Coupling a quantum dot, fermionic leads and a microwave cavity on-chip
M.R. Delbecq, V. Schmitt, F.D. Parmentier, N. Roch, J.J. Viennot, G. Fève, B. Huard, C. Mora, A. Cottet, T. Kontos
CCoupling a quantum dot, fermionic leads and a microwave cavity on-chip
M.R. Delbecq, V. Schmitt, F.D. Parmentier, N. Roch, J.J.Viennot, G. F`eve, B. Huard, C. Mora, A. Cottet and T. Kontos ∗ Laboratoire Pierre Aigrain, Ecole Normale Sup´erieure, CNRS UMR 8551,Laboratoire associ´e aux universit´es Pierre et Marie Curie et Denis Diderot,24, rue Lhomond, 75231 Paris Cedex 05, France (Dated: August 23, 2011)We demonstrate a hybrid architecture consisting of a quantum dot circuit coupled to a singlemode of the electromagnetic field. We use single wall carbon nanotube based circuits inserted insuperconducting microwave cavities. By probing the nanotube-dot using a dispersive read-out inthe Coulomb blockade and the Kondo regime, we determine an electron-photon coupling strengthwhich should enable circuit QED experiments with more complex quantum dot circuits.
PACS numbers: 73.23.-b,73.63.Fg
An atom coupled to a harmonic oscillator is one of themost illuminating paradigms for quantum measurementsand amplification[1]. Recently, the joint development ofartificial two-level systems and high finesse microwaveresonators in superconducting circuits has brought therealization of this model on-chip[2, 3]. This ”circuitQuantum Electro-Dynamics” architecture allows, at leastin principle, to combine circuits with an arbitrary com-plexity. In this context, quantum dots can also be usedas artificial atoms[4, 5]. Importantly, these systems oftenexhibit many-body features if coupled strongly to Fermiseas, as epitomized by the Kondo effect. Combining suchquantum dots with microwave cavities would thereforeenable the study of a new type of coupled fermionic-photonic systems.Cavity quantum electrodynamics[6] and its electroniccounterpart circuit quantum electrodynamics[1] addressthe interaction of light and matter in their most simpleform i.e. down to a single photon and a single atom(real or artificial). In the field of strongly correlated elec-tronic systems, the Anderson model follows the same pu-rified spirit[7]. It describes a single electronic level withonsite Coulomb repulsion coupled to a Fermi sea. Inspite of its apparent simplicity, this model allows to cap-ture non-trivial many body features of electronic trans-port in nanoscale circuits. It contains a wide spectrumof physical phenomena ranging from resonant tunnellingand Coulomb blockade to the Kondo effect. Thanksto progress in nanofabrication techniques, the Andersonmodel has been emulated in quantum dots made out oftwo dimensional electron gas[8], C60 molecules[9] or car-bone nanotubes[10]. Here, we mix the two above situa-tions. We couple a quantum dot in the Coulomb blockadeor in the Kondo regime to a single mode of the electro-magnetic field and take a step further towards circuitQED experiments with quantum dots. ∗ To whom correspondence should be addressed: [email protected]
FIG. 1: a. Schematics of the quantum dot embedded in themicrowave cavity. The transmitted microwave field has dif-ferent amplitude and phase from the input field as a result ofits interaction with the quantum dot inside the cavity. Thequantum dot is connected to ”wires” and capacitively coupledto a gate electrode in the conventional 3-terminal transportgeometry. b. Scanning electron microscope (SEM) picturein false colors of the coplanar waveguide resonator. Both thetypical coupling capacitance geometry of one port of the res-onator and the 3-terminals geometry are visible. c. Falsecolours SEM picture of a SWNT dot inside an on-chip cavityembedded in a schematics of the measurement setup.
