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Annalen der Physik

Quantum Optics Parity Effect on Generalized NOON States and Its Implications for Quantum Metrology

Agostino Migliore, ANTONINO MESSINA

Published Oct 11, 2022• 5 citations
Publisher

Abstract

Abstract NOON states are entangled states of great interest in the development of quantum technologies, because they enable high‐precision phase measurements in interferometric schemes relevant to quantum metrology and sensing. This study shows that two degenerate modes of a cavity field coupled to an atom or molecule that can be modeled as a two‐level system through two‐photon interactions, initially prepared in a (generalized) NOON state, experience a dynamical evolution that distinctly depends on the parity of the photon number. When the total photon number is even, the system dynamics is characterized by a time range in which the state of the system evolves near maximally entangled states that may be fed to an optical interferometer to make phase measurements essentially at the Heisenberg limit. Such entangled states cannot be achieved for odd photon numbers instead. The parity effect thus emerges as a potential control factor in quantum metrology, sensing, and lithography.

DOI
10.1002/andp.202200304

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Abstract NOON states are entangled states of great interest in the development of quantum technologies, because they enable high‐precision phase measurements in interferometric schemes relevant to quantum metrology and sensing. This study shows that two degenerate modes of a cavity field coupled to an atom or molecule that can be modeled as a two‐level system through two‐photon interactions, initially prepared in a (generalized) NOON state, experience a dynamical evolution that distinctly depends on the parity of the photon number. When the total photon number is even, the system dynamics is characterized by a time range in which the state of the system evolves near maximally entangled states that may be fed to an optical interferometer to make phase measurements essentially at the Heisenberg limit. Such entangled states cannot be achieved for odd photon numbers instead. The parity effect thus emerges as a potential control factor in quantum metrology, sensing, and lithography.
The rapidly increasing capability to modulate the physicochemical properties of atomic groups and molecules by means of their coupling to radiation, as well as the revolutionary potential of quantum computing for materials simulation and prediction, fuel the interest for non-classical phenomena produced by atom-radiation interaction in confined space. One of such phenomena is a “parity effect” that arises in the dynamics of an atom coupled to two degenerate cavity field modes by two-photon processes and manifests itself as a strong dependence of the field dynamics on the parity of the initial number of photons. Here we identify the physical origin of this effect in the quantum correlations that produce entanglement among the system components, explaining why the system evolution depends critically on the parity of the total number of photons. Understanding the physical underpinnings of the effect also allows us to characterize it within the framework of quantum information theory and to generalize it. Since a single photon addition/removal has dramatic effects on the system behavior, this effect may be usefully applied, also for amplification purposes, to optoelectronics and quantum information processing.
Photon number states are assigned a parity of +1 if their photon number is even and a parity of −1 if odd. The parity operator, which is minus one to the power of the photon number operator, is a Hermitian operator and thus a quantum mechanical observable although it has no classical analogue, the concept being meaningless in the context of classical light waves. In this paper we review work on the application of the parity operator to the problem of quantum metrology for the detection of small phase shifts with quantum optical interferometry using highly entangled field states such as the so-called N00N states, and states obtained by injecting twin Fock states into a beam splitter. With such states and with the performance of parity measurements on one of the output beams of the interferometer, one can breach the standard quantum limit, or shot-noise limit, of sensitivity down to the Heisenberg limit, the greatest degree of phase sensitivity allowed by quantum mechanics for linear phase shifts. Heisenberg limit sensitivities are expected to eventually play an important role in attempts to detect gravitational waves in interferometric detection systems such as LIGO and VIRGO.
It has been very recently shown that the dynamics of a two-level atom coupled to a bimodal degenerate cavity field by two-photon processes, is characterized by an interesting nonclassical dynamical behavior christened “parity effect”. This effect consists in the fact that if the cavity field is prepared leaving one mode in its vacuum state and exciting the other one in a generic linear combination of even number states only, or odd number states only, then there exists an appropriate intensity-dependent interval of time after which the bimodal cavity exhibits macroscopically different parity-dependent quantum features. We show that this nonclassical effect is at the origin of the possibility of generating Schrodinger cat like states of the bimodal field appropriately selecting its initial conditions.
Quantum entanglement is a fascinating physical resource that is central to the field of quantum information processing with important applications in such areas as quantum computing, quantum communication, quantum metrology, and quantum imaging. \nIt describes the physical phenomenon in which the quantum state of a multipartite system cannot be written as consisting of constituents that are independent of each other. In this case, the state of the system is called an entangled state. In this \ndissertation, the entanglement of photons--the elementary excitations of the quantized electromagnetic field--is studied. A focus is made on a particular type of entangled states of light called the d-mode N-photon NOON state, which consists of N photons with the N photons as an ensemble appearing at d orthogonal modes simultaneously and are inseparable. \nIn quantum metrology, a field that investigates the ultimate precision of measurements of unknown physical parameters, the two-mode NOON state has been studied extensively with respect to its ability to achieve super-resolution and super-sensitivity in the estimation of a single parameter. A lot of theoretical research has been conducted on the generation of two-mode NOON states, and two-mode NOON states with up to 4 photons have been experimentally and efficiently produced. On the other hand, recently there has been increasing interest in the study of the simultaneous estimation of multiple parameters. Accordingly, the multi-mode NOON state has been attracting more and more attention in view of its potential for enhanced efficiency as compared to using multiple two-mode NOON states separately to estimate the parameters. Nevertheless, no known generation method of multi-mode NOON states with more than 2 photons or 2 modes exists so far. \nIn this dissertation, several scalable generation methods of multi-mode NOON states are proposed. These methods take advantage of multi-photon quantum interference and they can theoretically be applied to produce NOON states with an \narbitrary number of modes and an arbitrary number of photons. The intrinsic generation probability for each method is calculated, and the methods are compared in regards to their feasibility and efficiency. The advantages of using multi-mode NOON \nstates in quantum metrology are also analyzed and discussed in detail.
We investigate the utility of parity detection to achieve Heisenberg-limited phase estimation for optical interferometry. We consider the parity detection with several input states that have been shown to exhibit sub-shot-noise interferometry with their respective detection schemes. We show that with parity detection, all these states achieve the sub-shot-noise limited phase uncertainty. Thus making the parity detection a unified detection strategy for quantum optical metrology. We also consider quantum states that are a combination of a NOON state and a dual-Fock state, which gives a great deal of freedom in the preparation of the input state, and is found to surpass the shot-noise limit.