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(1D, complex)]) – A list of piecewise-constant drives \(\{\gamma_\xi\}\) for
each of the tones of the global beam.
The number of drives must be the same as the number of tones of
the global beam, \(M\).
Drive values must be one-dimensional arrays and in rad/s. Drive durations must be
in seconds. All drives must have the same total duration, but can have different numbers
of segments.
lamb_dicke_parameters (np.ndarray) – A 4D array of parameters \(\{\eta_{\xi jkl}\}\) specifying
the laser-ion coupling strength. Its shape must be (M, 3, N, N)
where the first dimension \(\xi\) indicates the tone of
the global beam, the second dimension \(j\) indicates the axis,
the third dimension \(k\) indicates the collective
mode number, and the fourth dimension \(l\) indicates the ion.
relative_detunings (np.ndarray) – A 3D array \(\{\delta_{\xi jk} = \nu_{jk} - \delta_\xi \}\)
specifying the difference (in Hz) between each motional mode frequency
\(\nu_{jk}\) and the laser detunings for each tone,
\(\delta_\xi\) (the detuning from the qubit transition frequency
\(\omega_0\)). Its shape must be (M, 3, N) where the first
dimension \(\xi\) indicates the tone of the global beam,
the second dimension \(j\) indicates the axis and the third
dimension \(k\) indicates the collective mode number.
sample_times (np.ndarray, optional) – A 1D array of length \(T\) specifying the times \(\{t_i\}\) (in seconds) at which
this function calculates the relative phases.
If you omit it, this function calculates the phases only at the final time of the drives.
If provided, it must be ordered and contain at least one element.
name (str, 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 the time; the second and the third dimensions
indicate the ion. Otherwise, the shape is (N, N) where both dimensions indicate
the ion. The relative phases are stored as a strictly lower triangular matrix. See
the notes part for details.
- Return type
Tensor(real)
Notes
The interaction Hamiltonian for Mølmer–Sørensen-type
operations in the rotating frame for a multitone global beam is:
\[H_I(t) = i\hbar\sum_{j = 1}^N \sigma_{x, j} \sum_{p = 1}^{3N} \sum_{\xi = 1}^M
(-\beta_{\xi pj}^*(t)a_p + \beta_{\xi pj}(t) a_p^\dagger) ,\]
where \(\sigma_{x, j}\) is the Pauli \(X\) operator for the ion \(j\) and
\(\beta_{\xi pj}(t) = \eta_{\xi pj} \frac{\gamma_\xi(t)}{2} e^{i\delta_{\xi p} t}\),
indicating the coupling of the ion \(j\) to the motional mode \(p\).
The corresponding unitary operation is given by:
\[U(t) = \exp\left[ \sum_{j=1}^N \sigma_{x, j} B_j(t)
+ i\sum_{j=1}^N \sum_{k=1}^{j - 1} (\phi_{jk}(t) + \phi_{kj}(t))
\sigma_{x, j} \sigma_{x, k} \right] ,\]
where
\[ \begin{align}\begin{aligned}B_j(t) &\equiv \sum_{p = 1}^{3N} \sum_{\xi = 1}^M
\left(\eta_{\xi pj}\alpha_{\xi pj}(t)a_p^\dagger
- \eta_{\xi pj}^{\ast}\alpha_{\xi pj}^\ast(t)a_p \right) ,\\\phi_{jk}(t) &\equiv \mathrm{Im} \left[ \sum_{p=1}^{3N} \sum_{\xi_1 = 1}^M
\sum_{\xi_2 = 1}^M \int_{0}^{t} d \tau_1 \int_{0}^{\tau_1} d \tau_2
\beta_{\xi_1 pj}(\tau_1)\beta_{\xi_2 pk}^{\ast}(\tau_2) \right] ,\end{aligned}\end{align} \]
with \(\alpha_{\xi pj}(t) = \int_0^t d\tau \frac{\gamma_\xi(\tau)}{2}
e^{i \delta_{\xi p} \tau}\).
This function calculates the relative phases \(\phi_{jk}(t_i) + \phi_{kj}(t_i)\)
for all ions pairs at time \(t_i\) and stores the result in the form of a strictly
lower triangular matrix. That is, the \(jk\)-th element of
that matrix records the relative phase between the ion \(j\) and \(k\),
while the elements with indices \(j \leq k\) are zeros.