Questions tagged [coherent-states]

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Coherent States and ladder operators

I am working on a problem involving ladder operators and coherent states. I know that $$a|z\rangle=z|z\rangle$$ and $$\langle z|a^\dagger = \langle z|z^*.$$ I am wondering how I could figure out ...
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1answer
41 views

Creation and Annnihilator Operators: generality and meaning

I am studying my fisrst course in quantum mechanichs where we treated the example of the Harmonic Oscillator through the Weyl Heisenberg Spectrum Generating Algebra Method. In that context we ...
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0answers
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Diagonalisation of quasi-thermal state

I have the following density operator $$\frac{1}{t \pi N} \int_{\mathbb{C}} \mathrm{d}^2\gamma \exp \left[ -\frac{|\gamma+r\alpha|^2}{t^2 N} \right] |{\gamma}\rangle\langle{\gamma}|,$$ where $0\leq t,...
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1answer
61 views

What is the definition of functions of Grassmann numbers?

I understand there are some relevant questions, but none of them solves my issue. From Atland and Simons (Condensed Matter Field Theory), the definition of functions of Grassmann numbers are defined ...
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1answer
80 views

Show: $\langle n \vert \psi \rangle \langle \psi \vert n \rangle = \langle -\psi \vert n \rangle \langle n \vert \psi \rangle$ [closed]

The book (Altland and Simons, Condensed Matter Field Theory, Ch. 4.2) I am reading makes use of the identity \begin{equation} \langle n \vert \psi \rangle \langle \psi \vert n \rangle = \langle -\psi \...
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1answer
69 views

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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3answers
102 views

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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1answer
49 views

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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0answers
24 views

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
66 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
44 views

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
114 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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1answer
71 views

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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1answer
98 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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2answers
100 views

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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0answers
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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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3answers
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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
67 views

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
38 views

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
69 views

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
106 views

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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1answer
78 views

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
116 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
176 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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0answers
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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
97 views

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
49 views

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 ...
3
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2answers
208 views

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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0answers
87 views

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
109 views

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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4answers
111 views

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
95 views

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
166 views

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
134 views

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
154 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
85 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
115 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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65 views

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
76 views

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
158 views

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
221 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
171 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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99 views

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
340 views

$P$ 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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2answers
423 views

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 \...