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Trajectories 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. This approach is also known as the quantum jump method or stochastic unraveling of the master equation. Quantum trajectories the stochastic time evolution of individual quantum systems interacting with an environment or undergoing continuous measurement. J.|title=Quantum Measurement and Control|publisher=Cambridge University Press|year=2010|doi=10.1017/CBO9780511813948|url=https://www.cambridge.org/core/books/quantum-measurement-and-control/F78F445CD9AF00B10593405E9BAC6B9F This approach is also known as the quantum jump method or stochastic unraveling of the master equation. The density operator is recovered as an average over trajectories: Each trajectory corresponds to a possible physical realization of the system’s evolution.

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.^[2]

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.^[3]

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.^[2]

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.^[3]

Quantum information

Applied in:

quantum feedback
error correction
qubit monitoring^[1]

Numerical simulation

Trajectory methods are often more efficient than solving master equations directly.^[2]

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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-Each trajectory corresponds to a possible physical realization of the system’s evolution.<ref name="WisemanMilburn" />
+Each trajectory corresponds to a possible physical realization of the system’s evolution.<ref name="WisemanMilburn">{{cite book |last=Wiseman|first=H. M.|last2=Milburn|first2=G. J.|title=Quantum Measurement and Control|publisher=Cambridge University Press|year=2010|doi=10.1017/CBO9780511813948|url=https://www.cambridge.org/core/books/quantum-measurement-and-control/F78F445CD9AF00B10593405E9BAC6B9F
+}}</ref>
 == Connection to Lindblad equation ==
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 </math>
-can be represented as stochastic evolution of pure states.<ref name="PlenioKnight1998" />
+can be represented as stochastic evolution of pure states.<ref name="PlenioKnight1998">{{cite journal |last=Plenio |first=M. B. |last2=Knight |first2=P. L. |title=The quantum-jump approach to dissipative dynamics in quantum optics |journal=Reviews of Modern Physics |volume=70 |pages=101–144 |year=1998 |url=https://link.aps.org/doi/10.1103/RevModPhys.70.101 |doi=10.1103/RevModPhys.70.101}}</ref>
 === Physical meaning ===
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 </math>
-This produces non-unitary evolution.<ref name="Dalibard1992" />
+This produces non-unitary evolution.<ref name="Dalibard1992">{{cite journal |last=Dalibard |first=J. |last2=Castin |first2=Y. |last3=Mølmer |first3=K. |title=Wave-function approach to dissipative processes in quantum optics |journal=Physical Review Letters |volume=68 |pages=580–583 |year=1992 |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.68.580 |doi=10.1103/PhysRevLett.68.580}}</ref>
 === Jump process ===
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 {{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
-=References=
+= References =
 {{reflist|3}}

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