Abstract
Spin qubits have been successfully realized in electrostatically defined, lateral fewelectron quantumdot circuits^{1,2,3,4}. Qubit readout typically involves spin to charge information conversion, followed by a charge measurement made using a nearby biased quantum point contact^{1,5,6} (QPC). It is critical to understand the backaction disturbances resulting from such a measurement approach^{7,8}. Previous studies have indicated that QPC detectors emit phonons which are then absorbed by nearby qubits^{9,10,11,12,13}. We report here the observation of a pronounced backaction effect in multiple dot circuits, where the absorption of detectorgenerated phonons is strongly modified by a quantum interference effect, and show that the phenomenon is well described by a theory incorporating both the QPC and coherent phonon absorption. Our combined experimental and theoretical results suggest strategies to suppress backaction during the qubit readout procedure.
Main
The backaction process considered in this paper involves deleterious inelastic tunnelling events between two adjacent dots in a serial double or triple quantum dot (DQD, TQD). The energy difference Δ between the initial and final electronic dot states is provided by the absorption of a nonequilibrium acoustic phonon, which itself is generated by the quantum point contact (QPC) detector^{12}. Such an absorption process between adjacent dots is constrained by the energy conservation condition Δ=ℏqv_{ph} (v_{ph} is the sound velocity, q the phonon wave vector). More subtly, it is also sensitive to the difference in phase, Δ φ=d · q, of the associated phonon wave between the two dot positions, with d being the vector connecting the two dot centres^{14,15}. This qdependent (and hence Δdependent) phase difference controls the matrix element for phonon absorption because it determines whether the electron–phonon couplings in each of the two individual dots add constructively or destructively (Fig. 1 and ref. 16). The result is an oscillatory probability for inelastic electrontransfer events involving phonon absorption, with constructive interference occurring when Δ φ=(2n+1)π (where n is an integer).
Data showing a pronounced backaction effect are shown in Fig. 2a, which shows the stability diagram measured in charge detection for a fewelectron DQD without a voltage drop between its left and right leads. The charge configuration of the quantumdot structures influences the conductance of a nearby QPC because of the capacitive coupling between the dots and the QPC. To serve as a charge detector it is necessary to drive a current through the detector QPC, which, in turn, leads to the observed detector backaction. Multiple gates fabricated 85 nm above a highmobility twodimensional electron system (2DES) are used to define two dots and two QPCs (Fig. 2d). The differential transconductance dI_{QPC}/dV_{L} of the biased charge detector QPC (V _{QPC}=−1.2 mV) is plotted as a function of control gates V _{L},V _{R}. It shows local extrema at the boundaries between regions of different electronic ground states, yielding dark ‘charging’ and white ‘charge transfer’ lines. Specific ground state configurations are labelled (N_{L},N_{R}), where the integer N_{ξ} denotes the number of electrons in dot ξ=L (left) and R (right). As our DQD is cooled to T≃30 mK the unmeasured DQD is expected to be in its ground state.
Detector backaction manifests itself within a distinct triangularshaped region of deviations from the ground state configuration (1, 2), where a pronounced pattern of repeated, parallel stripes is present. It indicates an oscillating probability to find the DQD in the excited configuration (1, 1). The excitation process sketched in Fig. 2c includes an inelastic tunnelling transition (1,2)→(2,1) mediated by the absorption of a phonon, followed by an elastic (and therefore quick) tunnelling process (2,1)→(1,1). In our measurements, the tunnel barrier between the right dot and right lead is tuned to be almost closed (Fig. 2c,d). The direct transition (1,1)→(1,2) back into the ground state via an elastic tunnelling process from the right lead is consequently very slow and the excited configuration (1, 1) is metastable. The associated threelevel dynamics can result in average nonthermal occupations^{13}. In this way a metastable excited state is essential to directly observe detector backaction in a lowbandwidth stability diagram measurement. It requires asymmetric dot–lead tunnel couplings in the case of a DQD (Supplementary Information).
