Seminal work1 on the experimental discovery of quasiparticles that obey anyonic statistics in the electron Laughlin state ν = 1/3 (ref.2) opened fundamentally new prospects for studying the objects, the very existence of which has been discussed thus far merely hypothetically. Surprisingly, the anyonic statistics of FQHE quasiparticles were predicted immediately following Laughlin’s pioneering works3,4. However, it took over thirty years before there was reported the first direct experimental evidence in support of this remarkable physical phenomenon, whose practical application can offer entirely new perspectives in solid-state physics. As for the quasiparticle systems that obey anyonic statistics, there have been predicted a number of intriguing results, unachievable in boson and fermion systems5,6, while considering the non-abelian anyons in the FQHE state of 5/2 has already led to a beautiful theoretical idea of topological quantum computation7-9.

Although experimental information about the charged quasiparticles can be obtained using magneto-transport techniques, our understanding of neutral excitations in FQHE states, inactive in transport experiments, is relatively limited. Brilliant theoretical predictions about one type of neutral excitations, magneto-rotons in a single-mode approximation10, have been confirmed by microwave absorption experiments, both qualitatively and quantitatively, for several FQHE states with broken translation symmetry11. However, it is impossible to draw any conclusions about the statistical properties of magneto-rotons based on these experiments, as such excitations have a short lifetime11. Moreover, it is rather challenging to understand from the reported experiments11,12 whether magneto-rotons constitute the major excitation branch or there are other yet unknown excitations that might be of a great importance to the statistics and thermodynamics of FQHE states. The existing theories hypothesize about various properties of neutral excitations and even draw parallels between the magneto-roton complexes in FQHE (“magneto-gravitons”) and the gravitons in the general theory of relativity13-17. Most daring theories suggest that the physics of FQHE excitations helps broaden our outlook on the evolution of the Universe18,19.

The recipe for constructing quasi-equilibrium ensembles of magneto-rotons in two-dimensional electron systems under stationary excitation by the photons of a specific energy range, has already been developed for the integer Hall insulators20. It turns out that the ensembles can be formed provided that the physical characteristics of the electron system meet certain requirements. For example, the lowest-energy excitations should change the total spin of the electron system. In that case, the quasi-equilibrium ensembles of neutral excitations having a roton minimum in their dispersion are formed as a result of drastic retardation in the spin-flip process in magnetic field, with the simultaneous conservation of the large roton momentum20. Seemingly, this situation is unlikely to occur in the Laughlin state ν = 1/3 presumably analogous to the integer Hall ferromagnet ν = 1, though in terms of the composite fermions rather than electrons21 (for the Hall ferromagnet ν = 1, the aforementioned requirements are definitely not satisfied due to absence of the roton minimum in the dispersion of the spin exciton22). The single-mode approximation proven successful in describing the spin zero neutral excitations in the Laughlin state ν = 1/3 also predicts a monotonous dispersion dependency for spin one neutral excitations at ν = 1/3, similar to that of a spin exciton in the Hall ferromagnet ν = 1(ref.23).

Nevertheless, finding the exact solution to Schrodinger’s equation for a multiparticle electron system24 gave the first indication that in the Laughlin state ν = 1/3, the spin excitations with momentum on the order of inverse magnetic length can have lower energy than the minimal energy of the magneto-rotons. This result did not account for a substantial contribution to the energy of the excitations from the single-particle Zeeman energy that is linearly dependent on the magnetic field. Therefore, so far, it has not been clear whether physical systems with the spin one neutral excitations can, in practice, have lower energy than magneto-rotons.

In the presented work, we develop a numerical scheme of solving Schrodinger’s equation for the electron system with a finite number of particles in line with previous studies25 and investigate the dispersion properties of neutral excitations for the Laughlin state ν = 1/3 in GaAs/AlGaAs quantum wells. The single-mode approximation is found inapplicable for describing the spin one branches of the excitations in this case. Moreover, we confirm that for specific parameters of the electron system, the spin one neutral excitations are indeed the lowest-energy excitation branches (see Methods). Given the selected electron system with necessary parameters, by means of optical techniques (photoluminescence and photoinduced resonant reflection) (Fig.1), we succeeded in forming a macroscopic ensemble of neutral spin one excitations in the Laughlin state ν = 1/3 and devised a method for the direct measurement of their relaxation time. Our findings show that the ensemble of spin one neutral excitations exhibits the properties of a Bose system.

