Fig. 3

Theoretically computed evolution of reduced density matrices for a separable \(\left| {{\mathrm{\Psi }}^{sep}} \right\rangle\) = \(\frac{1}{{\sqrt 2 }}\left( {\left| {1_1,1_2} \right\rangle \pm \left| {1_2,1_1} \right\rangle } \right) \to \rho _{(1,2),(2,1)}\) = \(\left| {{\mathrm{\Psi }}^{sep}} \right\rangle \left\langle {{\mathrm{\Psi }}^{sep}} \right|\), b entangled \(\left| {{\mathrm{\Psi }}^{ent}} \right\rangle\) = \(\frac{1}{{\sqrt 2 }}\left( {\left| {1_1,1_1} \right\rangle + \left| {1_2,1_2} \right\rangle } \right) \to \rho _{(1,1),(2,2)}^{ent}\) = \(\left| {\Psi ^{ent}} \right\rangle \left\langle {\Psi ^{ent}} \right|\), and e incoherent \(\rho _{(1,2),(2,1)}^{inc}\) = \(\frac{1}{2}\left( {\left| {1_1,1_2} \right\rangle \left\langle {1_1,1_2} \right| + \left| {1_2,1_1} \right\rangle \left\langle {1_2,1_1} \right|} \right)\) bosons propagating in the noisy trimer shown in Fig. 1a. The simulated dephasing rates correspond to the experimental ones γ_exp = (γ₁, γ₂, γ₃) = (1.3012, 1.2365, 1.293) cm⁻¹. From c, d it is clear that at z = 12 cm, separable and entangled bosons are described by identical density matrices. Once in the steady state, e.g., at z = 100 cm, the density matrices exhibit three main peaks along the diagonal, indicating that particle bunching is the most probable outcome to occur d. In contrast, incoherent bosons exhibit a different behaviour where particle antibunching exhibits the highest probability as elucidated by the density matrices at z = 100 cm g. In h–j we depict density matrices for indistinguishable fermion pairs \(\left| {{\mathrm{\Psi }}^{fer}} \right\rangle\) = \(\frac{1}{{\sqrt 2 }}\left( {\left| {1_1,1_2} \right\rangle - \left| {1_2,1_1} \right\rangle } \right) \to \rho _{(1,2),(2,1)}^{fer}\) = \(\left| {1_1,1_2} \right\rangle \left\langle {1_1,1_2} \right|\) − \(\left| {1_1,1_2} \right\rangle \left\langle {1_2,1_1} \right|\) − \(\left| {1_2,1_1} \right\rangle \left\langle {1_1,1_2} \right|\) + \(\left| {1_2,1_1} \right\rangle \left\langle {1_2,1_1} \right|\). From this numerical results we see that in agreement with the Pauli exclusion principle, both fermions never occupy the same site (fermion antibunching) and the steady state contains off-diagonal entries demonstrating that some coherences survive the impact of dephasing