The periodic spatial modulation of a medium creates an artificial band structure with energy gaps and anomalous (i.e. negative) dispersion. If the particles repel each other, they may form a spatially localized state close to the within the energy gaps known as a gap soliton (GS). GSs are metastable states arising from the interplay between the anomalous dispersion and inter-particle interactions, which appear when the kinetic energy EK(r)~ −p2ħ2/(mbr2)] due to the localization of particles with a negative mass −mb within a radius r compensates the repulsive inter-particle interaction energy EI(r,N2d)= N2dg2d/r2 , where N2d is the particle density and g2d>0 their characteristic interaction energy. In general, solitons are interesting for applications in novel concepts for optical processing. While GSs in one-dimensional (1D) potentials have been extensively studied, GSs in 2D lattices have so far only been reported for purely photonic systems.
In this work, we demonstrate the formation and manipulation of GSs of polariton quantum condensates in a 2D tunable lattice. Polaritons result from the strong coupling of photons and quantum well (QW) excitons in a semiconductor microcavity (MC). Being bosonic light-matter quasi-particles, they combine features from both species. This is, the small mass arising from the photonic component allows them to condensate at low densities and high temperatures, while the inter-excitonic interactions provide a nonlinearity several orders of magnitude stronger than in purely photonic systems.
The GS studies were carried out on an (Al,Ga)As-based MC where a 170 × 170 mm2 sinusoidal square lattice is created by two interfering two SAWs [Fig. 1a)]. The MC consists of two Bragg reflectors (stacks of (Al,Ga)As l/4 layers with different index of refraction) embedding a l/2 cavity with three pairs of 15 nm-thick GaAs QWs (l is the wavelength of the confined photon). SAWs with wavelength lSAW = 8 mm, frequency of 370 MHz and propagation velocity of 3 mm/ns were excited by acoustihuc transducers deposited on the sample surface. The SAW strain field modulates the excitonic band gap and the microcavity optical resonance energy, thereby creating an effective lattice potential for polaritons given by V(x,y)=−FSAW[cos(kSAWx) + cos(kSAWy)], where kSAW = 2p/lSAW. The lattice amplitude FSAW can be controlled by the radio-frequency power Prf applied to the acoustic transducers (FSAW ∝√Prf). The polaritons were resonantly excited within a 70 mm spot using a single-mode, continuous-wave Gaussian pump laser delivering photons with energy Epump=1.5353 meV and in-plane momentum kpump = (kpx,kpy) = (0, 1.7) mm-1. They were probed by recording the steady-state photoluminescence (PL) emerging from the sample top surface (time integration of a few seconds) with spatial and angular (i.e. k) resolution.
The square lattice defined by V(x,y) creates mini-Brillouin-Zones (MBZs) of dimension kSAW separated by energy gaps [cf. inset Fig. 1b)]. Figure 1b) displays the single-particle band diagram along the X→ G→ M direction calculated for a shallow lattice. The curvature of the s band inverts close to the X and M giving rise to states with a negative mass. X is a saddle point with positive mass mp along X → M and negative mass −mb along X → Γ. M, in contrast, has a negative effective mass equal to −mb along both M → G and M → X. We have used a variational approach to solve the Gross-Pitaievskii equation for interacting polaritons in a square lattice potential and calculate metastable polaritonic states. One of the GS solutions has the energy E2d and spread in momentum indicated by the horizontal red line in Fig. 1b). The wave function of this mode (see |y|2 in Fig. 2c) is localized within a few lattice sites. The reciprocal space representation of y (cf. Fig. 1d) consists of an admixture of states around the M points of the lowest dispersion branch (s-band in Fig. 1b)). Note, in particular, that while |y|2 peaks at the minima of the lattice potential, its phase varies by 180o between adjacent minima.
A polariton condensate forms by excitation of a high density of particles in the lattice. Figure 2a) shows a time-integrated PL image of the condensate: the lines trace the path of the condensate wave function maxima (cf. Fig. 1a)) as they move with the acoustic velocity. The condensate emission extends over approximately three central lattice sites, in good agreement with the calculation in Fig. 1c). In the absence of acoustic excitation, polariton condensates are normally excited close to the bottom of the dispersion of Fig. 1a) (i.e., with with k=0). In a shallow lattice, in contrast, k-space PL map of Fig. 2b) shows that the condensate forms close to the negative-mass M-points [i.e. ksignal=(±0.5,±0,5)kSAW [cf. Fig. 1c)], in agreement with the calculation in Fig. 1d).
The preferential accumulation of particles at M, which triggers a GS condensate at M instead of a normal condensate at G, is attributed to the negative mass -mb. The latter provides an effective force, which counteracts diffusion of particles out of the M states. Furthermore, the preferential particle accumulation considerably reduces the optical threshold Pth for condensation, as indicated by the squares in Fig. 2c). This behavior can be understood by taking into account that a stable GS requires EK(r)+EI(r,N2d) < 0, which implies in N2d>N2d,min =pℏ2/(g2dmb). Since |mb| increases as the bands flattens at higher FSAW, N2d,min and, consequently, Pth decrease with FSAW. The solid line in Fig. 2c) shows that this behavior is also reproduced by the variational model.
E. A. Cerda-Méndez, D. Sarkar,D. N. Krizhanovskii, S. S. Gavrilov, K. Biermann, M. S. Skolnick, and P. V. Santos, “Exciton-polariton gap solitons in two-dimensional lattices”, Phys. Rev. Lett. 111, 166401 (2013)