week ending PRL 104, 186803 (2010) PHYSICAL REVIEW LETTERS 7 MAY 2010

Inelastic Backaction due to Quantum Point Contact Charge Fluctuations

C. E. Young and A. A. Clerk Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8 (Received 28 October 2009; published 4 May 2010) We study theoretically transitions of a double quantum-dot qubit caused by nonequilibrium charge fluctuations in a nearby quantum point contact (QPC) used as a detector. We show that these transitions are related to the fundamental Heisenberg backaction associated with the measurement, and use the uncertainty principle to derive a lower bound on the transition rates. We also derive simple expressions for the transition rates for the usual model of a QPC as a mesoscopic conductor, with screening treated at the RPA level. Finally, numerical results are presented which demonstrate that the charge noise and shot noise backaction mechanisms can be distinguished in QPCs having nonadiabatic potentials. The enhanced sensitivity of the charge noise to the QPC potential is explained in terms of interference contributions similar to those which cause Friedel oscillations.

DOI: 10.1103/PhysRevLett.104.186803 PACS numbers: 73.23.b

A fundamental consequence of the Heisenberg uncer- dot, which in turn drives transitions in the qubit. Several tainty principle is that the act of measurement necessarily recent experiments have attempted to investigate this in- perturbs the system being measured; these measurement- elastic backaction mechanism [14–17]. induced disturbances are termed ‘‘backaction.’’ Research Here we focus on an alternate mechanism for inelastic on quantum electronic systems has progressed to the point QPC backaction which involves its nonequilibrium charge where backaction effects, often near the limits imposed by fluctuations, as opposed to its current fluctuations. Recall the uncertainty principle, are of key relevance to experi- that the physics behind these two kinds of noise is very dif- ment [1]. They are particularly important in experiments ferent. Current fluctuations are related to the partitioning of involving solid state qubits, as they set a limit on how well electrons into transmitted and reflected streams by the QPC one can read out the qubits. In qubit-plus-detector systems, scattering potential; this results in a binomial distribution detector backaction can both dephase the qubit (i.e., T2 of the time-integrated current through the QPC [18]. In effects), as well as drive transitions between the qubit contrast, charge noise results from fluctuations in how long energy eigenstates (i.e., T1 effects, or ‘‘inelastic’’ back- electrons spend in the scattering region (i.e., dwell-time action). The study of backaction is also of purely funda- fluctuations) [19]. Importantly, the current noise mecha- mental interest, as it can shed light on the physics of the nism proposed in Ref. [13] can in principle be completely detector, which is often an interesting nonequilibrium suppressed by appropriate circuit design. In contrast, the quantum system in its own right. inelastic backaction arising from QPC charge fluctuations In this Letter, we focus on the inelastic backaction is the fundamental Heisenberg backaction: for a fixed associated with quantum point contact (QPC) charge de- measurement strength, it cannot be made arbitrarily small. tectors [2]. QPCs are among the most versatile and widely Using the uncertainty principle, we derive a rigorous used charge detectors in quantum electronics; having a lower bound on the charge noise inelastic backaction that complete understanding of their backaction is therefore exclusively involves directly measurable quantities. We highly desirable. They are the standard readout for both also derive simple expressions for transition rates valid charge and spin quantum-dot qubits [3,4], and have also for the usual model of a QPC as a coherent mesoscopic been used to make sensitive measurements of nanome- conductor. Our results suggest that the charge noise mecha- chanical oscillators [5]. The backaction of QPC detectors nism may already play a significant role in experiments. has been the subject of considerable study. Theoretically, Further, we show that charge fluctuations are more sensi- QPC-induced