Quantum jump method: Difference between revisions

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{{Use American English|date=January 2019}}{{Short description|Computational simulation method for open quantum systems}}

The '''quantum jump method''', also known as the '''[[Monte Carlo method|Monte Carlo]] wave function (MCWF) method''', is a technique in [[computational physics]] used for simulating [[open quantum system]]s and [[quantum dissipation]]. The quantum jump method was developed by [[Jean Dalibard|Dalibard]], Castin and [[Klaus Mølmer,|Mølmer]] withat a verysimilar time to the similar method alsoknown developedas by[[Quantum CarmichaelTrajectory inTheory]] thedeveloped sameby time[[Howard frameCarmichael|Carmichael]]. Other contemporaneous works on wave-function-based [[Monte Carlo method|Monte Carlo]] approaches to open quantum systems include those of Dum, [[Peter Zoller|Zoller]] and [[Helmut Ritsch|Ritsch]] and Hegerfeldt and Wilser.<ref name="MCD1993" /><ref name="PrimaryPapers">The associated primary sources are, respectively:

* {{cite journal|last=Dalibard|first=Jean|coauthors=Castin, Yvan; Mølmer, Klaus|title=Wave-function approach to dissipative processes in quantum optics|journal=Physical Review Letters|date=February 1992|volume=68|issue=5|pages=580–583|doi=10.1103/PhysRevLett.68.580|pmid=10045937|bibcode = 1992PhRvL..68..580D }}

* {{cite book |last=Carmichael |first=Howard |title=An Open Systems Approach to Quantum Optics |year=1993 |publisher=Springer-Verlag |isbn=9780387566344}}

* {{cite journal|last=Dum|first=R.|coauthors=Zoller, P.; Ritsch, H.|title=Monte Carlo simulation of the atomic master equation for spontaneous emission|journal=Physical Review A|year=1992|volume=45|issue=7|pages=4879–4887|doi=10.1103/PhysRevA.45.4879|pmid=9907570|bibcode = 1992PhRvA..45.4879D }}

* {{cite journal |last1=Hegerfeldt |first1=G. C. |last2=Wilser |first2=T. S. |year=1992 |title=Classical and Quantum Systems |journal= Proceedings of the Second International Wigner Symposium |publisher=World Scientific}}

</ref>

* {{cite journal|last=Dalibard|first=Jean|coauthorsauthor2=Castin, Yvan; |author3=Mølmer, Klaus |title=Wave-function approach to dissipative processes in quantum optics|journal=Physical Review Letters|date=February 1992|volume=68|issue=5|pages=580–583|doi=10.1103/PhysRevLett.68.580|pmid=10045937|bibcode = 1992PhRvL..68..580D |arxiv=0805.4002}}

== Method ==

* {{cite book |last=Carmichael |first=Howard |title=An Open Systems Approach to Quantum Optics |year=1993 |publisher=Springer-Verlag |isbn=9780387566344978-0-387-56634-4}}

* {{cite journal|last=Dum|first=R.|coauthorsauthor2=Zoller, P.; |author3=Ritsch, H. |title=Monte Carlo simulation of the atomic master equation for spontaneous emission|journal=Physical Review A|year=1992|volume=45|issue=7|pages=4879–4887|doi=10.1103/PhysRevA.45.4879|pmid=9907570|bibcode = 1992PhRvA..45.4879D }}

* {{cite journalbook |last1=Hegerfeldt |first1=G. C. |last2=Wilser |first2=T. S. |year=1992 |title=Classical and Quantum Systems |journalseries= Proceedings of the Second International Wigner Symposium |publisher=World Scientific|url=http://www.theorie.physik.uni-goettingen.de/~hegerf/collaps_gesamt.pdf|pages=104–105|chapter=Ensemble or Individual System, Collapse or no Collapse: A Description of a Single Radiating Atom|editor1=H.D. Doebner|editor2=W. Scherer|editor3=F. Schroeck, Jr.}}</ref>

== Method ==

[[File:Master equation unravelings.svg|thumb|An example of the quantum jump method being used to approximate the density matrix of a two-level atom undergoing damped [[Rabi oscillation]]s. The random jumps can clearly be seen in the top subplot, and the bottom subplot compares the fully simulated density matrix to the approximation obtained using the quantum jump method.]]

