20 votes
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Is the vacuum state a coherent state?

The coherent state $\vert \alpha\rangle$ is just a vacuum state $\vert 0\rangle$ translated in $x$ and $p$ space so $\alpha=x_0+ip_0$. Thus the vacuum state is a coherent state that has not been ...
ZeroTheHero's user avatar
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15 votes

Eigenstates of the creation operator

Well, here is a seat-of-the pants "lark" formal answer, going to "rigged" Fock spaces and places you (or anybody else) shouldn't really be at; except you may already have been ...
Cosmas Zachos's user avatar
15 votes
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Countable basis of coherent states used to express coherent states

A more detailed investigation of the completeness of an arbitrary countable subset $\lvert \alpha_i\rangle$ of coherent states can be found in "On the completeness of a system of coherent states" by ...
ACuriousMind's user avatar
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12 votes

What does coherent superposition mean?

The word "coherent" is used in Physics in a rather sloppy way. Your first state is a linear combination of harmonic oscillator eigenvectors that turns into a gaussian in momentum/position ...
QuantumBrick's user avatar
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11 votes
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Why are coherent states of harmonic oscillators called "coherent"?

Coherent states are eigenvectors for the (bosonic) annihilator,$$\hat a ~|\alpha\rangle = \alpha~|\alpha\rangle,$$and if we define the position and momentum quadratures as $\hat x = \hat a^\dagger + \...
CR Drost's user avatar
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11 votes

How are coherent states in the real world made?

Coherent states appear in nature because they're what you get when you drive a harmonic oscillator through a dipole interaction. Suppose we push on a harmonic oscillator with a time dependent force $F(...
DanielSank's user avatar
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10 votes
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What is the position wavefunction of coherent states?

Derivation from the eigenvalue condition The most straightforward approach is to start from the position representation of the annihilation operator $a$. I'll use the convention $a=\frac{1}{\sqrt2}(x+...
glS's user avatar
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9 votes

How are coherent states in the real world made?

Coherent states arise naturally in many quantum systems besides the free electromagnetic field. Many of them are mentioned in the following review by WM Zhang, which is actually dedicated to the same ...
David Bar Moshe's user avatar
9 votes

Countable basis of coherent states used to express coherent states

Nice question! It would surprise me that any coherent state basis would not be overcomplete, and I read your expansion a a very elegant proof of it. [Erratum: such bases exist, see the answer by ...
Frédéric Grosshans's user avatar
9 votes
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Why is laser light described by a coherent state?

A true but misleading identity Consider a single mode, and let $|n\rangle$ be the state with $n$ photons in that mode. A coherent state has the form $$ \newcommand{\la}{\langle} \newcommand{\ra}{\...
Chiral Anomaly's user avatar
8 votes

Why the name 'displacement' operator?

A coherent state is characterized by a complex number $\alpha \in \mathbb C$. Applying the displacement operator $D(\beta)$ to $|\alpha\rangle$ translates $\alpha$ in the complex plane by $\beta$, in ...
Noiralef's user avatar
  • 7,163
7 votes

How to compute expectation value $\langle e^{iH}\rangle$ for quadratic Hamiltonians?

There is a discussion of the evolution of a wavefunction under your general quadratic hamiltonian in the book by Guillemin and Sternberg "Symplectic methods in Physics." It's in their section on ...
mike stone's user avatar
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6 votes

Why are coherent states of harmonic oscillators called "coherent"?

In my opinion, the most intuitive way of explaining the meaning of harmonic oscillator coherent state is the following: An Harmonic Oscillator Coherent State (AOCS) is a solution of the time-...
AndreaPaco's user avatar
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6 votes

Completeness relation of coherent spaces

So let's start a step back because your coherent states are not normalized as I would normalize them. Coherent states The coherent states come from their response to the bosonic annihilator, $$\hat b |...
CR Drost's user avatar
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6 votes
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Representation of the displacement-operator in number basis

The displacement operator satisfies the identity $$ \hat{D}^{\dagger}(\alpha) = \hat{D}(-\alpha). $$ Therefore, when $m<n$, \begin{align*} ⟨m|\hat{D}(\alpha)|n⟩ &=\left(⟨n|\hat{D}^{\dagger}(\...
march's user avatar
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6 votes
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Is the density matrix corresponding to a state $|\alpha\rangle$ simply $\rho =| \alpha \rangle \langle \alpha \mid$?

