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{{Short description|Formulation of quantum mechanics}}
'''Quantum Trajectory Theory (QTT)''' is a formulation of [[Physics:Quantum mechanics|quantum mechanics]] used for simulating [[Physics:Open quantum system|open quantum system]]s, [[Physics:Quantum dissipation|quantum dissipation]] and single quantum systems.<ref name=":73">{{Cite journal|last=Ball|first=Phillip|date=28 March 2020|title=Reality in the making|url=https://www.newscientist.com/article/mg24532750-700-how-a-new-twist-on-quantum-theory-could-solve-its-biggest-mystery/|journal=New Scientist|pages=35–38}}</ref> It was developed by [[Biography:Howard Carmichael|Howard Carmichael]] in the early 1990s around the same time as the similar formulation, known as the [[Physics:Quantum jump method|quantum jump method]] or [[Monte Carlo method|Monte Carlo]] wave function (MCWF) method, developed by [[Biography:Jean Dalibard|Dalibard]], Castin and [[Biography:Klaus Mølmer|Mølmer]].<ref name="MCD1993">{{Cite journal|last1=Mølmer|first1=K.|last2=Castin|first2=Y.|last3=Dalibard|first3=J.|year=1993|title=Monte Carlo wave-function method in quantum optics|journal=Journal of the Optical Society of America B|volume=10|issue=3|page=524|bibcode=1993JOSAB..10..524M|doi=10.1364/JOSAB.10.000524|s2cid=85457742 }}</ref> Other contemporaneous works on wave-function-based [[Monte Carlo method|Monte Carlo]] approaches to open quantum systems include those of Dum, [[Biography:Peter Zoller|Zoller]] and [[Biography:Helmut Ritsch|Ritsch]], and Hegerfeldt and Wilser.<ref name="PrimaryPapers">The associated primary sources are, respectively:

* {{cite journal|last=Dalibard|first=Jean|author2=Castin, Yvan|author3=Mølmer, Klaus|date=February 1992|title=Wave-function approach to dissipative processes in quantum optics|journal=Physical Review Letters|volume=68|issue=5|pages=580–583|arxiv=0805.4002|bibcode=1992PhRvL..68..580D|doi=10.1103/PhysRevLett.68.580|pmid=10045937}}
* {{cite book|last=Carmichael|first=Howard|title=An Open Systems Approach to Quantum Optics|publisher=Springer-Verlag|year=1993|isbn=978-0-387-56634-4}}
* {{cite journal|last=Dum|first=R.|author2=Zoller, P.|author3=Ritsch, H.|year=1992|title=Monte Carlo simulation of the atomic master equation for spontaneous emission|journal=Physical Review A|volume=45|issue=7|pages=4879–4887|bibcode=1992PhRvA..45.4879D|doi=10.1103/PhysRevA.45.4879|pmid=9907570}}
* {{cite book|last1=Hegerfeldt|first1=G. C.|url=http://www.theorie.physik.uni-goettingen.de/~hegerf/collaps_gesamt.pdf|title=Classical and Quantum Systems|last2=Wilser|first2=T. S.|publisher=World Scientific|year=1992|editor1=H.D. Doebner|series=Proceedings of the Second International Wigner Symposium|pages=104–105|chapter=Ensemble or Individual System, Collapse or no Collapse: A Description of a Single Radiating Atom|editor2=W. Scherer|editor3=F. Schroeck, Jr.}}</ref>

QTT is compatible with the standard formulation of quantum theory, as described by the [[Physics:Schrödinger equation|Schrödinger equation]], but it offers a more detailed view.<ref name=":3">{{Cite web|last=Ball|first=Philip|title=The Quantum Theory That Peels Away the Mystery of Measurement|url=https://www.quantamagazine.org/how-quantum-trajectory-theory-lets-physicists-understand-whats-going-on-during-wave-function-collapse-20190703/|access-date=2020-08-14|website=Quanta Magazine|date=3 July 2019 |language=en}}</ref><ref name=":73" /> The Schrödinger equation can be used to compute the probability of finding a quantum system in each of its possible states should a measurement be made. This approach is fundamentally statistical and is useful for predicting average measurements of large ensembles of quantum objects but it does not describe or provide insight into the behaviour of individual particles. QTT fills this gap by offering a way to describe the trajectories of individual quantum particles that obey the probabilities computed from the Schrödinger equation.<ref name=":3" /><ref name=":8">{{Cite journal|title=Collaborating with the world's best to answer century-old mystery in quantum theory|url=https://doddwalls.ac.nz/wp-content/uploads/2020/03/DWC-2018-Annual-Report.pdf|journal=2019 Dodd-Walls Centre Annual Report|pages=20–21|access-date=2020-09-09|archive-date=2021-01-26|archive-url=https://web.archive.org/web/20210126032843/https://doddwalls.ac.nz/wp-content/uploads/2020/03/DWC-2018-Annual-Report.pdf}}</ref> Like the quantum jump method, QTT applies to open quantum systems that interact with their environment.<ref name=":73" /> QTT has become particularly popular since the technology has been developed to efficiently control and monitor individual quantum systems as it can predict how individual quantum objects such as particles will behave when they are observed.<ref name=":3" />

