Dispersive Readout

Dispersive readout is the workhorse measurement of superconducting quantum computing: the technique that turns "which state is the qubit in?" into a microwave phase you can digitize in a hundred nanoseconds. A transmon is capacitively coupled to a microwave resonator, and the two are deliberately placed far apart in frequency — detuned by much more than their coupling $g$ . In that regime they cannot trade real photons, but a second-order, virtual process survives: the resonator frequency is pulled up or down by a tiny amount $\chi$ that depends on whether the qubit sits in $|0\rangle$ or $|1\rangle$ . Send a weak probe tone through the resonator, measure the phase of what returns, and you have read the qubit without ever exchanging energy with it. The mechanism is the Jaynes–Cummings Hamiltonian seen through a Schrieffer–Wolff transformation. To leading order in $g/\Delta$ the exchange coupling is rotated away and replaced by a single term $\chi\,a^\dagger a\,\sigma_z$ — which has two faces: the resonator frequency becomes $\omega_r\pm\chi$ , or equivalently the qubit acquires a photon-number-dependent ac-Stark shift plus a Lamb shift. Because the effective Hamiltonian commutes with $\sigma_z$ , the measurement is quantum non-demolition: repeated reads agree, and the energy eigenstates are never scrambled. What the probe does destroy is coherence between $|0\rangle$ and $|1\rangle$ — information gain and measurement backaction are two halves of one photon flux. This article develops the dispersive Hamiltonian, the transmon's multilevel correction to $\chi$ , the critical photon number that bounds the drive, and the dephasing-versus-fidelity trade every readout engineer lives inside.

Key takeaways

Dispersive readout infers a superconducting qubit's state from the state-dependent frequency of a coupled microwave resonator, exchanging no energy with the qubit.
It lives in the dispersive limit $|\Delta|=|\omega_q-\omega_r|\gg g$ , where the Jaynes–Cummings exchange reduces to $\chi\,\sigma_z\,a^\dagger a$ and pulls the resonator by $\pm\chi$ .
The two-level pull is $\chi=g^2/\Delta$ ; the transmon's second level partly cancels it, giving $\chi=\chi_{01}-\tfrac12\chi_{12}=-(g^2/\Delta)(E_C/\hbar)/(\Delta-E_C/\hbar)$ .
The same photons cause measurement-induced dephasing $\Gamma_\varphi=8\chi^2\bar n/\kappa$ , which for a quantum-limited detector equals the rate at which which-state information is acquired — backaction is the information.
Homodyne detection separates the two pointer states with distinguishability growing as $\sqrt t$ ; a near-quantum-limited Josephson parametric amplifier delivers single-shot fidelities above 99 percent, but pushing beyond $n_{\mathrm{crit}}=\Delta^2/4g^2$ makes the qubit leak and loses the QND character.
Key idea: in the dispersive limit the qubit–resonator coupling becomes $\chi\,\sigma_z\,a^\dagger a$ — the resonator frequency shifts by $+\chi$ or $-\chi$ with the qubit state, so reading the resonator phase reads the qubit, QND and without energy exchange.

The core insight

In the dispersive limit the qubit–resonator coupling produces a term $\chi\,\sigma_z\,a^\dagger a$ : the resonator frequency shifts by $+\chi$ or $-\chi$ depending on the qubit state, so measuring the resonator phase measures the qubit — QND and without energy exchange.

Where this lives today

Dispersive readout is the measurement layer of every superconducting quantum computer now running. On processors from IBM, Google, and Rigetti, each transmon carries its own readout resonator, and dozens share one feedline through frequency multiplexing — one broadband probe, many qubits read at once, each amplified near the quantum limit by a Josephson parametric or traveling-wave amplifier. Single-shot fidelities above 99 percent in 50 to 300 nanoseconds are standard: exactly the projective measurement demanded by the DiVincenzo criteria. The stakes are highest in quantum error correction, where surface-code experiments read ancilla qubits over and over to extract syndromes. Here QND is a requirement, not a nicety — the ancilla must survive each measurement so successive rounds stay consistent, and the readout speed sets the code cycle time and the achievable logical error rate. The same fast, in-lifetime outcome enables active reset — a measured $|1\rangle$ triggers a conditional $\pi$ -pulse that prepares $|0\rangle$ far faster than passive $T_1$ decay — and measurement-based feedback, the backbone of remote entanglement and gate teleportation. In the strong-dispersive regime ( $\chi\gg\kappa$ ) the qubit becomes a photon-number-resolving detector ("number splitting"). Open problems remain pressing: suppressing measurement-induced state transitions at higher photon numbers and integrating quantum-limited amplifiers at scale.