The density matrix is indeed $|\alpha\rangle\langle\alpha|$. Remember that $\alpha \in \mathbb{C}$ so when you form $\langle\alpha|$ you must conjugate $\alpha$ (I always forget). So, $\rho_\alpha := |...
Mashy's user avatar
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6 votes
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Calculating free energy from coherent state path integral

The extra term you are missing can be interpreted as an error in the zero point energy. This error comes from the regularization of the infinite product for the partition function. Take for example ...
LPZ's user avatar
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6 votes

Finding the wavefunction of coherent state in 2D oscillator

@Abezhiko has done the bulk of the crucial combinatorics, up to the first line of his multi-level formula. Set $m\omega/\hbar= 1$, nondimensionalizing, and reinstate it only in the end if you must, ...
Cosmas Zachos's user avatar
5 votes
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Time evolution of squeezed states

It is a very good question with a subtle answer! Let me give you a short answer and then recommend a good paper that you can take a look. The time evolution of a squeezed states defined as you ...
Alex's user avatar
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5 votes
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Time Evolution of Coherent State (Gerry and Knight)

\begin{align} e^{-i\omega t/2}e^{-i\omega t \hat n}\vert\alpha\rangle&= e^{-i\omega t/2}\sum_{N}e^{-i\omega t \hat n}e^{-\vert\alpha\vert^2/2} \frac{\alpha^n}{\sqrt{n!}}\vert n\rangle\\ &=e^{-...
ZeroTheHero's user avatar
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5 votes

Why are coherent states of harmonic oscillators called "coherent"?

Coherent States of Harmonic Oscillator are Quantum Mechanical states which have definite phase and Minimum Uncertainty. The Quantum Mechanical states of Harmonic Oscillator do not have definite phase ...
FearlessVirgo's user avatar
5 votes

Definition of spatial and temporal coherence in QM?

To talk about lasers specifically we need to discuss the quantization of the electromagnetic field, i.e. quantum field theory (QFT). In order to avoid that subject for the moment and to build ...
Wintermute's user avatar
5 votes
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Why use coherent state path integral? What is its motivation or goal?

The coherent state path integral is basically a recipe for converting a Hamiltonian into a Lagrangian. In condensed matter, we often start with a "microscopic" Hamiltonian description of a material ...
tparker's user avatar
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5 votes

Basic question on holomorphic formalism in QM

The reason that you can pass from the holomorphic representation to the real representation and vice versa is the existence of a unitary transformation between the two representations. This ...
David Bar Moshe's user avatar
5 votes

Coherent states of Quantum harmonic oscillator

A good way to think of $\lambda$ is to consider the coherent state as a displaced harmonic oscillator ground state. Writing $\lambda=\lambda_r+i\lambda_i$ with $\lambda_{r,i}$ the real and imaginary ...
ZeroTheHero's user avatar
  • 45.4k
5 votes

Can we derive the classical beam-splitter theory from the quantum beam-splitter Fock state picture?

The cleanest way to understand beam splitters in quantum optics is via their action on the creation and annihilation operators, which is the same as their action on the classical field amplitudes: if ...
Emilio Pisanty's user avatar
5 votes

Laser power and coherent state amplitude

You can look at the details in the Wikipedia article on the quantization of the electromagnetic field, but I will sum up some relevant bits below. First, you seem to take the analogy with the massive ...
Frédéric Grosshans's user avatar
5 votes

Projective measurement using two mode squeezed state?

If you are happy with destructive measurements, this can be done in principle in the lab. Let $\rho$ be the input state, and write $|\xi\rangle = S|n,0\rangle$. Undo the squeezing transformation $S$, ...
Norbert Schuch's user avatar
5 votes
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Why don't the oscillator coherent states disperse in time?

The coherent structure of the state is not preserved by a general quantum evolution. It is preserved only in very special cases, i.e. when the evolution is generated by the second quantization of a ...
yuggib's user avatar
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5 votes
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It seems that expectation value of $H$ on coherent states is independent of time? But why?

As the Hamilton operator $H$ and the time-evolution operator $U(t)=e^{-iHt/\hbar}$ commute, the expectation value of $H$ in any state $|\psi(t)\rangle = U(t) | \psi(0)\rangle$ is independent of $t$. ...
Hyperon's user avatar
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