Notebook author: Nathan Shammah (nathan.shammah at gmail.com)
We use the Permutational Invariant Quantum Solver (PIQS) library, imported in QuTiP as $\texttt{qutip.piqs}$ to study the driven-dissipative open quantum dynamics of multiple two-level-system (TLS), or spin, ensembles.
We consider a system of two TLS ensembles with populations $N_1$ and $N_2$ with identical frequency $\omega_{0}$ with collective pumping and collective emission at identical rates, $\gamma_\text{CE}=(1+\bar{n})\gamma_0$ and $\gamma_\text{CP}=\bar{n}\gamma_0$, respectively, with $\bar{n}=\frac{1}{e^{\hbar\omega_0/k_\mathrm{B}T}-1}$ and
\begin{eqnarray} \dot{\rho} &=& -i\lbrack \omega_{0}\left(J_z^{(1)}+J_z^{(2)}\right),\rho \rbrack +\frac{\gamma_\text {CE}}{2}\mathcal{L}_{J_{-}^{(1)}+ J_{-}^{(2)}}[\rho] +\frac{\gamma_\text {CP}}{2}\mathcal{L}_{J_{+}^{(1)}+J_{+}^{(2)}}[\rho] \end{eqnarray}Ref. [2] has shown that for $N_1<N_2$, if the system is initialized in the state $|{\psi_0}\rangle=|{\downarrow\cdots\downarrow}\rangle_1\otimes|{\uparrow\cdots\uparrow}\rangle_2$, the system relaxes to a steady state for which the first subsystem is excited, i.e. $\langle J_z^{(1)}(\infty)\rangle>0$ and for some parameters $\frac{\langle J_z^{(1)}(\infty)\rangle}{(N_1/2)}\rightarrow 0.5$, also in the limit of zero temperature, $T\rightarrow 0$.
Notice that $\mathcal{L}_{J_{-}^{(1)}+ J_{-}^{(2)}}[\rho]\neq \mathcal{L}_{J_{-}^{(1)}}[\rho]+\mathcal{L}_{ J_{-}^{(2)}}[\rho]$, which is a case treated in Ref. [3] two obtain syncronized ensembles of atoms.
Here we explore what happens when to the master equation of Eq. (1) one adds also collective and local terms relative to single ensembles,
\begin{eqnarray} \dot{\rho} &=& -i\lbrack \omega_{0}\left(J_z^{(1)}+J_z^{(2)}\right),\rho \rbrack +\frac{\gamma_\text{CE}}{2}\mathcal{L}_{J_{-}^{(1)}+ J_{-}^{(2)}}[\rho] +\frac{\gamma_\text{CP}}{2}\mathcal{L}_{J_{+}^{(1)}+J_{+}^{(2)}}[\rho]\\ && +\frac{\gamma_\text{CEi}}{2}\mathcal{L}_{J_{-}^{(1)}}[\rho] +\frac{\gamma_\text{CEi}}{2}\mathcal{L}_{J_{-}^{(2)}}[\rho] +\sum_{n}^{N_1}\frac{\gamma_\text{E}}{2}\mathcal{L}_{J_{-,n}^{(1)}}[\rho]+\frac{\gamma_\text{D}}{2}\mathcal{L}_{J_{z,n}^{(1)}}[\rho]+\sum_{n}^{N_2}\frac{\gamma_\text{E}}{2}\mathcal{L}_{J_{-,n}^{(2)}}[\rho]+\frac{\gamma_\text{D}}{2}\mathcal{L}_{J_{z,n}^{(2)}}[\rho] \end{eqnarray}where $\gamma_\text {CEi}$ is the rate of superradiant decay for the individual ensembles of TLSs, $\gamma_\text{E}$ and $\gamma_\text{D}$ are the rates of local emission and dephasing.
Firstly, we will show how the collective dynamics of Eq. (1) can be investigated in a simple way using QuTiP's [4] $\texttt{jmat}$ function, which defines collective spins for maximally symmetric states in a Hilbert space of dimension $N_i+1$.
