Fermionization of Two Distinguishable Fermions

We study a system of two distinguishable fermions in a 1D harmonic potential. This system has the exceptional property that there is an analytic solution for arbitrary values of the interparticle interaction. We tune the interaction strength and compare the measured properties of the system to the theoretical prediction. For diverging interaction strength, the energy and square modulus of the wave function for two distinguishable particles are the same as for a system of two noninteracting identical fermions. This is referred to as fermionization. We have observed this phenomenon by directly comparing two distinguishable fermions with diverging interaction strength with two identical fermions in the same potential. We observe good agreement between experiment and theory. By adding more particles our system can be used as a quantum simulator for more complex systems where no theoretical solution is available.

Figure 1

Trap setup and sketch of the performed experiment. (a) Our trap consists of an optical potential created by a tight focus of a laser beam and a magnetic field gradient. (b) Deterministic preparation of two fermions in the ground state of a potential well. (c) We measure the tunneling dynamics through a potential barrier for a repulsively interacting system of two distinguishable fermions for various interaction energies. The mean interaction energy per particle is indicated by the parameter $U$. These results are then compared with the tunneling dynamics of two noninteracting identical fermions in the same potential.

Figure 2

Two particles in a 1D harmonic potential. (a) Relative wave function of two interacting fermions (blue [dark gray]) and two identical fermions (green [light gray]) in a 1D harmonic potential. For infinitely strong interaction ($-1/g_{\mid \uparrow \downarrow ⟩}\to 0$) the probability to find the two distinguishable fermions at the same position vanishes. In this case the square modulus of the total wave function of two distinguishable fermions is the same as for two identical fermions. (b) Kinetic energy of the relative motion. The blue [dark gray] and black curves show the energy of two interacting fermions in state $\mid \uparrow \downarrow ⟩$ depending on the coupling strength $g_{\mid \uparrow \downarrow ⟩}$ given in units of $a_{\parallel }=\sqrt{\hslash /\mu {\omega }_{\parallel }}$. The green [light gray] line shows the energy of two identical fermions in state $\mid \uparrow \uparrow ⟩$. The energy is plotted versus $-1/g_{\mid \uparrow \downarrow ⟩}$ for a better comparison with the experimental results. The experimentally studied region is indicated by the dashed rectangle.

Figure 3

Mean number of particles remaining in the potential well. After modifying the initial potential the particles can tunnel through a barrier of fixed height for a certain hold time. Subsequently, tunneling is switched off and the mean particle number left in the potential is recorded by averaging over many experimental realizations. Exponential fits to the data (solid lines) allow us to extract the tunneling time constants of two interacting distinguishable fermions for different interaction strengths (blue [dark gray]) and of two identical fermions (green [light gray]). Each data point is the average of about 70 measurements except for the first and the last data point in each series (about 230 realizations). The errors are the standard errors of the mean.

Figure 4

(a) Tunneling time constants for different values of the 1D coupling strength. The tunneling time constant of two repulsively interacting distinguishable fermions (blue [dark gray] curve) decreases by 2 orders of magnitude with increasing magnetic field. This is attributed to the gain in interaction energy when ramping across the CIR. The tunneling time constant of two noninteracting identical fermions (green [light gray] line) remains unaffected by the magnetic field within our experimental accuracy. At the magnetic field value where both curves cross we identify the fermionization of two distinguishable fermions. The errors are the statistical errors of the fits shown in Fig. 3. The blue [dark gray] line is a guide to the eye. (b) One-dimensional coupling constant $g_{\mid \uparrow \downarrow ⟩}$ with a CIR at $(783.4\pm 0.4)G$. For the calculation we used the perpendicular harmonic oscillator length $a_{\perp }=\sqrt{\hslash /\mu {\omega }_{\perp }}$ of the modified potential.

Figure 5

Interaction energy of two fermions for different interaction regimes. By using a WKB based calculation we can determine the energy of two distinguishable fermions at different interaction strengths (blue points) from the tunneling time constants presented in Fig. 4a. The blue curve shows the expected energy shift for a harmonically trapped system (dashed rectangle in Fig. 2).