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# Comment on “Quantum entangled dark solitons formed by ultracold atoms in optical lattices” 

Jacek Dziarmaga, 1 Piotr Deuar, 2 and Krzysztof Sacha 1 

$^{1}$Instytut Fizyki im. Mariana Smoluchowskiego and Mark Kac Complex Systems Research Center, Uniwersytet Jagiello´ nski, ul. Reymonta 4, PL-30-059 Krak´ ow, Poland 

$^{2}$Institute of Physics, Polish Academy of Sciences, Al. Lotnik´ ow 32/46, 02-668 Warsaw, Poland 

The recent Letter [1] describes full quantum simula- tions of a dark soliton in a Bose-Einstein condensate in a regime where the system cannot be described by the per- turbative approach. The authors argue, based on the fill- ing in of the two-point correlator g $^{(2)}$, that a photograph of a condensate would reveal a smooth atomic density without any localized dark soliton. This is in contrast to the perturbative regime where a photograph would show a dark soliton with a random position [2]. While we ad- mire the quantum simulations and other results in [1], we think that their conclusion about the outcome of a single experiment is not justified by this property of g $^{(2)}$. 

As the question of whether and when dark solitons fill in in single realisations has been a recurring and some- times confusing one in the field, it is important to pin down what conclusions can or cannot be made. With this aim, we provide the following counterexample in the non-perturbative regime (see also [3]). 

Let φ$_{q}$ ( x ) ∝ tanh[( x − q ) /ξ ] be a standard conden- sate wave function with a dark soliton at q . With ˆ a$_{q}$ = ∫ dx φ ∗ $_{q}$( x ) ˆ Ψ( x ), a state ( ˆ a † q ) N | 0 〉 is a condensate with a soliton at q , and let the N -particle state be a superposition of condensates with different q 

where ψ$_{0}$ ( q ) defines the superposition. 

After measurements of n atomic positions x$_{1}$, ..., x$_{n}$ the state (1) collapses to a conditional state 

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where ψ$_{n}$ ( q ) = φ$_{q}$ ( x$_{n}$ ) ..φ$_{q}$ ( x$_{1}$ ) ψ$_{0}$ ( q ). The ( n + 1)-st mea- surement will find a particle at x$_{n}$$_{+1}$ with a probability 

This is equivalent to simultaneous measurement of all x$_{i}$ . 

Using the methods of [2], we simulated measurement of all N = 5000 particles on a lattice of 31 sites, where x, q ∈ {− 15 , 15 } , assuming a soliton width ξ = 1 . 5, and a delocalized (uniform) superposition ψ$_{0}$ ( q ) ∝ 1 that is non-perturbatively wider than the soliton width. The in- set in the figure shows the ensemble average particle den- sity p$_{1}$ ( x ) and the main figure shows a generic histogram of particle positions x$_{1}$, .., x$_{N}$ measured in a single real- ization. Each single realization of the experiment finds a soliton localized at some definite but random q . 

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FIG. 1: (Color online) Histogram of measured atom positions in a single experiment from (1). Inset: single particle density p$_{1}$ ( x ) (dashed black line) and g$_{2}$ ( x ) (solid red line), normal- ized so that ∫ g$_{2}$ ( x ) dx = 1. 

What about the two-point correlator g$_{2}$ ( x ) = 〈 ψ$_{0}$ | ˆ Ψ $^{†}$(0) ˆ Ψ $^{†}$( x ) ˆ Ψ( x ) ˆ Ψ(0) | ψ$_{0}$ 〉 that is analyzed in [1]? This turns out nearly uniform (see the inset). Hence, the “filled in g$_{2}$ ” → “filled in soliton” line of reasoning is clearly incorrect. Of course, the soliton simulated in [1] may still be greying, but the point here is that one cannot answer such a question by analyzing g$_{2}$ ( x ). 

Our example demonstrates that, in some cases, a low order correlator like g$_{2}$ ( x ) is insufficient to draw conclu- sions on the outcome of a single experiment, and that the soliton in a BEC is such a case. Here, g$_{2}$ is equal to our p$_{2}$ ( x$_{2}$ ) after the first particle was measured at x$_{1}$ = 0. A measurement of only one particle is not enough to collapse the soliton position. If we want to infer the soliton position from a histogram of particle po- sitions, then the number of measured particles must be large enough to provide a histogram with a well-resolved soliton notch. This is a fundamental requirement for an accurate Bayesian inference of the soliton position. 

Support within Polish Government scientific funds (2008-2011 KS, 2009-2012 JD) as a research project is acknowledged. 

[1] R. V. Mishmash and L. D. Carr, Phys. Rev. Lett. 103 , 140403 (2009); see also: R. V. Mishmash et al. , Phys. Rev. A 80 , 053612 (2009). 

[2] J. Dziarmaga and K. Sacha, J. Phys. B 39 , 57 (2006); Phys. Rev. A 66 , 043620 (2002); J. Dziarmaga et al. , J. Phys. B 36 , 1217 (2003); Phys. Rev. A 66 , 043615 (2002). 

[3] J. Dziarmaga, Phys. Rev. A 70 , 063616 (2004). 