The physical picture

Picture the resonator as a bell and the qubit as a small magnet fixed beside it. Whether the magnet points up or down changes the bell's pitch by a hair. You never strike the bell hard enough to flip the magnet — you only listen for whether it rings a touch sharp or flat. That pitch shift is $2\chi$ , and "listening" is sending in a microwave tone and reading the phase of what returns. Because you never move the magnet, you can ring again and again and get the same answer: this is what makes the measurement quantum non-demolition. Energy conservation is the gatekeeper. With qubit and resonator far apart in frequency, $|\Delta|\gg g$ , a probe photon cannot hop onto the qubit — there is no resonant channel to absorb it. What survives is a virtual, second-order effect: field and qubit briefly borrow energy and hand it back, a frequency pull with no real absorption. That virtuality is exactly why the qubit's energy is left untouched.

Figure

Dispersive coupling pulls the resonator frequency by ±χ depending on the qubit state. For a two-level estimate with g/2π = 100 MHz and Δ/2π = 1 GHz, χ = g²/Δ gives χ/2π = 10 MHz, so the ground- and excited-state resonator responses sit 2χ/2π = 20 MHz apart

There is no free lunch. The probe field that carries away which-state information must, by complementarity, also dephase the qubit's superposition: the same $\bar n$ that sharpens the pointer separation broadens the qubit's phase. Information gain and backaction are two faces of one coin, both proportional to the photon flux through the cavity. And the real transmon is richer than a two-level atom: its $|1\rangle\!\to\!|2\rangle$ transition pulls the cavity with the opposite sign to $|0\rangle\!\to\!|1\rangle$ , so the net shift is a near-cancellation the device is engineered around. The second figure is what an experimentalist sees on a network analyzer — one cavity resonance sitting in one of two places.

Figure

The measured signal: cavity transmission peaks at ωr + χ for the qubit ground state (blue) and ωr − χ for the excited state (red), separated by 2χ/2π = 20 MHz with linewidth κ/2π = 5 MHz. A probe parked at ωr reads the two states as opposite phases; the drive must stay below n_crit = Δ²/4g² = 25 photons

The core equations

Everything starts from the Jaynes–Cummings Hamiltonian: a two-level qubit and a single resonator mode exchanging excitations at the vacuum Rabi rate $g$ , keeping only the energy-conserving terms $a^\dagger\sigma_-$ and $a\,\sigma_+$ .

H_{\mathrm{JC}}=\tfrac{1}{2}\hbar\omega_q\,\sigma_z+\hbar\omega_r\,a^\dagger a+\hbar g\,(a^\dagger\sigma_-+a\,\sigma_+) \qquad (1)

where: $\omega_q$ — qubit transition frequency; $\omega_r$ — bare resonator frequency; $a^\dagger,a$ — resonator photon creation and annihilation operators; $\sigma_z,\sigma_\pm$ — qubit Pauli and raising/lowering operators; $g$ — coupling (vacuum Rabi rate). Reading: the first two terms are the free qubit and field, the third swaps an excitation. Whether that swap actually happens is controlled by one dimensionless ratio — the small parameter every dispersive expansion is organized around.

\Delta\equiv\omega_q-\omega_r,\qquad \lambda\equiv\frac{g}{\Delta}\ll 1 \qquad (2)

where: $\Delta$ — qubit–resonator detuning; $\lambda$ — dispersive expansion parameter. Reading: $\lambda\ll1$ means real exchange is suppressed and can be rotated away order by order. Rotating the exchange away (Derivation 1) leaves an effective Hamiltonian in which qubit and resonator no longer swap energy but shift each other's frequency — the dispersive Hamiltonian.

H_{\mathrm{disp}}\approx\tfrac{1}{2}\hbar(\omega_q+\chi)\,\sigma_z+\hbar\big(\omega_r+\chi\,\sigma_z\big)a^\dagger a,\qquad \chi=\frac{g^2}{\Delta} \qquad (3)

where: $\chi$ — dispersive shift, equal to $g^2/\Delta$ for an ideal two-level system; $\chi\,\sigma_z\,a^\dagger a$ — the term that makes $\omega_r$ qubit-state-dependent; $\tfrac12\chi\,\sigma_z$ — the qubit Lamb shift. Reading: the qubit enters only through $\sigma_z$ , so $H_{\mathrm{disp}}$ commutes with the qubit energy — the seed of QND behavior. The single term $\chi\,\sigma_z\,a^\dagger a$ is the whole readout: the resonator frequency depends on the qubit sign, so the two computational states present two different cavities to the probe.

