Squeezed Displaced Cat States: A Signature of the PT‐Symmetry Phase Transition

We theoretical investigate the properties of the parity-time -symmetric trimer lattices in which each unit cell contains balanced gain and loss sites. The impacts of different parameters, such as intra-trimer couplings, inter-trimer couplings, and on-site energy, on the symmetry phase transition and topological phase transition of the trimer lattices are discussed. Our results show that the dynamics of the density distribution is significantly dependent on the difference between the on-site energy of the different sublattices. It is interesting that the topological edge states can be found only in the cases with broken symmetry phase. In principle, such trimer lattices may be established by a well-arranged optical waveguide array.

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.

We investigate on the right and left-handed (RH/LH) balanced gain and loss non-Hermitian electrical transmission lines (ETL) modeled using an imaginary resistor. The hamiltonian of each system is successfully derived in the framework of the tight-binding theory. We discuss the underlying symmetries and calculate the breaking thresholds of the Parity time (PT) and Anti PT (APT) symmetry phase transitions. Moreover, the modes dynamic characterization reveal the existence of critical points beyond which operation requires thresholdless transition and the eigen modes are either real or complex. We also present a self-consistent theory to investigate on their scattering properties. In particular, we demonstrate that the Anderson-like localized modes mostly emerge in the broken phase where eigen modes are complex with the localization length proportional to the cell numbers. Importantly, the mode localization is most likely to occur in structures consisting of a large cells number for RH ETL while LH ETL will confined modes even for infinitesimal small cell numbers. These results unveil embryonic applications in cryptography, switching and system control.

q-deformed photon states arise from generalized oscillator algebras in which the standard bosonic commutation relations are modified by a deformation parameter q. They effectively model realistic nonideal photon statistics, including deviations such as sub- and super-Poissonian distributions. We investigate quantum phase estimation in a Mach–Zehnder interferometer using q-deformed photon states, including q-coherent and q-cat states, which model realistic deviations from ideal light sources. By deriving closed-form photon count likelihoods via the Jordan–Schwinger mapping, we compute the quantum and classical Fisher information and perform Bayesian inference on simulated detector data. Our results show that photon counting remains an optimal measurement strategy even for deformed states, with classical and quantum Fisher information in exact agreement. Furthermore, the phase sensitivity improves with increasing q-deformation, indicating enhanced metrological performance driven by nonclassical photon statistics. These findings highlight the utility of q-deformed states in quantum sensing.

Quantum metrology has an important role in the fields of quantum optics and quantum information processing. Here we introduce a kind of non-Gaussian state, Laguerre excitation squeezed state as inputs of traditional Mach-Zehnder interferometer to examine phase estimation in realistic case. We consider the effects of both internal and external losses on phase estimation by using quantum Fisher information and parity detection. It is shown that the external loss presents a bigger effect than the internal one. The phase sensitivity and the quantum Fisher information can be improved by increasing the photon number and even surpass the ideal phase sensitivity by two-mode squeezed vacuum in a certain region of phase shift for realistic case. Our results can find significant practical applications in quantum metrology.

Quantum metrology is a subject of studying quantum measurement and quantum statistical deduction, and the precision of parameter estimation can be enhanced by quantum properties. In general, the process of parameter estimation includes four steps:preparation of probe state, parameterization process, measurement, and data processing. Of these four steps, the preparation of probe state is the most crucial. However, in practical applications, in the process of preparing quantum probe state, the probe system will couple to its environment, which will inevitably cause the quantum properties of the probe system to deteriorate, and thus reducing the precision of quantum parameter estimation. The dynamics of quantum Fisher information (QFI) for W state and Greenberger-Horne-Zeilinger (GHZ) state have been studied in decoherence channels. Because W state and GHZ state have different entanglement properties, the studies of the dynamics of QFI for the superposition of W state and GHZ state are of practical significance in quantum metrology field. In this paper, the dynamics of QFIs for the superposition of W state and GHZ state in three typical decoherence channels (depolarization channel, amplitude damping channel and phase damping channel) are studied. In the four steps of quantum parameter estimation, our major attention is paid to the first step (i.e., the preparation of probe state). For comparison, the QFIs of different probe states are studied, with the other three steps fixed, i.e., all the probe states will undergo the same parameterization, measurement and estimation process. The parameterization process involved here is a quantum spin operation (specified by the spin rotation direction), which is chosen to maximize the QFI of the probe state. The initial probe states under consideration are the superpositions of W state and GHZ state of three-particle and five-particle systems, and the QFI dynamics of those probe states are studied in the three different typical decoherence channels. By using the operator-sum (Kraus) representation of those three typical decoherence channels, the QFI dynamics of the probe state can be analytically derived in three different decoherence channels. The results show that in the depolarization channel, the maximum QFI of the probe state decreases with the decoherence evolving to zero in the end; in the amplitude damping channel, the QFI of the probe state decreases to the minimum with the decoherence evolution and then increases to the shot noise limit; in the phase damping channel, the QFI of the probe state decreases with the evolution of decoherence, but the final stable value is not zero. Further analyses show that W state component of the superposition plays a role in resisting phase damping and the GHZ state component plays a role in resisting amplitude damping. These results can help us to choose the optimal probe state for maximizing the estimation precision in practice.

