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 \(M\) tones of the global beam. Drive values must be in rad/s and durations must be in seconds. All drives must have the same total duration, but can have different segmentations.
lamb_dicke_parameters (np.ndarray) – The laser-ion coupling strength, \(\{\eta_{\xi jkl}\}\). Its shape must be
(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 \(\omega_0\). Its shape must be
(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, \(\{t_i\}\). If you provide it, it must be 1D, ordered, and contain at least one element. If you omit it, the phases are only calculated at the final time of the drives.
name (str or None, optional) – The name of the node.
- Returns:
Acquired phases \(\{\phi_{jk}(t_i) + \phi_{kj}(t_i)\}\) for all ion pairs. If you provide sample_times, the shape of the returned value is
(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)
See also
Graph.ions.ms_infidelityFinal operational infidelity of a Mølmer–Sørensen gate.
Graph.ions.ms_phasesCorresponding operation for single-tone individually-addressed beams.
boulderopal.ions.obtain_ion_chain_propertiesFunction 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 \(l\) and
\[\beta_{\xi jkl}(t) = \eta_{\xi jkl} \frac{\gamma_\xi(t)}{2} \exp(i 2 \pi \delta_{\xi jk} t)\]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
\[ \begin{align}\begin{aligned}B_l(t) &\equiv \sum_{\xi=1}^M \sum_{j=1}^{3} \sum_{k=1}^{N} \left(\eta_{\xi jkl}\alpha_{\xi jkl}(t)a_{jk}^\dagger - \eta_{\xi jkl}^{\ast}\alpha_{\xi jkl}^\ast(t)a_{jk} \right) ,\\\phi_{ln}(t) &\equiv \mathrm{Im} \left[ \sum_{\xi=1}^M \sum_{\chi=1}^M \sum_{j=1}^{3} \sum_{k=1}^{N} \int_{0}^{t} d \tau_1 \int_{0}^{\tau_1} d \tau_2 \beta_{\xi jkl}(\tau_1)\beta_{\chi jkn}^{\ast}(\tau_2) \right] ,\\\alpha_{\xi jkl}(t) &\equiv \int_0^t d\tau \frac{\gamma_\xi(\tau)}{2} \exp(i 2 \pi \delta_{\xi jk} \tau).\end{aligned}\end{align} \]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 \(l \leq n\).