\omega_r^{(0)}=\omega_r+\chi,\qquad \omega_r^{(1)}=\omega_r-\chi,\qquad 2\chi=\omega_r^{(0)}-\omega_r^{(1)} \qquad (4)

where: $\omega_r^{(0)},\omega_r^{(1)}$ — effective resonator frequency when the qubit is in $|0\rangle$ or $|1\rangle$ ; $2\chi$ — full separation of the two pointer responses. Reading: resolving the states requires this separation to beat the cavity linewidth, $2\chi\gtrsim\kappa$ . The dispersive picture has a hard ceiling: drive too many photons in and the perturbation series diverges. That scale is the critical photon number.

n_{\mathrm{crit}}=\frac{\Delta^2}{4g^2}=\frac{1}{4\lambda^2} \qquad (5)

where: $n_{\mathrm{crit}}$ — intracavity photon number at which the dispersive approximation breaks down. Reading: readout must obey $\bar n\ll n_{\mathrm{crit}}$ ; past it the qubit leaks out of the computational space. A transmon is not a two-level atom: its weakly anharmonic ladder lets the $|1\rangle\!\to\!|2\rangle$ transition pull the cavity oppositely to $|0\rangle\!\to\!|1\rangle$ , so the net shift is a difference of two contributions (Derivation 2).

\chi=\chi_{01}-\tfrac{1}{2}\chi_{12}=\frac{g^2}{\Delta}\,\frac{\alpha}{\Delta+\alpha}=-\frac{g_{01}^2}{\Delta_{01}}\,\frac{E_C/\hbar}{\Delta_{01}-E_C/\hbar} \qquad (6)

where: $\alpha=-E_C/\hbar$ — transmon anharmonicity (negative; $E_C$ is the charging energy); $g_{01}$ — coupling of the $0\!\to\!1$ transition; $\Delta_{01}$ — its detuning from the resonator. Reading: because $\alpha<0$ the two channels partly cancel, so $\chi$ is typically negative and smaller than the naive $g^2/\Delta$ . The probe photons that read the qubit also dephase it: photon-number fluctuations randomly ac-Stark-shift the qubit, washing out any $|0\rangle$ – $|1\rangle$ coherence.

\Gamma_\varphi=\frac{8\chi^2}{\kappa}\,\bar n \qquad (7)

where: $\Gamma_\varphi$ — measurement-induced dephasing rate; $\kappa$ — resonator energy decay (linewidth); $\bar n$ — mean intracavity photon number. Reading: dephasing is the backaction conjugate to information gain, growing with the very $\bar n$ that buys signal. To turn this into a signal, solve for the steady-state cavity field under a coherent probe: each qubit state produces a distinct coherent "pointer state" whose phase encodes the qubit.

\langle a\rangle_{0,1}=\frac{-i\,\varepsilon}{\kappa/2+i(\Delta_{rd}\mp\chi)} \qquad (8)

where: $\varepsilon$ — probe drive amplitude; $\Delta_{rd}=\omega_r-\omega_d$ — detuning of the resonator from the probe; $\mp\chi$ — the qubit-dependent shift. Reading: parked at $\Delta_{rd}=0$ , the two states give conjugate phases $\pm\arctan(2\chi/\kappa)$ that a homodyne detector reads directly. Distinguishing the two pointer states is Gaussian discrimination: the separation in units of noise is the SNR, which converts to fidelity through the complementary error function.

\mathrm{SNR}\;\propto\;\sqrt{\eta\,\kappa\,\bar n}\,\frac{2\chi}{\sqrt{\kappa^2+4\chi^2}}\,\sqrt{t},\qquad \mathcal{F}=1-\tfrac12\,\mathrm{erfc}\!\left(\frac{\mathrm{SNR}}{2\sqrt2}\right) \qquad (9)

where: $\eta\le1$ — detection (quantum) efficiency; $t$ — integration time; $\mathcal F$ — single-shot fidelity. Reading: distinguishability grows as $\sqrt t$ ; a quantum-limited amplifier pushes $\eta\to1$ , and finite $T_1$ caps $\mathcal F$ .

Derivation AThe dispersive shift from the Schrieffer–Wolff transformation

The goal is to show that a far-detuned qubit and resonator no longer exchange energy but shift each other's frequency, and to pin the shift to $\chi=g^2/\Delta$ . We split the Jaynes–Cummings Hamiltonian into a diagonal part and the exchange coupling, then apply a small unitary rotation chosen to cancel that coupling at first order. What survives at second order is a diagonal $a^\dagger a\,\sigma_z$ term — the dispersive coupling — plus a constant Lamb shift. This is the Schrieffer–Wolff transformation, the engine behind every result in this article.

Step 1. Write $H=H_0+V$ with the diagonal piece $H_0=\tfrac12\hbar\omega_q\sigma_z+\hbar\omega_r a^\dagger a$ and the off-diagonal exchange $V=\hbar g(a^\dagger\sigma_-+a\sigma_+)$ . Why: the transformation only needs to remove $V$ ; $H_0$ is already diagonal.