The stripe pattern constitutes the key signature of the coherent phononmediated backaction effect. It indicates that the probability to be in the excited configuration (1,1) oscillates as a function of the energy detuning Δ between the intermediate state (2,1) and the groundstate configuration (1,2) (Fig. 2b); each stripe is thus parallel to the white charge transfer line where these states are degenerate (that is, Δ=0; marked). The striped region is bounded by a line Δ=Δ_{max}, indicating that there is a maximum energy available to excite the DQD. By seeing how this boundary changes with increasing V _{QPC} (Fig. 2e,f), we find that Δ_{max}≃e V _{QPC}, consistent with the QPC indeed being the energy source for the initial DQD excitation.
The geometry of the backaction regions as well as the influences of temperature and the orbital excitation spectrum are discussed in the Supplementary Information. In short, the remaining boundaries of the triangularshaped regions of backaction correspond to energy thresholds for lead tunnelling. The width of each stripe is largely independent of temperature; this is indicative of an excitation process involving electron transfers between dots, without any involvement of lead electrons (Figs 2c and 3e,f). The regular spacing of the stripe features in both DQD and TQD (discussed below) experiments over so many stripes eliminates the possibility that they are due to resonances with orbital excitations of a dot, as there is no reason to expect such a uniform level spacing; furthermore, the energy spacing between the stripes is much smaller than would be expected for the average level spacing of the small dots studied here.
To quantify the interpretation of the stripe pattern in Fig. 2 in terms of interference and QPC backaction, we have developed a theoretical model that describes the generation of phonons by the nonequilibrium QPC charge fluctuations^{8} and their coherent absorption by the DQD. These fluctuations represent the fundamental backaction of the measurement—their magnitude is bounded from below by the rate at which information is obtained from the QPC via a Heisenberglike inequality^{8}. Given this, the backaction charge noise mechanism we describe must necessarily make a contribution to the observed oscillations. This mechanism is also consistent with the high visibility of the oscillations, as such visibility requires a highly localized source of hot phonons. Although we cannot completely rule out that other, less direct, backaction mechanisms also contribute (for example, generation of hot phonons in the QPC leads), it is not clear that such mechanisms would also yield such highvisibility oscillations. We describe bulk acoustic phonon modes of GaAs interacting with both electrons in the DQD, as well as with the fluctuating charge density of the biased QPC via a screened piezoelectric interaction. Using Keldysh perturbation theory, we can calculate the DQD state in the presence of backaction (Fig. 4 and Supplementary Information). The relevant part of the dot–phonon interaction (that is, terms that can cause transitions in the dot) take the form:
Here g〉 (e〉) denotes the DQD ground (excited) state, t_{c} is the pertinent interdot tunnel matrix element, () destroys (creates) a phonon of wavector q in branch μ, λ_{q,μ} is the effective matrix element (screened) for the interaction of phonons with a single dot and h.c. denotes the Hermitian conjugate. The first bracketed factor in the sum of equation (1) denotes the key interference of relevance: the two terms correspond to phonons interacting with electrons in either the left or right dot (which are centred at r_{L} and r_{R} respectively) (Fig. 1b).
Despite the explicit interference evident in equation (1), geometric averaging can still strongly suppress interference oscillations in observable quantities. Simply put, although the DQD groundexcited energy splitting fixes the magnitude of a phonon participating in an inelastic tunnelling event, it does not specify its direction; hence, the relative phase in the first term of equation (1) is not completely determined by Δ. This is typically the case in situations probing the emission of acoustic phonons by biased DQDs^{16}, where interference oscillations are observed, albeit with much smaller visibilities than seen here^{17,18}. In contrast, the simple geometric filtering depicted in Fig. 1a suggests that this averaging need not play a role in phonon absorption, as only phonons travelling from the QPC to the dots contribute. This is supported by our theoretical calculations, which also exhibit strong oscillations for realistic parameter values, and show a pronounced enhancement of interference oscillations when the DQD and QPC are all collinear (Fig. 4a–c).
The theory is also able to capture other aspects of the experimental data: in particular, the size of the backaction triangle grows with V _{QPC}, and the lowestenergy stripes (that is, the smallest values of Δ corresponding to long phonon wavelengths) are suppressed because of screening effects (see Fig. 4b). Using the fact that in the experiment the QPC and DQD are approximately collinear, the measured spacing of the interference parameter δ Δ=45 μeV in the DQD data of Fig. 2d yields a DQD separation d=h v_{s}/δ Δ≃250 nm; this is in good agreement with the separation estimated from scanning electron micrographs (Fig. 2d). The theory also shows that owing to the anisotropy of the electron–phonon matrix elements λ_{q,μ}, the overall magnitude of the phononinduced backaction is sensitive to the orientation of the dot–QPC axis with respect to crystallographic axes. This dependence on orientation is demonstrated in Fig. 4d. More details on the theoretical treatment is provided in Methods and Supplementary Information.
