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Diffstat (limited to 'figs/anim_laser/laser.jl')
| -rw-r--r-- | figs/anim_laser/laser.jl | 391 |
1 files changed, 391 insertions, 0 deletions
diff --git a/figs/anim_laser/laser.jl b/figs/anim_laser/laser.jl new file mode 100644 index 0000000..78cabff --- /dev/null +++ b/figs/anim_laser/laser.jl @@ -0,0 +1,391 @@ +using FFTW +using Printf +using SpecialFunctions +using LinearAlgebra + +# numerical values +hbar=6.58e-16 # eV.s +m=9.11e-31 # kg +V_=6.6 # eV +I=1e17 # W/m^2 +EF=3.3 # eV +omega_=1.5498*4 # eV + +# dimensionless quantities +V=1 +k=sqrt(EF/V_) +epsilon=sqrt(hbar^3*I/(8*m*V_*omega_^2)) +omega=omega_/V_ + +#epsilon=8*epsilon +#omega=4*omega + +nu=60 +ell=20 + +# for Crank-Nicolson +xmin_cn=-100 +xmax_cn=100 +nx_cn=4000 +tmax=10 +# mu=dt/dx^2 +mu=0.1 +nt=floor(Int,tmax/(mu*((xmax_cn-xmin_cn)/nx_cn)^2)) +# only save up to memt time steps +memt=10000 + + +# asymptotic value of \varphi_t(x) +function phi_asym(x,T,gkz,k,epsilon,omega,V,nu,ell) + # compute residues + res=residue(x,T,gkz,k,epsilon,omega,V,nu) + times=compute_times(omega,nu,ell) + N=(2*nu+1)*(2*ell+1) + fun=zeros(Complex{Float64},N) + for m in 1:N + for n in -nu:nu + fun[m]+=-1im*res[n+nu+1]*exp(-1im*(k*k+n*omega)*times[m]) + end + end + return fun +end +# asymptotic value of \partial\varphi_t(x) +function dphi_asym(x,T,gkz,k,epsilon,omega,V,nu,ell) + # compute residues + res=dresidue(x,T,gkz,k,epsilon,omega,V,nu) + times=compute_times(omega,nu,ell) + N=(2*nu+1)*(2*ell+1) + fun=zeros(Complex{Float64},N) + for m in 1:N + for n in -nu:nu + fun[m]+=-1im*res[n+nu+1]*exp(-1im*(k*k+n*omega)*times[m]) + end + end + return fun +end + +# return times +function compute_times(omega,nu,ell) + N=(2*nu+1)*(2*ell+1) + times=zeros(N) + for j in 1:N + times[j]=pi/(omega*(nu+0.5))*(j-1) + end + return times +end + +# \kappa_m^{(\sigma)} +function K(m,sigma,V,k,epsilon,omega) + return pow(V+2*epsilon*epsilon-k*k-sigma-m*omega,0.5,pi/2) +end +# mathfrak q_k +function Q(k,V) + return sqrt(V-k*k) +end + +# T_0 +function T0(k,V) + return 2im*k/(1im*k-sqrt(V-k*k)) +end + +# compute g +function g(kappa,epsilon,omega,N) + B=zeros(Complex{Float64},N) + # prepare vector of B's + for i in 1:N + t=2*pi/(omega*N)*(i-1) + B[i]=exp(1im*epsilon*epsilon/omega*sin(2*omega*t)-kappa*4*epsilon/omega*cos(omega*t)) + end + # take the fft + fft!(B) + # correct by a factor 1/N + for i in 1:N + B[i]=B[i]/N + end + return(B) +end + +# index of m-n +function mmn(n,nu) + return (n>=0 ? n+1 : n+4*nu+2) +end + +# compute residues at -ik^2-in\omega +function residue(x,T,gkz,k,epsilon,omega,V,nu) + # compute residue + out=zeros(Complex{Float64},2*nu+1) + if x>=0 + for n in -nu:nu + for m in -nu:nu + out[n+nu+1]+=1im*gkz[m+nu+1][mmn(n-m,nu)]*exp(-K(m,0,V,k,epsilon,omega)*x)*T[m+nu+1] + end + end + else + for n in -nu:nu + out[n+nu+1]=(n==0 ? 