Low frequency charge transport in quantum dots inthe Coulomb blockade or Kondo regime has been studiedwith exquisite details[10, 11]. However, their dynamicaspects have remained to a great extent unexplored sofar. Previous studies have tackled the problem in termsof photo-assisted electron tunnelling[12, 13]. Here, wefocus on the dispersive effect of the quantum dot on themicrowave field. In order to enhance the electron-photoninteraction which would be otherwise too small to be de- a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug tected, we place our quantum dot circuit inside an on-chip microwave cavity as depicted in figure 1a. One im-portant aspect of our approach is the implementation of”wires” which go inside the cavity (see figure 1). A source(S) and a drain (D) electrode are used to drive a DC cur-rent through the quantum dot. A gate electrode (G) isused to control in situ the position of the energy levels onthe dot. At the same time, a microwave continuous signalin the 4-8GHz range is sent to one port of the cavity andamplified through the other port. Both quadratures ofthe transmitted signal are measured. The temperature ofthe experiment is 1.5K. As shown in figure 1b and 1c, weuse single wall carbon nanotubes (SWNTs) embedded insuperconducting microwave on-chip cavities in order toimplement the model situation of figure 1a. SWNTs areideally suited to implement the kind of experiments wediscuss here. They can be contacted with normal[10],superconducting[14–16] or ferromagnetic[17, 18] materi-als to form various kinds of hybrid systems. Here, weinvestigate the most simple case i.e. a single quantumdot connected to two normal metal leads and capacitivelycoupled to a side gate electrode, as shown in figure 1c.However, our scheme can readily be generalized to morecomplex circuits like double quantum dots.The phase of the microwave signal transmitted throughthe cavity is particularly sensitive to the presence of thequantum dot circuit. Figure 2a displays the color scaleplot of the low frequency differential conductance of oneparticular device as a function of the source-drain volt-age V sd and the gate voltage V g . We observe the charac-teristic ”Coulomb diamonds” with resonant lines in the V sd − V g plane as well as the characteristic ”Kondo ridge”at zero bias from V g = − . V to V g = − . V , signallingthe emergence of the Kondo effect. As shown in figure 2cin black line, the conductance for V g = − . V peaks upto 0 . × e /h , which is close to its maximum possiblevalue 2 e /h . The corresponding variations of the phaseof the microwave signal in the vicinity of the cavity res-onance, at 4.976 GHz, are displayed as a function of V sd and V g in the color scale plot of figure 2b. Essentially allthe spectroscopic features observed in the conductanceare visible in the phase spectroscopy. In particular, asimilar peak at zero bias as in the DC conductance is ob-served as shown in red line in figure 2c. It corresponds toa variation of about 2 . − rad which is not proportionalto the DC conductance in general as shown in figure 2c.The observation of the Kondo resonance in the phase ofthe microwave signal shows that the fermionic and pho-tonic systems are coupled. Our Kondo dot-cavity sys-tem has to be described by an extension of the Ander-son model, known as the Anderson-Holstein model whichhas been devised to treat quantum impurities coupledto phonons. Our ”photonic” Anderson-Holstein hamilto-nian reads : H = H dot + H cav + ( λ K ˆ N K + λ K (cid:48) ˆ N K (cid:48) )(ˆ a +ˆ a † ) with λ K ( K (cid:48) ) and ˆ N K ( K (cid:48) ) respectively, the electron-photon coupling constant and the number of electrons for FIG. 2: a. Color scale plot of the differential conductancein units of 2 e /h measured along three charge states exhibit-ing the conventional transport spectroscopy. A Kondo ridgeis visible at zero bias around V g = − . V . b. Color scaleplot of the phase of the microwave signal at f = 4.976 GHz,measured simultaneously with the conductance of figure 2a. c. Differential conductance and phase of the transmitted mi-crowave signal at f = 4.976 GHz as a function of source drainbias V sd for V g = − . V . each K(K’) orbital of the nanotube-dot (which arise fromthe band structure of nanotubes), ˆ a being the photonfield operator. The coupling constants λ K ( K (cid:48) ) arise fromthe capacitive coupling of the nanotube energy levels tothe central conductor of the cavity. The terms H dot and H cav are the