Abstract
A quantum system in contact with a heat bath undergoes quantum transitions between energy levels upon absorption or emission of energy quanta by the bath. These transitions remain virtual unless the energy of the system is measured repeatedly, even continuously in time. Isolating the two indispensable mechanisms in competition, we describe in a synthetic way the main physical features of thermally activated quantum jumps. Using classical tools of stochastic analysis, we compute in the case of a two-level system the complete statistics of jumps and transition times in the limit when the typical measurement time is small compared to the thermal relaxation time. The emerging picture is that quantum trajectories are similar to those of a classical particle in a noisy environment, subject to transitions à la Kramer in a multi-well landscape, but with a large multiplicative noise.
Similar content being viewed by others
References
Nagourney W., Sandberg J., Dehmelt H.: Shelved optical electron amplifier: observation of quantum jumps. Phys. Rev. Lett. 56, 2797–2799 (1986)
Sauter Th., Neuhauser W., Blatt R., Toschek P.E.: Observation of quantum jumps. Phys. Rev. Lett. 57, 1696–1698 (1986)
Bergquist J.C., Hulet R.G., Itano W.M., Wineland D.J.: Observation of quantum jumps in a single atom. Phys. Rev. Lett. 57, 1699–1702 (1986)
Von Neumann J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1932)
Bohr N.: On the constitution of atoms and molecules. Philos. Mag. 26, 476 (1913)
Carmichael H.J.: An Open System Approach to Quantum Optics. Lecture Notes in Physics, vol. 18. Springer, Berlin (1993)
Dalibard J., Castin Y., Molner K.: Wave function approach to dissipative processes in quantum optics. Phys. Rev. Lett. 68, 580 (1992)
Dalibard, J., Castin, Y., Molner, K.: A Wave function approach to dissipative processes. arXiv:0805.4002
Wiseman H., Milburn G.: Quantum Measurement and Control. Cambridge University Press, Cambridge (2010)
Murch, K.W., Weber, S.J., Macklin, C., Siddiqi, I.: Observing single quantum trajectories of a superconducting qubit. arXiv:1305.7270
Wisemann, H.M., Gambetta, J.M.: Are dynamical quantum jumps detector dependent? Phys. Rev. Lett. 108, 220402 (2012)
Plenio, M.B., Knight, P.L.: The Quantum jump approach to dissipative dynamics in quantum. Rev. Mod. Phys. 70, 101–144 (1998) [and references therein]
Braginsky, V.B., Khalili, F.Y.: In: Thorne, K.S. (ed.) Quantum Measurement. Cambridge University Press, Cambridge (1992)
Grangier P., Levenson J.A., Poizat J.-P.: Quantum non-demolition measurements in optics. Nature 396, 537542 (1998)
Guerlin C. et al.: Progressive field-state collapse and quantum non-demolition photon counting. Nature 448, 889 (2007)
Gleyzes S. et al.: Quantum jumps of light recording the birth and death of a photon in a cavity. Nature 446, 297–300 (2007)
Barchielli A.: Measurement theory and stochastic differential equations in quantum mechanics. Phys. Rev. A 34, 1642 (1986)
Barchielli A., Gregoratti M.: Quantum Trajectories and Measurements in Continuous Time: The Diffusive Case. Lecture Notes in Physics, vol. 782. Springer, Berlin (2009)
Belavkin V.P.: Quantum continual measurements and a posteriori collapse on CCR. Commun. Math. Phys. 146, 611 (1992)
Barchielli A., Belavkin V.P.: Measurements continuous in time and a posteriori states in quantum mechanics. J. Phys. A 24, 1495 (1991)
Korotkov A.N., Averin D.V.: Continuous weak measurement of quantum coherent oscillations. Phys. Rev. B 64, 165310 (2001)
Goan H., Milburn G.J.: Dynamics of a mesoscopic charge quantum bit under continuous quantum measurement. Phys. Rev. B 64, 235307 (2001)
Gambetta J. et al.: Quantum trajectory approach to circuit QED: quantum jumps and the Zeno effect. Phys. Rev. A 77, 012112 (2008)
Jacobs K., Lougovski P., Blencowe M.P.: Continuous measurement of the energy eigenstates of a nanomechanical resonator without a nondemolition probe. Phys. Rev. Lett. 98, 147201 (2007)
Bauer M., Bernard D.: Convergence of repeated quantum nondemolition measurements and wave-function collapse. Phys. Rev. A 84, 044103 (2011)
Lindblad G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976)
Pellegrini C.: Existence, uniqueness and approximation of a stochastic Schrödinger equation: the diffusive case. Ann. Probab. 36, 2332–2353 (2008)
Bauer M., Benoist T., Bernard D.: Iterated stochastic measurements. J. Phys. A 45, 494020 (2012)
Caroli B., Caroli C., Roulet B.: Diffusion in a bistable potential: a systematic WKB treatment. J. Stat. Phys. 21, 415–437 (1979)
Freidlin M.I., Wentzell A.D.: Random Perturbations of Dynamical Systems. Springer, New York (1984)
Hudson R.L., Parthasarathy K.R.: Quantum Ito’s formula and stochastic evolutions. Commun. Math. Phys. 93, 301 (1984)
Jacod J., Protter Ph.: L’essentiel en théorie des probabilités. Cassini, Paris (2003)
Kallenberg O.: Foundations of Modern Probability, 2nd edn. Springer, Berlin (2000)
McKean H.P., Itô K.: Diffusion Processes and their Sample Paths. Classics in Mathematics. Springer, Berlin (1991)
Bauer, M., Bernard, D., et al.: in preparation
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bauer, M., Bernard, D. Real Time Imaging of Quantum and Thermal Fluctuations: The Case of a Two-Level System. Lett Math Phys 104, 707–729 (2014). https://doi.org/10.1007/s11005-014-0688-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-014-0688-z