What is Coherent state: Definition and 36 Discussions
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. 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 related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.
These states, expressed as eigenvectors of the lowering operator and forming an overcomplete family, were introduced in the early papers of John R. Klauder, e. g.
In the quantum theory of light (quantum electrodynamics) and other bosonic quantum field theories, coherent states were introduced by the work of Roy J. Glauber in 1963 and are also known as Glauber states.
The concept of coherent states has been considerably abstracted; it has become a major topic in mathematical physics and in applied mathematics, with applications ranging from quantization to signal processing and image processing (see Coherent states in mathematical physics). For this reason, the coherent states associated to the quantum harmonic oscillator are sometimes referred to as canonical coherent states (CCS), standard coherent states, Gaussian states, or oscillator states.
As I understand it, when the squeezing operator acts on an annihilation/creation operator, a function of sinh(r) and cosh(r) is produced, where r is the squeezing parameter. I've been reading some papers that say that up to '15 dB of squeezing' have been produced in a laboratory. Does this mean...
Given the usual raising & lowering operators ##A^{\dagger}## & ##A## for a quantum harmonic oscillator, consider a coherent state ##|\alpha\rangle \equiv e^{\alpha A^{\dagger} - \bar{\alpha} A} |0\rangle##. I first check that ##|\alpha\rangle## is an eigenvector of ##A##. I already proved that...
For example if I consider H = (a^†)b+a(b^†), how will it act on even coherent state i.e. |α⟩+|-α⟩?. I know that |α⟩ don't act on (a^†) because |α⟩ is a eigenstate of lowering operator.
I began this solution by assuming a = x+iy since a is a complex number.
So I wrote expressions of <a| and |a> in which |n><n| = I.
I got the following integral:
Σ 1/πn! ∫∫ dx dy exp[-(x^2 + y^2)] (x^2 + y^2)^n I
I tried solving it using Integration by Parts but got stuck in the (x^2 + y^2)^n...
Given the hamiltonian:
\hat{H} = \hbar \omega_{0} \hat{a}^{+}\hat{a} + \chi (\hat{a}^{+}\hat{a})^2,
where ##\hat{a}^{+}##, ##\hat{a}## are creation and annihilation operators.
Find evolution of the state ##|\psi(t) \rangle##, knowing that initial state ##|\psi(0)\rangle = |\alpha\rangle##...
Hey there, the task I'm working on is written below.
Find the quadrature distribution ρ(q), for an optical mode being in the coherent state |α>.
Hint: use ∑Hn(x)*(t^n)/(n!)
I really am struggling with this type of tasks :D
I tried to follow a solved example that I found in my workbook, but...
As far as I know we can express the position and momentum operators in terms of ladder operators in the following way
$${\begin{aligned}{ {x}}&={\sqrt {{\frac {\hbar }{2}}{\frac {1}{m\omega }}}}(a^{\dagger }+a)\\{{p}}&=i{\sqrt {{\frac {\hbar }{2}}m\omega }}(a^{\dagger }-a)~.\end{aligned}}.$$...
The definition of coherent state $$|\phi\rangle =exp(\sum_{i}\phi_i \hat{a}^\dagger_i)|0\rangle $$
How can I show that the state is eigenstate of annihilation operator a?
i.e.
$$\hat{a}_i|\phi\rangle=\phi_i|\phi\rangle$$
Dear all,
I am aware that a weakly driven dipole can be modeled as a damped driven simple harmonic oscillator.
If I have to model the dipole as being driven by a classical monochromatic electromagnetic wave, would the corresponding simple harmonic oscillator then be in coherent state ?
In...
Homework Statement
Show that the coherent state ##|c\rangle=exp(\int \frac{d^3p}{(2\pi)^3}c(\vec{p})a^{\dagger}_{\vec{p}})|0\rangle## is an eigenstate of the anhiquilation operator ##a_{\vec{p}}##. Express it in terms of the states of type ##|\vec{p}_1...\vec{p}_N\rangle##
Homework Equations...
I am studying Quantum Cryptography and I am quite new in Quantum area. I have read an article and I found this confusing statement:
My questions:
1. The three stage protocol implementing multiphoton. What is the meaning of coherent states of mean photon number?
2. How to describe the quantum...
So, I am doing my undergraduate research project in Quantum Cryptography, and I have some confusion in a few areas, especially in the topic of continuous variable quantum key distribution.
From what I understand,
Discrete Variable - Single photon. That is, for example, the BB84 protocol. Bob...
In a coherent state defined by |\alpha\rangle = \exp{\left(-\frac{|\alpha|^2}{2}\right)}\exp{\left(\alpha \hat{a}^\dagger\right)} |0\rangle there is a definite phase associated with the state by \alpha = |\alpha| \exp{\left(i\theta\right)} where the number operator and phase operator are...
