Circuit QED Overview

Circuit quantum electrodynamics, or circuit QED, couples quantized electromagnetic modes to fabricated quantum circuits. It borrows the language of atomic cavity QED but changes the physical implementation:

• a Josephson circuit replaces a natural atom;
• a coplanar, lumped-element, or three-dimensional microwave resonator replaces an optical cavity;
• coaxial lines and lithographic transmission lines replace free-space optical beams;
• phase-coherent microwave sources, usually at room temperature, drive the device through cold attenuators;
• quadrature receivers and parametric amplifiers replace optical photodetectors; and
• electromagnetic design makes many Hamiltonian parameters tunable rather than fixed by nature.

The analogy is powerful because the same bosonic modes, Rabi drives, Jaynes–Cummings interactions, dispersive shifts, input–output relations, and continuous measurements reappear. It is incomplete because an artificial atom is a nonlinear electrical oscillator with many levels, not an isolated two-level atom. Its control ports are also dissipation channels, its material interfaces can limit coherence, and its readout record is produced by an entire cryogenic receiver chain.

The discussion below focuses on transmon circuits and dispersive measurement.

Required background. Jaynes–Cummings model supplies excitation exchange; cavity QED supplies coupling and decay conventions; input–output theory supplies port amplitudes. Josephson effect explains the junction relations behind the circuit nonlinearity.

Translation from Atomic to Circuit QED

The correspondence is most useful when each abstract symbol is tied to a physical object.

Atomic or optical language	Circuit-QED realization	Important qualification
atomic transition	transition between eigenstates of a Josephson circuit	several nearby levels usually exist
optical cavity mode	standing microwave mode of a distributed or lumped resonator	package and slotline modes may also participate
dipole matrix element	charge- or flux-operator matrix element	set by circuit impedance and junction parameters
laser drive	phase-coherent microwave tone	synthesized, attenuated, filtered, and routed through a port
spontaneous emission	radiative and nonradiative relaxation	impedance and material participation matter
homodyne detection	microwave mixing against a local oscillator	amplification adds noise and finite bandwidth
cavity output coupler	coupling capacitor, pin, aperture, or transmission-line port	the same port can create Purcell decay
fixed species	fabricated circuit design	fabrication spread and drift require calibration

Natural atoms have precise selection rules and reproducible spectra. Artificial atoms offer larger electrical dipoles, lithographic integration, and tunability, but their finite anharmonicity makes leakage and drive-dependent dressing central engineering constraints.

From a Circuit to a Quantum Mode

Flux and charge as canonical variables

An electrical network becomes a quantum system after one identifies independent node-flux coordinates. For a node voltage $V(t)$, define

$$\Phi(t) = \int^t V(t')\,dt'.$$

The conjugate variable is the node charge $Q$. For one ideal $LC$ mode,

$$H_{LC} = \frac{Q^2}{2C} + \frac{\Phi^2}{2L}, \qquad [\Phi,Q] = i\hbar.$$

This is a harmonic oscillator with

$$\omega_r = \frac{1}{\sqrt{LC}}, \qquad Z_r = \sqrt{\frac{L}{C}}.$$

A convenient operator convention is

$$\begin{aligned} \Phi &= \Phi_{\mathrm{zpf}} \left( a+a^\dagger \right), \\ Q &= -iQ_{\mathrm{zpf}} \left( a-a^\dagger \right), \end{aligned}$$

where

$$\Phi_{\mathrm{zpf}} = \sqrt{\frac{\hbar Z_r}{2}}, \qquad Q_{\mathrm{zpf}} = \sqrt{\frac{\hbar}{2Z_r}}.$$

Thus impedance controls the size of vacuum flux and charge fluctuations. Large impedance enhances flux fluctuations; small impedance enhances charge fluctuations. This is the circuit counterpart of changing optical mode volume to alter a vacuum-field amplitude.

Why a linear circuit is not an atom

Every transition of an ideal $LC$ oscillator has the same frequency:

$$E_m = \hbar\omega_r \left( m+\frac12 \right).$$

A resonant control pulse therefore cannot select $|0\rangle\leftrightarrow |1\rangle$ without also addressing higher transitions. Frequency-selective control of just these two oscillator levels requires an anharmonic spectrum,

$$\omega_{12} \neq \omega_{01}.$$

In superconducting circuits, the Josephson junction supplies a low-loss nonlinear inductive energy

$$U_J(\phi) = -E_J\cos\phi,$$

where $\phi$ is the gauge-invariant phase difference and

$$E_J = \frac{\hbar I_c}{2e}.$$

Expanding the cosine begins with a quadratic inductance and then produces quartic and higher nonlinearities:

$$-E_J\cos\phi = -E_J + \frac{E_J}{2}\phi^2 - \frac{E_J}{24}\phi^4 + \cdots.$$

The quadratic term joins the linear circuit; the higher terms make the spectrum anharmonic.

