Physics:Quantum Trajectories

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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 describe stochastic evolution of individual quantum systems under measurement and environmental interaction.

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

dρdt=−iℏ[H^,ρ]+∑k(LkρLk†−12{Lk†Lk,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

H^eff=H^−iℏ2∑kLk†Lk.

This produces non-unitary evolution.[2]

Jump process

At random times:

|ψ⟩→Lk|ψ⟩‖Lk|ψ⟩‖.

The jump probability depends on ⟨Lk†Lk⟩.

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⟩=−iℏH^|ψ⟩dt+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]

They are essential for interpreting modern quantum experiments involving continuous observation.

See also

Table of contents (185 articles)

Index

Full contents

9. Quantum optics and experiments (5) ↑ Back to index
14. Plasma and fusion physics (8) ↑ Back to index
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 Wiseman, H. M.; Milburn, G. J. (2010). Quantum Measurement and Control. Cambridge University Press. doi:10.1017/CBO9780511813948. https://www.cambridge.org/core/books/quantum-measurement-and-control/F78F445CD9AF00B10593405E9BAC6B9F. 
  2. ↑ 2.0 2.1 2.2 Dalibard, J.; Castin, Y.; Mølmer, K. (1992). "Wave-function approach to dissipative processes in quantum optics". Physical Review Letters 68: 580–583. doi:10.1103/PhysRevLett.68.580. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.68.580. 
  3. ↑ 3.0 3.1 3.2 3.3 Plenio, M. B.; Knight, P. L. (1998). "The quantum-jump approach to dissipative dynamics in quantum optics". Reviews of Modern Physics 70: 101–144. doi:10.1103/RevModPhys.70.101. https://link.aps.org/doi/10.1103/RevModPhys.70.101. 
Author: Harold Foppele

Source attribution: Quantum Trajectories

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