dephasing has been studied in Refs. [6–8], tive to the nature of the QPC potential than current noise: and measured experimentally in seminal experiments on while the current noise is simply determined by the QPC mesoscopic interferometers [9]. Theory has also shown transmission T , the charge noise also depends on the that the backaction dephasing induced by a QPC can reach adiabaticity of the QPC potential. As a result, it should the fundamental quantum limit set by the Heisenberg be experimentally possible to distinguish the current and uncertainty principle [10–12]. Inelastic QPC backaction charge noise backaction mechanisms by their dependence has also received attention. Aguado and Kouwenhoven on T . We provide an explanation for this effect in terms of [13] discuss such effects in a system where a double interference contributions similar to those that give rise to quantum-dot is measured by a QPC. They focus exclu- Friedel oscillations. sively on a mechanism whereby QPC current flucutations Note that other aspects of QPC charge noise backaction (shot noise) lead to potential fluctuations across the double on qubits have been studied before. Bu¨ttiker and collabo-

0031-9007=10=104(18)=186803(4) 186803-1 Ó 2010 The American Physical Society week ending PRL 104, 186803 (2010) PHYSICAL REVIEW LETTERS 7 MAY 2010 rators have studied inelastic backaction due to equilibrium H^ coupling int changesR the average QPC current by an charge fluctuations, as well as dephasing due to nonequi- I^ t U dt0 t t0 ^ t0 ! amountR h ð Þi ¼ 2 ð Þh zð Þi, where ½ ¼ librium charge noise [10,20]. While the general form of the i i! 1 de h½I^ðÞ; Q^ ð0Þi is the finite-frequency gain of nonequilibrium charge noise has been derived [19], its @ 0 the detector. At ! ¼ 0, the gain ½0 is simply related to direct role in generating qubit transitions at weak coupling T has not to our knowledge been previously elucidated. , the change in the transmission of the QPC brought Related work has examined Fermi-edge physics in a about by changing the qubit state: 2 qubit-QPC system [21], as well as an indirect mechanism 2e Vqpc T U hI^i¼h^ i ½0h^ i: where QPC charge noise, in consort with spin-orbit cou- z h 2 2 z (2) pling, yields transitions in spin qubits [22]. ^ Model.—While our results are general, we focus on the Equation (2) concisely expresses the fact that, via Hint, the experimentally relevant case of a double quantum-dot QPC acts as a charge detector. This interaction is the (DQD) charge qubit electrostatically coupled to a QPC fundamental qubit-detector coupling: without it one cannot charge detector (see Fig. 1). The qubit Hamiltonian is use the QPC to measure ^ z. Consequently, the backaction ^ HDQD ¼ 2 ^ z þ tc^ x, where is the detuning and tc is due to Hint is the Heisenberg backaction associated with ^ the interdot tunneling amplitude. The eigenstates of ^ z the measurement. Fluctuations in Q both dephase the qubit correspond to an electron occupying either the left or right and induce transitions between its eigenstates: it is the dot. We start by considering a completely general model of inelastic backaction that we wish to describe. As Q^ is a a spin-degenerate, single channel QPC with applied bias weighted average of the QPC charge density, we refer to it 2e2 Vqpc. The QPC conductance is G ¼ h T , where T is the as the QPC ‘‘charge’’ in what follows. effective transmission. Before proceeding, note that one may also have a second ^ The QPC acts as a detector by virtue of the fact that the qubit-QPC interaction of the form Hint;2 ¼ e^ zU2ðtÞ, ^ ^ potential felt by QPC electrons depends on the qubit’s where U2½!ZT½!I½!, and ZT is a transimpedance charge state. The qubit-QPC interaction Hamiltonian that converts QPC current fluctuations into qubit potential therefore necessarily includes a term of the form fluctuations. Reference [13] provides a comprehensive description of the qubit transitions induced by this cou- Z ^ U ^ ^ Uðr~Þ pling; Ref. [23] furthers this analysis by considering the H int ¼ ^ zQ; Q dr~ ^ðr~Þ: (1) 2 U effects of a spatially dependent ZT. Note however that Hint;2 does not contribute to qubit detection: linear response ^ Here, Uðr~Þ is the change in the QPC scattering poten- tells us that Hint;2 cannot