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CritPt / Challenge_60_main / In this problem, we want to consider the resolution of single-lens imaging of…
Problem
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problem description
# Problem setup:
In this problem, we want to consider the resolution of single-lens imaging of thermal sources. The quantum state radiated by the thermal sources can be expressed as
where is the Sudarshan-Glauber representation and is an appropriate measure. Define
$\alpha=\left(\alpha_1, \ldots, \alpha_J\right)^{\top}$ as a column vector of complex field amplitudes for optical spatial modes on the source plane and as a multimode coherent state with amplitude .
For thermal sources, it is standard to assume to be a zero-mean complex Gaussian given by
where $\alpha^{\dagger}=\left(\alpha_1^*, \ldots, \alpha_J^*\right)$ denotes the complex transpose of . We consider a simple two point source case
\begin{equation}
[\Gamma_s]_{uv}=\epsilon_0\delta_{uv}[\delta_{uu_1}+\delta_{uu_2}]
\end{equation}
where , and , are the positions of the two point sources. If the quantum state passes through a single lens which has point spread function . We want to quantify the performance of a naive direct imaging method and the fundamental limit for resolution.
# Main problem:
Define , calculate the quantum Fisher information of estimating per each measured photon, express the answer using
,
.
Plain-text mathematical notation (without MathML)
# Problem setup:
In this problem, we want to consider the resolution of single-lens imaging of thermal sources. The quantum state radiated by the thermal sources can be expressed as
ρ_(s)=∫DαΦ(α)|α⟩⟨α|,
where Φ(α) is the Sudarshan-Glauber representation and Dα is an appropriate measure. Define $\alpha=\left(\alpha_1, \ldots, \alpha_J\right)^{\top}$ as a column vector of complex field amplitudes for J optical spatial modes on the source plane and |α⟩ as a multimode coherent state with amplitude α.
For thermal sources, it is standard to assume Φ to be a zero-mean complex Gaussian given by
Φ(α)=(1)/(det(πΓ_(s)))exp(−α^(†)Γ^(−1)α)
where $\alpha^{\dagger}=\left(\alpha_1^*, \ldots, \alpha_J^*\right)$ denotes the complex transpose of α. We consider a simple two point source case
\begin{equation}
[\Gamma_s]_{uv}=\epsilon_0\delta_{uv}[\delta_{uu_1}+\delta_{uu_2}]
\end{equation}
where ϵ₀→0, and u₁, u₂ are the positions of the two point sources. If the quantum state ρ_(s) passes through a single lens which has point spread function ψ(x). We want to quantify the performance of a naive direct imaging method and the fundamental limit for resolution.
# Main problem:
Define θ=(1)/(3)u₁+(2)/(3)u₂, calculate the quantum Fisher information of estimating θ per each measured photon, express the answer using
Δk²≡∫_(−∞)^(∞)dx[(∂ψ(x))/(∂x)]²,
γ≡∫_(−∞)^(∞)dx(∂ψ(x))/(∂x)ψ(x−u₂+u₁).
Original LaTeX notation
# Problem setup:
In this problem, we want to consider the resolution of single-lens imaging of thermal sources. The quantum state radiated by the thermal sources can be expressed as
$$
\rho_s=\int D \alpha \Phi(\alpha)|\alpha\rangle\langle\alpha|,
$$
where $\Phi(\alpha)$ is the Sudarshan-Glauber representation and $D \alpha$ is an appropriate measure. Define $\alpha=\left(\alpha_1, \ldots, \alpha_J\right)^{\top}$ as a column vector of complex field amplitudes for $J$ optical spatial modes on the source plane and $|\alpha\rangle$ as a multimode coherent state with amplitude $\alpha$.
For thermal sources, it is standard to assume $\Phi$ to be a zero-mean complex Gaussian given by
$$
\Phi(\alpha)=\frac{1}{\operatorname{det}(\pi \Gamma_s)} \exp \left(-\alpha^{\dagger} \Gamma^{-1} \alpha\right)
$$
where $\alpha^{\dagger}=\left(\alpha_1^*, \ldots, \alpha_J^*\right)$ denotes the complex transpose of $\alpha$. We consider a simple two point source case
\begin{equation}
[\Gamma_s]_{uv}=\epsilon_0\delta_{uv}[\delta_{uu_1}+\delta_{uu_2}]
\end{equation}
where $\epsilon_0\to 0$, and $u_1$, $u_2$ are the positions of the two point sources. If the quantum state $\rho_s$ passes through a single lens which has point spread function $\psi(x)$. We want to quantify the performance of a naive direct imaging method and the fundamental limit for resolution.
# Main problem:
Define $\theta=\frac{1}{3}u_1+\frac{2}{3}u_2$, calculate the quantum Fisher information of estimating $\theta$ per each measured photon, express the answer using
$
\Delta k^2 \equiv \int_{-\infty}^{\infty} d x\left[\frac{\partial \psi(x)}{\partial x}\right]^2
$,
$
\gamma \equiv \int_{-\infty}^{\infty} d x \frac{\partial \psi(x)}{\partial x} \psi\left(x-u_2+u_1 \right)
$.
code template
Code
import sympy as sp
Delta_k_sq, gamma = sp.symbols('Delta_k_sq gamma')
def answer(Delta_k_sq, gamma):
r"""
Return the expression of the quantum Fisher information in Sympy format.
Inputs
----------
Delta_k_sq: sympy.Symbol, $\Delta k^2 \equiv \int_{-\infty}^{\infty} d x\left[\frac{\partial \psi(x)}{\partial x}\right]^2$
gamma: sympy.Symbol, $\gamma \equiv \int_{-\infty}^{\infty} d x \frac{\partial \psi(x)}{\partial x} \psi\left(x-u_2+u_1 \right)$
Outputs
----------
QFI: sympy.Expr, quantum Fisher information of estimating $\theta=\frac{1}{3}u_1+\frac{2}{3}u_2$ per each measured photon,
$u_1$ and $u_2$ are the positions of the two point sources
"""
# ------------------ FILL IN YOUR RESULTS BELOW ------------------
QFI = ... # a SymPy expression of inputs
# ---------------------------------------------------------------
return QFIDiscussion
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