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arXiv:1411.3765

Temporal and spectral properties of quantum light

Birgit Stiller, Ulrich Seyfarth, Gerd Leuchs

Published May 18, 2017• 2 citations
arXivHTMLPDF

Abstract

Abstract This chapter starts from the classical optics description of a light field and its spectral densities, their measurement, and their interpretation, and discusses how these aspects are related to the quantum properties of a single light mode. It reviews ways to measure these quantum properties. Emphasis is placed on Gaussian states of a light mode, i.e. states for which the Wigner function has a two-dimensional Gaussian shape. The discussion continues with a description of more than one mode, and a unifying approach to quadratic Hamiltonians is presented, including phase conjugation, which is related to time reversal.

DOI
10.1093/oso/9780198768609.003.0004

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Abstract This chapter starts from the classical optics description of a light field and its spectral densities, their measurement, and their interpretation, and discusses how these aspects are related to the quantum properties of a single light mode. It reviews ways to measure these quantum properties. Emphasis is placed on Gaussian states of a light mode, i.e. states for which the Wigner function has a two-dimensional Gaussian shape. The discussion continues with a description of more than one mode, and a unifying approach to quadratic Hamiltonians is presented, including phase conjugation, which is related to time reversal.
Abstract In this chapter we collect a number of key elements of quantum optics relevant for quantum information theory. We begin by discussing the quantization of the electromagnetic field in terms of harmonic oscillators with ladder operators and the corresponding Fock space. Based on this description we introduce Glauber’s coherent states and examine their properties as proper quantum analogues of the classical harmonic oscillator. We then study the description of states of light in a quantum-mechanical phase space by employing quasi-probability distributions such as the Wigner function. With these tools at hand we then focus on the family of Gaussian states and their compact description in terms of covariance matrices and vectors of first moments. We review the formalism of Gaussian operations, symplectic operations, the Williamson normal form, and the Bloch-Messiah decomposition. Finally, we take a look at entanglement of Gaussian states.
Gaussian quantum states hold special importance in the continuous variable regime. In quantum information science, the understanding and characterization of central resources such as entanglement may strongly rely on the knowledge of the Gaussian or non-Gaussian character of the quantum state. However, the quantum measurement associated with the spectral photocurrent of light modes consists of a mixture of quadrature observables. Within the framework of two recent papers [Phys. Rev. A 88, 052113 (2013) and Phys. Rev. Lett. 111, 200402 (2013)], we address here how the statistics of the spectral photocurrent relates to the character of the Wigner function describing those modes. We show that a Gaussian state can be misidentified as non-Gaussian and vice versa, a conclusion that forces the adoption of tacit a priori assumptions to perform quantum state reconstruction. We experimentally analyze the light beams generated by the optical parametric oscillator operating above threshold to show that the data strongly supports the generation of Gaussian states of the field, validating the use of necessary and sufficient criteria to characterize entanglement in this system.
The language of statistical moments and covariance matrices we introduced in the previous chapter offers a compact and efficient formalism to deal with Gaussian states. A much more general approach may be taken to describe any quantum state in a setting which is reminiscent of classical phase space. This is the framework of quantum characteristic functions and quasi-probability distributions, which goes back to seminal work by Wigner on quantum corrections to classical statistical mechanics, and bloomed in the sixties with the rise of theoretical quantum optics and the emergence of a general unifying picture. Conceptually, the phase space description of quantum states hinges on the completeness of the set of displacement operators, which we shall prove in the form of the Fourier–Weyl relation between density matrices and characteristic functions. This relationship will constitute the bridge between phase space and Hilbert space descriptions, which will be useful in several applications in the context of quantum information. It will be convenient to handle most of the proofs and mathematical arguments concerning characteristic functions and quasi-probability distributions on a single mode of the system. Because displacement operators of multimode systems are just tensor products of local displacement operators, the extension of the formalism to systems with many degrees of freedom will be straightforward. Nevertheless, we shall always take care of explicitly linking the single-mode formulae which will appear in this chapter to the general multimode description adopted in the previous one.
In the present PhD work, we focus on a specific class of quantum states of light: the non-Gaussian states. These states have the particularity of exhibiting Wigner functions with some negative values. This quantum feature is a necessary condition to perform some quantum computation task; furthermore it is also useful for various other applications, including quantum communication and metrology. Different strategies can be used to generate these states. Here, we start from Gaussian states produced by optical parametric oscillators in the continuous wave regime, (i.e. single-mode and two-mode squeezed vacuum states). The non-Gaussian feature can only be obtained by non-linear phenomena (over-quadratic Hamiltonian). In our case, the non-linearity is induced by photon-counting-based measurements (also called non-Gaussian measurements). This study is mainly divided into two parts. First, the generation of non-classical states associated with two types of qubit encoding: the single-photon state, used for quantum information with discrete variables, and the coherent state superposition (the so-called optical Schrodinger cat state), used for quantum information with continuous variables. These two states have then been used to perform some quantum information protocols. The first one addresses the problem of single-photon entanglement witness, and the other the generation of entanglement between the two encodings (also called hybrid entanglement).
This review covers recent theoretical and experimental efforts to extend the application of the continuous-variable quantum technology of light beyond "Gaussian" quantum states, such as coherent and squeezed states, into the domain of "non-Gaussian" states with negative Wigner functions. Starting with basic Gaussian nonclassicality associated with single- and two-mode vacuum states produced by means of parametric down-conversion and applying a set of standard tools, such as linear interferometry, coherent state injection, and conditional homodyne and photon number measurements, one can implement a large variety of optical states and processes that are relevant in fundamental quantum physics as well as quantum optical information processing. We present a systematic review of these methods, paying attention to both fundamental and practical aspects of their implementation, as well as a comprehensive overview of the results achieved therewith.
This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.