change hIi at low frequency. The tial brought about by changing the qubit charge state, backaction mechanism of Ref. [13] is therefore not the and ^ is the electron density operator of the QPC. For Heisenberg backaction of the QPC. Moreover, the back- convenience, we introduce the parameter U represent- action due to Hint;2 can be made arbitrarily small by design- ing the average value of Uðr~Þ in the scattering region of ing the QPC-DQD circuit to suppress ZT. The backaction the QPC. For the electrostatic model in Fig. 1, and in the due to fluctuations in I^ can therefore be eliminated without 2Cc EC limit of weak coupling Cc C0,wehaveU ¼ C e affecting the sensitivity of the detector. 0 ^ where EC is the charging energy of the coupled right Heisenberg bound.—Fluctuations in Q can drive transi- ^ quantum dot. Via Hint, the QPC potential and hence the tions between the qubit eigenstates; for weak coupling, the conductance both depend on the qubit state; the QPC golden rule rates for excitation ( Q") and relaxation ( Q#) I^ U=@ 2 tc 2S current therefore yields information regarding the qubit are given byR Q"=#½ ¼ð Þ ð@Þ QQ½ , where S ! 2 dtei!t Q^ t Q^ 0 state. We assume throughout the realistic limit of weak QQ½ ¼ h ð Þ ð pÞiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiis the quantum noise spec- coupling in which linear response theory applies. Thus the ^ 2 2 tral density of Q and @ ¼ þ 4tc is the qubit level splitting. As this backaction is due to the coupling used to make the measurement, one can use the uncertainty prin- ciple to rigorously derive lower bounds on the transition rates without having to specify a QPC model or the cou- pling potential Uðr~Þ. Using the Heisenberg inequality which bounds the symmetric-in-frequency noise of a linear detector [24,25], as well the relation between the asymmetric-in-frequency noise and dissipative response coefficients, we find      t 2 U 2 eU 2 R c j ½ j ð Þ Re K : Q"=# 2 (3) FIG. 1 (color online). Electrostatic model of the coupled @ 4SII½ 2@ Zin½ charge qubit-QPC system. Inset: Schematic showing the detun- R h S ! 1 S ! S ! ing of the DQD. Here, K e2 and II½ ¼2 ð II½ þ II½ Þ is the 186803-2 week ending PRL 104, 186803 (2010) PHYSICAL REVIEW LETTERS 7 MAY 2010 symmetrized QPC current noise. The input impedance cussed in Ref. [13], where one must also estimate the value Z ! of theR QPC as seen by the qubit is in½ of a transimpedance that is difficult to measure. Simple 1 i! ^ ^ @=ð! 0 de h½QðÞ; Qð0ÞiÞ.AsZin is directly related estimates using Eq. (4) and typical experimental values to the QPC charge-charge susceptibility, it can be mea- (e.g., Ref. [16]) suggest the expected charge noise back- sured by driving Rabi oscillations in the DQD and mea- action is at least comparable in magnitude to the shot noise suring the induced oscillations in the QPC charge Q^ using a mechanism in recent experiments. We stress that screening T Z second QPC, similar to the experiment of Ref. [26]. The only enters through the measurable quantities and in magnitude and phase of the resulting oscillations in Q^ (see also Ref. [10]). Charge noise backaction signature.—Next we outline a provide a measure of Zin½!. The bound in Eq. (3) is completely general, and does not way to distinguish the charge and current noise backaction rely on any assumptions regarding the QPC. Most impor- mechanisms in experiment. Here we note that the shot noise mechanism of Ref. [13] yields a backaction excita- tantly, all of the parameters on the right-hand side are S T 1 T intrinsic properties of the QPC and directly measurable. tion rate I" II ð Þ. Thus an observed depen- Note also that RefZ1g vanishes as !2 when ! ! 0, dence of " on the QPC transmission of the form in T 1 T T 1=2 reflecting the capacitive coupling between qubit and QPC. ð Þ (and hence a maximum at ¼ ) is often Backaction transition rates.