Although we have focused so far on backaction in DQDs, the mechanism we describe is extremely general, and is in fact even more ubiquitous in systems with more than two dots. As discussed, a key requirement to see the effect is the existence of a longlived metastable excited state. Such a situation occurs rather naturally in serial TQD structures^{19,20,21,22,23}, as the centre dot is effectively decoupled from one of the leads whenever either one of the other two dots (left, right) is in Coulomb blockade. This directly yields a metastable excited state in which the charge of the middle dot is unable to relax. As a consequence deviations from the ground state configuration are often observed along the charging line of the centre dot^{20} and backaction effects occur naturally in the stability diagram. We study detector backaction in a TQD in Fig. 3 by successively increasing e V _{QPC}. Already at relatively small bias V _{QPC}≤300 μV (Fig. 3a,b), a triangularshaped region of telegraph noise is observed along the central charging line^{20}. It indicates slowly fluctuating deviations from the ground state configuration, which can be caused by external noise or detector backaction^{10}. The underlying excitation processes, sketched in Fig. 3e,f, are similar to the one discussed above for the DQD. Indeed, the population of the right dot does not fluctuate; it plays the same role as the closed barrier in the case of the DQD, namely to block charge exchange between the centre dot and the right lead. Further increasing V _{QPC} to 500 μV in (Fig. 3c) reveals the familiar pattern of equally spaced stripes, both within the (1,0,0) and (0,1,1) regions. As the bias is increased even more to V _{QPC}=700 μV (Fig. 3d) the striped regions expand further, revealing the V _{QPC} dependence also observed in the case of the DQD (Fig. 2f).
By considering experimental data on both DQD and TQD systems, we have demonstrated that interference can strongly affect the phononmediated backaction generated by a QPC in quantumdot circuits. Furthermore, we have shown that this effect is well described by a basic theoretical model incorporating the generation of phonons by the QPC detector and their coherent absorption by the dots. Our study suggests the possibility of mitigating backaction effects by making use of this interference. One could, for example, endeavour to first tune the DQD/TQD to an operating point where destructive interference suppresses phonon absorption, and only then energize the QPC to make a measurement. More complex schemes that also incorporate the anisotropy of the electron–phonon interaction with respect to crystallographic axes could potentially yield even greater backaction reduction. Because the piezoelectric coupling to inplane phonons is maximized in the 〈110〉 directions^{24}, by aligning the QPC–DQD axis away from these directions, one could appreciably decrease the phononmediated backaction excitation discussed here (for example, Fig. 4a versus Fig. 4d).
Methods
Experiment.
The samples were fabricated from GaAs/AlGaAs heterostructures containing 2DESs 100 nm (TQD) and 85 nm (DQD) below the surface, respectively. The 2DESs are characterized at cryogenic temperatures by carrier densities of n_{S}=2.1×10^{11} cm^{−2} and n_{S}=1.9×10^{11} cm^{−2} with mobilities of μ=1.72×10^{6} cm^{2} Vs^{−1} and μ=1.19×10^{6} cm^{2} Vs^{−1} for the TQD and DQD, respectively. Metallic gate electrodes were fabricated on the sample surface by electronbeam lithography and standard evaporation/liftoff techniques. Negative voltages applied to these electrodes are used to locally deplete the 2DESs and thereby define the quantum dot and QPC structures. All measurements were performed in dilution refrigerators at cryogenic temperatures below 100 mK. To detect the charge configuration of the TQD, the voltage of one gate of the TQD was slightly modulated and the detector differential transconductance was measured in linear response (a.c. setup). A constant voltage was also applied across the QPC to enhance detector backaction. In the case of the DQD only a constant voltage was applied across the QPC and the direct current I_{QPC} flowing through the QPC was measured to detect the charge configuration of the DQD (d.c. setup). The differential transconductance dI_{QPC}/dV _{L} was then computed numerically. Both methods result in the differential transconductance and their interpretation is identical.
