Questions tagged [coherent-states]
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 obeys the minimal uncertainty relation in Heisenberg's uncertainty relationship.
239
questions
6
votes
2
answers
639
views
When is a state entangled?
I have read from What's the difference between an entangled state, a superposed state and a cat state? that an entangled state is one that cannot be expressed as product state. Suppose we have the ...
1
vote
1
answer
79
views
"Deriving" Poisson bracket from commutator
This Q/A shows that deriving P.B.s from commutators is subtle. Without going into deep deformation quantization stuff, Yaffe manages to show that $$\lim_{\hbar \to 0}\frac{i}{\hbar}[A,B](p,q)=\{a(p,q),...
- 785
0
votes
2
answers
84
views
Free evolution of coherent states
Is there a closed formula to express the time evolution of coherent states in absence of the potential term (only kinetic energy)?
The coherent state $|\alpha \rangle$ is defined by
$$\hat a|\alpha \...
- 9
0
votes
0
answers
72
views
Coherent State as Eigenvector for some Observable?
A coherent state $|\alpha\rangle$ is an eingenvector of the operator $\hat{a}$, but this is not an observable (i.e., not an hermitian operator). But every vector is eigenvector of a complete set of ...
3
votes
3
answers
186
views
Prove that integrating a displacement operator with a Gaussian gives $\int d^2\gamma e^{-|\gamma|^2/2}D(\gamma)=\pi|0⟩\!⟨0|$
I'm looking for "nice" ways to prove the following identity for displacement operators:
$$\int d^2\gamma e^{-|\gamma|^2/2}D(\gamma)=\pi|0⟩\!⟨0|,$$
with $|0\rangle$ the vacuum state and $D(\...
- 14.8k
1
vote
0
answers
76
views
Majorana Boson Coherent States
Consider $a$ be a bosonic operator, and we define $\Phi = a+a^{\dagger}$ and it is clear that $\Phi^{\dagger}=\Phi$ that implies "Majorana Boson". Now, i want to find the coherent states for ...
- 177
0
votes
0
answers
19
views
Integration over the complex plane and the completeness relation of the coherent states [duplicate]
I am studying some of the properties of coherent states using the book "Introductory Quantum Optics" by C. Gerry & L. Knight. (C. Gerry & L. Knight, Chapter 3, Section 5) And when I ...
2
votes
2
answers
94
views
Grassmann variables and orthogonality of coherent fermionic states
Let a coherent fermionic state
$$
\left|\phi\right> := \left|0\right> + \left|1\right> \phi,\tag{0}
$$
where $\phi$ is a Grassmann number (i.e. it anticommutes with other Grassmann numbers). ...
- 1,611
1
vote
1
answer
53
views
Question about coherent population trapping
In coherent population trapping, if we denote the ground states in a $\Lambda$-like system as $|0\rangle$ and $|1 \rangle$ and the excited state $|2 \rangle$, there is a linear combination $|d \...
- 581
1
vote
0
answers
24
views
How Does Laser Light Maintain Coherence Amid Photon-Atom Entanglement?
Laser light is known to produce "coherent state light," which consists of a superposition of different photon numbers. However, wouldn't the entanglement between the atoms and the light ...
- 2,193
1
vote
2
answers
66
views
Coherent creation operator: unitary or not?
In Quantum Mechanics, for coherent states $|z\rangle$ it can be prooved that if $|0\rangle$ is the vacuum state for an harmonic oscillator, therefore:
\begin{equation}
|z\rangle=e^{za^{\dagger}-z^*a}|...
1
vote
0
answers
33
views
Coherent spin state (CSS) for an electron with spin
Standard definition for the spin coherent state (CSS) for the system of $N$ identical particles reads
$$
|\theta, \phi\rangle = \bigotimes\limits_{k=1}^{N} \left[ \cos\frac{\theta}{2} |0\rangle_k + e^{...
