Steady state is the long-time limit of an open quantum system: $\rho_{\text{ss}}$ where $d\rho_{\text{ss}}/dt = 0$. The steady state satisfies the master equation with zero time derivative, making it the fixed point of open system dynamics.
For dissipative systems (non-Hermitian effective Hamiltonian), the steady state is unique and stable. All trajectories converge to it. For conservative systems (no decay), the steady state may not exist.
Finding steady states is useful for understanding long-time behavior, designing steady-state cooling, and optimizing dissipative state preparation.
QuTiP's steadystate() solves for $\rho_{\text{ss}}$ by finding the null space of the Liouvillian:
from qutip import * import numpy as np H = 0.5 * sigmaz() c_ops = [0.1 * sigmam()] # Compute steady state rho_ss = steadystate(H, c_ops) print(rho_ss) # Density matrix at t → ∞ # Verify it's steady L = liouvillian(H, c_ops) print((L * rho_ss).full()) # Should be ~zero
Steady state is where decay and energy drive balance. A qubit with driving and decay reaches a steady mixture of ground and excited states. A dissipative cooler reaches its ground state via continuous measurement and feedback.
For pure decay (no driving), the steady state is always the ground state. With driving, the steady state is a mixture—its composition depends on detuning and coupling strength.
Steady states describe the long-time limit. For short-time dynamics, use mesolve.