Artificial Atoms

The transmon Hamiltonian

The transmon is a capacitively shunted Josephson junction or split-junction loop. Its standard single-mode Hamiltonian is

$$H_{\mathrm t} = 4E_C \left( n-n_g \right)^2 - E_J\cos\phi,$$

where $e^{i\phi}$ raises the integer Cooper-pair number:

$$[n,e^{i\phi}]=e^{i\phi}.$$

The familiar $[\phi,n]=i$ is a local phase-coordinate relation; a compact angle does not define a globally canonical self-adjoint pair on all periodic states.

Here:

• $n$ counts excess Cooper-pair charge in units of $2e$;
• $n_g$ is the dimensionless offset charge;
• $E_C=e^2/(2C_\Sigma)$ is the charging energy for total island capacitance $C_\Sigma$; and
• $E_J$ is the Josephson energy.

For an isolated transmon island, integer Cooper-pair charge gives a periodic phase representation. The charge-basis matrix is

$$\langle n|H_{\mathrm t}|m\rangle =4E_C(n-n_g)^2\delta_{nm} -\frac{E_J}{2}(\delta_{n,m+1}+\delta_{n,m-1}).$$

This tridiagonal eigenproblem avoids approximating the cosine. In the transmon regime,

$$\frac{E_J}{E_C} \gg 1,$$

wavefunctions are localized near a minimum of the cosine potential. Charge dispersion becomes exponentially small in $\sqrt{E_J/E_C}$, making the transition frequency comparatively insensitive to offset-charge noise.

Weakly anharmonic spectrum

Expanding near $\phi=0$ and treating the quartic term perturbatively gives, to leading order,

$$\frac{\omega_{01}}{2\pi} \simeq \frac{\sqrt{8E_JE_C}-E_C}{h},$$

and

$$\frac{\alpha}{2\pi} \equiv \frac{\omega_{12}-\omega_{01}}{2\pi} \simeq -\frac{E_C}{h}.$$

The transmon is therefore only weakly anharmonic:

$$|\alpha| \ll \omega_{01}.$$

That is both a strength and a cost. Large $E_J/E_C$ suppresses charge sensitivity, while a small relative anharmonicity makes short control pulses more likely to populate $|2\rangle$ and higher states.

Worked transmon spectrum

Suppose

$$\frac{E_J}{h} = 20.0\ \mathrm{GHz}, \qquad \frac{E_C}{h} = 0.250\ \mathrm{GHz}.$$

Then

$$\frac{E_J}{E_C} = 80,$$

and the leading transmon approximation gives

$$\begin{aligned} \frac{\omega_{01}}{2\pi} &\simeq \sqrt{ 8(20.0)(0.250) } - 0.250 \\ &= 6.075\ \mathrm{GHz}. \end{aligned}$$

The anharmonicity and next transition are approximately

$$\frac{\alpha}{2\pi} \simeq -0.250\ \mathrm{GHz},$$ $$\frac{\omega_{12}}{2\pi} \simeq 5.825\ \mathrm{GHz}.$$

A measured spectrum should be compared with a numerical diagonalization, not only these asymptotic formulas. Junction asymmetry, additional modes, and coupling-induced Lamb shifts can move the transitions.

A qubit is a selected subspace

The computational states are usually the two lowest eigenstates,

$$|0\rangle \equiv |g\rangle, \qquad |1\rangle \equiv |e\rangle.$$

They do not exhaust the Hilbert space. A reliable control model retains enough levels to estimate leakage and drive-induced frequency shifts:

$$H_{\mathrm{atom}} = \sum_j \hbar\omega_j |j\rangle\langle j|.$$

The charge operator has adjacent-level matrix elements

$$n_{j,j+1} = \langle j|n|j+1\rangle$$

that grow approximately as $\sqrt{j+1}$ in the weakly anharmonic limit. Consequently, a tone aimed at $|0\rangle\leftrightarrow|1\rangle$ can off-resonantly drive $|1\rangle\leftrightarrow|2\rangle$.

Tunability and sweet spots

Replacing one junction by a superconducting loop containing two junctions makes an effective $E_J$ flux dependent. This permits frequency tuning and parametric interactions. Tunability also couples the transition frequency to flux noise. At a first-order sweet spot,

$$\frac{\partial\omega_{01}}{\partial\Phi_{\mathrm{ext}}} = 0,$$

small flux fluctuations contribute only at second and higher order. A frequency-tunable device therefore carries a control-versus-dephasing tradeoff absent from an ideal fixed atom.

Microwave Resonators

Distributed coplanar resonators

A coplanar-waveguide resonator is a section of superconducting transmission line bounded by capacitive or inductive discontinuities. For phase velocity $v_p$ and physical length $\ell$, a half-wave mode has approximately

$$\omega_m \simeq (m+1) \frac{\pi v_p}{\ell}, \qquad m=0,1,2,\ldots.$$

The exact spectrum depends on boundary capacitances, bends, kinetic inductance, coupling structures, and loading by qubits. Distributed resonators offer spatially separated voltage antinodes, multiple ports, and bus modes that can connect several circuits.

Lumped-element resonators

An interdigitated or parallel-plate capacitor joined to a spiral or meandered inductor approximates a compact $LC$ mode. Lumped resonators can reduce footprint and make impedance easier to engineer, but stray capacitance and package coupling become a larger fraction of the intended circuit.