1im*(exp(1im*k*x)-exp(-1im*k*x)) : 0) + for m in -nu:nu + out[n+nu+1]+=1im*T[m+nu+1]*gkz[m+nu+1][mmn(n-m,nu)]*exp(-1im*x*pow(k*k+n*omega,0.5,-pi/2)) + end + end + end + + return out +end +# compute \partial residues at -ik^2-in\omega +function dresidue(x,T,gkz,k,epsilon,omega,V,nu) + # compute residue + out=zeros(Complex{Float64},2*nu+1) + if x>=0 + for n in -nu:nu + for m in -nu:nu + out[n+nu+1]+=-K(m,0,V,k,epsilon,omega)*1im*gkz[m+nu+1][mmn(n-m,nu)]*exp(-K(m,0,V,k,epsilon,omega)*x)*T[m+nu+1] + end + end + else + for n in -nu:nu + out[n+nu+1]=(n==0 ? -k*(exp(1im*k*x)+exp(-1im*k*x)) : 0) + for m in -nu:nu + out[n+nu+1]+=pow(k*k+n*omega,0.5,-pi/2)*T[m+nu+1]*gkz[m+nu+1][mmn(n-m,nu)]*exp(-1im*x*pow(k*k+n*omega,0.5,-pi/2)) + end + end + end + + return out +end + + +# matching condition for residue +function residue_matching(k,epsilon,omega,V,nu) + # preliminary: g^{(\kappa_m^{(0)})} + # g^{(\kappa_m^{(0)})}_n=gkz[m+nu+1][n+1] if n>=0 and gkz[m+nu+1][n+4*nu+2] if n<0 + gkz=Array{Array{Complex{Float64}},1}(undef,2*nu+1) + for m in -nu:nu + gkz[m+nu+1]=g(K(m,0,V,k,epsilon,omega),epsilon,omega,4*nu+1) + end + + # solve matching condition + G=zeros(Complex{Float64},2*nu+1,2*nu+1) + v=zeros(Complex{Float64},2*nu+1) + for n in -nu:nu + for m in -nu:nu + G[n+nu+1,m+nu+1]=((K(m,0,V,k,epsilon,omega)-1im*pow(k*k+n*omega,0.5,-pi/2))*gkz[m+nu+1][mmn(n-m,nu)]-epsilon*(gkz[m+nu+1][mmn(n-m+1,nu)]-gkz[m+nu+1][mmn(n-m-1,nu)])) + end + end + v[nu+1]=-2im*k + + return (G\v,gkz) +end + + +# fractional power with an arbitrary branch cut +function pow(x,a,cut) + if(angle(x)/cut<=1) + return(abs(x)^a*exp(1im*angle(x)*a)) + else + return(abs(x)^a*exp(1im*(angle(x)-sign(cut)*2*pi)*a)) + end +end + +# current +function J(phi,dphi,x,epsilon,omega,t) + if x>=0 + return(2*imag(conj(phi)*(dphi+2im*epsilon*sin(omega*t)*phi))) + else + return(2*imag(conj(phi)*dphi)) + end +end + +# compute \psi_t(x) using Crank-Nicolson +function psi_cn(V,omega,epsilon,k,xs,ts,memt) + nx=length(xs) + nt=length(ts) + + # length of output vector + len=min(nt,memt) + # how often to write to output + freq=ceil(Int,nt/memt) + + # initialize + psis=zeros(Complex{Float64},nx,len) + times=zeros(Float64,len) + + # previous time step + psi_prev=zeros(Complex{Float64},nx) + + # init + for i in 1:nx + times[1]=0 + psis[i,1]=(xs[i]<0 ? exp(1im*k*xs[i])+(T0(k,V)-1)*exp(-1im*k*xs[i]) : T0(k,V)*exp(-Q(k,V)*xs[i])) + psi_prev[i]=psis[i,1] + end + # the boundary condition is that psi at the boundary is constant (nothing needs to be done for this) + + # matrix structure of the Crank-Nicolson algorithm + # diagonals of M + M0=zeros(Complex{Float64},nx-2) + Mp=zeros(Complex{Float64},nx-3) + Mm=zeros(Complex{Float64},nx-3) + v=zeros(Complex{Float64},nx-2) + + for j in 1:nt-1 + # print progress + progress(j,nt-1,1000) + + # t_j + t=ts[j] + # t_{j+1} + tp=ts[j+1] + dt=tp-t + for mu in 2:nx-1 + # x_mu + x=xs[mu] + dx2=(xs[mu+1]-xs[mu])*(xs[mu]-xs[mu-1]) + # tridiagonal matrix + M0[mu-1]=1im/dt-1/dx2-(x<0 ? 0 : (V-2*epsilon*omega*cos(omega*tp)*x)/2) + if mu<nx-1 + Mp[mu-1]=1/(2*dx2) + end + if mu>2 + Mm[mu-2]=1/(2*dx2) + end + # right side + v[mu-1]=1im*psi_prev[mu]/dt-(psi_prev[mu+1]+psi_prev[mu-1]-2*psi_prev[mu])/(2*dx2)+(x<0 ? 