#!/usr/bin/env python # coding: utf-8 # ## Superradiant light emission (Dicke superradiance) # # Notebook author: Nathan Shammah (nathan.shammah at gmail.com) # # We consider a system of $N$ two-level systems (TLSs) with identical frequency $\omega_{0}$, which can emit collectively at a rate $\gamma_\text{CE}$ [1], and suffer from dephasing and local losses at rates $\gamma_\text{D}$ and $\gamma_\text{E}$, respectively. The dynamics can be written as # \begin{eqnarray} # \dot{\rho} &=&-i\lbrack \omega_{0}J_z,\rho \rbrack # +\frac{\gamma_\text {CE}}{2}\mathcal{L}_{J_{-}}[\rho] # +\sum_{n=1}^{N}\frac{\gamma_\text{D}}{2}\mathcal{L}_{J_{z,n}}[\rho] # +\frac{\gamma_\text{E}}{2}\mathcal{L}_{J_{-,n}}[\rho] # \end{eqnarray} # # When $\gamma_\text{E}=\gamma_\text{D}=0$ this dynamics is the classical superradiant master equation. # In this limit, a system initially prepared in the fully-excited state undergoes superradiant light emission whose peak intensity scales proportionally to $N^2$. # # This dynamics has been studied in Refs. [2-4] and implemented in various quantum optical platforms, including in the solid state [5-10]. A discussion of the difference between superradiant phase transition (typical of the Dicke model) and Dicke superradiance is present in Refs. [3] and [11]. # # Below, using PIQS [4] and QuTiP [12], we investigate the time evolution of the collective dynamics for an ensemble initialized in different initial quantum states. # # Note that in the table above and in $\texttt{qutip.piqs}$ functions, the Lindbladian $\mathcal{L}[\rho]$ is written with a factor 1/2 with respect to $\mathcal{L}_{A}[\rho]$ reported in the LaTeX math equations, in order to have the Lindbladian and full Liouvillian matrix consistently defined by the rates $\gamma_\alpha$. # In[1]: from time import clock import matplotlib.pyplot as plt import scipy from qutip import * from qutip.piqs import * from scipy.sparse import load_npz, save_npz # In[2]: N = 20 ntls = N nds = num_dicke_states(N) print("The number of Dicke states is", nds) [jx, jy, jz] = jspin(N) jp = jspin(N, "+") jm = jspin(N, "-") system = Dicke(N = N) # In[3]: a = 1/np.sqrt(2) b = 1/np.sqrt(2) css_symmetric = css(N, a, b) css_antisymmetric = css(N, a,-b) excited_ = dicke(N, N/2,N/2) superradiant = dicke(N,N/2,0) subradiant = dicke(N,j_min(N),-j_min(N)) ground_ = dicke(N,N/2,-N/2) ghz_ = ghz(N) # In[4]: # here we set the initial coefficients gE = 0 # local emission gD = 0 # local dephasing gP = 0 # local pumping gCE = 1 # collective emission gCD = 0 # collective dephasing gCP = 0 # collective pumping w0 = 1 # bare frequency wi = 0 # coherent drive frequency # spin hamiltonian h0 = w0 * jz hint = wi * jx h = h0 #+ hint #set initial conditions for spins by initializing the system and building the Liouvillian matrix system = Dicke(hamiltonian = h, N = N, emission = gE, pumping = gP, dephasing = gD, collective_emission = gCE, collective_pumping = gCP, collective_dephasing = gCD) clock_t0 = clock() lind = system.lindbladian() liouv = system.liouvillian() clock_tf = clock() dt_clock = clock_tf - clock_t0 print("Time (in seconds) to generate the Liouvillian for N = 20 TLSs:", dt_clock) # In[5]: ## Solution of the dynamics for different initial conditions # parameters for the time integration of the dynamics nt = 1001 td0 = np.log(N)/(N*gCE) # delay time is used as a reference tmax = 10 * td0 t = np.linspace(0, tmax, nt) # initial states rho01 = excited_ rho02 = superradiant rho03 = css_symmetric rho04 = css_antisymmetric rho05 = subradiant rho06 = ghz_ #Excited clock_t0 = clock() result1 = mesolve(liouv, rho01, t, [], e_ops = [jz, jp*jm, jz**2], options = Options(store_states=True)) rhot1 = result1.states jz_t1 = result1.expect[0] jpjm_t1 = result1.expect[1] jz2_t1 = result1.expect[2] clock_tf = clock() dt_clock = clock_tf - clock_t0 print("Elapsed time (in seconds) for this run: ", dt_clock) #Superradiant clock_t0 = clock() result2 = mesolve(liouv, rho02, t, [], e_ops = [jz, jp*jm, jz**2], options = Options(store_states=True)) rhot2 = result2.states jz_t2 = result2.expect[0] jpjm_t2 = result2.expect[1] jz2_t2 = result2.expect[2] clock_tf = clock() dt_clock = clock_tf - clock_t0 print("Elapsed time (in seconds) for this run: ", dt_clock) #CSS Symmetric clock_t0 = clock() result3 = mesolve(liouv, rho03, t, [], e_ops = [jz, jp*jm, jz**2], options = Options(store_states=True)) rhot3 = result3.states jz_t3 = result3.expect[0] jpjm_t3 = result3.expect[1] jz2_t3 = result3.expect[2] clock_tf = clock() dt_clock = clock_tf - clock_t0 print("Elapsed time (in seconds) for this run: ", dt_clock) #CSS Antisymmetric clock_t0 = clock() result4 = mesolve(liouv, rho04, t, [], e_ops = [jz, jp*jm, jz**2], options = Options(store_states=True)) rhot4 = result4.states jz_t4 = result4.expect[0] jpjm_t4 = result4.expect[1] jz2_t4 = result4.expect[2] clock_tf = clock() dt_clock = clock_tf - clock_t0 print("Elapsed time (in seconds) for this run: ", dt_clock) #Subradiant clock_t0 = clock() result5 = mesolve(liouv, rho05, t, [], e_ops = [jz, jp*jm, jz**2], options = Options(store_states=True)) rhot5 = result5.states jz_t5 = result5.expect[0] jpjm_t5 = result5.expect[1] jz2_t5 = result5.expect[2] clock_tf = clock() dt_clock = clock_tf - clock_t0 print("Elapsed time (in seconds) for this run: ", dt_clock) #GHZ clock_t0 = clock() result6 = mesolve(liouv, rho06, t, [], e_ops = [jz, jp*jm, jz**2], options = Options(store_states=True)) rhot6 = result6.states jz_t6 = result6.expect[0] jpjm_t6 = result6.expect[1] jz2_t6 = result6.expect[2] clock_tf = clock() dt_clock = clock_tf - clock_t0 print("Elapsed time (in seconds) for this run: ", dt_clock) # #### Visualization # In[6]: jmax = (0.5 * N) j2max = (0.5 * N + 1) * (0.5 * N) label_size = 20 label_size2 = 20 label_size3 = 20 label_size4 = 15 plt.rc('text', usetex = True) fig1 = plt.figure() plt.plot(t/td0, jz_t1/jmax, '-.', label = r"$|\frac{N}{2},\frac{N}{2}\rangle$") plt.plot(t/td0, jz_t2/jmax, '-', label = r"$|\frac{N}{2},0\rangle$") plt.plot(t/td0, jz_t3/jmax, '-', label = r"$|+\rangle_\mathrm{CSS}$") plt.plot(t/td0, jz_t4/jmax, '-', label = r"$|-\rangle_\mathrm{CSS}$") plt.plot(t/td0,jz_t5/jmax, '-', label = r"$|0,0\rangle$") plt.plot(t/td0,jz_t6/jmax, '--', label = r"$|\mathrm{GHZ}\rangle$") #plt.ylim([-1,1]) plt.xlabel(r'$t/t_\mathrm{D}$', fontsize = label_size3) plt.ylabel(r'$\langle J_{z} \rangle(t)$', fontsize = label_size3) plt.xticks([0,(tmax/2)/td0,tmax/td0]) plt.yticks([-1,0,1]) plt.legend(fontsize = label_size4, ncol = 2) # plot plt.title(r'Total Inversion', fontsize = label_size3) plt.show() plt.close() fig2 = plt.figure() plt.plot(t/td0, jpjm_t1/j2max, '-.', label = r"$|\frac{N}{2},\frac{N}{2}\rangle$") plt.plot(t/td0, jpjm_t2/j2max, '-', label = r"$|\frac{N}{2},0\rangle$") plt.plot(t/td0, jpjm_t3/j2max, '-', label = r"$|+\rangle_\mathrm{CSS}$", linewidth = 2) plt.plot(t/td0, jpjm_t4/j2max, '-', label = r"$|-\rangle_\mathrm{CSS}$") plt.plot(t/td0,jpjm_t5/j2max, '-', label = r"$|0,0\rangle$") plt.plot(t/td0,jpjm_t6/j2max, '--', label = r"$|\mathrm{GHZ}\rangle$") #plt.ylim([0,1]) plt.xticks([0,(tmax/2)/td0,tmax/td0]) plt.yticks([0,0.5,1]) plt.xlabel(r'$t/t_\mathrm{D}$', fontsize = label_size3) plt.ylabel(r'$\langle J_{+}J_{-}\rangle(t)$', fontsize = label_size3) plt.legend(fontsize = label_size4, ncol = 2) # plot plt.title(r'Light Emission', fontsize = label_size3) plt.show() plt.close() fig3 = plt.figure() djz1 = (jz2_t1- jz_t1**2)/jmax**2 djz2 = (jz2_t2- jz_t2**2)/jmax**2 djz3 = (jz2_t3- jz_t3**2)/jmax**2 djz4 = (jz2_t4- jz_t4**2)/jmax**2 djz5 = (jz2_t5- jz_t5**2)/jmax**2 djz6 = (jz2_t6- jz_t6**2)/jmax**2 plt.plot(t/td0, djz1, '-.', label = r"$|\frac{N}{2},\frac{N}{2}\rangle$") plt.plot(t/td0, djz2, '-', label = r"$|\frac{N}{2},0\rangle$") plt.plot(t/td0, djz3, '-', label = r"$|+\rangle_\mathrm{CSS}$") plt.plot(t/td0, djz4, '-', label = r"$|-\rangle_\mathrm{CSS}$") plt.plot(t/td0, djz5, '-', label = r"$|0,0\rangle$") plt.plot(t/td0, djz6, '--', label = r"$|\mathrm{GHZ}\rangle$") #plt.ylim([-1,1]) plt.xticks([0,(tmax/2)/td0,tmax/td0]) plt.yticks([0,0.5,1]) plt.xlabel(r'$t/t_\mathrm{D}$', fontsize = label_size3) plt.ylabel(r'$\Delta J_{z}^2(t)$', fontsize = label_size3) plt.rc('xtick', labelsize=label_size) plt.rc('ytick', labelsize=label_size) plt.legend(fontsize = label_size4, ncol = 2) plt.title(r'Second-moment Collective Correlations', fontsize = label_size3) plt.show() plt.close() # We have found very different time evolutions for different quantum states that experience the same dynamics, leading to superradiant light emission, subradiance, delayed superradiant decay (superfluorescence), and macroscopic quantum correlations. # # #### References # # [1] R. 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