Three-dimensional cavities

A machined superconducting enclosure supports microwave modes with large physical volume and potentially small surface participation. A chip-mounted qubit couples through an antenna. Three-dimensional cavities can provide long-lived storage and clean mode structure, but they are less naturally dense than planar layouts and require careful package-mode accounting.

Readout, bus, and storage roles

Not every resonator should have the same linewidth.

• A readout resonator is deliberately coupled to an output line so its state-dependent field leaves quickly enough to measure.
• A bus resonator mediates coherent interactions between circuits and may be only virtually excited during a gate.
• A storage resonator is engineered for long photon lifetime and small internal loss.
• A Purcell filter is an additional frequency-selective network that passes readout photons while suppressing qubit-frequency radiation.

Calling every mode “the cavity” hides these distinct functions.

Linewidth and quality-factor conventions

Let $\kappa$ be the resonator energy-decay rate, so that

$$\frac{d}{dt} \langle a^\dagger a\rangle = -\kappa \langle a^\dagger a\rangle$$

without a drive at zero bath occupation. At finite occupation the equation is $d\langle n\rangle/dt=-\kappa(\langle n\rangle-\bar n_{\mathrm{th}})$. The coherent field amplitude decays at $\kappa/2$. For a resonance with ordinary frequency $f_r=\omega_r/(2\pi)$ and full width at half maximum $\Delta f$, this convention gives

$$\Delta f = \frac{\kappa}{2\pi}, \qquad Q = \frac{\omega_r}{\kappa} = \frac{f_r}{\Delta f}.$$

Separate the total rate into ports and internal loss:

$$\kappa = \kappa_{\mathrm{in}} + \kappa_{\mathrm{out}} + \kappa_{\mathrm{int}} + \cdots.$$

The loaded, coupling, and internal quality factors obey

$$\frac{1}{Q_L} = \frac{1}{Q_{\mathrm{in}}} + \frac{1}{Q_{\mathrm{out}}} + \frac{1}{Q_i}.$$

One must state whether a quoted linewidth describes energy or field decay and whether it is angular or ordinary frequency.

A cryogenic electromagnetic environment

Microwave photons near several gigahertz have energies far below optical photons. Their equilibrium occupation at temperature $T$ is

$$\bar n_{\mathrm{th}} = \frac{1}{ \exp\!\left( \hbar\omega_r/k_{\mathrm B}T \right)-1 }.$$

For $f_r=6.0\ \mathrm{GHz}$,

$$\frac{hf_r}{k_{\mathrm B}} \simeq 0.288\ \mathrm K.$$

The equilibrium occupations are approximately

$$\begin{array}{c|c} T & \bar n_{\mathrm{th}}\\ \hline 20\ \mathrm{mK} & 5.6\times10^{-7}\\ 50\ \mathrm{mK} & 3.16\times10^{-3}\\ 100\ \mathrm{mK} & 5.95\times10^{-2} \end{array}$$

A dilution refrigerator base temperature does not prove that a mode has the corresponding occupation. Warm radiation can enter through a control line; attenuators can be poorly thermalized; nonequilibrium quasiparticles can excite a qubit; and amplifier backaction can propagate toward the device. Effective mode temperature is an inferred property, not a label copied from a thermometer.

Attenuation, filtering, and isolation

An input chain usually distributes attenuation among temperature stages so that room-temperature thermal noise is absorbed and re-emitted at progressively lower temperatures. Low-pass, infrared, and absorptive filters limit out-of-band radiation. Output isolators or circulators reduce noise sent back from amplifiers.

These components have finite insertion loss, bandwidth, magnetic-field sensitivity, and thermalization. A wiring diagram should therefore report where attenuation occurs, which ports are monitored, and which losses occur before the first amplifier.

Internal loss and participation

Resonator loss is frequently organized by participation ratios:

$$\frac{1}{Q_i} \simeq \sum_k p_k\tan\delta_k + \frac{1}{Q_{\mathrm{other}}},$$

where $p_k$ is the fraction of the total electric stored energy in dielectric region $k$, and $\tan\delta_k$ is its effective loss tangent. Using the fraction of total electric-plus-magnetic energy instead would introduce a factor of two for a resonant lossless reference mode. Interfaces, dielectrics, radiation, seams, vortices, and quasiparticles may all contribute. The model is a useful budget, but fitted participation and loss tangent can be correlated. Geometry changes and independent material controls are needed to identify a mechanism.

Circuit-QED Interaction

Multilevel coupling

For a transmon capacitively coupled to one resonator, a useful retained model is

$$\frac{H}{\hbar} ={} \omega_r a^\dagger a + \sum_j \omega_j |j\rangle\langle j| + \sum_{i,j} g_{ij} \left( a+a^\dagger \right) |i\rangle\langle j|.$$

Choose the resonator phase so its voltage quadrature is proportional to $a+a^\dagger$. Hermiticity requires $g_{ji}=g_{ij}^*$. The coefficients contain a circuit participation factor, the resonator zero-point voltage, and the artificial-atom charge matrix element. In a simplified capacitive picture,

$$g_{ij} \propto \frac{2e}{\hbar} \beta V_{\mathrm{zpf}} \langle i|n|j\rangle,$$

where $\beta$ is a voltage-divider factor. This explains why circuit-QED couplings can be large: the electrical dipole is engineered and the mode voltage can be concentrated at the qubit.

