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Quantum Theory of Light: The State of Quantized Field and Photon
Published in Yanhua Shih, An Introduction to Quantum Optics, 2020
In this chapter, we study the quantum state of quantized field. To analyze and to solve a problem of quantum optics, we need to know the state of the quantized field, or the state of a photon, or the state of a group of photons. We first study the photon number state and coherent state. Photon number state representation, especially the single-photon state representation, and the coherent state representation are widely used to specify the state of a radiation field. The concept of density operator or density matrix is introduced in the process of calculating the expectation value of a quantum observable. We then introduce and distinguish the pure state and mixed state. The quantum states of composite system, especially the state of two-photon field, are studied by introducing an 2-D Hilbert space constructed as the direct or tensor product of the Hilbert spaces of two subsystems. A simple model for the creation of single-photon state and multi-photon state is given with details. The concepts of product state, entangled state, and mixed state are introduced and distinguished in the last section of this chapter.
Molecular Vibrational Imaging by Coherent Raman Scattering
Published in Shoogo Ueno, Bioimaging, 2020
Yasuyuki Ozeki, Hideaki Kano, Naoki Fukutake
To derive a quantum-mechanical picture of the CRS processes, we introduce the density matrix ρ(t), and describe the interaction between light and molecules [136]. The density matrix can address mixed states, i.e., an ensemble of molecules or statistics of quantum states. Specifically, the diagonal elements of the density matrix provide populations (probability) of the corresponding eigenstate and the off-diagonal elements provide the coherence (or degree of superposition) of different eigenstates. The density matrix obeys the Liouville variant (quantum Liouville equation) of the Schrödinger equation as follows: ∂ρ∂t=1iℏ[H(t),ρ],
Quantum Cascade Lasers: Electrothermal Simulation
Published in Joachim Piprek, Handbook of Optoelectronic Device Modeling and Simulation, 2017
Song Mei, Yanbing Shi, Olafur Jonasson, Irena Knezevic
The simulations of electronic transport fall into two camps depending on how the electron single-particle density matrix is treated. The diagonal elements of the density matrix represent the occupation of the corresponding levels and off-diagonal elements represent the “coherence” between two levels. Transport is semiclassical or incoherent when the off-diagonal coherences are much smaller than the diagonal terms and can be approximated as proportional to the diagonal terms times the transition rates between states [50]. In that case, the explicit calculation of the off-diagonal terms is avoided and only the diagonal elements are tracked, which simplifies the simulation considerably. However, when the off-diagonal terms are appreciable, transport is partially coherent and has to be addressed using quantum transport techniques.
Efficient transfer of inversion doublet populations in deuterated ammonia using adiabatic rapid passage
Published in Molecular Physics, 2022
S. Herbers, Y. M. Caris, S. E. J. Kuijpers, J.-U. Grabow, S. Y. T. van de Meerakker
Coherent phenomena can be described by the time-dependent Schrödinger equation. Related to this equation but able to track the population of multiple states at once is the von Neumann equation in the interaction picture: The density matrix contains the populations of the levels as diagonal elements. The time is denoted as t, with t = 0 representing the start of the excitation pulse. The rovibrational Hamiltonian contains the energies of the levels as diagonal elements. The matrix is used to transform into the interaction frame. It contains a reference energy level on all diagonal elements for the lower energy levels. Typically the lowest energy level of the system is chosen for this purpose. For all upper energy levels the diagonal elements are with ω being the circle-frequency of the radiation. The transition dipole matrix contains all transition dipole moments, and ϵ represents the amplitude of the raditation field. The notation is the commutator. The dimension of the matrices in this equation is equal to the number of levels involved in the transition scheme, i.e. neglecting non-involved states.
Pre-Born–Oppenheimer molecular structure theory
Published in Molecular Physics, 2019
A possible resolution of the quantum-classical molecular structure puzzle will start out from the description of the molecule as an open quantum system being in interaction with an environment [124,125]. According to decoherence theory pointer states are selected by the continuous monitoring of the environment. As a result, the system's reduced density matrix (after tracing out the environmental degrees of freedom from the world's density matrix) written in this pointer basis evolves in time so that its off-diagonal elements decay exponentially with some decoherence time, characteristic to the underlying microscopic interaction process with the environment (radiation or matter). This decay of the off-diagonal elements leads to the suppression of the interference terms between different pointer states, and results in a (reduced) density matrix the form of which corresponds to that of mixed states. Hence, this result can be interpreted as the emergence of the classical features in a quantum mechanical treatment.4 So, decoherence theory allows us to identify pointer states, which are selected and remain stable as a result of the molecule's interaction with its environment.
Unravelling open-system quantum dynamics of non-interacting Fermions
Published in Molecular Physics, 2018
Decoherence, dephasing and dissipation in large open quantum systems are important phenomena in a broad variety of fields, such as nonadiabatic processes in chemistry and materials science, [1–7], quantum biology [8,9] and quantum information [10,11]. They are commonly described using the concept of the density matrix (DM), which generalises the notion of a wave function as the quantum state descriptor. Despite great success in atomic physics, DM approaches have not found extensive application in the field of large electronic systems, except in cases of small systems, where it is sufficient and possible to address only a small number of electronic states [12–18]. For describing the quantum dynamics of open systems having a large number of electrons and electronic states a different approach is probably needed. Here, it is natural to consider time-dependent (current) density functional theory (TDDFT), based on the Runge–Gross (RG) theorem [19] which simplifies the treatment of the dynamics of interacting electrons by mapping them onto non-interacting Fermions. Extensions of the RG theorem to open systems have indeed appeared [20–25], but the follow-up progress has yet to be achieved, and the main cause for delay is the fact that non-interacting Fermions develop an interaction through the coupling with the bath.1