ms_phases

ions.ms_phases(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 when single-tone individually-addressed laser beams are used.

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 or None ]) – The piecewise-constant drives, $\{\gamma_j\}$ , one for each of the $N$
lamb_dicke_parameters (np.ndarray) – The laser-ion coupling strength, $\{\eta_{jkl}\}$ (3, N, N), where the dimensions indicate, respectively, axis, collective mode, and ion.
relative_detunings (np.ndarray) – The difference $\{\delta_{jk} = \nu_{jk} - \delta\}$ (in Hz) between each motional mode frequency and the laser detuning from the qubit transition frequency $\omega_0$ (3, N), where the dimensions indicate, respectively, 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.

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)

Notes

The internal and motional Hamiltonian of $N$

H_0 = \sum_{j=1}^{3} \sum_{k=1}^{N} \hbar\nu_{jk} \left(a_{jk}^\dagger a_{jk} + \frac{1}{2}\right) + \sum_{l=1}^N \frac{\hbar \omega_0}{2} \sigma_{z,l} ,

where $j$ indicates axis dimension ( $x$ , $y$ , or $z$ ), $k$ indicates collective mode, $a_{jk}$ is the annihilation operator, and $\sigma_{z,l}$ is the Pauli $Z$ operator for ion $l$

The interaction Hamiltonian for Mølmer–Sørensen-type operations in the rotating frame with respect to $H_0$

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

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

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

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

The corresponding unitary operation is given by ¹

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_{j=1}^{3} \sum_{k=1}^{N} \left(\eta_{jkl}\alpha_{jkl}(t)a_{jk}^\dagger - \eta_{jkl}^{\ast}\alpha_{jkl}^\ast(t)a_{jk} \right) ,

\phi_{ln}(t) \equiv \mathrm{Im} \left[ \sum_{j=1}^{3} \sum_{k=1}^{N} \int_{0}^{t} d \tau_1 \int_{0}^{\tau_1} d \tau_2 \beta_{jkl}(\tau_1)\beta_{jkn}^{\ast}(\tau_2) \right] ,

\alpha_{jkl}(t) \equiv \int_0^t d\tau \frac{\gamma_l(\tau)}{2} \exp(i 2 \pi \delta_{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 $l \leq n$

References

[1] C. D. B. Bentley, H. Ball, M. J. Biercuk, A. R. R. Carvalho, M. R. Hush, and H. J. Slatyer, Adv. Quantum Technol. 3, 2000044 (2020).

Examples

Refer to the How to optimize error-robust Mølmer–Sørensen gates for trapped ions user guide to find how to use this and related nodes.

Parameters

Returns

Return type

SEE ALSO

Notes

References

Examples