Jaynes–Cummings reduction

If only $|0\rangle$ and $|1\rangle$ are relevant and counter-rotating terms are negligible, the interaction reduces to

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

The detuning convention used here is

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

On resonance, coherent exchange occurs at a rate set by $g$, and the one-excitation vacuum-Rabi splitting is $2g$ in angular-frequency units. The same spectroscopy and time-domain logic used in atomic cavity QED applies.

Where the analogy fails

The two-level Jaynes–Cummings model can fail because:

• the transmon has a nearby $|2\rangle$ level;
• a strong drive changes the dressed spectrum;
• several chip or package modes couple appreciably;
• counter-rotating terms matter when $g/\omega$ is not small;
• a tunable coupler adds its own levels and nonlinearities;
• dissipation is frequency dependent rather than a single constant rate;
• quasiparticle parity or slow classical noise produces nonstationary spectra; or
• the circuit parameters drift during the calibration interval.

The shared Hamiltonian is a controlled approximation, not an ontological claim that a transmon is literally an atom.

Coupled modes and quantum signatures

An avoided crossing with independently known bare modes identifies coherent coupling when linewidths are included. The weak-probe spectrum can also be reproduced by classical coupled oscillators. Quantized excitation structure requires additional observations, such as photon-number-dependent transitions, antibunching, or entanglement; cavity-QED platforms discusses that distinction.

The Dispersive Regime

Ideal two-level result

When

$$|\Delta|\gg g\sqrt{n+1},$$

real excitation exchange is suppressed. A perturbative transformation of the ideal two-level model gives

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

with

$$\chi_{\mathrm{2L}} = \frac{g^2}{\Delta}.$$

The resonator frequencies conditioned on the qubit state differ by

$$\omega_r^{(e)}-\omega_r^{(g)} = 2\chi.$$

Conversely, the qubit frequency depends on photon number:

$$\omega_q(n) \simeq \omega_q + \chi + 2\chi n.$$

This is the ac Stark shift per photon together with the vacuum Lamb shift in the stated convention.

Multilevel transmon correction

For a weakly anharmonic transmon, set $\alpha=\omega_{12}-\omega_{01}<0$ and use the bare resonator frequency in $\Delta=\omega_{01}-\omega_r^{(0)}$. Let $g=g_{01}$ and $g_{12}\simeq\sqrt2g$. Second-order perturbation theory gives

$$\delta\omega_r^{(g)}=-\frac{g^2}{\Delta},\qquad \delta\omega_r^{(e)}= \frac{g^2}{\Delta}-\frac{|g_{12}|^2}{\Delta+\alpha}.$$

Hence half the conditional frequency separation is

$$\chi=\frac{\delta\omega_r^{(e)}-\delta\omega_r^{(g)}}2 \simeq\frac{g^2\alpha}{\Delta(\Delta+\alpha)}.$$

The common resonator shift is $-|g_{12}|^2/[2(\Delta+\alpha)]$. Absorb it into the midpoint frequency $\omega_r$ used in the readout equations below, so the two frequencies are $\omega_r\pm\chi$. The qubit vacuum shift is not obtained by blindly substituting this transmon $\chi$ for the two-level Lamb shift.

Outside the straddling interval, the two virtual paths partly cancel. If $0<\Delta<|\alpha|$, they add; perturbation theory still requires both adjacent transitions to remain well separated from their coupling matrix elements. Blais et al. and Koch et al. derive this multilevel structure.

Worked dispersive shift

Take ordinary-frequency parameters

$$\frac{g}{2\pi} = 100\ \mathrm{MHz}, \qquad \frac{\Delta}{2\pi} = 1.00\ \mathrm{GHz}, \qquad \frac{\alpha}{2\pi} = -200\ \mathrm{MHz}.$$

The common factor of $2\pi$ cancels in the rational expression, giving

$$\begin{aligned} \frac{\chi}{2\pi} &\simeq \frac{ (100\ \mathrm{MHz})^2 (-200\ \mathrm{MHz}) }{ (1000\ \mathrm{MHz}) (800\ \mathrm{MHz}) } \\ &= -2.50\ \mathrm{MHz}. \end{aligned}$$

The separation of the two weak-probe resonances is therefore

$$\frac{2|\chi|}{2\pi} = 5.00\ \mathrm{MHz}.$$

The perturbative ratios are

$$\frac{g}{|\Delta|} = 0.100, \qquad \frac{|g_{12}|}{|\Delta+\alpha|} \simeq \frac{\sqrt2(100)}{800} = 0.177.$$

These are small enough for a first estimate, not a guarantee of high-accuracy dispersive behavior under a strong readout pulse.

