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Maxwell–Bloch Equations
The Maxwell–Bloch equations, also called the optical Bloch equations describe the dynamics of a two-state quantum system interacting with the electromagnetic mode of an optical resonator. They are analogous to (but not at all equivalent to) the Bloch equations which describe the motion of the nuclear magnetic moment in an electromagnetic field. The equations can be derived either semiclassically or with the field fully quantized when certain approximations are made. Semi-classical formulation The derivation of the semi-classical optical Bloch equations is nearly identical to solving the two-state quantum system (see the discussion there). However, usually one casts these equations into a density matrix form. The system we are dealing with can be described by the wave function: : \psi = c_g\psi_g + c_e\psi_e : \left, c_g\^2 + \left, c_e\^2 = 1 The density matrix is : \rho = \begin\rho_ & \rho_ \\ \rho_ & \rho_\end = \beginc_e c_^* & c_e c_^* \\ c_g c_^* & c_g c_^* \end (ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-state Quantum System
In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit. Two-state systems are the simplest quantum systems that are of interest, since the dynamics of a one-state system is trivial (as there are no other states the system can exist in). The mathematical framework required for the analysis of two-state systems is that of linear differential equations and linear algebra of two-dimensional spaces. As a result, the dynamics of a two-state system can be solved analytically without any approximation. The generic behavior of the system is that the wavefunction's amplitude oscillates between the two states. A very well known example of a two-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent State
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems.J.R. Klauder and B. Skagerstam, ''Coherent States'', World Scientific, Singapore, 1985. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well (for an early reference, see e.g. Schiff's textbook). The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be relate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiconductor Bloch Equations
The semiconductor Bloch equations Lindberg, M.; Koch, S. W. (1988). "Effective Bloch equations for semiconductors". ''Physical Review B'' 38 (5): 3342–3350. do10.1103%2FPhysRevB.38.3342/ref> (abbreviated as SBEs) describe the optical response of semiconductors excited by coherent classical light sources, such as lasers. They are based on a full quantum theory, and form a closed set of integro-differential equations for the quantum dynamics of microscopic polarization and charge carrier distribution. Schäfer, W.; Wegener, M. (2002). ''Semiconductor Optics and Transport Phenomena''. Springer. . Haug, H.; Koch, S. W. (2009). ''Quantum Theory of the Optical and Electronic Properties of Semiconductors'' (5th ed.). World Scientific. p. 216. . The SBEs are named after the structural analogy to the optical Bloch equations that describe the excitation dynamics in a two-level atom interacting with a classical electromagnetic field. As the major complication beyond the atomic approac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorenz System
The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. In popular media the " butterfly effect" stems from the real-world implications of the Lorenz attractor, namely that in a chaotic physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable. The shape of the Lorenz attractor itself, when plotted in phase space, may also be seen to resemble a butterfly. Overview In 1963, Edward Lorenz, with the help of Ellen Fetter who was responsible for the numerical s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atomic Electron Transition
Atomic electron transition is a change (or jump) of an electron from one energy level to another within an atom or artificial atom. It appears discontinuous as the electron "jumps" from one quantized energy level to another, typically in a few nanoseconds or less. It is also known as an electronic (de-)excitation or atomic transition or quantum jump. Electron transitions cause the emission or absorption of electromagnetic radiation in the form of quantized units called photons. Their statistics are Poissonian, and the time between jumps is exponentially distributed. The damping time constant (which ranges from nanoseconds to a few seconds) relates to the natural, pressure, and field broadening of spectral lines. The larger the energy separation of the states between which the electron jumps, the shorter the wavelength of the photon emitted. History Danish physicist Niels Bohr first theorized that electrons can perform quantum jumps in 1913. Soon after, James Franck and Gustav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotating-wave Approximation
The rotating-wave approximation is an approximation used in atom optics and magnetic resonance. In this approximation, terms in a Hamiltonian that oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radiation is near resonance with an atomic transition, and the intensity is low. Explicitly, terms in the Hamiltonians that oscillate with frequencies \omega_L + \omega_0 are neglected, while terms that oscillate with frequencies \omega_L - \omega_0 are kept, where \omega_L is the light frequency, and \omega_0 is a transition frequency. The name of the approximation stems from the form of the Hamiltonian in the interaction picture, as shown below. By switching to this picture the evolution of an atom due to the corresponding atomic Hamiltonian is absorbed into the system ket, leaving only the evolution due to the interaction of the atom with the light field to consider. It is in this picture that the rapidly oscillating terms mentioned previou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindblad Superoperator