== Method ==

In QTT [[Physics:Open quantum system|open quantum system]]s are modelled as [[Physics:Scattering|scattering]] processes, with classical external fields corresponding to the inputs and classical [[Stochastic process|stochastic process]]es corresponding to the outputs (the fields after the measurement process).<ref name=":0">{{Cite web|title=Howard Carmichael – Physik-Schule|url=https://physik.cosmos-indirekt.de/Physik-Schule/Howard%20Carmichael|access-date=2020-08-14|website=physik.cosmos-indirekt.de|language=de}}</ref> The mapping from inputs to outputs is provided by a quantum [[Stochastic|stochastic]] process that is set up to account for a particular measurement strategy (e.g., [[Physics:Photon counting|photon counting]], [[Homodyne detection|homodyne]]/[[Heterodyne|heterodyne]] detection, etc.).<ref name=":9">{{Cite web|title=Dr Howard Carmichael - The University of Auckland|url=https://unidirectory.auckland.ac.nz/profile/h-carmichael|access-date=2020-08-14|website=unidirectory.auckland.ac.nz|archive-date=2021-05-11|archive-url=https://web.archive.org/web/20210511224344/https://unidirectory.auckland.ac.nz/profile/h-carmichael}}</ref> 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|density matrix]] as a function of time may be calculated by averaging over many simulated trajectories.

Like other Monte Carlo approaches, QTT provides an advantage over direct master-equation approaches by reducing the number of computations required. For a Hilbert space of dimension N, the traditional master equation approach would require calculation of the evolution of N<sup>2</sup> atomic density matrix elements, whereas QTT only requires N calculations. This makes it useful for simulating large open quantum systems.<ref name=":1">{{Cite journal|date=1991|title=Quantum optics. Proceedings of the XXth Solvay conference on physics, Brussels, November 6–9, 1991|url=https://www.sciencedirect.com/journal/physics-reports/vol/219|journal=Physics Reports}}</ref>

The idea of monitoring outputs and building measurement records is fundamental to QTT. This focus on measurement distinguishes it from the quantum jump method which has no direct connection to monitoring output fields. When applied to direct photon detection the two theories produce equivalent results. Where the quantum jump method predicts the quantum jumps of the system as photons are emitted, QTT predicts the "clicks" of the detector as photons are measured. The only difference is the viewpoint.<ref name=":1" />

QTT is also broader in its application than the quantum jump method as it can be applied to many different monitoring strategies including direct [[Physics:Photodetection|photon detection]] and [[Heterodyne|heterodyne]] detection. Each different monitoring strategy offers a different picture of the system dynamics.<ref name=":1" />

== Applications ==
There have been two distinct phases of applications for QTT. Like the quantum jump method, QTT was first used for computer simulations of large quantum systems. These applications exploit its ability to significantly reduce the size of computations, which was especially necessary in the 1990s when computing power was very limited.<ref name="MCD1993" /><ref>{{Cite journal|last=L. Horvath and H. J. Carmichael|date=2007|title=Effect of atomic beam alignment on photon correlation measurements in cavity QED|journal=Physical Review A|volume=76, 043821|issue=4|article-number=043821|doi=10.1103/PhysRevA.76.043821|arxiv=0704.1686|bibcode=2007PhRvA..76d3821H |s2cid=56107461}}</ref><ref>R. Chrétien (2014) "[https://orbi.uliege.be/handle/2268/212746 Laser cooling of atoms: Monte-Carlo wavefunction simulations]" Masters Thesis.</ref>

