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Civil War Book Review

New strategies for phase estimation in quantum optics

Yang Gao

Published May 31, 2010• 0 citations
Publisher

Abstract

One important problem in quantum optics is to resolve an extremely small change of phase shift. The complementarity between photon number and phase sets an ultimate limit, the so-called Heisenberg limit, on the phase measurement sensitivity. The precise phase estimation has many technological applications, such as optical gyroscopes, gravitational wave detection, quantum imaging and sensing. In this thesis I show that the utilization of the parity measurement in the optical interferometry is actually applicable to a wide range of quantum entangled input states. Comparison of the performance of the various quantum states then can be made within such a unified output measurement scheme. Based on such a universal detection scheme, we present a comparison of the phase sensitivity reduction for various quantum states of light in the presence of photon loss. I also provide a simple condition that could be used to check whether an arbitrary state can achieve the Heisenberg limit independently of the detection scheme. It implies that the fidelity between the two output states with zero phase and minimal detectable phase applied respectively should significantly different from unity as the minimal phase shift scaling as 1/N, whereas the average number of input photons N goes to infinity. Next I give several measures to characterize the which-way information in the interference experiment. We define a new distinguishability associated with the fidelity between two density matrices to measure the which-way information. We demonstrate that the changes of mutual entropy as well as the entanglement of formation of the whole system, i.e. the physical system plus the which-way detector can also be used to describe the which-way information. With such quantities, we show that as the fringe visibility of the interference pattern gets larger, the less which-way information is obtained. Finally, I show that coherent light coupled with photon number resolving detectors can provide a super-resolution much below the Rayleigh diffraction limit, with sensitivity no worse than shot-noise in terms of the detected photon power. This scheme would have applications to laser radar, given the difficulty in making entangled states of light, as well as their susceptibility to atmospheric absorption.