[[File:MC-ensemble average.gif|thumb|Animation of the Monte Carlo prediction (blue) for the population of a coherently-driven, damped two-level system as more trajectories are added to the ensemble average, compared to the master equation prediction (red).]]

The quantum jump method is an approach which is much like the [[Lindblad equation|master-equation treatment]] except that it operates on the wave function rather than using a [[density matrix]] approach. The main component of the method is evolving the system's wave function in time with a pseudo-Hamiltonian; where at each [[time step]], a quantum jump (discontinuous change) may take place with some probability. The calculated system state as a function of time is known as a [[Quantum stochastic calculus#Quantum trajectories|quantum trajectory]], and the desired density matrix as a function of time may be calculated by averaging over many simulated trajectories. For a Hilbert space of dimension N, the number of wave function components is equal to N while the number of density matrix components is equal to N². Consequently, for certain problems the quantum jump method offers a performance advantage over direct master-equation approaches.<ref name=MCD1993>{{Cite doi|10.1364/JOSAB.10.000524}}</ref>

The quantum jump method is an approach which is much like the [[Lindblad equation|master-equation treatment]] except that it operates on the wave function rather than using a [[density matrix]] approach. The main component of thethis method is evolving the system's wave function in time with a pseudo-Hamiltonian; where at each [[time step]], a quantum jump (discontinuous change) may take place with some probability. The calculated system state as a function of time is known as a [[Quantum stochastic calculus#Quantum trajectories|quantum trajectory]], and the desired [[density matrix]] as a function of time may be calculated by averaging over many simulated trajectories. For a Hilbert space of dimension N, the number of wave function components is equal to N while the number of density matrix components is equal to N². Consequently, for certain problems the quantum jump method offers a performance advantage over direct master-equation approaches.<ref name=MCD1993>{{Cite doijournal | last1 = Mølmer | first1 = K. | last2 = Castin | first2 = Y. | last3 = Dalibard | first3 = J. | doi = 10.1364/JOSAB.10.000524 | title = Monte Carlo wave-function method in quantum optics | journal = Journal of the Optical Society of America B | volume = 10 | issue = 3 | pages = 524 | year = 1993 |bibcode = 1993JOSAB..10..524M }}</ref>

<!-- Sections to be written: Algorithm; Equivalence to master equation treatment (maybe); Applications -->

== Further reading ==

* A recent review is {{cite journal|last=Plenio|first=M. B.|author2=Knight, P. L. |title=The quantum-jump approach to dissipative dynamics in quantum optics|journal=Reviews of Modern Physics|date=1 January 1998|volume=70|issue=1|pages=101–144|doi=10.1103/RevModPhys.70.101|bibcode=1998RvMP...70..101P|arxiv = quant-ph/9702007 }}

== References ==

{{Reflist}}

== Further reading ==

* A recent review is {{cite journal|last=Plenio|first=M. B.|author2=Knight, P. L. |title=The quantum-jump approach to dissipative dynamics in quantum optics|journal=Reviews of Modern Physics|date=1 January 1998|volume=70|issue=1|pages=101–144|doi=10.1103/RevModPhys.70.101|bibcode=1998RvMP...70..101P|arxiv = quant-ph/9702007 |s2cid=14721909 }}

== External links ==

* [http://qutip.org/docs/latest/guide/dynamics/dynamics-monte.html mcsolve] {{Webarchive|url=https://web.archive.org/web/20230930194128/https://qutip.org/docs/latest/guide/dynamics/dynamics-monte.html |date=2023-09-30 }} Quantum jump ([[Monte Carlo method|Monte Carlo]]) solver from [[QuTiP]] for [[Python_(programming_language)|Python]].

* [https://qojulia.org QuantumOptics.jl] the quantum optics toolbox in [[Julia (programming language)|Julia]].

* [https://qo.phy.auckland.ac.nz/toolbox/ Quantum Optics Toolbox] for [[MATLAB|Matlab]]

[[Category:Quantum mechanics]]

[[Category:Computational physics]]

[[Category:Monte Carlo methods]]

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