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arXiv:1208.0657 · New Journal of Physics

Fundamental constraints on particle tracking with optical tweezers

Michael A. Taylor, Joachim Knittel, Warwick P. Bowen

Published Feb 11, 2013• 29 citations
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Abstract

A general quantum limit to the sensitivity of particle position measurements is derived following the simple principle of the Heisenberg microscope. The value of this limit is calculated for particles in the Rayleigh and Mie scattering regimes, and with parameters which are relevant to optical tweezers experiments. The minimum power required to observe the zero-point motion of a levitating bead is also calculated, with the optimal particle diameter always smaller than the wavelength. We show that recent optical tweezers experiments are within two orders of magnitude of quantum limited sensitivity, suggesting that quantum optical resources may soon play an important role in high sensitivity tracking applications.

DOI
10.1088/1367-2630/15/2/023018

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A general quantum limit to the sensitivity of particle position measurements is derived following the simple principle of the Heisenberg microscope. The value of this limit is calculated for particles in the Rayleigh and Mie scattering regimes, and with parameters which are relevant to optical tweezers experiments. The minimum power required to observe the zero-point motion of a levitating bead is also calculated, with the optimal particle diameter always smaller than the wavelength. We show that recent optical tweezers experiments are within two orders of magnitude of quantum limited sensitivity, suggesting that quantum optical resources may soon play an important role in high sensitivity tracking applications.
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Particle sensing in optical tweezers systems provides information on the position, velocity, and force of the specimen particles. The conventional quadrant detection scheme is applied ubiquitously in optical tweezers experiments to quantify these parameters. In this paper, we show that quadrant detection is nonoptimal for particle sensing in optical tweezers and propose an alternative optimal particle sensing scheme based on spatial homodyne detection. A formalism for particle sensing in terms of transverse spatial modes is developed and numerical simulations of the efficacies of both quadrant and spatial homodyne detection are shown. We demonstrate that 1 order of magnitude improvement in particle sensing sensitivity can be achieved using spatial homodyne over quadrant detection.
Tracking small transverse displacements of an optical beam with ultra-high accuracy is a fundamental problem underlying numerous important applications ranging from pointing, acquisition and tracking for establishing a lasercom link, to atomic force microscopy for imaging with atomic-scale resolution. Determining what is the optimal quantum-optical probe and the best achievable sensitivity of measuring a small transverse optical beam displacement, is the fundamental question central to these sensing schemes. By mapping this problem to an array of nested Mach-Zehnder interferometers, we explicitly construct the optimal probe state. It is entangled across the spatial modes allowed within the Fresnel number product of the propagation geometry, and entangled across the temporal modes within the time-bandwidth product of the optical probe. We show that the optimal sensitivity of measuring the beam displacement achieves a Heisenberg limited scaling over both the number of temporal modes and the average number of photons transmitted per mode. Surprisingly, we discover a sub-Heisenberg limited scaling over the number of available spatial modes. To qualify the quantum enhancement, we also establish the optimal sensitivity of a classical-light probe, which gives shot-noise limit over both the number of temporal modes and the number of photons per mode, and Heisenberg limited scaling over the number of spatial modes. Finally, we construct an explicit design for quantum-enhanced beam tracking, which uses a Gaussian (multi-mode-entangled squeezed-state) probe and a Gaussian (multi-mode homodyne) receiver.
Quantum Optics The radiation pressure from a focused laser beam used to cool and trap particles is the basis for advanced optical tweezing methods that have found a diverse range of applications in biology, chemistry, and physics. When a trapped particle is subjected to a force, it shifts its position in the trap, thereby providing the possibility of sensing that force. By confining a 50-nm-diameter silica bead in an ultrahigh vacuum, Jain et al. isolated the particle from environment-induced jiggling and effectively restricted the particle's motion to that due to quantum optical forces. They then showed that they can measure the recoil rate of photons from the particle, providing the possibility of exploiting the effect for highly sensitive force measurements. Phys. Rev. Lett. 116 , 243601 (2016).