The second phase of application has been catalysed by the development of technologies to precisely control and monitor single quantum systems. In this context QTT is being used to predict and guide single quantum system experiments including those contributing to the development of quantum computers.<ref name=":73" /><ref name=":10" /><ref>{{Cite book|last=Wiseman|first=H.|title=Quantum Measurement and Control|publisher=Cambridge University Press|year=2011}}</ref><ref>{{Cite journal|last=K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi|date=2014|title=Observing single quantum trajectories of a superconducting quantum bit|journal=Nature|volume=502|issue=7470|pages=211–214|doi=10.1038/nature12539|pmid=24108052|arxiv=1305.7270|s2cid=3648689}}</ref><ref>{{Cite journal|last=N. Roch, M. Schwartz, F. Motzoi, C. Macklin, R. Vijay, A. Eddins, A. Korotkov, K. Whaley, M. Sarovar, and I. Siddiqi|date=2014|title=Observation of measurement-induced entanglement and quantum trajectories of remote superconducting qubits|journal=Physical Review Letters|volume=112, 170501-1-4, 2014.|issue=17|article-number=170501|doi=10.1103/PhysRevLett.112.170501|pmid=24836225|arxiv=1402.1868|bibcode=2014PhRvL.112q0501R |s2cid=14481406|via=American Physical Society}}</ref><ref>{{Cite journal|last=P. Campagne-Ibarcq, P. Six, L. Bretheau, A. Sarlette, M. Mirrahimi, P. Rouchon, and B. Huard|date=2016|title=Observing quantum state diffusion by heterodyne detection of fluorescence|journal=Physical Review X|volume=6|issue=1 |article-number=011002 |doi=10.1103/PhysRevX.6.011002|arxiv=1511.01415 |bibcode=2016PhRvX...6a1002C |s2cid=53548243|doi-access=free}}</ref><ref name=":8" />

It has also been shown that quantum trajectories have full and universal quantum computational power.<ref>{{Cite journal |last1=Santos |first1=M. F. |last2=Terra Cunha |first2=M. |last3=Chaves |first3=R. |last4=Carvalho |first4=A. R. R. |date=2012-04-24 |title=Quantum Computing with Incoherent Resources and Quantum Jumps |url=https://link.aps.org/doi/10.1103/PhysRevLett.108.170501 |journal=Physical Review Letters |volume=108 |issue=17 |article-number=170501 |doi=10.1103/PhysRevLett.108.170501|pmid=22680844 |arxiv=1111.1319 |bibcode=2012PhRvL.108q0501S |hdl=10072/342738 |hdl-access=free }}</ref>

== Quantum measurement problem ==
QTT addresses one aspect of the [[Physics:Measurement problem|measurement problem]] in quantum mechanics by providing a detailed description of the intermediate steps through which a quantum state approaches the final, measured state during the so-called "[[Physics:Wave function collapse|collapse of the wave function]]". It reconciles the concept of a [[Physics:Quantum jump|quantum jump]] with the smooth evolution described by the [[Physics:Schrödinger equation|Schrödinger equation]]. The theory suggests that "quantum jumps" are not instantaneous but happen in a coherently driven system as a smooth transition through a series of [[Physics:Quantum superposition|superposition states]].<ref name=":8" /> This prediction was tested experimentally in 2019 by a team at [[Organization:Yale University|Yale University]] led by [[Biography:Michel Devoret|Michel Devoret]] and Zlatko Minev, in collaboration with Carmichael and others at [[Organization:Yale University|Yale University]] and the [[Organization:University of Auckland|University of Auckland]]. In their experiment they used a [[Physics:Superconductivity|superconducting]] artificial atom to observe a quantum jump in detail, confirming that the transition is a continuous process that unfolds over time. They were also able to detect when a quantum jump was about to occur and intervene to reverse it, sending the system back to the state in which it started.<ref name=":10">{{Cite web|last=Ball|first=Philip|title=Quantum Leaps, Long Assumed to Be Instantaneous, Take Time|url=https://www.quantamagazine.org/quantum-leaps-long-assumed-to-be-instantaneous-take-time-20190605/|access-date=2020-08-27|website=Quanta Magazine|date=5 June 2019 |language=en}}</ref> This experiment, inspired and guided by QTT, represents a new level of control over quantum systems and has potential applications in correcting errors in quantum computing in the future.<ref name=":10" /><ref name=":2">{{Cite web|last=Shelton|first=Jim|date=3 June 2019|title=Physicists can predict the jumps of Schrödinger's cat (and finally save it)|url=https://www.sciencedaily.com/releases/2019/06/190603124621.htm|access-date=2020-08-25|website=ScienceDaily|language=en}}</ref><ref>{{Cite web|last=Dumé|first=Isabelle|date=7 June 2019|title=To catch a quantum jump|url=https://physicsworld.com/a/to-catch-a-quantum-jump/|access-date=2020-08-25|website=Physics World|language=en-GB}}</ref><ref>{{Cite web|last=Lea|first=Robert|date=2019-06-03|title=Predicting the leaps of Schrödinger's Cat|url=https://medium.com/swlh/predicting-the-leaps-of-schr%C3%B6dingers-cat-advanced-warning-of-randomness-in-quantum-mechanics-c8071ca3a662|access-date=2020-08-25|website=Medium|language=en}}</ref><ref name=":8" /><ref name=":73" />

== References ==
{{Reflist}}

== 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) solver from [[Software:QuTiP|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 [[Software:MATLAB|Matlab]]

[[Category:Quantum mechanics]]
[[Category:Computational physics]]
[[Category:Monte Carlo methods]]

{{Sourceattribution|Quantum Trajectory Theory}}

Physics:Quantum Trajectory Theory - Revision history

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