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Questions tagged [coherent-states]

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Why are coherent states not linearly independent?

From the completeness relation one can see that, $$|\psi \rangle = \int \frac{d^2 \alpha}\pi \langle \alpha | \psi \rangle |\alpha\rangle.$$ And if $|\psi\rangle = |\beta \rangle$ (which is another ...
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What are the eigenvector's of the $\hat a^2$ operator?

Since $\hat a^2$ and $\hat a$ commute, then one of the eigenvectors of $\hat a^2$ will be, the coherent state $|\alpha\rangle$. Are there others states as well?
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Coherent states in QFT and Poisson distribution

Consider a scalar field interacting with an external source $J(x)$ for a finite period of time. The vacua for before and after are defined by $$ a_\mathbf{p} \left| 0_- \right> = 0$$ $$ A_\mathbf{p}...
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Numerical method to find the roots of the expected value of a spin 2j state

Currently I am working with finding the solutions for the following problem: I have a unit sphere in which I have n points defined by their polar and azimuthal angles: $\theta_n , \phi_n$. I then do ...
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1answer
62 views

Coherent state under Kerr evolution

I have a bosonic mode associated to the usual operators $a$, $a^\dagger$. I'm interested in knowing the evolution of a coherent state $\vert \alpha \rangle = e^{\alpha a^\dagger - \alpha^\ast a}\vert ...
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1answer
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Perelomov coherent states for an arbitrary Hamiltonian

I'm reading about Perelomov coherent states, but I'm not sure if I'm getting it right. From this question and some Perelomov papers I understand the following: The Perelomov coherent states are ...
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2answers
108 views

Why don't the oscillator coherent states disperse in time?

A Gaussian wavepacket is made of a continuum of frequencies (or energies) and stretches in time due to the phenomenon of dispersion: the different plane wave components with different frequencies ...
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Plotting quadrature uncertainties in phase space

In most books like in the picture given below, the uncertainties regarding quantum states like coherent and squeezed states are represented in phase space plot by some area enclosed within a circle or ...
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84 views

Coherent state of second quantized hamiltonian

In my preparation for the exam I tried to solve the exercise 2.4 in Coleman's Introduction to Many-Body Physics. I like diagonalizing Hamiltonian, so I picked this problem. Also to learn more about ...
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Creation operator acting on a coherent state. Occupation number operator

For a coherent state $$|\alpha\rangle=e^{-\frac{|\alpha|^{2}}{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}(a^{\dagger})^n}{n!}|0\rangle$$ I want to find a simplified expression for $a^{\dagger}|\alpha\...
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Some questions on coherent states and corresponding Hilbert spaces. Reproducing kernal

I have a few questions related to coherent states. I use this source https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_5.pdf. Using standart inner product $\langle\cdot|\cdot\rangle$ ...
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Finding the uncertainty of coherent states using operators [duplicate]

The coherent state is defined such that $a|\alpha\rangle =\alpha|\alpha\rangle $. We can calculate the uncertainty using $$\sqrt{\langle x^2\rangle-\langle x\rangle ^2}\sqrt{\langle p^2\rangle-\...
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1answer
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Scalar product of squeezed coherent states

Consider two states of the type $|\alpha,\xi \rangle = \hat{D}(\alpha) \hat{S}(\xi) |0\rangle$, where $D$ and $S$ are the displacement and squeeze operators, respectively, and $|0\rangle$ is a 1D ...
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Expansion of an arbitrary density matrix in terms of coherent states?

It is well-known that any pure state can be expanded in terms of coherent states namely $$\left|\psi\right>=\frac{1}{\pi}\int d^2\alpha\left<\alpha|\psi\right>\left|\alpha\right>$$ due to ...
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1answer
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What is the closure relation for multimode coherent state?

How does the closure relation for multimode coherent state $| \{ \alpha_\lambda \} \rangle $ look like? I suppose it should be some generalization of the closure relation for singlemode state $$\frac{...
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1answer
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How is the complex integration done for the Wigner function in coherent state representation?

$$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{Tr}\left[ \hat{\rho}e^{\lambda\hat{a}^\dagger} e^{-\lambda^* \hat{a}} \right] e^{-\frac{|\lambda|^2}{2}} \, d^2\lambda....
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1answer
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Projective measurement using two mode squeezed state?