Secondly, we will exploit the permutational invariance of the local processes in Eq. (2) to investigate the exact dynamics using the Dicke basis, $\rho = \sum_{j,m,m'}p_{jmm'}|j,m\rangle\langle j,m'|$ [1], where $p_{jmm'}$ is a probability density. We will do so numerically using the PIQS library [1].
In the following we might use in plots thefollowing equivalent notation $\gamma_\text {CE}=\gamma_\Downarrow$ (gCE), $\gamma_\text {CP}=\gamma_\Uparrow$ (gCP), $\gamma_\text {E}=\gamma_\downarrow$ (gE), $\gamma_\text {P}=\gamma_\uparrow$ (gP), and $\gamma_\text {D}=\gamma_\phi$ (gD).
In [1]:
from qutip import *
from qutip.piqs import *
import matplotlib.pyplot as plt
from scipy import constants
In [2]:
# Number of TLSs in the two ensembles
N1 = 1
N2 = 4
N = N1 + N2
# TLSs bare frequency
w0 = 1
# Bose-Einstein distribution determines the occupation number
giga = 10**(6)
frequency_hertz = w0*10*giga
temperature_kelvin = 10**(2)
x = (frequency_hertz / temperature_kelvin) * (constants.hbar / constants.Boltzmann)
n0 = 1/(np.exp(x)-1)
print("n0 =",n0)
# set collective pumping and collective emission rates (coupled ensembles)
g0 = 1
gCE = g0 * (1 + n0)
gCP = g0 * n0
print("gCE =", gCE)
print("gCP =", gCP)
# define identity operators and norms in the tensor space
dim1_mat = N1 + 1
dim2_mat = N2 + 1
id1_mat = qeye(dim1_mat)
id2_mat = qeye(dim2_mat)
norm2 = id2_mat.tr()
norm1 = id1_mat.tr()
# build collective spin operators for N1 and N2
jx1_mat = jmat(N1/2,"x")
jx2_mat = jmat(N2/2,"x")
jy1_mat = jmat(N1/2,"y")
jy2_mat = jmat(N2/2,"y")
jz1_mat = jmat(N1/2,"z")
jz2_mat = jmat(N2/2,"z")
jm1_mat = jmat(N1/2,"-")
jm2_mat = jmat(N2/2,"-")
# place collective spin operators in tensor space (N1 + N2)
jz1_tot = tensor(jz1_mat, id2_mat)
jz2_tot = tensor(id1_mat, jz2_mat)
jx12_mat = tensor(jx1_mat, id2_mat) + tensor(id1_mat, jx2_mat)
jy12_mat = tensor(jy1_mat, id2_mat) + tensor(id1_mat, jy2_mat)
jz12_mat = tensor(jz1_mat, id2_mat) + tensor(id1_mat, jz2_mat)
jm12_mat = tensor(jm1_mat, id2_mat) + tensor(id1_mat, jm2_mat)
jp12_mat = jm12_mat.dag()
# define Hamiltonian
h1_mat = w0 * jz1_mat
h2_mat = w0 * jz2_mat
htot = tensor(h1_mat, id2_mat) + tensor(id1_mat, h2_mat)
# build Liouvillian using QuTiP
collapse_operators = [np.sqrt(gCE)*jm12_mat, np.sqrt(gCP)*jp12_mat]
L_collective = liouvillian(htot, collapse_operators)
In [3]:
#Check the algebra of the spin operators in the tensor space
print(jp12_mat*jm12_mat - jm12_mat*jp12_mat == 2*jz12_mat)
print(jx12_mat*jy12_mat - jy12_mat*jx12_mat == 1j*jz12_mat)
In [4]:
# set superradiant delay time for the excited ensemble (N2)
td0 = np.log(N2)/(N2*gCE)
tmax = 30 * td0
nt = 1001
t = np.linspace(0, tmax, nt)
#set initial tensor state for spins (Use QuTiP's jmat() basis)
excited1 = np.zeros(jz1_mat.shape)
excited2 = np.zeros(jz2_mat.shape)
ground1 = np.zeros(jz1_mat.shape)
ground2 = np.zeros(jz2_mat.shape)
excited1[0,0] = 1
