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posit.c
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posit.c
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/* Copyright (c) 2014, Giuseppe Argentieri <giuseppe.argentieri@ts.infn.it>
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* 1. Redistributions of source code must retain the above copyright notice,
* this list of conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright notice,
* this list of conditions and the following disclaimer in the documentation
* and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
* LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
* CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
* SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
* INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
* CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
* POSSIBILITY OF SUCH DAMAGE.
*
*/
/*
*
*
* Filename: posit.c
*
* Description: Determine the positivity violation of the Redfield dynamics
* on a given initial pure state
*
* Version: 1.0
* Created: 20/07/2014 09:59:35
* Revision: none
* License: BSD
*
* Author: Giuseppe Argentieri (ga), giuseppe.argentieri@ts.infn.it
* Organization: Università degli Studi di Trieste
*
*
*/
#include "initial.h"
#include "funcs.h"
#include <stdlib.h>
#include <gsl/gsl_complex.h>
#include <gsl/gsl_complex_math.h>
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_blas.h>
/*
* FUNCTION
* Name: main
* Description:
*
*/
int main ( int argc, char *argv[] )
{
double o_eff = gsl_hypot(OMEGA, D);
/* sigma^z eigenstates, Bloch vectorial form */
double z_minus[] = { 1, 0, -OMEGA/o_eff, -D/o_eff };
double z_plus[] = { 1, 0, OMEGA/o_eff, D/o_eff };
gsl_vector* z_m = gsl_vector_calloc(4);
gsl_vector* z_p = gsl_vector_calloc(4);
int i;
for ( i = 0; i < 4; i++ ) {
gsl_vector_set( z_m, i, z_minus[i] );
gsl_vector_set( z_p, i, z_plus[i] );
}
/* parameters */
double r, theta;
return EXIT_SUCCESS;
} /* ---------- end of function main ---------- */
/*
* FUNCTION
* Name: state_matrix
* Description: return the density patrix of the pure state |psi><psi|
*
*/
gsl_matrix_complex* state_matrix ( double r, double theta )
{
gsl_matrix_complex* psi = gsl_matrix_complex_calloc (2, 2);
gsl_matrix_complex_set ( psi, 1, 1, gsl_complex_rect(r*r, 0) );
gsl_matrix_complex_set ( psi, 1, 2, gsl_complex_rect(r*sqrt(1-r*r)*cos(theta),
r*sqrt(1-r*r)*sin(theta)) );
gsl_matrix_complex_set ( psi, 2, 1, gsl_complex_rect(r*sqrt(1-r*r)*cos(theta),
-r*sqrt(1-r*r)*sin(theta)) );
gsl_matrix_complex_set ( psi, 2, 2, gsl_complex_rect(1-r*r, 0) );
return psi;
} /* ----- end of function state_matrix ----- */
/*
* FUNCTION
* Name: rotate
* Description: return the density matrix in the rotated frame of reference
* of an angle OMEGA*t
*
*/
gsl_matrix_complex* rotate ( gsl_matrix_complex* m, double t )
{
/* copy m into a working matrix M */
gsl_matrix_complex* M = gsl_matrix_complex_calloc ( 2, 2);
gsl_matrix_complex_memcpy( M, m );
/* rotation matrix */
gsl_matrix_complex* Rot = gsl_matrix_complex_calloc( 2, 2 );
gsl_matrix_complex_set ( Rot, 1, 1, gsl_complex_rect( cos(OMEGA*t/2), 0) );
gsl_matrix_complex_set ( Rot, 1, 2, gsl_complex_rect( -sin(OMEGA*t/2), 0));
gsl_matrix_complex_set ( Rot, 2, 1, gsl_complex_rect( sin(OMEGA*t/2), 0) );
gsl_matrix_complex_set ( Rot, 2, 2, gsl_complex_rect( cos(OMEGA*t/2), 0) );
/* inverse of Rot:
*
* first perform the LU decomposition of Rot: */
gsl_permutation* perm = gsl_permutation_calloc(2);
int signum;
gsl_matrix_complex* Rot_lu = gsl_matrix_complex_calloc( 2, 2 );
gsl_matrix_complex_memcpy ( Rot_lu, Rot );
gsl_linalg_complex_LU_decomp ( Rot_lu, perm, &signum);
/* then calculate Rot_inv: */
gsl_matrix_complex* Rot_inv = gsl_matrix_complex_calloc( 2, 2 );
gsl_linalg_complex_LU_invert ( Rot_lu, perm, Rot_inv );
/* Finally calculate Rot * m * Rot_inv */
gsl_blas_zgemm ( CblasNoTrans, CblasNoTrans, gsl_complex_rect(1.,0),
Rot, M, gsl_complex_rect(0,0), M );
gsl_blas_zgemm ( CblasNoTrans, CblasNoTrans, gsl_complex_rect(1.,0),
M, Rot_inv, gsl_complex_rect(0,0), M );
return (M);
} /* ----- end of function rotate ----- */
/*
* FUNCTION
* Name: D_psi
* Description: the matrix D[|psi><psi|]
*
*/
gsl_matrix_complex* D_psi ( gsl_matrix_complex* dpsi )
{
gsl_matrix_complex* dp = gsl_matrix_complex_calloc(2, 2);
return (dp);
} /* ----- end of function D_psi ----- */