Quantum Fisher Information Probing a Quantum-Gas Cavity QED

The quantum Fisher information, the quantum analog of the classical Fisher information, is a central quantity in quantum metrology and quantum sensing due to its connection to parameter estimation and fidelity susceptibility. Using numerically exact methods applied to a paradigmatic open quantum system, the spin-boson model, we calculate both static and dynamical quantum Fisher information matrix elements with respect to spin-bath couplings and magnetic field strengths. As the spin-bath interaction increases, we first show that the coupling-coupling matrix elements relative to the ground state of the Hamiltonian are linked to the entanglement growth and signal the Berezinskii-Kosterlitz-Thouless quantum phase transition through their nonmonotonic behavior. We also point out that the static quantum Fisher information exhibits a nonperturbative behavior in the zero-coupling limit, which we justify with an analytic argument. Furthermore, we demonstrate that the time-dependent matrix elements can reveal non-Markovian effects as well as the transition from the coherent to incoherent regime at the Toulouse point, remaining robust under pure dephasing noise. Nonmonotonic signatures of the quantum Fisher information matrix reflect changes in quantum resources such as entanglement and coherence, quantify non-Markovian behavior, and enable criticality-enhanced quantum sensing, thereby shedding light on key features of open quantum systems.

Dicke model describes a collective interaction between the two-level atoms and the light cavity and has been predicted to show a peculiar quantum phase transition, which is a second-order phase transition from a normal phase (in a weak-coupling strength) to a superradiant phase (in a strong-coupling strength). The model plays an important role in illustrating the quantum ground-state properties of many-body macroscopic quantum states. In the experiment, Dicke quantum phase transition in an open system could be formed by a Bose-Einstein condensate coupled to a high-finesse optical cavity. This experiment on the Bose-Einstein condensate trapped in the optical cavity have opened new frontiers, which could combine the cold atoms with quantum optics and makes it possible to enter into the strongly coupled regime of cavity quantum electrodynamics. In strong coupled regime, the atoms exchange the photons many times before spontaneous emission and cavity losses set in. It has become a hot research topic in recent years and plays an important role in many fields of modern physics, such as condensed matter physics, nuclear physics, etc. It can be applied to the manipulation of the geometric phase and entanglement in quantum information and computing. Quantum phase transition has been widely studied for the Dicke model as a typical example. Many different research methods about the mean-field approximation have been used to analyze the ground state properties of the Dicke model. In this paper, we study the ground state properties of two-component Bose-Einstein condensate in a single-mode cavity. Meanwhile, the associated quantum phase transition is described by the spin-coherent-state variational method, whose advantage is that the ground state energy and wave function can be obtained without the thermodynamic limit. By taking the average in the boson coherent state, we obtain an equivalent effective pesudospin Hamiltonian, which will be diagonalized by using the spin coherent state. Finally, we can obtain the energy functional, which is the basics of the variation to obtain the numerical solution of photon number and the expression of the atomic number and the ground state energy. This paper presents a rich phase diagram, which can be manipulated by changing the atom-field coupling imbalance between two components and the atom-field frequency detuning. While in the single-mode Dicke model there exist only the normal phase and the superradiation phase. When the frequency of one component atom is zero or the frequency of the two component atoms are equal in optical cavity, the system returns to the standard Dicke model, in which there occurs the second-order phase transition from the normal phase to the superradiant phase by adjusting the atom-field coupling. In conclusion, we discover that the stimulated radiation comes from the collective state of atomic population inversion, which does not exist in the single-mode Dicke model. Meanwhile, the new stimulated-radiation state S and S, which can only be produced by one component of the atom, are observed in the two-component Bose-Einstein condensates in the single-mode optical cavity. By adjusting the atom-field coupling imbalance and the atom-field frequency detuning (the blue or red detuning), the order of the superradiation state and the stimulated-radiation states can be exchanged between the two components of the atom.

We compute the quantum Fisher information (QFI) associated with two different local operators in the Kitaev honeycomb model, and find divergent behaviour in the second derivatives of these quantities with respect to the driving parameter at the quantum phase transition between the gapped and gapless phases for both fully anti-ferromagnetic and fully ferromagnetic exchange couplings. The QFI associated with a local magnetization operator behaves differently from that associated with a local bond operator depending on whether the critical point is approached from the gapped or gapless side. We show how the behaviour of the second derivative of the QFI at the critical point can be understood in terms of diverging length scales in the correlators of the local generators.