excited2[0,0] = 1
ground1[-1,-1] = 1
ground2[-1,-1] = 1
excited1 = Qobj(excited1)
excited2 = Qobj(excited2)
ground1 = Qobj(ground1)
ground2 = Qobj(ground2)
sdp = tensor(excited1, excited2)
sdap = tensor(ground1, excited2)
ground12 = tensor(ground1, ground2)
rho0 = sdap
In [5]:
#solve using qutip (using QuTiP's jmat() basis)
result = mesolve(L_collective, rho0, t, [],
e_ops = [jz12_mat, jz1_tot, jz2_tot],
options = Options(store_states=True))
rhot = result.states
jzt = result.expect[0]
jz1t = result.expect[1]
jz2t = result.expect[2]
In [6]:
# plot jz1t, jz2t, jz12t
j2max = (0.5 * N + 1) * (0.5 * N)
jmax = 0.5 * N
j1max = 0.5 * N1
j2max = 0.5 * N2
label_size = 20
plt.rc('text', usetex = True)
plt.rc('xtick', labelsize = label_size)
plt.rc('ytick', labelsize = label_size)
fig_size = (12, 6)
lw = 2
fig1 = plt.figure(figsize = fig_size)
plt.plot(t/td0, jzt/jmax, '-', label = r"$\langle J_{z,\mathrm{ tot}}\rangle$", linewidth = 2*lw)
plt.plot(t/td0, jz1t/j1max, '--', label = r"$\langle J_{z,1}\rangle$", linewidth = lw)
plt.plot(t/td0, jz2t/j2max, '-.', label = r"$\langle J_{z,2}\rangle$", linewidth = lw)
plt.xlabel(r'$t/t_\text{D}$', fontsize = label_size)
plt.ylabel(r'$\langle J_z(t)\rangle$', fontsize = label_size)
plt.xticks([0, (tmax/2)/td0, tmax/td0])
plt.legend(fontsize = label_size)
plt.show()
plt.close()
In [7]:
# check partial traces
print(jz12_mat.ptrace(0)/norm2 == jz1_mat)
print(jz12_mat.ptrace(1)/norm1 == jz2_mat)
rho1pt = rho0.ptrace(0)
rho2pt = rho0.ptrace(1)
In [8]:
# Number of TLSs in the two ensembles
N1 = 5
N2 = 15
N = N1 + N2
# local-collective simulations with this system size take approx 5 minutes on a MacBook Pro for time integration
# TLSs bare frequency
w0 = 1
# Bose-Einstein distribution determines the occupation number
# low temperature limit
frequency_hertz = 10**(13)
temperature_kelvin = 10**(1)
x = (frequency_hertz / temperature_kelvin) * (constants.hbar / constants.Boltzmann)
n0 = 1/(np.exp(x) -1)
print("occupation number, n0 = ",n0)
# set collective pumping and collective emission rates (coupled ensembles)
g0 = 1
gCE = g0 * (1 + n0)
gCP = g0 * n0
# Local rates
gE = 1
gD = 1
# Collective rates of the single ensembles
gCEi = 1
# Algebra in the Dicke basis
[jx1_dicke, jy1_dicke, jz1_dicke] = jspin(N1)
jp1_dicke = jspin(N1,"+")
jm1_dicke = jp1_dicke.dag()
[jx2_dicke, jy2_dicke, jz2_dicke] = jspin(N2)
jp2_dicke = jspin(N2,"+")
jm2_dicke = jp2_dicke.dag()
# Bulding the tensor space for N1 + N2
dim1_dicke = num_dicke_states(N1)
dim2_dicke = num_dicke_states(N2)
id1_dicke = qeye(dim1_dicke)
id2_dicke = qeye(dim2_dicke)
norm2_dicke = id2_dicke.tr()
norm1_dicke = id1_dicke.tr()
# Place operators of a single ensemble (N1 or N2) in the tensor space
jz1_dicke_tot = tensor(jz1_dicke, id2_dicke)
jz2_dicke_tot = tensor(id1_dicke, jz2_dicke)
# Place operators of two ensemble (N1 + N2) in the tensor space
jx12_dicke = tensor(jx1_dicke, id2_dicke) + tensor(id1_dicke, jx2_dicke)