- 227
2
votes
0
answers
63
views
Paradox when expressing an operator in terms of creation/annihilation operators [duplicate]
I'm trying to expand an arbitrary operator using creation/annihilation operators following this post, where $|m\rangle \langle n|$ is expressed as
$$
|n\rangle \langle m|~=~\sum_{k\in\mathbb{N}_0} c^{...
- 695
1
vote
0
answers
31
views
Existence of Glauber-Sudardhan $P$-representation of arbitrary given density operator for light field
In all textbooks on quantum optics I can reach (Scully, Leonhardt, Walls, etc), the Glauber-Sudardhan $P$-representation $P(\alpha)$ is introduced in the following two ways:
Fourier transform of $\...
- 695
2
votes
1
answer
145
views
Operator acting on product of coherent states
My problem
Find $O_\phi|\psi\rangle$, where the state $|\psi\rangle$ is defined on a composite space $\mathcal H_A\otimes \mathcal H_B$ as
$$|\psi\rangle = \left(\bigotimes_{k=1}^N|\alpha_k'\rangle\...
0
votes
1
answer
96
views
Identity of bosonic coherent states
I have a short question about the meaning of the identity of the bosonic coherent states.
Before I ask the question I will explain some background.
The eigenstate of the bosonic annihilation operator $...
- 237
1
vote
0
answers
22
views
Can we treat Gazeau-Klauder coherent states for infinite potential well as a superposition of Fock states?
If we define coherent states of infinite potential well based on Gazeau-Klauder coherent states. Can we use ladder operators and bosonic algebra for them which we use for Glauber coherent states?
- 781
0
votes
1
answer
74
views
Applying a phase shift to a coherent state vs a phase-space rotation (Mach-Zhender)
I'm having some trouble with the physical implementation of a phase vs rotation in phase-space for a coherent state.
Say I have a laser pulse which yields a coherent state $|\alpha\rangle$, I then put ...
- 53
0
votes
0
answers
45
views
Prepration of a Gaussian modulated coherent state
In Continuous Variable -Quantum Key Distribution (CVQKD), usually Gaussian modulated coherent states are sent. This means both quadratures of a coherent state are chosen from two normal distribution. ...
- 1
2
votes
3
answers
245
views
Finding the wavefunction of coherent state in 2D oscillator
Suppose I have a two-dimensional harmonic oscillator, $H= \hbar\omega(a_x^{\dagger}a_x+a_y^{\dagger}a_y)$. We define the operator $b=\frac{1}{\sqrt{2}}(a_x+ia_y)$.
If eigenkets of the hamiltonian are $...
- 81
1
vote
3
answers
242
views
It seems that expectation value of $H$ on coherent states is independent of time? But why?
Let's say the particle is in the state $| \psi(0) \rangle = \exp(-i\alpha p/\hbar) |0 \rangle$, where $p$ is the momentum operator.
I have to show that $| \psi(0) \rangle$ is a coherent state and to ...
- 81
0
votes
0
answers
52
views
What is the state vector of a displaced (single-mode) squeezed vacuum state in the quadrature basis?
I've been hunting through the quantum optics literature for the displaced squeezed state written in the $q$-quadrature basis ($p$-quad would be fine too, since it's just a Fourier transform), but it ...
- 215
1
vote
0
answers
114
views
Phase distribution of coherent states
I am studying the phase distribution for coherent states, as is defined in quantum optics. (See, for example, Introductory Quantum Optics by Gerry and Knight, pages 46–48).
In this situation, we seek ...
2
votes
0
answers
93
views
Proof of coherent state displacement operator solution
In $3D$ space I have two $2\times2$ non-Hermitian matrix operators, $A$ and $A^\dagger$, of the form:
$$A=\begin{pmatrix}
A_{11}(x_j,\partial_j) & A_{12}(x_j,\partial_j)\\
A_{21}(x_j,\partial_j) &...
