Emergent light crystal from frustration and pump engineering

Matteo Biondi, Gianni Blatter, and Sebastian Schmidt

Phys. Rev. B 98, 104204 – Published 17 September 2018

Abstract

We demonstrate how pump engineering drives the emergence of frustration-induced quasi-long-range order in a low-dimensional photonic cavity array. We consider a Lieb chain of nonlinear cavities as described by the Bose-Hubbard model and featuring a photonic flatband in the single-particle spectrum. Incoherent pumping of the Lieb lattice leads to a photonic density wave which manifests an algebraic decay of correlations with twice the period of the lattice unit cell. This work opens up other directions for the emergence of strongly correlated phases in quantum optical frustrated systems through pump design.

Received 5 March 2018

DOI:https://doi.org/10.1103/PhysRevB.98.104204

Physics Subject Headings (PhySH)

Exotic phases of matter Quantum optics

1-dimensional systems Quantum cavities Strongly correlated systems

Approximation methods for many-body systems Lattice models in condensed matter Numerical techniques

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Matteo Biondi, Gianni Blatter, and Sebastian Schmidt

Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland

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Vol. 98, Iss. 10 — 1 September 2018

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Figure 1
Schematic view of the driven-dissipative quasi-one-dimensional Lieb lattice. The lattice has a three-site unit cell with basis sites $A, B, C$ (gray). Here, the lattice displays $N = 5$ cells and terminates with $A$ and $B$ sites for symmetry. Photons can hop between neighboring sites with amplitude $J$ (green) and experience an on-site repulsion $U$ (red), as described by the Bose-Hubbard Hamiltonian [see Eq. (1)]. The Lieb lattice hosts a set of single-particle (photon number $n = 1$ ) plaquette states localized on one $C$ site and two neighboring $A$ sites (top-left inset); these originate from destructive quantum interference of amplitudes at site $A$ [see Eq. (2)]. Product states of nonoverlapping plaquettes are many-body eigenstates of the Hamiltonian with zero interaction energy. The lattice is pumped homogeneously with strength $P_{e}$ (orange light cones) and loses photons to the environment with rate $κ$ . The nonequilibrium dynamics is described by the master equation (4). Due to the energy-dependent spectral density of the pump, the steady state is a density wave where only every second $C$ site is occupied (blue light cones) giving rise to frustration-induced quasi-long-range order [see Eq. (7)].
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Figure 2
(a) Energy levels $ω_{n, α} / J$ of the quasi-1D Lieb lattice (1) with $N = 5$ unit cells, i.e., $3 N + 2$ sites, as a function of excitation (photon) number $n$ , while the index $α$ labels energies at fixed $n$ . The levels $ω_{n, α}$ are scaled to the hopping amplitude $J$ and have been obtained via exact diagonalization (see Appendix pp2). Frustration in the Lieb chain leads to a set of single-particle ( $n = 1$ ) plaquette states at zero energy [see Eq. (2)]; product states of nonoverlapping plaquettes are also zero-energy eigenstates with $n > 1$ up to the density-wave state [see Eq. (3)], with $n_{dw} = (N + 1) / 2$ , i.e., $n_{dw} = 3$ in (a). This manifold is marked by the blue lines (including the vacuum with $n = 0$ ). The top-left inset shows the single-particle levels $ω_{n = 1, α}$ in momentum space (crosses), including lines for large $N$ (note the zero-energy flatband). The band structure asymmetry is due to a finite offset $ω_{A} \neq ω_{B}$ . (b) The zero-energy manifold is pumped (see text) using a Lorentzian spectral density. Occupation of other levels is suppressed if the width $Γ$ is smaller than the minimal excitation gap $Δ$ to higher bands with $n = n_{dw} + 1$ (bottom panel). The gap decreases with system size [see top panel in (b)]. The blue dashed line is an exponential fit, $Δ / J \propto exp (- γ N)$ , $γ = 1.31 \pm 0.67$ . (c) Occupation of the density wave $ρ_{n_{dw}}$ from the numerical solution of the master equation (4) as a function of the Lorentzian width $Γ / Δ$ and pump strength $P_{e} / κ$ for the lattice with $N = 5$ unit cells as in (a). For small $Γ$ and large $P_{e}$ the steady state of the Lieb chain is a quasipure density wave. (d) Populations within the flatband manifold, as obtained from the analytical solution of the master equation (4) in the projected Hilbert space (6) (bars), versus numerical results (symbols), for $Γ / Δ = 1.3$ and $P_{e} / κ = 1$ (top) and $P_{e} / κ = 10^{2}$ [bottom; cf. points marked by triangles in (c)] showing excellent agreement supporting the validity of the analytic study. Other parameters chosen as $U / J = 2.5$ , $κ / Γ = 10^{- 3}$ , $ω_{B} = ω_{A} + 2 J$ .
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Figure 3
(a) Density-density correlator of $C$ sites $g_{1, j}^{(2)}$ showing QLRO with period doubling. The result obtained from the projected Hilbert space for a lattice with $N = 15$ unit cells (line) agrees well with the exact numerics for a smaller system of $N = 7$ unit cells (symbols). The pump strength is $P_{e} / κ = 75$ and we have chosen $Γ / Δ = 1.5$ in our numerical analysis, with $Δ / J \approx 3 \times 10^{- 4}$ the excitation gap for $N = 7$ . A fit $g_{1, 2 k}^{(2)} = 1 - 1 / k^{β}$ to the envelope at even $j = 2 k$ (dashed) decays algebraically with an exponent $β \approx 1.1$ . The inset shows results for smaller sizes providing an extrapolated value $β \approx 1.06$ . (b) Average filling $ν = \bar{n} / N_{s}$ (left) and compressibility $K = (\sum_{n} n^{2} ρ_{n} - {\bar{n}}^{2}) / \bar{n}$ (right) as a function of pump strength $P_{e} / κ$ for fixed values of the Lorentzian width $Γ / Δ = 0.13, 1.3$ . Here, $\bar{n} = \sum_{n} n ρ_{n}$ denotes the population of the manifold with photon number $n$ and $ρ_{n} = \sum_{α} ρ_{n, α}$ . The analytical solution of the master equation (4) in the projected Hilbert space (6) (lines) agrees well with the exact numerics for small values of the Lorentzian width $Γ / Δ = 0.13$ (filled symbols, on top of the line). Deviations occur when the spectral width $Γ / Δ = 1.3$ (open symbols) is comparable with the excitation gap (see Fig. 2) at large pump strengths. The black upward (red downward) triangles are obtained including up to $n = n_{dw} + 1$ ( $n = n_{dw} + 2$ ) excitations in the full lattice Hilbert space, with good convergence in the data.
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