Nonequilibrium and local detection of the normal fraction of a trapped two-dimensional Bose gas

We propose a method to measure the normal fraction of a two-dimensional Bose gas, a quantity that generally differs from the noncondensed fraction. The idea is based on applying a spatially oscillating artificial gauge field to the atoms. The response of the atoms to the gauge field can be read out either mechanically from the deposited energy into the cloud or optically from the macroscopic optical properties of the atomic gas. The local nature of the proposed scheme allows one to reconstruct the spatial profile of the superfluid component; furthermore, the proposed method does not require having established thermal equilibrium in the gas in the presence of the gauge field. The theoretical description of the system is based on a generalization of the Dum-Olshanii theory of artificial gauge fields to the interacting many-body context. The efficiency of the proposed measurement scheme is assessed by means of classical field numerical simulations. An explicit atomic level scheme minimizing disturbing effects such as spontaneous emission and light shifts is proposed for ${}^{87}\mathrm{Rb}$ atoms.

Figure 1

Scheme of the setup under consideration. (Left panel) Generic sketch of the $\Lambda$ configuration of atomic levels and laser beams involved in the optical processes. (Center panel) View from above of the two-dimensional atomic pancake (lying within the dashed circle) and of the laser beams (the hatched disk is the spot of the probe beams in the $xy$ plane). (Right panel) Side view. ${\mathbf{k}}_c$ and ${\mathbf{k}}_p^{\pm }$ are the wave vectors of the coupling and probe beams, respectively. In practice, ${\mathbf{k}}_p^{\pm }={(k_p^2-q^2/4)}^{1/2}{\mathbf{e}}_z\pm (q/2){\mathbf{e}}_x$ with $q\ll k_p$, where ${\mathbf{e}}_i$ is the unit vector along axis $i$.

Figure 2

Quantum Bogoliubov prediction (31) for the effective normal fraction $f_n^{\mathrm{eff}}$. In (a), dependence of $f_n^{\mathrm{eff}}$ on the switch-off rate $\gamma$ of the gauge field for three different values of the gauge-field spatial-modulation wave vector $\mathbf{q}=(2\pi /L){\mathbf{e}}_x$ (solid line, $q\xi \simeq 0.1$), $\mathbf{q}=7\times (2\pi /L){\mathbf{e}}_x$ (dotted line, $q\xi \simeq 0.7$) and $\mathbf{q}=10\times (2\pi /L){\mathbf{e}}_x$ (dashed line, $q\xi \simeq 1$). System parameters: square box (with periodic boundary conditions) of size $L/\xi \simeq 63$ containing $N\simeq 40000$ particles with interaction constant $g=0.1{\hslash }^2/m$ at a temperature $T/T_d=0.1$. $k_B{T}_d=2\pi {\hslash }^2\rho /m$ is the degeneracy temperature. The healing length $\xi$ of the gas is defined by ${\hslash }^2/\left(m{\xi }^2\right)=\rho g$ and the Bogoliubov sound velocity is defined as $c_s={(\rho g/m)}^{1/2}$. The dashed curve corresponds to the same value of $q\xi \simeq 1$ as in Fig. 4 and the vertical dotted lines indicate the values of $\gamma /\left(c_{s}q\right)$ considered in that figure. The thin horizontal line is the prediction (34) of the quantum Bogoliubov theory in the thermodynamic limit. Interestingly, for the considered atom number, the finite-size thermodynamic formula [Eq. (39)] evaluated with the quantum Bogoliubov theory gives already the same value at the scale of the figure [35]. The quadratic rather than linear dependence of $f_n^{\mathrm{eff}}$ on $\gamma$ for small values of $\gamma$ is a finite-size effect. The fact that in (a), the values of $f_n^{\mathrm{eff}}$ for the dotted curve are close to the arithmetic mean of the solid and dashed curves, for common values of $\gamma /\left(c_{s}q\right)$, suggests that $f_n^{\mathrm{eff}}$ is roughly an affine function of ${\left(q\xi \right)}^2$ for fixed $\gamma /\left(c_{s}q\right)$. This guess is substantiated in panel (b) where the dependence of the effective normal fraction on the modulation wave vector $\mathbf{q}=q{\mathbf{e}}_x$ is shown for $\gamma /\left(c_{s}q\right)=0.2$ [the crosses are the Bogoliubov predictions and the line is a linear fit of $f_n^{\mathrm{eff}}$ as a function of $q^2$].

Figure 3

Classical field simulation of the normal fraction $f_n$ for a finite-size two-dimensional, spatially homogeneous interacting Bose gas as a function of temperature. The calculation has been performed using the thermodynamic expression (39) for the normal fraction. The coupling constant is $g=0.1{\hslash }^2/m$. The different curves refer to simulations performed with different system sizes $L$, resulting in different numbers of grid points $M={16}^2$ (dotted green line), ${32}^2$ (dashed red line), and ${64}^2$ (solid black line) [the grid spacing is $b\simeq 0.99\lambda$, where $\lambda$ is the thermal de Broglie wavelength; see the text]. The number of classical field realizations is $n_{\mathrm{real}}=1000$. The dot-dashed line is the classical field prediction for the normal fraction $f_n$ of an ideal gas in the thermodynamic limit: As discussed in the text, the decrease at high temperatures is an artifact of the classical field model. The blue circle indicates the result of a numerical simulation of the deposited energy scheme as in Fig. 4 for a homogeneous system with an infinite beam waist $w=\infty$ (in which case we simply dropped the nonmodulated term in the gauge field), a number of grid points $M={32}^2$, a gauge-field modulation wave vector $\mathbf{q}=(2\pi /L){\mathbf{e}}_x\simeq (0.2/\xi ){\mathbf{e}}_x$, and including a careful extrapolation of ${\epsilon }_{\mathrm{gauge}}\to 0$ [numerics down to ${\epsilon }_{\mathrm{gauge}}/{\left(m{k}_B{T}_d\right)}^{1/2}=0.01$] and of $\gamma \to 0$ [numerics down to $\gamma /\left(c_{s}q\right)=0.05$, where $c_s={(\rho g/m)}^{1/2}$ is the Bogoliubov sound velocity].

