# ms_phases_multitone

ions.ms_phases_multitone(drives, lamb_dicke_parameters, relative_detunings, sample_times=None, *, name=None)

Calculate the relative phases for all pairs of ions described by a Mølmer–Sørensen-type interaction where the ions are being addressed by a multitone global beam.

Use this function to calculate the acquired phases for all ion pairs at the final time of the drives, or at the sample times that you provide.

### Parameters

• drives (list [Pwc ]) – The piecewise-constant drives, $\{\gamma_j\}$, one for each of the $M$
• lamb_dicke_parameters (np.ndarray) – The laser-ion coupling strength, $\{\eta_{\xi jkl}\}$(M, 3, N, N), where the dimensions indicate, respectively, tone of the global beam, axis, collective mode, and ion.
• relative_detunings (np.ndarray) – The difference $\{\delta_{\xi jk} = \nu_{jk} - \delta_\xi \}$ (in Hz) between each motional mode frequency and the laser detunings for each tone from the qubit transition frequency $\omega_0$(M, 3, N), where the dimensions indicate, respectively, tone of the global beam, axis, and collective mode.
• sample_times (np.ndarray or None , optional) – The times (in seconds) at which to calculate the relative phases, $\{t_i\}$
• name (str or None , optional) – The name of the node.

### Returns

Acquired phases $\{\phi_{jk}(t_i) + \phi_{kj}(t_i)\}$(T, N, N), where the first dimension indicates time, and the second and third dimensions indicate ions. Otherwise, the shape is (N, N), with the outer time dimension removed. The relative phases are stored as a strictly lower triangular matrix. See the notes part for details.

### Return type

Tensor(real)

Graph.ions.ms_infidelity : Final operational infidelity of a Mølmer–Sørensen gate.

Graph.ions.ms_phases : Corresponding operation for single-tone individually-addressed beams.

boulderopal.ions.obtain_ion_chain_properties : Function to calculate the properties of an ion chain.

## Notes

The interaction Hamiltonian for Mølmer–Sørensen-type operations in the rotating frame for a multitone global beam is

$H_I(t) = i\hbar \sum_{\xi=1}^M \sum_{j=1}^{3} \sum_{k=1}^{N} \sum_{l=1}^N \sigma_{x,l} \left(-\beta_{\xi jkl}^*(t)a_{jk} + \beta_{\xi jkl}(t) a_{jk}^\dagger\right) ,$

where $\sigma_{x,l}$ is the Pauli $X$ operator for the ion $l$

$\beta_{\xi jkl}(t) = \eta_{\xi jkl} \frac{\gamma_\xi(t)}{2} \exp(i 2 \pi \delta_{\xi jk} t)$

indicates the coupling between ion $l$ and motional mode $(\xi,j,k)$

The corresponding unitary operation is given by

$U(t) = \exp\left[ \sum_{l=1}^N \sigma_{x,l} B_l(t) + i\sum_{l=1}^N \sum_{n=1}^{l-1} (\phi_{ln}(t) + \phi_{nl}(t)) \sigma_{x,l} \sigma_{x,n} \right] ,$

where

$B_l(t) \equiv \sum_{\xi=1}^M \sum_{j=1}^{3} \sum_{k=1}^{N} \left(\eta_{\xi jkl}\alpha_{\xi jkl}(t)a_{jk}^\dagger - \eta_{\xi jkl}^{\ast}\alpha_{\xi jkl}^\ast(t)a_{jk} \right) ,$ \phi_{ln}(t) \equiv \mathrm{Im} \left[ \sum_{\xi=1}^M \sum_{\chi=1}^M \sum_{j=1}^{3} \sum_{k=1}^{N} \int_{0}^{t} d \tau_1 \int_{0}^{\tau_1} d \tau_2 \beta_{\xi jkl}(\tau_1)\beta_{\chi jkn}^{\ast}(\tau_2) \right] , $\alpha_{\xi jkl}(t) \equiv \int_0^t d\tau \frac{\gamma_\xi(\tau)}{2} \exp(i 2 \pi \delta_{\xi jk} \tau).$

This function calculates the relative phases for all ions pairs at sample times $\{t_i\}$

$\Phi_{ln}(t_i) = \phi_{ln}(t_i) + \phi_{nl}(t_i),$

and stores them in a strictly lower triangular matrix. That is, $\Phi_{ln}(t_i)$ with $l > n$ gives the relative phase between ions $l$ and $n$, while $\Phi_{ln}(t_i) = 0$ for $l \leq n$