Homework Statement
In a coherent state ##|\alpha\rangle##, letting ##P(n)## denote the probability of finding ##n^{\text{th}}## harmomic oscillator state. Show that
$$\displaystyle{\langle\hat{n}\rangle \equiv \sum\limits_{n}n\ P(n)=|\alpha|^{2}}$$
Homework Equations
The Attempt at a...
I’m confused about how you find the vector |s;s⟩ to use in the general equation
|θ,ϕ⟩=exp(−iϕS3) * exp(−iθS2) |s;s⟩
For spin Coherent states (From http://www.scholarpedia.org/article/Coherent_state_(Quantum_mechanics)#4._Spin_Coherent_States
Eq 12)
Or
how you find the vector |j,m=j⟩ to use...
I've found online that the coherent state of the harmonic oscillator is
|\alpha \rangle = c \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} | n\rangle
where
|n\rangle = \frac{(a^\dagger)^n}{\sqrt{n!}} |0\rangle
and |0> is called the initial state.
I've some code where I need to have this...
How does one write a Lagrangian of a coherent state of vector fields (of differing energy levels) in terms of the the individual Lagrangians?
I desperately need to know how to know to do this, for a theory of mine to make any progress.
Please stick with me, if I didn't make sense just ask...
I am confused. In harmonic oscillator problem in quantum mechanics eigenstates of operator ##\hat{a}## are called coherent states. So we practically get Gaussian. In problem of quantum linear harmonic oscillator we have phonons ##|n \rangle##. In case of laser we have photons ##|n \rangle##...
My question is, how does one get a wave function for a 'combined' spin-1 and a spin-0 field? How is this possible? I have only been able to find combined states for equal spin identical particles.
If you don't understand my question, I'll be glad to reword it.
Homework Statement
I ended up solving the problem as I was typing it up, I am posting what I did anyway as it took so long to type and might be useful to someone else.
I am trying to figure out the position representation of a coherent state and it's time evolution. I should be getting a...
Hi, I am trying to find the wavefunction of a coherent state of the harmonic oscillator ( potential mw2x2/2 ) with eigenvalue of the lowering operator: b.
I know you can do this is many ways, but I cannot figure out why this particular method does not work.
It can be shown (and you can find...
Homework Statement
A particle in harmonic potential $$H=\hbar \omega (a^\dagger a+1/2)$$ is at ##t=0## in coherent state $$a|\psi>=2|\psi>$$ a) Calculate expected value of energy and it's uncertainty at ##t=0##
b) With what probability do we at ##t=0## measure the energy less than ##3\hbar...
Hi, there. The following is copied from a book " atom-photon interaction " by Prof. Claude Cohen-Tannoudji, Page 415.
I do not understand it at all. I do know some thing about the coherent state.
such as
##a |\alpha\rangle = \alpha |\alpha\rangle##
and
##|\alpha\rangle =...
So a coherent state in quantum mechanics is "the most classical" quantum state (A Gaussian wave packet), which satisfies the Heisenberg uncertainty relation with an equality. This allows the wave packet to travel in space in a more localized fashion (like a classical particle) because its...
". . . a coherent state remains unchanged by the detection (or annihilation) of field excitation or, say, a particle."
— Wikipedia's entry on "Coherent States"
I do not understand this. Is it saying that if a system in a coherent state, like a quantum harmonic oscillator, emits a...
Do photons used to produce the Sagnac effect (interference) have to have their phase in a coherent state or will a collection of random photons have the same property.
Along the same lines, if you purchase a simple laser pointer are the photon produced here in a coherent state?
Finally, is...
Dear all:
I knew the coherent state is the eigenstate of annihilation operator with a complex eigenvalue, which seriously disturbed me of thinking it. Can anyone give me a simple and understandable picture of what is it, thanks a lot in advance.
As I understand, it is postulated that only coherent states in LQG correspond to classical spacetimes. Is the ground state of LQG a coherent state? Otherwise, what principle selects that the universe should be in a coherent state?
Hi all,
the annihilation operator satisfies the equation \hat{a}|n>=\sqrt{n}|n-1> and \hat{a}|0>=0
so the matrix of \hat{a} should be
http://www.tuchuan.com/a/2010020418032158925.jpg
and zero is the only eigenvalue of this matrix.
The coherent state is defined by...
Coherent state... "Re"?
So we're talking about coherent state of harmonic oscillator... and for |psi(x,t)|^2, we came up with an equation... that has a term with "Re" in it. Does anyone know what I'm talking about or should I type up the whole equation? What does the "Re" mean?>?>
Greetings,
I´m taking contact with femtochemistry. As you know, the goal of this area is the real time following of the molecular dinamics, providing the first real experimental data from transition states, resonances, mechanisms of energy redistribution.. and so on.
The key to study...