Step 2. Introduce the anti-Hermitian generator $S=\lambda(a^\dagger\sigma_--a\sigma_+)$ with $\lambda=g/\Delta$ , and demand $[S,H_0]=-V$ . A direct commutator confirms this cancels the coupling at first order in the frame $\tilde H=e^{S}He^{-S}$ . Why: this is exactly what makes the linear-in- $g$ term vanish.

Step 3. Expand $\tilde H=H+[S,H]+\tfrac12[S,[S,H]]+\cdots$ . The first-order term $[S,H_0]+V$ vanishes by construction, leaving $\tilde H\approx H_0+\tfrac12[S,V]$ to second order. Why: with the first-order piece killed, the leading correction is this half-commutator.

Step 4. Evaluate the surviving commutator using $[a,a^\dagger]=1$ , $\sigma_+\sigma_-=\tfrac12(1+\sigma_z)$ , and $\sigma_-\sigma_+=\tfrac12(1-\sigma_z)$ , which gives $\tfrac12[S,V]=\hbar\dfrac{g^2}{\Delta}\big(a^\dagger a+\tfrac12\big)\sigma_z$ . Why: the algebra collapses the operator product into a photon-number operator times $\sigma_z$ — purely diagonal.

Step 5. Collect terms: $\tilde H=\hbar\omega_r a^\dagger a+\tfrac12\hbar(\omega_q+\chi)\sigma_z+\hbar\chi\,a^\dagger a\,\sigma_z$ with $\chi=g^2/\Delta$ . The piece $\chi\,a^\dagger a\,\sigma_z$ shifts the resonator by $\pm\chi$ ; the piece $\tfrac12\chi\sigma_z$ is the qubit's Lamb shift. Why: reading the same term as a shift of $\omega_r$ or of $\omega_q$ makes readout and ac-Stark spectroscopy the same physics. *Note: the $+\tfrac12$ inside $(a^\dagger a+\tfrac12)$ correctly yields the vacuum Lamb shift — the constant zero-point contribution that shifts the qubit frequency even with no photons present.*

Go deeper · Derivation B · 1 minThe transmon multilevel dispersive shift

Derivation BThe transmon multilevel dispersive shift

The goal is to correct the two-level result for a real transmon, whose nearly harmonic ladder means the resonator is pulled not only by $0\!\to\!1$ but also by $1\!\to\!2$ , with opposite sign. We compute each transition's pull separately, weight the $1\!\to\!2$ channel by the fact that it acts only when the qubit is in $|1\rangle$ , and combine them. The result explains why measured transmon shifts are typically negative and a few megahertz — not the tens a naive estimate predicts — and why designers tune $\Delta$ and $E_C$ so carefully.

Step 1. Assign each transition its own dispersive pull, $\chi_{01}=g_{01}^2/\Delta_{01}$ and $\chi_{12}=g_{12}^2/\Delta_{12}$ , with the transmon relations $g_{12}\approx\sqrt2\,g_{01}$ (harmonic-oscillator matrix element) and $\Delta_{12}=\Delta_{01}+\alpha$ , where $\alpha=-E_C/\hbar$ is the anharmonicity. Why: each pair of levels contributes a shift, and the $\sqrt2$ enhancement of $g_{12}$ makes the second transition far from negligible.

Step 2. Combine them into the effective qubit shift $\chi=\chi_{01}-\tfrac12\chi_{12}$ . The factor $\tfrac12$ and the minus sign come from projecting the multilevel Stark shifts onto the $\{|0\rangle,|1\rangle\}$ subspace: the $1\!\to\!2$ pull raises the $|1\rangle$ cavity frequency where $0\!\to\!1$ lowered it. Why: only the difference of the two pulls is observable, so the contrast is a competition, not a sum.

Step 3. Substitute the transmon relations: $\chi=\dfrac{g_{01}^2}{\Delta_{01}}-\dfrac12\cdot\dfrac{2g_{01}^2}{\Delta_{01}+\alpha}=\dfrac{g_{01}^2}{\Delta_{01}}\dfrac{\alpha}{\Delta_{01}+\alpha}=-\dfrac{g_{01}^2}{\Delta_{01}}\dfrac{E_C/\hbar}{\Delta_{01}-E_C/\hbar}$ . Why: the algebra collapses to one ratio in $\alpha$ (equivalently $E_C$ ), negative for a typical transmon with $|\Delta_{01}|>E_C/\hbar$ .

