Adiabatic passage transfers population between quantum states by changing the Hamiltonian slowly, so that the system follows an instantaneous eigenstate from the initial to the target state. Its great strength is robustness. The result does not depend on the precise pulse area, timing or amplitude — only on the change being slow enough and on the eigenstate connecting the right endpoints. Rapid adiabatic passage in two-level systems and stimulated Raman adiabatic passage (STIRAP) in three-level systems are workhorses of atomic, molecular and magnetic-resonance physics, and of quantum technologies.

The adiabatic theorem

If \(H(t)\) changes slowly compared with the energy gap separating an eigenstate from the rest of the spectrum, a system prepared in that eigenstate remains in it [1, 2]. Adiabatic passage chooses \(H(t)\) so that the relevant eigenstate is the initial state at the start and the target state at the end.

Rapid adiabatic passage (two levels)

Sweep the detuning of a drive through resonance, \(H(t) = \tfrac{\hbar}{2}\big(\Delta(t)\,\sigma_z + \Omega(t)\,\sigma_x\big)\), from large negative to large positive \(\Delta\). The instantaneous eigenstates rotate from \(|0\rangle\) to \(|1\rangle\), so a slow sweep inverts the population. The Landau–Zener formula quantifies “slow” [3]. For a linear sweep \(\Delta = \alpha t\) with constant coupling \(\Omega\), the probability of a diabatic transition (failure) is

$$P_{\rm dia} = \exp\!\Big(-\frac{\pi\,\Omega^2}{2\,|\alpha|}\Big),$$

exponentially small for slow sweeps. In NMR and MRI, frequency-swept adiabatic pulses invert spins uniformly over large bandwidths, despite inhomogeneous radio-frequency fields [4]; see Spins — NMR, ESR & colour centres.

STIRAP (three levels)

To transfer population from \(|1\rangle\) to \(|3\rangle\) via a lossy intermediate state \(|2\rangle\), use a pump field \(\Omega_P\) on \(1\leftrightarrow2\) and a Stokes field \(\Omega_S\) on \(2\leftrightarrow3\). On two-photon resonance the Hamiltonian has a dark state

$$|D\rangle = \cos\theta\,|1\rangle - \sin\theta\,|3\rangle, \qquad \tan\theta = \frac{\Omega_P(t)}{\Omega_S(t)},$$

with no component on \(|2\rangle\). Applying the Stokes pulse before the pump — the “counterintuitive” order — rotates \(\theta\) from \(0\) to \(\pi/2\), and the population moves from \(|1\rangle\) to \(|3\rangle\) without ever occupying the lossy intermediate level [5]. The transfer is robust provided the pulses are adiabatic, typically requiring a pulse area well above \(\pi\) (a common rule of thumb is \(\Omega_{\rm rms}T \gtrsim 10\)). STIRAP is used for molecules, atoms, ions, solid-state spins and superconducting circuits — for example across the three lowest levels of a transmon [6] — and has many extensions to multi-level systems [7, 8, 9].

Adiabatic quantum computation

In adiabatic quantum computing, the ground state of a simple Hamiltonian is slowly deformed into the ground state of a problem Hamiltonian that encodes the answer [10]. The run time is set by the minimum gap. Adapting the speed locally to the gap, rather than sweeping at a constant rate, recovers the quadratic Grover speed-up for unstructured search [11]. See Variational algorithms & QAOA as control.

Making it faster

The weakness of adiabatic passage is its duration, which competes with decoherence. Ways to speed it up:

  • Shortcuts to adiabaticity add counterdiabatic corrections or use invariants to reach the adiabatic result in finite time;
  • dressed-state schemes harness non-adiabatic transitions with corrections that require no extra couplings, while minimizing population in the intermediate level of STIRAP [12];
  • optimal control shapes the pulses for a given duration and error budget, often starting from an adiabatic protocol. Robustness can be kept by optimizing over parameter ensembles; see Robust & ensemble optimal control.

Strengths and limitations

Strengths

  • Insensitive to pulse area, amplitude and exact timing; broadband.
  • STIRAP avoids populating lossy intermediate states.

Limitations

  • Slow: needs large pulse areas and long durations.
  • Sensitive to two-photon detuning (STIRAP) and to decoherence during the long passage.
  • Phases of the final state are less robust than populations, which matters for gates.