Entanglement is at the heart of quantum technologies such as quantum information and quantum metrology. Providing larger quantum Fisher information (QFI), entangled systems can be better resources than separable systems in quantum metrology. However the effects on the entanglement dynamics such as decoherence usually decrease the QFI considerably. On the other hand, Dzyaloshinskii-Moriya (DM) interaction has been shown to excite entanglement. Since an increase in entanglement does not imply an increase in QFI, and also there are cases where QFI decreases as entanglement increases, it is interesting to study the influence of DM interaction on quantum metrology. In this work, we study the QFI of thermal entanglement of two-qubit and three-qubit Heisenberg models with respect to SU(2) rotations. We show that even at high temperatures, DM interaction excites QFI of both ferromagnetic and antiferromagnetic models. We also show that QFI of the ferromagnetic model of two qubits can surpass the shot-noise limit of the separable states, while QFI of the antiferromagnetic model in consideration can only approach to the shot-noise limit. Our results open new insights in quantum metrology with Heisenberg models.

Quantum entanglement lies at the heart of quantum information and quantum metrology. In quantum metrology, with a colossal amount of quantum Fisher information (QFI), entangled systems can be ameliorated to be a better resource scheme. However, noisy channels affect the QFI substantially. This research work seeks to investigate how QFI of N-qubit Greenberger-Horne-Zeilinger (GHZ) state is affected when subjected to decoherence channels: bit-phase flip (BPF) and generalize amplitude damping (GAD) channels, which can be induced experimentally. We determine the evolution under these channels, deduce the eigenvalues, and then derive the QFI. We found that when there is no interaction with the environment, the Heisenberg limit can be achieved via rotations along the z direction. It has been shown that in BPF channel, the maximal mean QFI of the N-qubit GHZ state ({bar{F}}_{max}) dwindles as decoherence rate (pB) increases due to flow of information from the system to the environment, until pB = 0.5, then revives to form a symmetric around pB = 0.5. Thus, pB > 0.5 leads to a situation where more noise yields more efficiency. We found that in GAD channel, at finite temperature, QFIs decay more rapidly than at zero temperature. Our results also reveal that QFI can be enhanced by adjusting the temperature of the environment.

Quantum metrology exploits quantum correlations to make precise measurements with limited particle numbers. By utilizing inter- and intra- mode correlations in an optical interferometer, we find a state that combines entanglement and squeezing to give a 7-fold enhancement in the quantum Fisher information (QFI) -- a metric related to the precision -- over the shot noise limit, for low photon numbers. Motivated by practicality we then look at the squeezed cat-state, which has recently been made experimentally, and shows further precision gains over the shot noise limit and a 3-fold improvement in the QFI over the optimal Gaussian state. We present a conceptually simple measurement scheme that saturates the QFI, and we demonstrate a robustness to loss for small photon numbers. The squeezed cat-state can therefore give a significant precision enhancement in optical quantum metrology in practical and realistic conditions.

The purpose of quantum teleportation is to achieve perfect transmission of quantum information from one site to another distant site. In the teleportation process, the quantum system is inevitably affected by its surrounding environment, causing the system to lose its coherence, which will result in distortion of the transmitted information. In recent years, weak measurement and measurement reversal have been proposed to suppress the decoherence of quantum entanglement and protect some quantum states. On the other hand, quantum Fisher information (QFI) is an important physical quantity in quantum metrology, which can give the optimal value estimating the accuracy of parameters. As is well known, QFI is highly susceptible to environmental noise and can lead its measurement accuracy to decrease. Therefore, it is of great importance to examine how to protect QFI from being influenced by the external circumstance during the teleportation procedure. In this paper, we study how to improve the QFI of teleporting a single-qubit state via a Greenberger-Horne-Zeilinger state in a finite temperature environment with the technique of weak measurement and weak measurement reversal. According to different qubit transmission cases of three quantum teleportation schemes, we consider their respective QFIs in detail. After constructing the quantum logic circuit of each teleportation scheme, we first analyze the variance trend of QFI against the generalized amplitude damping noise parameters. Then by introducing weak measurement and measurement reversal on each noise particle of the three schemes, we optimize the related partial measurement parameters and explore the corresponding improved QFI, namely, the difference between the QFI with optimal partial measurements and that without partial measurements. We find that optimizing partial measurements can efficiently enhance the QFI of the teleported state for the three kinds of teleportation schemes at finite temperature. Moreover, with the value of p fixed, the lower the environment temperature, the larger the value of the improved QFI is. Our results could be useful in further understanding the applications of weak measurement and measurement reversal to the quantum communication process and may shed light on estimating some relevant quantum parameters and implementing quantum information tasks.

