## Abstract

Systems consisting of few interacting fermions are the building blocks of matter, with atoms and nuclei being the most prominent examples. We have created a few-body quantum system with complete control over its quantum state using ultracold fermionic atoms in an optical dipole trap. Ground-state systems consisting of 1 to 10 particles are prepared with fidelities of ∼90%. We can tune the interparticle interactions to arbitrary values using a Feshbach resonance and have observed the interaction-induced energy shift for a pair of repulsively interacting atoms. This work is expected to enable quantum simulation of strongly correlated few-body systems.

The exploration of naturally occurring few-body quantum systems such as atoms and nuclei has been extremely successful, largely because they could be prepared in well-defined quantum states. Because these systems have limited tunability, researchers created quantum dots—“artificial atoms”—in which properties such as particle number, interaction strength, and confining potential can be tuned (*1*, *2*). However, quantum dots are generally strongly coupled to their environment, which hindered the deterministic preparation of well-defined quantum states.

In contrast, ultracold gases provide tunable systems in a highly isolated environment (*3*, *4*). They have been proposed as a tool for quantum simulation (*5*, *6*), which has been realized experimentally for various many-body systems (*7*–*10*). Achieving quantum simulation of few-body systems is more challenging because it requires complete control over all degrees of freedom: the particle number, the internal and motional states of the particles, and the strength of the interparticle interactions. One possible approach to this goal is using a Mott insulator state of atoms in an optical lattice as a starting point. In this way, systems with up to four bosons per lattice site have been prepared in their ground state (*11*, *12*). Recently, single lattice sites have been addressed individually (*13*).

In single isolated trapping geometries, researchers could suppress atom number fluctuations by loading bosonic atoms into small-volume optical dipole traps (*14*–*18*). However, these experiments were not able to gain control over the system’s quantum state.

We prepare few-body systems consisting of 1 to 10 fermionic atoms in a well-defined quantum state, making use of Pauli’s principle, which states that each single-particle state cannot be occupied by more than one identical fermion. Therefore, the occupation probability of the lowest-energy states approaches unity for a degenerate Fermi gas, and we can control the number of particles by controlling the number of available single-particle states. We realize this by deforming the confining potential such that quantum states above a well-defined energy become unbound. This approach requires a highly degenerate Fermi gas in a trap whose depth can be controlled with a precision much higher than the separation of its energy levels.

To fulfill these requirements, we use a small-volume optical dipole trap with large level spacing. This microtrap is created by the focus of a single laser beam (Fig. 1) with a waist of *w*_{0} ≲ 1.8 μm and measured radial and axial trapping frequencies (ω_{r},ω_{a}) = 2π × (14.0 ± 0.1,1.487 ± 0.010) kHz (*19*). We load the microtrap from a reservoir of cold atoms. The reservoir consists of a two-component mixture of ^{6}Li atoms in the two lowest-energy Zeeman substates |*F* = 1/2, *m _{F}* = +1/2〉 and |

*F*= 1/2,

*m*= −1/2〉 (labeled state |1〉 and |2〉) in a large-volume optical dipole trap. The reservoir has a degeneracy of

_{F}*T*/

*T*

_{F}≈ 0.5 (

*19*), where

*T*

_{F}is the Fermi temperature. We superimpose the microtrap with the reservoir and transfer about 600 atoms into the microtrap. After removal of the reservoir, the degeneracy of the system is determined by

*T*

_{F}≈ 3 μK in the microtrap and the temperature

*T*≲ 250 nK of the reservoir (

*20*). Assuming thermal equilibrium between the microtrap and the reservoir, this corresponds to

*T*/

*T*

_{F}≲ 0.08. According to Fermi-Dirac statistics, this yields an occupation probability for the lowest state exceeding 0.9999, which is large enough not to constrain our preparation scheme.

To spill the excess atoms from the microtrap, we add a linear potential in the axial direction by applying a magnetic field gradient. To obtain the same potential for both components, we apply the gradient at a large magnetic offset field where the difference in the magnetic moments of states |1〉 and |2〉 is negligible. A particular magnetic field is then chosen so that the interaction strength between atoms in the different states vanishes because of a nearby Feshbach resonance (fig. S2). By varying the depth of the microtrap and the strength of the magnetic field gradient, we can control the number of bound states in the potential (Fig. 2A). If fewer than 10 bound states remain, the system is essentially one-dimensional because of the approximate 1:10 aspect ratio of the trap; consequently, each energy level is occupied by one atom per spin state. During the spilling process, we adiabatically tilt the potential, wait to let atoms in unbound states escape, and then ramp the potential back up (*19*).

To probe the prepared systems, it is necessary to measure the number of atoms in the microtrap with single-atom resolution and near-unity fidelity. We achieve this by releasing the atoms from the microtrap, recapturing them in a compressed magneto-optical trap, and then recording their fluorescence with a charge-coupled device (CCD) camera (Fig. 1). With this technique, we can count the total number of atoms in the magneto-optical trap with a fidelity exceeding 99% for 1 to 10 atoms (fig. S1).

