Permutational Invariance¶
Permutational Invariant Quantum Solver (PIQS)¶
The Permutational Invariant Quantum Solver (PIQS) is a QuTiP module that allows to study the dynamics of an open quantum system consisting of an ensemble of identical qubits that can dissipate through local and collective baths according to a Lindblad master equation.
The Liouvillian of an ensemble of \(N\) qubits, or twolevel 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.
Consider a system evolving according to the equation
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 superoperators 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 [28], the Liouvillian \(\mathcal{D}_\text{TLS}(\rho)\) can be built as a blockdiagonal 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) emissioncollective_dephasing
defines \(\gamma_\text{CD}\), collective dephasingcollective_pumping
defines \(\gamma_\text{CP}\), collective pumping.emission
defines \(\gamma_\text{E}\), incoherent emission (losses)dephasing
defines \(\gamma_\text{D}\), local dephasingpumping
defines \(\gamma_\text{P}\), incoherent pumping.
Then the system.lindbladian()
creates the total TLS Lindbladian 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)
For more example of use, see the “Permutational Invariant Lindblad Dynamics” section in the tutorials section of the website, http://qutip.org/tutorials.html.
Operators 
Command 
Description 

Collective spin algebra \(J_x,\ J_y,\ J_z\) 

The collective spin algebra \(J_x,\ J_y,\ J_z\) for \(N\) TLSs 
Collective spin \(J_x\) 

The collective spin operator \(Jx\). Requires \(N\) number of TLSs 
Collective spin \(J_y\) 

The collective spin operator \(J_y\). Requires \(N\) number of TLSs 
Collective spin \(J_z\) 

The collective spin operator \(J_z\). Requires \(N\) number of TLSs 
Collective spin \(J_+\) 

The collective spin operator \(J_+\). 
Collective spin \(J_\) 

The collective spin operator \(J_\). 
Collective spin \(J_z\) in uncoupled basis 

The collective spin operator \(J_z\) in the uncoupled basis of dimension \(2^N\). 
Dicke state \(j,m\rangle\) density matrix 

The density matrix for the Dicke state given by \(j,m\rangle\) 
Excitedstate density matrix in Dicke basis 

The excited state in the Dicke basis 
Excitedstate density matrix in uncoupled basis 

The excited state in the uncoupled basis 
Groundstate density matrix in Dicke basis 

The ground state in the Dicke basis 
GHZstate density matrix in the Dicke basis 

The GHZstate density matrix in the Dicke (default) basis for N number of TLS 
Collapse operators of the ensemble 

The collapse operators for the ensemble can be called by the c_ops method of the Dicke class. 
Note that the mathematical object representing the density matrix of the full system that is manipulated (or obtained from steadystate) in the Dickebasis formalism used here is a representative of the density matrix. This representative object is of linear size N^2, whereas the full density matrix is defined over a 2^N Hilbert space. In order to calculate nonlinear functions of such density matrix, such as the Von Neumann entropy or the purity, it is necessary to take into account the degeneracy of each block of such blockdiagonal density matrix. Note that as long as one calculates expected values of operators, being Tr[A*rho] a linear function of rho, the representative density matrix give straightforwardly the correct result. When a nonlinear function of the density matrix needs to be calculated, one needs to weigh each degenerate block correctly; this is taken care by the dicke_function_trace in qutip.piqs, and the user can use it to define general nonlinear functions that can be described as the trace of a Taylor expandable function. Two nonlinear functions that use dicke_function_trace and are already implemented are purity_dicke, to calculate the purity of a density matrix in the Dicke basis, and entropy_vn_dicke, which can be used to calculate the Von Neumann entropy.
More functions relative to the qutip.piqs module can be found at API documentation. Attributes to the qutip.piqs.Dicke
and qutip.piqs.Pim
class can also be found there.