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arXiv:1810.06932 · Physical Review A

Unidimensional time-domain quantum optics

Stéphane Virally, Bertrand Reulet

Published Aug 21, 2019• 15 citations
arXivHTMLPDF

Abstract

Choosing the right first-quantization basis in quantum optics is critical for the interpretation of experimental results. The usual frequency basis is, for instance, inappropriate for short, subcycle waveforms. Deriving first quantization in the time domain shows that the electromagnetic field is not directly proportional, nor even causally related, to the photonic field (the amplitude probability of a photon detection). We derive the relation between the two and calculate the statistics of the electromagnetic field for specific states in the time domain, such as the single photon Fock state. We introduce the dual of the Hamiltonian in the time domain and extend the concept of quadratures to all first quantization bases.

DOI
10.1103/physreva.100.023833

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Choosing the right first-quantization basis in quantum optics is critical for the interpretation of experimental results. The usual frequency basis is, for instance, inappropriate for short, subcycle waveforms. Deriving first quantization in the time domain shows that the electromagnetic field is not directly proportional, nor even causally related, to the photonic field (the amplitude probability of a photon detection). We derive the relation between the two and calculate the statistics of the electromagnetic field for specific states in the time domain, such as the single photon Fock state. We introduce the dual of the Hamiltonian in the time domain and extend the concept of quadratures to all first quantization bases.
In a concise, but accessible for the first acquaintance form the procedure for the quantization of linear oscillator is set out. By analogy with this procedure the procedure of quantization (second quantization) of classical Maxwell’s electrodynamics is set up. The physical sense of the wave functions arguments of transverse electromagnetic field and its Fourier transformation is set up. One pay attention as for quantum coherent (almost a classic) states of the electromagnetic field and for photonics Fock states. Attention is drawn to the fact of absence the power of the universal content of such concepts as field amplitude, phase and number of particles (photons), which are used by experimenter’s to describe the states of a quantized field. The semi quantitative description the interaction processes of a quantum electromagnetic field with substance is set up. Specified situations are shown in which the discrepancy between the predictions of classical and quantum electrodynamics is noticeable at the macroscopic level.
We revisit the quantum oscillator model of the electromagnetic field and conclude that, while the nonlocal positive and negative frequency ladder operators generate a photon Fock basis, the Hermitian field operators obtained by second quantization of real Maxwell fields describe photon-antiphoton pairs that couple locally to Fermionic matter and can be modeled classically. Their commutation relations define a scalar product that can be the basis of a first quantized theory of single photons. Since a one-photon state collapses to a zero-photon state when the photon is counted, the field describing it must be interpreted as a probability amplitude.
We employ here the enhancement of the vector potential amplitude quantization at a single photon state. The analysis of the general solution of the vector potential, obtained by resolving Maxwell’s equations, implies that the amplitude is proportional to the angular frequency. The photon vector potential function αkλ(r,t) can be written in the plane wave representation satisfying the classical wave propagation equation, Schrödinger’s equation for the energy with the relativistic massless field Hamiltonian and a linear time-dependent equation for the vector potential amplitude operator. Thus, the vector potential αkλ(r,t) with the quantized amplitude may play the role of a real wave function for the photon in a nonlocal representation that can be suitably normalized. We then deduce that the amplitudes of the electric and magnetic fields, respectively, of a single free photon are proportional to the square of the angular frequency. This might open perspectives for the development of nondestructive photon detection methods based on the influence of the electric and/or magnetic fields of photons on the energy levels of atoms and molecules.
Quantum light is considered to be one of the key resources of the coming second quantum revolution expected to give rise to groundbreaking technologies and applications. If the spatio-temporal and polarization structure of modes is known, the properties of quantum light are well understood. This information provides the basis for contemporary quantum optics and its applications in quantum communication and metrology. However, thinking about quantum light at the most fundamental timescale, namely the oscillation cycle of a mode or the inverse frequency of an involved photon, we realize that the corresponding picture has been missing until now. For instance, how to comprehend and characterize a single photon at this timescale? To fill this gap, we demonstrate theoretically how local quantum measurements allow to reconstruct and visualize a quantum field under study at subcycle scales, even when its temporal mode structure is a priori unknown. In particular, generation and tomography of ultrabroadband squeezed states as well as photon-subtracted states derived from them are described, incorporating also single-photon states. Our results set a cornerstone in the emerging chapter of quantum physics termed time-domain quantum optics. We expect this development to elicit new spectroscopic concepts for approaching e.g. fundamental correlations and entanglement in the dynamics of quantum matter, overcoming the temporal limitation set by the oscillation cycles of both light and elementary excitations.
We consider an approach in which the usual wave function ${\ensuremath{\psi}}_{j}({x}_{j})$ in the quadrature representation of mode jjj of the electromagnetic field is further quantized to produce a field operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}}_{j}({x}_{j}).$ Since the electromagnetic field is already second quantized, this corresponds to an additional or third quantization. The third-quantization approach can be used to perform certain quantum optics calculations in the Heisenberg picture that could only be performed in the Schr\"odinger picture when using the conventional second-quantized theory. This approach also allows an interesting generalization of quantum optics and quantum electrodynamics that is analogous to symmetry breaking in elementary particle theory. The predictions of the generalized theory could be tested using a proposed photon scattering experiment.
Abstract One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.