—We now specialize and regarded as a signature of the shot noise mechanism treat the QPC as a coherent scatterer, described by a [14,17]. From Eq. (4), we find that, in general, Q" will T scattering matrix s with both time-reversal and inversion not have the same dependence on : the only case where it T symmetries. The effects of screening in the QPC are de- does is when T ð1T Þ is a constant that does not change as scribed using the method developed by Bu¨ttiker and col- T and T are simultaneously varied by changing gate laborators [10,20,27], which combines the circuit model in voltages. As noted in previous work [10,11], this condition Fig. 1 with the random-phase approximation (RPA). is satisfied if and only if the QPC potential is adiabatic [31] Within this model, the spatial variations in the QPC charge and so is the qubit-QPC coupling [i.e., the variation of and internal potential are taken into account by using a Uðr~Þ is smooth compared to the Fermi wavelength F]. If capacitive network to describe them [28,29]. these conditions are violated, Q"ðT Þ will not have its In the presence of screening, the total (screened) change maximum at T ¼ 1=2. This provides a way to distinguish in QPC potential Utotðr~Þ, resulting from a change in the the charge and current noise backaction mechanisms. qubit state, differs from the bare (i.e., unscreened) potential To test this assertion, we have numerically studied the change Uðr~Þ appearing in Hint.ItisUtotðr~Þ, not Uðr~Þ, charge noise associated with a model two-dimensional which determines the change in the QPC transmission QPC saddle point potential using a recursive Green func- T . Screening also modifies SQQ. Importantly, within tion approach extended to include the calculation of local the RPA, one finds that T and SQQ are both determined properties [32]. We consider the potential by the same parametricR change of the scattering matrix, given by s ¼ dr~Utotðr~Þ½s=Uðr~Þ, where =Uðr~Þ 2 2 U x; y We4x =w y2; is a functional derivative with respect to the local QPC ð Þ¼ (6) potential [30]. Using this approach [29], we find that, even in the presence of screening, one can relate the qubit where W is the quantity modified by the coupling to the transition rates to T without needing to know the precise DQD. The gating effect of the qubit is therefore modeled as form of Utotðr~Þ. In the low-temperature limit kBT a shift Uðx; yÞþUðx; yÞ¼½1 þ U=ða2WÞUðx; yÞ, feVqpc; @g, and for @ eVqpc, we find where a is the lattice spacing. Note that our goal is not to   advocate a particular coupling model, but rather to study t 2 T 2 eV @ c ð Þ qpc the dependence of the charge noise on the QPC adiabatic- Q"½¼ ; (4) @ T ð1 T Þ 4@ ity. In Eq. (6), the adiabaticity of both the potential and the coupling is set by the barrier width w.     1 t 2 eU 2 R Figure 2 shows Q as a function of transmission. In the c ð Þ Re K : " Q#½ ¼ Q"½ þ 2 (5) limit of an adiabatic potential and coupling (i.e., w F), @ @ Zin½ Q" is to a good approximation proportional to T ð1 T Þ, Here we have taken s and s to be approximately energy and therefore to SII. In contrast, in the nonadiabatic case independent within the energy ranges of interest. w F, the T dependence of Q" is markedly different: it Equations (4) and (5) are the central results of this Letter. is strongly skewed towards T > 1=2. Observation of this They allow one to make a simple prediction for the charge- asymmetry offers a way to distinguish the charge noise and noise-induced inelastic backaction using only experimen- current noise backaction. tally accessible QPC quantities. The excitation rate Q" has Role of Friedel oscillations.—The increased charge a particularly simple form that depends only on the QPC noise backaction for T > 1=2 is due to interference con- bias Vqpc, transmission T , and sensitivity T . This is in tributions that are the nonequilibrium analogue of Friedel stark contrast to the current noise backaction rates dis- oscillations [33]. In one dimension, the continuity equation 186803-3 week ending PRL 104, 186803 (2010) PHYSICAL REVIEW LETTERS 7 MAY 2010

backaction, the charge noise backaction need not be maxi- mal at QPC transmission T ¼ 1=2. This work was supported by NSERC, FQRNT, and CIFAR.

