""" 1 TLS - Decay in infinite waveguide =================================== This is a basic example of a single two-level system (TLS) decaying into an infinite waveguide. All the examples are in units of the TLS total decay rate, gamma. Hence, in general, gamma=1. It covers two cases: 1. Symmetrical coupling into the waveguide 2. Chiral coupling, where the TLS is only coupled to the right channel of the waveguide. """ #%% # Imports #-------- import QwaveMPS as qmps import matplotlib.pyplot as plt import numpy as np import time as t #%% # Symmetrical Solution #--------------------- # #^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ #Choose the simulation parameters #^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # #Setup of the bin size, coupling and input parameters: # # * Size of each system bin (d_sys), this is the TLS Hilbert subspace, and the total system bin (d_sys_total) containing all the emitters. # For a single TLS, d_sys1=2 and d_sys_total=np.array([d_sys1]). # # * Size of the time bins (d_t_total). This contains the field Hilbert subspace at each time step. In this case we allow one photon per time step and per right (d_t_r) # and left (d_t_l) channels. Hence, the subspace is d_t_total=np.array([d_t_l,d_t_r]) # # * Choice of coupling. Here, it is first calculated with symmetrical coupling, \gamma_l,gamma_r=qmps.coupling('symmetrical',gamma=1) # and then with chiral coupling, gamma_l,gamma_r=qmps.coupling('chiral_r',gamma=1) # # * Input parameters (input_params). Define the data parameters that will be used in the calculation: # # * Time step (delta_t) # * Maximum time (tmax) # * d_sys_total (as defined above) # * d_t_total (as defined above) # * Maximum bond dimension (bond). bond >=d_t_total(number of excitations). # Starting with the TLS excited and field in vacuum, 1 excitation enough with bond=4 #Choose the bins: d_t_l=2 #Time right channel bin dimension d_t_r=2 #Time left channel bin dimension d_t_total=np.array([d_t_l,d_t_r]) #Total field bin dimensions d_sys1=2 # tls bin dimension d_sys_total=np.array([d_sys1]) #total system bin (in this case only 1 tls) #Choose the coupling: gamma_l,gamma_r=qmps.coupling('symmetrical',gamma=1) # same as gamma_l, gamma_r = (0.5,0.5) #Define input parameters (dataclass) input_params = qmps.parameters.InputParams( delta_t=0.05, # Time step of the simulation tmax = 8, # Maximum simulation time d_sys_total=d_sys_total, d_t_total=d_t_total, gamma_l=gamma_l, gamma_r = gamma_r, bond_max=4 # Maximum bond dimension, simulation parameter that adjusts truncation of entanglement information ) #Make a tlist for plots: tmax=input_params.tmax delta_t=input_params.delta_t tlist=np.arange(0,tmax+delta_t,delta_t) #%% #^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ #Choose the initial state and Hamiltonian #^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ #* Choice the system initial state. Here, initially excited. # #* Choice of the waveguide initial state. Here, starting in vacuum, # and considering that there is vacuum before the interaction until tmax. # #* Selection of the corresponding Hamiltonian. """ Choose the initial state""" sys_initial_state=qmps.states.tls_excited() #TLS initially excited #waveguide initially vacuum for as long as calculation (tmax) wg_initial_state = qmps.states.vacuum(tmax,input_params) #wg_initial_state = None # Another equivalent way to set the initial vacuum state #To track computational time start_time=t.time() """Choose the Hamiltonian""" hm=qmps.hamiltonian_1tls(input_params) # Create the Hamiltonian for a single TLS #%% #^^^^^^^^^^^^^^^^^^^^^^^^^^^^ #Calculate the time evolution #^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # #Time evolution calculation in the Markovian regime: """Calculate time evolution of the system""" bins = qmps.t_evol_mar(hm,sys_initial_state,wg_initial_state,input_params) #%% #^^^^^^^^^^^^^^^^^^^^^^^^^^^ #Choose Relevant observables #^^^^^^^^^^^^^^^^^^^^^^^^^^^ # #* Get the TLS population with the tls_pop_op = qmps.tls_pop() #* Get bosonic fluxes. This can be doe in two different ways: # # * Using the boson operator: # # b_pop_l = qmps.b_dag_l(input_params) @ qmps.b_l(input_params) # # b_pop_r = qmps.b_dag_r(input_params) @ qmps.b_r(input_params) # # * Using population operators directly: # # b_pop_l = qmps.b_pop_l(input_params) # # b_pop_r = qmps.b_pop_r(input_params) """Choose Observables""" # Calculate the two level system population tls_pop_op = qmps.tls_pop() # Calculate the fluxes out of the