\(T_{2}^*\) Ramsey

This experiment provides a more accurate estimate of the qubit's resonance frequency, \(\omega_q\), and an estimate of the coherence time, \(T_2^*\).

Description

The qubit resonance frequency, \(\omega_q\), determined using Pulsed qubit spectroscopy tends to be slightly detuned from its actual value due to finite pulse duration, power broadening, environmental noise, and imperfect fitting of the Lorentzian function. While this serves as a good first guess for \(\omega_q\), the Ramsey experiment uses the phase evolution of the qubit to obtain a much more precise value.

Additionally, this experiment allows us to extract the dephasing time, \(T_2^*\), which characterises the lifetime of the superposition state.

The pulse sequence for the Ramsey experiment consists of a \(\frac{\pi}{2}\)-pulse followed by a \(-\frac{\pi}{2}\)-pulse, with a free evolution of the qubit between the two pulses.

\(\frac{\pi}{2}\)-pulse

The qubit begins in the ground state and the first \(\frac{\pi}{2}\)-pulse prepares the qubit in the superposition state \(\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\).

Free evolution

The qubit freely evolves for a time \(t\). A relative phase difference, \(\delta\), accumulates between the \(|0\rangle\) and \(|1\rangle\) states, where (in the rotating frame)

\[ \delta(t) = (\omega_{d} - \omega_{q})t, \]

and \(\omega_{d}\) and \(\omega_{q}\) are the microwave drive frequency and qubit resonance frequency, respectively. This corresponds to a rotation of the qubit state around the \(z\)-axis (in the \(xy\)-plane). The final state of the qubit at time \(t\) is given by

\[ \frac{1}{\sqrt{2}}(|0\rangle + e^{-t/T_2^*}e^{i \delta(t)}|1\rangle). \]

Here, the term \(e^{-t/T_2^*}\) represents the loss of coherence (dephasing) over time due to environmental noise.

\(-\frac{\pi}{2}\)-pulse

A \(-\frac{\pi}{2}\)-pulse is applied after time \(t\), rotating the state by \(\frac{\pi}{2}\) around the \(y\)-axis in the opposite direction to the first pulse. This projects the accumulated phase information onto the \(z\)-axis (population basis).

This pulse sequence is then repeated for different values of \(t\). Plotting the amplitude of the resonator signal against \(t\) yields damped oscillations. The frequency of these oscillations corresponds to the detuning \(\Delta = \omega_{d}-\omega_{q}\), and the decay envelope corresponds to \(T_2^*\).

Experiment steps

1. A \(\frac{\pi}{2}\)-pulse (\(R_y(\frac{\pi}{2})\)) is applied, which prepares the qubit in the superposition state \(\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\).

2. A time \(t\) is waited.

3. A \(-\frac{\pi}{2}\)-pulse (\(R_y(-\frac{\pi}{2})\)) is applied.

4. The resonator transmission is measured.

5. Steps 1 to 4 are repeated for different values of \(t\).

Analysis steps

1. Signal Processing: The amplitude of the readout resonator's signal is calculated as \(\sqrt{I^2 + Q^2}\), where \(I\) and \(Q\) are the in-phase and quadrature components of the transmitted signal. Alternatively, the signal can be rotated into a single axis to represent the population probability \(P_{|1\rangle}\).

2. Fitting: The data is plotted against the free evolution time \(t\) and fitted to a damped sinusoidal function:

   \[ f(t) = A e^{-t/T_2^*} \cos(2\pi \Delta t + \phi) + C \]

   where:

   • \(A\): The amplitude of the oscillations.
   • \(\Delta\): The oscillation frequency, which represents the detuning \((\omega_d - \omega_q) / 2\pi\).
   • \(T_2^*\): The dephasing time (decay constant).
   • \(\phi\): The phase offset.
   • \(C\): The vertical offset (background signal).

3. Extraction:

   • Frequency correction: The qubit resonance frequency is updated using the fitted detuning \(\Delta\):

     $$ \omega_{q, \text{new}} = \omega_{d} - 2\pi\Delta $$ Note: To distinguish the sign of \(\Delta\), the experiment is often run with an intentional "artificial" detuning applied to the drive frequency.

   • Coherence time: The parameter \(T_2^*\) is extracted directly from the exponential decay term, providing a measure of the qubit's coherence limit including inhomogeneous broadening.