Signatures of Fractional Exclusion Statistics in the Spectroscopy of Quantum Hall Droplets

Abstract

We show how spectroscopic experiments on a small Laughlin droplet of rotating bosons can directly demonstrate Haldane fractional exclusion statistics of quasihole excitations. The characteristic signatures appear in the single-particle excitation spectrum. We show that the transitions are governed by a "many-body selection rule" which allows one to relate the number of allowed transitions to the number of quasihole states on a finite geometry. We illustrate the theory with numerically exact simulations of small numbers of particles.PACS numbers: 73.43.Cd, 05.30.Pr One of the most dramatic features of strongly correlated phases is the emergence of quasiparticle excitations with unconventional quantum statistics. The archetypal example is the fractional, "anyonic", quantum statistics predicted for the quasiparticles of the fractional quantum Hall phases [1,2]. While experiments on semiconductor devices have shown that these quasiparticles have fractional charges [3][4][5], a direct observation of the fractional statistics has remained lacking.In this Letter we show how precision spectroscopy measurements of rotating droplets of ultracold atoms could be used to demonstrate the Haldane fractional exclusion statistics [6] of quasiholes in the Laughlin state of bosons. By involving only spectroscopic signatures of the rotating droplet, our proposal plays to the strengths of atomic physics experiments. We show that evidence of the fractional exclusion statistics appears in counting the numbers of lines in the radio-frequency (RF) absorption spectrum. In this sense, the method is conceptually similar to classic evidence of quantum statistics, as appearing in the rotational levels of homonuclear diatomic molecules (e.g. the Fermi statistics of the proton causing the rotational levels of H 2 to depend on whether the spins of the nuclei are in singlet or triplet state). Our method differs substantially from proposals to measure the fractional braiding statistics of quasiholes [7][8][9], notably by not requiring local time-dependent potentials for the adiabatic manipulation of the positions of the quasiholes.We have in mind a fast rotating gas of identical bosonic atoms, initially in a single internal (hyperfine) state ⇑, and confined to a quasi-2D layer with oscillator length a z . The gas is subjected to a tight circularly symmetric harmonic trap of frequency ω 0 , with ω 0 V 0 , in which V 0 ≡ 2 π as az ω 0 is a characteristic interaction energy for atoms with scattering length a s . Hence, the interactions leave the particles in the lowest Landau level (LLL) [10].In addition, we shall consider a weak quartic potential -weak compared to both ω 0 and V 0 -for reasons to be described below. Specifically, we shall consider an initial state of N i atoms which has been spun up to the angular momentum L i = N i (N i − 1). Then, for the case of contact repulsive interactions relevant in typical cold gas experiments, the groundstate is the (exact) ν = 1/2 Laughlin state. Furthermore, for the case of contact interactions, the...

Generation and spectroscopic signatures of a fractional quantum Hall liquid of photons in an incoherently pumped optical cavity

“…However, as it was pointed out in [21] not all these transitions have a significant matrix element. Losing a photon from a Laughlin state can in fact be seen as resulting in the creation of two quasi-holes, which imposes the constraint that the largest single-particle angular momentum in the distribution of the extra angular momentum be 2 .…”

“…The same condition can be obtained noting that the largest single-particle angular momentum in the final (N − 1)-particle state cannot exceed that of the initial Laughlin state with N particles. As discussed in detail in [21], this many-body selection rule is not exact; still the matrix elements for non-allowed transitions are orders of magnitude smaller and practically negligible.…”

“…As a first step in this ambitious program, we propose here a relatively simple measurement of the angular-momentum-resolved emission spectrum. Our proposal is inspired by the recent work [21] where RF spectroscopy was proposed as a way to extract information on a small FQH droplet of rotating bosonic atoms from the number of allowed transitions lines. In the optical case under investigation here, the role of RF spectroscopy is played by the spontaneous emission of light due to radiative photon losses through the non-perfect cavity mirrors.…”

“…[21] Fig. 7 shows the N = 3-particle Laughlin state population P (3) FQH as a function of Γ p and Γ e for fixed Γ l /∆ = π × 10 −5 and ω at = ω 0 in the presence of a step potential with radius R/ = 4.15.…”

“…As a second remarkable result, we show how informa-tion on the microscopic physics of the quantum Hall liquid of light can be extracted from an angular-momentumresolved spectroscopy of the light emitted by the cavity: Taking inspiration from a related proposal for rotating atomic gases [21], we show how clear signatures of the fractional exclusion statistics can be obtained by looking at the spectral distribution of the emission lines and at their unique multiplicity structure. The paper is organized as follows.…”