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We note that the definition, Eq. 6, for the momentum distribution is strongly dependent on the considered size L in the case of a trapped system, where all the particles are confined in a finite region of space and L can far exceed this region. Throughout the paper we specify the size of the overall lattice used to host the trapped system, and in any case this size has always been chosen so as to be of the order of the trapped atomic cloud.
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In fact, Δ Ep-h overestimates in principle the interaction energy, because it contains the interaction of an extra particle with a background of N particles, instead of N-1 as in a system with fixed particle number N.
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In fact, Δ Ep-h overestimates in principle the interaction energy, because it contains the interaction of an extra particle with a background of N particles, instead of N-1 as in a system with fixed particle number N.
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In this respect it is worth noticing that, in the bulk system, low-energy particle-hole excitations can correspond to the transfer of a quasiparticle between two arbitrarily distant points. This is best understood in the classical limit J→0, where low-energy particle-hole excitations simply correspond to displacing a particle between two sites at quasidegenerate values of the local chemical potential. The periodicity of a truly incommensurate local chemical potential is by definition infinity, so that pairs of sites which are closest in energies can find themselves at opposite ends of the system. Nonetheless, other accidental, nonsystematic quasidegeneracies at shorter distances (from the quasiperiod up) can be expected, either associated with the external potential alone (possibly including the trapping one) or with the sum of the potential and of the interparticle interactions.
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In this respect it is worth noticing that, in the bulk system, low-energy particle-hole excitations can correspond to the transfer of a quasiparticle between two arbitrarily distant points. This is best understood in the classical limit J→0, where low-energy particle-hole excitations simply correspond to displacing a particle between two sites at quasidegenerate values of the local chemical potential. The periodicity of a truly incommensurate local chemical potential is by definition infinity, so that pairs of sites which are closest in energies can find themselves at opposite ends of the system. Nonetheless, other accidental, nonsystematic quasidegeneracies at shorter distances (from the quasiperiod up) can be expected, either associated with the external potential alone (possibly including the trapping one) or with the sum of the potential and of the interparticle interactions.
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More quantitatively, a one-dimensional quasicondensate should behave as a Luttinger liquid, for which the coherent fraction n (k=0) scales with the number of particles as N1-K/2, where K is the Luttinger exponent; the well-known upper bound for K in a disordered potential, K<2/3, imposes that the coherent fraction cannot grow slower than N2/3. Hence finding a coherent fraction n (k=0) N2/3 is a strong indication of the absence of quasicondensation. In the considered example, we have N 127, N 2/3 25, and n (k=0) 2.2 (see Fig. 20).
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