CritPt / Challenge_27_main / Consider a one-axis twisting interaction described by the Hamiltonian…

# Problem setup:
Consider a one-axis twisting interaction described by the Hamiltonian (H)^=χ(S)^^(z)(S)^^(z), where (S)^^(z)=∑_(j=1)^(N)(S)^_(j)^(z) is the total spin operator constructed by summing over N spin-1/2 operators. Apart from the one-axis twisting interaction, we also add dissipation terms: single-particle dephasing terms described by Lindblad jump operator (L)^_(z,j)=√(γ_(z)) (S)^_(j)^(z) and spin-particle spin-flip terms described by Lindblad jump operators (L)^_(+,j)=√(γ)(S)^_(j)^(+) and (L)^_(−,j)=√(γ)(S)^_(j)^(−). The Wineland spin-squeezing parameter is defined as ξ²≡Nmin⟨ΔS_(⊥)²⟩/|⟨(S)^⟩|², where (S)^ is the total spin vector, and min⟨ΔS_(⊥)²⟩ denotes the minimum variance in directions perpendicular to the Bloch vector.

# Main problem:

Considering the initial state as a coherent spin state pointing in the +x direction, calculate the Wineland spin-squeezing parameter optimized over time t (labelled by $\xi^2_{\rm opt}$) based on the following parameters: N=10⁶, χ=1.0×10^(−6), γ=0.01, γ_(z)=0.01. You should express the numerical result in terms of decibels and keep up to three significant figures.

# Problem setup:
Consider a one-axis twisting interaction described by the Hamiltonian $\hat{H}=\chi\hat{S}^z\hat{S}^z$, where $\hat{S}^z=\sum_{j=1}^N \hat{S}^z_j$ is the total spin operator constructed by summing over $N$ spin-$1/2$ operators. Apart from the one-axis twisting interaction, we also add dissipation terms: single-particle dephasing terms described by Lindblad jump operator $\hat{L}_{z,j}=\sqrt{\gamma_z}\,\hat{S}^z_j$ and spin-particle spin-flip terms described by Lindblad jump operators $\hat{L}_{+,j}=\sqrt{\gamma}\hat{S}^+_j$ and $\hat{L}_{-,j}=\sqrt{\gamma}\hat{S}^-_j$. The Wineland spin-squeezing parameter is defined as $\xi^2\equiv N \min \langle\Delta S_{\perp}^2\rangle/|\langle\hat{\mathbf{S}}\rangle|^2$, where $\hat{\mathbf{S}}$ is the total spin vector, and $\min \langle\Delta S_{\perp}^2\rangle$ denotes the minimum variance in directions perpendicular to the Bloch vector.

# Main problem:

Considering the initial state as a coherent spin state pointing in the $+x$ direction, calculate the Wineland spin-squeezing parameter optimized over time $t$ (labelled by $\xi^2_{\rm opt}$) based on the following parameters: $N=10^6$, $\chi=1.0\times 10^{-6}$, $\gamma=0.01$, $\gamma_z=0.01$. You should express the numerical result in terms of decibels and keep up to three significant figures.