User Guide

The Permutational Invariant Quantum Solver (PIQS) is an open-source Python solver to study the exact Lindbladian dynamics of open quantum systems consisting of identical qubits. It is integrated in QuTiP and can be imported as as a model.

Using this library, the Liouvillian of an ensemble of \(N\) qubits, or two-level systems (TLSs), \(\mathcal{D}_{TLS}(\rho)\), can be built using only polynomial – instead of exponential – resources. This has many applications for the study of realistic quantum optics models of many TLSs and in general as a tool in cavity QED [1].

Consider a system evolving according to the equation

\[ \begin{align}\begin{aligned}\dot{\rho} = \mathcal{D}_\text{TLS}(\rho)=-\frac{i}{\hbar}\lbrack H,\rho \rbrack +\frac{\gamma_\text{CE}}{2}\mathcal{L}_{J_{-}}[\rho] +\frac{\gamma_\text{CD}}{2}\mathcal{L}_{J_{z}}[\rho] +\frac{\gamma_\text{CP}}{2}\mathcal{L}_{J_{+}}[\rho]\\+\sum_{n=1}^{N}\left( \frac{\gamma_\text{E}}{2}\mathcal{L}_{J_{-,n}}[\rho] +\frac{\gamma_\text{D}}{2}\mathcal{L}_{J_{z,n}}[\rho] +\frac{\gamma_\text{P}}{2}\mathcal{L}_{J_{+,n}}[\rho]\right)\end{aligned}\end{align} \]

where \(J_{\alpha,n}=\frac{1}{2}\sigma_{\alpha,n}\) are SU(2) Pauli spin operators, with \({\alpha=x,y,z}\) and \(J_{\pm,n}=\sigma_{\pm,n}\). The collective spin operators are \(J_{\alpha} = \sum_{n}J_{\alpha,n}\) . The Lindblad super-operators are \(\mathcal{L}_{A} = 2A\rho A^\dagger - A^\dagger A \rho - \rho A^\dagger A\).

The inclusion of local processes in the dynamics lead to using a Liouvillian space of dimension \(4^N\). By exploiting the permutational invariance of identical particles [2-8], the Liouvillian \(\mathcal{D}_\text{TLS}(\rho)\) can be built as a block-diagonal matrix in the basis of Dicke states \(|j, m \rangle\).

The system under study is defined by creating an object of the Dicke class, e.g. simply named system, whose first attribute is

• system.N, the number of TLSs of the system \(N\).

The rates for collective and local processes are simply defined as

• collective_emission defines \(\gamma_\text{CE}\), collective (superradiant) emission
• collective_dephasing defines \(\gamma_\text{CD}\), collective dephasing
• collective_pumping defines \(\gamma_\text{CP}\), collective pumping.
• emission defines \(\gamma_\text{E}\), incoherent emission (losses)
• dephasing defines \(\gamma_\text{D}\), local dephasing
• pumping defines \(\gamma_\text{P}\), incoherent pumping.

Then the system.lindbladian() creates the total TLS Linbladian superoperator matrix. Similarly, system.hamiltonian defines the TLS hamiltonian of the system \(H_\text{TLS}\).

The system’s Liouvillian can be built using system.liouvillian(). The properties of a Piqs object can be visualized by simply calling system. We give two basic examples on the use of PIQS. In the first example the incoherent emission of N driven TLSs is considered.

from piqs import Dicke
from qutip import steadystate
N = 10
system = Dicke(N, emission = 1, pumping = 2)
L = system.liouvillian()
steady = steadystate(L)

• Superradiant Light Emission
• Superradiance: Qubits in a cavity
• Spin squeezing

PIQS functions

Operators	Command	Description
Collective spin Jx	jspin(N, "x")	The collective spin operator Jx. Requires N number of TLS
Collective spin J+	jspin(N, "+")	The collective spin operator J+.
Collective spin J-	jspin(N, "-")	The collective spin operator Jz.
Collective spin Jx in uncoupled basis	jspin(N, "z", basis='uncoupled')	The collective spin operator Jz in the uncoupled basis
Dicke state |j, m>	dicke(N, j, m)	A Dicke state given by |j, m>
Excited state in uncoupled basis	excited(N, basis="uncoupled")	The excited state in the uncoupled basis
GHZ state in the Dicke basis	ghz(N)	The GHZ state in the Dicke (default) basis for N number of TLS
Collapse operators of the ensemble	Dicke.c_ops()	The collapse operators for the ensemble can be called by the c_ops method of the dicke class.