Figure 2 shows the mean atom number and its variance as a function of the minimum microtrap depth during the spilling process. The atom number shows a step-like dependence on the trap depth with plateaus for even atom numbers. These plateaus appear at trap depths where the potential barrier for atoms in the uppermost level becomes so low that these atoms leave the trap on a time scale that is shorter than the duration of the spilling process. A simple estimation (*21*) shows that the lifetime of this state can be up to three orders of magnitude shorter than the lifetime of the lower states. When an appropriate trap depth is chosen, the fluctuations in the atom number are as low as *var*/〈*N*〉 = σ^{2}/〈*N*〉 = 0.017 for eight atoms, corresponding to a suppression of 18 dB compared to a system obeying Poissonian statistics. We can then calculate an upper bound for the degree of degeneracy in the microtrap of *T*/*T*_{F} < 0.19 by assuming that all fluctuations result from holes in the Fermi distribution; this provides a complementary method to probe the degeneracy of the lowest-energy states of an ultracold Fermi gas, which is conceptually related to recent studies of antibunching in degenerate Fermi gases (*22*, *23*).

To estimate the probability of finding the system in its ground state after the spilling process, we bin the measured fluorescence signal into a histogram (Fig. 3B). For the preparation of systems consisting of two fermions, we obtain a fidelity of 96(1)%. The error is the statistical error calculated by assuming that the occurrence of samples with undesired atom number follows a Poissonian distribution. From combinatorial considerations (*19*), we deduce that only a negligible fraction of the prepared two-particle systems are not in the ground state before we ramp the potential back up at the end of the preparation process. To check whether we create excitations in the system by ramping up the potential, we perform the spilling process a second time. After the second spilling process, we measure a fidelity of 92(2)% for preparing two atoms. This yields an upper bound of 6(2)% for the excitation probability during the potential ramps (*19*). If we assume the same excitation probability for ramping up and down, we get an estimated fidelity of 93(2)% to prepare the system in its ground state after ramping the potential back up after the first spilling process. For eight atoms, we find a ground-state preparation probability of 84(2)%. By varying the time between the two spilling processes, we found the 1/*e*-lifetime of the prepared two-particle system in its ground state to be ∼60 s, which shows the high degree of isolation from the environment.

To realize configurations with an arbitrary imbalance in the number of atoms in state |1〉 and |2〉, we prepare balanced systems and perform a second spilling process that only removes atoms in state |1〉. We do this by changing the value of the magnetic offset field to 40 G where atoms in state |2〉 have negligible magnetic moment and are therefore unaffected by the magnetic field gradient (fig. S2). Using this technique, we have created imbalanced systems with fidelities similar to those in the balanced case (*19*).

Precise control over the trapping potential is not only essential to prepare few-body samples, it is also an effective tool to probe strongly interacting systems. We use it to explore one of the simplest nontrivial few-body systems: two nonidentical fermions with repulsive interactions in the ground state of a one-dimensional harmonic trap. We first prepare a noninteracting pair of atoms in states |1〉 and |2〉 in the ground state of the trap. Then we lower the potential barrier such that the two atoms slowly escape the trap on a time scale of τ = 630 ± 120 ms, which we measure by recording the decrease in the mean atom number as a function of hold time (Fig. 4B).

To repeat the measurement for two atoms with repulsive interaction, we tune the scattering length *a* to a large positive value, *a* = 4100 *a*_{0}, where *a*_{0} is the Bohr radius, using a Feshbach resonance. In this case, we observe a much higher initial loss rate followed by a slow decay. This fast initial loss can be explained by the energy shift *U* of the ground state due to repulsive interactions, which effectively decreases the height of the potential barrier (Fig. 4A). Because the system is quasi–one-dimensional for the lowest 10 axial states, one has to consider the radial confinement (*24*) for the calculation of *U* (*25*). Given the trap parameters, one expects a shift *U* on the order of half the axial level spacing per particle. After one of the atoms has left the trap, the interaction energy drops to zero, leaving the remaining atom in the unperturbed ground state of the potential. Within our measurement accuracy, we measure an equal probability for the remaining atom to be in state |1〉 and |2〉. By developing theoretical models for these interaction-induced dynamics, one can use this method to quantitatively study strongly interacting systems.

The system we created is well suited for quantum simulation with fully controlled few-body systems. For attractive interactions, it can be used to study Bardeen-Cooper-Schrieffer (BCS)–like pairing in finite systems, which is a model used for the description of nuclei (*26*). Splitting a trap containing a repulsively interacting pair of atoms into a double well creates entangled pairs of neutral atoms, which can be used for quantum information processing (*27*, *28*). By transferring the prepared ground-state samples into a periodic potential (*29*), our system can be used to overcome one of the current major challenges of studying quantum many-body physics with ultracold atoms: preparing systems with sufficiently low entropy to explore phenomena such as quantum magnetism.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/332/6027/336/DC1

Materials and Methods

Figs. S1 and S2

References

## References and Notes

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**Acknowledgments.**We thank M. G. Raizen for inspiring discussions and M. Weidemüller for the loan of a fiber laser. This work was supported by the Helmholtz Alliance HA216/EMMI and the Heidelberg Center for Quantum Dynamics. We thank J. Ullrich and his group for their support. G.Z. and A.N.W. acknowledge support by the IMPRS-QD.