Improved mirror position estimation using resonant quantum smoothing
- Trevor A Wheatley^{1, 2}Email author,
- Mankei Tsang^{3, 4},
- Ian R Petersen^{1} and
- Elanor H Huntington^{1, 2, 5}
DOI: 10.1140/epjqt/s40507-015-0026-0
© Wheatley et al.; licensee Springer. 2015
Received: 26 February 2015
Accepted: 23 April 2015
Published: 20 May 2015
Abstract
Quantum parameter estimation, the ability to precisely obtain a classical value in a quantum system, is very important to many key quantum technologies. Many of these technologies rely on an optical probe, either coherent or squeezed states to make a precise measurement of a parameter ultimately limited by quantum mechanics. We use this technique to theoretically model, simulate and validate by experiment the measurement and precise estimation of the position of a cavity mirror. In non-resonant systems, the achieved estimation enhancement from quantum smoothing over optimal filtering has not exceeded a factor two, even when squeezed state probes were used. Using a coherent state probe, we show that using quantum smoothing on a mechanically resonant structure driven by a resonant forcing function can result significantly greater improvement in parameter estimation than with non-resonant systems. In this work, we show that it is possible to achieve a smoothing improvement by a factor in excess of three times over optimal filtering. By using intra-cavity light as the probe we obtain finer precision than has been achieved with the equivalent quantum resources in free-space.
Keywords
quantum smoothing quantum parameter estimation cavity mirror positionPACS Codes
42.50.Dv 03.65.Ta 03.67.-a1 Background
1.1 Introduction
The field of quantum metrology can be described as using quantum resources to enhance measurement precision beyond that achievable with purely classical resources. There are a number of resources that are available such as entanglement [1], superposition [2] and squeezing [3]. There are also tools such as adaptive feedback [4] and quantum smoothing [5] to further exploit the quantum enhancement. Quantum parameter estimation (QPE) is a related discipline that is focussed more specifically on precisely estimating the classical parameters of a quantum system. The importance of QPE to fields such as gravitational wave detection [6], quantum metrology [7, 8], quantum control [9] and opto-mechanical force [10, 11] sensing has been well established. Technological evolution in recent times has seen an increase in the range of pertinent architectures where quantum mechanical effects have become relevant [11–13]. There have also been experimental demonstrations of key advances in QPE. For example, in optical phase estimation we saw successive lowering of achievable mean square estimation error by the use of adaptive feedback [14] and adaptive feedback was combined with smoothing [15] to achieve a further reduction. With the addition of phase quadrature squeezing the limit was once more lowered [16]. A recent extension of these QPE techniques to a more macroscopic domain uses an optical probe beam to obtain an estimate of the position, momentum and force acting on a free-space mirror [17].
As was shown in [5], an increase in estimation precision relative to filtering is expected when quantum smoothing is used. Previous work here has only considered first-order forcing noise processes with non-resonant interactions between the forcing functions and the system. For such setups, results to date have yet to show a greater than two improvement of the smoothed estimate over the filtered equivalent. An interesting open question therefore remains as to whether this factor of two improvement is an upper limit for more complicated systems. So in this work we consider a higher order forcing function that is Lorentzian in frequency. Additionally, we consider resonant interactions between the forcing function and the system (mirror) with the centre frequency of the Lorentzian aligned with the peak of a mechanical resonance. Here our theory suggests that for more realistic resonant systems driven by less benign processes, the factor of two improvement with smoothing can be improved on significantly. We present theory and simulations results showing a greater than two smoothing improvement over the equivalent optimal filtered estimate obtained. The results of the simulations both verify and extend beyond the theoretical analysis and we present experimental results to verify the simulations.
1.2 Theory - optics
To date the experimental demonstrations of smoothing have focussed on systems where the probe beam (even when quantum enhanced) interactions are in free-space. It is relatively well known that optical cavities can be used to enhance measurement precision. In the context of this work the strong intra-cavity field in an optical cavity provides more photon interaction with the parameter to be estimated. As each photon potentially probes the parameter many times the cumulative effect gives higher sensitivity without need for extra photon resources. Because the experimental validation makes use of the enhancements in sensitivity achievable by the use of optical cavities, the theory and simulation assume an intra-cavity probe.
1.3 Theory - smoother
1.4 Plant and forcing function
System parameters for simulation and experimental validation of the simulator.
