Noninvasive measurement of dynamic correlation functions

Philipp Uhrich, Salvatore Castrignano, Hermann Uys, and Michael Kastner

Phys. Rev. A 96, 022127 – Published 21 August 2017

Abstract

The measurement of dynamic correlation functions of quantum systems is complicated by measurement backaction. To facilitate such measurements we introduce a protocol, based on weak ancilla-system couplings, that is applicable to arbitrary (pseudo)spin systems and arbitrary equilibrium or nonequilibrium initial states. Different choices of the coupling operator give access to the real and imaginary parts of the dynamic correlation function. This protocol reduces disturbances due to the early-time measurements to a minimum, and we quantify the deviation of the measured correlation functions from the theoretical, unitarily evolved ones. Implementations of the protocol in trapped ions and other experimental platforms are discussed. For spin- $1 / 2$ models and single-site observables we prove that measurement backaction can be avoided altogether, allowing for the use of ancilla-free protocols.

Received 24 November 2016
Revised 16 January 2017

DOI:https://doi.org/10.1103/PhysRevA.96.022127

Physics Subject Headings (PhySH)

Quantum measurements Trapped ions

Nonequilibrium lattice models Strongly correlated systems

Quantum correlations, foundations & formalism

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Philipp Uhrich^1,2, Salvatore Castrignano³, Hermann Uys^2,4, and Michael Kastner^1,2,*

¹National Institute for Theoretical Physics (NITheP), Stellenbosch 7600, South Africa
²Institute of Theoretical Physics, Department of Physics, University of Stellenbosch, Stellenbosch 7600, South Africa
³Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
⁴Council for Scientific and Industrial Research, National Laser Centre, Pretoria, Brummeria, 0184, South Africa

^*kastner@sun.ac.za

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Vol. 96, Iss. 2 — August 2017

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Images

Figure 1
Error analysis for the example of Sec. 4 for parameter values $(t_{1}, t_{2}) = (1, 10), α_{1} = α_{2} = π / 3$ , and $(θ_{1}, θ_{2}) = (π / 7, π / 5)$ . All plots show relative errors ${\tilde{ε}}_{sys} = ε_{sys} / | C |$ , and similarly for statistical and total errors. Left: Exact systematic error ${\tilde{ε}}_{sys}$ (red curve starting from the origin) and estimates (29) of ${\tilde{ε}}_{stat}$ for sample sizes $n = 10^{2}, 10^{3}, 10^{4}$ (solid, dashed, dotted black curves, respectively). The functional dependence of the bounds on the coupling time $λ$ is clear, with the systematic error increasing with $λ$ , while the stochastic error is inversely proportional to $λ$ . Center: Estimate for ${\tilde{ε}}_{tot}$ (line) together with numerically measured values for sample size $n = 10^{4}$ (dots). The estimated minimum error is $33 %$ at $λ^{*} = 0.42$ . The error estimate captures the qualitative behavior of the numerical data and is consistently larger. Right: Log-log plot of the estimated minimum error ${\tilde{ε}}_{tot} (λ^{*})$ (dots) and its position $λ^{*}$ (squares) as a function of sample size $n$ . Straight lines indicate that the performance of the noninvasive measurement protocol improves like a power law with $n$ .
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Figure 2
Predicted upper bounds on the relative error ${\tilde{ε}}_{tot}$ for measurements of correlations $C (t_{i}, t_{j})$ within the cNIMP for times $(t_{1}, t_{2}, t_{3}) = (0, 1, 10)$ , initial system state parameters as in Fig. 1, and sample size $n = 10^{5}$ . Left: Estimated total relative error for measurements of $C_{n}^{λ} (t_{1}, t_{2})$ . The black curves are included to guide the reader's eye, and their intersection indicates the minimum error of $37 %$ at $(λ_{1}^{*}, λ_{2}^{*}) = (0.40, 0.41)$ . Right: Corresponding prediction for measurements of $C_{n}^{λ} (t_{1}, t_{3})$ , exhibiting a minimum error of $25 %$ at $(λ_{1}^{*}, λ_{2}^{*}) = (0.37, 0.00)$ . Although measurement of this correlation is performed in the cNIMP by observing states of only the first ancilla (coupled to lattice site 1 at $t_{1}$ ) and the spin at site 2, the additional coupling of the second ancilla to site 2 at intermediate time $t_{2}$ increases the systematic error, which causes the total error to increase with $λ_{2}$ . This reflects, and is due to, the fact that the cNIMP perturbs the system's dynamics more strongly.
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Figure 3
Minimum total error ${min}_{λ} {\tilde{ε}}_{tot} (λ)$ as a function of sample size $n$ . Times and system parameters are as in Fig. 2. Left: For $C_{n}^{λ} (t_{1}, t_{2})$ the minimum error of the sNIMP (lower black line) decreases faster than that of the cNIMP (upper red line). As a result, to construct an estimator with a total error of $10 %$ or less, sample sizes in the sNIMP must be at least $10^{6}$ , which is about two orders of magnitude smaller than for the cNIMP. Right: For measurements of $C_{n}^{λ} (t_{1}, t_{3})$ the minimum error of either protocol decreases at the same rate with the sample size; however the errors of the cNIMP (upper red line) are consistently larger than those of the sNIMP (lower black line). Results for $C_{n}^{λ} (t_{2}, t_{3})$ are similar (not shown).
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