Characterization of a microwave frequency resonator via a nearby quantum dot
T. Frey, P. J. Leek, M. Beck, K. Ensslin, A. Wallraff, T. Ihn
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Characterization of a microwave frequency resonator via a nearby quantumdot
T. Frey, a) P. J. Leek, M. Beck, K. Ensslin, A. Wallraff, and T. Ihn Solid State Physics Laboratory, ETH Z¨urich, 8093 Z¨urich, Switzerland Institute for Quantum Electronics, ETH Z¨urich, 8093 Z¨urich, Switzerland (Dated: 22 November 2018)
We present measurements of a hybrid system consisting of a microwave transmission-line resonator and alateral quantum dot defined on a GaAs heterostructure. The two subsystems are separately characterizedand their interaction is studied by monitoring the electrical conductance through the quantum dot. Thepresence of a strong microwave field in the resonator is found to reduce the resonant conductance throughthe quantum dot, and is attributed to electron heating and modulation of the dot potential. We use thisinteraction to demonstrate a measurement of the resonator transmission spectrum using the quantum dot.The interaction of light and matter is one of the mostfundamental processes in physics. One way to explorethis area is to use artificial atoms such as quantum dotswhich offer e.g. the possibility to tune the energy spacingof the individual electronic states. Using this possibil-ity the resonant absorption of photons by electrons ina quantum dot has been investigated in transport mea-surements of photon assisted tunneling . Cavity quan-tum electrodynamics (QED), the study of the couplingof matter to light confined in a cavity , is traditionallystudied with atoms but also with solid state systems suchas self-assembled quantum dots . Furthermore the re-alization of circuit QED , in which a single microwavephoton is trapped in an on-chip cavity and coherentlycoupled to a quantum two-level system, has led to sig-nificant progress in control and coupling of microwavephotons and superconducting qubits. The study of theinteraction between the electromagnetic field of such aresonator and a semiconductor quantum dot marks animportant step toward realizing a hybrid quantum infor-mation processor , in which the advantages of differentsystems, such as a long relaxation time of the individualqubit and interaction between distant qubits , could beexploited in one device.The sample, shown in Fig. 1 (a), consists of a laterallydefined quantum dot positioned at an antinode of theelectric field of a microwave transmission-line resonator.The dot is realized on an Al x Ga − x As heterostructurewith a two-dimensional electron gas (2DEG) residing atthe heterointerface about 35 nm below the surface. Thedevice is fabricated by three stages of optical lithogra-phy followed by local anodic oxidation (LAO) with anatomic force microscope (AFM) to define the quantumdot. In the first of the three lithography steps the mesafor the quantum dot (dark gray parts, labeled M in Fig. 1(a)) is wet etched. Ohmic contacts (labeled C in Fig. 1(a)) are then used to contact the 2DEG. Finally, the mi-crowave resonator and its ground plane (labeled R andGND in Fig. 1 (a)) are defined in a lift off process by a) Electronic mail: [email protected]
FIG. 1. (Color online): (a) Optical micrograph of a microwaveresonator (R) with an integrated quantum dot, (GND):ground plane of the resonator, (C): ohmic contact, (M): 2DEGmesa. (b) Magnified view of a coupling capacitor, location onthe chip marked with rectangles in (a). (c) Enlarged view ofthe 2DEG mesa, location on the chip marked with a circlein (a). Edge of the mesa highlighted with a dashed line. (d)AFM picture of the measured quantum dot, realized on the2DEG mesa. depositing a bilayer of 3 nm Ti and 200 nm Al. The min-imum distance from the mesa edge to the resonator cen-ter conductor is around 2 µ m. The coplanar waveguideresonator is designed to have a fundamental frequency f ≈ of Q ext ≈ FIG. 2. (Color online) (a) Charge stability diagram of thequantum dot without applied microwave power. White ar-rows indicate excited states. (b) Conductance through thequantum dot as a function of the PG voltage, for differentmicrowave powers applied to the resonator in the vicinity ofpoint A in (a). (c) Conductance change ∆ G of the quantumdot as a function of the microwave frequency applied to theresonator. mission of the resonator in the microwave regime.Coulomb blockade diamonds of the quantum dot withno microwave power applied, measured via lock-in tech-niques, are shown in Fig. 2 (a). The charging energy ofthe quantum dot is found to be ∆ E C ≈ E C = e /C Σ = e / ǫ ǫ GaAs d to be about115 nm. Excited state resonances are observable, indi-cated by white arrows. The typical single-particle levelspacing that was resolved is about 350 µ eV. Using theconstant density of states of the 2D system , the single-particle level spacing can be used to estimate the di-ameter of the quantum dot to be about 110 nm. Thevalue is in good agreement with the dot size found us-ing the disc capacitor model. The electron temperatureextracted by fitting a thermally broadened Coulomb res-onance is T e .
