Dynamical complexity of non-Gaussian many-body systems with dissipation

Abstract

We characterize the dynamical state of many-body bosonic and fermionic many-body models with inter-site Gaussian couplings, on-site non-Gaussian interactions and local dissipation comprising incoherent particle loss, particle gain, and dephasing. We first establish that, for fermionic systems, if the dephasing noise is larger than the non-Gaussian interactions, irrespective of the Gaussian coupling strength, the system state is a convex combination of Gaussian states at all times. Furthermore, for bosonic systems, we show that if the particle loss and particle gain rates are larger than the Gaussian inter-site couplings, the system remains in a separable state at all times. Building on this characterization, we establish that at noise rates above a threshold, there exists a classical algorithm that can efficiently sample from the system state of both the fermionic and bosonic models. Finally, we show that, unlike fermionic systems, bosonic systems can evolve into states that are not convex-Gaussian even when the dissipation is much higher than the on-site non-Gaussianity. Similarly, unlike bosonic systems, fermionic systems can generate entanglement even with noise rates much larger than the inter-site couplings.

Figures (4)

• Figure 1: Sketch of the model, with $n$ sites on a lattice, where each site contains $L$ modes. There are Gaussian couplings between the different sites, while the non-Gaussian interactions are only onsite. Nonlocal couplings are allowed. In the bosonic case, interactions of the form $n_{i,\sigma}^2$ are also allowed.
• Figure 2: Phase diagram for both bosonic and fermionic systems in the presence of generic noise. (a) For fermionic system the state remains convex-Gaussian at all times for error rates $\kappa_3 \geq 2U$. (b) In bosonic systems the state remains separable at all times for error rates $\kappa_1, \kappa_2 \geq 2J$.
• Figure 3: Schematic depiction of the Trotterization schemes in the proof of Theorem \ref{['theorem:fermions_high_noise']} (fermionic systems with weak non-Gaussianity). For simplicity, we only depict a 1D setting, with each site containing $3$ modes ($L=3$). A single Trotter step consists of a Gaussian channel (orange rectangles) that includes the combined effect of $H_g(t)$ and the particle gain and loss dissipators, followed by non-Gaussian gates (blue rectangles) interspersed with the dephasing dissipator (gray curved rectangles). Crucially, a non-Gaussian gate followed by sufficiently strong dephasing can be written as a convex combination of Gaussian channels.
• Figure 4: Schematic depiction of the Trotterization schemes in the proof of Theorem \ref{['theorem:weak_couplings']} (bosonic systems with weak inter-site couplings). For simplicity, we only depict a 1D setting, with each site containing $3$ modes ($L=3$). A single Trotter step consists of a layer of single-site channels which include the Hamiltonian terms acting on that site and the dephasing dissipator, followed by 2-site gates interspersed with particle gain and loss dissipators (in gray circles). Crucially, a 2-site gate followed by sufficiently strong noise can be written as a convex combination of single-site channels

Theorems & Definitions (2)

• Theorem 1
• Theorem 2