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Quantum noise spectroscopy (QNS)

Theory

This notebook measures the noise spectrum using CPMG (Carr-Purecell-Meiboom-Gill)/ periodic dynamical decoupling (DD) sequence. A CPMG sequence is

\[(\pi/2) - [\tau - (\pi) - \tau]^{\times \text{N}} - (\pi/2)\]

Under such a sequence, the off digonal elments of density matrix decay as:

\[W(T) = e^{-\chi(T)},\]

where \(W(T) = \rho_{\text{off}} (T) / \rho_{\text{off}} (0)\).

For larger number of \(\pi\) pulses, i.e., \(T \to \infty\)

\[\chi(T) = \frac{1}{2\pi} \int_0^\infty |F(\omega, T)|^2 S(\omega) d\omega\]

where

\(S(\omega) =\) Power spectral density (PSD) which encodes effect of noise

\(F(\omega, T) =\) Filter function which encodes effect of the DD sequence

For \(T \to \infty\),

\[F(\omega, T) \approx 2\pi T \sum_{-\infty}^{\infty} |C_{\omega_m}|^2 \delta (\omega - \omega_m)\]

with \(\omega_m = m\pi/2\tau\) and \(C_{\omega_m} = \frac{1}{T} \int_0^T f(t) e^{-i\omega_m t} dt\), where \(f(t)\) is the time-domain filter function. It leads to

\[\chi(T) = T \sum_{m = 0}^{\infty} |C_{\omega_m}|^2 S(\omega_m)\]

For CPMG/ periodic DD sequences, \(|C_{\omega_m}|^2 = \frac{4}{\pi^2 m^2} \delta_{m, 2l+1}\) with \((l = 0,1,2....)\),

Description

This notebook measures the dephasing noise spectrum using a simple sequence of equidistant \(\pi\)-pulses, such as the Carr-Purcell-Meiboom-Gill (CPMG) sequence1. A CPMG sequence is defined as \((\pi/2) - [\tau - (\pi) - \tau]^{\times N} - (\pi/2)\), where \(\tau\) is the delay time, and \(N\) is an integer defining the number of repetitions of the interpulse.

It is quite common to find that noise spectrum \(S(\omega)\) decreases rapidly with \(\omega\) in solid state settings, for example \(S(\omega) \propto 1/\omega^\beta\) for \(\beta>1\). In these cases, we can ignore the contribution from the higher harmonics of the filter function leading to (single-\(\delta\) approximation)

For CPMG, the coefficients \(C_m\) are expressed as \(|C_m|^2 = \frac{4}{\pi^2 m^2} \delta_{m, 2l+1}\) with \((l = 0, 1, 2, \ldots)\). Consequently, we obtain \(\frac{1}{T_2} = \frac{4}{\pi^{2}}\sum_{l = 0}^{\infty} \frac{1}{(2l+1)^{2}} S(\omega\_m)\). In particular, if the noise spectrum \(S(\omega)\) decreases rapidly with \(\omega\), then

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

In this manner, by measuring various \(T_2\) values under the CPMG sequence with different delays, one can scan the entire noise spectrum.

Experiment and Analysis Steps

  1. Applying the CPMG pulse sequence for a given delay \(\tau\) (delay):

  2. Fix a interpulse delay between \(\pi\) pulses

  3. Apply enough number \((N)\) of \(\pi\) pulses so that filter function is close to delta function and measure \(W(T)\). Estimate \(\chi(T) = -\ln [W(T)]\)

  4. Change \(\tau\) and repeat last two steps. Like this we generate an ensemble of \(\chi(T)\) for various \(\tau\)

  5. Use \(S(\pi/2\tau) = \frac{\pi^2}{4T} \chi(T)\) to calculate PSD

     1. Applying a $\pi/2$-pulse (`RXGate(x/2)`).

 2. Applying `order`-times the following sequence:

    I. Waiting for a time $\tau$  (`delay`).

    II. Applying a $\pi$-pulse (`RXGate(x)`).

    III. Waiting for a time $\tau$  (`delay`).

 5. Applying a $\pi/2$-pulse (`RXGate(x/2)`).
    1. Repeating 1. for different increasing values of order.

  6. Repeating 1. for different values of delay.

  7. Computing the resonator's amplitude (amplitude) signal as a function of delay. The amplitude is calculated as \(I^2 + Q^2\).

  8. Determining \(T^{}{2}\) (T2) by fitting a stretched function to the experimental trace (amplitude versus delay). The fitting function is \(f(x) = A ,\text{exp}(-(x / \theta) ^ \alpha) + \varphi_{\text{off}}\), with \(\theta = T^{}_{2}\).

  9. Getting the spectral noise density as a function of computing 1/T_2 as a function of frequency. The frequency is computed as frequency = omega / 2 / np.pi, with omega = np.pi / (2 * delay).

  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. ↩