Critical photon scale and breakdown

For the ideal two-level model, a common estimate of the photon number where dispersive mixing is no longer small is

$$n_{\mathrm{crit}} = \frac{\Delta^2}{4g^2}.$$

The worked parameters give

$$n_{\mathrm{crit}} = 25.$$

This is a scale, not a hard threshold. A transmon has level-dependent detunings, and readout can induce transitions below this estimate or remain useful beyond it for selected observables. Accurate modeling may require numerical diagonalization and a driven open-system simulation with several transmon levels.

Near the straddling regime, the resonator frequency lies between adjacent transmon transitions. Dispersive contributions can add rather than cancel, producing a larger $|\chi|$, but proximity to another transition also narrows the regime where simple perturbation theory is reliable.

Dispersive Readout

State-dependent pointer fields

Drive the readout resonator through an input port at frequency $\omega_d$. In a frame rotating at the drive, let

$$\Delta_r = \omega_d-\omega_r.$$

For a qubit fixed in eigenstate $s\in\{g,e\}$, the coherent amplitude obeys

$$\dot\alpha_s = - \left[ \frac{\kappa}{2} - i \left( \Delta_r-\chi_s \right) \right] \alpha_s + \sqrt{\kappa_{\mathrm{in}}}\, \beta_{\mathrm{in}}(t),$$

where

$$\chi_g=-\chi, \qquad \chi_e=+\chi$$

for the Hamiltonian convention above. For a constant drive, the steady pointer amplitudes are

$$\alpha_s^{\mathrm{ss}} = \frac{ \sqrt{\kappa_{\mathrm{in}}}\, \beta_{\mathrm{in}} }{ \kappa/2 - i \left( \Delta_r-\chi_s \right) }.$$

Readout works because $\alpha_g$ and $\alpha_e$ produce different outgoing complex amplitudes.

Amplification and loss

Cold attenuation reduces incident thermal noise. Output isolators suppress amplifier noise reaching the chip; a first-stage parametric amplifier then raises the signal above subsequent receiver noise. A phase-preserving amplifier must add noise, whereas a phase-sensitive amplifier can ideally amplify one selected quadrature without that added-noise penalty.

Loss before the first amplifier reduces information even if later gain is arbitrarily large. Homodyne and heterodyne detection explains the quadrature and auxiliary-mode conventions.

Input–output relation

For one monitored port, a common phase convention is

$$b_{\mathrm{out}} = b_{\mathrm{in}} - \sqrt{\kappa_{\mathrm{out}}}\,a.$$

Another convention uses a plus sign. Observable scattering amplitudes are unchanged when the internal phase definitions are transformed consistently. The output pointer separation is proportional to

$$\sqrt{\kappa_{\mathrm{out}}} \left( \alpha_e-\alpha_g \right).$$

Loss before the first amplifier reduces the efficiency and cannot be undone by later gain.

Signal-to-noise and measurement efficiency

For a chosen filter $w(t)$ and measured quadrature $V(t)$, define

$$S = \int w(t)V(t)\,dt.$$

Repeated preparations produce conditional distributions

$$p(S|g), \qquad p(S|e).$$

Under a vacuum-noise homodyne convention, write the record as

$$dY_s=2\sqrt{\eta\kappa_{\mathrm{out}}}\, \operatorname{Re}[e^{-i\vartheta(t)}\alpha_s(t)]\,dt+dW, \qquad \mathbb E[dW^2]=dt.$$

Here $\eta$ includes loss and receiver inefficiency after the selected output port; the monitored fraction is already in $\kappa_{\mathrm{out}}$. For the filtered record, define $\mathrm{SNR}=|\mu_e-\mu_g|/(2\sigma)$ for equal conditional variance. Matching the filter to the mean difference and tracking its quadrature gives

$$\mathrm{SNR}^2= \eta\kappa_{\mathrm{out}}\int|\alpha_e-\alpha_g|^2\,dt.$$

A fixed quadrature uses its projected separation instead and can lose information. Relaxation during a pulse creates non-Gaussian histograms; then this ideal Gaussian SNR does not determine assignment error alone.

Backaction and approximate nondemolition character

The qubit and pointer field become entangled:

$$c_g|g\rangle|\alpha_g\rangle + c_e|e\rangle|\alpha_e\rangle.$$

For an initially common vacuum cavity, a qubit superposition, coherent drive, and vacuum Markov output channels, tracing over both the cavity and emitted fields gives

$$\frac{|\rho_{ge}(t)|}{|\rho_{ge}(0)|} =\exp\left[-\frac12|\alpha_e(t)-\alpha_g(t)|^2 -\frac{\kappa}{2}\int_0^t|\alpha_e-\alpha_g|^2\,dt'\right].$$

The first term is information still in the cavity; the integral is information already carried away. In a steady state the first term is constant and the decay rate is

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

During ring-up or ring-down this expression alone omits the derivative of the intracavity overlap. Other intrinsic dephasing or relaxation must be added separately under their stated models (Gambetta et al.).

