Improved position measurement of nanoelectromechanical systems using cross correlations
We consider position measurements using the cross-correlated output of two tunnel junction position detectors. Using a fully quantum treatment, we calculate the equation of motion for the density matrix of the coupled detector-detector-mechanical oscillator system. After discussing the presence of a bound on the peak-to-background ratio in a position measurement using a single detector, we show how one can use detector cross correlations to overcome this bound. We analyze two different possible experimental realizations of the cross correlation measurement and show that in both cases the maximum cross-correlated output is obtained when using twin detectors and applying equal bias to each tunnel junction. Furthermore, we show how the double-detector setup can be exploited to drastically reduce the added displacement noise of the oscillator.
It is expected that, in the near future, position measurements of nanomechanical systems will reach the quantum limit. Experimental progress in this direction is very fast and displacement sensitivities near the standard quantum limit have already been demonstratedKnobel and Cleland (2003); LaHaye et al. (2004); Naik et al. (2006).
In the current generation of experiments, the coupling between the resonator and the mesoscopic detector is typically very weak. The position measurement can therefore not be seen as a strong projective measurement. It is better described within the framework of weak measurement theory that was recently developed in the context of solid-state quantum computingAverin (2003); Korotkov and Averin (2001); pilgram2002 (); Clerk et al. (2003). This theory describes a continuous measurement process where the information about the measured object can be extracted, for instance, from the spectral density of the detector (and not simply from its average output). An important result in this theory is the Korotkov-Averin bound, which puts an upper limit of 4 to the ratio of the contribution of the measured state to the detector’s spectral density, and the intrinsic background detector noise, for any linear detector measuring a two-level system.
Since a quantum position measurement by a mesoscopic detector can be described within the same theoretical framework as a qubit measurement, one might ask if such a bound also exists in the case of a position measurement. In this article, we first show that, for fixed system parameters, the peak-to-background ratio in the spectral density of a position detector weakly coupled to an oscillator is also bounded from above. This result is obtained by considering the example of a single tunnel-junction detector, a simple detector that has been thoroughly studied theoreticallyBocko et al. (1988); Yurke and Kochanski (1990); Schwabe et al. (1995); Mozyrsky and Martin (2002); Clerk and Girvin (2004); Wabnig et al. (2005, 2007) and realized experimentallyCleland et al. (2002); Flowers-Jacobs et al. (2007).
Besides showing that the peak-to-background ratio is bounded in the typical single-detector position measurement, we also propose, in this article, two simple experimental configurations (Fig. 1) where, by using the cross correlations between two detectors, the bound on the peak-to-background ratio can be overcome. As the oscillator-independent parts of the output signal of the two detectors are uncorrelated, the background noise in these configurations is zero and therefore the peak-to-background ratio diverges. In the context of qubit readout, this idea has already been proposed in an insightful work by Jordan and Büttiker Jordan and Büttiker (2005) and was shown experimentally to improve readout fidelityBuehler et al. (2003). Experimentally, position measurements should hence also profit from using cross-correlated detector outputs. We analyze in detail the two configurations presented in Fig. 1 and obtain analytical results for the optimal cross-correlated signal as a function of different detector parameters.
Previous studiesClerk (2004) of the position measurement problem focused on finding the conditions for quantum-limited detector sensitivity, under which one minimizes the total detector contribution to the output displacement noise. We show that the double-detector setup proposed here can in fact be used to almost totally get rid off the added displacement noise of the oscillator due to detector back-action. This is a remarkable result that nicely complements the general single-detector analysis made in Ref. Clerk, 2004.
The article is organized as follows: in Section II, we introduce the formalism used in the rest of the paper, viz., a master equation for the -resolved density matrix, where is the number of charges that have passed through the detector. This equation of motion allows us to find expressions for the combined moments of charge (detector) and oscillator quantities. In Section III, the formalism is applied to the case of one position detector coupled to the oscillator.Clerk and Girvin (2004) We analyze the peak-to-background ratio and show that this quantity is always bounded from above in the single-detector case. This bound cannot be made arbitrarily large simply by increasing the detector sensitivity. Section IV generalizes this treatment to a configuration with two detectors and demonstrates that measuring the current cross correlations of the two detectors allows one to get arbitrarily high values of the peak-to-background ratio: i.e., it is possible to eliminate the bound that exist in the single-detector case. In Section V, we demonstrate how the proposed setup can be used to diminish the added position noise of the oscillator induced by the presence of the detector, allowing position measurement beyond the standard quantum limit derived for a single detector.