We study the quantum critical effect enhanced spin-nematic squeezing and quantum Fisher information (QFI) in the spin-1 dipolar atomic Bose-Einstein condensate. We show that the quantum phase transitions can improve the squeezing and QFI in the nearby regime of critical point, and the Heisenberg-limited high-precision metrology can be obtained. The different properties of the ground squeezing and entanglement under even and odd number of atoms are further analyzed, by calculating the exact analytical expressions.We also demonstrate the squeezing and entanglement generated by the spin-mixing dynamics around the phase transition point. It is shown that the steady squeezing and entanglement can be obtained, and the Bogoliubov approximation can well describe the dynamics of spin-nematic squeezed vacuum state.

We investigate the quantum phase transition (QPT) in the XXZ central spin model, which can be described as a spin-$\frac{1}{2}$ particle coupled to $N$ bath spins. In general, the QPT is supposed to occur only in the thermodynamical limit. In contrast, we present that the central spin model exhibits a normal-to-superradiant phase transition in the limit where the ratio of the transition frequency of the central spin to that of the bath spins and the number of bath spins tend to infinity. We give the low-energy effective Hamiltonian analytically in the normal phase and the superradiant phase, and we find that the longitudinal interaction $\mathrm{\ensuremath{\Delta}}$ can significantly influence the excitation number and the coherence of the ground state. These two quantities are remarkably enhanced for the negative longitudinal interaction while suppressed for the positive longitudinal interaction. In addition, the finite-size effect on the central spin model is also illustrated through the mean-field analysis. We further exploit the quantum Fisher information to characterize the QPT and propose a measurement scheme that can be applied in practice. This work builds a connection between the qubit-spin systems and the qubit-field systems, which provides a possibility for the realization of criticality-enhanced quantum sensing in central spin systems.

We consider a process of parameter estimation in a spin-j system surrounded by a quantum-critical spin chain. Quantum Fisher information lies at the heart of the estimation task. We employ Ising spin chain in a transverse field as the environment which exhibits a quantum phase transition. Fisher information decays with time almost monotonously when the environment reaches the critical point. By choosing a fixed time or taking the time average, one can see the quantum Fisher information presents a sudden drop at the critical point. Different initial states of the environment are considered. The phenomenon that the quantum Fisher information, namely, the precision of estimation, changes dramatically can be used to detect the quantum criticality of the environment. We also introduce a general method to obtain the maximal Fisher information for a given state.

Summary form only given. The photon model of the quantum electromagnetic (EM) field has recently been developed further for the areas of cavity QED and quantum optics in dielectric media. The basic problem is that the natural EM field modes to use (Fox-Li modes) are eigenfunctions of a non-Hermitian operator, unlike the normal modes that satisfy a Helmholtz equation. Such eigenfunctions have complex eigenvalues, and unlike the situation for normal modes, orthogonality occurs with a related set of adjoint mode functions, satisfying eigenvalue equations involving the adjoint operator. The standard canonical quantization procedure based on expanding the vector potential in terms of modefunctions then runs into difficulties, and the desired outcome of showing the EM field is equivalent to a set of quantum harmonic oscillators (QHO) is hard to achieve, as this paper shows.

We analyze the bipartite and multipartite entanglement for the ground state of the one-dimensional $XY$ model in a transverse magnetic field in the thermodynamical limit. We explicitly take into account the spontaneous symmetry breaking in order to explore the relation between entanglement and quantum phase transitions. As a result we show that while both bipartite and multipartite entanglement can be enhanced by spontaneous symmetry breaking deep into the ferromagnetic phase, only the latter is affected by it in the vicinity of the critical point. This result adds to the evidence that multipartite, and not bipartite, entanglement is the fundamental indicator of long-range correlations in quantum phase transitions.

