LETTERS Gain without inversion in semiconductor nanostructures M. D. FROGLEY 1 , J. F. DYNES 1 , M. BECK 2 , J. FAIST 2 AND C. C. PHILLIPS 1 * 1 Physics Department, Imperial College, Prince Consort Road, London SW7 2AZ, UK 2 Institute of Physics, University of Neuch ˆ atel, Neuch ˆ atel CH-2000, Switzerland *e-mail: chris.phillips@imperial.ac.uk Published online: 19 February 2006; doi:10.1038/nmat1586 W hen Einstein showed that light amplification needed a collection of atoms in ‘population inversion’ (that is, where more than half the atoms are in an excited state, ready to emit light rather than absorb it) he was using thermodynamic arguments 1 . Later on, quantum theory predicted 2,3 that matter–wave interference effects inside the atoms could, in principle, allow gain without inversion (GWI). The coherent conditions needed to observe this strange effect have been generated in atomic vapours 4 , but here we show that semiconductor nanostructures can be tailored to have ‘artificial atom’ electron states which, for the first time in a solid, also show GWI. In atomic experiments, the coherent conditions, typically generated either by coupling two electron levels to a third with a strong light beam 2,3 or by tunnel coupling both levels to the same continuum (Fano effect 5 ), are also responsible for the observation of ‘electromagnetically induced transparency’ (EIT) 6 . In turn, this has allowed observations of markedly slowed 7 and even frozen 8 light propagation. Our ‘artificial atom’ GWI effects are rooted in the same phenomena and, from an analysis of the absorption changes, we infer that the light slows to ∼c/40 over the spectral range where the optical gain appears. The strength of the interaction between an optical coupling beam (of electric field amplitude E c ) and an |i〉→|j 〉 transition of energy E ij and transition dipole z ij , is measured by a Rabi frequency Ω Rabi =[Δ 2 ij + ( ez ij E c ) 2 ] 1/2 , where Δ ij = E ij − ¯ hω c is the detuning, ¯ hω c is the coupling photon energy and ¯ h is the reduced Planck’s constant. Rabi oscillations, where the electron population cycles coherently between states |i〉 and |j 〉, can be seen with resonant (Δ ij = 0) coupling, but this needs intense and uniform beams 6 , so that the oscillation period, τ Rabi = 2π/Ω Rabi , becomes shorter than the transition dephasing time. This is equivalent to saying, in the energy domain, that the splitting of the new ‘dressed’ energy levels 9 (Fig. 1a) needs to be larger than their linewidths ( ¯ hΩ Rabi > 1/γ ij , where γ ij denotes the lifetime broadening caused by dephasing processes) so that they can be resolved in an experiment. Using off-resonant coupling (Δ ij > ez ij E c ) gives larger Rabi frequencies, which are less sensitive to variations in coupling beam |1〉 |2〉 |1〉 |2〉 |3〉 hω 23 hω – hω 12 hω – |1,t 〉 |1,s〉 |2,t 〉 |2,s〉 |2,t 〉 |2,s〉 |3,t 〉 |3,s〉 |1〉 a b hω gain = 185 meV – hω hω c = 155 meV hω – Figure 1 Schematic of the ‘dressing’ of electron energy levels by a strong coupling beam 9 .a, When driven at ¯ h ω c ∼ E 12 , a two-level system evolves into two doublets, each split by the Rabi energy, ¯ hΩ Rabi , denoted by the red arrows, which itself increases monotonically with the coupling beam intensity. b, The three-level system whose ‘bare’ transition energies are denoted by the solid black arrows. Most (∼80%) of the electrons remaining in state |1〉 and the gain arising from coherences appearing in the upper two states, which have been split into Rabi doublets by the off-resonant coupling beam. Upward green arrows represent the coupling photons, the downward dashed arrows are the transitions generating the optical gain and the blue arrows denote the frequency, ¯ h ω 0 , where extra loss appears. intensity, but it is only useful, for example, for GWI applications if the detuning is chosen to produce gain in spectral regions where there is useful oscillator strength 10 . We are able to choose the energies of the synthetic electron states in our semiconductor nanostructure ‘artificial atoms’ (Fig. 2) to engineer this. In our case the coupling beam is being weakly absorbed by the electrons in state |1〉, but at the same time it is dressing each of states |2〉 and |3〉 into a Rabi doublet (Fig. 1b). This dressing increases the absorption at the lower dressed state transition energy nature materials VOL 5 MARCH 2006 www.nature.com/naturematerials 175 Nature Publishing Group ©2006