200 mK. Using the maximum of theCoulomb resonance at position A in Fig. 2 (a), the totaltunnel coupling of the quantum dot state to the leadsis estimated to be smaller than the thermal energy ofthe electrons. The measured Coulomb resonances do nothave simple thermally broadened line shapes, indicatingthat the quantum dot is not deep in the single-level trans-port regime.Before studying the interaction between the quantumdot and the microwave field, the transmission of the res-onator is characterized using a standard network ana-lyzer. A loaded quality factor of Q L ≈ f ≈ .
878 GHz.The highest quality factor, that we have reached on bare GaAs with an under-coupled resonator , is approx-imately 10 .In a next set of experiments the effect on the conduc-tance of the quantum dot circuit of driving the microwaveresonator is investigated. The quantum dot is swept atzero bias through the Coulomb resonance (position Ain Fig. 2 (a)) by changing the voltage on the PG andthe dot conductance is recorded via lock-in techniques.The measurement is repeated with the addition of a mi-crowave tone applied to the resonator at f for a rangeof different microwave powers (Fig. 2 (b)). The appliedpowers are specified at the output of the microwave gen-erator, and the total damping of the microwave from thegenerator to the sample is estimated to be about −
30 dB.A reduction in the conductance of the quantum dot anda broadening of the Coulomb resonance with increasingmicrowave power are observed. We have performed thesame measurement on another Coulomb resonance andfound similar behavior.We now investigate the conductance of the quantumdot at the Coulomb resonance (position A, Fig. 2 (a)),while sweeping the frequency of the signal applied tothe resonator. In Fig. 2 (c) the change of the conduc-tance ∆ G at the Coulomb resonance is plotted as a func-tion of the applied microwave frequency f at a power of −
27 dBm. The change ∆ G is measured relative to thevalue obtained when the applied microwave frequencyis detuned by several GHz from an eigenfrequency ofthe resonator. Sharp minima in the conductance sig-nal of the quantum dot are observed at frequencies of6 .
878 GHz, 13 .
773 GHz, 20 .
658 GHz, 27 .
517 GHz, la-beled with 1 to 4 in Fig. 2 (c). They correspond to thefundamental frequency ( f ) and the first three harmon-ics ( f , f , f ) of the resonator within an relative error of E n = ( f n − n · f ) /f n ≈ . µ V in the vicinity of f . The frequency applied to the feed lines of the res-onator is swept and the conductance through the dot ismeasured with the quantum dot tuned to position A. Torelate the influence of the strength of the electromagneticfield to the quantum dot signal, the measurement was re-peated for microwave powers ranging from −
55 dBm to −
20 dBm. In Fig. 3 (a) the change in conductance ∆ G ofthe quantum dot is plotted versus the frequency appliedto the resonator. The minimum of the conductance isfound at the resonance frequency of the resonator and thechange of the conductance ∆ G increases with microwavepower. In Fig. 3 (b) the minima of ∆ G , extracted fromdatasets as shown in Fig. 3 (a), are plotted as a functionof the applied microwave power to establish a relation FIG. 3. (Color online) (a) Conductance measurement atCoulomb resonance A versus frequency in the vicinity of thefundamental mode, for indicated microwave drive powers. (b)Maximum conductance change ∆ G versus microwave power.(c) Resonator spectrum extracted from the dot conductancesignal (blue dots) fitted with a Lorentzian line shape (greenline), see text. The dark circles are from a network analyzermeasurement and are fitted with a Lorentzian line shape aswell (red curve). The two curves are offset for clarity. between the two quantities. The fact that the curve inFig. 3 (b) levels off for small microwave power is ascribedto the finite sensitivity of the quantum dot. Note thatthe smallest input power that is clearly detectable as achange in conductance is P min = −