jy12_dicke = tensor(jy1_dicke, id2_dicke) + tensor(id1_dicke, jy2_dicke)
jz12_dicke = tensor(jz1_dicke, id2_dicke) + tensor(id1_dicke, jz2_dicke)
jm12_dicke = tensor(jm1_dicke, id2_dicke) + tensor(id1_dicke, jm2_dicke)
jp12_dicke = jm12_dicke.dag()
h1_dicke = w0 * jz1_dicke
h2_dicke = w0 * jz2_dicke
htot = tensor(h1_dicke, id2_dicke) + tensor(id1_dicke, h2_dicke)
# Build the collective Liovillian (Hamiltonian + collective Lindbladian)
L_collective_dicke = liouvillian(htot,[np.sqrt(gCE)*jm12_dicke, np.sqrt(gCP)*jp12_dicke])
In [9]:
# check algebra relations in tensor space
print(jp12_dicke * jm12_dicke - jm12_dicke * jp12_dicke == 2*jz12_dicke)
print(jx12_dicke * jy12_dicke - jy12_dicke * jx12_dicke == 1j*jz12_dicke)
In [10]:
## Define Piqs objects
# case 1: only collective coupled processes (already defined above)
system1 = Dicke(N = N1)
system2 = Dicke(N = N2)
# case 2: collective coupled processes + dephasing
system1gD = Dicke(N = N1)
system2gD = Dicke(N = N2)
system1gD.dephasing = gD
system2gD.dephasing = gD
# case 3: collective coupled processes + local emission
system1gE = Dicke(N = N1)
system2gE = Dicke(N = N2)
system1gE.emission = gE
system2gE.emission = gE
# case 4: collective coupled processes + collective emission of single ensembles
system1gCEi = Dicke(N = N1)
system2gCEi = Dicke(N = N2)
system1gCEi.collective_emission = gCEi
system2gCEi.collective_emission = gCEi
# Define identity operators in tensor space
id_tls1 = to_super(qeye(dim1_dicke))
id_tls2 = to_super(qeye(dim2_dicke))
###Build the Lindbladians
## case 1
L1_local_dicke = system1.liouvillian()
L2_local_dicke = system2.liouvillian()
print("case 1")
# Build local Lindbladians in tensor space
L_local_dicke = super_tensor(L1_local_dicke, id_tls2) + super_tensor(id_tls1, L2_local_dicke)
# Total local-collective Liouvillian in tensor space
L_dicke_tot = L_collective_dicke + L_local_dicke
## case 2
L1gD_local_dicke = system1gD.liouvillian()
L2gD_local_dicke = system2gD.liouvillian()
print("case 2")
# Build local Lindbladians in tensor space
LgD_local_dicke = super_tensor(L1gD_local_dicke, id_tls2) + super_tensor(id_tls1, L2gD_local_dicke)
# Total local-collective Liouvillian in tensor space
LgD_dicke_tot = L_collective_dicke + LgD_local_dicke
## case 3
L1gE_local_dicke = system1gE.liouvillian()
L2gE_local_dicke = system2gE.liouvillian()
print("case 3")
# Build local Lindbladians in tensor space
LgE_local_dicke = super_tensor(L1gE_local_dicke, id_tls2) + super_tensor(id_tls1, L2gE_local_dicke)
# Total local-collective Liouvillian in tensor space
LgE_dicke_tot = L_collective_dicke + LgE_local_dicke
## case 4
L1gCEi_local_dicke = system1gCEi.liouvillian()
L2gCEi_local_dicke = system2gCEi.liouvillian()
# Build local Lindbladians in tensor space
LgCEi_local_dicke = super_tensor(L1gCEi_local_dicke, id_tls2) + super_tensor(id_tls1, L2gCEi_local_dicke)