We use supersymmetry transformations to design transparent and one-way reflectionless (thus unidirectionally invisible) complex crystals with balanced gain and loss profiles. The scattering coefficients are investigated using the transfer matrix approach. It is shown that the amount of reflection from the left can be made arbitrarily close to zero whereas the reflection from the right is enhanced arbitrarily (or vice versa).

Harnessing parity–time symmetry with balanced gain and loss profiles has created a variety of opportunities in electronics from wireless energy transfer to telemetry sensing and topological defect engineering. However, existing implementations often employ ad hoc approaches at low operating frequencies and are unable to accommodate large-scale integration. Here we report a fully integrated realization of parity–time symmetry in a standard complementary metal–oxide–semiconductor process technology. Our work demonstrates salient parity–time symmetry features such as phase transition as well as the ability to manipulate broadband microwave generation and propagation beyond the limitations encountered by existing schemes. The system shows 2.1 times the bandwidth and 30% noise reduction compared to conventional microwave generation in the oscillatory mode, and displays large non-reciprocal microwave transport from 2.75 to 3.10 GHz in the non-oscillatory mode due to enhanced nonlinearities. This approach could enrich integrated circuit design methodology beyond well-established performance limits and enable the use of scalable integrated circuit technology to study topological effects in high-dimensional non-Hermitian systems.

The reduced density operator of a single qubit coupled to a quantum oscillator is time-periodic even for stationary Hamiltonians. We construct some quantum master equations for such density operator and show that they can be expressed in the Lindblad form. Although the qubit is treated as an open system that exchanges information with the oscillator (environment), the dynamics of the entire system is unitary and such that no information is lost at any time. The time-evolution of the qubit is therefore developed with balanced gain and loss profile. Some subtleties arise since the appropriate master equation must include not only the decay (diffusion) process but also the excitation of the qubit. The advances reported in this work are addressed to cover the decay process only.KeywordsQuantum master equationJaynes–Cummings modelLindblad superoperatorMathematics Subject Classification (2010)Primary 81S22; 81R15; Secondary 81V80

Quantum Fisher information is a central quantity in quantum metrology. We discuss an alternative representation of quantum Fisher information for unitary parametrization processes. In this representation, all information of parametrization transformation, i.e., the entire dynamical information, is totally involved in a Hermitian operator . Utilizing this representation, quantum Fisher information is only determined by and the initial state. Furthermore, can be expressed in an expanded form. The highlights of this form is that it can bring great convenience during the calculation for the Hamiltonians owning recursive commutations with their partial derivative. We apply this representation in a collective spin system and show the specific expression of . For a simple case, a spin-half system, the quantum Fisher information is given and the optimal states to access maximum quantum Fisher information are found. Moreover, for an exponential form initial state, an analytical expression of quantum Fisher information by operator is provided. The multiparameter quantum metrology is also considered and discussed utilizing this representation.

We study the preparation of coherent quantum states in a two-photon micromaser for applications in quantum metrology. While this setting can be in principle realized in a host of physical systems, we consider atoms interacting with the field of a cavity. We focus on the case of the interaction described by the Jaynes-Cummings Hamiltonian, which cannot be achieved by the conventional approach with three-level atoms coupled to the cavity field at two-photon resonance. We find that additional levels are required in order to cancel Stark shifts emerging in the leading order. Once this is accomplished, the dynamics of the cavity features a degenerate stationary state manifold of pure states. We derive the analytic form of these states and show that they include Schr\"odinger cat states with a tunable mean photon number. We also confirm these states can be useful in phase estimation protocols with their quantum Fisher information exceeding the standard limit. To account for realistic imperfections, we consider single-photon losses from the cavity, finite lifetime of atom levels, and higher order corrections in the far-detuned limit, which result in metastability of formerly stationary cavity states and long-time dynamics with a unique mixed stationary state. Despite being mixed, this stationary state can still feature quantum Fisher information above the standard limit. Our work delivers a comprehensive overview of the two-photon micromaser dynamics with particular focus on application in phase estimation and, while we consider the setup with atoms coupled to a cavity, the results can be directly translated to optomechanical systems.