A resonant-midpoint readout drive acts for $0<\kappa t<4$, with $\chi/\kappa=1/2$ and drive amplitude $\sqrt{\kappa_{\mathrm{in}}}\beta_{\mathrm{in}}/\kappa=1$. Top: the pointer separation persists after switch-off. Bottom: including the intracavity overlap gives the actual reduced-qubit coherence; the output-only expression approaches it only as the cavity empties.

Dispersive readout is approximately quantum nondemolition when the measured observable is close to the dressed qubit energy and readout-induced transitions are negligible. It is not perfectly nondemolition. Strong driving, dressed dephasing, leakage, residual qubit–resonator exchange, counter-rotating terms, and quasiparticle events can change the state during measurement.

Two different metrics must be separated:

• assignment fidelity asks whether the reported outcome matches the prepared state;
• QND fidelity asks whether repeated measurements preserve and repeat the state.

A detector can assign accurately while disturbing the state, or preserve the state while adding too much noise to assign accurately.

Purcell decay and filtering

The readout port also gives a qubit-like excitation a route to escape. For a weak, spectrally smooth output bath and $|\Delta|\gg g,\kappa$, the ideal two-level dispersive estimate is

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

For

$$\frac{\kappa}{2\pi} = 5.0\ \mathrm{MHz}, \qquad \frac{g}{\Delta} = 0.10,$$

the estimate is

$$\frac{\Gamma_{\mathrm P}}{2\pi} \simeq 0.050\ \mathrm{MHz},$$

or

$$T_{1,\mathrm P} = \frac{1}{\Gamma_{\mathrm P}} \simeq 3.18\ \mu\mathrm s.$$

This deliberately simple result shows the design conflict: a broad resonator speeds field extraction but may shorten $T_1$. A Purcell filter engineers the output impedance so it is transmissive near the readout frequency and suppressive near the qubit frequency. Multilevel matrix elements and the complete frequency-dependent admittance are needed for a predictive device model.

Assignment matrices

For binary readout, write

$$\mathbf m = A\mathbf p,$$

where $\mathbf p=(p_g,p_e)^{\mathsf T}$ is the premeasurement population and $\mathbf m$ is the observed assignment frequency. One convention is

$$A = \begin{pmatrix} P(g_{\mathrm m}|g) & P(g_{\mathrm m}|e)\\ P(e_{\mathrm m}|g) & P(e_{\mathrm m}|e) \end{pmatrix}.$$

A stable calibrated matrix can infer initial populations even when its conditional response includes relaxation during readout. It does not physically restore the state or make the measurement nondemolition. Unmodeled leakage, incorrect calibration preparations, or a changed relaxation process bias the inferred populations.

Coherent Control

Single-qubit drives

A microwave line or resonator applies a time-dependent charge drive. In the qubit subspace and a rotating frame, a resonant control Hamiltonian has the form

$$\frac{H_d(t)}{\hbar} = \frac{1}{2} \left[ \Omega_x(t)\sigma_x + \Omega_y(t)\sigma_y \right] + \frac{\delta(t)}{2}\sigma_z.$$

For fixed drive phase and zero detuning, pulse area sets the rotation. If the rotation axis changes, the Hamiltonians need not commute and the full time-ordered evolution is required. Actual calibration must handle:

• amplitude-to-Rabi-rate conversion;
• carrier detuning and ac Stark shifts;
• line distortion and reflections;
• finite anharmonicity and leakage;
• phase synchronization across channels;
• microwave crosstalk; and
• drift over the calibration interval.

Derivative-quadrature pulse shaping can suppress leading leakage and phase error by adding a quadrature proportional to the envelope derivative. It does not eliminate the need for multilevel simulation and experimental calibration.

Virtual Z rotations

A frame update changes the phase assigned to later control pulses and is equivalent to a $Z$ rotation in the computational frame. It can be nearly instantaneous because no physical pulse is sent, but phase tracking must remain consistent across drives, measurements, couplers, and compiler layers.

Resonator-mediated interactions

Several artificial atoms can couple to one bus mode:

$$\frac{H}{\hbar} = \omega_r a^\dagger a + \sum_i \left[ \frac{\omega_i}{2}\sigma_z^{(i)} + g_i \left( a\sigma_+^{(i)} + a^\dagger\sigma_-^{(i)} \right) \right].$$

In a dispersive regime, virtual photons mediate qubit–qubit interactions. Direct capacitive couplings, tunable couplers, parametric modulation, and microwave-activated interactions provide other routes. The resonator may be a bus, a spectator, a leakage channel, or all three depending on the pulse.

For two qubits weakly dispersive to a common bus, the leading exchange is

$$\frac{H_{\mathrm{ex}}}{\hbar} =J(\sigma_+^{(1)}\sigma_-^{(2)}+\mathrm{h.c.}),\qquad J=\frac{g_1g_2}{2}\left(\frac1{\Delta_1}+\frac1{\Delta_2}\right)$$

after choosing real couplings. A large qubit–qubit detuning makes this exchange off resonant even when $J\ne0$. Higher levels introduce additional conditional shifts and leakage pathways.

