Zelong's Wiki
Introduction

Transmon-resonator dispersive regime

A readout resonator turns a transmon's quantum state into a microwave field displacement. Large detuning suppresses real energy exchange, leaving a state-dependent resonator frequency that can be measured.

Introduction

A transmon stores a qubit in two weakly anharmonic energy levels. A resonator makes that hidden state visible. The resonator is a microwave mode with a narrow linewidth; the transmon is an electric dipole placed near a voltage antinode of that mode.

If the two are resonant, an excitation sloshes back and forth. If they are far detuned, the sloshing becomes virtual. The qubit then pulls the resonator frequency up or down, while the resonator photon number shifts the qubit frequency. This is the dispersive regime.

Readout is the practical payoff. Send a microwave pulse through the resonator, let the outgoing field acquire a qubit-dependent amplitude and phase, then demodulate that field into an IQ point. The experimenter never measures island charge directly; they measure the microwave shadow cast by the qubit.

The whole design is a tradeoff: make 2∣χ∣2|\chi| large enough to separate states before decay, keep κ\kappa large enough for speed, keep photon number below the breakdown scale, and avoid letting the same port become a strong qubit-loss channel.

Physical Picture

CinCLCgCshJJinput lineLC resonatortransmon
A transmon capacitively coupled to an LC resonator, driven and read out through the input line.

Readout design map

?
probe detuning from fr (MHz) response integrated IQ plane
χ/2π-0.88 MHz
2|χ|/κ1.47
ncrit62.7
TPurcell209 us

Dispersive and readable.

Python Script

Show calculation script Dispersive shift, resonator pulls, and IQ pointers
Download Python script 
import numpy as np
import matplotlib.pyplot as plt

wr  = 7.10      # GHz
wq  = 5.20      # GHz
g   = 0.120     # GHz
a   = -0.250    # GHz, transmon anharmonicity
kap = 0.0012    # GHz
n   = 8

d     = wq - wr
chi   = g*g*a/(d*(d + a))
ncrit = d*d/(4*g*g)
fg    = wr - chi
fe    = wr + chi
xs    = np.linspace(wr - 0.015, wr + 0.015, 600)

def lor(f):
    return 1/np.sqrt(1 + (2*(xs - f)/kap)**2)

def ptr(f):
    return np.sqrt(n)*(kap/2)/(kap/2 + 1j*(wr - f))

ag, ae = ptr(fg), ptr(fe)

print(f"Delta/2pi = {d*1e3:.1f} MHz")
print(f"chi/2pi   = {chi*1e3:.2f} MHz")
print(f"2chi/kappa= {2*abs(chi)/kap:.2f}")
print(f"ncrit     = {ncrit:.1f}")

fig, ax = plt.subplots(1, 2, figsize=(8, 3.2), layout="constrained")
ax[0].plot(xs, lor(fg), label="qubit |g>")
ax[0].plot(xs, lor(fe), label="qubit |e>")
ax[0].set(xlabel="probe frequency (GHz)", ylabel="|S21|")
ax[0].legend()
ax[1].scatter([ag.real, ae.real], [ag.imag, ae.imag])
ax[1].plot([ag.real, ae.real], [ag.imag, ae.imag])
ax[1].set(xlabel="I", ylabel="Q", aspect="equal")
plt.show()

Hamiltonian

H^JCℏ=ωra^†a^+ωq2σ^z+g(a^†σ^−+a^σ^+)\frac{\hat H_\mathrm{JC}}{\hbar}=\omega_r \hat a^\dagger \hat a+\frac{\omega_q}{2}\hat\sigma_z+g\left(\hat a^\dagger\hat\sigma_-+\hat a\hat\sigma_+\right)Δ=ωq−ωr,∣Δ∣≫gn+1\Delta=\omega_q-\omega_r,\qquad |\Delta|\gg g\sqrt{n+1}H^dispℏ≃(ωr+χσ^z)a^†a^+12(ωq+χ)σ^z\frac{\hat H_\mathrm{disp}}{\hbar}\simeq\left(\omega_r+\chi\hat\sigma_z\right)\hat a^\dagger\hat a+\frac{1}{2}\left(\omega_q+\chi\right)\hat\sigma_zχ2lvl=g2Δ,χtmon≃g2αΔ(Δ+α)\chi_\mathrm{2lvl}=\frac{g^2}{\Delta},\qquad \chi_\mathrm{tmon}\simeq\frac{g^2\alpha}{\Delta(\Delta+\alpha)}

The Jaynes-Cummings model keeps only energy-conserving exchange: a resonator photon can become a qubit excitation and back again. The rate gg is the vacuum coupling set by the resonator’s zero-point voltage and the qubit dipole matrix element.

Large detuning lets a Schrieffer-Wolff transformation remove the exchange term to leading order. What remains is diagonal in the qubit state: ∣g⟩|g\rangle and ∣e⟩|e\rangle see different resonator frequencies. The resonator, in turn, gives the qubit an ac Stark shift proportional to photon number.

For a true two-level atom, χ=g2/Δ\chi=g^2/\Delta. A transmon is multilevel, so the nearby ∣e⟩→∣f⟩|e\rangle\to|f\rangle transition partially cancels or enhances the shift. The compact estimate χ≃g2α/[Δ(Δ+α)]\chi\simeq g^2\alpha/[\Delta(\Delta+\alpha)] is often the first design calculation.

The ideal measurement is QND because [H^disp,σ^z]=0[\hat H_\mathrm{disp},\hat\sigma_z]=0. In the lab, QND quality is limited by qubit T1T_1, photon shot noise, leakage out of the two-level subspace, dressed dephasing, and any photons or quasiparticles left behind after the measurement pulse.

Readout And Measurement

αg,e(ωd)=−iϵκ/2+i(ωr∓χ−ωd),SNR∝ηκt ∣αe−αg∣\alpha_{g,e}(\omega_d)=\frac{-i\epsilon}{\kappa/2+i(\omega_r\mp\chi-\omega_d)},\qquad \mathrm{SNR}\propto\sqrt{\eta\kappa t}\,|\alpha_e-\alpha_g|ncrit≃Δ24g2,ΓP≃κ(gΔ)2n_\mathrm{crit}\simeq\frac{\Delta^2}{4g^2},\qquad \Gamma_P\simeq\kappa\left(\frac{g}{\Delta}\right)^2

The readout pulse is usually a shaped microwave tone near one of the pulled resonator frequencies. During ring-up the two qubit states drag the cavity toward two different coherent amplitudes. The outgoing field passes through isolators, a near-quantum-limited amplifier, a HEMT amplifier, mixers, filters, and a digitizer.

Homodyne or heterodyne demodulation produces an IQ trace. A single-shot classifier integrates a chosen quadrature, or a two-dimensional matched filter, and assigns the point to the calibrated ground or excited cloud. Continuous measurement keeps the whole time trace and updates a state estimate as information arrives.

The useful knobs are tightly coupled. Increasing κ\kappa shortens the cavity memory and speeds ring-up, but increases the bare Purcell rate. Increasing photon number improves separation, but approaching ncritn_\mathrm{crit} invalidates the simple dispersive Hamiltonian and can cause transitions or leakage.

Modern readout hardware works because the first amplifier is superconducting and close to the device. Without parametric amplification, the HEMT noise temperature would bury a microsecond-scale single-shot signal. With good efficiency, the measurement backaction is mostly the intended projection plus the photon shot-noise dephasing implied by the same pointer separation.

Reference