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{{Quantum book backlink|Open quantum systems}}
'''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.<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><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>
This approach is also known as the '''quantum jump method''' or '''stochastic unraveling''' of the master equation.<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>

[[File:Quantum_trajectories.jpg|thumb|400px|Quantum trajectories describe stochastic evolution of individual quantum systems under measurement and environmental interaction.]]

=Quantum Trajectories=
== Basic idea ==

Instead of evolving the density operator <math>\rho</math>, quantum trajectories describe the evolution of a state vector <math>|\psi(t)\rangle</math> subject to stochastic processes.

=== Ensemble interpretation ===

The density operator is recovered as an average over trajectories:

<math>
\rho(t) = \mathbb{E}\left[|\psi(t)\rangle \langle \psi(t)|\right].
</math>

Each trajectory corresponds to a possible physical realization of the system’s evolution.<ref name="WisemanMilburn" />

== Connection to Lindblad equation ==

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

=== Unraveling ===

The Lindblad equation

<math>
\frac{d\rho}{dt}
=
-\frac{i}{\hbar}[\hat{H},\rho]
+
\sum_k
\left(
L_k \rho L_k^\dagger
-\frac{1}{2}\{L_k^\dagger L_k,\rho\}
\right)
</math>

can be represented as stochastic evolution of pure states.<ref name="PlenioKnight1998" />

=== 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

<math>
\hat{H}_{\mathrm{eff}} =
\hat{H}
-
\frac{i\hbar}{2}
\sum_k L_k^\dagger L_k.
</math>

This produces non-unitary evolution.<ref name="Dalibard1992" />

=== Jump process ===

At random times:

<math>
|\psi\rangle \rightarrow \frac{L_k |\psi\rangle}{\|L_k |\psi\rangle\|}.
</math>

The jump probability depends on <math>\langle L_k^\dagger L_k \rangle</math>.

== 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.<ref name="WisemanMilburn" />

== Diffusive trajectories ==

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

=== Stochastic Schrödinger equation ===

<math>
d|\psi\rangle = -\frac{i}{\hbar}\hat{H}|\psi\rangle dt + \text{noise terms}.
</math>

These describe continuous monitoring processes.<ref name="PlenioKnight1998" />

== 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.<ref name="Dalibard1992" />

=== Quantum information ===

Applied in:

* quantum feedback
* error correction
* qubit monitoring<ref name="WisemanMilburn" />

=== Numerical simulation ===

Trajectory methods are often more efficient than solving master equations directly.<ref name="PlenioKnight1998" />

== 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.<ref name="WisemanMilburn" />

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

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Trajectories|1}}

Physics:Quantum Trajectories - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template