# Quantum amplification of boson-mediated interactions [(link)](https://www.nature.com/articles/s41567-021-01237-9.pdf)
[local](paper.pdf)
2021-07-26 PTB Journal Club
## Yellow
### general formalism
* *amplified boson-mediated interaction*
* realizable by many physical systems
#### 1st two terms: unamplified
* $\hat S$ : collective degree of freedom $\hat S = \sum_i \beta_i \hat s_i$
* $\hat a, \hat a^{\dagger}, \omega$ : harmonic oscillator mode (coupled)
* $\delta$ : system dependant offset
* $\Omega_0$ : coupling strength
#### 2nd term: parametric modulation of *boson channel*
* $g$ : modulation strength
* $\theta$ : relative phase, modulation - coupling interaction
## Blue
### 1 Trapped ion qubits:
* $\hat a, \hat a^{\dagger}$ : *normal mode* of motion
* $\hat s_i$ : Pauli matrix $\hat\sigma_j^i$ for the $i$-th ion
* $\delta$ : detuning of spin-motion coupling drive from
* $\omega$ : phonon mode implementing interaction
* $\beta_i$ : participation of $i$-th ion
### 2 OOP mode coupling; *g* = 0
* $\hat S = \hat \sigma_x^1 - \hat \sigma_x^2$
* $|+-\rangle$ and $|+-\rangle$ displaced, loops in phase space (see fig. 1a)
* $H_M$ $\leadsto$ $\Phi$ (enclosed area) acquired geometric phase
* $\tau = 2\pi/\delta$
* $\leadsto$ returns oscillator to initial state
* disentangles motion from spin state
* $\hat U = \exp\bigg(i \frac{\Phi}{4}\hat S^2\bigg)$ , $\Phi = 2\pi (\Omega_0/\delta)^2$
* effective spin-spin interaction
* $\leadsto$ $\Phi$-rotation in Bloch sphere with $|\uparrow\uparrow\rangle$ and $|\downarrow\downarrow\rangle$ at the poles
* $\delta = 2\Omega_0 \curvearrowright\Phi = \pi/2$ $\leadsto$ maximally entangled state
* $t_{\text{gate}} \propto \frac{1}{\Omega_0}$ ?
### 3 squeezing
* (see fig. 1b)
* normal mode *transformation*
* interaction strength amplification
* $\Phi = 0 \leadsto$ maximal amplification
## Magenta
### 1 description of their trap
* $^{25}\text{Mg}^+$ ions
* hyperfine qubits
* $30\ \mu \text{m}$ above surface
* surface-electrode radio-frequency trap
* 15 K (cryo)
* OOP mode @ 5.9 MHz
* $|\downarrow\rangle$,$|\uparrow\rangle$ within electronic ground state
* 21.3 mT
* microwaves
### 2 quantum operations in their trap
* init in $|\downarrow\downarrow\rangle$
* resolved SB cooling to $\bar n \approx 0.3$
* state dependant fluorescence detection
* microwave MS Gate
* gate time 342 $\mu\text{s}$
* oscillating potential into trap RF drive
### 3 Experiments