50 dBm, correspond-ing to a resonator population of n ≈ · photons.Hence a stronger coupling between the two systems wouldbe required to realize a more sensitive detection poten-tially at the single-photon level. The conductance scalefor the dataset with P = −
20 dBm in Fig. 3 (a) is nowconverted into a power scale using Fig. 3 (b) and thennormalized. A Lorentzian line shape is obtained, bluepoints in Fig. 3 (c). The coupling between the dot andthe resonator is assumed to be the same over the narrowfrequency range covered in the dataset. Also shown withopen circles in Fig. 3 (c) is the S signal measured witha network analyzer, offset from the other curve by − Q L . The ob-tained values are Q L = 2896 ±
22 for the network analyzermeasurement, and Q L = 2890 ±
30 for the dot conduc-tance based measurement, in very good agreement witheach other, supporting the assumption of the constantcoupling between resonator and dot.We now discuss the potential coupling scenarios be-tween the quantum dot and the microwave resonator.Due to the large extension of the resonator in comparisonto the quantum dot there is not only a coupling of themicrowave to the quantum dot itself but also to the sur-rounding 2DEG areas. The direct coupling capacitancebetween the resonator and the dot in the present geome-try is of the order of 1 aF, much smaller than the coupling capacitances between the resonator and the leads, whichare in the range of 0 . − which can arise due toheating of the 2DEG by the microwave. In addition theline shapes of the conductance resonances for the highermicrowave powers ( −
24 dBm and −
21 dBm, Fig. 2 (c))indicate that the peak form is a superposition of two res-onances. This line shape is consistent with a modulationof the voltage on one or more of the gates and leads ofthe quantum dot by the microwave. The coupling of thecavity to the quantum dot may therefore be explained asa combination of heating and gate modulation.In conclusion we have fabricated a quantum dot anda microwave resonator integrated on one chip. Couplingbetween the two systems could be observed and harmonicmodes of the cavity over a frequency range of around30 GHz could be detected with the quantum dot. Usingthe quantum dot conductance signal, the quality factorof the fundamental mode of the resonator was extracted.We thank P. Studerus, Th. M¨uller, B. K¨ung andC. R¨ossler for technical support. The research wasfunded by the EU IP SOLID and ETH Zurich. T. H. Oosterkamp, T. Fujisawa, W.G. van der Wiel, K. Ishibashi,R.V. Hijman, S. Tarucha, and L. P. Kouwenhoven, Nature ,873 (1998). T. H. Oosterkamp, L. P. Kouwenhoven, A. E. A. Koolen, N. C.van der Vaart, and C. J. P. M. Harmans, Phys. Rev. Lett. S. Haroche, J.-M. Raimond,
Exploring the Quantum: Atoms,Cavities, and Photons (OUP Oxford, 2006). J. P. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn, S.Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke,A. Forchel, Nature , 197 (2004). T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs,G. Rupper, C. Ell, O. B. Shchekin, D. G. Deppe, Nature ,200 (2004). A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. -S. Huang,J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature
162 (2004). M. Trif, V. N. Golovach, D. Loss, Phys. Rev. B S. Amasha, K. MacLean, I. P. Radu, D. M. Zumbuhl, M. A.Kastner, M. P. Hanson, A. C. Gossard, Phys. Rev. Lett. J. Majer, J.M. Chow, J.M. Gambetta, J. Koch, B. R. Johnson, J.A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff,A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf,Nature
443 (2007). see e.g., S. Gustavsson, R. Leturcq, T. Ihn, K. Ensslin, and A.C. Gossard, J. Appl. Phys. , 122401 (2009). M. Goeppl, A. Fragner, M. Baur, R. Bianchetti, S. Filipp, J. M.Fink, P. J. Leek, G. Puebla, L. Steffen, A. Wallraff Journal ofApplied Physics , 113904 (2008).12