To interpret the observed backaction in terms of the energy detuning Δ between different charge configurations an accurate conversion from gate voltages to units of energy is necessary. Such a linear transformation has been performed, following the methods described elsewhere^{25}. The conversion relation reads Δ=(α_{gL}^{R}−α_{gL}^{L})V _{L}+(α_{gR}^{R}−α_{gR}^{L})V _{R}, with the following set of conversion factors determined for the red symbols in Fig. 2f: α_{gL}^{R}=(54±5) meV/V , α_{gR}^{R}=(105±4) meV/V , α_{gR}^{L}=(62±4) meV/V , α_{gL}^{L}=(90±5) meV/V . The conversion factors related to the blue symbols in Fig. 2f read α_{gL}^{R}=(65±6) meV/V , α_{gR}^{R}=(109±7) meV/V , α_{gR}^{L}=(61±6) meV/V , α_{gL}^{L}=(97±7) meV/V .
Theory.
The fluctuating nonequilibrium QPC charge density operator is modelled as , where the total charge operator is described by scattering theory (see refs 8, 26). Note that as we are interested in a singlechannel QPC, the spatial profile f(r) of the fluctuating QPC charge density is fixed; for simplicity, we take it to be a Gaussian of width r_{QPC}. This fluctuating QPC charge density is coupled to acoustic phonons via the standard piezoelectric interaction (using parameters appropriate for GaAs (ref. 24)). We calculate the Keldysh Green functions of the acoustic phonons in the presence of this coupling to the QPC, working to leading order in the electron–phonon interaction, and using scattering theory to calculate the QPC Keldysh Green functions. We then use these ‘dressed’ phonon Green functions to calculate the Fermi golden rule excitation rate of the DQD via the coupling described in equation (1). This excitation rate is finally incorporated into a master equation describing the occupation probability of the three relevant DQD states (Fig. 2c). In addition to the excitation rate (top panel of Fig. 2c), there is a rate Γ_{fast} describing the tunnelling from the excited state to the metastable auxiliary state (middle panel of Fig. 2c) and a rate Γ_{slow} describing the slow decay back to the true ground state (bottom panel of Fig. 2c). We take Γ_{fast}=1 GHz and Γ_{slow}=10 kHz; in this regime of Γ_{fast}≫Γ_{slow}, the nonground state population of the DQD is independent of Γ_{fast}, whereas Γ_{slow} determines the overall magnitude of the interference oscillations. By using the master equation to calculate the stability diagram as a function of gate voltages, one can obtain the DQD charge susceptibility d(〈n_{R}〉+ε〈n_{L}〉)/dV _{L}, which is proportional to the measured transconductance. The parameter ε∼0.4 characterizes the QPC’s asymmetric response to charge in the R versus L dot. Further details about the explicit form of λ_{q,μ} (including the role of screening and dimensionality) are provided in the Supplementary Information.
References
Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in fewelectron quantum dots. Rev. Mod. Phys. 79, 1217–1265 (2007).
Fujisawa, T., Hayashi, T. & Sasaki, S. Timedependent singleelectron transport through quantum dots. Rep. Prog. Phys. 69, 759–796 (2006).
Petersson, K. D., Petta, J. R., Lu, H. & Gossard, A. C. Quantum coherence in a oneelectron semiconductor charge qubit. Phys. Rev. Lett. 105, 246804 (2010).
Gaudreau, L. et al. Coherent control of threespin states in a triple quantum dot. Nature Phys. 8, 54–58 (2011).
Field, M. et al. Coulomb blockade as a noninvasive probe of local density of states. Phys. Rev. Lett. 77, 350–353 (1996).
Elzerman, J. M. et al. Fewelectron quantum dot circuit with integrated charge read out. Phys. Rev. B 67, 161308 (2003).
Aguado, R. & Kouwenhoven, L. P. Double quantum dots as detectors of highfrequency quantum noise in mesoscopic conductors. Phys. Rev. Lett. 84, 1986–1989 (2000).
Young, C. E. & Clerk, A. A. Inelastic backaction due to quantum point contact charge fluctuations. Phys. Rev. Lett. 104, 186803 (2010).
Khrapai, V. S., Ludwig, S., Kotthaus, J. P., Tranitz, H. P. & Wegscheider, W. Doubledot quantum ratchet driven by an independently biased quantum point contact. Phys. Rev. Lett. 97, 176803 (2006).
