Circuit QED

Circuit quantum electrodynamics, or circuit QED, couples superconducting artificial atoms to quantized microwave resonators and transmission lines. For open-system physics it is one of the cleanest platforms: the resonator is both a quantum degree of freedom and a measurement port, microwave loss channels are engineered with known rates, and continuous voltage records can be modeled with input–output theory and stochastic master equations.

We follow a readout signal from qubit-dependent resonator motion to the measured voltage, keeping track of the dephasing and relaxation caused by the same coupling. Circuit QED Overview explains the hardware and device design.

Qubit, Resonator, and Loss Channels

The simplest idealized circuit-QED Hamiltonian is the Jaynes–Cummings model,

$$\frac{H_{\mathrm{JC}}}{\hbar} = \omega_r a^\dagger a + \frac{\omega_q}{2}\sigma_z + g \left( a^\dagger\sigma_- + a\sigma_+ \right),$$

where $a$ annihilates a resonator photon, $\omega_r$ is the resonator frequency, $\omega_q$ is the qubit transition frequency, and $g$ is the transverse coupling. For the closed-system spectrum, see Jaynes–Cummings Model.

The minimal open-system version adds resonator decay, qubit relaxation, and qubit dephasing:

$$\dot\rho = - \frac{i}{\hbar} [H,\rho] + \kappa\mathcal D[a]\rho + \gamma_1\mathcal D[\sigma_-]\rho + \frac{\gamma_\phi}{2} \mathcal D[\sigma_z]\rho + \cdots .$$

Here $\kappa$ is the resonator energy-decay rate, $\gamma_1=1/T_1$ is qubit relaxation, and $\gamma_\phi$ is pure dephasing. The ellipsis may include thermal photons, leakage levels, additional ports, quasiparticle loss, flux noise, or engineered dissipation.

Dispersive Regime

Most superconducting qubit readout operates in the dispersive regime, where the detuning

$$\Delta = \omega_q-\omega_r$$

is large compared with the photon-enhanced coupling over the occupied photon-number range:

$$\lvert\Delta\rvert\gg g\sqrt{n+1}.$$

For an ideal two-level qubit, a second-order transformation gives the approximate dispersive Hamiltonian

$$\frac{H_{\mathrm{disp}}}{\hbar} = \left( \omega_r+\chi\sigma_z \right) a^\dagger a + \frac{\omega_q+\chi}{2}\sigma_z,$$

with

$$\chi \simeq \frac{g^2}{\Delta}$$

in the simplest two-level approximation. For a weakly anharmonic transmon with $\alpha=\omega_{12}-\omega_{01}<0$, the leading expression is $\chi\simeq g^2\alpha/[\Delta(\Delta+\alpha)]$. Both relevant transitions must remain dispersive. The two-level estimate also fails as photon number approaches its critical scale $n_{\mathrm{crit}}\sim\Delta^2/(4g^2)$.

The physical content is that the resonator frequency depends on the qubit state:

$$\omega_r^{(g)} \simeq \omega_r-\chi, \qquad \omega_r^{(e)} \simeq \omega_r+\chi.$$

A microwave probe near the resonator therefore acquires a qubit-dependent phase and amplitude. Measuring the outgoing field becomes an indirect measurement of $\sigma_z$.

Resonator Readout

Input–output theory supplies the connection between the intracavity field and the measured traveling microwave field. For a single port with no other cavity loss,

$$b_{\mathrm{out}}(t) = b_{\mathrm{in}}(t) + \sqrt{\kappa}\,a(t),$$

up to the phase convention used for the port. The same coupling gives the dissipator $\kappa\mathcal D[a]\rho$ when the output field is not retained. See Input–Output Theory.

During a dispersive readout pulse, the resonator relaxes toward different coherent amplitudes depending on the qubit state:

$$\alpha_g(t), \qquad \alpha_e(t).$$

The outgoing field contains information about which pointer state the resonator occupies. The distinguishability of $\alpha_g$ and $\alpha_e$ controls both the measurement signal and the dephasing caused by information leaking into the output field.