TLS to the left and right b_pop_l = qmps.b_dag_l(input_params) @ qmps.b_l(input_params) b_pop_r = qmps.b_dag_r(input_params) @ qmps.b_r(input_params) photon_pop_ops = [b_pop_l, b_pop_r] #%% #^^^^^^^^^^^^^^^^^^^^^^^^^^ #Calculate the observables #^^^^^^^^^^^^^^^^^^^^^^^^^^ # #Get time dependent expectation values by acting on the relevant states (system/field) with #your operators. # #Here, we calculate population dynamics, including the TLS population, #photon fluxes, the integrated fluxes over time, and total quanta to #check quanta conservation. """Calculate population dynamics""" # Can calculate a single observable to get a time ordered ndarray of expectation values # Use the system_states to calculate observables having to do with the emitter system tls_pop = qmps.single_time_expectation(bins.system_states, tls_pop_op) # Can also calculate a list of observables on the same states # Use output_field_states to calculate observables of the outgoing field photon_fluxes = qmps.single_time_expectation(bins.output_field_states, photon_pop_ops) # Net photons propagating in each direction is the cumulatively integrated fluxes over time net_flux_l = np.cumsum(photon_fluxes[0]) * delta_t net_flux_r = np.cumsum(photon_fluxes[1]) * delta_t # Add the integrated flux leaving the system with the TLS population for total quanta total_quanta = tls_pop + net_flux_l + net_flux_r print("--- %s seconds ---" %(t.time() - start_time)) #%% #^^^^^^^^^^^^^^^^ #Plot the results #^^^^^^^^^^^^^^^^ # #Example plot containing, #* Integrated photon flux traveling to the right #* Integrated photon flux traveling to the left #* TLS population #* Conservation check (for one excitation it should be 1) """Plotting the results""" plt.plot(tlist,np.real(net_flux_r),linewidth = 3,color = 'orange',linestyle='-',label=r'$N^{\rm out}_{R}$') # Photons propagating to the right channel plt.plot(tlist,np.real(net_flux_l),linewidth = 3,color = 'brown',linestyle='--',label=r'$N^{\rm out}_{L}$') # Photons propagating to the left channel plt.plot(tlist,np.real(tls_pop),linewidth = 3, color = 'k',linestyle='-',label=r'$n_{TLS}$') # TLS population plt.plot(tlist,np.real(total_quanta),linewidth = 3,color = 'g',linestyle='-',label='Total') # Conservation check (for one excitation it should be 1) plt.legend() plt.xlabel(r'Time, $\gamma t$') plt.ylabel('Populations') plt.grid(True, linestyle='--', alpha=0.6) plt.ylim([0.,1.05]) plt.xlim([0.,tmax]) plt.show() #%% # Right Chiral Solution #---------------------- # #Similar example but now for a chiral TLS #with an updated coupling to be coupled only to the right channel #%% #^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ #Update the simulation coupling #^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ gamma_l,gamma_r=qmps.coupling('chiral_r',gamma=1) input_params.gamma_l=gamma_l input_params.gamma_r=gamma_r #%% #^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ #Update Hamiltonian with new coupling #^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ hm=qmps.hamiltonian_1tls(input_params) #%% #^^^^^^^^^^^^^^^^^^^^^^^^^^^^ #Calculate the time evolution #^^^^^^^^^^^^^^^^^^^^^^^^^^^^ """Calculate time evolution of the system""" bins = qmps.t_evol_mar(hm,sys_initial_state,wg_initial_state,input_params) #%% #^^^^^^^^^^^^^^^^^^^^^^^^^^ #Calculate the observables #^^^^^^^^^^^^^^^^^^^^^^^^^^ """Calculate population dynamics""" tls_pop_ch = qmps.single_time_expectation(bins.system_states, tls_pop_op) photon_fluxes_ch = qmps.single_time_expectation(bins.output_field_states, photon_pop_ops) net_fluxes = np.cumsum(photon_fluxes_ch, axis=1) * delta_t total_quanta_ch = tls_pop_ch + np.sum(net_fluxes, axis=0) #%% #^^^^^^^^^^^^^^^^ #Plot the results #^^^^^^^^^^^^^^^^ """Plotting the results""" plt.plot(tlist,np.real(net_fluxes[1]),linewidth = 3,color = 'orange',linestyle='-',label=r'$N^{\rm out}_{R}$') # Photons propagating to the right channel plt.plot(tlist,np.real(net_fluxes[0]),linewidth = 3,color = 'brown',linestyle='--',label=r'$N^{\rm out}_{L}$') # Photons propagating to the left channel plt.plot(tlist,np.real(tls_pop_ch),linewidth = 3, color = 'k',linestyle='-',label=r'$n_{TLS}$') # TLS population plt.plot(tlist,np.real(total_quanta_ch),linewidth = 3,color = 'g',linestyle='-',label='Total') # Conservation check (for one excitation it should be 1) plt.legend() plt.xlabel(r'Time, $\gamma t$') plt.ylabel('Populations') plt.grid(True, linestyle='--', alpha=0.6) plt.ylim([0.,1.05]) plt.xlim([0.,tmax]) plt.tight_layout() plt.show()