We study the quantum phase transition driven by a point impurity in a chain seamed with ring frustration. With strong coupling and light impurity, the system is in a topological extended-kink (TEK) phase, which exhibits gapless excitations in the bulk. With strong coupling and heavy impurity, the system is in a gapped kink bound state (KBS) phase. Two-point bulk and impurity correlations are defined to characterize the two phases. In the TEK phase, both the bulk and impurity correlations are long range and factorizable so that scaling functions can be parsed. The scaling functions relies on the distance scaled by the system's size. An impurity correlation length can be extracted from the impurity correlation. In the transition from TEK to KBS, the scaling function of the bulk correlation undergoes an abrupt steplike change. Meanwhile, the impurity correlation length decreases from a divergent value to a finite one. The ground state of the TEK phase retains a relatively high value of entanglement entropy due to the absence of symmetry breaking. However, spontaneous symmetry breaking occurs in the KBS phase, which induces antiferromagnetic order in the bulk and entangled spin configuration near the impurity.

According to our scheme to construct quantum phase transitions (QPTs) in spin chain systems with matrix product ground states, we first successfully combine matrix product state (MPS) QPTs with spontaneous symmetry breaking. For a concrete model, we take into account a kind of MPS QPTs accompanied by spontaneous parity breaking, though for either side of the critical point the GS is typically unique, and show that the kind of MPS QPTs occur only in the thermodynamic limit and are accompanied by the appearance of singularities, diverging correlation length, vanishing energy gap and the entanglement entropy of a half-infinite chain not only staying finite but also whose first derivative discontinuous.

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.

In this work, the concept of quantum Fisher information (QFI) is used to characterize the quantum transitions and factorization transitions in one-dimensional anisotropic XY models with periodic coupling interaction and quasiperiodic one. For the periodic-two model, it is found that the Ising transition and anisotropic transition can be distinctively illustrated by the evolution of QFI and its first-order derivatives, confirmed additionally by the scaling behavior. For the quasiperiodic Fibonacci chain, the number of quantum phase transitions increases from one to the lth Fibonacci number $$F_{l}$$Fl when the anisotropic parameter $$\gamma $$? approaches zero. The phase diagram for the approximant $$F_{l}=8 $$Fl=8 is derived as an example. In addition, the factorization transition in the two models can be marked by the correlation quantity defined from the QFI. The present work demonstrates the implication of the QFI as a general fingerprint to characterize the quantum transitions and factorization transitions.

Path integral approach to the full Dicke model

Quantum sensing is one of the arenas that exemplifies the superiority of quantum technologies over their classical counterparts. Such superiority, however, can be diminished due to unavoidable noise and decoherence of the probe. Thus, metrological strategies to fight against or profit from decoherence are highly desirable. This is the case of certain types of decoherence-driven many-body systems supporting dissipative phase transitions, which might be helpful for sensing. Boundary time crystals are exotic dissipative phases of matter in which the time-translational symmetry is broken, and long-lasting oscillations emerge in open quantum systems at the thermodynamic limit. We show that the transition from a symmetry unbroken into a boundary time crystal phase, described by a second-order transition, reveals quantum-enhanced sensitivity quantified through quantum Fisher information. We also determine the critical exponents of the system and establish their relationship. Our scheme is indeed a demonstration of harnessing decoherence for achieving quantum-enhanced sensitivity. From a practical perspective, it has the advantage of being independent of initialization and can be captured by a simple measurement.

We carefully investigate the comprehensive impact of atom-cavity interaction and artificial magnetic fields on quantum phase transitions of anti-Jaynes-Cummings triangle model in the infinite frequency limit. We discover that ground states of the optical field can be a gapped normal phase (NP) or three kinds of gapless superradiant phases with infinite degeneracy. When the atom-cavity coupling is weak, the optical field is in a NP, which is a vacuum with no photons. Otherwise, it will stay at one of the superradiant phases: a normal superradiant phase without photon currents and another two chiral superradiant phases with opposite photon currents. The former only breaks the continuous U(1) symmetry and its gapless excitations are normal Goldstone modes. Nevertheless, the latter, mainly induced by an external synthetic gauge field, break both the continuous U(1) symmetry and chiral symmetry, thereby corresponding gapless excitations are chiral Goldstone modes. In addition, we also propose a detecting scheme to distinguish these superradiant phases.