Step 4. Sanity-check the limit $|\alpha|\to\infty$ (a true two-level atom, no second level): $\chi\to g_{01}^2/\Delta_{01}$ , recovering Derivation 1. For finite anharmonicity the magnitude is reduced. Why: the transmon must reduce to the two-level result when the higher level is pushed away — and it does. *Numerically, $g_{01}/2\pi=100$ MHz, $\Delta_{01}/2\pi=1$ GHz, $E_C/h=200$ MHz give $\chi_{01}/2\pi=10$ MHz, $\chi_{12}/2\pi=25$ MHz, and $\chi=10-\tfrac12(25)=-2.5$ MHz — note the naive form $-(g^2/\Delta)\alpha/(\Delta-\alpha)$ gives the wrong sign and is not used here.*

Go deeper · Derivation C · 1 minMeasurement-induced dephasing and the information–backaction identity

Derivation CMeasurement-induced dephasing and the information–backaction identity

The goal is to prove that learning the qubit's state through the cavity necessarily destroys any superposition of $|0\rangle$ and $|1\rangle$ , and to compute the rate. A qubit superposition drives the cavity into a state-dependent coherent field; the two fields are distinguishable, so the joint state is entangled, and tracing out the leaking cavity reduces the qubit coherence by their overlap. This yields $\Gamma_\varphi=8\chi^2\bar n/\kappa$ and the deep statement that, at the quantum limit, the dephasing rate equals the rate at which which-state information is extracted.

Step 1. Prepare $|\psi\rangle=c_0|0\rangle|\alpha_0\rangle+c_1|1\rangle|\alpha_1\rangle$ , where $|\alpha_{0,1}\rangle$ are the steady-state coherent pointer states from equation (8). Why: evolving under $\chi\sigma_z a^\dagger a$ correlates each qubit branch with a distinct field — which-path entanglement.

Step 2. The reduced qubit coherence is proportional to the coherent-state overlap, $\rho_{01}\propto\langle\alpha_1|\alpha_0\rangle=\exp[-\tfrac12|\alpha_0-\alpha_1|^2]$ . Why: coherent-state overlap shrinks exponentially with separation, so the more distinguishable the pointers the more coherence is lost.

Step 3. On resonance the steady-state separation is $|\alpha_0-\alpha_1|^2\approx 16\chi^2\bar n/\kappa^2$ , obtained from equation (8) with $\Delta_{rd}=0$ in the limit $\chi\ll\kappa$ . Why: the two cavities respond with conjugate phases, and their difference scales with $\chi/\kappa$ and $\sqrt{\bar n}$ .

Step 4. Photons leak at rate $\kappa$ , each carrying a snapshot of the separation, so coherence decays at $\Gamma_\varphi=\tfrac{\kappa}{2}|\alpha_0-\alpha_1|^2=8\chi^2\bar n/\kappa$ . Why: the rate of information leaving the cavity is set by $\kappa$ times the per-photon distinguishability, which is exactly the dephasing rate.

Step 5. Identify $\Gamma_\varphi=\Gamma_{\mathrm{meas}}$ for an ideal, quantum-limited detector: the dephasing rate equals the measurement rate. Why: you cannot gain which-state information without dephasing by exactly the corresponding amount — complementarity made quantitative, and why $\eta<1$ detectors dephase more than they inform. *The coefficient $8\chi^2\bar n/\kappa$ follows directly from $|\alpha_0-\alpha_1|^2=16\chi^2\bar n/\kappa^2$ and $\Gamma_\varphi=\tfrac{\kappa}{2}|\alpha_0-\alpha_1|^2$ .*

Limiting cases

The dispersive regime is a window between two failures — too close in frequency and the qubit hybridizes with the photon; too weak a pull and the pointer states blur. The knobs are $\lambda=g/\Delta$ and the drive $\bar n$ .

Case	Result	Meaning
Resonant / strong coupling	$\vert\Delta\vert\lesssim g$	Vacuum Rabi splitting: qubit and photon hybridize and swap energy at rate $g$ . No dispersive readout — a measurement scrambles the qubit's energy.
Dispersive readout regime	$g\ll\vert\Delta\vert$ , $\ \bar n\ll n_{\mathrm{crit}}=\Delta^2/4g^2$	The effective coupling $\chi\sigma_z a^\dagger a$ is valid; the QND pull $\pm\chi$ is the operating point of standard readout.
Deep dispersive, weak pull	$g\ll\vert\Delta\vert$ , $\ \chi\ll\kappa$	The pull is smaller than the cavity linewidth: the two pointer states overlap in frequency, SNR collapses, and readout becomes slow and low-fidelity.
Strong dispersive	$\chi\gg\kappa$	The qubit spectrum splits into resolved peaks spaced by $2\chi$ , one per photon number — number splitting, a photon-counting resource.
Overdriven	$\bar n\gtrsim n_{\mathrm{crit}}$	The dispersive approximation fails; higher-order terms drive measurement-induced state transitions (leakage) and the QND character is lost.

