Calculates the properties necessary for the simulation of an ion chain.
Use this function to obtain the mode frequencies, mode eigenvectors, and
Lamb–Dicke parameters for an ion chain with a single ion species. You can
use this information to simulate the evolution of the ion chain, or to
obtain optimized controls for it.
- Parameters:
atomic_mass (float) – The atomic mass \(m\) of the ions of the chain in atomic units.
This function assumes that all the ions in the chain are from the same
atomic species.
ion_count (int) – The number of ions in the chain, \(N\).
radial_x_center_of_mass_frequency (float) – The radial center-of-mass trapping frequency in Hertz for the
x-direction (\(\tilde \nu_1\)), corresponding to the unit vector
\((1, 0, 0)\).
radial_y_center_of_mass_frequency (float) – The radial center-of-mass trapping frequency in Hertz for the
y-direction (\(\tilde\nu_2\)), corresponding to the unit vector
\((0, 1, 0)\).
axial_center_of_mass_frequency (float) – The axial center-of-mass trapping frequency in Hertz for the
z-direction (\(\tilde\nu_3\)), corresponding to the unit vector
\((0, 0, 1)\).
radial_x_wave_number (float) – The component \(k_1\) of the laser difference angular wave vector
(in units of \(\textrm{rad}/\textrm{m}\)) in the radial
x-direction, corresponding to the unit vector \((1, 0, 0)\).
radial_y_wave_number (float) – The component \(k_2\) of the laser difference angular wave vector
(in units of \(\textrm{rad}/\textrm{m}\)) in the radial
y-direction, corresponding to the unit vector \((0, 1, 0)\).
axial_wave_number (float) – The component \(k_3\) of the laser difference angular wave vector
(in units of \(\textrm{rad}/\textrm{m}\)) in the axial z-direction,
corresponding to the unit vector \((0, 0, 1)\).
- Returns:
The result of a calculation of ion chain properties.
- Return type:
qctrl.dynamic.types.ion_chain_properties.Result
Notes
This function uses the equations of motion of the ion chain to find the
properties of its collective modes of oscillation. In a Mølmer–Sørensen
gate , these modes mediate the entanglement between qubits.
The ions interact among themselves via electric forces, besides being
subject to a harmonic trapping potential defined by the center-of-mass
frequencies \(\tilde\nu_1\), \(\tilde\nu_2\), and \(\tilde\nu_3\) that you
provided. In terms of coordinates \(x_{jp}\), which represent the
position of the \(p\)-th ion along the \(j\)-th axis, the total potential
energy of the ion chain is:
\[V = \sum_{j=1}^3 \sum_{p=1}^N \frac{m (2\pi \tilde\nu_j)^2 x_{jp}^2}{2} +
\sum_{p=1}^N \sum_{q\ne p} \frac{1}{8\pi \varepsilon_0}
\frac{e^2}{\left[\sum_{j=1}^3 \left(x_{jp}-x_{jq}\right)^2
\right]^{1/2}},\]
where \(e\) is the electron charge and where \(\varepsilon_0\) is the
vacuum permittivity. In the formula for the potential energy, the index
\(j\) labels the axes, while the indices \(p\) and \(q\) label the ions.
A power expansion of the potential around the equilibrium positions of
the ions \(\{x_{jp}^{(0)}\}\) (so that the zeroth and first order terms
vanish) allows the definition of a matrix \(\mathcal{V}\) whose
eigenvalues and eigenvectors give the normal modes of oscillation:
\[\begin{split}\begin{align*}
V = & \frac{1}{2} \sum_{j,l=1}^3 \sum_{p,q=1}^N \left.
\frac{\partial^2 V}{\partial x_{jp} \partial x_{lq}} \right|_0
(x_{jp}-x_{jp}^{(0)}) (x_{lq}-x_{lq}^{(0)}) + \ldots \\
= & \frac{1}{2} \sum_{j,l=1}^3 \sum_{p,q=1}^N \mathcal{V}_{jp,lq}
(x_{jp}-x_{jp}^{(0)}) (x_{lq}-x_{lq}^{(0)}) + \ldots
\end{align*}\end{split}\]
In the set of coordinates that diagonalizes \(\mathcal{V}\), the equations
of motion simplify to independent harmonic oscillators. For this reason,
the diagonalization of \(\mathcal{V}\) provides the frequencies
\(\nu_{jp}\) of the modes of oscillation through the formula:
\[\nu_{jp} = \frac{\omega_{jp}}{2\pi},\]
where \(m \omega_{jp}^2\) is the eigenvalue of \(\mathcal{V}\)
corresponding to the \(p\)-th mode of oscillation in the direction of the
\(j\)-th axis. The corresponding eigenvector
\((b_{jp1}, b_{jp2}, \ldots, b_{jpN})\) appears in the formula
for the Lamb–Dicke parameters \(\eta_{jpn}\):
\[\eta_{jpn} = b_{jpn} k_j \sqrt{\frac{\hbar}{4\pi m \nu_{jp}}}.\]
Note that the function orders the modes of oscillation for a given axis
in increasing order of frequency. In this way, the last of the radial
modes have frequencies that correspond to the radial center-of-mass
frequencies that you provided (\(\tilde \nu_1 = \nu_{1N}\) and
\(\tilde \nu_2 = \nu_{2N}\)), while the first of the axial modes
has the same frequency as the axial center-of-mass frequency that you
provided (\(\tilde \nu_3 = \nu_{31}\)).
This function returns to you the frequencies of the normal modes of
oscillation, the eigenvectors, and the Lamb–Dicke parameters. You can
use this information to simulate and optimize Mølmer–Sørensen gates
that use these collective oscillations to entangle the qubits.
In particular, the Lamb–Dicke parameters are present in the Hamiltonian
for the Mølmer–Sørensen interaction .
References
Examples
See the How to optimize error-robust Mølmer–Sørensen gates for trapped ions
user guide.