File size: 2,493 Bytes
9755170 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | """Symplectic/Bogoliubov formulas with a FIXED reference vacuum.
Expected pair count equals per-mode photon number for the homogeneous +/-k pair.
Total photons in that pair are twice this number. Not a dispersive QED solver.
"""
from __future__ import annotations
import math
import numpy as np
from scipy.special import eval_chebyu
def bogoliubov(M:np.ndarray,omega:float=1.)->tuple[complex,complex]:
M=np.asarray(M,dtype=float)
if M.shape!=(2,2) or not np.all(np.isfinite(M)) or not math.isfinite(omega) or omega<=0:
raise ValueError('2x2 real M and fixed positive reference frequency required.')
a,b,c,d=M.ravel()
return complex((a+d)/2,(c/omega-omega*b)/2),complex((a-d)/2,(c/omega+omega*b)/2)
def _log_sinh(x:float)->float:
if x<=0:raise ValueError('Positive argument required.')
return math.log(math.sinh(x)) if x<20 else x-math.log(2)+math.log1p(-math.exp(-2*x))
def log_photon_number(M:np.ndarray,N:int=1,omega:float=1.)->float:
"""Logarithmic expected pair count; avoids overflow for large hyperbolic N.
Returns -infinity at exact zero. Input M must be real determinant one;
near a band edge, floating-point classification is not an interval certificate.
"""
if not isinstance(N,int) or N<1:raise ValueError('Positive integer N required.')
_,beta=bogoliubov(M,omega);b=abs(beta)
if b==0:return -math.inf
D=float(np.trace(M))/2
if abs(D)>1:
L=math.acosh(abs(D));logU=_log_sinh(N*L)-_log_sinh(L)
elif abs(D)==1:
logU=math.log(N)
else:
value=abs(float(eval_chebyu(N-1,D)))
if value==0:return -math.inf
logU=math.log(value)
return 2*(math.log(b)+logU)
def photon_number(M:np.ndarray,N:int=1,omega:float=1.)->float:
value=log_photon_number(M,N,omega)
return math.exp(value) if value<=math.log(np.finfo(float).max) else math.inf
def squeezing(M:np.ndarray,N:int=1,omega:float=1.)->float:
half=log_photon_number(M,N,omega)/2
if half==-math.inf:return 0.
if half<0:return math.asinh(math.exp(half))
return half+math.log1p(math.sqrt(1+math.exp(-2*half)))
def hyperbolic_prefactor(M:np.ndarray,omega:float=1.)->float:
D=abs(float(np.trace(M)))/2
if D<=1:raise ValueError('Strictly hyperbolic matrix required.')
_,beta=bogoliubov(M,omega)
return abs(beta)**2/(D*D-1)
def finite_cycle_rate_error(K0:float,N:int)->float:
if K0<1 or N<1:raise ValueError('K0 >= 1 and N >= 1 required.')
return math.log(N*(K0+1))/N
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