Quantum-fluctuation-induced time-of-flight correlations of an interacting trapped Bose gas

We investigate numerically the momentum correlations in a two-dimensional, harmonically trapped interacting Bose system at $T=0$ temperature, by using a particle number preserving Bogoliubov approximation. Interaction-induced quantum fluctuations of the quasicondensate lead to a large anticorrelation dip between particles of wave numbers $\mathbf{k}$ and $-\mathbf{k}$ for $\left|\mathbf{k}\right|\sim 1/R_c$, with $R_c$ the typical size of the condensate. The anticorrelation dip found is a clear fingerprint of coherent quantum fluctuations of the condensate. In contrast, for larger wave numbers, $\left|\mathbf{k}\right|\gg 1/R_c$, a weak positive correlation is found between particles of wave numbers $\mathbf{k}$ and $-\mathbf{k}$, in accordance with the Bogoliubov result for homogeneous interacting systems.

Figure 1

Sketch of the origin of quantum-fluctuation-induced quasiparticle correlations in a trap. Even at $T=0$, interaction-induced quantum fluctuations of the condensate induce virtual quasiparticle excitations, and amount to fluctuations and correlations, measurable through ToF experiments. The pair structure of excitations induces positive correlation between particles with opposite wave numbers $\mathbf{k}$ and $-\mathbf{k}$.

Figure 2

Radial part of the dimensionless Bogoliubov eigenfunctions $R_c{u}_{n,m}\left(\mathbf{x}\right), R_c{v}_{n,m}\left(\mathbf{x}\right)$ plotted as a function of the dimensionless radial coordinate $y=\left|\mathbf{x}\right|R_c$ for $(n,m)=(50,0)$ (top) and $(n,m)=(40,20)$ (bottom), corresponding to excitation energies ${\varepsilon }_{50,0}/\mu =1.6$ and ${\varepsilon }_{40,20}/\mu =1.5$, respectively. Here $R_c=\sqrt{2\mu /\left(m{\omega }^2\right)}$ is the typical size of the condensate, ${\zeta }^{-1}=2\mu /\left(\hslash \omega \right)=100$, and $\mu R_c^2/g=1250$, corresponding to $N=1962$ particles and $\langle \delta \widehat{N}\rangle =608$. In the top figure, the dimensionless single mode condensate wave function ${\varphi }_0$ is also displayed. The anomalous part $v_{n,m}$ is nonzero only in the regime of the condensate, while the normal part $u_{n,m}$ of the wave function can be more extended. For $m\ne 0$ both $u_{n,m}\to 0$ and $v_{n,m}\to 0$ at the center of the trap.

Figure 3

Radial part of the dimensionless Fourier-transformed Bogoliubov eigenfunctions $u_{n,m}\left(\mathbf{k}\right)/R_c, v_{n,m}\left(\mathbf{k}\right)/R_c$ as a function of the dimensionless wave number $\left|\mathbf{k}\right|R_c$ for $(n,m)=(50,0)$ and $(n,m)=(40,20)$, corresponding to excitation energies ${\varepsilon }_{50,0}/\mu =1.6$ and ${\varepsilon }_{40,20}/\mu =1.5$, respectively. Here $R_c=\sqrt{2\mu /\left(m{\omega }^2\right)}$ is the typical size of the condensate, ${\zeta }^{-1}=2\mu /\left(\hslash \omega \right)=100$, and $\mu R_c^2/g=1250$, corresponding to $N=1962$ particles and $\langle \delta \widehat{N}\rangle =608$. The anomalous component $v_{n,m}\left(\mathbf{k}\right)$ has a well-defined peak at wave number $|{\mathbf{k}}_{\mathrm{peak}}|$ and vanishes for lower $\left|\mathbf{k}\right|$, while the normal part $u_{n,m}\left(\mathbf{k}\right)$ is extended in momentum space.

Figure 4

Dimensionless expectation values $\langle {\widehat{n}}_{\mathbf{k}}\rangle /l_0^2$ as a function of $\left|\mathbf{k}\right|l_0$ for $N=1962$ and for dimensionless interaction strengths $\tilde{g}=1$ and 4, corresponding to $\langle \delta \widehat{N}\rangle =145$ and 608. Dotted lines represent contributions of noncondensed particles $\langle \delta {\widehat{n}}_{\mathbf{k}}\rangle /l_0^2$, with $l_0=\sqrt{\hslash /\left(m\omega \right)}$, multiplied by a factor of 50 for better visibility. The extension of the condensate increases with increasing $\tilde{g}$, and the peak in $\langle {\widehat{n}}_{\mathbf{k}}\rangle$ gets narrower. The long tail quasiparticle contributions $\langle \delta {\widehat{n}}_{\mathbf{k}}\rangle$ get more pronounced with increasing $\tilde{g}$.

