Example of an infinite volume Lindblad system

"Infinite volume Lindblad system" doesn't make much sense to me. Apparently, what you are asking for is an example of a Lindblad system with steady state $rho_{infty}$: if $L(rho_{infty})=0$, then this state is invariant under the action of the dynamical semigroup $phi(t)=exp L t$ driving the open system dynamics.

The simplest example we may think of is a single bosonic mode $a$ with frequency $omega$ (e.g. a monochromatic laser in the quantum regime with very few photons) immersed in the electromagnetic field at zero temperature. The open system dynamics is then given by the following master equation [1]: $$ L(rho)=-i[H_S,rho]+gamma_0 left(a rho a^dagger-frac{1}{2}{a^dagger a,rho}right), $$ where $H_S=hbaromega a^dagger a$ and $gamma_0$ is the standard emission coefficient. You can easily verify that $rho_infty=|0rangle!langle 0|$ is the unique steady state of the open dynamics; physically, this means that the bosonic mode is decaying toward the vacuum state.

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As for an open quantum system spread over an infinite volume, we may consider a chain of $2n+1$ harmonic oscillators with bosonic annihilation operators ${a_j}_{j=-n}^n$, and then let $n$ goes to infinity. The system Hamiltonian is $H_S=sum_{j=-n}^n hbaromega_j a_j^dagger a_j$, and we consider a single common bosonic bath acting all over the chain. The bath Hamiltonian is $H_B=sum_k hbarOmega_k b_k^dagger b_k$, and we take it at zero temperature, i.e. the state of the bath is $rho_B=bigotimes_k |0rangle_klangle 0|$. We connect the harmonic oscillators and the thermal bath through the interaction Hamiltonian $H_I=sum_{k} g_k left(sum_{j=-n}^n a_j b_k^dagger +h.c.right)$. Following the standard derivation of the master equation in the Markovian limit [1], you can see that the GKLS equation driving the dynamics of the state of the system of harmonic oscillators $rho_S$ reads: $$ L(rho_S)=-i[H_S,rho_S]+sum_{j=-n}^nsum_{j'=-n}^ngamma_0 left( a_j rho a_{j'}^dagger-frac{1}{2}{a_{j'}^dagger a_j,rho}right). $$ Once again, you can easily verify that $rho_{infty}=bigotimes_{j=-n}^n |0rangle_jlangle 0|$ (the ground state of the system Hamiltonian) is the steady state of the dynamics. This is because the action of the thermal bath (expressed by the collective jump operator $sum_{j=-n}^n a_j$, i.e. annihilation of all the oscillator modes) consists in completely absorbing the energy of the system.

[1] H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford University Press, 2002).