Common trapMore probe photons always mean better readout

It is tempting to treat the $\pm\chi$ pull as exact and pour in more photons for signal — SNR does grow as $\sqrt{\bar n}$ . But the dispersive result is only the leading term of an expansion in $\lambda=g/\Delta$ , valid while $\bar n\ll n_{\mathrm{crit}}=\Delta^2/4g^2$ . As $\bar n$ approaches $n_{\mathrm{crit}}$ , higher-order Jaynes–Cummings terms wake up: the qubit is driven up its ladder and out of the computational subspace ("measurement-induced state transitions"), the two-level picture dissolves, and the read stops being QND — the very property that made it useful. Real readout lives in a narrow sweet spot: enough photons for a clean single-shot, few enough to stay dispersive and QND.

Worked examples

1 · Two-level shift and critical photon number (confidence builder)

A qubit couples to its readout resonator with $g/2\pi=100$ MHz, detuned by $\Delta/2\pi=1$ GHz. Treating it as two-level, find $\chi$ , the pointer separation $2\chi$ , and the critical photon number.

Step 1. The dispersive shift is $\chi=g^2/\Delta$ : in ordinary frequency, $\chi/2\pi=(100\ \mathrm{MHz})^2/(1000\ \mathrm{MHz})=\mathbf{10\ MHz}$ .

Step 2. The responses sit at $\omega_r\pm\chi$ , so their separation is $2\chi/2\pi=\mathbf{20\ MHz}$ . With linewidth $\kappa/2\pi=5$ MHz, $2\chi>\kappa$ and the peaks are well resolved — the situation in the second figure.

Step 3. The critical photon number is $n_{\mathrm{crit}}=\Delta^2/4g^2=(1000)^2/(4\cdot100^2)=\mathbf{25}$ . Readout must keep $\bar n$ a fraction of this to stay dispersive and QND. χ/2π = 10 MHz, 2χ/2π = 20 MHz, n_crit = 25.

2 · The transmon correction, fully worked

Same coupling and detuning, but now a real transmon with $E_C/h=200$ MHz (anharmonicity $\alpha/2\pi=-200$ MHz). Find the true dispersive shift and compare with the two-level estimate.

Step 1. The $0\!\to\!1$ pull is the two-level value, $\chi_{01}/2\pi=g_{01}^2/\Delta_{01}=(100)^2/1000=\mathbf{10\ MHz}$ .

Step 2. The $1\!\to\!2$ pull uses $g_{12}=\sqrt2\,g_{01}$ and $\Delta_{12}=\Delta_{01}+\alpha=800$ MHz: $\chi_{12}/2\pi=2\cdot(100)^2/800=\mathbf{25\ MHz}$ — larger than $\chi_{01}$ , since the $\sqrt2$ enhancement and smaller detuning both help.

Step 3. Combine with the projection weight: $\chi=\chi_{01}-\tfrac12\chi_{12}$ , so $\chi/2\pi=10-\tfrac12(25)=\mathbf{-2.5\ MHz}$ . The second level overturns the sign and shrinks the magnitude fourfold.

Step 4. Cross-check the compact form: $\chi=(g^2/\Delta)\,\alpha/(\Delta+\alpha)=10\cdot(-200)/800=-2.5$ MHz. The measurable separation is $2|\chi|/2\pi=\mathbf{5\ MHz}$ — four times smaller than the naive 20 MHz, which is why transmon readout resonators are engineered with care. χ/2π = −2.5 MHz, 2|χ|/2π = 5 MHz (vs. 20 MHz two-level).

3 · Dephasing budget and single-shot fidelity

A readout runs at $\chi/2\pi=2.5$ MHz, $\kappa/2\pi=5$ MHz (so $\chi=\kappa/2$ , near SNR-optimal), and $\bar n=5$ . Estimate the measurement-induced dephasing rate and the single-shot fidelity at a homodyne SNR of 6.

Step 1. The dephasing rate is $\Gamma_\varphi=8\chi^2\bar n/\kappa$ . As a frequency, $\Gamma_\varphi/2\pi=8(2.5)^2(5)/5=\mathbf{50\ MHz}$ , i.e. $\Gamma_\varphi\approx3.1\times10^8\ \mathrm{s^{-1}}$ and $T_\varphi=1/\Gamma_\varphi\approx\mathbf{3.2\ ns}$ .

Step 2. This is the point of QND readout, not a bug: the superposition dephases in nanoseconds, yet the populations of $|0\rangle$ and $|1\rangle$ are untouched because $H_{\mathrm{disp}}$ commutes with $\sigma_z$ . You lose the phase, keep the answer, and can read again.

