Quantum Trajectory Theory

In QTT open quantum systems are modelled as scattering processes, with classical external fields corresponding to the inputs and classical stochastic processes corresponding to the outputs (the fields after the measurement process).{{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}} The mapping from inputs to outputs is provided by a quantum stochastic process that is set up to account for a particular measurement strategy (e.g., photon counting, homodyne/heterodyne detection, etc.).{{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|url-status=dead}} The calculated system state as a function of time is known as a quantum trajectory, and the desired 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 N² atomic density matrix elements, whereas QTT only requires N calculations. This makes it useful for simulating large open quantum systems.{{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}}

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.

QTT is also broader in its application than the quantum jump method as it can be applied to many different monitoring strategies including direct photon detection and heterodyne detection. Each different monitoring strategy offers a different picture of the system dynamics.

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.{{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|page=043821|doi=10.1103/PhysRevA.76.043821|arxiv=0704.1686|bibcode=2007PhRvA..76d3821H |s2cid=56107461}}R. Chrétien (2014) "[https://orbi.uliege.be/handle/2268/212746 Laser cooling of atoms: Monte-Carlo wavefunction simulations]" Masters Thesis.

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.{{Cite book|last=Wiseman|first=H.|title=Quantum Measurement and Control|publisher=Cambridge University Press|year=2011}}{{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}}{{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|page=170501|doi=10.1103/PhysRevLett.112.170501|pmid=24836225|arxiv=1402.1868|bibcode=2014PhRvL.112q0501R |s2cid=14481406|via=American Physical Society}}{{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 |page=011002 |doi=10.1103/PhysRevX.6.011002|arxiv=1511.01415 |bibcode=2016PhRvX...6a1002C |s2cid=53548243|doi-access=free}}

It has also been shown that quantum trajectories have full and universal quantum computational power.{{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 |pages=170501 |doi=10.1103/PhysRevLett.108.170501|pmid=22680844 |arxiv=1111.1319 |bibcode=2012PhRvL.108q0501S |hdl=10072/342738 |hdl-access=free }}

QTT addresses one aspect of the 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 "collapse of the wave function". It reconciles the concept of a quantum jump with the smooth evolution described by the 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 superposition states. This prediction was tested experimentally in 2019 by a team at Yale University led by Michel Devoret and Zlatko Minev, in collaboration with Carmichael and others at Yale University and the University of Auckland. In their experiment they used a 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.{{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|language=en}} 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.{{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}}{{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}}{{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}}

{{Reflist}}

• [http://qutip.org/docs/latest/guide/dynamics/dynamics-monte.html mcsolve] Quantum jump (Monte Carlo) solver from QuTiP for Python.
• [https://qojulia.org QuantumOptics.jl] the quantum optics toolbox in Julia.
• [https://qo.phy.auckland.ac.nz/toolbox/ Quantum Optics Toolbox] for Matlab

Category:Quantum mechanics

Category:Computational physics

Category:Monte Carlo methods