- 147
2
votes
2
answers
168
views
Non-classicality of coherent state and squeezed states
Recently I have started studying about the coherent state and squeezed states of light. But I have a question about why do we call these states non classical? What are the things that deny their ...
0
votes
0
answers
69
views
Is there any way to write eigenstates of infinite square well in terms of eigenstates of harmonic oscillator?
I wanted to find Husimi Q function using expression $Q= \langle \alpha|\rho|\alpha \rangle$, where $|\alpha \rangle $ is coherent states of harmonic oscillator. I want to consider system $\rho=|u_n\...
- 781
1
vote
0
answers
117
views
Finding an operator in terms of creation and annihilation operators that satisfy some conditions
I have a problem where I'm looking to find the following Hermitian operator $\hat{A}$ written in terms of the operators $\hat{a}^{\dagger}\hat{a}$, $\hat{a}^2$, $\hat{a}^{\dagger 2}$, $\hat{a}$, $\hat{...
- 21
1
vote
1
answer
110
views
$U(1)$ symmetry transformations in second quantization confusion
I'm reading Chapter 18 (BEC and Superfluidity) of Girvin and Yang and ran into some confusion.
Let $|\alpha\rangle = e^{-|\alpha|^2}e^{\alpha b^†_0}|0\rangle $, where $\alpha$ is just a complex ...
1
vote
0
answers
141
views
Large alpha coherent state almost eigenstate of creation operator?
I’m surveying the relation between quantum and classical mechanics. My interest is how a quantum coherent state approaches a classical state when the wave becomes bigger.
Regarding this, I have a ...
- 11
0
votes
0
answers
68
views
Coherent States and Temperature for Scalar QFT with Source
This is a follow-up question on a question I previously asked, namely Coherent states and thermal properties. The authors of the article I am referring to in the previous question (Thermodynamics of ...
- 3,992
0
votes
0
answers
129
views
Coherent states and thermal properties
I am reading a paper called Thermodynamics of Coherent States and Black Hole Entropy, written by Bashkirov and Sukhanov. If I understand correctly, they define a coherent state by the equation
$$a|d\...
- 3,992
0
votes
2
answers
191
views
What defines "minimal coherence" as a condition for the emergence of stationary interference in a chaotic wave field?
Consider the following observations:
A superposition of two electromagnetic waves with different frequencies will never produce visible interference patterns. Such waveforms will produce ...
- 181
3
votes
1
answer
222
views
Advantage of coherent path integral
I think(?) I am quite familiar with path integral over phase space, but not familiar with the coherent state path integral. What is the advantage of this coherent path integral besides the usual path ...
- 3,622
0
votes
0
answers
73
views
Homodyne Detection of Photon Number States
I'm currently trying to understand why you can detect the signal of single photons with homodyne detection. I found that the difference current $i_{34}$ is given by
$$i_{34}\sim -2 ⟨\Psi|_1⟨\alpha|_2\...
- 23
0
votes
0
answers
141
views
Evolution of Quantum Harmonic Oscillator into coherent state
Why does a quantum harmonic oscillator that is driven by an electromagnetic wave in cosine form with its frequency equal to the resonance frequency of the oscillator evolve from its groundstate into a ...
- 95
0
votes
1
answer
721
views
Coherent state basis
I'm learning about coherent states in a more in depth lesson the the quantum harmonic oscillator. Coherent states are eigenstates of the lowering operator. In my head this is just saying: since any ...
- 719
9
votes
1
answer
663
views
Calculating free energy from coherent state path integral
Edit: It turns out that problem encountered in this question is not limited to BdG Hamiltonians.
I am having trouble in using the coherent state path integral approach to calculate the free energy. ...
- 660
2
votes
2
answers
1k
views
Driven Quantum Harmonic Oscillator
Consider the Hamiltonian
$$
H = \frac{p^2}{2} + \frac{ x^2}{2} - F(t) x.
$$
This is essentially a time dependent shifted harmonic oscillator, which can be represented as
$$
H' = \frac{p^2}{2} + \frac{...
