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Quantum noise spectroscopy (QNS)

This experiment measures the dephasing noise spectrum of a qubit using a Carr–Purcell–Meiboom–Gill (CPMG) sequence1.

Description

A CPMG sequence is a type of periodic dynamical decoupling sequence consisting of an initial \(\frac{\pi}{2}\) pulse, a train of evenly spaced \(\pi\) pulses, and a final \(\frac{\pi}{2}\) pulse:

\[(\frac{\pi}{2}) - [\tau - (\pi) - \tau]^N - (\frac{\pi}{2}) .\]

Here, \(\tau\) is the interpulse delay and \(N\) is an integer defining the number of repetitions of the \(\pi\) pulse. It's assumed that the \(\pi\) and \(\frac{\pi}{2}\) pulse durations are much shorter than \(\tau\), so the total free-evolution time is \(T = 2N\tau\).

The first \(\frac{\pi}{2}\) pulse prepares the qubit in a coherent superposition state:

\[ |0\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle). \]

During each delay (\(\tau\)), the qubit accumulates phase noise from the environment. Every \(\pi\) pulse flips the qubit state, reversing the sign of phase accumulation and cancelling the effect of slow (low-frequency) noise. Repeating the \([\tau - (\pi) - \tau]\) block \(N\) times produces a sequence of sign flips that suppresses noise at low \(\omega\), while making the qubit sensitive to noise near the characteristic frequency, given by:

\[ \omega = \frac{m\pi}{2\tau} \qquad (m = 1,3,5,\ldots). \]

Mathematically, the off-diagonal elements of the qubit density matrix, \(\rho_\text{off}(T)\), decay during the sequence, and the normalised coherence, \(W(T)\) is given by:

\[W(T) = \frac{\rho_{\text{off}} (T) }{\rho_{\text{off}} (0)} = e^{-\chi(T)},\]

where \(\chi(T)\) quantifies the accumulated dephasing. For large \(N\) (the long-sequence limit), \(\chi(T)\) grows linearly with the evolution time so that

\[ W(T) = e^{-\frac{T}{T_2}} \]

defines the dephasing time \(T_2\).

In many solid-state systems the noise spectrum \(S(\omega)\) decreases rapidly with \(\omega\); for example, \(S(\omega) \propto 1/\omega^\beta\) for \(\beta>1\). In this regime, the higher harmonics of the CPMG filter can be neglected (we only keep \(m=1\)) and we obtain the approximation that the qubit is only sensitive to dephasing noise at \(\omega = \frac{\pi}{2\tau}\):

\[ \frac{1}{T_2} = \frac{4}{\pi^2} S(\frac{\pi}{2\tau}). \]

By performing the sequence for different values of \(\tau\) and measuring \(T_2\), it's therefore possible to reconstruct the entire dephasing noise spectrum \(S(\omega) = S(\frac{\pi}{2\tau})\).

Experiment steps

  1. The CPMG pulse sequence is applied for a given delay, \(\tau\), with large enough \(N\) to be in the long-sequence limit.

  2. After the final \(\frac{\pi}{2}\) pulse, the qubit is read out and the \(I\!-\!Q\) data are recorded.

  3. Steps 1 and 2 are repeated for various values of \(\tau\).

Analysis steps

  1. The qubit state at the end of each sequence is determined from the \(I\!-\!Q\) data by applying the discriminator trained in the Readout discriminator training experiment and the corresponding ground state population, \(P_0(T)\), is calculated.

  2. Since the final \(\frac{\pi}{2}\) pulse maps the coherence onto the population difference, the normalised coherence can be calculated as \(W(T) = 2P_0(T) - 1\).

  3. The dephasing time \(T_2\) for each chosen \(\tau\) is extracted from \(\chi(T) = -\ln W(T) = \frac{T}{T_2}\).

  4. The noise spectral density at \(\omega = \frac{\pi}{2\tau}\) is obtained from \(S(\omega) = \frac{\pi^2}{4T_2}\).

  5. Steps 1-4 are repeated for each value of \(\tau\) to obtain the complete noise spectrum.

  1. Tatsuro Yuge, Susumu Sasaki, and Yoshiro Hirayama. Measurement of the noise spectrum using a multiple-pulse sequence. Phys. Rev. Lett., 107:170504, Oct 2011. doi:10.1103/PhysRevLett.107.170504. ↩