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.Àb 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 ÁU dt0 t t0 ^ t0 ! amountR h ð Þi ¼ 2 ð À Þh zð Þi, where ½ ¼ librium charge noise [10,20]. While the general form of the i i! À 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 ÁT 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 ÁT ÁU 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 ^ ÁU ^ ^ ÁUðr~Þ pling; Ref. [23] furthers this analysis by considering the H int ¼ ^ zQ; Q dr~ ^ðr~Þ: (1) 2 ÁU effects of a spatially dependent ZT. Note however that Hint;2 does not contribute to qubit detection: linear response ^ Here, ÁUðr~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 ÁU represent- action due to Hint;2 can be made arbitrarily small by design- ing the average value of ÁUðr~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,wehaveÁU 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 À À conductance both depend on the qubit state; the QPC golden rule rates for excitation ( Q") and relaxation ( Q#) I^ À ÁU=@ 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.