In steady state, for the ideal linear dispersive model with vacuum loss channels, the qubit coherence decays at the measurement-induced rate

$$\Gamma_\phi^{\mathrm{meas}} = \frac{\kappa}{2} \left| \alpha_e-\alpha_g \right|^2.$$

During ring-up and ring-down, qubit coherence is also reduced by distinguishability still stored inside the resonator. For an initially common coherent cavity state, no qubit flips, and total vacuum linewidth $\kappa$, define $\delta\alpha(t)=\alpha_e(t)-\alpha_g(t)$. The magnitude is

$$\frac{|\rho_{ge}(t)|}{|\rho_{ge}(0)|} =\exp\left[-\frac{|\delta\alpha(t)|^2}{2} -\frac{\kappa}{2}\int_0^t|\delta\alpha(s)|^2\,ds\right].$$

The first term is the intracavity overlap; the second is the accumulated output-field overlap. Therefore $\kappa|\delta\alpha|^2/2$ alone is not the complete instantaneous reduced-qubit dephasing rate during a transient. Collection inefficiency reduces accessible information, not the dephasing already caused by all output ports. Gambetta et al. treat the dispersive dynamics.

Measurement Backaction

Dispersive readout is approximately a quantum nondemolition measurement of $\sigma_z$ when the Hamiltonian commutes with $\sigma_z$ and unwanted transitions are negligible. In that limit, the measurement dephases superpositions of $\lvert g\rangle$ and $\lvert e\rangle$ without directly changing their populations.

If the resonator field becomes correlated with the qubit,

$$c_g\lvert g\rangle\lvert\alpha_g\rangle + c_e\lvert e\rangle\lvert\alpha_e\rangle,$$

then tracing out the intracavity field gives an overlap factor $\langle\alpha_e|\alpha_g\rangle$ in the qubit coherence; emitted fields supply an additional overlap factor. The overlap of the field records decreases as the two pointer amplitudes separate. Thus readout backaction is not a mysterious extra effect; it is the loss of phase coherence caused by information about $\sigma_z$ becoming available outside the qubit.

This is the platform version of Measurement Backaction and Diffusive Trajectories.

Effective Qubit Trajectories

When the resonator is fast compared with the qubit dynamics, one often eliminates it and writes an effective continuous measurement of $\sigma_z$. A schematic homodyne stochastic master equation is

$$d\rho_c = \mathcal L\rho_c\,dt + \sqrt{\eta\Gamma_m}\, \mathcal H[\sigma_z]\rho_c\,dW_t,$$

with record

$$dY_t = 2\sqrt{\eta\Gamma_m}\, \langle\sigma_z\rangle_c\,dt + dW_t.$$

Choose $\mathcal L$ to include $\Gamma_m\mathcal D[\sigma_z]$. Its off-diagonal decay rate is $2\Gamma_m$, so in the steady dispersive limit above $\Gamma_m=\kappa|\alpha_e-\alpha_g|^2/4$. Here $\eta$ includes collection and detector loss, the homodyne phase is aligned with the pointer separation, and $dW_t^2=dt$. Averaging over records recovers this unconditional dephasing.

This reduced equation is powerful, but it hides the resonator dynamics. It is least reliable when the readout pulse is short, the resonator is not adiabatically eliminated, the photon number is high, or dressed-state transitions matter.

Purcell Decay

The resonator also provides a decay channel for the qubit. In the dispersive regime, the qubit-like excitation has a small resonator component. If the resonator decays at rate $\kappa$, this admixture gives the approximate Purcell decay rate

$$\gamma_{\mathrm P} \simeq \kappa \left( \frac{g}{\Delta} \right)^2$$

for an ideal two-level model far from resonances and filters. Real devices use Purcell filters, multilevel corrections, and frequency-dependent impedances, so this expression is a scaling estimate, not a complete design rule.

The conceptual lesson is open-system: an engineered measurement port can also be an unwanted relaxation bath. Increasing readout bandwidth can improve measurement speed while worsening radiative decay unless the environment is shaped carefully.

Noise Channels

Circuit-QED models often organize noise into a few experimentally meaningful rates:

Channel	Model term	Typical physical sources
resonator decay	$\kappa\mathcal D[a]\rho$	external coupling, internal loss
qubit relaxation	$\gamma_1\mathcal D[\sigma_-]\rho$	dielectric loss, Purcell loss, quasiparticles
pure dephasing	$\gamma_\phi\mathcal D[\sigma_z]\rho/2$	flux noise, charge noise, photon-number fluctuations
thermal excitation	$\gamma_\uparrow\mathcal D[\sigma_+]\rho$	hot modes, nonequilibrium quasiparticles
leakage	multilevel dissipators	finite transmon anharmonicity, strong drives

The familiar relation

$$\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\phi}$$

is a rate-level summary, not a microscopic derivation. It assumes Markovian exponential decay and a clean separation between relaxation and pure dephasing.