Step 3. Discrimination of two equal-variance Gaussians gives $\mathcal F=1-\tfrac12\,\mathrm{erfc}(\mathrm{SNR}/2\sqrt2)$ . At SNR $=6$ , $\mathrm{SNR}/2\sqrt2=2.12$ and $\mathrm{erfc}(2.12)=2.7\times10^{-3}$ , so $\mathcal F=\mathbf{0.9986}$ .

Step 4. Since SNR grows as $\sqrt t$ , reaching SNR $=6$ takes an integration time set by $\eta$ , $\kappa$ , $\bar n$ ; a near-quantum-limited amplifier ( $\eta\to1$ ) hits it in tens to hundreds of nanoseconds, while finite $T_1$ sets the ceiling. Γ_φ ≈ 3.1×10⁸ s⁻¹ (T_φ ≈ 3.2 ns); fidelity ≈ 99.86% at SNR = 6.

Physics of the portrait

Dispersive readout is the confluence of two traditions: the cavity QED of atoms and photons, born in the 1960s and perfected into single-photon-resolved measurement by the 1990s, and the superconducting circuits that in 2004 put qubit and resonator on one chip. The through-line is a single concept — a virtual, non-destructive, state-dependent frequency pull.

Origins · 1 mindispersive readout and how this was discovered

dispersive readout

state-dependent cavity pull χ = g²/Δ · virtual photon exchange · circuit QED, 2004–present · measurement in superconducting quantum computing

Dispersive readout has no single inventor; it is the circuit-QED heir of a much older idea. Edwin Jaynes and Fred Cummings wrote down the atom–field model that bears their name in 1963, and the dispersive limit of that model — a frequency shift instead of an exchange — was long understood as the basis of QND detection in atomic physics. Serge Haroche's group in Paris turned it into an art through the 1990s, using the dispersive phase shift a single atom imparts to a microwave cavity to count photons one by one without absorbing them, work recognized with the 2012 Nobel Prize. The decisive translation to solid state came in 2004, when Alexandre Blais, Ren-Shou Huang, Andreas Wallraff, Steven Girvin, and Robert Schoelkopf proposed circuit quantum electrodynamics — a superconducting qubit strongly coupled to an on-chip resonator — and identified the dispersive shift as the readout mechanism, with Wallraff et al. observing strong coupling the same year. Jens Koch and coworkers introduced the transmon and its multilevel shift in 2007, and Jay Gambetta and collaborators (2006) quantified the measurement-induced dephasing that ties information to backaction. Near-quantum-limited Josephson parametric amplification, arriving around 2010–2011, made single-shot high-fidelity readout routine. What began as a way to watch a single atom nudge a cavity is now the measurement that reads out every superconducting quantum processor.

History

1963

— Jaynes and Cummings write down the two-level-atom-plus-single-mode model

1999

— Nakamura, Pashkin, and Tsai demonstrate coherent control of a superconducting charge qubit

2004

— Blais, Huang, Wallraff, Girvin, and Schoelkopf propose circuit QED and dispersive readout; Wallraff et al. observe strong coupling

2007

— Koch et al. introduce the transmon and its multilevel dispersive shift

2011

— near-quantum-limited Josephson parametric amplifiers enable single-shot QND readout and the observation of quantum jumps

Going further

Texts and reviews. Blais, Grimsmo, Girvin, Wallraff, "Circuit quantum electrodynamics," Rev. Mod. Phys. 93, 025005 (2021) — the definitive review · Koch et al., Phys. Rev. A 76, 042319 (2007) — the transmon and its multilevel $\chi$ · Blais, Huang, Wallraff, Girvin, Schoelkopf, Phys. Rev. A 69, 062320 (2004) — the founding circuit-QED proposal · Wallraff et al., Nature 431, 162 (2004) — strong coupling observed · Gambetta et al., Phys. Rev. A 74, 042318 (2006) — measurement-induced dephasing and number splitting · Nielsen & Chuang, Quantum Computation and Quantum Information · Preskill, Lecture Notes on Quantum Computation. The key results to keep in mind: the Schrieffer–Wolff transformation gives $\chi=g^2/\Delta$ and the Lamb shift from $H_{\mathrm{JC}}$ ; the critical photon number is $n_{\mathrm{crit}}=\Delta^2/4g^2$ ; the transmon multilevel shift is $\chi=\chi_{01}-\tfrac12\chi_{12}=(g^2/\Delta)\alpha/(\Delta+\alpha)=-(g^2/\Delta)(E_C/\hbar)/(\Delta-E_C/\hbar)$ , with the worked value $\chi/2\pi=-2.5$ MHz for $g/2\pi=100$ MHz, $\Delta/2\pi=1$ GHz, $E_C/h=200$ MHz; the dephasing rate is $\Gamma_\varphi=8\chi^2\bar n/\kappa$ from the pointer-state overlap; and the Gaussian discrimination fidelity is $\mathcal F=1-\tfrac12\,\mathrm{erfc}(\mathrm{SNR}/2\sqrt2)$ , giving $\mathcal F\approx0.9986$ at SNR $=6$ . Device-specific figures (99 percent fidelities, 50–300 ns) are quoted only at order of magnitude.

