用schwinger态矢求转动矩阵d m,m′ j （β）

Naftali R. Smith, Pierre Le Doussal, Satya N. Majumdar +1 more

We study N N spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index \beta β . In the fermion model \beta β controls the strength of the interaction, \beta=2 β = 2 corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions N_D N D in a domain D D of macroscopic size in the bulk of the Fermi gas. We predict that for general \beta β the variance of N_D N D grows as A_{\beta} \log N + B_{\beta} A β log N + B β for N \gg 1 N ≫ 1 and we obtain a formula for A_\beta A β and B_\beta B β . This is based on an explicit calculation for \beta\in\left\{ 1,2,4\right\} β ∈ { 1 , 2 , 4 } and on a conjecture that we formulate for general \beta β . This conjecture further allows us to obtain a universal formula for the higher cumulants of N_D N D . Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter K = 2/\beta K =

洪义 范

Harish Parthasarathy

Let [ a , a = 1. Let https://www.w3.org/1998/Math/MathML" display="block"> ρ ( t ) = Z ( t ) exp − β 1 ( t ) a ∗ a + β 2 ( t ) a 2 + β ¯ 2 ( t ) a ∗ 2 − β 3 ( t ) a − β ¯ 3 ( t ) a ∗ where β 1 ( t ) is real, Z ( t ), β 1 ( t ), β 2 ( t ), β 3 ( t ) are constrained by the conditions https://www.w3.org/1998/Math/MathML" display="block"> T r ( ρ ( t ) ) = and ρ ( t ) satisfies the GKSL equation https://www.w3.org/1998/Math/MathML" display="block"> ρ ′ ( t ) = − i [ H , ρ ( t ) ] − ( 1 / 2 ) L L ∗ ρ ( t ) + ρ ( t ) L L ∗ − 2 L ∗ ρ ( t ) L with https://www.w3.org/1998/Math/MathML" display="block"> H = ω a ∗ a + λ a + λ ¯ a ∗ L = α a + β a ∗ , α , β , λ ∈ ℂ

万影

文方 解, Bao Cheng-guang

J. D. Franson

Two classical, coincident light pulses propagating through two different dispersive media will become broadened and their degree of coincidence will be reduced. The width of the coincidence curve can be shown to be proportional to ( β 1 2 + β 2 2 ) x 2 , where β1, and β2 are the dispersion coefficients in the two media and x is the distance traveled. When entangled photon pairs from parametric down conversion are considered instead, it is found that the width of the coincidence curve is then proportional to (β1 + β2)2x2, which gives an interference term of 2β1β1. If β1 = β2, the dispersion experienced by one photon can exactly cancel the dispersion experienced by the other in such a way that their coincidence is maintained.