Quasinormal modes (QNM) are the modes of
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
dissipation of a perturbed object or field, ''i.e.'' they describe perturbations of a field that decay in time.
Example
A familiar example is the perturbation (gentle tap) of a wine glass with a knife: the glass begins to ring, it rings with a set, or superposition, of its natural frequencies — its modes of sonic energy dissipation. One could call these modes ''
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
'' if the glass went on ringing forever. Here the amplitude of oscillation decays in time, so we call its modes ''quasi-normal''. To a high degree of accuracy, quasinormal ringing can be approximated by
:
where
is the amplitude of oscillation,
is the frequency, and
is the decay rate. The quasinormal
frequency is described by two numbers,
:
or, more compactly
:
:
Here,
is what is commonly referred to as the
quasinormal mode frequency. It is a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
with two pieces of information: real part is the temporal oscillation; imaginary part is the temporal,
exponential decay
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
.
::
In certain cases the amplitude of the wave decays quickly, to follow the decay for a longer time one may plot
::
Mathematical physics
In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, a quasinormal mode is a formal solution of linearized
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s (such as the linearized equations of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
constraining perturbations around a
black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
solution) with a complex
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
(
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
).
Black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
s have many quasinormal modes (also: ringing modes) that describe the exponential decrease of asymmetry of the black hole in time as it evolves towards the perfect spherical shape.
Recently, the properties of quasinormal modes have been tested in the context of the
AdS/CFT correspondence
In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter s ...
. Also, the asymptotic behavior of quasinormal modes was proposed to be related to the
Immirzi parameter in
loop quantum gravity, but convincing arguments have not been found yet.
Electromagnetism and photonics
There are essentially two types of resonators in optics. In the first type, a high-
Q factor optical microcavity
An optical microcavity or microresonator is a structure formed by reflecting faces on the two sides of a spacer layer or optical medium, or by wrapping a waveguide in a circular fashion to form a ring. The former type is a standing wave cavity, and ...
is achieved with lossless dielectric optical materials, with mode volumes of the order of a cubic wavelength, essentially limited by the diffraction limit. Famous examples of high-Q microcavities are micropillar cavities, microtoroid resonators, photonic-crystal cavities. In the second type of resonators, the characteristic size is well below the diffraction limit, routinely by 2-3 orders of magnitude. In such small volumes, energies are stored for a small period of time. A plasmonic nanoantenna supporting a localized
surface plasmon
Surface plasmons (SPs) are coherent delocalized electron oscillations that exist at the interface between any two materials where the real part of the dielectric function changes sign across the interface (e.g. a metal-dielectric interface, such ...
quasinormal mode essentially behaves as a poor antenna that radiates energy rather than stores it. Thus, as the optical mode becomes deeply sub-wavelength in all three dimensions, independent of its shape, the Q-factor is limited to about 10 or less.
Formally, the resonances (i.e., the quasinormal mode) of an open (non-Hermitian) electromagnetic micro or nanoresonators are all found by solving the time-harmonic source-free Maxwell’s equations with a complex
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, the real part being the resonance frequency and the imaginary part the damping rate. The damping is due to energy loses via leakage (the resonator is coupled to the open space surrounding it) and/or material absorption. Quasinormal-mode solvers exist to efficiently compute and normalize all kinds of modes of plasmonic nanoresonators and photonic microcavities. The proper normalisation of the mode leads to the important concept of mode volume of non-Hermitian (open and lossy) systems. The mode volume directly impact the physics of the interaction of light and electrons with optical resonance, e.g. the local density of electromagnetic states,
Purcell effect
Henry Purcell (, rare: September 1659 – 21 November 1695) was an English composer.
Purcell's style of Baroque music was uniquely English, although it incorporated Italian and French elements. Generally considered among the greatest Eng ...
,
cavity perturbation theory In mathematics and electronics, Cavity perturbation theory describes methods for derivation of perturbation formulae for performance changes of a cavity resonator.
These performance changes are assumed to be caused by either introduction of a sma ...
,
strong interaction
The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the n ...
with quantum emitters,
superradiance
In physics, superradiance is the radiation enhancement effects in several contexts including quantum mechanics, astrophysics and relativity.
Quantum optics
In quantum optics, superradiance is a phenomenon that occurs when a group of ''N'' emit ...
.
Biophysics
In computational biophysics, quasinormal modes, also called quasiharmonic modes, are derived from diagonalizing the matrix of equal-time correlations of atomic fluctuations.
See also
*
Resonance (quantum field theory).
References
{{Reflist
Theoretical physics
Waves
Biophysics