Quantum-friction.md

@@ -10,4 +10,103 @@ The quantum friction potential is given by:
 ```
 where $`\rho_0=\frac{k_F^3}{6\pi^2}`$ is reference density. By construction, this potential removes any irrotational currents. Thus it provides a convenient method of removing phonon excitations from the system.
 
- 
\ No newline at end of file
+# Usage
+The quantum friction is controlled via `input` file via tags:
+```bash
+# --------------- QUANTUM FRICTION ------------------
+# qfalpha                 0.0     # alpha parameter for quantum friction term
+# qfstart                 0.0     # start time for evolving with quantum friction [eF]
+# qfstop                  0.0     # stop time for evolving with quantum friction [eF]
+# qfswitch                0.0     # the friction will be activated and deactivated gradually over this period of time [eF] 
+```
+*Notes*:
+* `qfalpha ~ 1` looks to be reasonable choice,
+* too large value of `qfalpha` may lead to instability of the code, typically it manifests via growing of the energy during the evolution.  
+
+# Example
+Consider application a time-dependent potential:
+```math
+V_{\textrm{ext}}(x,y,t)=s(t,t_{\textrm{start}}, t_{\textrm{stop}})\exp\left[-\frac{x^2}{2\sigma_x^2}-\frac{y^2}{2\sigma_y^2}\right]
+```
+ to the unitary Fermi gas, being initially in the uniform state, where $`s(t,t_{\textrm{start}}, t_{\textrm{stop}})`$ is (smooth) step function that acquires 1 in time interval $`[t_{\textrm{start}}, t_{\textrm{stop}}]`$, otherwise is 0. Implemamntion of this potential is following (`problem-definition.h`):
+```c
+__device__ __host__ inline double switch_function(double t, double T, double alpha)
+{
+    return 0.5*( 1.0+tanh( alpha*tan( M_PI_2*( 2.0*t/T-1.0 ) ) ) );
+}
+
+__device__ __host__ inline double smooth_step(double t, double step_start, double step_stop, double T, double alpha)
+{
+    if(t<=step_start || t>=step_stop) return 0.0; 
+    if(t>=step_start+T && t<=step_stop-T) return 1.0;
+    if(t>step_start && t<step_start+T) return switch_function(t-step_start, T, alpha);
+    else return 1.0-switch_function(t-step_stop+T, T, alpha);
+}
+
+__device__ double v_ext(int ix, int iy, int iz, int it, int spin, double *params, size_t extra_data_size, void *extra_data)
+{
+    double x = DX*(ix-NX/2);
+    double y = DY*(iy-NY/2);     // for 1d code iy will be always 0
+    double z = DZ*(iz-NZ/2);     // for 1d and 2d codes iz will be always 0
+    double t = dc_t0 + dc_dt*it; // time
+
+    // ADD HERE FORMULA FOR V_ext(r)
+    double AMPLITUDE = params[0];
+    double AX=params[1];
+    double AY=params[2];
+    double AZ=params[3];
+    
+    double T_START=params[4];
+    double T_STOP=params[5];
+    double T_SWITCH=params[6];
+    
+    double gauss = AMPLITUDE * 
+                   smooth_step(t, T_START, T_STOP, T_SWITCH, 1.0) *
+                   exp(-1.*AX*x*x -1.*AY*y*y -1.*AZ*z*z );
+                   
+    double V_ext = gauss;
+
+    return V_ext; 
+}
+
+extern "C" void process_params(double *params, double *kF, double *mu, size_t extra_data_size, void *extra_data)
+{
+    // PROCESS INPUT FILE PARAMETERS 
+    double eF = 0.5*kF[0]*kF[0];
+    params[0]*=eF;
+    
+    // sigmas
+    int i;
+    for(i=1; i<=3; i++) if(params[i]>1.0e-9) params[i] = 1.0/(2.0*params[i]*params[i]);
+    
+    // times
+    for(i=4; i<=6; i++) params[i]/=eF;
+}
+```
+
+When executing this code for lattice (`predefines.h`):
+```c
+#define NX 24
+#define NY 24
+#define NZ 8
+#define FUNCTIONAL ASLDA
+```
+with input file parmaters:
+```bash
+params0     0.25 # gauss amplitude [eF]
+params1     2.0 # width in x direction
+params2     1.9 # width in y direction
+params4     0.0 # turn-on
+params5     50.0 # turn-off
+params6     10.0 # switch-time
+# ...
+qfalpha                 1.0 # or 0.0 (=deactivated)
+qfstart                 0.0
+qfstop                  250.0
+qfswitch                1.0
+# ...
+```
+the resulting evolution of the total energy looks like this:
+![qf](uploads/0a927351fb183a211fb554689fdd30cb/qf.png)
+
+It is observed, that time-dependent potential excites the system. Selected potential does not introduce angular momentum to the system, and only phonons are induced. When evolving this state with the quantum friction (`qfalpha=1.0`) we see that the system returns after some time to its ground state. The origin for energy fluctuations for `qfalpha=0.0` and for $`t\varepsilon_F>50`$ see [here](Regularization schemes of the pairing field).
\ No newline at end of file