Correspondence on "Single-shot simulations of dynamic quantum many-body systems"

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by  M. K. Olsen

Correspondence on “Single-shot simulations of dynamic quantum many-body
systems”
M. K. Olsen1, J. F. Corney1, R. J. Lewis-Swan2, and A. S. Bradley3
1School of Mathematics and Physics, University of In so far as the predictions of the TWM are accu-
Queensland, Brisbane, QLD 4072, Australia. rate,thejointprobabilitydistributionprovidesahidden-
2Department of Physics, University of Colorado, Boul- variable explanation for symmetrically ordered observ-
der, Colorado 80309-0440, USA. ables. It is then consistent to assign an element of real-
3Department of Physics, Centre for Quantum Science, ity to individual trajectories. For other types of observ-
7 andDodd-WallsCentreforPhotonicandQuantumTech- ables, the interpretation is not so straightforward, but
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nologies, University of Otago, Dunedin, New Zealand. the corrections to calculated averages are simple. For
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In a recent letter to Nature Physics, Sakmann and large enough N these contributions become negligible.
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Kasevich claim to solve the many-body time depen- The interpretation of single trajectory TWM (or clas-
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dent Schr¨odinger equation to simulate single experimen- sical field method) simulations as possible experimental
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tal runs of interacting quantum systems [1]. In a sub- outcomes has succesfully described vortex formation in
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sequent comment, Drummond and Brand pointed out trapped condensates [6]. Via appropriate noise in the
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that the authors had, in two of their three results, only initial conditions, it yields insight into interesting non-
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solved a truncated version of the full equation [2]. Fur- mean field physics captured in the deterministic evolu-

] ther to these criticisms, we add some further comments tion of a single trajectory [5]. Lewis-Swan et al. also
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particularlyinreferencetothetruncatedWignermethod show that TWM trajectories quantitatively reproduce
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(TWM) as applied to condensed Bose gases [3]. the number distributions for a wide and well defined
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t The authors claim that the MCTDHB (multiconfig- range of states [7]. For a Bose-Hubbard model the re-
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urational time-dependent Hartree method for bosons) sults compare well with the exact values. These results
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u method can perform single-shot simulations of many- support claims that individual trajectories have useful
q body quantum experiments, whereas the TWM cannot and practical relationship to experiment.
[ because it is restricted to positive distributions. How- The arguments provided by Sakmann and Kasevich

ever, this argument overlooks the point that the imple- concern the actual Wigner function, and are thus not
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v mentation of the MCTDHB as described in the paper is relevanttothepositiveapproximationtothissampledby
2 also not exact. For example, the first two results in [1] the TWM. If the TWM describes a particular physical
8 consider only two or four terms in expansions which for- scenario, i.e. the operator moments are accurate, then
2 mally require N terms for an N-body system. The fact assigningphysicalsignificancetothequadraturevaluesof
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that the authorsmanagedto reproducethe exactresults individualtrajectoriesrequiresnofurtherapproximation.
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for 10 atoms says little about the accuracy of their basis Insummary,MCTDHBandTWMarepowerfulmeth-
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2 truncation for either 100 or 10,000 atoms. ods with different limitations. Their domains of applica-
0 The issue of non-Gaussian pure states and positive bility are complimentary, with MCTDHB demonstrably
7 Wignerfunctionsraisedin[1]hasalreadybeendiscussed working well for small N, whereas TWM improves as N
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in [4]. Essentially,the TWM providesa jointprobability increases. Althoughcaremustbetakenintheinterpreta-
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v distribution that is often a good approximation to the tionofsingletrajectoriesofTWM,theiruseinsimulating
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X true Wigner distribution, and generally becomes more experimental shots has some justification and has more-
accuratefor largerN. It cangive anaccuratecalculation over proven insightful in understanding the physics of a
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a ofthe momentsfor unitaryevolutionevenwhenthe true number of experimental situations.
Wigner function is not positive definite [4], and achieves The authors declare that they have no competing fi-
this without introducing mixedness. nancial interests.
[1] K.SakmannandM.Kasevich,Nat.Phys.12,451(2016). Lett. 94, 040401 (2005).
[2] P.D. Drummond and J. Brand, arXiv:1610.07633v1. [6] C. N. Weiler et al. , Nature455, 948 (2008).
[3] M.J. Steel et al. , Phys. Rev.A 58, 4824 (1998). [7] R. J. Lewis-Swan, M.K. Olsen, and K.V. Kheruntsyan,
[4] J.F. Corney and M.K. Olsen, Phys. Rev. A 91, 023824 Phys. Rev.A 94, 033814 (2016).
(2015).
[5] A. A. Norrie, R. J. Ballagh, C. W. Gardiner, Phys. Rev.