The two-mode relative phase associated with Gaussian states plays an important role in quantum information processes in optical, atomic, and electronic systems. In this work, the origin and structure of the two-mode relative phase in pure Gaussian states is studied in terms of its dependences on the quadratures of the modes. This is done by constructing local canonical transformations to an associated two-mode squeezed state. The results are illustrated by studying the time dependence of the phase under a nonlocal unitary model evolution containing correlations between the modes. In a more general context, this approach may allow the two-mode phase to be studied in situations sensitive to different physical parameters within experimental configurations relevant to quantum information processing tasks.
The two-mode relative phase associated with Gaussian states plays an important role in quantum information processes in optical, atomic, and electronic systems. In this work, the origin and structure of the two-mode relative phase in pure Gaussian states is studied in terms of its dependences on the quadratures of the modes. This is done by constructing local canonical transformations to an associated two-mode squeezed state. The results are illustrated by studying the time dependence of the phase under a nonlocal unitary model evolution containing correlations between the modes. In a more general context, this approach may allow the two-mode phase to be studied in situations sensitive to different physical parameters within experimental configurations relevant to quantum information processing tasks.
The entanglement of general asymmetric pure Gaussian two-mode states is examined in terms of the coefficients of the quadrature components of the wave function. The entanglement criterion and the entanglement of formation are directly evaluated as a function of these coefficients, without the need for local unitary transformations. These reproduce results for the special case of symmetric pure states which employ a relation between squeezed states and Einstein-Podolsky-Rosen correlations. The modification of the quadrature coefficients and the corresponding entanglement due to application of various optical elements is also derived.
Quantum coherence plays an important role in quantum information processing. In this Letter, we experimentally demonstrate the conversion of local and correlated Gaussian quantum coherence in the process of converting two squeezed states into an entangled state. We also investigate the relationship among total, local, and correlated coherence and show that the total coherence of a two-mode Gaussian state is the sum of local quantum coherence of each mode and the correlated quantum coherence between two modes. Our results highlight the connection of different quantum coherence in a two-mode Gaussian system and provide references for potential application.
Extending the scheme developed for a single mode of the electromagnetic field in the preceding paper [F. Dell'Anno, S. De Siena, and F. Illuminati, Phys. Rev. A 69, 033812 (2004)], we introduce two-mode nonlinear canonical transformations depending on two heterodyne mixing angles. They are defined in terms of Hermitian nonlinear functions that realize heterodyne superpositions of conjugate quadratures of bipartite systems. The canonical transformations diagonalize a class of Hamiltonians describing nondegenerate and degenerate multiphoton processes. We determine the coherent states associated with the canonical transformations, which generalize the nondegenerate two-photon squeezed states. Such heterodyne multiphoton squeezed states are defined as the simultaneous eigenstates of the transformed, coupled annihilation operators. They are generated by nonlinear unitary evolutions acting on two-mode squeezed states. They are non-Gaussian, highly nonclassical, entangled states. For a quadratic nonlinearity the heterodyne multiphoton squeezed states define two-mode cubic phase states. The statistical properties of these states can be widely adjusted by tuning the heterodyne mixing angles, the phases of the nonlinear couplings, as well as the strength of the nonlinearity. For quadratic nonlinearity, we study the higher-order contributions to the susceptibility in nonlinear media and we suggest possible experimental realizations of multiphoton conversion processes generating the cubic-phase heterodyne squeezed states.
We introduce a formalism of nonlinear canonical transformations for general systems of multiphoton quantum optics. For single-mode systems the transformations depend on a tunable free parameter, the homodyne local-oscillator angle; for n-mode systems they depend on n heterodyne mixing angles. The canonical formalism realizes nontrivial mixing of pairs of conjugate quadratures of the electromagnetic field in terms of homodyne variables for single-mode systems, and in terms of heterodyne variables for multimode systems. In the first instance the transformations yield nonquadratic model Hamiltonians of degenerate multiphoton processes and define a class of non-Gaussian, nonclassical multiphoton states that exhibit properties of coherence and squeezing. We show that such homodyne multiphoton squeezed states are generated by unitary operators with a nonlinear time evolution that realizes the homodyne mixing of a pair of conjugate quadratures. Tuning of the local-oscillator angle allows us to vary at will the statistical properties of such states. We discuss the relevance of the formalism for the study of degenerate (up-)down-conversion processes. In a companion paper [F. Dell'Anno, S. De Siena, and F. Illuminati, 69, 033813 (2004)], we provide the extension of the nonlinear canonical formalism to multimode systems, we introduce the associated heterodyne multiphoton squeezed states, and we discuss their possible experimental realization.