Figure 5

Scaling collapse of $\langle \delta {\widehat{n}}_{\mathbf{k}}\rangle /R_c^2$, plotted as a function of $\mathbf{k}{\xi }_h$ for different $\zeta =\hslash \omega /\left(2\mu \right)$'s, while keeping $\tilde{g}=4$ and $\rho \left(0\right)$ constant. Here $R_c=\sqrt{2\mu /\left(m{\omega }^2\right)}$ is the typical size of the condensate, ${\xi }_h=\hslash /\sqrt{m\mu }$ is the healing length with $\mu =g\rho \left(0\right)$, and we used ${\zeta }^{-1}=25, {\zeta }^{-1}=50$, and ${\zeta }^{-1}=100$, corresponding to $(N,\langle \delta \widehat{N}\rangle )=(121,34), (N,\langle \delta \widehat{N}\rangle )=(489,145)$, and $(N,\langle \delta \widehat{N}\rangle )=(1962,608)$, respectively. The homogeneous momentum distribution, Eq. (17), is also plotted for comparison, yielding good agreement with the common envelope function traced out by $\langle \delta {\widehat{n}}_{\mathbf{k}}\rangle /R_c^2$ as $\omega$ decreases. Inset: Noncondensed contribution $\langle \delta {\widehat{n}}_{\mathbf{k}}\rangle /R_c^2$, plotted as a function of $\mathbf{k}{R}_c$ for $\tilde{g}=4$ and ${\zeta }^{-1}=25$, using the logarithmic scale on both axes. Homogeneous distribution, Eq. (17), is also shown. For large wave numbers $\left|\mathbf{k}\right|\gg 1/{\xi }_h$, the universal power law decay $\sim 1/{\left|\mathbf{k}\right|}^4$ is recovered.

Figure 6

Different contributions to dimensionless diagonal and off-diagonal correlation functions $C(\mathbf{k},\mathbf{k})/l_0^4$ and $C(\mathbf{k},-\mathbf{k})/l_0^4$, plotted as a function of dimensionless wave number $\left|\mathbf{k}\right|l_0$ for fixed $N=1962$ and for two different interaction strengths $\tilde{g}=1$ and 4. Here $l_0=\sqrt{\hslash /\left(m\omega \right)}$ is the oscillator length, and the interaction values correspond to $\langle \delta \widehat{N}\rangle =138$ and 608, respectively. The condensate-quasiparticle contribution $C^{\left(1\right)}$ gives a positive peak in diagonal correlations, but gets negative in the off diagonal, expressing that quantum fluctuations deplete the condensate. As in a homogeneous system, the quasiparticle-quasiparticle correlation $C^{\left(2\right)}$ is positive both in the diagonal and in the off diagonal. However, this contribution is much smaller than $C^{\left(1\right)}$ for wave numbers of the order of $1/R_c$. The amplitude of the correlations $C^{\left(1\right)}$ and $C^{\left(2\right)}$ increases with increasing interaction strength, as the hybridization of the condensate with virtual excitations gets more pronounced.

Figure 7

Different contributions to dimensionless off-diagonal correlation function $C(\mathbf{k},-\mathbf{k})/l_0^4$, plotted as a function of dimensionless wave number $\left|\mathbf{k}\right|l_0$ for particle number $N=1962$ and interaction strength $\tilde{g}=4$, using the logarithmic scale on the vertical axis. Here $l_0=\sqrt{\hslash /\left(m\omega \right)}$ is the oscillator length, and the interaction corresponds to $\langle \delta \widehat{N}\rangle =608$. The background signal $\langle {\widehat{n}}_{\mathbf{k}}\rangle \langle {\widehat{n}}_{-\mathbf{k}}\rangle /l_0^4$ shows a steep decrease due to the disappearance of the condensate wave function, followed by a slower decay as an effect of noncondensed particles. The condensate-quasiparticle contribution $C^{\left(1\right)}$ is constrained to the regime of the single mode condensate, and converges to zero rapidly for $\left|\mathbf{k}\right|\gg 1/R_c$. The quasiparticle-quasiparticle correlation $C^{\left(2\right)}$ gives a slowly decaying tail, dominating the correlation function for $\left|\mathbf{k}\right|\gg 1/R_c$.

Figure 8

Dimensionless correlation functions $C^{\left(1\right)}(\mathbf{k},{\mathbf{k}}^{\prime })/l_0^4$ and $C^{\left(2\right)}(\mathbf{k},{\mathbf{k}}^{\prime })/l_0^4$ plotted as a function of dimensionless wave number $\mathbf{k}{l}_0$, for fixed values of ${\mathbf{k}}^{\prime }$. Here $l_0=\sqrt{\hslash /\left(m\omega \right)}$ is the oscillator length, and we have used ${\zeta }^{-1}=100$ and $\tilde{g}=4$, corresponding to $N=1962$ particles and $\langle \delta \widehat{N}\rangle =608$. First row: ${\mathbf{k}}^{\prime }{l}_0=(0.16,0)$. The condensate-quasiparticle correlation $C^{\left(1\right)}$ is positive if $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$ point to the same direction, and gives negative contribution in the ${\mathbf{k}}^{\prime }\approx -\mathbf{k}$ regime. The positive correlation $C^{\left(2\right)}$ is concentrated to small $\mathbf{k}{l}_0$ wave numbers, due to subleading corrections to condensate-quasiparticle correlations contained in $C^{\left(1\right)}$. Second row: ${\mathbf{k}}^{\prime }{l}_0=(2,0)$. The dominant contribution here is the quasiparticle-quasiparticle correlation $C^{\left(2\right)}$, giving negative values for small wave numbers, and narrow positive peaks around $\mathbf{k}={\mathbf{k}}^{\prime }$ and $-{\mathbf{k}}^{\prime }$, expressing correlations in the noncondensed fraction of the gas.