# Total local-collective Liouvillian in tensor space
LgCEi_dicke_tot = L_collective_dicke + LgCEi_local_dicke
print("case 4")
In [11]:
## Initial conditions
# set superradiant delay time for the excited ensemble (N2)
td0 = np.log(N2)/(N2*gCE)
tmax = 30 * td0
nt = 1001
t = np.linspace(0, tmax, nt)
# set initial tensor state for spins (Use QuTiP's jmat() basis)
excited1_dicke = excited(N1)
excited2_dicke = excited(N2)
ground1_dicke = ground(N1)
ground2_dicke = ground(N2)
sdp_dicke = tensor(excited1_dicke, excited2_dicke)
sdap_dicke = tensor(ground1_dicke, excited2_dicke)
ground12_dicke = tensor(ground1_dicke, ground2_dicke)
rho0_dicke = sdap_dicke
In [12]:
## Solve using qutip (using the Dicke basis)
# case 1
result_0 = mesolve(L_dicke_tot, rho0_dicke, t, [],
e_ops = [jz12_dicke, jz1_dicke_tot, jz2_dicke_tot],
options = Options(store_states=True))
rhot_0 = result_0.states
jzt_0 = result_0.expect[0]
jz1t_0 = result_0.expect[1]
jz2t_0 = result_0.expect[2]
print("case 1")
# case 2
result_gD = mesolve(LgD_dicke_tot, rho0_dicke, t, [],
e_ops = [jz12_dicke, jz1_dicke_tot, jz2_dicke_tot],
options = Options(store_states=True))
rhot_gD = result_gD.states
jzt_gD = result_gD.expect[0]
jz1t_gD = result_gD.expect[1]
jz2t_gD = result_gD.expect[2]
print("case 2")
# case 3
result_gE = mesolve(LgE_dicke_tot, rho0_dicke, t, [],
e_ops = [jz12_dicke, jz1_dicke_tot, jz2_dicke_tot],
options = Options(store_states=True))
rhot_gE = result_gE.states
jzt_gE = result_gE.expect[0]
jz1t_gE = result_gE.expect[1]
jz2t_gE = result_gE.expect[2]
print("case 3")
# case 4
result_gCEi = mesolve(LgCEi_dicke_tot, rho0_dicke, t, [],
e_ops = [jz12_dicke, jz1_dicke_tot, jz2_dicke_tot],
options = Options(store_states=True))
rhot_gCEi = result_gCEi.states
jzt_gCEi = result_gCEi.expect[0]
jz1t_gCEi = result_gCEi.expect[1]
jz2t_gCEi = result_gCEi.expect[2]
print("case 4")
In [13]:
## Plots jz1t, jz2t, jz12t in the Dicke basis for different parameter values
#spin normalization constants
j2_max = (0.5 * N + 1) * (0.5 * N)
jmax = 0.5 * N
j1max = 0.5 * N1
j2max = 0.5 * N2
#plot graphics properties
plt.rc('text', usetex = True)
label_size = 20
fig_size = (14, 7)
lw = 2
lw1 = 1*lw
lw2 = 1*lw
lw3 = 1*lw
fig1 = plt.figure(figsize=(7,4))
plt.rc('xtick', labelsize = label_size)
plt.rc('ytick', labelsize = label_size)
plt.plot(t/td0, jz1t_0/j1max, '-k', label = r"$\gamma_\Downarrow$ Only", linewidth = lw)
plt.plot(t/td0, jz2t_0/j2max, '-r', linewidth = lw)
plt.plot(t/td0, jz1t_gE/j1max, '--k', label = r"$\gamma_\downarrow=\gamma_\Downarrow$", linewidth = lw2)
plt.plot(t/td0, jz2t_gE/j2max, '--r', linewidth = lw2)
plt.rcParams['text.latex.preamble']=[r"\usepackage{xcolor}"]
plt.xlabel(r'$t/t_\text{D}$', fontsize = label_size)
#make double label y-axis - STARTS
left = -5.5
center = 0
yshift = -0.4
#label Jz1
plt.text(left, center+yshift,r'$\langle J_{z}^{(1)}(t)\rangle$,',
horizontalalignment = 'right',