- 427
0
votes
1
answer
214
views
Is the wave function of a coherent state just a Gaussian density? [closed]
The formula giving the wavefunction of a coherent state looks pretty complicated, but am I correct in saying it is just a Gaussian distribution function? i.e.
$$\psi(x) = \frac{1}{\sigma \sqrt{2\pi} } ...
2
votes
1
answer
174
views
Non-uniqueness of Glauber-Sudarshan $P$-function
For a state $\rho$ acting on single bosonic mode with coherent states $|\alpha\rangle$, one can always define a $P$-function to furnish a diagonal representation of the state in the coherent-state ...
- 5,041
1
vote
0
answers
86
views
Trouble proving Wigner function identity [closed]
I am trying to prove $$\int d^2 \alpha W(\alpha)=1$$ where $W(\alpha)$ represents the Wigner funcion. However, I have trouble solving it. I tried solving it as follows but I think I have done some ...
- 157
4
votes
1
answer
161
views
Application of Schur's lemma to proving the completeness of coherent states
I am studying many-body path integral through Altland & Simons's textbook called "Condensed Matter Field Theory," and the book states the completeness of the coherent states as below:
$$\...
- 340
1
vote
0
answers
199
views
Overcompleteness of coherent states
I am trying to show that the basis of coherent states,
$$ |\phi\rangle =e^{-\sum_{\alpha'}\phi_{\alpha'}a^\dagger_{\alpha'}}|0\rangle, \quad\text{ where }\quad a_\alpha|\phi\rangle=\phi_\alpha|\phi\...
- 980
2
votes
0
answers
57
views
Could one call eigenstates of $ \hat{a} = \hat{x} + i\hat{p}$ coherent states for other potentials than the harmonic oscillator?
Let's say I look at the quantum system of a particle in one dimension, subject to any other potential than the one of the harmonic oscillator, and I define $\hat{a}$ as stated above. I would find the ...
- 6,763
0
votes
1
answer
75
views
Fermionic coherent state in Fock representation
The notes I follow define a Fermionic coherent state $|c\rangle$ as
\begin{equation}
\hat{c}|c\rangle=c|c\rangle
\end{equation}
where $\hat{c}$ is the Fermionic annihilation operator and $c$ is a ...
- 155
1
vote
1
answer
227
views
SPDC One Arm vs Very Weak Coherent state
I know that SPDC(Spontaneous Parametric Down Conversion) is a method to generate heralded single photon source.
So If we do homodyne tomography of single photon fock state, another arm of SPDC is used ...
- 153
2
votes
0
answers
386
views
Quantum Harmonic Oscillator density matrix in coherent states base [closed]
I was trying to calculate matrix elements of the density operator for a 1D QHO (with Hamiltonian $\mathcal H = \hbar\omega a^\dagger a $) in the base of coherent states $\{\vert\alpha\rangle\}$ and ...
- 21
0
votes
0
answers
27
views
Normal ordering of an exponential [duplicate]
I would like to recalculate Eq.(2.4) in PRA, 31,4,(1985), which expresses the exponential of operators as a normal ordering form. This equation reads
\begin{equation}
D=e^{\alpha K_{+} - \alpha^{*} K_{...
- 374
1
vote
1
answer
171
views
Decomposing a coherent state? [closed]
Can you decompose a coherent state $|\alpha\rangle$ into $p|p\rangle+q|q\rangle$, where $|p\rangle$ and $|q\rangle$ are eigenstates of the $P$ and $Q$ operators respectively with eigenvalues p and q?
...
- 43
1
vote
1
answer
128
views
Confusion with two-mode bra-ket notation [closed]
Let us consider some abstract two-mode bosonic model with a conserved total number of quanta (i.e. eigenvalue $N$ of the operator $\hat{N} = a^\dagger a + b^\dagger b$ remains constant with ...
- 227