Enhancement of many-body quantum interference in chaotic bosonic systems supplementary material

Enhancement of many-body quantum interference in chaotic bosonic systems supplementary material

Peter Schlagheck, ¹ Denis Ullmo, ² Juan Diego Urbina, ³ Klaus Richter, ³ and Steven Tomsovic ^{3, 4}

1

CESAM research unit, University of Liege, 4000 Li` ege, Belgium

2

LPTMS, CNRS, Univ. Paris-Sud, Universit´ e Paris-Saclay, 91405 Orsay, France

3

Institut f¨ ur Theoretische Physik, Universit¨ at Regensburg, D-93040 Regensburg, Germany

4

Department of physics and astronomy, Washington State University, Pullman, WA USA

In this supplementary material we provide some of the details leading to Eq. (6). First, note that the prefactor can be written as

A γ (q f , q i ) = e ^iπα/4 (2π ~ ˜ ) ^L/2

det − ∂ ² R γ

∂q ^l _f ∂q ^l _i

(q f , q i )

!

1/2

(1) with α the index of inertia of −∂q i /∂ p _i [1]. Next, start- ing from Eq. (5), assume that the contributions of non- identical pairs of orbits cancel out because of the time average, and that only short chords, i.e. points such that q ^′ _i ≃ q ^′′ _i and q ^′ _f ≃ q ^′′ _f contribute significantly. In such a case, the variables Q _f,i ≡ ( q ^′′ _f,i + q ^′ _f,i )/2 and δ q _f,i ≡ ( q ^′′ _f,i − q ^′ _f,i ) can be introduced. The key is to expand in the “small” variables δ q _f,i , which means more specifically keeping only the zero’th order term for the smooth function A γ but expanding the action R γ to first order. With γ ^′ = γ ^′′ = γ, and using the property p ^(γ) _f = ∂R γ /∂q f ; p ^(γ) _i = −∂R γ /∂ q _i (with p ^(γ) _i and p ^(γ) _f the initial and final “momenta” of the trajectory γ), gives

h f ˆ i(t) diag = Z

dQ i dQ f dδq _f dδq _i hq ^′′ _i |ˆ ρ 0 |q ^′ _i i hq ^′ _f | f ˆ |q ^′′ _f i.

× X

γ

|A γ (Q f Q _i )| ² exp i

˜

~

p ^(γ) _f δq _f − p ^(γ) _i δq _i

= Z

dQ i dQ f

X

γ

|A γ (Q f Q _i )| ² [ρ 0 ] W (Q i , p ^(γ) _i )

×[f ] W ( Q _f , p ^(γ) _f ) , (2)

with ˆ ρ 0 ≡ |Φ 0 ihΦ 0 | the initial density. Note that

−∂ ² R γ /∂Q f ∂Q i = ∂P i /∂ Q _{f γ} . Therefore, the determi- nant in Eq. (1) is just the Jacobian of the transformation from the final “position” Q f to the initial “momentum”

P _i . Thus, P

γ

R dQ f |A γ | ² 7→ (2π ˜ ~ ) ^L R

dP i leading to Eq. (6).

[1] R. G. Littlejohn, J. Stat. Phys. 68 , 7 (1992).