Abstract
We consider the Wigner quasi-probability distribution function of a single mode to address the question of whether a stochastic sampling and binning of the absolute square of the complex field amplitude can yield a distribution \(\tilde{P}_{n}\) that closely approximates the true particle number probability distribution \(P_{n}\) of the underlying quantum state. By providing an operational definition of the binned distribution \(P_{n}\) in terms of the Wigner function, we explicitly calculate the overlap between \(\tilde{P}_{n}\) and \(P_{n}\) and hence quantify the statistical distance between the two distributions. We find that there is indeed a close quantitative correspondence between \(\tilde{P}_{n}\) and \(P_{n}\) for a wide range of quantum states that have smooth and broad Wigner function relative to the scale of oscillations of the Wigner function for the relevant Fock state. However, we also find counterexamples, including states with high mode occupation, for which \(\tilde{P}_{n}\) does not closely approximate \(P_{n}\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We clarify our terminology here by noting that evolution of stochastic trajectories for a phase-space variable from some initial state (defined appropriately by a corresponding initial Wigner function) under a particular Hamiltonian, is completely equivalent to directly sampling this variable from the known Wigner function of the final state after said evolution.
References
Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)
Moyal, J.E.: Quantum Mechanics as a Statistical Theory. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 45, pp. 99–124. Cambridge University Press, Cambridge (1949)
Leonhardt, U.: Essential Quantum Optics. Cambridge University Press, Cambridge (2010)
Schleich, W.P.: Quantum Optics in Phase Space. Wiley, New York (2011)
Walls, D.F., Milburn, G.: Quantum Optics. Springer Study Edition. Springer, New York (1995)
Drummond, P.D., Hardman, A.D.: Simulation of quantum effects in Raman-active waveguides. EPL (Europhy. Lett.) 21, 279 (1993)
Werner, M.J., Raymer, M.G., Beck, M., Drummond, P.D.: Ultrashort pulsed squeezing by optical parametric amplification. Phys. Rev. A 52, 4202–4213 (1995)
Steel, M.J., et al.: Dynamical quantum noise in trapped Bose-Einstein condensates. Phys. Rev. A 58, 4824 (1998)
Sinatra, A., Castin, Y., Lobo, C.: A Monte Carlo formulation of the Bogoliubov theory. J. Mod. Opt. 47, 2629 (2000)
Sinatra, A., Lobo, C., Castin, Y.: Classical-field method for time dependent Bose-Einstein condensed gases. Phys. Rev. Lett. 87, 210404 (2001)
Sinatra, A., Lobo, C., Castin, Y.: The truncated Wigner method for Bose-condensed gases: limits of validity and applications. J. Phys. B 35, 3599 (2002)
Gardiner, C.W., Anglin, J.R., Fudge, T.I.A.: The stochastic Gross-Pitaevskii equation. J. Phys. B 35, 1555 (2002)
Polkovnikov, A.: Evolution of the macroscopically entangled states in optical lattices. Phys. Rev. A 68, 033609 (2003)
Norrie, A.A., Ballagh, R.J., Gardiner, C.W.: Quantum turbulence in condensate collisions: an application of the classical field method. Phys. Rev. Lett. 94, 040401 (2005)
Norrie, A.A., Ballagh, R.J., Gardiner, C.W.: Quantum turbulence and correlations in Bose-Einstein condensate collisions. Phys. Rev. A 73, 043617 (2006)
Ruostekoski, J., Isella, L.: Dissipative quantum dynamics of bosonic atoms in a shallow 1d optical lattice. Phys. Rev. Lett. 95, 110403 (2005)
Isella, L., Ruostekoski, J.: Nonadiabatic dynamics of a Bose-Einstein condensate in an optical lattice. Phys. Rev. A 72, 011601 (2005)
Isella, L., Ruostekoski, J.: Quantum dynamics in splitting a harmonically trapped bose-einstein condensate by an optical lattice: truncated wigner approximation. Phys. Rev. A 74, 063625 (2006)
Deuar, P., Drummond, P.D.: Correlations in a BEC collision: first-principles quantum dynamics with 150 000 atoms. Phys. Rev. Lett. 98, 120402 (2007)
Polkovnikov, A.: Phase space representation of quantum dynamics. Ann. Phys. 325, 1790–1852 (2010)
Corney, J.F., Olsen, M.K.: Non-Gaussian pure states and positive wigner functions. Phys. Rev. A 91, 023824 (2015)
Hudson, R.L.: When is the Wigner quasi-probability density non-negative? Rep. Math. Phys. 6, 249–252 (1974)
Blakie, P.B., Bradley, A.S., Davis, M.J., Ballagh, R.J., Gardiner, C.W.: Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques. Adv. Phys. 57, 363–455 (2008)
Martin, A.D., Ruostekoski, J.: Quantum and thermal effects of dark solitons in a one-dimensional Bose gas. Phys. Rev. Lett. 104, 194102 (2010)
Witkowska, E., Deuar, P., Gajda, M., Rzazewski, K.: Solitons as the early stage of quasicondensate formation during evaporative cooling. Phys. Rev. Lett. 106, 135301 (2011)
Karpiuk, T., et al.: Spontaneous solitons in the thermal equilibrium of a quasi-1d Bose gas. Phys. Rev. Lett. 109, 205302 (2012)
Javanainen, J., Ruostekoski, J.: Emergent classicality in continuous quantum measurements. New J. Phys. 15, 013005 (2013)
Lee, M.D., Ruostekoski, J.: Classical stochastic measurement trajectories: Bosonic atomic gases in an optical cavity and quantum measurement backaction. Phys. Rev. A 90, 023628 (2014)
Olsen, M.K., Bradley, A.S.: Numerical representation of quantum states in the positive-P and Wigner representations. Opt. Commun. 282, 3924–3929 (2009)
Olsen, M.K., Bradley, A.S., Cavalcanti, S.B.: Fock-state dynamics in Raman photoassociation of Bose-Einstein condensates. Phys. Rev. A 70, 033611 (2004)
Bhattacharyya, A.: On a measure of divergence between two statistical populations defined by their probability distributions. Bull. Calcutta Math. Soc. 35, 99–109 (1943)
Loudon, R., Knight, P.L.: Squeezed light. J. Mod. Opt. 34, 709–759 (1987)
Yuen, H.P.: Two-photon coherent states of the radiation field. Phys. Rev. A 13, 2226–2243 (1976)
Gilliland, D.C.: Integral of the bivariate normal distribution over an offset circle. J. Am. Stat. Assoc. 57, 758–768 (1962)
Schleich, W., Wheeler, J.A.: Oscillations in photon distribution of squeezed states and interference in phase space. Nature 326, 574–577 (1987)
Marshall, T.W.: Random electrodynamics. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 276, 475–491 (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Lewis-Swan, R.J. (2016). On the Relation of the Particle Number Distribution of Stochastic Wigner Trajectories and Experimental Realizations. In: Ultracold Atoms for Foundational Tests of Quantum Mechanics. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-41048-7_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-41048-7_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-41047-0
Online ISBN: 978-3-319-41048-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)