standard Anderson hamiltonian of a singleenergy level coupled to fermionic reservoirs and the stan-dard hamiltonian of a single photon mode coupled to aphotonic bath. As shown in figure 3a, the capacitive cou-pling between the cavity and the dot induces oscillationsof the electronic level. There is also an indirect couplingthrough oscillations of the bias between source and drain,as indicated by the dashed lined edges of the Fermi seas infigure 3a. The resonator allows to probe both the disper-sive and the dissipative response of the dot. Therefore,both the frequency and the width in energy of the bosonicmode are affected by the mutual interaction between theelectronic and photonic systems. Since our cavity con-tains a large number of photons (about 10000 at − dBm of input power), it is justified to use classical electrody-namics to describe the coupled systems. The circuit el-ement corresponding to the quantum dot has a complexadmittance Y dot ( ω ), following the spirit of the scatteringtheory of AC transport in mesoscopic circuits[19, 20]. Toleading order with respect to the energy scales of thedot, one gets Y dot ( ω ) ≈ α/R dot + jC dot ω . The dissipa-tive part is proportional to the differential conductance1 /R dot of the dot and stems from the residual asymmetricAC coupling of the leads S and D to the cavity. The re-active part C dot corresponds to a capacitance. As shownin figure 3a, we model the resonator as a discrete LC-circuit with a damping resistor R (in red) coupled viacoupling capacitors (in green) to external leads. Thecorresponding frequency broadening and frequency shiftread δf D ≈ α/ (2 C res R dot ) and δf R ≈ − C dot f / (2 C res )respectively, where f is the resonance frequency and C res is the capacitance of the resonator. Figure 3b showshow to directly measure δf D and δf R . The top paneldisplays the expected variations of the phase close to asingle resonance when a finite δf D or δf R are included(in red and blue respectively). The reference curve (for δf D = δf R = 0) is in black dashed lines. The lowerpanel shows that, subtracting the reference curve, a finite δf D affects the odd part of the phase contrast curve (inred) whereas δf R affects its even resonant part (in blue).From these curves, δf R and δf D can be directly mea-sured from the area of the blue curve and the area of halfof the red curve, respectively. The corresponding exper-imental curves are shown in figure 3c for V g = − . V (on the Kondo ridge), taking the point V g = − . V and Vsd=0mV as a reference. We observe a resonanceat 4 . GHz with a quality factor of about 160 for theeven part in blue. The oscillations of the odd part in redcorrespond to residual imperfections of our amplificationline. We measure directly δf R and δf D by integrating thewhole blue curve and half of the red curve (the positivepart) FIG. 3: a. Capacitive coupling of the quantum dot to thecavity. Both the fermionic leads and the quantum dot arecoupled to the resonator, resulting in an AC modulation ofboth V sd and V g (shadings). b. Upper panel, the signatureof dispersion and dissipation to frequency dependence of themicrowave signal phase for a standard resonance. Referenceresonance (black dotted), shifted by δf R (blue) as a result ofdispersion and broadened by δf D (red) as a result of dissi-pation. Lower panel : the even part (in blue) and odd part(in red) as a function of frequency. The area under the bluecurve is proportional to δf R and the area of half the red oneis proportional to δf D . c. Even and odd parts of the phasecontrast δφ as a function of frequency on the coulomb peakat V g = − . V on the spectroscopy of figure 2. The evenpart exhibits a resonance centered on f = 4 . GHz . Theodd part shows residual modulation due to imperfection inthe measurement lines.
We now focus on C dot . This quantity is a direct mea-surement of the charge susceptibility of the electronicsystem. For a single particle resonance with width Γ ,the scattering theory[19, 20] predicts C dot = 2 e /π Γ at resonance, which amounts to re-expressing the spectraldensity of the single energy level coupled to the fermionicleads in terms of a quantum capacitance. If electroncorrelations are present, the situation changes. In theCoulomb blockade regime as well as in the Kondo regime,on expects a reduction of the capacitance on a peak withrespect to that of a single particle resonance with thesame width [24, 25].
