A Central Limit Theorem in Many-Body Quantum Dynamics

Arous, Gérard Ben; Kirkpatrick, Kay; Schlein, Benjamin

doi:10.1007/s00220-013-1722-1

Cited by 56 publications

(98 citation statements)

References 40 publications

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“…In [29], Bogoliubov transformations play an important role to get a norm approximation for the dynamics of N -particle systems (adapting techniques developed in the timeindependent setting in [30]). A slightly different point of view is presented in [3,4], where fluctuations around mean field Hartree dynamics are described by time-dependent Bogoliubov transformations. Similarly, as explained above, also in [8,23,24], one can think that the effect of the Bogoliubov transformation consists of canceling certain terms from the generator of the fluctuation dynamics.…”

Section: Bogoliubov Transformations and The Gross-pitaevskii Regimementioning

confidence: 99%

“…However, it is important to stress the difference with respect to our work. In the mean field regime studied in [3,4,8,23,24,29], the effect of the correlations is of lower order; the Bogoliubov transformation is only needed to get second-order corrections. In the Gross-Pitaevskii regime that we are considering, on the other hand, because of the singularity of the interaction the correlations are a leading-order effect.…”

Section: Bogoliubov Transformations and The Gross-pitaevskii Regimementioning

confidence: 99%

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Quantitative Derivation of the Gross‐Pitaevskii Equation

Benedikter

Oliveira

Schlein

2014

Comm Pure Appl Math

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130

218

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Starting from first principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles N . The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one particle reduced density, the form of the initial data is preserved by the many-body evolution, up to a small error which vanishes as N −1/2 in the limit of large N .

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Section: Bogoliubov Transformations and The Gross-pitaevskii Regimementioning

confidence: 99%

Section: Bogoliubov Transformations and The Gross-pitaevskii Regimementioning

confidence: 99%

Quantitative Derivation of the Gross‐Pitaevskii Equation

Benedikter

Oliveira

Schlein

2014

Comm Pure Appl Math

Self Cite

130

218

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show abstract

“…In the work [14], it is proved that the fluctuations around the limiting dynamics given by the Hartree equation satisfy a central limit theorem. This approach builds on previous central limit theorems in a quantum setting [63,64,91,99,101,105,113].…”

Section: Previously Known Resultsmentioning

confidence: 99%

“…We note that, due to the presence of only one random parameter ω ∈ , the norm in (14) [41,42,92,107,108]. The results of Theorems 2 and 3 show that these Duhamel expansions converge to zero in a class of density-matrices in a low-regularity space with a random component.…”

Section: Remark 12mentioning

confidence: 93%

Randomization and the Gross–Pitaevskii Hierarchy

Sohinger

Staffilani

2015

Arch Rational Mech Anal

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We study the Gross-Pitaevskii hierarchy on the spatial domain T 3 . By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is α > 3 4 . It was shown in our previous joint work with Gressman (J Funct Anal 266(7): 2014) that the range α > 1 is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon (Commun Math Phys 279(1): [169][170][171][172][173][174][175][176][177][178][179][180][181][182][183][184][185] 2008), where the spacetime estimate is known to hold whenever α ≥ 1. The goal of our paper is to extend the range of α in this class of estimates in a probabilistic sense. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.

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“…In quantum setting, apart from the i.i.d. scenario [8][9][10][11], in recent years we have seen a plethora of CLT-type results mainly in the context of quantum many-body dynamics [12][13][14][15][16]. Yet, the main focus in these studies are the properties of quantum states and observables without counting the measurement effects.…”

Section: Introductionmentioning

confidence: 99%

On the central limit theorem for unsharp quantum random variables

Dimić

Dakić

2018

New J. Phys.

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We study the weak-convergence properties of random variables generated by unsharp quantum measurements. More precisely, for a sequence of random variables generated by repeated unsharp quantum measurements, we study the limit distribution of relative frequency. We provide a representation theorem for all separable states, showing that the distribution can be well approximated by a mixture of normal distributions. Furthermore, we investigate the convergence rates and show that the relative frequency can stabilize to some constant at best at the rate of order N 1 for all separable inputs. On the other hand, we provide an example of a strictly unsharp quantum measurement where the better rates are achieved by using entangled inputs. This means that in certain cases the noise generated by the measurement process can be suppressed by using entanglement. We deliver our result in the form of quantum information task where the player achieves the goal with certainty in the limiting case by using entangled inputs or fails with certainty by using separable inputs.

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A Central Limit Theorem in Many-Body Quantum Dynamics

Cited by 56 publications

References 40 publications

Quantitative Derivation of the Gross‐Pitaevskii Equation

Quantitative Derivation of the Gross‐Pitaevskii Equation

Randomization and the Gross–Pitaevskii Hierarchy

On the central limit theorem for unsharp quantum random variables

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