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arXiv:1206.2265 · Physical Review A

Optimal measurement precision of a nonlinear interferometer

Juha Javanainen, Han Chen

Published Jun 4, 2012• 7 citations
arXivHTMLPDF

Abstract

We study the best attainable measurement precision when a double-well trap with bosons inside acts as an interferometer to measure the energy difference of the atoms on the two sides of the trap. We introduce time-independent perturbation theory as the main tool in both analytical arguments and numerical computations. Nonlinearity from atom-atom interactions will not indirectly allow the interferometer to beat the Heisenberg limit, but in many regimes of the operation the Heisenberg limit scaling of measurement precision is preserved in spite of added tunneling of the atoms and atom-atom interactions, often even with the optimal prefactor.

DOI
10.1103/physreva.85.063605

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We study the best attainable measurement precision when a double-well trap with bosons inside acts as an interferometer to measure the energy difference of the atoms on the two sides of the trap. We introduce time-independent perturbation theory as the main tool in both analytical arguments and numerical computations. Nonlinearity from atom-atom interactions will not indirectly allow the interferometer to beat the Heisenberg limit, but in many regimes of the operation the Heisenberg limit scaling of measurement precision is preserved in spite of added tunneling of the atoms and atom-atom interactions, often even with the optimal prefactor.
Quantum metrology studies the use of entanglement and other quantum resources to improve precision measurement. An interferometer using N independent particles to measure a parameter χ can achieve at best the "standard quantum limit" (SQL) of sensitivity δχ ∝ N-1/2. The same interferometer using N entangled particles can achieve in principle the "Heisenberg limit" δχ ∝ N-1, using exotic states, e.g., NooN states. Recent theoretical work argues that interactions among particles may be a valuable resource for quantum metrology, allowing scaling beyond the Heisenberg limit. Specifically, a k-particle interaction will produce sensitivity δχ ∝ N-kwith appropriate entangled states and δχ ∝ N-(k-1/2)even without entanglement. Here we demonstrate this "super-Heisenberg" scaling in a nonlinear, non-destructive measurement of the magnetisation of an atomic ensemble.
Quantum precision measurement is based on the basic principle of quantum mechanics by using the interaction between light, atoms and magnetism to measure physical quantities, also known as precision measurements based on microscopic particle systems and their quantum states. As an important means of quantum precision measurement, interferometer precision measurement technology has great application value in quantum communication. The linear interferometer measures the magnitude of the physical quantity by using the phase change obtained from the measurements, but measurement accuracy is limited and unable to meet the requirements of today's scientific problems for the precision measurement of some physical quantities. On this basis, nonlinear interferometer is able to take advantage of the quantum entangled state, that is, using the two light fields of quantum correlation characteristics to realize quantum enhanced precision measurement, thus greatly improving the measurement sensitivity, Therefore, the scope of application is wider, but the preparation of quantum entangled states has many limitations in practical manipulation. With the maturity of experimental conditions and technology, how to use both of these interferometers to further improve the measurement accuracy of the phase signal so as to break the limitation to shot noise, breaking the standard quantum limit and even approaching to the Heisenberg limit has become a frontier research topic . In this paper, we introduce several methods to improve the accuracy of parameter evaluation in the measurement process by using linear (including an atomic/photon interferometer) and nonlinear interferometer to call quantum resources at different stages. High-precision measurement can be achieved by inputting non-classical states into the interferometer, such as compressed state, bi-fock state, and NOON state. And we also introduce the weak measurement developed for the direct observation of quantum states and its application to non-Hermitian systems, and the multiparameter measurement proposed to eliminate the accuracy balance between parameters. Compared with the first two measurement methods, weak measurement method is based on the weak value amplification principle of an indirect measurement. Measurements are performed virtually without perturbing the quantum system, which does not lead the wave function to collapse, the weak value of the real and virtual part have different physical significance, The combination of weak measurement theory and non-Hermitian system also further improves the measurement sensitivity. Multi-parameter measurement uses quantum entanglement, quantum control and other quantum resources to make the measurement progress reach the Heisenberg limit, which is the current research hotspot in the field of precision measurement. Furthermore, we present a conjecture whether there will be multi-atomic mixing measurements based on atomic spin effects or ultra-high sensitivity measurement instruments with precision of fT or even aT by using other particles detection. Finally, several measurement methods are analyzed and compared with each other, and the development prospect of quantum precision measurement is forecasted.
We consider an optical interferometer with coherent light in one input and a squeezed vacuum in another. Such an interferometer is known to beat the standard quantum limit of sensitivity to the difference of phase shifts in its arms. We find the ultimate limit of precision for such an interferometer by calculating the quantum Fisher information of the mixed quantum state in one of the interferometer's outputs about the difference phase. We show that, in the vicinity of the black fringe, this information is asymptotically close to the quantum Fisher information about this phase for the two-mode readout. We conclude that the single-mode readout is optimal for phase estimation in squeezing-enhanced interferometry and allows for the Heisenberg scaling of precision. We also show that the optimal local measurement in the vicinity of the black fringe consists of amplifying the output field in a phase-sensitive way and measuring its photon number.
Interferometry with ultracold atoms promises the possibility of ultraprecise and ultrasensitive measurements in many fields of physics, and is the basis of our most precise atomic clocks. Key to a high sensitivity is the possibility to achieve long measurement times and precise readout. Ultracold atoms can be precisely manipulated at the quantum level and can be held for very long times in traps; they would therefore be an ideal setting for interferometry. In this paper, we discuss how the nonlinearities from atom–atom interactions, on the one hand, allow us to efficiently produce squeezed states for enhanced readout and, on the other hand, result in phase diffusion that limits the phase accumulation time. We find that low-dimensional geometries are favorable, with two-dimensional (2D) settings giving the smallest contribution of phase diffusion caused by atom–atom interactions. Even for time sequences generated by optimal control, the achievable minimal detectable interaction energy Δ E min is of the order of 10 −4 μ, where μ is the chemical potential of the Bose–Einstein condensate (BEC) in the trap. From these we have to conclude that for more precise measurements with atom interferometers, more sophisticated strategies, or turning off the interaction-induced dephasing during the phase accumulation stage, will be necessary.