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arXiv:1611.08123 · Physical Review A

Noninvasive measurement of dynamic correlation functions

Philipp Uhrich, Salvatore Castrignano, Hermann Uys, Michael Kästner

Published Aug 21, 2017• 25 citations
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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.

DOI
10.1103/physreva.96.022127

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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-1/21/21/2 models and single-site observables we prove that measurement backaction can be avoided altogether, allowing for the use of ancilla-free protocols.
ENGLISH ABSTRACT : Dynamic correlations hψ| O1(t1)O2(t2)|ψi of quantum observables are useful quantities for the study of quantum dynamics. Attempts at measuring these correlations are however complicated, due to the measurement backaction (wave function collapse) incurred during measurements at the early time t1. We propose a noninvasive measurement protocol, based on a weak ancilla target coupling, which reduces this backaction at t1. We show that both real and imaginary parts of the desired correlation can be extracted through appropriate choices of the initial ancilla state and of the ancilla target coupling Hamiltonian. The protocol is applicable to arbitrary (pseudo)spin systems with arbitrary (non)equilibrium initial states. Errors arising in experimental implementations are analysed, and we show that deviations from the desired correlation can be minimised through an optimal choice of the ancilla target coupling time. Implementation in linear ion trap experiments is discussed. We derive the positive-operator-valued measure which describes the noninvasive measurement at t1. For dynamic correlations of single-site spin-1/2 observables, this operator formalism shows that measurement backaction is of no concern. Real parts can be obtained with projective measurements of the target at t1 and t2. Imaginary parts are obtained by performing a local rotation of the target at t1, followed by a projective measurement at t2. These ancilla-free protocols are theoretically simpler than the noninvasive measurement protocol, but remain experimentally challenging. Rotations and projections performed at t1 may be subject to noise, which propagates into the measured correlation. We use Lieb-Robinson theory to bound the size of the resulting error terms. An analysis of the spatio-temporal behaviour of these errors provides guidance for experimental implementation of the ancilla-free measurement protocols.
We introduce a protocol that gives access to out-of-time-ordered correlation functions in many-body quantum systems. Unlike other such protocols, our proposal, which can be applied to arbitrary initial states, neither requires ancilla degrees of freedom to the quantum system of interest, nor has the need for randomized measurements. Nontrivial experimental capabilities required to implement the protocol are single-site measurements, single-site rotations, and backwards time evolution. To exemplify the implementation of the protocol, we put forward a strategy for Hamiltonian sign inversion $H\ensuremath{\rightarrow}\ensuremath{-}H$ in arrays of Rydberg-dressed atoms. In this way, a complete and practical toolbox is obtained for the measurement of out-of-time-ordered correlations in equilibrium and nonequilibrium situations.
Experimental access to many-body quantum systems is often limited by measurement backaction, and key dynamical properties are typically obtained by perturbing a system and measuring its response. Here we replace this active paradigm with a minimally invasive protocol based on a pair of weak quantum measurements that leverages measurement backaction as a strength. By correlating time-separated measurements with the first detecting fluctuations -- of any sort -- and the second tracking their time evolution, our method directly measures dynamical correlation functions without external perturbation. We demonstrate this technique in an atomic Bose-Einstein condensate using phase-contrast imaging to obtain the two-time density-density correlation function known as the Van Hove function and, through its Fourier transform, the dynamical structure factor. Due to the role of spatial correlations in scattering, these quantities underpin neutron and X-ray scattering and atomic Bragg spectroscopy. This approach is broadly applicable, providing access to correlation functions between any pair of observables amenable to weak measurement, thereby going beyond the capabilities of conventional strong measurements. We further isolate the role of quantum backaction through Aharonov's post-selection-based quantum weak values.
When measuring quantum spins at two or more different times, the later measurements are affected by measurement backaction occurring due to the earlier measurements. This makes the measurement of temporal quantum correlation functions challenging. In this paper, I propose a measurement protocol that mitigates the effect of measurement backaction by exploiting spin selection rules. I show that, under suitable conditions, the effect of measurement backaction on two-time quantum correlations becomes negligible when probing a system consisting of spins with large spin quantum numbers $l\ensuremath{\gg}s$ by coupling it to a spin-sss ancilla degree of freedom. A potential application of such a measurement protocol is the probing of an array of Bose-Einstein condensates by light.
Generalized quantum measurement schemes are described by positive operator-valued measures going beyond the projection postulate, which predicts the instantaneous collapse of the systems wave function. This allows to take the noninvasive limit and investigate the correlations of such weak measurements which facilitate the observation of non-commuting observables within the same system. We propose a scheme in which the detector is coupled to the measured system for a finite time, as it is the case in many real setups. This leads to non-Markovian effects appearing by memory functions which are related to symmetric and antisymmetric correlators of the detector variables. We investigate these functions addressing the role of equilibrium and non-equilibrium detectors and how they differ from and could realize the standard Markovian measurement respectively. The latter scheme leads to the symmetrized operator order (aka Keldysh ordering), which is widely used in quantum measurement discussions. We show that the non-Markovian measurement scheme yields information beyond the standard approach, allowing e.g. for the prove of the non classical nature of a system (quasiprobability statistics) by second-order correlation functions.
We consider two protocols for the measurement of the temporal correlation functions of a dichotomic variable QQQ appearing in Leggett-Garg-type inequalities. The protocols measure solely whether QQQ has the same or a different sign at the end of a given time interval, thereby measuring no more than is required for determination of the correlation function. They are inspired, in part, by a decoherent histories analysis of the two-time histories of QQQ, which yields a number of useful insights, although the protocols are ultimately expressed in macrorealistic form independent of quantum theory. The first type involves an ancilla coupled to the system with two sequential controlled-not (cnot) gates, and the two-time histories of the system (whose probabilities yield the correlation function) are determined in a single final time measurement of the ancilla. It is noninvasive for special choices of initial system states and partially invasive for more general choices. Modified Leggett-Garg-type inequalities which accommodate the partial invasiveness are discussed. The quantum picture of the protocol shows that for certain choices of the primary system initial state, the final state is unaffected by the two cnot gate interactions, hence the protocol is undetectable with respect to final system-state measurements, although it is still invasive at intermediate times. This invasiveness can be reduced with different choices of ancilla states and the protocol is then similar in flavor to a weak measurement. The second type of protocol is based on the fact that the behavior of QQQ over a time interval can be determined from knowledge of the dynamics together with a measurement of certain initial (or final) data. Its quantum version corresponds to the known fact that when sets of histories are decoherent, their probabilities may be expressed in terms of a record projector, hence the two-time histories in which QQQ has the same or a different sign can be determined by a single projective measurement. The resulting protocol resembles the decay-type protocol proposed by Huelga and collaborators (which is noninvasive but requires a stationarity assumption).