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Ramsey (1-2 states)

This experiment provides a more accurate estimate of the qubit's resonance frequency between the first and second excited states, \(\omega_{q12}\).

Description

The qubit resonance frequency between the first and second excited states, \(\omega_{q12}\), determined using Pulsed Qubit Spectroscopy (1-2 states) tends to be slightly detuned from its actual value due to finite pulse duration, power broadening, environmental noise, and imperfect fitting of the Lorentzian function1. While this serves as a good first guess for \(\omega_{q12}\), the Ramsey experiment uses the phase evolution of the qubit to obtain a much more precise value.

This experiment is very similar to Ramsey (0-1 states); however, prior to the standard Ramsey pulse sequence, a \(\pi\)-pulse is applied to prepare the qubit in the first excited state \(|1\rangle\). The Ramsey sequence, which 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, is then applied.

\(\pi\)-pulse

The \(\pi\)-pulse firstly prepares the qubit in the first excited state \(|1\rangle\).

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

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

Free evolution

The qubit freely evolves for a time \(t\). A relative phase difference, \(\delta\), accumulates between the \(|1\rangle\) and \(|2\rangle\) states, where (in the rotating frame)

\[ \delta = (\omega_{d} - \omega_{q12})t, \]

and \(\omega_{d}\) and \(\omega_{q12}\) are the microwave drive frequency and qubit resonance frequency between the first and second excited states, respectively. The final state of the qubit at time \(t\) is given by

\[ \frac{1}{\sqrt{2}}(|1\rangle + e^{i \delta}|2\rangle). \]

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

A \(-\frac{\pi}{2}\)-pulse is applied after time \(t\), returning the qubit to the measurement basis. However, due to the accumulated phase, \(\delta\), it does not return to \(|1\rangle\). When the readout pulse is applied, the probability of the qubit being in the \(|1\rangle\) or \(|2\rangle\) state is given by the projection onto the \(z\)-axis.

This pulse sequence is then repeated for different values of \(t\). Plotting the amplitude of the resonator signal against \(t\) yields damped oscillations with frequency \(\delta=\omega_{d}-\omega_{q12}\) (see figure below). These can be described by

\[ f(t) = A \cos((\omega_{d}-\omega_{q12}) t + \phi) e^{-\Gamma t} + C , \]

where \(A\) and \(\phi\) are the initial amplitude and phase of the oscillations, respectively, and \(C\) is an offset constant. The decay rate, \(\Gamma\), arises from additional dephasing of the superposition due to the qubit interacting with its surrounding environment. \(\omega_{q12}\) can be extracted from a fit of \(f(t)\) to the data.

Experiment steps

  1. A \(\pi\)-pulse with frequency \(\omega_q\) is applied to the qubit to prepare it in the first excited state \(|1\rangle\).

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

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

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

  5. The resonator transmission is measured.

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

Analysis steps

  1. The amplitude of the filter 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, respectively.

  2. This amplitude is plotted against \(t\) (free evolution time), and the function, \(f(t)\), which describes damped oscillations, is fit to the data. \(\omega_{q12}\) is then extracted from the fit.

Note

Using the value of \(\omega_q\) obtained from the Ramsey (0-1 states) experiment with the newly-obtained \(\omega_{q12}\), the anharmonicity \(\alpha\) can also be calculated.

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  1. Mahdi Naghiloo. Introduction to experimental quantum measurement with superconducting qubits. 2019. arXiv:1904.09291. ↩