Gaussian quantum states of bosonic systems are an important class of states. In particular, they play a key role in quantum optics as all processes generated by Hamiltonians up to second order in the field operators (i.e. linear optics and quadrature squeezing) preserve Gaussianity. A powerful approach to calculations and analysis of Gaussian states is using phase-space variables and symplectic transformations. The purpose of this note is to serve as a concise reference for performing phase-space calculations on Gaussian states. In particular, we list symplectic transformations for commonly used optical operations (displacements, beam splitters, squeezing), and formulae for tracing out modes, treating homodyne measurements, and computing fidelities.
Quantum continuous variable systems are quantum systems obeying the canonical commutation relations. As we have seen in the previous chapter, such systems always require the adoption of infinite-dimensional Hilbert spaces, even when a finite number of degrees of freedom is considered. Besides making their dynamics potentially intractable, this fact also implies that the study of properties related to the information and correlation content of continuous variable systems turns out to be particularly difficult. A widely known way to make continuous variable systems more tractable is to consider the restriction to Gaussian states. Already well trodden in older research on quantum optics, the study of Gaussian systems has more recently become a central theme of continuous variable quantum information. If open quantum systems, coupled to an external environment, are considered, the Gaussian regime applies when the system couples linearly to its environment and when system and environment are governed by Hamiltonians which are at most quadratic in the canonical operators (or, equivalently, when the overall Hamiltonian of system and environment is at most quadratic). These conditions may seem rather restrictive but, as we shall point out in the course of the book, are actually ordinarily met by a vast number of existing experimental set-ups in the areas of quantum optics, trapped ions, optomechanics, atomic ensembles and certain superconducting degrees of freedom (all of these platforms are reviewed in Chapter 9 ). What we will refer to as Gaussian measurements – POVMs that preserve the Gaussian character of initial states – are also customarily carried out with comparatively high efficiency, while the description of quantum noise can be incorporated seamlessly into the Gaussian picture. What is perhaps even more important to note is that the restriction to the Gaussian realm, while not exhaustive of the full wealth of dynamics and situations allowed in the whole Hilbert space, still lets one include most of the processes relevant to quantum technologies. Squeezing, whereby certain canonical quadratures have uncertainties below the vacuum state noise, and can hence be used in precision measurements, quantum entanglement, and cooling, where the entropy is drained out of a system in order to reach a pure quantum state and thus initialise information protocols or low noise experiments, may all be described within the Gaussian regime. Within the quantum information discourse, the Gaussian restriction has been often criticised on the grounds that no real genuinely quantum effects can ever be observed without leaving it at some point, since the statistics obtained from Gaussian states through Gaussian measurements can be reproduced by classical probability distributions. While technically correct, the reader should be reminded that such critiques often disregard the fact that the object described by a Gaussian state does entail a genuinely quantum description in a Hilbert space. Hence, for instance, the preparation of a pure Gaussian quantum state is the preparation of a pure vector of the Hilbert space. Likewise, a highly squeezed Gaussian state prepared for a metrological protocol is a state with very low noise in a certain physical observable, regardless of whether it can be mimicked by a classical distribution or not. The testing and exploitation of quantum non-locality (stronger than classical correlations that imply the presence of quantum entanglement) with Gaussian states does instead require a departure from Gaussian measurements. However, as we will see, certain non-Gaussian measurements are customarily implemented with current technology, and it is hence still relevant to study in detail the creation of such a resource for Gaussian states. This chapter will introduce Gaussian states from a somewhat unusual angle: as ground and thermal states of at most second-order Hamiltonians. Our treatment will highlight the privileged role played by quadratic canonical ( i.e. , ‘symplectic’) transformations in this context, which we will take care to describe in full generality, and will hinge on the reduction of Gaussian states into their normal modes, which allows for their systematic diagonalisation. We will then proceed to explore an equivalent characterisation in terms of statistical moments of the canonical operators, and to introduce the implications of non-commutativity on such statistics in the form of the Robertson–Schrödinger uncertainty relation.