Dispersive shift¶

In this section we present the dispersive shift routines provided by Qibocal.

Theory¶

A system with a qubit and a resonator, interacting with each other, is described by the James-Cummings Hamiltonian. Restricting the qubit states to the first two levels, we get

\[\hat{H}_{\text{JC}} = \hbar \omega_r \hat{a}^\dagger \hat{a} + \frac{\hbar \omega_q}{2} \hat{\sigma}_z + \hbar g (\hat{a}^\dagger \hat{\sigma}_- + \hat{a} \hat{\sigma}_+)\]

where \(\omega_r\) and \(\omega_q\) are respectively the resonator and the qubit frequencies, and \(g\) is the coupling constant between the qubit and the resonator. In the dispersive regime \(g \ll \lvert \omega_r - \omega_q \rvert\), the Hamiltonian can be rewritten as

(1)¶\[\hat{H}_{\text{eff}} = \hbar \hat{a}^\dagger \hat{a} (\omega_r - \chi \hat{\sigma}_z) + \frac{\hbar}{2} (\omega_q + \chi) \hat{\sigma}_z\]

where we introduced the dispersive shift

\[\chi = \frac{g^2}{\lambda}.\]

Equation (1) shows that the resonator frequency is \(\omega_{r,0} = \omega_r - \chi\) (\(\omega_{r,1} = \omega_r + \chi\)) when the qubit is in the ground (excited) state. The separation between the two freqiencies is \(\lvert 2 \chi \rvert\).

Protocol description¶

After collecting the data from the two spectroscopies, for each readout frequency the distance of the centers of the blobs for \(\ket{0}\) and \(\ket{1}\) states are evaluated. The best readout frequency is the one maximizing the distance between the two blobs.

Parameters¶

class qibocal.protocols.dispersive_shift.dispersive_shift.DispersiveShiftParameters(frequency: tuple[float, float, float] | tuple[Literal['linspace'], float, float, int] | tuple[Literal['window'], float, float, float] | tuple[Literal['linwindow'], float, float, int] | tuple[Literal['center'], float, float] | tuple[Literal['lincenter'], float, int] | tuple[Literal['asym'], tuple[float, float], float] | tuple[Literal['linasym'], tuple[float, float], int] | None = None, freq_width: int | None = None, freq_step: int | None = None)[source]

Dispersive shift inputs.

frequency: tuple[float, float, float] | tuple[Literal['linspace'], float, float, int] | tuple[Literal['window'], float, float, float] | tuple[Literal['linwindow'], float, float, int] | tuple[Literal['center'], float, float] | tuple[Literal['lincenter'], float, int] | tuple[Literal['asym'], tuple[float, float], float] | tuple[Literal['linasym'], tuple[float, float], int] | None = None: Frequency range [Hz].

freq_width: int | None = None: Width [Hz] for frequency sweep relative to the readout frequency [Hz].

freq_step: int | None = None: Frequency step for sweep [Hz].

frequency_range(center: float = 0.0) → tuple[float, float, float][source]

nshots: int: Number of executions on hardware.

relaxation_time: float: Wait time for the qubit to decohere back to the ground state.

Example¶

It follows an example of the experiment parameters.

- id: dispersive shift qt
  operation: dispersive_shift_qutrit
  parameters:
    freq_step: 200000
    freq_width: 1000000

After running qq run, the experiment is executed and the result will looks like the following picture.

Requirements¶

Rabi experiments