Quantum coherence in momentum space of light-matter condensates

We show that the use of momentum-space optical interferometry, which avoids any spatial overlap between two parts of a macroscopic quantum state, presents a unique way to study coherence phenomena in polariton condensates. In this way, we address the longstanding question in quantum mechanics: “Do two components of a condensate, which have never seen each other, possess a definitive phase?” [P. W. Anderson, Basic Notions of Condensed Matter Physics (Benjamin Cummings, Menlo Park, CA, 1984)]. A positive answer to this question is experimentally obtained here for light-matter condensates, created under precise symmetry conditions, in semiconductor microcavities, taking advantage of the direct relation between the angle of emission and the in-plane momentum of polaritons.

Figure 1

(a) Sketch of the excitation and relaxation processes to form propagating polariton wave packets (WPs) on a background showing the energy vs $k_x$ emission obtained under nonresonant, low power excitation conditions. The gray ellipse depicts the excitation laser at 1.545 eV and $k_x\sim 0$. The dashed lines indicate the energy relaxation of excitons into polariton WPs. Polariton WPs, propagating with $k_x\approx \pm 1.6$ $\mu {\mathrm{m}}^{-1}$ (slightly displaced for the sake of clarity), are depicted with circles, coded in colors explained in (b). The emission intensity is coded in a logarithmic, false color scale. (b) Sketch in real space of the experimental configuration. A laser beam is split into two arms, A and B, distanced by $d$. They create four propagating polariton WPs, coded in different colors, $n_{1,2}^A$ (magenta, blue) and $n_{1,2}^B$ (red, green), moving along the $x$ axis of a microcavity ridge in the direction depicted by the arrows.

Figure 2

(a) Emission in real space, along the $x$ axis of the ridge, vs time. Gray circles at $x=\pm 35$ $\mu m$ indicate the spatial location of the A and B laser beams; the trajectories of the four WPs, $n_1^A$, $n_2^A$, $n_1^B$, and $n_2^B$, are indicated by the dashed arrows. (b) Momentum-space emission, along $k_x$, vs time. The gray circle indicates that the laser beams A and B excite the ridge at $k_x\sim 0$. The dashed, black arrows indicate the acceleration of the condensates $n_1^{\text{coh}}$ and $n_2^{\text{coh}}$, as well as the deceleration of the WPs $n_1^B$ and $n_2^A$. Intensity is coded in a normalized, logarithmic false color scale.

Figure 3

(a) Momentum distribution $n\left(\mathbf{k}\right)$, at 35 ps after the excitation, showing the condensates $n_1^{\text{coh}}$ and $n_2^{\text{coh}}$ at $k_x=\mp 1.6$ $\mu {\mathrm{m}}^{-1}$, respectively. (b) Corresponding $n\left(\mathbf{r}\right)$ distribution showing WPs $n_1^A$, $n_2^A$, $n_1^B$, and $n_2^B$. (c) Fourier transform of $n\left(\mathbf{k}\right)$, obtaining a frequency at $\Delta X=d=70$ $\mu m$. (d) Momentum distribution $n\left(\mathbf{k}\right)$ at 66 ps showing $n_1^B$ and $n_2^A$ at $k_x=\mp 1.6$ $\mu {\mathrm{m}}^{-1}$, respectively. (e) Real-space distribution $n\left(\mathbf{r}\right)$ showing the interferences of $n_{12}$ at $x=0$, created by the overlapping in real space of ${\psi }_1^B$ and ${\psi }_2^A$. The white dashed rectangle marks the region of interest where the interference occurs. (f) Fourier transform restricted to the region of interest in $n\left(\mathbf{r}\right)$, showing a frequency at $\Delta K_x=\kappa =3.2$ $\mu {\mathrm{m}}^{-1}$. (g) Momentum distribution $n\left(\mathbf{k}\right)$ at 108 ps, showing the interferences $n_{12}$ at $k_x\sim 0$. (h) Corresponding $n\left(\mathbf{r}\right)$ distribution showing $n_1^B$ and $n_2^A$. (i) Fourier transform of $n\left(\mathbf{k}\right)$, obtaining a frequency at $\Delta X=d_{12}=60$ $\mu m$. Intensities in the false color scales for momentum, real, and Fourier spaces are normalized to unity. The tilt in all panels originates from the orientation of the ridge with respect to the entrance slit of the spectrometer. The white dashed arrows mark the distances in real and momentum space between WPs. The solid arrows show these distances in the corresponding Fourier transform. Supplemental Video S1 (S2) shows the time evolution of the emission in real (momentum) space [20].