DOI
10.31390/gradschool_dissertations.1616

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Civil War Book Review•Apr 14, 2010•0 citations
One important problem in quantum optics is to resolve an extremely small change of phase shift. The complementarity between photon number and phase sets an ultimate limit, the so-called Heisenberg limit, on the phase measurement sensitivity. The precise phase estimation has many technological applications, such as optical gyroscopes, gravitational wave detection, quantum imaging and sensing. In this thesis I show that the utilization of the parity measurement in the optical interferometry is actually applicable to a wide range of quantum entangled input states. Comparison of the performance of the various quantum states then can be made within such a unified output measurement scheme. Based on such a universal detection scheme, we present a comparison of the phase sensitivity reduction for various quantum states of light in the presence of photon loss. I also provide a simple condition that could be used to check whether an arbitrary state can achieve the Heisenberg limit independently of the detection scheme. It implies that the fidelity between the two output states with zero phase and minimal detectable phase applied respectively should significantly different from unity as the minimal phase shift scaling as 1/N, whereas the average number of input photons N goes to infinity. Next I give several measures to characterize the which-way information in the interference experiment. We define a new distinguishability associated with the fidelity between two density matrices to measure the which-way information. We demonstrate that the changes of mutual entropy as well as the entanglement of formation of the whole system, i.e. the physical system plus the which-way detector can also be used to describe the which-way information. With such quantities, we show that as the fringe visibility of the interference pattern gets larger, the less which-way information is obtained. Finally, I show that coherent light coupled with photon number resolving detectors can provide a super-resolution much below the Rayleigh diffraction limit, with sensitivity no worse than shot-noise in terms of the detected photon power. This scheme would have applications to laser radar, given the difficulty in making entangled states of light, as well as their susceptibility to atmospheric absorption.
Phase measurement using a lossless Mach–Zehnder interferometer with certain entangled N-photon states can lead to a phase sensitivity of the order of 1/N, the Heisenberg limit. However, previously considered output measurement schemes are different for different input states to achieve this limit. We show that it is possible to achieve this limit just by the parity measurement for all the commonly proposed entangled states. Based on the parity measurement scheme, the reductions of the phase sensitivity in the presence of photon loss are examined for the various input states.
Quantum optical metrology deals with estimation of an unknown parameter by exploiting the non-classical properties of the light. The unknown parameter that we are trying to estimate is the optical phase. Precise optical phase measurement has been a well-known problem and has many applications, most notably the gravitational wave detection. In this thesis we investigate the interferometric measurement schemes. We consider the parity detection for a class of input states that have been shown to exhibit sub-shot noise limited phase estimate with their respective detection schemes. Our results indicate that the parity detection applies to all these strategies with various input states and thus acts as a unified detection scheme towards the goal of interferometric phase estimates beyond the shot-noise limit. We also consider the performance of the so-called optimal state with the canonical phase measurement scheme that was proposed by Sanders and Milburn [Phys. Rev. Lett. 75, 2944 (1995)] in presence of photon loss. The model for photon loss is a generic fictitious beam splitter and the analytical treatment requires density matrix approach rather than the state-vector formalism. We present full density-matrix calculations. Our results indicate that, for a given amount of loss, the phase estimate saturates but does not diverge as one would expect with increasing the loss. Finally, we study the continuous measurement and feedback scheme with optical homodyne detection for a single optical qubit. We found a protocol that speeds up the rate of increase of the average purity of the system and generates a deterministic evolution for the purity in the limit of strong feedback.
Phase estimation has a wide range of applications. Over the years, several strategies have been studied to improve precision in phase estimation. These strategies include using exotic quantum states to quantum detection schemes. This dissertation summarizes my effort in improving the precision of phase estimation with a linear and nonlinear interferometer. Chapter 1 introduces quantum optics and quantum metrology. I introduce all relevant quantum states of light used. We also look into tools and terminologies of quantum metrology such as Fisher information, shot-noise limit, Heisenberg limit, etc., along with examples of phase estimation with a Mach-Zehnder interferometer. In Chapter 2, I discuss multiple phase estimation using a multimode interferometer. Building upon previous work, our scheme consists of a multimode interferometer with single-photon inputs. By using a quantum Fisher information analysis, we show that our scheme gives a constant improvement over other schemes. We also show that our scheme with photon-number-resolving detection approaches the quantum Cram\'er-Rao bound. Moreover, we also consider the probabilistic nature of photon emission at the input, and we study its effect on phase sensitivity. I discuss phase estimation with SU(1,1) interferometer in Chapter 3. We look at phase sensitivity in this interferometer with different input states. Namely, we consider two different phase estimation scheme, one using thermal and squeezed states, and others using coherent and displaced squeezed states with parity and on-off as a detection scheme. We also look into the effect of photon loss inside the interferometer. In Chapter 4, we revisit phase estimation in SU(1,1) interferometer from the perspective of quantum Fisher information. I discuss in detail a longstanding confusion regarding the use of quantum Fisher information in SU(1,1) interferometer. We show that phase averaging or quantum Fisher information matrix method is needed in general for calculating the phase sensitivity which resolves inconsistencies reported in previously published articles.
Quantum-enhanced interferometers utilize non-classical states of light in order to surpass the limitations imposed by classical physics on phase measurement sensitivity and resolution. We review key scientific developments in this growing field of research and derive fundamental metrology concepts, namely the Cram\'{e}r-Rao bound and the Fisher information, which are used to analyze the metrological performance of quantum states of light. We propose a supersensitive quantum interferometry protocol that can be amplified to the macroscopic realm (i.e. the mean number of photons of the probe state can reach approximately 100,000 photons) using contemporary techniques of spontaneous parametric down conversion. The parity measurement on the output field asymptotically saturates the quantum Cram\'{e}r-Rao bound, which scales like the Heisenberg limit. We uncover the role of photon statistics on phase sensitivity and rule out the necessity of mode entanglement for quantum-enhanced interferometry. Based on these insights we outline a practical metrology technique (independent of mode entanglement) that measures a phase delay of a single-mode anti-squeezing operation. This scheme can also be scaled macroscopically using contemporary techniques and is shown to significantly surpass the shot-noise limit (SNL) in the presence of realistic losses. Finally, the physical resources that enable quantum-enhanced interferometry is studied by analyzing the quantum Fisher information (QFI) using the first quantization and the second quantization formalisms of quantum mechanics. It is shown that increasing the intra-mode correlations of an interferometer (as quantified by the second order Glauber coherence function) can be conducive to attaining a quantum advantage in phase estimation; whereas, introducing mode entanglement can reduce phase sensitivity. Using the first quantization description, we derive a formula for the QFI that shows explicitly how the Heisenberg scaling term depends on particle entanglement.
There has been much recent interest in quantum optical interferometry for applications to metrology, subwavelength imaging, and remote sensing such as in quantum laser radar (LADAR). For quantum LADAR, atmospheric absorption rapidly degrades any quantum state of light, so that for high-photon loss the optimal strategy is to transmit coherent states of light, which suffer no worse loss than the Beer law for classical optical attenuation, and which provides sensitivity at the shot-noise limit. We show that coherent light coupled with photon-number-resolving detectors can provide a super-resolution much below the Rayleigh diffraction limit, with sensitivity no worse than shot noise in terms of the detected photon power.
In this thesis we will theoretically investigate three potentially useful physical systems, after first developing the theoretical framework necessary for studying them. First, we will study the multiphoton absorption properties of maximally path entangled number (N00N) states. This is directly relevant to quantum lithography, and beating the Rayleigh diffraction limit. Next, we will develop a new scheme for quantum interferometry: dubbed coherent-light boosted super-sensitive quantum interferometry, which has the potential to reach below the shot noise limit for high photon fluxes, and requires no esoteric detection protocol, or technological elements that have yet to be developed. Finally we propose a method to perform parity detection on the output modes of a Mach-Zehnder interferometer that has been fed with two-mode squeezed vacuum. This detection scheme relies on a double homodyning technique, that makes intensity correlation measurements at a series of chosen bias phases. Sub-Heisenberg sensitivity scaling is expected for this setup.