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Quantum trajectories the stochastic time evolution of individual quantum systems interacting with an environment or undergoing continuous measurement. A representation of open quantum dynamics in terms of random pure-state evolutions instead of deterministic density matrix evolution.^[1]^[2] This approach is also known as the quantum jump method or stochastic unraveling of the master equation.^[3]

Quantum Trajectories

Basic idea

Instead of evolving the density operator $ρ$ , quantum trajectories describe the evolution of a state vector $| ψ (t) ⟩$ subject to stochastic processes.

Ensemble interpretation

The density operator is recovered as an average over trajectories:

$ρ (t) = 𝔼 [| ψ (t) ⟩ ⟨ ψ (t) |] .$

Each trajectory corresponds to a possible physical realization of the system’s evolution.^[1]

Connection to Lindblad equation

Quantum trajectories provide an equivalent formulation of the Lindblad master equation.

Unraveling

The Lindblad equation

$\frac{d ρ}{d t} = - \frac{i}{ℏ} [\hat{H}, ρ] + \sum_{k} (L_{k} ρ L_{k}^{†} - \frac{1}{2} {L_{k}^{†} L_{k}, ρ})$

can be represented as stochastic evolution of pure states.^[3]

Physical meaning

continuous evolution → effective non-Hermitian Hamiltonian
jumps → discrete stochastic events

Together they reproduce the ensemble dynamics.

Quantum jump method

Effective Hamiltonian

Between jumps, the system evolves under

${\hat{H}}_{e f f} = \hat{H} - \frac{i ℏ}{2} \sum_{k} L_{k}^{†} L_{k} .$

This produces non-unitary evolution.^[2]

Jump process

At random times:

$| ψ ⟩ \to \frac{L_{k} | ψ ⟩}{‖ L_{k} | ψ ⟩ ‖} .$

The jump probability depends on $⟨ L_{k}^{†} L_{k} ⟩$ .

Continuous measurement

Quantum trajectories arise naturally in continuous measurement theory.

Measurement interpretation

Each trajectory corresponds to a measurement record.

Examples:

photon counting
homodyne detection
weak measurement

This links stochastic evolution to experimental observations.^[1]

Diffusive trajectories

In some cases, evolution is continuous rather than involving jumps.

Stochastic Schrödinger equation

$d | ψ ⟩ = - \frac{i}{ℏ} \hat{H} | ψ ⟩ d t + noise terms .$

These describe continuous monitoring processes.^[3]

Relation to decoherence

Decoherence emerges from averaging over trajectories:

individual trajectories remain pure
ensemble average produces mixed states

This explains loss of coherence in open systems.

Applications

Quantum optics

Used to model photon emission and detection processes.^[2]

Quantum information

Applied in:

quantum feedback
error correction
qubit monitoring^[1]

Numerical simulation

Trajectory methods are often more efficient than solving master equations directly.^[3]

Physical significance

Quantum trajectories provide a detailed picture of open quantum dynamics at the level of individual realizations. They unify stochastic processes, measurement theory, and quantum evolution.^[1]

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