Spontaneous formation of a macroscopically extended coherent state

It is a straightforward result of electromagnetism that dipole oscillators radiate more strongly when they are synchronized and that if there are $N$ dipoles, then the overall emitted intensity scales with $N^2$. In atomic physics, such an enhanced radiative property appears when coherence among two-level identical atoms is established and is well known as “superradiance.” In superfluorescence, atomic coherence develops via a self-organization process stemming from the common radiated field, starting from an incoherently prepared population inversion. In this work we establish the experimental conditions for formation of a macroscopic dipole via superfluorescence, involving the remarkable number of $4\times {10}^{12}$ atoms. Self-driven atom dynamics, without the mediation of cavity QED nor quantum dots or quantum well structures, is observed in a cryogenically cooled rare-earth doped material. We present clear evidence of a decay rate that is enhanced by more than one million times compared to that of independently emitting atoms.

Figure 1

(a) Experimental setup. A 0.1% concentration, $4\times 5\times 6.2{\mathrm{mm}}^3$-volume Er:YSO crystal is immersed in LHe and maintained at 1.6 K. Er ions are indirectly pumped with a Ti:sapphire laser (stabilized ring cavity, 10-MHz linewidth) to the metastable level ${}^{4}I_{13/2}$ shown in (c). The formation of the macroscopic dipole is accompanied by coherent emission in the forward direction. The incident laser beam is focused by lens L1 and its polarization can be rotated by means of half-wave plate $\lambda /2$. A dichroic mirror (DM) reflects the pump laser light toward sensor G1 allowing for contextual laser absorption measurements. The forward light transmitted through DM is attenuated by a set of calibrated neutral density filters (NDF). Longpass filters (F) are used to remove stray laser light. Though optical window W we can monitor isotropic fluorescence or investigate off-axis SF emission. (b) Superfluorescent beam propagation and diffraction in the pencil-shaped sample geometry. The beam divergence angle ${\theta }_d$ is obtained by measuring with the knife-edge technique the forward beam radius at several distances from the crystal center, and the obtained ${\theta }_d$ value is compatible with diffraction from an aperture whose radius coincides with the measured laser beam waist ${\omega }_0=163\mu \mathrm{m}$ at the crystal position ($\pm 1\mathrm{cm}$). [(c) and (d)] Emission spectral filtering. Above the threshold $P_{\mathrm{th}}$, superfluorescence emission takes place through the highest branching ratio transition, represented in the ${\mathrm{Er}}^{3+}$ energy levels scheme by the magenta dotted line arrow. The corresponding magenta line in the recorded $P>P_{\mathrm{th}}$ spectrum is about ${10}^5$ times stronger than the overall emission related to the blue lines, whose intensity is instead comparable to that of the magenta line in the $P<P_{\mathrm{th}}$ spectrum.

Figure 2

Spontaneously generated superfluorescent pulses at different pump photon flux, corresponding to pump intensity values of 67 and $192\mathrm{W}/{\mathrm{cm}}^2$. (a) Representative individual time traces of photon bunches at different atom number $N\sim 5.4\times {10}^{12}$ (pink), $N\sim 4.6\times {10}^{12}$ (yellow), and $N\sim 2.6\times {10}^{12}$ (green). They have been recorded under identical excitation conditions and time shifted in the plot for representation purposes. The time dependence of the signal amplitude is very well fitted by a squared hyperbolic secant function. (b) Pulse duration ${\tau }_R$ versus peak photon rate $R_p$ for pumping rates. The black lines are fits in the form given by Eq. (3), giving an average $\mu =(2.4\pm 0.1)\times {10}^{-6}$, compatible with diffraction from an aperture of diameter $2{\omega }_0=326\mu \mathrm{m}$. (c) Nominal area of the fit function given by Eq. (2) versus integrated pulse area. Linear regression with unitary slope demonstrates pure SF dynamics for each recorded photon bunch. (d) Peak photon rate versus observed atom number. The recorded data scale superlinearly, as shown by the $N^2$ trend represented by the black line.

Figure 3

(a) Absorption laser spectroscopy for the transition ${}^{4}I_{15/2}\to {}^{4}I_{9/2}$ for $\pi$ and $\sigma$ polarization with and without 370-mT magnetic field. (b) Forward (d) and off-axis average intensity of the superfluorescent emission for different longitudinal profiles of the inverted population density at absorption coefficients 1.6 and $4.6{\mathrm{cm}}^{-1}$, corresponding to centered (809.0025-nm) and slightly off-resonance (809.0015-nm) wavelengths, respectively [blue and red dots in Fig. 3]. The dotted lines indicate when the off-axis emission is so important as to reverse the growth trend in the forward emission. The black lines are a parabolic fit to the data. (c) Data agreement with the steady-state rate equations analysis. Deviation from the initial linearity of the population inversion with pump absorption is due to the superfluorescence emission, which is more pronounced in the data recorded at resonance (red dot data).

Figure 4

Detection of transverse photon bunches with a fast photodiode (10-ns rise time, 0.8-${\mathrm{mm}}^2$ area) set in front of optical window at $\gtrsim 3$ cm distance from W with no input coupling optics.

Figure 5

Temporal dynamics recorded at an ultrafast (25-ps rise time) InGaAs photodiode.