Figure 4

Classical field simulation of the deposited energy measurement scheme. Initial thermal equilibrium state at $T/T_d=0.1$ with $g=0.1{\hslash }^2/m$ and a system size $L/\xi \simeq 63$ (the number of grid points is $M={64}^2$, and $n_{\mathrm{real}}=1000$ independent realizations of the classical field are used). Gauge-field modulation wave vector $\mathbf{q}\simeq (1/\xi ){\mathbf{e}}_x$. The real-time evolution is followed during $\tau =3/\gamma$. The complete form Eq. (21) of the gauge field is considered, with ${\mathbf{r}}_0$ located in the center of the system, and a beam waist $w=30\xi$. For each value of the gauge-field intensity ${\epsilon }_{\mathrm{gauge}}$ of Eq. (25), two calculations are performed to extract the energy change $\Delta E_2$ due to the spatially modulated component, from which an approximate value of the normal fraction $f_n$ is obtained (see text): the first calculation with $|{\Omega }_p^{+}{|}^2\left(0^{+}\right)/{\left|{\Omega }_c\right|}^2={\epsilon }_{\mathrm{gauge}}/\left(\hslash k_c\right)$ and ${\Omega }_p^{-}\equiv 0$ and the second one with $|{\Omega }_p^{\pm }{|}^2\left(0^{+}\right)/{\left|{\Omega }_c\right|}^2={\epsilon }_{\mathrm{gauge}}/\left(2\hslash k_c\right)$. (Open squares with error bars) Spatially homogeneous system of size $L$ (with periodic boundary conditions), which corresponds to $N\simeq 40000$ atoms. (Solid circles with error bars) System in a circular potential well (see text), with a total atom number $N\simeq 30000$ adjusted to have the same central density $\rho$, hence, the same degeneracy temperature $T_d$ and healing length $\xi$ as in the homogeneous case. The suggested experimental values of Table correspond to the circled point indicated by the oblique arrow and require the measurement of a relative energy change of $\simeq$0.7$\%$. (Thin lines) Quadratic fit (with no linear term; see the text) of the effective $f_n$ as a function of ${\epsilon }_{\mathrm{gauge}}$, over the interval $0.05\le {\epsilon }_{\mathrm{gauge}}/{\left(m{k}_B{T}_d\right)}^{1/2}<0.2$ (dashed lines for the spatially homogeneous system and solid lines for the system in the potential well). Black, red, and blue lines and points (from bottom to top) correspond to an excitation sequence with $\gamma /\left(c_{s}q\right)=0.4$, 0.2, and $0.1$, respectively, with $c_s={(\rho g/m)}^{1/2}$ the Bogoliubov sound velocity. The extrapolated zero-${\epsilon }_{\mathrm{gauge}}$ values of the effective normal fraction are indicated in the figure. The region between the horizontal dotted lines indicates the confidence interval of the thermodynamic prediction shown in Fig. 3. The residual $14\%$ deviation of $f_n$ from the thermodynamic value (after linear extrapolation to $\gamma =0$) is a finite $q\xi$ effect (see text).

Figure 5

Scheme of the scattering geometry under examination in Sec. 5b. Probe light is incident at wave vector ${\mathbf{k}}_p=k_p{\mathbf{e}}_z$ and the scattered light is collected at a wave vector ${\mathbf{k}}_{\mathrm{sc}}={(k_p^2-Q^2)}^{1/2}{\mathbf{e}}_z+\mathbf{Q}$.

Figure 6

Scheme of the internal levels of ${}^{87}\mathrm{Rb}$ atom involved in the $D_1$ transition ($J=1/2\to J^{\prime }=1/2$). The hyperfine splitting of the ground (excited) state is ${\Delta }_{\mathrm{hf},g}=2\pi 6.834$ GHz (${\Delta }_{\mathrm{hf},e}=2\pi 814$ MHz). The natural linewidth of the excited state is $\Gamma =2\pi 5.75$ MHz. Thin (thick) lines indicate transitions that are induced by the probe (coupling) beams. Solid (dashed) lines indicate the desired (main undesired) transitions. The relevant dipole matrix elements are shown in units of the reduced dipole element $\langle J=1/2\left|\right|er\left|\right|J^{\prime }=1/2\rangle$ of the $D_1$ line, in bold for the desired transitions and in italic for the undesired ones. Our first choice for the $\Lambda$ system $|a\rangle ,|e\rangle ,|b\rangle$ states is shown, in the left (a) panel, with $|a\rangle =|F=1,m_F=-1\rangle$, $|e\rangle =|F^{\prime }=2,m_{F^{\prime }}=-1\rangle$, $|b\rangle =|F=2,m_F=-2\rangle$. The second choice is shown in the right (b) panel, with $|a\rangle =|F=2,m_F=-2\rangle$, $|e\rangle =|F^{\prime }=2,m_{F^{\prime }}=-2\rangle$, $|b\rangle =|F=1,m_F=-1\rangle$. Note that we have taken the $y$ axis as the quantization axis of angular momenta.