Let us look at the photoluminescence first as namely this technique is used to form a quasi-equilibrium ensemble of neutral excitations (see Methods). The photoluminescence signal at the formation of the Laughlin state ν = 1/3 undergoes significant changes. At low temperatures, the photoluminescence intensity from the upper spin sublevel is an order of magnitude higher than that from the lowest spin sublevel filled with electrons in the equilibrium (Fig. 2). Moreover, at higher temperatures (above 1 K), when both spin sublevels are filled in the equilibrium and the Laughlin state ν = 1/3 is destroyed, the photoluminescence intensity from the lowest spin Landau sublevel exceeds that from the upper spin sublevel, which corresponds to the probabilities of optical transitions for analogous electronic systems26. The effect of enormous gain in the photoluminescence signal in the Laughlin state ν = 1/3 produced by the excited spin Landau sublevel has universal nature, i.e., it is independent of the photon energy exciting the electron system (Fig. 3). Such an amplification of the photoluminescence signal generated by the excited spin Landau sublevel is nothing else but the collective response of the quasi-equilibrium excitations to a valence hole, analogous to that already observed in the photoluminescence signal of another condensate (the condensate of spin-triplet magneto-rotons demonstrating properties of a Bose system) in the integer quantum Hall insulator ν = 2 (ref.20). Therefore, based on measured photoluminescence spectra, we can draw a qualitative conclusion that under cw photoexcitation, there is formed a condensate of neutral excitations, exhibiting collective response to a photoexcited hole in the valence band. Furthermore, analyzing life times for different spin one excitations, it can be concluded that spin one “magneto-gravitons” are the most likely candidates to form this condensate (see Methods.)

Our further study is devoted to determining the statistical properties of the condensate. To accomplish this, we employed the technique of photoinduced resonant reflection – the photoinduced elastic scattering of light. In essence, it means using one radiation source to form an ensemble of excitations, whereas the other provides the probing photons that are elastically scattered on the neutral excitations (Fig.1). By varying the excitation wavelength of the former at a fixed wavelength and radiation power level of the latter, we can compare the effectiveness of forming neutral excitations by the photons of different energies. Conversely, varying the wavelength of the probing radiation from the second radiation source while keeping constant the wavelength and the power level of the first one makes it possible to investigate how effective the different channels of elastic light scattering are, based on the photon scattering efficiency from the fixed quantity of the excitations created by the first radiation source.

In the presence of a macroscopic ensemble of excitations, a standard signal of photoinduced resonant reflection has a negative sign (Fig. 4), demonstrating the filling of the phase space on the upper spin Landau sublevel at the appearance of the electrons incorporated in the excitations. This, in turn, makes the electrons less likely to transfer from the valence band to the upper spin sublevel. Accordingly, the depth of the minimum in the scattering signal dependency on the excitation photon energy characterizes the effectiveness of pumping the excitations with photons from the first radiation source as a function of the photon energy.

Apart from the standard channel of elastic light scattering, the spectrum of resonant reflection evidences the development of another scattering channel. The scattering efficiency in this channel increases with the rising number of neutral excitations. Its energy exceeds that of the major scattering channel, forming an electron-hole pair, consisting of a valence hole and an electron at the upper spin sublevel of the zero Landau level of the conductance band, by the calculated energy of a spin one “magneto-graviton” with zero momentum (1.4 meV) (Fig. 5). As the quantity of excitations in the electron system increases, the amplitude of the scattering signal in this channel is amplified nearly by order of magnitude. Note that in this case, the radiation power is raised only in the first radiation source used to pump the neutral excitations, whereas that of the probing source remains unchanged (Fig. 5). Therefore, the more excitations are present in the electron system, the more likely photon backscattering is – a particular property of the Bose statistics27. One of the plausible scenarios for such a scattering channel can be as follows. A photon from the probing radiation source is absorbed in a quantum well, yielding an electron-hole pair and a spin one “magneto-graviton” with zero momentum (Inset to Fig. 1). After that, the electron-hole pair recombines, absorbing the newly created spin one “magneto-graviton” or another spin one “magneto-graviton” occupying the same quantum state, which leads to a scattering signal with the energy and longitudinal momentum of the probing photon (the transverse component to the quantum well plane is not preserved as the translation symmetry in the direction of the heterostructure growth is broken). Given the Bose statistics of spin one “magneto-gravitons,” the probability of this process is proportional to the number of spin one “magneto-gravitons” occupying the same quantum state (scattering probabilities for similar processes are discussed in ref.15, besides the complex valence band structure may facilitate the scattering efficiency in this channel, see Supplementary). Presumably, there are possible alternative descriptions of the observed scattering process, though all of them would be related to forming the system of neutral anyon complexes with Bose statistics27.

The most striking outcome of the resonant reflection measurements concerns the relaxation time of the ensemble of neutral excitations. To measure the relaxation time directly, the first laser source is cut off by an external mechanical shutter (trigger time of 10 µs), and the amplitude change in the resonance reflection signal due to the second (probing) source is registered as a function of the time elapsed from the triggering of the shutter. Despite the constant level of probing radiation power, the signal of resonant reflection is reduced with time. The decay time of the resonance reflection signal exceeds 10 seconds (Fig. 6), which is an absolute record among all the relaxation times of excitations in quantum Hall states known to this day20. Such a prolonged relaxation time provides the grounds for considering this ultra-long-life quasi-equilibrium ensemble (a condensate of neutral anyon complexes with spin one) as a new state of anyon matter. An experimental realization of such a condensate presented in this paper opens up intriguing possibilities of directly manipulating the quasiparticles of this exotic state of matter in real-time.