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Time-dependent response and correlation functions are studied in random quantum systems composed of infinitely many parts without mutual interaction and defined with statistically independent random matrices. The latter are taken within the three Wigner-Dyson universality classes. In these systems, the response functions are shown to be exactly given by statistical averages over the random-matrix ensemble. Analytical results are obtained for the time dependence of the mean response and correlation functions at zero and positive temperatures. At long times, the mean correlation functions are shown to have a power-law decay for GOE at positive temperatures, but for GUE and GSE at zero temperature. Otherwise, the decay is much faster in time. In relation to these power-law decays, the associated spectral densities have a dip around zero frequency. The diagrammatic method is developed to obtain higher-order response functions and the third-order response function is explicitly calculated. The response to impulsive perturbations is also considered. In addition, the quantum fluctuations of the correlation function in individual members of the ensemble are characterised in terms of their probability distribution, which is shown to change with the temperature.

We undertake extensive analysis of English Premier League data over the period 2009/10 to 2017/18 to identify and rank key factors affecting the economic and footballing performances of the teams. Alternative end-of-season league tables are generated by re-ranking the teams based on five different descriptors - total expenditure, total funds spent on players, total funds spent on foreign players, the ratio of foreign to British players and the overall profit. The unequal distribution of resources and expenditure between the clubs is analyzed through Lorenz curves. A comparative analysis of the differences between the alternative tables and the conventional end-of-season league table establishes the most likely factors to influence the performances of the teams that we also rank using Principal Component Analysis. We find that the top teams in the league are also those that tend to have the highest expenditure overall, for all players, including foreign players; they also have the highest ratios of foreign to British players. Our statistical and machine learning study also indicates that successful performance on the field may not guarantee healthy profits at the end of the season., Comment: Graphical Abstract & 8 figures in the main text, 3 Appendices with additional figures and tables

Quantum dynamics of a collection of atoms subjected to phase modulation has been carefully revisited. We present an exact analysis of the evolution of a two-level system (represented by a spinor) under the action of a time-dependent matrix Hamiltonian. The dynamics is shown to evolve on two coupled potential energy surfaces, one of them binding while the other one scattering type. The dynamics is shown to be quasi-integrable with nonlinear resonances. The bounded dynamics with intermittent scattering at random moments presents the scenario reminiscent to Anderson and dynamical localization. We believe that a careful analytical investigation of a multi-component system which is classically non-integrable is relevant to many other fields, including quantum computation with multi-qubit system., Comment: 18 pages, 4 figures

A quantum engine with n qubits performing thermodynamic cycles with two thermal reservoirs is presented. While such constructions have been aplenty, here we show the existence of what we term as "limit cycle" at a purely quantum level of description owing to the properties of superoperators governing the evolution of states. It is shown that the limit cycle is the same under forward and reverse protocol of cycle operations. This limit cycle becomes the basis of the quantum engine. One dimensional Ising model has been used to illustrate these ideas.

We construct surface codes corresponding to genus greater than one in the context of quantum error correction. The architecture is inspired by the topology of invariant integral surfaces of certain non-integrable classical billiards. Corresponding to the fundamental domains of rhombus and square torus billiard, surface codes of genus two and five are presented here. There is significant improvement in encoding rates and code distance, in addition to immunity against noise.

For a three-level system monitored by an ancilla, we show that quantum Zeno effect can be employed to control quantum jump for error correction. Further, we show that we can realize cNOT gate, and effect dense coding and teleportation. We believe that this work paves the way to generalize the control of a qudit.

We construct a double-toric surface code by exploiting the planar tessellation using a rhombus-shaped tile. With n data qubits, we are able to encode at least n/3 logical qubits or quantum memories. By a suitable arrangement of the tiles, the code achieves larger distances, leading to significant error-correcting capability. We demonstrate the robustness of the logical qubits thus obtained in the presence of external noise. We believe that the optimality of the code presented here will pave the way for design of efficient scalable architectures.

We employ the stochastic path-integral formalism and action principle for continuous quantum measurements - the Chantasri-Dressel-Jordan (CDJ) action formalism [1, 2] - to understand the stages in which quantum Zeno effect helps control the states of a simple quantum system. The detailed dynamics of a driven two-level system subjected to repeated measurements unravels a myriad of phases, so to say. When the detection frequency is smaller than the Rabi frequency, the oscillations slow down, eventually coming to a halt at an interesting resonance when measurements are spaced exactly by the time of transition between the two states. On the other hand, in the limit of large number of repeated measurements, the dynamics organizes itself in a rather interesting way about two hyperbolic points in phase space whose stable and unstable directions are reversed. Thus, the phase space flow occurs from one hyperbolic point to another, in different ways organized around the separatrices. We believe that the systematic treatment presented here paves the way for a better and clearer understanding of quantum Zeno effect in the context of quantum error correction.