Let me define two mode squeezed states as $ \left | \xi \right>_n=\exp\left(\xi \hat{a}^\dagger \hat{b}^\dagger-\hat{a} \hat{b} \xi^\star\right)\left | n,0 \right>$ where $\left|n,0\right>$ ...
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does squeezed coherent state form an overcomplete basis

Coherent states form an over complete bases, but how about squeezed coherent state? Do they form an over complete basis as well?
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2answers
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Work definition does not make any sense for coherent states? TPM (two projective measurement) scheme

I'm talking about the relation that if we are in a isolated, quantum system that only allows for work exchange with the surrounding system we know from the first law of thermodynamics that $\Delta E = ...
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1answer
100 views

Laser power and coherent state amplitude

How it is possible to relate the power $P$ of a laser light with frequency $\omega$, to the amplitude $\alpha$ of its description as a coherent state $\vert \alpha \rangle$ ? For a massive particle in ...
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1answer
154 views

Eigenstates of the creation operator

We know that coherent states $\vert\alpha\rangle$ are eigenvectors of the annihilation operator $\hat{a}$, i.e. $$ \hat{a} \vert\alpha\rangle = \alpha \vert\alpha\rangle $$ while the creation operator ...
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Spread of the (smeared) field observable under time-evolution

Setup: Essentially, I'm interested in performing an analysis which is completely standard in QM, but I've never seen the analogue in QFT: Given I measure a system to have some value of its canonical ...
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1answer
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Constructing Gravitational waves with gravitons

Suppose I want to construct a gravitational wave as a coherent sum of many gravitons. It's easy to think of what the frequency distribution of the gravitons should be, as all the LIGO discoveries more ...
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2answers
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Experimental aspects of squeezing of photons and phonons

I know that photons have been squeezed to 15 dB and phonons have been squeezed by 7.2 dB. Spins have been squeezed by 20 dB. Why is it so hard to squeeze the states of particles? If I wanted to ...
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Fock states and beam splitters

A classical beam splitter divides incoming light into two parts according to the reflection and transmission coefficients. A quantum beam splitter on the other hand, can be modelled using Fock states,...
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Majorana Fermion Coherent States

I was wondering if there are coherent states for Majorana operators, so, states that fulfill the relation \begin{align} \hat{\gamma}_A |a,b\rangle &= a |a,b\rangle \\ \hat{\gamma}_B |a,b\...
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1answer
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Coherent states of the form $|{-\alpha}\rangle$

I've a brief question about coherent states in quantum mechanics. As everyone knows, a coherent state is just the proper state of the anhilitation operator $a$, thus they're defined with the ...
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Coherent state vector notation

I'm reading a paper which says that since $\alpha_k$ is real $$|-\alpha_k> = |(-\alpha_k,0)^T> = |(0, -\alpha_k)^T> $$ I'm confused about the notation and don't understand what the zeroes ...
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4answers
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What is the electric field due to a coherent radiation? Is it polarized?

What is the time-structure (i.e., how does the magnitude and direction change with time) of the expectation value of the electric field $\langle\textbf{E}\rangle(t)$ of a radiation field described by ...
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2answers
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Are Bi-photons and Fock States (with $N=2$) the same?

I've been researching about quantum states of light, and I came across with both terms. For what I understand of both, I think they are the same (entangled individual photons with same position and ...
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1answer
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Exponential of ladder operators acting on vacuum state [closed]

How would I solve expressions of the following nature: $$<0|e^{Vt(a+a^\dagger)}|0>$$ and $$<0|e^{\omega aa^\dagger t}|0>~?$$ My intuition is that I have to expand the exponent as a ...
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1answer
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Displacement operator action on exponential of ladder operators

How do I evaluate the following: $$D_{\alpha}^{-1} e^{-i\Omega aa^{+}t} D_{\alpha},$$ where $D_a$ is the usual displacement operator for coherent states $$D_{\alpha} = e^{\alpha a^{+} - \alpha^* a}.$$ ...
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1answer
143 views

Time evolution of operators with explicit time dependence in case of time dependent Hamiltonian

In case of a time dependent Hamiltonian of the sort $$H=\frac{p^2}{2m}+\frac{1}{2}m \omega(t) x^2$$ I have solved for the time evolution operator using the Schrodinger equation and got $U(t,0)$. If, I ...
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1answer
83 views

Operator in coherent state basis

I am reading "Introductory to Quantum Optics" by Christopher C. Gerry and Peter L. Knight but I don't understand a solution from which you can obtain the matrix elements of an operator in the number ...
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1answer
104 views

