Gaussian States in Quantum Information

These notes originated out of a set of lectures in Quantum Optics and Quantum Information given by one of us (MGAP) at the University of Napoli and the University of Milano. A quite broad set of issues are covered, ranging from elementary concepts to current research topics, and from fundamental concepts to applications. A special emphasis has been given to the phase space analysis of quantum dynamics and to the role of Gaussian states in continuous variable quantum information.

Nonlinearity Induced Entanglement Stability in a Qubit-Oscillator System

We consider a system composed of a qubit interacting with a quartic (undriven) nonlinear oscillator (NLO) through a conditional displacement Hamiltonian. We show that even a modest nonlinearity can enhance and stabilize the quantum entanglement dynamically generated between the qubit and the NLO. In contrast to the linear case, in which the entanglement is known to oscillate periodically between zero and its maximal value, the nonlinearity suppresses the dynamical decay of the entanglement once it is established. While the entanglement generation is due to the conditional displacements, as noted in several works before, the suppression of its decay is related to the presence of squeezing and other complex processes induced by two- and four-phonon interactions. Finally, we solve the respective Markovian master equation, showing that the previous features are preserved also when the system is open.

Optical quantum memory for polarization qubits with V-type three-level atoms

We investigate an optical quantum memory scheme with V-type three-level atoms based on the controlled reversible inhomogeneous broadening (CRIB) technique. We theoretically show the possibility to store and retrieve a weak light pulse interacting with the two optical transitions of the system. This scheme implements a quantum memory for a polarization qubit - a single photon in an arbitrary polarization state - without the need of two spatially separated two-level media, thus offering the advantage of experimental compactness overcoming the limitations due to mismatching and unequal efficiencies that can arise in spatially separated memories. The effects of a relative phase change between the atomic levels, as well as of phase noise due to, for example, the presence of spurious electric and magnetic fields are analyzed.

More bang for your buck: Super-adiabatic quantum engines

The practical untenability of the quasi-static assumption makes any realistic engine intrinsically irreversible and its operating time finite, thus implying friction effects at short cycle times. An important technological goal is thus the design of maximally efficient engines working at the maximum possible power. We show that, by utilising shortcuts to adiabaticity in a quantum engine cycle, one can engineer a thermodynamic cycle working at finite power and zero friction. Our findings are illustrated using a harmonic oscillator undergoing a quantum Otto cycle.

Cavity-aided quantum parameter estimation in a bosonic double-well Josephson junction

We describe an apparatus designed to make non-demolition measurements on a Bose-Einstein condensate (BEC) trapped in a double-well optical cavity. This apparatus contains, as well as the bosonic gas and the trap, an optical cavity. We show how the interaction between the light and the atoms, under appropriate conditions, can allow for a weakly disturbing yet highly precise measurement of the population imbalance between the two wells and its variance. We show that the setting is well suited for the implementation of quantum-limited estimation strategies for the inference of the key parameters defining the evolution of the atomic system and based on measurements performed on the cavity field. This would enable {\it de facto} Hamiltonian diagnosis via a highly controllable quantum probe.

Thermometry precision in strongly correlated ultracold lattice gases

The precise knowledge of the temperature of an ultracold lattice gas simulating a strongly correlated
system is a question of both fundamental and technological importance. Here, we address such
question by combining tools from quantum metrology together with the study of the quantum
correlations embedded in the system at finite temperatures. Within this frame we examine the spin-
1 2 XY chain, first estimating, by means of the quantum Fisher information, the lowest attainable
bound on the temperature precision. We then address the estimation of the temperature of the sample
from the analysis of correlations using a quantum non demolishing Faraday spectroscopy method.
Remarkably, our results show that the collective quantum correlations can become optimal
observables to accurately estimate the temperature of our model in a given range of temperatures.

Single- and two-mode quantumness at a beam splitter

In the context of bipartite bosonic systems, two notions of classicality of correlations can be defined: P-classicality, based on the properties of the Glauber-Sudarshan P-function; and C-classicality, based on the entropic quantum discord. It has been shown that these two notions are maximally inequivalent in a static (metric) sense -- as they coincide only on a set of states of zero measure. We extend and reinforce quantitatively this inequivalence by addressing the dynamical relation between these types of non-classicality in a paradigmatic quantum-optical setting: the linear mixing at a beam splitter of a single-mode Gaussian state with a thermal reference state. Specifically, we show that almost all P-classical input states generate outputs that are not C-classical. Indeed, for the case of zero thermal reference photons, the more P-classical resources at the input the less C-classicality at the output. In addition, we show that the P-classicality at the input -- as quantified by the non-classical depth -- does instead determine quantitatively the potential of generating output entanglement. This endows the non-classical depth with a new operational interpretation: it gives the maximum number of thermal reference photons that can be mixed at a beam splitter without destroying the output entanglement.

Nonclassicality and criticality in symmetry protected magnetic phases

Quantum and global discord in a spin-1 Heisenberg chain subject to single-ion anisotropy (uniaxial field) are studied using exact diagonalisation and the density matrix renormalisation group (DMRG). We find that these measures of quantum non-classicality are able to detect the quantum phase transitions confining the symmetry protected Haldane phase and show critical scaling with universal exponents. Moreover, in the case of thermal states, we find that quantum discord can increase with increasing temperature.

