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arXiv:2406.00176

Topological weak-measurement-induced geometric phases revisited

Graciana Puentes

Published May 31, 2024• 0 citations
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Abstract

We present an analytical and numerical study of a class of geometric phase induced by weak measurements. In particular, we analyze the dependence of the geometric phase on the winding ($W$) of the polar angle ($φ$), upon a sequence of $N$ weak measurements of increased magnitude ($c$), resulting in the appearance of a multiplicity of critical measurement-strength parameters where the geometric phase becomes stochastic. Adding to the novelty of our approach, we not only analyze the weak-measurement induced geometric phase by a full analytic derivation, valid in the quasicontinuous limit ($N \rightarrow \infty$), but also we analyze the induced geometric phase numerically, thus enabling us to unravel the finite-$N$ interplay of the geometric phase with the measurement strength parameter, and its stability to perturbations in the measurements protocol.

DOI
10.48550/arxiv.2406.00176

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We present an analytical and numerical study of a class of geometric phase induced by weak measurements. In particular, we analyze the dependence of the geometric phase on the winding (WWW) of the polar angle (φφφ), upon a sequence of NNN weak measurements of increased magnitude (ccc), resulting in the appearance of a multiplicity of critical measurement-strength parameters where the geometric phase becomes stochastic. Adding to the novelty of our approach, we not only analyze the weak-measurement induced geometric phase by a full analytic derivation, valid in the quasicontinuous limit (N→∞N \rightarrow \inftyN→∞), but also we analyze the induced geometric phase numerically, thus enabling us to unravel the finite-NNN interplay of the geometric phase with the measurement strength parameter, and its stability to perturbations in the measurements protocol.
Abstract We present an analytic and numerica study of a class of geometric phase induced by weak measurements. In particular, we analyze the dependence of the geometric phase on the winding W of the polar angle φ, upon a sequence of N weak measurements of increased magnitude (c), resulting in the appearance of a multiplicity of critical measurement-strength parameters where the geometric phase becomes stochastic. Adding to the novelty of our approach, we not only analyze the weak-measurement-induced geometric phase by a full analytic derivation, valid in the quasi-continuous limit (N → ∞), but also we analyze the induced geometric phase numerically, thus enabling us to unravel the finite-N interplay of the geometric phase with the measurement strength parameter, and its stability to perturbations in the measurements protocol.
We present an analytical and numerical study of a class of geometric phase induced by weak measurements. In particular, we analyze the dependence of the geometric phase on the winding ( W ) of the polar angle ( φ ), upon a sequence of N weak measurements of increased magnitude (c) , resulting in the appearance of a multiplicity of critical measurement-strength parameters where the geometric phase makes a |π| discrete jump. Adding to the novelty of our approach, we not only analyze the weak-measurement-induced geometric phase by a full analytic derivation, valid in the quasi-continuous limit (N→∞) , but also we analyze the induced geometric phase numerically, thus enabling us to unravel the finite- N interplay of the geometric phase with the measurement-strength parameter, and its stability to perturbations in the measurements protocol.
Coherent steering of a quantum state, induced by a sequence of weak measurements, has become an active area of theoretical and experimental study. For a closed steered trajectory, the underlying phase factors involve both geometrical and dynamical terms. Furthermore, considering the reversal of the order of the measurement sequence, such a phase comprises a symmetric and an antisymmetric term. Superseding common wisdom, we show that the symmetric and the antisymmetric components do not correspond to the dynamical and geometrical parts, respectively. Addressing a broad class of measurement protocols, we further investigate the dependence of the induced phases on the measurement parameters (e.g., the measurement strength). We find transitions between different topologically distinct sectors, defined by integer-valued winding numbers, and show that the transitions are accompanied by diverging dephasing. We propose experimental protocols to observe these effects.
Coherent steering of a quantum state, induced by a sequence of weak measurements, has become an active area of theoretical and experimental study. For a closed steered trajectory, the underlying phase factors involve both geometrical and dynamical terms. Furthermore, considering the reversal of the order of the measurement sequence, such a phase comprises a symmetric and an antisymmetric term. Superseding common wisdom, we show that the symmetric and the antisymmetric components do not correspond to the dynamical and geometrical parts, respectively. Addressing a broad class of measurement protocols, we further investigate the dependence of the induced phases on the measurement parameters (e.g., the measurement strength). We find transitions between different topologically distinct sectors, defined by integer-valued winding numbers, and show that the transitions are accompanied by diverging dephasing. We propose experimental protocols to observe these effects.
Measurement plays a quintessential role in the control of quantum systems. Beyond initialization and readout which pertain to projective measurements, weak measurements, in particular through their back action on the system, may enable various levels of coherent control. The latter ranges from observing quantum trajectories to state dragging and steering. Furthermore, just like the adiabatic evolution of quantum states that is known to induce the Berry phase, sequential weak measurements may lead to path-dependent geometric phases. Here we measure the geometric phases induced by sequences of weak measurements and demonstrate a topological transition in the geometric phase controlled by measurement strength. This connection between weak measurement-induced quantum dynamics and topological transitions reveals subtle topological features in measurement-based manipulation of quantum systems. Our protocol could be implemented for classes of operations (e.g., braiding) that are topological in nature. Furthermore, our results open new horizons for measurement-enabled quantum control of many-body topological states.
The state of a quantum system, adiabatically driven in a cycle, may acquire a measurable phase depending only on the closed trajectory in parameter space. Such geometric phases are ubiquitous and also underline the physics of robust topological phenomena such as the quantum Hall effect. Equivalently, a geometric phase may be induced through a cyclic sequence of quantum measurements. We show that the application of a sequence of weak measurements renders the closed trajectories, hence the geometric phase, stochastic. We study the concomitant probability distribution and show that, when varying the measurement strength, the mapping between the measurement sequence and the geometric phase undergoes a topological transition. Our finding may impact measurement-induced control and manipulation of quantum states-a promising approach to quantum information processing. It also has repercussions on understanding the foundations of quantum measurement.