FIG. 4: a. Color scale plot of the even part of the phasecontrast δφ of two Coulomb peaks as a function of the gatevoltage V g and the frequency of the microwave signal in thevicinity of the cavity resonance. δφ is taken with respect to areference phase in the empty orbital at V g = 2 . V . b. Gatedependence of the reactive (blue dots) and the dissipative (reddots) parts of the dot response extracted respectively from thearea under the even part (figure 4a) and the area under halfof the corresponding odd part. Formulae of the main text for δf R (blue line) and δf D (orange line)give C = 18 aF , α =0 . c. Color scale plot of theeven part of the phase contrast δφ of the Kondo spectroscopyshown in figure 2 as a function of the gate voltage V g andthe frequency f. The line cut corresponds to the curves offigure 3c. d. Gate dependence of the reactive (blue dots) andthe dissipative (red dots) parts of the dot response extractedrespectively from the area under the even part (figure 4c) andthe area under half of the corresponding odd part. Formulaeof the main text for δf R (blue line) and δf D (orange line) give C = 22 aF , α = 0 . The measured even part of the phase contrast as afunction of frequency and gate voltage are presented infigure 4a and c in color scale. We investigate both theCoulomb blockade (left panels) and the Kondo regime(right panels) for the same device by tuning it in differentgate regions. The point at V g = 2 . V ( V g = − . V ) andVsd=0mV is our phase reference for the Coulomb block-ade and the Kondo regime respectively. The Coulombblockade peaks (transport spectroscopy not shown) arevisible as two elongated pink spots in the f − V g planecentered at 4.976GHz which span over 50 MHz. Themeasured δf R and δf D are shown in figure 4b in blueand red dots respectively. They modulate like Coulombblockade peaks up to 15 kHz and 5 kHz respectively. Thedispersive shift δf R can be directly translated into a ca-pacitance from f = 4 . GHz and C res = 0 . pF , whichare known from our setup. A comparison with the scaledconductance is shown in blue line using the expression C f / (2 C res ) × dI/dV × h/ e for δf R with C = 18 aF .Whereas the Coulomb peaks are well taken into account,this empirical formula fails to account for the Coulombvalley. The electron-photon coupling strength can be di-rectly evaluated from these measurements. Indeed, theexpected capacitance change for the quantum dot canbe calculated using an Equation of Motion technique(EOM) for the Green’s functions. It can also be eval-uated using the Bethe Ansatz on the Coulomb peaksat T = 0. Therefore, the measured capacitance change∆ C dot of the dot is directly related to the calculated C th one by ∆ C dot = α AC C th [25]. The couplings λ K ( K (cid:48) ) in our on-chip Anderson-Holstein Hamiltonian can becalculated from λ K ( K (cid:48) ) = eα AC V rms . In the above ex-pression, V rms corresponds to the rms voltage of a sin-gle photon in the cavity mode[2] and e to the elemen-tary charge. As shown on figure 4b, the EOM theory, ingreen dashed lines, accounts well for our measurementsand agrees well on the peaks with the Bethe ansatz re-sult [25]. From this, we extract α AC ≈ .
3, which leads to λ K ( K (cid:48) ) ≈ M Hz . The Kondo ridge of figure 2 is visibleas two merged elongated pink spots. The correspondingmeasured δf R and δf D are shown in figure 4d. Theyboth modulate up to 30 kHz and 60 kHz respectively asthe gate voltage sweeps the energy levels of the dot. Inparticular, we extract C dot of 16aF for the Kondo ridge at V g = − . V . This allows us to provide another estimatefor λ K ( K (cid:48) ) from λ K ( K (cid:48) ) ≈ eV rms (cid:112) C dot /C Kondo . We use C Kondo = 4 e /πT K ≈ aF as the upper bound of thecapacitance expected for the Kondo ridge, T K being thefull width at half maximum of the Kondo peak as mea-sured from figure 2C. Consistently with the previous es-timate, we get λ K ( K (cid:48) ) ≈ M Hz . As expected[23], δf D is well accounted for with α/ (2 R dot C res ), with α = 0 . dI/dV = 1 /R dot (see orange linein figure 4d, we present a similar curve in figure 4b for α = 0 .
