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arXiv:2006.07028 · Physical Review A

Ancilla-assisted probing of temporal quantum correlations of large spins

Michael Kästner

Published Aug 13, 2020• 2 citations
arXivHTMLPDF

Abstract

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-$s$ ancilla degree of freedom. A potential application of such a measurement protocol is the probing of an array of Bose-Einstein condensates by light.

DOI
10.1103/physreva.102.022212

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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≫sl\ensuremath{\gg}sl\ensuremath≫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.
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.
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.
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.
Measurement plays a crucial role in a quantum system beyond just learning about the system state: it changes the post-measurement state and hence influences the subsequent time evolution; further, measurement can even create entanglement in the post-measurement conditional state. In this work, we study how careful choice of parameters for a typical measurement process on cold atoms systems -- phase contrast imaging -- has a strong impact on both what the experimentalist observes but also on the backaction the measurement has on the system, including the creation and diffusion of quasiparticles emerging from the quantum many-body dynamics. We focus on the case of a Bose-Einstein-condensate array, in the low-temperature and low-momentum limit. Our theoretical investigation reveals regimes where the imaging light probes either the bare particle or quasiparticle dynamics. Moreover, we find a path to selectively measuring quasiparticle modes directly, as well as controlling over the measurement-induced creation and diffusion of quasiparticles into different momentum states. This lays a foundation for understanding the effects of both experimental approaches for probing many-body systems, but also more speculative directions such as observable consequences of `spontaneous collapse' predictions from novel models of quantum gravity on aspects of the Standard Model.
We consider a small F=1F=1F=1 spinor condensate inside an optical cavity driven by an optical probe field, and subject the output of the probe to a homodyne detection, with the goal of investigating the effect of measurement backaction on the spin dynamics of the condensate. Using the stochastic master equation approach, we show that the effect of backaction is sensitive to not only the measurement strength but also the quantum fluctuation of the spinor condensate. The same method is also used to estimate the atom numbers below which the effect of backaction becomes so prominent that extracting spin dynamics from this cavity-based detection scheme is no longer practical.
We suggest a simple experimental method for probing antiferromagnetic spin correlations of two-component Fermi gases in optical lattices. The method relies on a spin selective Raman transition to excite atoms of one spin species to their first excited vibrational mode where the tunneling is large. The resulting difference in the tunneling dynamics of the two spin species can then be exploited, to reveal the spin correlations by measuring the number of doubly occupied lattice sites at a later time. We perform quantum Monte Carlo simulations of the spin system and solve the optical lattice dynamics numerically to show how the timed probe can be used to identify antiferromagnetic spin correlations in optical lattices.