Common Approximations

Circuit-QED calculations often rely on:

• rotating-wave approximation for the qubit-resonator coupling;
• dispersive expansion in $g/\Delta$;
• two-level truncation of a weakly anharmonic transmon;
• Markovian treatment of transmission-line ports;
• adiabatic elimination of a fast resonator;
• Gaussian or white-noise approximation for selected noise sources;
• semiclassical treatment of strong readout drives.

Each approximation has a failure mode. Strong drives can populate levels outside the qubit subspace. Small detuning can invalidate dispersive elimination. Slow flux noise is not a broadband Markov bath. Residual thermal photons can dephase an apparently idle qubit. A careful model records which approximation is being used rather than hiding all imperfections inside fitted $T_1$ and $T_2$ values.

Exercises

Dispersive Frequency Pull

Starting from the Jaynes–Cummings model with detuning $\Delta=\omega_q-\omega_r$, estimate the resonator frequency shift in the dispersive regime for an ideal two-level qubit.

Solution

For $|\Delta|\gg g$, second-order perturbation theory gives energy shifts of order $g^2/\Delta$. The effective dispersive Hamiltonian is

$$\frac{H_{\mathrm{disp}}}{\hbar} = \left( \omega_r+\chi\sigma_z \right)a^\dagger a + \frac{\omega_q+\chi}{2}\sigma_z,$$

with

$$\chi\simeq\frac{g^2}{\Delta}$$

for the ideal two-level model. Therefore the resonator frequency is shifted to approximately $\omega_r-\chi$ for $\lvert g\rangle$ and $\omega_r+\chi$ for $\lvert e\rangle$.

Measurement-Induced Dephasing

If a readout pulse produces pointer amplitudes $\alpha_g$ and $\alpha_e$ and the resonator decays at rate $\kappa$, what is the steady-state measurement-induced dephasing rate in the ideal dispersive model? Why is a short pulse different?

Solution

The outgoing field carries information about the qubit through the separation of the two pointer amplitudes. In steady state,

$$\Gamma_\phi^{\mathrm{meas}} = \frac{\kappa}{2} \left| \alpha_e-\alpha_g \right|^2.$$

Thus dephasing grows with the rate at which distinguishable information about the qubit leaves through the resonator port. If $\alpha_e=\alpha_g$, the output field does not distinguish the qubit states and this contribution vanishes. During a short pulse, the changing intracavity overlap must also be included, as in the transient expression above.

Purcell Scaling

Explain why the Purcell decay rate scales as $\kappa(g/\Delta)^2$ in the dispersive limit.

Solution

In the dispersive regime, a mostly qubit-like excitation contains a small resonator-like admixture of amplitude of order $g/\Delta$. Decay through the resonator port is proportional to the resonator weight, which is the squared amplitude:

$$\left( \frac{g}{\Delta} \right)^2.$$

Multiplying by the resonator decay rate gives the scaling estimate

$$\gamma_{\mathrm P} \simeq \kappa \left( \frac{g}{\Delta} \right)^2.$$

Combining Relaxation and Dephasing

A qubit has $T_1=40\,\mu\mathrm{s}$ and pure dephasing time $T_\phi=80\,\mu\mathrm{s}$. Under the Markovian rate model, what is $T_2$?

Solution

Use

$$\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\phi}.$$

Substituting,

$$\frac{1}{T_2} = \frac{1}{80\,\mu\mathrm{s}} + \frac{1}{80\,\mu\mathrm{s}} = \frac{1}{40\,\mu\mathrm{s}}.$$

Therefore

$$T_2=40\,\mu\mathrm{s}.$$

References

• A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, “Circuit quantum electrodynamics,” Reviews of Modern Physics 93, 025005 (2021). arXiv:2005.12667.
• A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Physical Review A 69, 062320 (2004).
• A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Reviews of Modern Physics 82, 1155-1208 (2010).
• J. Gambetta, A. Blais, D. I. Schuster, A. Wallraff, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Qubit-photon interactions in a cavity: Measurement-induced dephasing and number splitting,” Physical Review A 74, 042318 (2006).
• J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Charge-insensitive qubit design derived from the Cooper pair box,” Physical Review A 76, 042319 (2007).
• A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162-167 (2004).