Ii Equation of motion for the density matrix
Approaches based on quantum master equations have proven useful in the study of nanomechanical systems Utami et al. (2004); Rodrigues and Armour (2005a). By writing the equation of motion for the density matrix of the full (detector and oscillator) system and tracing out the detector degrees of freedom, one can obtain an equation of motion for the reduced density matrix describing the evolution of the oscillator taking into account the coupling to the detector. In order to investigate electronic transport in the coupled system, it is useful to refine this approach to keep track of , the number of charges that passed through the detector. This allows one to calculate an equation of motion for the -resolved density-matrixClerk and Girvin (2004); Rammer et al. (2004); Wabnig et al. (2005), a quantum equivalent to the -resolved master equation approach widely used in the study of transport properties of classical nanomechanical systemsArmour (2004); Doiron et al. (2004).
To study the current cross correlations between two tunnel junction position detectors coupled to an oscillator, we use such a fully quantum approach. We label the detectors with the index and model each of them as a pair of metallic leads with constant density of states (in the energy range relevant to tunneling) coupled via the tunneling Hamiltonian . The Hamiltonian for one detector can therefore be written as a sum of a bath Hamiltonian describing the leads of junction and a tunneling Hamiltonian
where is a wave-vector in the right(left) lead. The coupling between the detector and the position of the oscillator is modeled by a linear dependence of the tunneling amplitude
In this equation, is the bare (oscillator-independent) tunneling amplitude of detector , is the position operator of the oscillator and is the part of the full tunneling amplitude detector that depends on the position of the oscillator. We allow for a general relative phase , describing the details of the coupling between the tunnel junction and the oscillator. Such a phase can in principle be controlled by a magnetic flux penetrating an extended tunnel junction consisting of a loop containing two junctions, one of which couples to the oscillator. Note that in our notation is dimensionless and has dimensions of one over length and that we assume for simplicity that the tunneling amplitudes do not depend on the single particle energies . The operator decreases (increases) , the number of charges that tunneled through junction . Its presence in the tunneling Hamiltonian allows one to keep track of the transport processes that occur during the evolution of the system.
We are interested in calculating the equation of motion for the reduced, -resolved, density matrix
where is the reduced density matrix that is obtained by tracing out the leads’ degrees of freedom from the full system density matrix. Within a Born-Markov approximation, the equation of motion of can be expressed as
where is the total tunneling Hamiltonian, the trace is on both pairs of leads, is the coupled density matrix of the two sets of leads and
with . In our system, the Born approximation corresponds to assuming that tunneling in both tunnel junctions is weak enough so that it can be treated using second-order perturbation theory. The Markov approximation, on the other hand, is valid as long as the typical correlation times in the leads () are much shorter than , i.e. the typical evolution time of the oscillator. In practice, this limits the applicability of the following results to the strongly biased case . This is experimentally feasible since typical oscillator frequencies are between 10 – 100 MHz and the measurements are done at a much larger bias voltage than these frequencies. Knobel and Cleland (2003); LaHaye et al. (2004); Naik et al. (2006); Flowers-Jacobs et al. (2007)
Since the leads of detector 1 are totally independent of those of detector 2, can be written as a tensor product of the density matrices describing each pair of leads . Also, as has no diagonal contribution in the basis that diagonalizes , the trace over leads of a quantity that is linear in vanishes. As a result of those two properties, the trace in Eq. (6) can be rewritten as a sum over two traces, each involving only one pair of leads
This effectively makes the two-detector problem two single-detector problems. The trace over the leads’ degrees of freedom is then carried out in the standard way. blum1996 ()
As mentioned above, we are interested in calculating the time-evolution of the -resolved density matrix. Thus, we have to calculate . We use the relations
as well as the equivalent identities for detector 2 in Eq. (6) to find the equation of motion for .
