer between two phase qubits viaa resonant cavity,” Nature , 438–442 (2007). M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak,E. Lucero, M. Neeley, A. D. O/’Connell, H. Wang, J. M. Martinis,and A. N. Cleland, “Generation of fock states in a superconduct-ing quantum circuit,” Nature , 310–314 (2008). A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R. Huang, J. Ma-jer, S. Kumar, S. Girvin, and R. Schoelkopf, “Strong coupling ofa single photon to a superconducting qubit using circuit quantumelectrodynamics,” Nature , 162–167 (2004). J. Gao, J. Zmuidzinas, B. A. Mazin, H. G. LeDuc, and P. K. Day,“Noise properties of superconducting coplanar waveguide mi-crowave resonators,” Applied Physics Letters , 102507 (2007). R. Pound, “Electronic frequency stabilization of microwave os-cillators,” Review of Scientific Instruments , 490 (1946). R. Drever, “Laser phase and frequency stabilization using an op-tical resonator,” Applied Physics B: Photophys. Laser Chem. ,97–105 (1983). Pound-locking is sometimes also referred to as Pound-Drever-Halllocking. E. Rubiola,
Phase Noise and Frequency Stability in Oscillators (Cambridge University Press, 2009). R. Barends, H. L. Hortensius, T. Zijlstra, J. J. A. Baselmans,S. J. C. Yates, J. R. Gao, and T. M. Klapwijk, “Noise in nbtin,al, and ta superconducting resonators on silicon and sapphiresubstrates,” , 936–939 (2009). E. D. Black, “An introduction to pound–drever–hall laser frequency stabilization,”American Journal of Physics , 79–87 (2001). T. Lindstr¨om, J. E. Healey, M. S. Colclough, C. M.Muirhead, and A. Y. Tzalenchuk, “Properties of super-conducting planar resonators at millikelvin temperatures,”Phys. Rev. B , 132501 (2009). J. Anstie,
A 50K Dual-Mode sapphire oscillator and whisperingspherical mode oscillators , Ph.D. thesis, University of WesternAustralia (2006). S. Chang,
Ultrastable cryogenic microwave sapphire resonatoroscillator , Ph.D. thesis, University Wester Australia (2000). E. Rubiola, Private communications. T. Lindstr¨om, J. Burnett, M. Oxborrow, and A. Y. Tzalenchuk,In preparation. W. Riley, “Handbook of frequency stability analysis,” Tech. Rep.1065 (2008). T. Witt and D. Reymann, “Using power spectra and allan vari-ances to characterise the noise of zener-diode voltage standards,”IEEE Proc. Sci. Meas. Technol.147