Prerequisites. Circuit Quantum Electrodynamics · The Transmon Qubit · Superconducting Qubits · DiVincenzo Criteria. Leads to. Josephson Parametric Amplifier · Quantum Error Correction · Measurement-Based Feedback.

Ask the tutor: Want to see how tuning the detuning Δ and charging energy E_C trades readout speed against Purcell decay and the sign of χ? Ask me to walk through a real transmon readout design.

Does dispersive readout actually touch the qubit's energy?

No — and that is the point. Because the qubit and resonator are detuned by far more than their coupling, energy conservation forbids a probe photon from being absorbed; there is no resonant process. What remains is a virtual, second-order pull with no real exchange. The effective Hamiltonian $\chi\sigma_z a^\dagger a$ depends on the qubit only through $\sigma_z$ , so it commutes with the qubit's energy: the measurement projects onto $|0\rangle$ or $|1\rangle$ without flipping them, and repeated reads agree. That is exactly what "quantum non-demolition" means.

If it is QND, why does it dephase the qubit at all?

QND refers to the energy eigenbasis $\{|0\rangle,|1\rangle\}$ : the populations survive. But a superposition is different. The cavity field takes a slightly different amplitude for each qubit branch, so the qubit and the leaking photons become entangled; once those photons escape to the amplifier carrying which-state information, the phase coherence is gone. This measurement-induced dephasing, $\Gamma_\varphi=8\chi^2\bar n/\kappa$ , is not a flaw but the backaction required by complementarity. You may learn the answer and keep it; you may not learn it and keep the superposition.

Why is the transmon dispersive shift so much smaller than g²/Δ?

Because the transmon is not a two-level atom. Its weakly anharmonic ladder means the resonator is pulled by the $1\!\to\!2$ transition as well as $0\!\to\!1$ , in opposite directions. With $g_{12}=\sqrt2\,g_{01}$ and a detuning only $E_C/\hbar$ different, the $1\!\to\!2$ contribution is actually larger, so the net shift $\chi=\chi_{01}-\tfrac12\chi_{12}$ is a near-cancellation — typically negative and a few megahertz, not the tens a naive estimate gives. This is why $\chi$ is engineered: designers tune $\Delta$ and $E_C$ to balance fast readout against Purcell decay and dephasing.

What exactly is the critical photon number, and what happens past it?

The critical photon number $n_{\mathrm{crit}}=\Delta^2/4g^2=1/4\lambda^2$ is where the dispersive expansion in $\lambda=g/\Delta$ stops converging. Below it, the $\pm\chi$ pull is an excellent description and the read is QND. As $\bar n$ approaches $n_{\mathrm{crit}}$ , higher-order Jaynes–Cummings terms drive the qubit up its ladder and out of the computational subspace — "measurement-induced state transitions" — and the two-level picture collapses. The read stops being QND, with errors no signal processing can undo. Practical readout uses only a fraction of $n_{\mathrm{crit}}$ .

Why measure the phase of the probe rather than its amplitude?

Parked at the bare resonator frequency, the two qubit states shift the cavity by $+\chi$ and $-\chi$ , giving transmitted fields of equal magnitude but opposite phase, $\pm\arctan(2\chi/\kappa)$ . The information sits purely in the phase quadrature, and a homodyne detector — which mixes the signal against a reference "local oscillator" and reads one quadrature — extracts it with the best signal-to-noise. Amplitude readout works if you park on a peak's flank, but phase-sensitive homodyne detection at the symmetric point is the standard, most robust choice, and what quantum-limited amplifiers are built to boost.

How does a Josephson parametric amplifier help?

Single-shot readout means distinguishing the two pointer states in one shot, before the qubit decays — from signals of only a few photons over tens of nanoseconds. Conventional semiconductor amplifiers add far more noise than that, washing out the distinction. A Josephson parametric amplifier (JPA), and its broadband cousin the traveling-wave amplifier, amplify with noise near the quantum limit, pushing the detection efficiency $\eta$ toward 1. Since SNR $\propto\sqrt\eta$ , this is the difference between a blurry multi-microsecond read and a crisp sub-microsecond single-shot above 99 percent fidelity.