Following a counting-statistics approachBelzig (2003); Blanter (2006), it is particularly useful to express the equation of motion in terms of a counting field , the conjugate quantity to the transfered charge . Indeed, Fourier-transforming in the transfered-charge indices ,
leads to an equation of motion from which the time-dependence of all moments of (for example, ) can be determined. The current-current correlations can then be obtained by taking successive derivatives with respect to of the equation of motion of .
In the regime of weak coupling between the oscillator and the detectors, we can write the equation of motion of as
Since , taking corresponds to completely tracing out the charge degrees of freedom. In this case, one finds that is of Caldeira-Leggett formCaldeira and Leggett (1983a, b). We can thus identify the constants and as, respectively, the diffusion and damping constants induced by forward () or backward () propagating currents in detector . We can also identify as the average back-action force exerted on the oscillator by detector . We find explicitly
where the two inelastic tunneling rates are given by
involving a transfer of energy from the oscillator to the lead electron. We denote by the forward tunneling rate, i.e. the rate at which electrons tunnel in the direction favored by the voltage bias. The backward rate corresponds to the reverse process. In Eqs. (II) and (II), is the Fermi distribution function describing the local thermal equilibrium of the right lead of detector and is the same for the left lead.
Comparing these relations with the one derived in the single-detector caseClerk and Girvin (2004) shows that the full damping and diffusion coefficients governing the evolution of the oscillator are the sum of two single-detector contributions.
The Caldeira-Leggett form of Eq. (11) allows us to include the effect of direct coupling of the oscillator to the environment by adding detector-independent contributions and (where is the extrinsic quality factor of the mode) to the previously derived diffusion and damping constants. The evolution of the oscillator is then governed by the two constants and . For the specific case where the electronic temperature is zero and where , current will only be possible along the () direction, and both and will be zero. In this case one can also show that and that the diffusion parameters are given by .
The equation of motion for different moments of the oscillator can be evaluated by taking the trace of . More generally, equations of motion for combined moments of charge and oscillator quantities can be obtained by also considering derivatives with respect to the counting fields
Iii Single-detector case: Bound on the peak-to-background ratio
One of the main motivations for studying position measurements using cross-correlated detector outputs is to remove the bound on the peak-to-background ratio that appears in the single-detector case, just like in the case of a weak measurement of a two-level systemJordan and Büttiker (2005). In this section, we first review the results of Clerk and Girvin (CG) [Clerk and Girvin, 2004] for the single-detector configuration, in the case where one considers the dc-biased, , tunnel junction where the dependent tunneling phase is . We then carefully analyze the peak-to-background ratio and show that this quantity is bounded from above in the single-detector case, for finite bias voltage and oscillator displacement.
Using the single-detector analogue of Eq. (11), CG showed that, under the conditions mentioned above and to first non-vanishing order in , the current noise of a tunnel junction position detector is given by
where is the average of in the ground state of the (quantum) harmonic oscillator and
its power spectrum. The full current noise is the sum of the usual frequency-independent Poissonian (shot) noise and the contribution of interest due to the coupling of the junction to the oscillator. This second part is itself expressed as the difference of a classical part (which is proportional to ) and a quantum correction (which is proportional to ).
A relevant figure of merit of such detectors is the peak-to-background ratio : the ratio of the contribution of the oscillator to the full current noise at frequency over the unavoidable frequency-independent intrinsic detector noise. This ratio is maximal at and, in the case where one only considers the contribution in Eq. (18), was shown to be given by
where we used and introduced the dimensionless sensitivity parameter .foot_sens () At this point, one should proceed with care when maximizing with respect to the sensitivity parameter, as depends on through . Writing out explicitely all terms in that depend on , one finds that
is a non-monotonic function of the sensitivity parameter . For a given , one can then find an optimal value
for which is maximal
We can examine this result in two different limits. The first is when the damping is mainly detector-independent (), like in the case where the extrinsic quality factor of the resonator is low, . In this case, the maximal peak-to-background ratio,
is reached when the sensitivity parameter is extremely large. However, since the rightmost term of Eq. (24) is by definition small in this limit, the peak-to-background ratio cannot become extremely large when the extrinsic resonator damping dominates the detector-induced one.