003 ). Interestingly, the empirical formula shownin blue line for C = 22 aF is in better agreement with themeasured δf R in the Kondo regime than in the Coulombblockade regime. Even though this might arise from non-universal features of the Anderson Hamiltonian, the ob-servation of a finite C dot is consistent with the partic-ipation of the K and K’ orbitals which naturally leadto the high Kondo temperature observed here. Like forsingly occupied closed double quantum dots [26], a fi-nite capacitance resembling the conductance is expectedif λ K (cid:54) = λ K (cid:48) due to the finite orbital susceptibility of thedot in the Kondo regime[27].In conclusion, our method can be generalized to manyother types of hybrid quantum dot circuits[28–30]. Themeasured coupling is similar to the ones demonstrated recently in superconducting circuits and can readily beused to probe the quantum regime for the microwavecavities. Generally, our findings pave the way to circuitquantum electrodynamics with complex open quantumcircuits. They could be used for example to ”simulate”on-chip other aspects of the Anderson-Holstein hamilto-nian like polaronic effects.We are indebted with M. Devoret, A. Clerk, R. Lopez,R. Aguado, P. Simon, A. Levy Yeyati and G. Zarand fordiscussions. We gratefully acknowledge the Meso groupof LPS Orsay for illuminating discussions on resonantdetection techniques and P. Bertet and L. Dumoulin forhelping us to fabricate the superconducting resonators.The devices have been made within the consortium SalleBlanche Paris Centre. This work is supported by theANR contracts DOCFLUC, HYFONT and SPINLOC. [1] A.A Clerk, M.H. Devoret, S.M. Girvin, et al., Rev. Mod.Phys. 61, 2472 (2010).[2] A. Wallraff, D.I. Schuster, A. Blais, et al, Nature 431,162 (2004).[3] I. Chiorescu, P. Bertet, K. Semba, et al, Nature 431, 159(2004).[4] T. Yoshie, A. Scherer, J. Hendrickson, et al, Nature 432,200 (2004).[5] , J.P. Reithmaier, G. S¸ek, A. L¨offler, et al, Nature 432,197 (2004).[6] J.-M. Raimond, M. Brune, and S. Haroche, Rev. Mod.Phys. 73, 565 (2001).[7] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,Rev. Mod. Phys. 68, 13 (1996).[8] D. Goldhaber-Gordon, D.H. Shtrikman, D. Mahalu etal., Nature 391, 156 (1998).[9] N. Roch, S. Florens, V. Bouchiat et al. , Nature 453, 633(2008).[10] D. Cobden and J. Nyg˚ard, Phys Rev. Lett. 89, 046803(2002).[11] , M. Grobis, I. G. Rau, R. M. Potok, et al. , Phys Rev.Lett. 100, 246601 (2008).[12] J.M. Elzerman, S. De Franceschi, D. Goldhaber-Gordon,et al., J. Low Temp.Phys. 118, 375 (2000).[13] A. Kogan, S. Amasha, and M.A. Kastner, Science 304,1293 (2004).[14] M.R. Buitelaar , T. Nussbaumer and C. Sch¨oneneberger,Phys. Rev. Lett. 89, 256801 (2002).[15] J.P. Cleuziou, W. Wernsdorfer, V. Bouchiat et al., NatureNanotech. 1, 53 (2006).[16] L.G. Herrmann, F. Portier, A. Levy Yeyati et al., Phys.Rev. Lett. 104, 026801 (2010).[17] A. Cottet, T. Kontos T., S. Sahoo et al., Semicond. Sci.and Technol. 21, S78, (2006).[18] C. Feuillet-Palma, T. Delattre, P. Morfin et al., Phys.Rev. B 81, 115414 (2010).[19] M. B¨uttiker, A. Prˆetre and H. Thomas, Phys. Lett. A180, 364 (1993).[20] A. Prˆetre, H. Thomas and M. B¨uttiker, Phys. Rev. B 54,8130 (1996).[21] P. Jarillo-Herrero, J. Kong, H.S.J. van der Zant, et al.,[1] A.A Clerk, M.H. Devoret, S.M. Girvin, et al., Rev. Mod.Phys. 61, 2472 (2010).[2] A. Wallraff, D.I. Schuster, A. Blais, et al, Nature 431,162 (2004).[3] I. Chiorescu, P. Bertet, K. Semba, et al, Nature 431, 159(2004).[4] T. Yoshie, A. Scherer, J. Hendrickson, et al, Nature 432,200 (2004).[5] , J.P. Reithmaier, G. S¸ek, A. L¨offler, et al, Nature 432,197 (2004).[6] J.-M. Raimond, M. Brune, and S. Haroche, Rev. Mod.Phys. 73, 565 (2001).[7] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,Rev. Mod. Phys. 68, 13 (1996).[8] D. Goldhaber-Gordon, D.H. Shtrikman, D. Mahalu etal., Nature 391, 156 (1998).[9] N. Roch, S. Florens, V. Bouchiat et al. , Nature 453, 633(2008).[10] D. Cobden and J. Nyg˚ard, Phys Rev. Lett. 89, 046803(2002).[11] , M. Grobis, I. G. Rau, R. M. Potok, et al. , Phys Rev.Lett. 100, 246601 (2008).[12] J.M. Elzerman, S. De Franceschi, D. Goldhaber-Gordon,et al., J. Low Temp.Phys. 118, 375 (2000).[13] A. Kogan, S. Amasha, and M.A. Kastner, Science 304,1293 (2004).[14] M.R. Buitelaar , T. Nussbaumer and C. Sch¨oneneberger,Phys. Rev. Lett. 89, 256801 (2002).[15] J.P. Cleuziou, W. Wernsdorfer, V. Bouchiat et al., NatureNanotech. 1, 53 (2006).[16] L.G. Herrmann, F. Portier, A. Levy Yeyati et al., Phys.Rev. Lett. 104, 026801 (2010).[17] A. Cottet, T. Kontos T., S. Sahoo et al., Semicond. Sci.and Technol. 21, S78, (2006).[18] C. Feuillet-Palma, T. Delattre, P. Morfin et al., Phys.Rev. B 81, 115414 (2010).[19] M. B¨uttiker, A. Prˆetre and H. Thomas, Phys. Lett. A180, 364 (1993).[20] A. Prˆetre, H. Thomas and M. B¨uttiker, Phys. Rev. B 54,8130 (1996).[21] P. Jarillo-Herrero, J. Kong, H.S.J. van der Zant, et al.,