Momentum-space Landau levels in driven-dissipative cavity arrays

We theoretically study the driven-dissipative Harper-Hofstadter model on a two-dimensional square lattice in the presence of a weak harmonic trap. Without pumping and loss, the eigenstates of this system can be understood, in certain limits, as momentum-space toroidal Landau levels, where the Berry curvature, a geometrical property of an energy band, acts like a momentum-space magnetic field. We show that key features of these eigenstates can be observed in the steady state of the driven-dissipative system under a monochromatic coherent drive and present a realistic proposal for an optical experiment using state-of-the-art coupled cavity arrays. We discuss how such spectroscopic measurements may be used to probe effects associated both with the off-diagonal elements of the matrix-valued Berry connection and with the synthetic magnetic gauge.

Figure 1

Top: Zero-point-energy error, with (solid curve; ${\eta }_{\text{zpe}}^{\text{nab}}$) and without (dashed line; ${\eta }_{\text{zpe}}$) the shift from the off-diagonal matrix elements of the Berry connection. Including just the first term in the sum, Eq. (10), gives a curve identical to the one for ${\eta }_{\text{zpe}}^{\text{nab}}$. The chosen trap strength is $\kappa =0.02J$. Bottom: Level-spacing error, for the same trap strength, considering $\beta =0,1$ (solid curve), $\beta =1,2$ (dashed curve), $\beta =2,3$ (dashed-dotted curve), and $\beta =3,4$ (dotted curve).

Figure 2

Intensity spectra for different pumping conditions: (a) pumping the single site (5,5), (b) pumping with a Gaussian profile centered at site (5,5) with width $\sigma =1$, (c) homogeneous pumping across all lattice sites, and (d) pumping with a random phase across all lattice sites. These results were obtained by numerical solving Eq. (15) for the steady state in a lattice of $N\times N=45\times 45$ sites, with $\kappa =0.02J, \gamma =0.001J$, and $\alpha =1/11$. Solid black curves correspond to using the Landau gauge; dashed (green) curves, the symmetric gauge. Dotted vertical lines (with labels indicating the value of $\beta$) mark the states which were selected for later analysis. Spectra in (a) and (d) are identical for both gauges.

Figure 3

Real-space reconstruction of the states $\beta =3$, 6, 15, and 26 in Fig. 2.

Figure 4

Momentum-space reconstruction of the eigenstates. Top row: States corresponding to $\beta =0$, 2, 4, and 6 in Fig. 2, using the Landau gauge. Bottom row: States corresponding to $\beta =0$, 1, 9, and 20 in Fig. 2, using the symmetric gauge.

Figure 5

Momentum-space reconstruction of the eigenstates in the full BZ, using the symmetric gauge and homogeneous pumping with a random on-site phase. Top row: States corresponding to $\beta =9$, 20, and 30. Bottom row: States corresponding to $\beta =38$, 59, and 99. Parameters are the same as in Fig. 2.

Figure 6

Momentum-space reconstruction of the eigenstates in the full BZ, using the Landau (top row) and symmetric gauge (middle and bottom rows) and a spatially homogeneous pump with a random on-site phase. We have considered the state $\beta =4$ for different trap positions $(m_0,n_0)$. For the top and middle rows, we have (from left to right) (0,0) (trap in the center), (2,0), (5.5,0), and (11,0), whereas for the bottom row we chose the positions (0,2), (0,5.5), (0,11), and (11,11). Parameters are the same as in Fig. 2.

Figure 7

Top row: Intensity spectrum for a small lattice of $11\times 11$ sites, with $\gamma =0.05J, \kappa =0.2J$, and $\alpha =1/7$. Dashed vertical (orange) lines show the first ladder of eigenstates of $\mathcal{H}$ from Eq. (1). The second ladder of states is indicated by dash-dotted vertical (green) lines. Second row: Profile of the $\beta =7$ state in real space (left), momentum space (center), and the corresponding population over bands in the MBZ (right) for a conservative system with no pumping or decay. Third row: Reconstruction of the $\beta =7$ mode wave function in real space (left), momentum space (center), and the corresponding population over bands in the MBZ (right) obtained in the driven-dissipative system by setting the pump frequency at ${\omega }_0=-1.63J$ on resonance with the desired mode (dotted vertical black line in the top panel). The $\delta$-like pump at (0,5) is visible as a dark square in the left panel. Bottom row: Slice along the $p_x^0=0$ line in the MBZ (solid black line) compared to the analytical prediction of Eq. (8), $|{\chi }_7(0,p_y){|}^2$ [dotted (blue) line], and to the population over bands for the nondissipative system [dashed (orange) line].