arXiv subjects

Zhan Zheng

Publications and source records attributed to Zhan Zheng.

4 recordsLinked to original sources

arXiv2025

Approximate universality and large measurement gain of Rabi model in a linear potential under strong Doppler broadening

Harnessing quantum resources in the atomic external degrees of freedom, particularly matter-wave states with large momentum broadening, holds significant potential for enhancing the sensitivity of Kasevich-Chu atom gravimeters at the standard quantum limit. However, a fully quantum-mechanical investigation of the critical Doppler effect inherent to this approach remains lacking. Employing SU(2) Lie group theory, we derive a generic scalar Riccati equation governing the unitary dynamics of the Rabi model within a linear potential and analyze the Doppler effect's impact on Rabi oscillations because of the strong coupling between the internal and external states. Furthermore, by integrating Fisher information theory, we demonstrate the approximate universality and high metrological gain of phase-rotation measurement protocols under strong Doppler broadening induced by large-momentum width. This theoretical work provides insightful implications for broader generalization, such as extensions to finite-temperature scenarios or multi-pulse sequences, exemplified by the $π/2-π-π/2$ pulse sequence characteristic of Kasevich-Chu atom gravimeters. Thus this study lays a theoretical foundation for developing high-sensitivity, noise-resistant atom gravimeters that leverage external-state quantum resources.

quant-ph↗

arXiv2023

On the convergence conditions of Laplace importance sampling with randomized quasi-Monte Carlo

The study further explores randomized QMC (RQMC), which maintains the QMC convergence rate and facilitates computational efficiency analysis. Emphasis is laid on integrating randomly shifted lattice rules, a distinct RQMC quadrature, with IS,a classic variance reduction technique. The study underscores the intricacies of establishing a theoretical convergence rate for IS in QMC compared to MC, given the influence of problem dimensions and smoothness on QMC. The research also touches on the significance of IS density selection and its potential implications. The study culminates in examining the error bound of IS with a randomly shifted lattice rule, drawing inspiration from the reproducing kernel Hilbert space (RKHS). In the realm of finance and statistics, many problems boil down to computing expectations, predominantly integrals concerning a Gaussian measure. This study considers optimal drift importance sampling (ODIS) and Laplace importance sampling (LapIS) as common importance densities. Conclusively, the paper establishes that under certain conditions, the IS-randomly shifted lattice rule can achieve a near $O(N^{-1})$ error bound.

math.NA↗

arXiv2022

On the error rate of importance sampling with randomized quasi-Monte Carlo

Importance sampling (IS) is valuable in reducing the variance of Monte Carlo sampling for many areas, including finance, rare event simulation, and Bayesian inference. It is natural and obvious to combine quasi-Monte Carlo (QMC) methods with IS to achieve a faster rate of convergence. However, a naive replacement of Monte Carlo with QMC may not work well. This paper investigates the convergence rates of randomized QMC-based IS for estimating integrals with respect to a Gaussian measure, in which the IS measure is a Gaussian or $t$ distribution. We prove that if the target function satisfies the so-called boundary growth condition and the covariance matrix of the IS density has eigenvalues no smaller than 1, then randomized QMC with the Gaussian proposal has a root mean squared error of $O(N^{-1+ε})$ for arbitrarily small $ε>0$. Similar results of $t$ distribution as the proposal are also established. These sufficient conditions help to assess the effectiveness of IS in QMC. For some particular applications, we find that the Laplace IS, a very general approach to approximate the target function by a quadratic Taylor approximation around its mode, has eigenvalues smaller than 1, making the resulting integrand less favorable for QMC. From this point of view, when using Gaussian distributions as the IS proposal, a change of measure via Laplace IS may transform a favorable integrand into unfavorable one for QMC although the variance of Monte Carlo sampling is reduced. We also give some examples to verify our propositions and warn against naive replacement of MC with QMC under IS proposals. Numerical results suggest that using Laplace IS with $t$ distributions is more robust than that with Gaussian distributions.

math.NA↗

arXiv2013

Do entangled photons induce two-photon two-atom transitions more efficiently than other states of light ?

We study in this paper the efficiency of different two-photon states of light to induce the simultaneous excitation of two atoms of different kinds when the sum of the energies of the two photons matches the sum of the energies of the two atomic transitions, while no photons are resonant with each individual transition. We find that entangled two-photon states produced by an atomic cascade are indeed capable of enhancing by a large factor the simultaneous excitation probability as compared to uncorrelated photons, as predicted some years ago by Muthukrishnan et al, but that several non-entangled, separable, correlated states, produced either by an atomic cascade or parametric down conversion, or even appropriate combinations of coher- ent states, have comparable efficiencies. We show that the key ingredient for the increase of simultaneous excitation probability is the presence of strong frequency anti-correlation and not time correlation nor time-frequency entanglement.

quant-ph↗