Quantum Monte Carlo study of a one-dimensional phase-fluctuating condensate in a harmonic trap

C. Gils, L. Pollet, A. Vernier, F. Hebert, G. G. Batrouni, and M. Troyer

Phys. Rev. A 75, 063631 – Published 29 June 2007

Abstract

We study numerically the low-temperature behavior of a one-dimensional Bose gas trapped in an optical lattice. For a sufficient number of particles and weak repulsive interactions, we find a clear regime of temperatures where density fluctuations are negligible but phase fluctuations are considerable, i.e., a quasicondensate. In the weakly interacting limit, our results are in very good agreement with those obtained using a mean-field approximation. In coupling regimes beyond the validity of mean-field approaches, a phase-fluctuating condensate also appears, but the phase-correlation properties are qualitatively different. It is shown that quantum depletion plays an important role.

Received 22 January 2007

DOI:https://doi.org/10.1103/PhysRevA.75.063631

Authors & Affiliations

C. Gils¹, L. Pollet¹, A. Vernier², F. Hebert², G. G. Batrouni², and M. Troyer¹

¹Institut für Theoretische Physik, ETH Zürich, CH-8093 Zürich, Switzerland
²Institute Non-Linéaire de Nice, UMR CNRS 6618, Université de Nice-Sophia Antipolis, 1361 route des Lucioles, F-06560 Valbonne, France

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Vol. 75, Iss. 6 — June 2007

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Images

Figure 1
(Color online) (i) Thermal regime: As a result of both density and phase fluctuations, the widths of the density profile, $L_{n}$ , and the momentum profile, $L_{k}$ , depend on $β$ with $L_{n} (β) > L_{n}^{sat}$ , $L_{k} (β) > L_{k}^{sat}$ . (ii) Quasicondensate: Density fluctuations are negligible ( $L_{n} = L_{n}^{sat}$ ; $L$ is the half size of the condensate cloud for $β \geq β_{QC}$ ). However, the phase fluctuates, as seen for $L_{k} (β) > L_{k}^{sat}$ , the phase-correlation function $C (0, 0.8 L) < 1$ . (iii) True condensate: Phase fluctuations also disappear as $L_{k} (β)$ approaches $L_{k}^{sat}$ and $C (0, 0.8 L)$ approaches 1. The numerical results are $β_{QC} \approx 0.03$ and $β_{TC} = 0.115 (5)$ (analytical estimate ${\tilde{β}}_{TC} = 0.116$ ).Reuse & Permissions
Figure 2
(Color online) $N = 2000$ , $U = 0.4$ , $V_{t} = 0.3$ $(γ = 0.003)$ . Density fluctuations are suppressed for $β \geq β_{QC} \approx 0.03$ and the density profiles $n_{j} = ⟨ {\hat{n}}_{j} ⟩$ $(\circ)$ are identical for $β \geq β_{QC}$ (b)–(d). In (a), the line is a fit to a Gaussian profile, while in (b)–(d) the lines are fits to a TF profile $n_{0} (1 - j^{2} ∕ L^{2})$ , where $L = 12.6$ $(L_{TF} = 12.6)$ . Error bars in the figures are always smaller than the symbol size.Reuse & Permissions
Figure 3
(Color online) $N = 2000$ , $U = 0.4$ , $V_{t} = 0.3$ . The density profiles $\sqrt{n_{0} n_{j}}$ $(\circ)$ are virtually independent of temperature for $β \geq β_{QC} \approx 0.03$ (b)–(d), while the Green’s function $G (0, j)$ $(▵)$ varies with $β$ . The lines are the analytical estimates, a Gaussian fit for $\sqrt{n_{0} n_{j}}$ in (a), a TF profile $n_{0} \sqrt{1 - j^{2} ∕ L^{2}}$ for $\sqrt{n_{0} n_{j}}$ in (b)–(d) and $\sqrt{n_{0} n_{j}} \exp {- α ∣ \ln [(L - j) ∕ (L + j)] ∣}$ [see Eq. (2)] for $G (0, j)$ .Reuse & Permissions
Figure 4
(Color online) The size $s = β_{TC} ∕ β_{QC}$ of the QC regime depending on the number of particles $N$ [(a)–(c), $U = 0.5$ , $V_{t} = 0.01$ ] and coupling constant $U$ [(d)–(f), $N = 2000$ , $V_{t} = 0.4$ ]. $L_{n} ∕ L_{n}^{sat}$ $(\circ)$ , $L_{k} ∕ L_{k}^{sat}$ $(▵)$ , $C (0, 0.8 L)$ $(\nabla)$ . (a) $γ = 0.08$ , ${\tilde{β}}_{TC} = 3.5$ , $β_{TC} \approx 3.2 (4)$ , $β_{QC} \approx 2.0$ , $s \approx 1.6$ . (b) $γ = 0.04$ , ${\tilde{β}}_{TC} = 2.5$ , $β_{TC} \approx 2.6 (4),$ $β_{QC} \approx 1.0$ , $s \approx 2.6$ . (c) $γ = 0.02$ , ${\tilde{β}}_{TC} = 1.64$ , $β_{TC} \approx 1.6 (1)$ , $β_{QC} \approx 0.3$ , $s \approx 5.3$ . (d) $γ = 0.001$ , ${\tilde{β}}_{TC} = 0.06$ , $β_{TC} \approx 0.06 (2)$ , $β_{QC} \approx 0.04$ , $s \approx 1.5$ . (e) $γ = 0.006$ , ${\tilde{β}}_{TC} = 0.15$ , $β_{TC} \approx 0.15 (5)$ , $β_{QC} \approx 0.03$ , $s \approx 5$ . (f) $γ = 0.17$ , ${\tilde{β}}_{TC} = 0.82$ , NO $β_{TC}$ , $β_{QC} \approx 0.01$ , $s \approx β_{QC}^{- 1} = 100$ . Comparison of (a)–(c), and (d)–(f), respectively, shows that $s$ increases with increasing $N$ and $U$ .Reuse & Permissions
Figure 5
(Color online) $N = 2000$ , $U = 10.0$ , $V_{t} = 0.4$ $(γ = 0.17)$ . The density profile $\sqrt{n_{0} n_{j}}$ $(\circ)$ is invariant for $β \geq β_{QC} \approx 0.1$ , while the Green’s function $G (0, j)$ varies throughout the QC regime. However, $G (0, j)$ does not approach $\sqrt{n_{0} n_{j}}$ (as in Fig. 3). The line is a TF fit to $\sqrt{n_{0} n_{j}}$ with $L = 33.3$ $(L_{TF} = 33.5)$ .Reuse & Permissions
Figure 6
(Color online) Ground state $\sqrt{n_{0} n_{j}}$ $(\circ)$ and $G (0, j)$ $(▵)$ for different $γ$ . The lines are fits to a TF profile with (a) $L = 35.2$ $(L_{TF} = 35.6)$ and (b) $L = 48.0$ $(L_{TF} = 48.3)$ . For increasing $γ$ , the correlation hole becomes more pronounced, and thus the quantum depletion increases.Reuse & Permissions

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