Thermodynamics of trajectories of a quantum harmonic oscillator coupled to N baths

We undertake a thorough analysis of the thermodynamics of the trajectories followed by a quantum harmonic oscillator coupled to $N$ dissipative baths by using a new approach to large-deviation theory inspired by phase-space quantum optics. As an illustrative example, we study the archetypal case of a harmonic oscillator coupled to two thermal baths, allowing for a comparison with the analogous classical result. In the low-temperature limit, we find a significant quantum suppression in the rate of work exchanged between the system and each bath. We further show how the presented method is capable of giving analytical results even for the case of a driven harmonic oscillator. Based on that result, we analyse the laser cooling of the motion of a trapped ion or optomechanical system, illustrating how the emission statistics can be controllably altered by the driving force.

Macroscopic Quantum Resonators (MAQRO): 2015 update

Do the laws of quantum physics still hold for macroscopic objects - this is at the heart of Schrödinger’s cat paradox - or do gravitation or yet unknown effects set a limit for massive particles? What is the fundamental relation between quantum physics and gravity? Ground-based experiments addressing these questions may soon face limitations due to limited free-fall times and the quality of vacuum and microgravity. The proposed mission Macroscopic Quantum Resonators (MAQRO) may overcome these limitations and allow addressing such fundamental questions. MAQRO harnesses recent developments in quantum optomechanics, high-mass matter-wave interferometry as well as state-of-the-art space technology to push macroscopic quantum experiments towards their ultimate performance limits and to open new horizons for applying quantum technology in space. The main scientific goal is to probe the vastly unexplored ‘quantum-classical’ transition for increasingly massive objects, testing the predictions of quantum theory for objects in a size and mass regime unachievable in ground-based experiments. The hardware will largely be based on available space technology. Here, we present the MAQRO proposal submitted in response to the 4th Cosmic Vision call for a medium-sized mission (M4) in 2014 of the European Space Agency (ESA) with a possible launch in 2025, and we review the progress with respect to the original MAQRO proposal for the 3rd Cosmic Vision call for a medium-sized mission (M3) in 2010. In particular, the updated proposal overcomes several critical issues of the original proposal by relying on established experimental techniques from high-mass matter-wave interferometry and by introducing novel ideas for particle loading and manipulation. Moreover, the mission design was improved to better fulfill the stringent environmental requirements for macroscopic quantum experiments.

Divisible quantum dynamics satisfies temporal Tsirelson's bound

We show that divisibility of qubit quantum processes implies temporal Tsirelson's bound. We also prove that the classical bound of the temporal Bell's inequality holds for dynamics that can be described by entanglement-breaking channels---a more general class of dynamics than that allowed by classical physics.

Complete framework for efficient characterisation of non-Markovian processes

A non-Markovian process is one that retains `memory' of its past. A systematic understanding of these processes is necessary to fully describe and harness a vast range of complex phenomena; however, no such general characterisation currently exists. This long-standing problem has hindered advances in understanding physical, chemical and biological processes, where often dubious theoretical assumptions are made to render a dynamical description tractable. Moreover, the methods currently available to treat non-Markovian quantum dynamics are plagued with unphysical results, like non-positive dynamics. Here we develop an operational framework to characterise arbitrary non-Markovian quantum processes. We demonstrate the universality of our framework and how the characterisation can be rendered efficient, before formulating a necessary and sufficient condition for quantum Markov processes. Finally, we stress how our framework enables the actual systematic analysis of non-Markovian processes, the understanding of their typicality, and the development of new master equations for the effective description of memory-bearing open-system evolution.

Non-equilibrium thermodynamics of harmonically trapped bosons

We apply the framework of non-equilibrium quantum thermodynamics to the physics of quenched small-size bosonic quantum gases in a harmonic trap. By studying the temporal behaviour of the Loschmidt echo and of the atomic density profile within the trap, which are informative of the non-equilibrium physics and the correlations among the particles, we establish a link with the statistics of (irreversible) work done on the system. This highlights interesting connections between the degree of inter-particle entanglement and the non-equilibrium thermodynamics of the system.

Work Fluctuations in Bosonic Josephson Junctions

We calculate the first two moments and full probability distribution of the work performed on a system of bosonic particles in a two-mode Bose-Hubbard Hamiltonian when the self-interaction term is varied instantaneously or with a finite-time ramp. In the instantaneous case, we show how the irreversible work scales differently depending on whether the system is driven to the Josephson or Fock regime of the bosonic Josephson junction. In the finite-time case, we use optimal control techniques to substantially decrease the irreversible work to negligible values. Our analysis can be implemented in present-day experiments with ultracold atoms and we show how to relate the work statistics to that of the population imbalance of the two modes.

Nonequilibrium critical scaling from quantum thermodynamics

The emerging field of quantum thermodynamics is contributing important results and insights into archetypal many-body problems, including quantum phase transitions. Still, the question whether out-of-equilibrium quantities, such as fluctuations of work, exhibit critical scaling after a sudden quench in a closed system has remained elusive. Here, we take a novel approach to the problem by studying a quench across an impurity quantum critical point. By performing density matrix renormalization group computations on the two-impurity Kondo model, we are able to establish that the irreversible work produced in a quench exhibits finite-size scaling at quantum criticality. This scaling faithfully predicts the equilibrium critical exponents for the crossover length and the order parameter of the model, and, moreover, implies a new exponent for the rescaled irreversible work. By connecting the irreversible work to the two-impurity spin correlation function, our findings can be tested experimentally.