ms_displacements

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

Calculate the displacements for each mode and ion combination where ions are described by a Mølmer–Sørensen-type interaction.

Use this function to calculate the displacements for each ion and each mode 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, {γj}\{\gamma_j\}, one for each of the NN
  • lamb_dicke_parameters (np.ndarray) – The laser-ion coupling strength, {ηjkl}\{\eta_{jkl}\}(3, N, N), where the dimensions indicate, respectively, axis, collective mode, and ion.
  • relative_detunings (np.ndarray) – The difference {δjk=νjk−δ}\{\delta_{jk} = \nu_{jk} - \delta\} (in Hz) between each motional mode frequency and the laser detuning from the qubit transition frequency ω0\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 displacements, {ti}\{t_i\}
  • name (str or None , optional) – The name of the node.

Returns

Displacements {ηjklαjkl(ti)}\{\eta_{jkl}\alpha_{jkl}(t_i)\}(T, 3, N, N_d), where the dimensions indicate, respectively, time, axis, collective mode, and ion. Otherwise, the shape is (3, N, N_d), with the outer time dimension removed. Here, N_d is the number of addressed ions (drives that are not None); the displacements of undriven ions are not returned (as they are zero).

Return type

Tensor(complex)

SEE ALSO

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

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

Notes

This function calculates, at sample times {ti}\{t_i\}, the contribution to the displacement of mode kk in dimension jj from ion ll, namely ηjklαjkl(ti)\eta_{jkl}\alpha_{jkl}(t_i)

αjkl(t)≡∫0tdτγl(τ)2exp⁡(i2πδjkτ). \alpha_{jkl}(t) \equiv \int_0^t d\tau \frac{\gamma_l(\tau)}{2} \exp(i 2 \pi \delta_{jk} \tau) .

You can calculate the state-dependent displacement by summing over the contributions from all ions. That is, using the displacement superoperator Djk\mathcal{D}_{jk}, the displacement in phase space for mode (j,k)(j,k)

Djk(∑l=1Nσx,lηjklαjkl(ti)). \mathcal{D}_{jk} \left(\sum_{l=1}^N \sigma_{x,l}\eta_{jkl}\alpha_{jkl}(t_i) \right) .

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.

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