0 : (V-2*epsilon*omega*cos(omega*t)*x)*psi_prev[mu]/2) + # correct for boundary conditions + # assumes the boundary condition is constant in time! + if mu==2 + v[mu-1]-=psi_prev[1]/(2*dx2) + end + if mu==nx-1 + v[mu-1]-=psi_prev[nx]/(2*dx2) + end + end + + M=Tridiagonal(Mm,M0,Mp) + + # copy to psi_prev + psi_prev[2:nx-1]=M\v + + # write to output + if j%freq==0 + psis[:,Int(j/freq)+1]=psi_prev + times[Int(j/freq)+1]=tp + end + end + + return (times,psis) +end + +# returns the times in b that are closest to those in a +# assumes the vectors are ordered +function nearest_times(a,b) + out=zeros(Int,length(a)) + pointer=1 + for i in 1:length(a) + if pointer==length(b) + out[i]=length(b) + end + for j in pointer:length(b)-1 + if b[j+1]>a[i] + out[i]=(abs(b[j+1]-a[i])<abs(b[j]-a[i]) ? j+1 : j) + pointer=out[i] + break + end + # all remaining points are beyond the range + if j==length(b)-1 + out[i]=length(b) + pointer=out[i] + end + end + end + return out +end + +# space derivative using Crank Nicolson +function deriv_cn(phi,i,xs) + if i==1 + return (phi[i+1]-phi[i])/(xs[i+1]-xs[i]) + end + if i==length(phi) + return (phi[i]-phi[i-1])/(xs[i]-xs[i-1]) + end + return (phi[i+1]-phi[i])/(xs[i+1]-xs[i])/2 + (phi[i]-phi[i-1])/(xs[i]-xs[i-1])/2 +end + +################################################ + + +# print progress +function progress(j,tot,freq) + if (j-1)%ceil(Int,tot/freq)==0 + if j>1 + @printf(stderr,"\r") + end + @printf(stderr,"%d/%d",j,tot) + end + if j==tot + @printf(stderr,"%d/%d\n",j,tot) + end +end + +# print animation data using Crank Nicolson and compare to Laplace +function print_anim_cn_cmp() + @printf(stderr,"epsilon=% .8e omega=% .8e k=% .8e V=% .8e z=% .8e T=% .8e gamma=% .8e\n",epsilon,omega,k,V,4*sqrt(nu)*epsilon/sqrt(omega),2*pi/(omega_/hbar)*1e15,sqrt((V-k^2)/(4*epsilon^2))) + xmin=-10 + xmax=10 + nx=200 + + # array of times for the asymptote + times=compute_times(omega,nu,ell) + + # positions at which to compute phi + xs=Array(0:nx_cn)*(xmax_cn-xmin_cn)/nx_cn.+xmin_cn + # times at which to compute phi + ts=Array(0:nt-1)*tmax/nt + + # compute matching condition + (T,gkz)=residue_matching(k,epsilon,omega,V,nu) + + # compute wave function + (t_psi,psi)=psi_cn(V,omega,epsilon,k,xs,ts,memt) + + # nearest times using Crank Nicolson + indices_t=nearest_times(times,t_psi) + indices_x=nearest_times(Array(0:nx-1)*(xmax-xmin)/nx.+xmin,xs) + + ps_cn=Array{Array{Complex{Float64},1}}(undef,nx) + ps_asym=Array{Array{Complex{Float64},1}}(undef,nx) + dps_cn=Array{Array{Complex{Float64},1}}(undef,nx) + dps_asym=Array{Array{Complex{Float64},1}}(undef,nx) + for i in 1:nx + progress(i,nx,1000) + x=xmin+(xmax-xmin)*(i-1)/nx + ps_cn[i]=Array{Complex{Float64}}(undef,length(times)) + dps_cn[i]=Array{Complex{Float64}}(undef,length(times)) + for j in 1:length(times) + # phase change in magnetic gauge + phase=(x<0 ? 1 : exp(-2im*epsilon*sin(omega*times[j])*x)) + ps_cn[i][j]=phase*psi[indices_x[i],indices_t[j]] + dps_cn[i][j]=phase*deriv_cn(psi[:,indices_t[j]],indices_x[i],xs)-(xs[indices_x[i]]<0 ? 0 : 2im*epsilon*sin(omega*times[j])*psi[indices_x[i],indices_t[j]]) + end + ps_asym[i]=phi_asym(x,T,gkz,k,epsilon,omega,V,nu,ell) + dps_asym[i]=dphi_asym(x,T,gkz,k,epsilon,omega,V,nu,ell) + end + + # print values at each time + for j in 1:length(times) + if times[j]>tmax + break + end + for i in 1:nx + x=(xmin+(xmax-xmin)*i/nx) + # remove point at 0 + if i!=Int(nx/2) + @printf("% .8e % .8e % .8e % .8e % .8e % 8e\n",times[j]*hbar/V_*1e15,x*sqrt(hbar^2*1.6e-19/(2*m*V_))*1e9,abs(ps_asym[i][j])^2,abs(ps_cn[i][j])^2,J(ps_asym[i][j],dps_asym[i][j],x,epsilon,omega,times[j]),J(ps_cn[i][j],dps_cn[i][j],x,epsilon,omega,times[j])) + end + end + print('\n') + end +end + + +print_anim_cn_cmp() |