Parameter | Simulation | Experiment | Description |
---|---|---|---|
h(ω) | \(\frac{c_{1}s + c_{2}\omega_{m}}{s^{2} + \beta s + \omega_{m}^{2}} e^{-s\tau}\) | \(\frac{c_{1}s + c_{2}\omega_{m}}{s^{2} + \beta s + \omega_{m}^{2}}e^{-s\tau}\) | Plant transfer function |
\(S_{f}(\omega)\) | \(\frac{Q}{2}[\frac{1}{(\omega-\omega_{i})^{2} + \gamma ^{2}} + \frac{1}{(\omega+\omega_{i})^{2} + \gamma^{2}}]\) | \(\frac{Q}{2}[\frac{1}{(\omega-\omega_{i})^{2} + \gamma^{2}} + \frac {1}{(\omega +\omega_{i})^{2} + \gamma^{2}}]\) | Forcing function PSD |
R | 7.7 × 10^{−11} | 7.7 × 10^{−11} | Measurement noise magnitude term where \(R\delta(t-t') = \sigma(\eta(t),\eta(t))\), η(t) is white Gaussian noise |
Q | 7.4 × 10^{−2} | 7.4 × 10^{−2} | Forcing function magnitude term where \(Q\delta(t-t') = \sigma(\xi(t),\xi(t))\), ξ(t) is white Gaussian noise |
γ | 1,333 | 1,333 | Forcing function damping factor |
\(\omega_{m}\) | 2π⋅7,930 | 2π⋅7,930 | Frequency of PZT resonance |
\(\omega_{i}\) | 2π⋅7,930 | 2π⋅7,930 | Frequency of forcing function resonance |
\(c_{1}\) | 131 | 131 | Constant |
\(c_{2}\) | 196 | 196 | Constant |
β | 2,494 | 2,494 | PZT resonance damping factor |
τ | 0 and 18.5 × 10^{−6} | system | Time delay |
F | 250 kS/s | 250 kS/s | Sample rate |
N | 2^{15} | 2^{16} | Number of samples |
Averages | 21 | 5 | Number of data sets averaged |
2 Results and discussion
2.1 Simulation
We developed a numerical simulator to test the theory and provide a baseline against which an experimental testbed can be compared. The numerical results can also be used to inform subsequent experiments. The simulation was done using Simulink and the parameter values are shown in Table 1. The input and measurement noise processes were entered as floating point arrays from the workspace. The plant (\(h(t)\)) and controller (\(h_{c}(t)\)) transfer functions were implemented using transfer function blocks with the numerator and denominator coefficients extracted from the workspace. The Simulink model provides the parameter \(v_{z}(t)\) (see Figure 2) for the smoother. In the experimental validation discussed later a cut-down version of the Simulink model was used to process the experimental data to obtain \(v_{z}(t)\) from the recorded experimental values of \(v_{y}(t)\) and \(v_{c}(t)\). The smoothing (both simulation and experimental) was implemented in the frequency domain using the Fourier transforms of the relevant parameters from the workspace and the Simulink model. The goal of the simulation was to find whether there exists a range of parameters (preferably experimentally feasible) that allow for a greater than two improvement over the optimal filtered estimate.
It is quite clear in Figure 6 that there is good agreement between simulation and theory, with most data points agreeing within error bars. It is noted that the model assumes that the optical cavity remains linear and as such does not account for large values of the non-linear detuning term (Δ in equation (1)). The theory and simulation show that a greater than two smoothing enhancement for a resonant process acting on a mechanically resonant structure is achievable for a wide range of parameters.
2.2 Experiment
3 Conclusions
We have developed theory describing resonance enhanced mirror position estimation of a cavity mirror using quantum smoothing. This theory has been used to design a numerical simulation model, which we have experimentally validated. We have demonstrated that performing quantum smoothing on a mechanically resonant structure when driven by a resonant forcing function gives greater enhancement in precision when compared to non-resonant systems. When driven by a Lorentzian process we achieved a simulated improvement in precision of greater than two times better than the equivalent optimal filter, which is consistent with theory. We have also experimentally validated the simulation using an experimental testbed. The simulations have identified a good parameter regime where greater improvement should be possible in future experiments. With future improvements in the system we expect to see further precision enhancements. In future work it should be possible to demonstrate further improvement in precision by the incorporation of quantum enhancement using a phase squeezed probe beam. These results demonstrate the advantage of resonances when performing quantum parameter estimation. This is an initial proof of concept that may have applications in areas where mechanical systems are being measured in quantum limited domains.
Declarations
Acknowledgements
This work was supported financially by the Australian Research Council, Grant No. CE110001029, DP1094650, FL110100020 and DP109465.
MT acknowledges support from the Singapore National Research Foundation under NRF Grant No. NRF-NRFF2011-07.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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