We present a systematic treatment of a 0-\texorpdfstring{$\pi$}{Lg} qubit in the presence of a time-dependent external flux. A gauge-invariant Lagrangian and the corresponding Hamiltonian is obtained. The effect of the flux noise on the qubit relaxation is obtained using perturbation theory. Under a time-dependent drive of sinusoidal form, the survival probability, and transition probabilities have been studied for different strengths and frequencies. The driven qubit is shown to possess coherent oscillations among two distinct states for a weak to moderate strength close to resonant frequencies of the unperturbed qubit. The parameters can be chosen to prepare the system in its ground state. This feature paves the way to prolong the lifetime by combining ideas from weak measurement and quantum Zeno effect. We believe that this is an important variation of a topologically protected qubit which is tunable.

We show that quantized superconducting circuits are non-integrable at the classical level of description, adorned by nonlinear resonances amidst stochastic sea. The spectral fluctuations of these quasi-integrable systems exhibit intermediate behaviour between regularity and chaos. The distribution function of ratios of adjacent spacings, and, nearest-neighbour spacing distribution functions attest to the occurrence of "mild chaos". Based on these features, we propose criteria for protection of qubits from decoherence which amounts to choosing the parameters of the system in a way that the system resides as close as possible to the elliptic point of the primary nonlinear resonance of the corresponding classical system., Comment: 7 pages, 4 figures

We present the relativistic analogue of Anderson localization in one dimension. We use Dirac equation to calculate the transmission probability for a spin-$\frac{1}{2}$ particle incident upon a rectangular barrier. Using the transfer matrix formalism, we numerically compute the transmission probability for the case of a large number of identical barriers spread randomly in one dimension. The particular case when the incident particle has three component momentum and shows spin-flip phenomena is also considered. Our calculations suggest that the incident relativistic particle shows localization behaviour similar to that of Anderson localization. A number of results which are generalizations of the non-relativistic case are also obtained., Comment: 14 pages, 11 figures

Data on nuclear level densities extracted from transmission data or gamma energy spectrum store the basic statistical information about nuclei at various temperatures. Generally this extracted data goes through model fitting using computer codes like CASCADE. However, recently established semiclassical methods involving no adjustable parameters to determine the level density parameter for magic and semi-magic nuclei give a good agreement with the experimental values. One of the popular ways to paramaterize the level density parameter which includes the shell effects and its damping was given by Ignatyuk. This damping factor is usually fitted from the experimental data on nuclear level density and it comes around 0.05 $MeV^{-1}$. In this work we calculate the Ignatyuk damping factor for various nuclei using semiclassical methods., Comment: 9 Figures, 1 Table

We establish a connection between macroscopic "heating or cooling" of a finite many-body quantum system and the non-adiabatic Landau-Zener-St\"{u}ckelberg transitions between its quantum states. We have considered the well-known Nilsson model for describing the single-particle states of nuclei and subject the system to a random walk in the deformation space. This subsumes modelling of an evolving many-body system where the dynamics is chaotic. We discover a universality in the distribution of final "temperatures", beginning with a canonical equilibrium at some temperature $T$. The quantum system is thrown out of equilibrium where free energy and work undergo fluctuations. These fluctuations are shown to respect Jarzynski inequality, and, the Bochkov-Kuzovlev equalities. We believe that this study will pave the way towards understanding non-equlibrium phenomena in other finite quantum systems like metallic clusters, quantum dots, and others., Comment: 5 Figures, 2 Tables

We study the phenomenon of neutrino spin-flavor oscillations due to solar magnetic fields. This allows us to examine how significantly the electron neutrinos produced in the solar interior undergo a resonant spin-flavor conversion. We construct analytical models for the solar magnetic field in all the three regions of the Sun. Neutrino spin-flavor oscillations in these magnetic fields are examined by studying the level crossing phenomenon and comparing the two cases of zero and non-zero vacuum mixing respectively. Results from the Borexino experiment are used to place an upper limit on the magnetic field in the solar core. Related phenomena such as effects of matter on neutrino spin transitions and differences between Dirac and Majorana transitions in the solar magnetic fields are also discussed., Comment: Published in Res.Astron.Astrophys

Coloring the faces of 2-dimensional square lattice with $k$ distinct colors such that no two adjacent faces have the same color is considered by establishing connection between the $k$ coloring problem and a generalized vertex model. Associating the colors with $k$ distinct species of particles with infinite repulsive force between nearest neighbors of the same type and zero chemical potential $\mu$ associated with each species, the number of ways $[W(k)]^N$ for large $N$ is related to the entropy of the {\it{hard square lattice gas}} at close packing of the lattice, where $N$ is the number of lattice sites. We discuss the evaluation of $W(k)$ using transfer matrix method with non-periodic boundary conditions imposed on at least one dimension and show the characteristic Toeplitz block structure of the transfer matrix. Using this result, we present some analytical calculations for non-periodic models that remain finite in one dimension. The case $k=3$ is found to approach the exact result obtained by Lieb for the residual entropy of ice with periodic boundary conditions. Finally, we show, by explicit calculation of the contribution of subgraphs and the series expansion of $W(k)$, that the genenralized Pauling type estimate(which is based on mean field approximation) dominates at large values of $k$. We thus also provide an alternative series expansion for the chromatic polynomial of a regular square graph., Comment: 16 pages, 8 figures, Some major revisions done