verticalalignment='center',
color = "k", rotation='vertical',fontsize = label_size)
#label Jz2
plt.text(left, center-yshift, r' $\langle J_{z}^{(2)}(t)\rangle$',
horizontalalignment='right', verticalalignment='center',
color = "r", rotation='vertical',fontsize = label_size)
#make double label y-axis - ENDS
plt.xticks([0, (tmax/2)/td0, tmax/td0])
plt.yticks([-1, -0.5, 0, 0.5, 1])
plt.legend(fontsize = label_size)
plt.title(r'Two ensembles', fontsize = label_size)
plt.show()
plt.close()
## Second Figure
plt.rc('xtick', labelsize = label_size)
plt.rc('ytick', labelsize = label_size)
fig2 = plt.figure(figsize=(7,4))
plt.plot(t/td0, jz1t_gCEi/j1max, '-.k', label = r"$\gamma_{\Downarrow,i}=\gamma_\Downarrow$",
linewidth = lw3)
plt.plot(t/td0, jz2t_gCEi/j2max, '-.r', linewidth = lw3)
plt.plot(t/td0, jz1t_gD/j1max, ':k', label = r"$\gamma_\phi=\gamma_\Downarrow$", linewidth = lw1)
plt.plot(t/td0, jz2t_gD/j2max, ':r',linewidth = lw1)
plt.rcParams['text.latex.preamble']=[r"\usepackage{xcolor}"]
plt.xlabel(r'$t/t_\text{D}$', fontsize = label_size)
#make double label y-axis - STARTS
#label Jz1
plt.text(left, center+yshift,r'$\langle J_{z}^{(1)}(t)\rangle$,',
horizontalalignment = 'right',
verticalalignment='center',
color = "k", rotation='vertical',fontsize = label_size)
#label Jz2
plt.text(left, center-yshift, r' $\langle J_{z}^{(2)}(t)\rangle$',
horizontalalignment='right', verticalalignment='center',
color = "r", rotation='vertical',fontsize = label_size)
#make double label y-axis - ENDS
plt.xticks([0, (tmax/2)/td0, tmax/td0])
plt.yticks([-1, -0.5, 0, 0.5, 1])
plt.legend(fontsize = label_size)
plt.title(r'Two ensembles', fontsize = label_size)
plt.show()
plt.close()
We have studied the dissipative dynamics of two ensembles of TLSs, exploring the possibility of the systems to undergo local dephasing, collective emission of the single ensembles, collective emission of the two ensembles coupled to the same reservoir and local de-excitations. We have found that in the general casse spin exchange between antisymmetrically prepared ensemble is transient [1].
[1] N. Shammah, S. Ahmed, N. Lambert, S. De Liberato, and F. Nori, https://arxiv.org/abs/1805.05129 Open quantum systems with local and collective incoherent processes: Efficient numerical simulation using permutational invariance
[2] Y. Hama, W.J. Munro, and K. Nemoto, Phys. Rev. Lett. 120, 060403 (2018) Relaxation to Negative Temperatures in Double Domain Systems
[3] M. Xu, D.A. Tieri, E.C. Fine, J.K. Thompson, and M.J. Holland, Phys. Rev. Lett. 113, 154101 (2014) Synchronization of Two Ensembles of Atoms
[4] B.A. Chase and J.M Geremia, Phys. Rev. A 78, 052101 (2010) Collective processes of an ensemble of spin-1/2 particles
[5] J.R. Johansson, P.D. Nation, and F. Nori, Comp. Phys. Comm. 183, 1760 (2012) http://qutip.org
In [14]:
qutip.about()
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