Coherent States and their existence

In my Quantum Mechanic class, I have learned that to solve for any quantum system, we solve the time independent Schrodinger equation(for time independent Hamiltonian) and then apply the time ...
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Determining the Greens function

I understand the basics of greens function. But here in this paper, i am not able to understand how have they solved for the Greens function. In particular, I don't get what $G_{R}$ and $G_{A}$ are....
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1answer
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Commutation relation coherent states

I am reading p. 159, chapter 4 of Condensed Matter Field Theory and I don't really get this commutation relation: They want to show that $\left[\hat{a}_i,\hat{a}_j^\dagger\right] = \delta_{ij}$. The ...
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Classical analogy for why driving a quantum harmonic oscillator creates a coherent state?

If we drive a quantum harmonic oscillator (e.g. starting from its ground state) at its resonance frequency then it will not just create an excitation in the first excited state but will create a ...
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3answers
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Does every quantum system have coherent states?

I am taking a course on Quantum Mechanics and last class we saw the coherent states of the harmonic oscillator, namely states $ | \lambda \rangle $ such that $$ a | \lambda \rangle = \lambda | \lambda ...
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3answers
152 views

Spin coherent state for general spin $S$

In Wen's book on Many-body QFT, he claimed that the coherent state for a spin-$S$ particle can be written as a tensor product of $2S$ spin-1/2 coherent states: $$|\hat{n}\rangle=|z\rangle\otimes|z\...
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1answer
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Is the superfluid state a coherent state?

In the normal to a superfluid phase transition, U(1) symmetry related to particle number conservation is spontaneously broken which seems to imply that the superfluid state is a state in which there ...
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2answers
151 views

Adjoint of creation and anihilation operators and problem with expected values of $\hat a$ and $\hat a^{\dagger}$ for coherent state

The definition of the adjoint operator of an operator $\hat A$ is \begin{equation} (\vec x|\hat A \vec y) = (\hat A \vec x| \vec y) \quad \forall x, y \in \mathcal{H} \end{equation} where $(\cdot|\...
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What is the main requirement for two arbitrary quantum states to be able to serve as qubit states?

Does it need to be orthogonal ? If now the state is not arbitrary but a coherent state $\lvert\alpha\rangle$, if we use $\lvert0\rangle=\lvert\alpha\rangle$ and $\lvert1\rangle=\lvert-\alpha\rangle$, ...
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Don't get the same result for an inner product doing it in different ways

I'm having a problem when trying to work out an inner product. First I have the inner product $\left<0|\psi\right>$, where $0$ is the fundamental state and $\left|\psi\right>=(a^\dagger - \...
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2answers
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$P$ representation of the thermal density operator

I'm trying to derive the P representation for the thermal state $$ \rho = \sum_{n=0}^\infty \frac{\mathrm{e}^{-\beta \omega n}}{Z} |n\rangle \langle n | $$ where $\beta$ is the inverse temperature, $...
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Coherent states of Quantum harmonic oscillator

Coherent states of Quantum harmonic oscillator . The Hamiltonian of Quantum harmonic oscillator is $H=(a^+ a+\frac{1}{2})\hbar \omega$,$a=\sqrt{\frac{m \omega}{2 \hbar}}(\hat{x}+\frac{i \hat{p}}{m \...
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Can you create an actual coherent state?

If I understood correctly, a coherent state $\lvert\alpha\rangle $ is an eigenstate of the creation and annihilation operators, meaning that adding or removing a particles does not change it. ...
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1answer
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Connection between Hubbard-Stratonovich and (generalized) coherent states

A simple-minded mean-field approximation for the Bose-Hubbard model consists in writing operators as $\hat{a}_i = \alpha_i + \hat{\delta \alpha}_i, \alpha_i \in \mathbb{C}$ and only include terms up ...
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Can we define the spin coherent state for spin half operator $\vec{S}$?

Here I meet two kinds of definitions for the spin coherent state: $\vec{\widehat{S}}|\vec{n}\rangle=S\vec{n}|\vec{n}\rangle$ (in my quantum field theory course); $\vec{n}\cdot\vec{\widehat{S}}|\vec{n}...
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Example of a quantum-mechanical theory with nontrivial classical limit

I am looking for a toy model example of a well defined quantum-mechanical theory with the following properties: It can be constructed via canonical quantization starting from some classical theory ...