Nonclassicality in quantum sensors can improve sensitivity, but often increases susceptibility to noise. Thus, modeling physically relevant noise sources and analyzing their effect on quantum metrology are both of importance to the field of quantum sensing. In this dissertation, I demonstrate that local noise sources, which are present in almost all many-spin systems, can be tractably modeled when assuming permutation symmetry of the noise, and we show that many common local noise sources can be mapped to a Fokker-Planck equation on quantum phase space. We apply this description of noise to study quantum sensing using noisy probe states and establish a previously unknown connection between loss of Fisher information for noisy states and discontinuities in the Fisher information as a function of a continuous family of measurements. Finally, we extend the phase space description of local noise by developing a new spin Wigner function, which we call the solid spin Wigner function. We develop this new Wigner function by noting that the permutationally symmetric operator space used to describe symmetric local noise resembles the symmetric subspace of an ensemble of SU(3) particles. Using this, it is possible to embed the SU(2) group structure of the spin ensemble within this larger group, and describe the system using an SU(3) Wigner function.
Nonclassicality in quantum sensors can improve sensitivity, but often increases susceptibility to noise. Thus, modeling physically relevant noise sources and analyzing their effect on quantum metrology are both of importance to the field of quantum sensing. In this dissertation, I demonstrate that local noise sources, which are present in almost all many-spin systems, can be tractably modeled when assuming permutation symmetry of the noise, and we show that many common local noise sources can be mapped to a Fokker-Planck equation on quantum phase space. We apply this description of noise to study quantum sensing using noisy probe states and establish a previously unknown connection between loss of Fisher information for noisy states and discontinuities in the Fisher information as a function of a continuous family of measurements. Finally, we extend the phase space description of local noise by developing a new spin Wigner function, which we call the solid spin Wigner function. We develop this new Wigner function by noting that the permutationally symmetric operator space used to describe symmetric local noise resembles the symmetric subspace of an ensemble of SU(3) particles. Using this, it is possible to embed the SU(2) group structure of the spin ensemble within this larger group, and describe the system using an SU(3) Wigner function.
Phase space methods, particularly Wigner functions, provide intuitive tools for representing and analyzing quantum states. We focus on systems with SU(2) dynamical symmetry, which naturally describes spin and a wide range of two-mode quantum models. We present a unified phase space framework tailored to these systems, highlighting its broad applicability in quantum optics, metrology, and information. After reviewing the core SU(2) phase-space formalism, we apply it to states designed for optimal quantum sensing, where their nonclassical features are clearly revealed in the Wigner representation. We then extend the approach to systems with an indefinite number of excitations, introducing a generalized framework that captures correlations across multiple SU(2)-invariant subspaces. These results offer practical tools for understanding both theoretical and experimental developments in quantum science.
This thesis explores the interference of nonclassical states of light, particularly Fock and Gaussian states, in noisy linear interferometers, with applications to quantum information and quantum sensing. Using the phase-space formalism, analytical tools based on generating functions are developed to describe quantum optical interference in a unified way. For multiphoton Fock states, new zero probability events (suppression laws) are identified beyond the previously derived symmetry permutation principle, revealing rich interference structures that is degraded with photon distinguishability. For Gaussian states, the Hafnian-based description of Gaussian Boson Sampling is extended to include partial distinguishability via the overlap matrix of the internal state of the photons. Finally, the link between these interference effects and quantum multiparameter estimation is examined for the simultaneous estimation of phase and loss. This study shows that while probe incompatibility can vanish for optimized non-Gaussian states and some two-mode Gaussian states, at high photon number, measurement incompatibility remains a fundamental constraint even in this limit.
This dissertation serves as a general introduction to Wigner functions, phase space, and quantum metrology but also strives to be useful as a how-to guide for those who wish to delve into the realm of using continuous variables, to describe quantum states of light and optical interferometry. We include many of the introductory elements one needs to appreciate the advantages of this treatment as well as show many examples in an effort to make this dissertation more friendly. In the initial segment of this dissertation, we focus on the advantages of Wigner functions and their use to describe many quantum states of light. We focus on coherent states and squeezed vacuum with a Mach Zehnder Interferometer for many of our examples, also used by experiments such as advanced LIGO. Later, we will also analyze this setup in more detail with a full example including the effects of many noise sources such as phase drift, photon loss, inefficient detectors, and thermal noise. In this setup, we also show the optimal measurement scheme, which is currently not employed in experiment. Throughout our metrology discussions, we will also discuss various quantum limits and use quantum Fisher information to show optimal bounds. When applicable, we also discuss the use of quantum Gaussian information and how it relates to our Wigner function treatment. The remainder of our discussion focuses on investigating the effects of photon addition and subtraction to various states of light and analyze the nondeterministic nature of this process. We use examples of m photon additions to a coherent state as well as discuss the properties of an m photon subtracted thermal state. We also provide an argument that this process must always be a nondeterministic one, or the ability to violate quantum limits becomes apparent. We show that using phase measurement as one's metric is much more restrictive, which limits the usefulness of photon addition and subtraction. When we consider SNR however, we show improved SNR statistics, at the cost of increased measurement time. In this case of SNR, we also quantify the efficiency of the photon addition and subtraction process.
The SU(2) Wigner function represents a quantum state of a spin-J as a real-valued function on the surface of a 2-sphere. For an ensemble of N spin-1/2 particles, this representation is useful when the dynamics is restricted to a single SU(2) irrep, e.g., the symmetric subspace with J=N/2. Physically relevant noise sources tend to be local, such as spontaneous emission, depolarizing, and incoherent optical pumping, all of which transfer the state outside of the initial irrep, and as such the SU(2) Wigner function is no longer a useful representation. In this work, we address this issue by encoding a noisy spin ensemble in an SU(3) irrep, and evaluating the SU(3) Wigner function for that irrep. We find that physical constraints enforced by the noise eliminate all but three real parameters from the input to the Wigner function, which can then be interpreted as a polar, azimuthal, and radial component. This interpretation leads us to refer to the resulting Wigner function as the solid spin Wigner function, visualized on a solid ball rather than a hollow sphere.
We analyse the phase space representation of the optimal measurement of a phase shift in an interferometer with equal photon loss in both its arms. In the local phase estimation scenario with a fixed number of photons, we identify features of the spin Wigner function that warrant sub-shot-noise precision, and discuss their sensitivity to losses. We derive the asymptotic form of an integral kernel describing the process of photon loss in the phase space in the limit of large photon numbers. The analytic form of this kernel allows one to assess the ultimate precision limit for a lossy interferometer. We also provide a general lower bound on the quantum Fisher information in terms of spin Wigner functions.
We show how the decoherence that occurs in an entangling atomic spin–light interface can be simply modeled as the dynamics of a bosonic mode. Although one seeks to control the collective spin of the atomic system in the permutationally invariant (symmetric) subspace, diffuse scattering and optical pumping are local, making an exact description of the many-body state intractable. To overcome this issue we develop a generalized Holstein–Primakoff approximation for collective states which is valid when decoherence is uniform across a large atomic ensemble. In different applications the dynamics is conveniently treated as a Wigner function evolving according to a thermalizing diffusion equation, or by a Fokker–Planck equation for a bosonic mode decaying in a zero-temperature reservoir. We use our formalism to study the combined effect of Hamiltonian evolution, local and collective decoherence, and measurement backaction in preparing nonclassical spin states for application in quantum metrology.