[1] M. H. Devoret and R. J. Schoelkopf, Nature (London) 406, 1039 (2000). [2] M. Field et al., Phys. Rev. Lett. 70, 1311 (1993). [3] J. R. Petta et al., Phys. Rev. Lett. 93, 186802 (2004). [4] J. M. Elzerman et al., Nature (London) 430, 431 (2004). FIG. 2 (color online). Numerical results for the zero-frequency [5] M. Poggio et al., Nature Phys. 4, 635 (2008). [6] Y. Levinson, Europhys. Lett. 39, 299 (1997). qubit excitation rate Q as a function of QPC transmission T . " [7] I. L. Aleiner, N. S. Wingreen, and Y. Meir, Phys. Rev. Lett. We use a lattice spacing a ¼ 6:2nm, giving a hopping energy 2 2 79, 3740 (1997). t ¼ @ =ð2m a Þ¼14:5 meV, where m ¼ 0:068me in GaAs. [8] S. A. Gurvitz, Phys. Rev. B 56, 15215 (1997). For EF ¼ 16 meV, the Fermi wavelength is F ¼ 6a. The [9] E. Buks et al., Nature (London) 391, 871 (1998). dashed (red) line is for an adiabatic barrier with w ¼ 10F, [10] S. Pilgram and M. Bu¨ttiker, Phys. Rev. Lett. 89, 200401 while the solid (blue) line is for a nonadiabatic barrier with w ¼ (2002). F; each curve is scaled by its maximum value. In the non- [11] A. A. Clerk, S. M. Girvin, and A. D. Stone, Phys. Rev. B adiabatic case, Q is asymmetric and has its maximum at T > " 67, 165324 (2003). 1=2. Inset: the corresponding transmission curves as a function [12] A. N. Korotkov and D. V. Averin, Phys. Rev. B 64, 165310 of W. (2001).   [13] R. Aguado and L. P. Kouwenhoven, Phys. Rev. Lett. 84, d 1 Z d 1986 (2000). Q^ ¼ dx UðxÞ I^ðx; tÞ (7) dt U dx [14] E. Onac et al., Phys. Rev. Lett. 96, 176601 (2006). [15] V.S. Khrapai et al., Phys. Rev. Lett. 97, 176803 (2006). relates the charge operator Q^ to a weighted average of the [16] S. Gustavsson et al., Phys. Rev. Lett. 99, 206804 (2007). current flowing into the scattering region of the QPC. This [17] S. Gustavsson et al., Phys. Rev. B 78, 035324 (2008). is distinct from and independent of the transport current [18] L. S. Levitov, H. Lee, and G. B. Lesovik, J. Math. Phys. 37 (flowing through the QPC) used to define the QPC con- (N.Y.) , 4845 (1996). S I^ [19] M. H. Pedersen, S. A. van Langen, and M. Bu¨ttiker, Phys. ductance and shot noise II. The current operator of a Rev. B 57, 1838 (1998). coherent conductor contains spatially varying terms that [20] M. Bu¨ttiker and M. L. Polianski, J. Phys. A 38, 10 559 oscillate with a wavelength F=2. These terms give rise to (2005). Friedel oscillations in the density-density correlation func- [21] I. Snyman and Y.V. Nazarov, Phys. Rev. Lett. 99, 096802 tion, but do not contribute to, e.g., the average conductance (2007). or low-frequency shot noise [34]. In contrast, we find that [22] M. Borhani, V.N. Golovach, and D. Loss, Phys. Rev. B 73, they can enhance the nonequilibrium charge noise. Their 155311 (2006). 80 contributions to SQQ½! via Eq. (7) correspond to interfer- [23] A. L. Chudnovskiy, Phys. Rev. B , 081309(R) (2009). ences between incoming and transmitted waves, and thus [24] V.B. Braginsky and F. Y. Khalili, Quantum Measurement (Cambridge University Press, Cambridge, England, 1992). T Uðr~Þ scale as .If varies slowly on the scale of F, then [25] A. A. Clerk et al., arXiv:0810.4729v1. the spatial averaging described by Eq. (7) strongly sup- [26] S. Lu¨scher et al., Phys. Rev. Lett. 98, 196805 (2007). presses these terms. In the opposite limit, no such averag- [27] S. Pilgram and M. Bu¨ttiker, Phys. Rev. B 67, 235308 ing occurs; the terms thus enhance the charge noise at (2003). higher values of T . Note that the same interferences yield [28] M. Bu¨ttiker and T. Christen, in Mesoscopic Electron the surprising result that the total charge in a perfect (i.e., Transport, edited by L. L. Sohn et al. (Kluwer, T ¼ 1), zero temperature one-dimensional wire of length Dordrecht, 1997), Vol. 345, p. 259. L fluctuates; at ! ¼ 0, the spectral density is SQQ½0¼ [29] See supplementary material at http://link.aps.org/ 3 2 supplemental/10.1103/PhysRevLett.104.186803 for dis- ðe V@=4Þ½sinðkFLÞ=EF , where V EF=e is the bias cussion of Eqs. (4) and (5), the quantum limit, and our and EF is the Fermi energy. screening model. Conclusion.—The fundamental Heisenberg backaction [30] M. Bu¨ttiker, J. Math. Phys. (N.Y.) 37, 4793 (1996). of QPC qubit detectors is due to nonequilibrium charge [31] L. Glazman and R. Sheckhter, JETP 94, 292 (1998). fluctuations. We have shown that the corresponding inelas- [32] G. Metalidis and P. Bruno, Phys. Rev. B 72, 235304 tic transition rates in the qubit can be expressed exclusively (2005). in terms of quantities that are directly measurable in ex- [33] J. Friedel, Nuovo Cimento 7, 287 (1958). periment. We have also shown that, unlike the current noise [34] M. Bu¨ttiker, Phys. Rev. B 45, 3807 (1992).

186803-4