Superconducting qubits develops processor-level gates and correlated errors; open-system circuit QED develops conditional measurement dynamics; cavity and circuit QED frontiers connects these models to current research. A single idle $T_1$ or one assignment number does not determine a multi-qubit operation.

Exercises

1. Quantize an LC mode

An ideal resonator has $L=8.0\ \mathrm{nH}$ and $C=80\ \mathrm{fF}$.

1. Find its resonance frequency.
2. Find its impedance.
3. Explain whether increasing $L/C$ at fixed $LC$ enhances zero-point flux or charge fluctuations.

Solution

The angular frequency is

$$\omega_r = \frac{1}{\sqrt{LC}}.$$

With

$$LC = (8.0\times10^{-9}) (80\times10^{-15}) = 6.4\times10^{-22}\ \mathrm{s^2},$$

we obtain

$$\omega_r = 3.95\times10^{10}\ \mathrm{s^{-1}},$$

and

$$f_r = \frac{\omega_r}{2\pi} = 6.29\ \mathrm{GHz}.$$

The impedance is

$$Z_r = \sqrt{\frac{L}{C}} = \sqrt{\frac{8.0\ \mathrm{nH}}{80\ \mathrm{fF}}} = 316\ \Omega.$$

At fixed $LC$, increasing $L/C$ increases $Z_r$. Since

$$\Phi_{\mathrm{zpf}} \propto \sqrt{Z_r}, \qquad Q_{\mathrm{zpf}} \propto \frac{1}{\sqrt{Z_r}},$$

zero-point flux increases and zero-point charge decreases.

2. Converge the charge-basis spectrum

For $E_J/h=20.0\,\mathrm{GHz}$, $E_C/h=0.250\,\mathrm{GHz}$, and $n_g=0$, compare the leading oscillator approximation with a charge-basis calculation. Use Python with NumPy:

import numpy as np
for cutoff in (12, 18, 24):
    n = np.arange(-cutoff, cutoff + 1)
    H = np.diag(4 * 0.250 * n**2)
    H += np.diag(np.full(len(n)-1, -20.0/2), 1)
    H += np.diag(np.full(len(n)-1, -20.0/2), -1)
    E = np.linalg.eigvalsh(H)   # energies divided by h, in GHz
    f01, f12 = np.diff(E[:3])
    print(cutoff, f01, f12, f12-f01)

Solution

The converged values are

$$f_{01}=6.063469\,\mathrm{GHz},\qquad f_{12}=5.786084\,\mathrm{GHz},\qquad \frac{\alpha}{2\pi}=-0.277386\,\mathrm{GHz}.$$

The leading $f_{01}=6.074555\,\mathrm{GHz}$ estimate is close, while the leading anharmonicity $-0.250\,\mathrm{GHz}$ differs by about 11% relative to that approximation. Subtracting adjacent transitions amplifies the importance of higher corrections. Converging the charge cutoff addresses numerical truncation; it does not include omitted circuit modes or coupling to a resonator.

3. Microwave thermal occupation

Calculate the equilibrium thermal occupation of a $6.0\ \mathrm{GHz}$ resonator at $20\ \mathrm{mK}$ and $100\ \mathrm{mK}$. Use

$$\frac{hf}{k_{\mathrm B}} = 0.288\ \mathrm K$$

for this frequency. What experimental conclusion does the comparison support?

Solution

The Bose occupation is

$$\bar n_{\mathrm{th}} = \frac{1}{ \exp(hf/k_{\mathrm B}T)-1 }.$$

At $20\ \mathrm{mK}$,

$$\frac{hf}{k_{\mathrm B}T} = \frac{0.288}{0.020} = 14.4,$$

so

$$\bar n_{\mathrm{th}} \simeq 5.6\times10^{-7}.$$

At $100\ \mathrm{mK}$,

$$\frac{hf}{k_{\mathrm B}T} = 2.88,$$

and

$$\bar n_{\mathrm{th}} \simeq 5.95\times10^{-2}.$$

The occupation is extremely temperature sensitive in this range. Even when the refrigerator base is cold, a mode coupled weakly to a warmer line can have enough residual photons to affect dephasing and initialization. Measure or bound the effective occupation rather than inferring it solely from the base thermometer.

4. Multilevel dispersive shift

A transmon and readout resonator have

$$\frac{g}{2\pi} = 100\ \mathrm{MHz}, \quad \frac{\Delta}{2\pi} = 1.00\ \mathrm{GHz}, \quad \frac{\alpha}{2\pi} = -200\ \mathrm{MHz}.$$

Using

$$\chi \simeq \frac{g^2\alpha}{ \Delta(\Delta+\alpha) },$$

find $\chi/(2\pi)$ and the separation between the two conditional resonator frequencies. Compare with the ideal two-level result.