Taubert, D. et al. Telegraph noise in coupled quantum dot circuits induced by a quantum point contact. Phys. Rev. Lett. 100, 176805 (2008).
Gasser, U. et al. Statistical electron excitation in a double quantum dot induced by two independent quantum point contacts. Phys. Rev. B 79, 035303 (2009).
Schinner, G. J., Tranitz, H. P., Wegscheider, W., Kotthaus, J. P. & Ludwig, S. Phononmediated nonequilibrium interaction between nanoscale devices. Phys. Rev. Lett. 102, 186801 (2009).
Harbusch, D., Taubert, D., Tranitz, H. P., Wegscheider, W. & Ludwig, S. Phononmediated versus coulombic backaction in quantum dot circuits. Phys. Rev. Lett. 104, 196801 (2010).
Miller, A. & Abrahams, E. Impurity conduction at low concentrations. Phys. Rev. 120, 745–755 (1960).
Imry, Y. in Tunnelling Phenomena in Solids (eds Burstein, E. & Lundqvsit, S.) 563–576 (Plenum, 1969).
Brandes, T. Coherent and collective quantum optical effects in mesoscopic systems. Phys. Rep. 408, 315–474 (2005).
Fujisawa, T. et al. Spontaneous emission spectrum in double quantum dot devices. Science 282, 932–935 (1998).
Roulleau, P. et al. Coherent electron–phonon coupling in tailored quantum systems. Nature Commun. 2, 239 (2011).
Gaudreau, L. et al. Stability diagram of a fewelectron triple dot. Phys. Rev. Lett. 97, 036807 (2006).
Schröer, D. et al. Electrostatically defined serial triple quantum dot charged with few electrons. Phys. Rev. B 76, 075306 (2007).
Rogge, M. C. & Haug, R. J. Twopath transport measurements on a triple quantum dot. Phys. Rev. B 77, 193306 (2008).
Rogge, M. C. & Haug, R. J. The three dimensionality of triple quantum dot stability diagrams. New J. Phys. 11, 113037 (2009).
Granger, G. et al. Threedimensional transport diagram of a triple quantum dot. Phys. Rev. B 82, 075304 (2010).
Jasiukiewicz, C. Acoustic phonon emission by hot 2D electrons: The angular distribution of the emitted phonon power. Semicond. Sci. Technol. 13, 537–547 (1998).
Taubert, D., Schuh, D., Wegscheider, W. & Ludwig, S. Determination of energy scales in fewelectron double quantum dots. Rev. Scient. Inst. 82, 123905 (2011).
Pedersen, M., van Langen, S. & Büttiker, M. Charge fluctuations in quantum point contacts and chaotic cavities in the presence of transport. Phys. Rev. B 57, 1838–1846 (1998).
Acknowledgements
S.L. and D. T. acknowledge financial support by the German Science Foundation via SFB 631, LU 819/41, and the German Excellence Initiative via the ‘Nanosystems Initiative Munich’ (NIM). G.G. acknowledges funding from the NRC–CNRS collaboration. A.S.S. and A.A.C. acknowledge funding from NSERC and CIFAR.
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Contributions
D.T. fabricated the DQD samples, performed the DQD experiments and analysed the data. D.H. performed preliminary experiments on another DQD sample. A.K. fabricated the TQD sample. L.G., G.G. and S.S. performed the TQD experiments and analysed the data. P.Z. assisted in these experiments. D.S. and W.W. grew the heterostructures for the DQD samples; Z.R.W. optimized and grew the heterostructure for the TQD sample. D.T., L.G., C.E.Y., A.A.C., A.S.S. and S.L. wrote the paper. C.E.Y. and A.A.C. developed the theoretical model and supported both experimental groups. A.S.S. and S.L. supervised the experimental collaboration from Ottawa and Munich. The experiments have been supervised collaboratively by A.S.S. and S.L.; the theoretical modelling was surpervised by A.A.C.
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Granger, G., Taubert, D., Young, C. et al. Quantum interference and phononmediated backaction in lateral quantumdot circuits. Nature Phys 8, 522–527 (2012). https://doi.org/10.1038/nphys2326
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DOI: https://doi.org/10.1038/nphys2326
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