In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation, named after Vittorio Gorini, Andrzej Kossakowski, George Sudarshan and Göran Lindblad), master equation in Lindblad form, quantum Liouvillian, or Lindbladian is one of the general forms of Markovian and time-homogeneous master equations describing the (in general non-unitary) evolution of the density matrix that preserves the laws of quantum mechanics (i.e., is trace-preserving and completely positive for any initial condition). The Schrödinger equation is a special case of the more general Lindblad equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation. The Schrödinger equation deals with state vectors, which can only describe pure quantum states and are thus less general than density matrices, which can describe mixed states as well. Motivation In the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cavity Quantum Electrodynamics
Cavity quantum electrodynamics (cavity QED) is the study of the interaction between light confined in a reflective cavity and atoms or other particles, under conditions where the quantum nature of photons is significant. It could in principle be used to construct a quantum computer. The case of a single 2-level atom in the cavity is mathematically described by the Jaynes–Cummings model, and undergoes vacuum Rabi oscillations , e\rangle, n-1\rangle\leftrightarrow, g\rangle, n\rangle, that is between an excited atom and n-1 photons, and a ground state atom and n photons. If the cavity is in resonance with the atomic transition, a half-cycle of oscillation starting with no photons coherently swaps the atom qubit's state onto the cavity field's, (\alpha, g\rangle+\beta, e\rangle), 0\rangle\leftrightarrow, g\rangle(\alpha, 0\rangle+\beta, 1\rangle), and can be repeated to swap it back again; this could be used as a single photon source (starting with an excited atom), or as an interfa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pauli Matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in connection with isospin symmetries. \begin \sigma_1 = \sigma_\mathrm &= \begin 0&1\\ 1&0 \end \\ \sigma_2 = \sigma_\mathrm &= \begin 0& -i \\ i&0 \end \\ \sigma_3 = \sigma_\mathrm &= \begin 1&0\\ 0&-1 \end \\ \end These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together with the iden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lowering Operator
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum. Terminology There is some confusion regarding the relationship between the raising and lowering ladder operators and the creation and annihilation operators commonly used in quantum field theory. The creation operator ''a''''i''† increments the number of particles in state ''i'', while the corresponding annihilation operator ''ai'' decrements the number of particles in state ''i''. This clearly satisfies the requirements of the above definition of a ladder operator: the incrementing or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jaynes–Cummings Model
The Jaynes–Cummings model (sometimes abbreviated JCM) is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity (or a bosonic field), with or without the presence of light (in the form of a bath of electromagnetic radiation that can cause spontaneous emission and absorption). It was originally developed to study the interaction of atoms with the quantized electromagnetic field in order to investigate the phenomena of spontaneous emission and absorption of photons in a cavity. The Jaynes–Cummings model is of great interest to atomic physics, quantum optics, solid-state physics and quantum information circuits, both experimentally and theoretically. It also has applications in coherent control and quantum information processing. Historical development 1963: Edwin Jaynes & Fred Cummings The model was originally developed in a 1963 article by Edwin Jaynes and Fred Cummings to elucidate the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bloch Equations
In physics and chemistry, specifically in nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI), and electron spin resonance (ESR), the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization M = (''M''''x'', ''M''''y'', ''M''''z'') as a function of time when relaxation times ''T''1 and ''T''2 are present. These are phenomenological equations that were introduced by Felix Bloch in 1946. Sometimes they are called the equations of motion of nuclear magnetization. They are analogous to the Maxwell–Bloch equations. In the laboratory (stationary) frame of reference Let M(''t'') = (''Mx''(''t''), ''My''(''t''), ''Mz''(''t'')) be the nuclear magnetization. Then the Bloch equations read: :\frac = \gamma ( \mathbf (t) \times \mathbf (t) ) _x - \frac :\frac = \gamma ( \mathbf (t) \times \mathbf (t) ) _y - \frac :\frac = \gamma ( \mathbf (t) \times \mathbf (t) ) _z - \frac where γ is the gyromagnetic ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