Solution

Using ordinary-frequency units consistently,

$$\begin{aligned} \frac{\chi}{2\pi} &\simeq \frac{ (100)^2(-200) }{ (1000)(800) } \ \mathrm{MHz} \\ &= -2.50\ \mathrm{MHz}. \end{aligned}$$

The two conditional resonator frequencies are separated by

$$\frac{2|\chi|}{2\pi} = 5.00\ \mathrm{MHz}.$$

The ideal two-level formula would give

$$\frac{\chi_{\mathrm{2L}}}{2\pi} = \frac{(100\ \mathrm{MHz})^2}{ 1000\ \mathrm{MHz} } = 10.0\ \mathrm{MHz}.$$

It has the wrong magnitude and, under the stated Hamiltonian convention, the opposite sign. The nearby $|1\rangle\leftrightarrow|2\rangle$ transition cannot be ignored.

5. Critical photons and Purcell tradeoff

For $g/\Delta=0.10$ and $\kappa/(2\pi)=5.0\ \mathrm{MHz}$:

1. estimate $n_{\mathrm{crit}}$ in the ideal two-level model;
2. estimate the Purcell-limited $T_1$; and
3. describe why simply reducing $\kappa$ is not a complete solution.

Solution

The critical photon scale is

$$n_{\mathrm{crit}} = \frac{\Delta^2}{4g^2} = \frac{1}{4(0.10)^2} = 25.$$

The Purcell rate estimate is

$$\frac{\Gamma_{\mathrm P}}{2\pi} \simeq \frac{\kappa}{2\pi} \left( \frac{g}{\Delta} \right)^2 = (5.0\ \mathrm{MHz})(0.01) = 0.050\ \mathrm{MHz}.$$

Therefore

$$T_{1,\mathrm P} = \frac{1}{2\pi(0.050\ \mathrm{MHz})} = 3.18\ \mu\mathrm s.$$

Reducing $\kappa$ suppresses this simple Purcell channel but slows resonator ring-up and ring-down, which can lengthen measurement and reset. A frequency-selective Purcell filter can keep a broad readout channel near $\omega_r$ while reducing environmental admittance near $\omega_q$.

6. Separate relaxation and dephasing

A qubit has $T_1=60\ \mu\mathrm s$ and Ramsey $T_2=40\ \mu\mathrm s$. Under the exponential Markovian rate model, find $T_\phi$. Then give two reasons the inferred value might not identify a single microscopic noise mechanism.

Solution

Use

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

Thus

$$\begin{aligned} \frac{1}{T_\phi} &= \frac{1}{40\ \mu\mathrm s} - \frac{1}{120\ \mu\mathrm s} \\ &= \frac{1}{60\ \mu\mathrm s}, \end{aligned}$$

so

$$T_\phi = 60\ \mu\mathrm s.$$

This number is an effective rate under the stated fit. Slow flux noise can produce nonexponential Ramsey decay, and residual photons, critical-current noise, charge dispersion, or telegraph fluctuators can contribute simultaneously. In addition, both $T_1$ and the qubit frequency can drift during data collection.

7. Correct a binary assignment model

Use the convention

$$\mathbf m = A\mathbf p, \qquad A = \begin{pmatrix} 0.97 & 0.08\\ 0.03 & 0.92 \end{pmatrix}.$$

An experiment observes

$$\mathbf m = \begin{pmatrix} 0.62\\ 0.38 \end{pmatrix}.$$

Estimate $\mathbf p$ by matrix inversion. State one reason not to regard the corrected values as exact.

Solution

The determinant is

$$\det A = (0.97)(0.92)-(0.08)(0.03) = 0.8900.$$

Therefore

$$A^{-1} = \frac{1}{0.8900} \begin{pmatrix} 0.92 & -0.08\\ -0.03 & 0.97 \end{pmatrix}.$$

Applying it,

$$p_g = \frac{ 0.92(0.62)-0.08(0.38) }{ 0.8900 } \simeq 0.607,$$ $$p_e = \frac{ -0.03(0.62)+0.97(0.38) }{ 0.8900 } \simeq 0.393.$$

The correction inherits finite-sample and calibration uncertainty. It is also biased if the prepared calibration states were imperfect, if leakage is omitted, if the assignment probabilities drift, or if relaxation during measurement differs from that included in the calibrated response.

8. Noise from a warm attenuator

A passive attenuator has power transmission $\tau$ and thermal occupation $n_T$ at the signal frequency. Its unused bath mode is independent of the input. Derive the output mean occupation, then compare a 20dB attenuator at 4K and at 20mK for a 6GHz input with $n_{\mathrm{in}}=1000$.

Solution

The bosonic loss channel is $b_{\mathrm{out}}=\sqrt{\tau}b_{\mathrm{in}}+\sqrt{1-\tau}b_T$. For independent zero-mean thermal inputs,

$$n_{\mathrm{out}}=\tau n_{\mathrm{in}}+(1-\tau)n_T.$$

Here $\tau=0.01$. At 4K, $n_T=(e^{0.288/4}-1)^{-1}=13.39$, giving $n_{\mathrm{out}}\simeq23.3$. At 20mK, $n_T\simeq5.6\times10^{-7}$, giving $n_{\mathrm{out}}\simeq10.0$. The same attenuation at a warm stage adds much more noise. A second cold 20dB stage reduces the latter value to about 0.1, still far above its equilibrium vacuum occupation. Real stage placement also has to accommodate dissipated drive power and thermal anchoring.

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