Crossover (Quantum)
Level Repulsion and the 2g Gap
Whenever two quantum states are brought into resonance by tuning a control parameter, what happens at the meeting point depends entirely on whether a coupling term connects them. With no coupling the levels are diabatic; they slide straight through one another and a genuine crossover occurs. Introduce a coupling Hamiltonian element g and the picture changes: the eigenvalues can no longer become equal, and instead of touching, the two branches bend away in a hyperbolic anticrossing. This level repulsion is a direct consequence of diagonalizing a 2×2 Hermitian matrix whose off-diagonal element is nonzero, so it appears identically for two coupled qubits, a qubit coupled to a cavity photon, or a qubit hybridizing with a stray two-level defect.
The defining feature is the size of the gap. At the bias where the bare detuning Δ = E1 − E2 passes through zero, the separation between the dressed levels reaches its minimum value of exactly 2g, and each eigenstate is an equal-weight superposition of the original basis states. Away from the crossover the branches asymptotically rejoin the bare energies, so a spectroscopy scan shows two nearly straight lines that smoothly avoid each other in the middle. For superconducting hardware the splitting is read off in frequency units: a g/2π of 50 MHz produces a 100 MHz anticrossing, comfortably resolvable against typical transmon linewidths of a few hundred kilohertz.
Adiabatic Versus Diabatic Passage
How a system behaves while it is driven through the crossover is governed by the sweep speed relative to the gap. A slow flux ramp keeps the state pinned to one eigenvalue branch, so it adiabatically follows the curve around the gap and ends up in the other bare state; this is how controlled-phase entangling gates are realized between tunable superconducting qubits. A fast ramp drives a diabatic (Landau-Zener) transition in which the system continues along its original bare level as though the crossing were unavoided, which is exactly what designers want when shuttling a qubit frequency past a parasitic resonance without leaving its computational state.
Governing Relations
E± = (E1 + E2)/2 ± ½√(Δ2 + 4g2) , Δ = E1 − E2
Minimum gap at Δ = 0:
ΔEmin = 2g
Landau-Zener diabatic (crossing) probability:
Pdiab = exp( −2πg2 / (ℏ × |dΔ/dt|) )
Vacuum Rabi splitting (qubit-resonator):
Δf ≈ 2g/2π , g/2π = (Cc/Cq) × √(fq fr) / 2
Where g = coupling strength, Δ = bare detuning, ℏ = reduced Planck constant, Cc = coupling capacitance, Cq = qubit capacitance, fq, fr = qubit and resonator frequencies. Example: g/2π = 75 MHz → observed anticrossing ≈ 150 MHz.
Crossover Regimes in Circuit QED
| Coupled system | Typical g/2π | Observed gap (2g) | Tuning knob | Crossover use |
|---|---|---|---|---|
| Transmon ↔ readout cavity | 50 to 150 MHz | 100 to 300 MHz | Qubit flux bias | Vacuum Rabi / dispersive shift cal |
| Tunable transmon ↔ transmon | 5 to 30 MHz | 10 to 60 MHz | Flux on one qubit | CZ entangling gate |
| Flux qubit ↔ flux qubit | 20 to 100 MHz | 40 to 200 MHz | Persistent-current bias | Coherent state swap |
| Qubit ↔ TLS defect | 0.1 to 10 MHz | 0.2 to 20 MHz | Frequency sweep | Defect characterization, avoidance |
| Qubit ↔ bus resonator | 30 to 100 MHz | 60 to 200 MHz | Qubit flux bias | Resonator-mediated 2-qubit gate |
Frequently Asked Questions
What is the difference between a crossover and an avoided crossing?
A true crossover (diabatic crossing) happens when two levels intersect because nothing couples them, so the states pass through each other unchanged. An avoided crossing occurs when a coupling term g links the two states, lifting the degeneracy so the branches repel instead of touching. At the bias where the bare levels would cross, the dressed eigenstates are split by exactly 2g and each is an equal superposition of the original states. In superconducting qubits, a spectroscopic feature that looks like a crossing but opens into a gap on closer inspection is direct evidence of coherent coupling.
How do you measure the coupling strength g from an avoided crossing?
Sweep a tuning parameter, usually external flux on a tunable transmon or flux qubit, while running two-tone spectroscopy to map the transition frequencies. Near the crossover the two branches bend away from each other, and the minimum vertical separation equals 2g. With g/2π typically 20 to 300 MHz, the splitting appears as a 40 to 600 MHz gap. Fitting both branches to E = (E1+E2)/2 ± ½√(Δ2+4g2) extracts g, the detuning slope, and the crossover bias. The vacuum Rabi splitting is this same physics for a qubit and a single cavity photon.
Why does sweep speed through a crossover determine whether a state transfers or stays?
The Landau-Zener formula gives the probability of staying on the original bare state as P = exp(−2πg2/(ℏ|dΔ/dt|)), where dΔ/dt is how fast the detuning is swept through the crossover. Sweeping slowly relative to g keeps the system on the adiabatic eigenstate, so it follows the lower branch around the gap and the state transfers. Sweeping fast makes the passage diabatic, so the state continues as if the levels crossed. Controlled-phase gates use the adiabatic limit, while fast flux pulses use the diabatic limit to protect a qubit during frequency excursions.