Indeed, the real maximum of is reached when one considers the opposite limit of a very high resonator -factorClerk and Girvin (2004), . For , the peak-to-background ratio can be shown to obey
In the single-detector case and for given system parameters ( and ), the peak-to-background ratio is therefore always bounded whatever the strength of the coupling and the bound does not depend on and . As can be seen from Eq. (22), the peak-to-background ratio is in this second case maximal in the limit of vanishing coupling. While the optimal can be increased by increasing the bias voltage, we stress that our bound on denotes the optimal value of the peak-to-background reachable for a set of fixed system parameters.
The nature of the true bound on (i.e., the one found in the case ) is very similar to the Korotkov-Averin bound that arises in the context of a weak measurements of a qubit. To make this more apparent, we can derive this bound following the linear-response approach that has been used to derive the bound on in the measurement of two-levels systems, treating the detector as a position-to-current linear amplifier with responsivity (dimensionful gain) . As noted by CG, considering only the dominant term in Eq. (18) corresponds to writing . At resonance, the power spectrum is inversely proportional to the damping rate , in the same way that the response of the detector measuring a qubit is inversely proportional to the dephasing rate due to the measurement device. Moreover, in both cases one can show that the dephasing (damping rate) is proportional to the fluctuations of the bare input of the detectors. For a position detector in the high effective temperature limit , the detector-induced damping is indeed proportional to the symmetrized detector force noiseClerk (2004) , such that . Also, since for a tunnel junction detector there is no reverse gain and the real part of the cross-correlator vanishesClerk and Girvin (2004), the condition on quantum-limited efficiency of the position measurementClerk (2004)
becomes exactly the one used to derive the Korotkov-Averin bound . We then find that . Using in the tunnel junction system, this result corresponds exactly to Eq. (25), the bound previously derived using the equation-of-motion approach.
Iv Peak-to-background ratio in current cross correlations
Extending ideas from the qubit measurement problemJordan and Büttiker (2005), we now demonstrate how to eliminate the bound on the peak-to-background ratio in a position measurement. Calculating the current-current correlations between two tunnel-junction position detectors, we show that for cross correlation measurements, diverges. We also obtain analytical results for the cross correlations in two typical cases.
To calculate the current cross correlations, we use the generalized MacDonald formulaRodrigues and Armour (2005b); MacDonald (1962), a general result (valid for stationary processes) that provides a way, in the present case, to relate the symmetrized cross correlations to the Fourier sine-transform of the time-derivative of the covariance of and , the number of charges that tunneled through each junction. The generalized MacDonald formula reads
where we defined
In this last equation corresponds to and represents the coupled moment of and at time .
To proceed further, we restrict ourselves to the case of zero electronic temperature and dc-bias. In the following subsections, we analyze in detail the two different cases depicted schematically in Fig. 1. We have in mind that a realization of the setup shown in Fig. 1 is made in a similar way as the single-detector setup in Ref. Flowers-Jacobs et al., 2007. This means that the tunnel junctions correspond to atomic point contacts (formed by electromigration) which are separated by about 1 nm from the oscillator. In contrast, the two detectors are assumed to be separated from each other by at least 20 nm. Therefore, capacitive cross-talking between the detectors will play a negligible role. foot_cap ()
iv.1 In-phase configuration
We will first consider the case where both , the case where both tunnel junctions are located on the same side of the oscillator, cf. Fig. 1 a). To calculate the cross correlations, we use Eq. (17) (with ), to find that
and therefore that in this case is given by
where the double bracket denotes the covariance of two quantities: . This means that, to lowest order in , the full cross-correlated output of the detectors is given in this configuration by
The cross-correlated signal does not contain any oscillator-independent contribution. Using Eq. (17), a closed system of differential equations involving and can be generated. This system can be solved, using the boundary conditions and assuming that all averages that do not contain are time-independent and can therefore be evaluated in the stationary () limitArmour (2004).
Solving for the different covariances, we find that the current cross correlations can be written as
where we introduced the gains . Evidently, the cross-correlated output of the detectors (31) does not contain any frequency-independent background noise. The peak-to-background ratio therefore diverges for all values of , not because of an increased signal but due to the absence of background noise in this configuration.
For this type of measurement, a relevant figure of merit of the detection system is the ratio of the cross-correlated output over the frequency-independent noise power of individual detectors : , where . For our position detector, we find
where we used . From this inequality, we see that the maximal cross-correlated output is found for (i) twin-detectors (where ) and (ii) equal bias voltages . Also, like in the single-detector case, is maximal in the limit where there is no extrinsic oscillator damping and where the correction to the average current due to the coupling to the oscillator vanishes ().
Once again it is instructive to compare our value of for twin detectors with the equivalent result in the case of a weak measurement of a qubit using cross correlationsJordan and Büttiker (2005). In the latter case, the cross-correlated output was shown to be limited to of the single-detector signal due the increased (doubled) detector-induced dephasing. This is the same here.
iv.2 Out-of-phase detection
We can also analyze the case where one detector couples to and the other to , as would happen if the two detectors were located on opposite sides of the resonator (see Fig. 1). In terms of the tunneling phases , this corresponds to taking and . Using Eq. (27), the cross correlations are then given by
As the coupling between detector 1 and the oscillator is the same as in the previous case remains unchanged in this second configuration. The covariance on the other hand changes sign (but keeps the same norm) in this new configuration. Equation (34) then yields
The cross correlations in the second configuration are the same as in the first one, but of negative sign. From an amplifier point of view, this is easily explained since putting corresponds to transforming in . Finally, note that this configuration was analyzed for two single-electron transistor position detectors coupled to a classical oscillator, in Ref. [Rodrigues and Armour, 2005b] by Rodrigues and Armour. In their article, these authors only explicitly calculated zero-frequency cross correlations between the currents in both detectors, but they conjectured that, at the resonance frequency of the oscillator, this detector-configuration (corresponding to in our approach) should yield strong negative cross correlations, just like the ones predicted here.
V Bound on the added displacement noise
As shown in Sec. III, to derive the equivalent of the Korotkov-Averin bound in a position measurement, one needs to consider the full current noise, where no distinction is made between the signal due to the intrinsic equilibrium fluctuations of the oscillator and the remainder of the signal . This second contribution contains, amongst other things, the added signal due to heating of the oscillator by the detector. When trying to measure precisely the equilibrium fluctuations of a nanomechanical oscillator however, it is important to consider the two contributions separately: is exactly what you would like to measure while limits the sensitivity of the measurement. When using a single linear detector like the tunnel junction, this measurement sensitivity is quantum-mechanically bounded from belowClerk (2004).
When discussing this bound on added noise, one usually considers the added displacement noise, that corresponds to the added current noise referred back to the oscillator. We therefore introduce the total displacement noise , defined as
where is the -to- gain of the detector, is the part of the full displacement spectrum that arises due to the presence of the detector. In the relevant limit of a detector with a high power gain (), it was shown using general arguments that : the best possible detector therefore adds exactly as much noise as a zero-temperature bath of frequency Clerk (2004); caves1982 ().
Before discussing the limit on the added displacement noise in a cross correlation setup, it is helpful to describe how the quantum limit on is reached in a single-detector configuration. Let’s consider for definitiveness the experimentally relevant configuration where . For a measurement to be quantum limited, the effective temperature of the oscillator must not be dramatically higher than . This is natural, since added fluctuations due to the higher effective temperature are, by definition, unwanted back-action noise. In this regime, one therefore cannot expect to be close to the quantum limit unless . The regime of in which quantum-limited displacement sensitivity can be achieved is therefore very different from the one where the bound on the peak-to-background ratio can be reached.
Using the expression for the full current noise derived earlier (Eq. (18)), we write the full position noise as
where in the last line we introduced the oscillator’s response function . Splitting the second term into a detector dependent and independent part, we find
This way of writing the equilibrium fluctuations implies that we consider in , in agreement with our previous assumption that . The added noise contains three contributions, corresponding to the detector shot noise, the detector-induced heating of the oscillator and a correction () arising from the cross correlation between the detector output noise and the back-action force, , respectively. Explicitly, taking ,111The derived bound is therefore valid up to a positive correction of order . we obtain
For a fixed bias voltage, the relevant tunable parameter is directly the detector-oscillator coupling (and not the dimensionless sensitivity parameter , since is independent of ).222In principle, we could use the bias voltage as an optimization parameter. In this case, we would find that for ; there is no limit on the added position noise in the low power gain regime (