## Not Very Many

In physics, the behavior of a system sometimes becomes easier to grasp when the number of particles is large and statistics begin to matter, but knowing how large the system needs to be for that to happen is a challenging computational problem. **Wenz et al.** (p. 457) used a one-dimensional trapped gas of

^{6}Li atoms to study this crossover from few to many. To simplify the problem, they worked with one “impurity” atom that was in a state unlike the other—“majority”—atoms. For weak and intermediate interactions, the system approached the many-body limit with as few as four majority atoms.

## Abstract

Knowing when a physical system has reached sufficient size for its macroscopic properties to be well described by many-body theory is difficult. We investigated the crossover from few- to many-body physics by studying quasi–one-dimensional systems of ultracold atoms consisting of a single impurity interacting with an increasing number of identical fermions. We measured the interaction energy of such a system as a function of the number of majority atoms for different strengths of the interparticle interaction. As we increased the number of majority atoms one by one, we observed fast convergence of the normalized interaction energy toward a many-body limit calculated for a single impurity immersed in a Fermi sea of majority particles.

The ability to connect the macroscopic properties of a many-body system to the microscopic physics of its individual constituent particles is one of the great achievements of physics. This connection is usually made using the assumption that the number of particles tends to infinity. Then a transition from discrete to continuous variables can be made, which greatly simplifies the theoretical description of large systems. When does a system become large enough for this approximation to be valid? This is a difficult question to answer because most calculations based on a microscopic description become prohibitively complex before their predictions approach the many-body solution. Experimentally, this question has been studied in the context of helium droplets (*1*) and nuclear physics (*2*) by measuring the emergence of superfluidity for increasing system size. We addressed this question with the use of ultracold lithium atoms, which have already been used to study few-particle systems with tunable interactions (*3*–*5*).

In our experiments, we control the system size on the single-particle level while maintaining full control over the interparticle interactions. We achieve this by deterministically preparing few-particle systems of ultracold ^{6}Li atoms (*6*) whose interparticle interaction can be tuned using Feshbach resonances (*7*, *8*). This allows us to explore the crossover from few- to many-body physics by studying the fermionic quantum impurity problem, where a single impurity atom interacts with a number of fermionic majority atoms. The majority atoms do not interact with each other because of the Pauli principle. In this system the impurity acts as a test particle, which we use to probe the majority component. The limit of a single majority particle (Fig. 1A) has been thoroughly investigated both theoretically (*9*) and experimentally (*3*, *10*). For a large number of majority atoms, the system can be described as a single impurity interacting with a Fermi sea (Fig. 1C). Similar physics has been experimentally explored by introducing a small fraction of impurity atoms into a large Fermi sea of ultracold atoms (*11*–*13*).

We prepare (*N* + 1)-particle systems consisting of one impurity atom and *N* majority atoms in their *N*-particle ground state trapped in an elongated optical dipole potential (Fig. 2A) (*6*). We create these quasi–one-dimensional few-particle systems using ultracold fermionic ^{6}Li atoms in different hyperfine states, where we label the minority particle as and the atoms of the majority component as . This system is very well described by the Hamiltonian (1)where *x*_{0} is the position of a single impurity atom that interacts with *N* atoms of the majority component, which are located at the coordinates *x*_{1} to *x _{N}*, through a contact interaction of strength

*g*

_{1D};

*ħ*is Planck’s constant divided by 2π;

*m*is the mass of a

^{6}Li atom; and ω

_{||}is the axial frequency of the harmonic trapping potential (

*14*). This Hamiltonian can be solved analytically for the two limiting cases of a single majority particle (

*N*= 1) (

*9*) and an infinite number of majority particles in a homogeneous system (ω

_{||}→ 0) (

*15*).

To probe the system, we measure the interaction energy between the impurity and the majority component as a function of the number of majority particles. We do this by changing the internal state of the single impurity atom from an initial hyperfine state to a different hyperfine state by means of a radio-frequency (RF) pulse (*16*, *17*). If no majority atoms are present, the transition occurs at a frequency ν_{0} corresponding to the hyperfine splitting between the initial and final states of the impurity atom. In the presence of *N* majority atoms, the interactions between the impurity and the majority atoms lead to an interaction shift Δν(*N*) of the transition frequency.

Using such RF spectroscopy measurements, we study systems with weak (*g*_{1D} = 0.36), intermediate (*g*_{1D} = 1.14), and strong (*g*_{1D} = 2.80) repulsive interactions, where *g*_{1D} is given in units of *a*_{||}*ħ*ω_{||} and is the harmonic oscillator length. We perform these measurements for systems containing one to five majority atoms. As an example, the resulting RF spectra for strong interaction are shown in Fig. 2B. Note that a trapped system has discrete eigenstates and thus all RF spectra consist of discrete transitions. Because the resolution of our RF spectroscopy is much higher than the level spacing *ħ*ω_{||} of our trap, we resolve these discrete transitions and therefore observe sharp transitions with symmetric line shapes. Hence, we can fit all these transitions with Gaussians and obtain the interaction energies Δ*E*(*N*) = *h*Δν(*N*) for different coupling strengths *g*_{1D} (Fig. 2C).

The addition of each majority atom increases the number of particles with which the impurity atom can interact, and thus the interaction energy rises for increasing *N*. For weak interactions, it was shown that (*18*), and hence Δ*E* diverges for *N* → ∞. Therefore, we rescale the interaction energy by the natural energy scale of the system, the Fermi energy of the majority atoms *E*_{F}, to obtain the dimensionless interaction energy E = Δ*E/E*_{F}. In a trapping potential, the Fermi energy is determined by the energy of the lowest single-particle level not occupied by a majority atom. Because we consider only the gain in interaction energy, we neglect the zero-point motion, which means that *E*_{F} = *Nħ*ω_{||} for the case of a harmonic trap. For our optical trap, which is slightly anharmonic, we determine the Fermi energy for each *N* by precise measurements of the level spacings and Wentzel-Kramers-Brillouin calculations (*10*, *14*).

To compensate for the change in density caused by an increase in the number of particles, we rescale *g*_{1D} with the line density of the majority atoms. For the trapped systems, we approximate the line density by its peak density given by the Fermi wave vector , which has been shown to be very accurate even for low particle numbers (*18*). As a result, we obtain a dimensionless interaction parameter γ = (π*m/ħ*^{2})*g*_{1D}*/k*_{F} (*19*).

To investigate how the measured interaction energy of our system approaches the many-body limit, we compare our data with theoretical predictions for the two limiting cases of one and an infinite number of majority particles. The normalized interaction energy ε_{2} = Δ*E*_{2}*/E*_{F} of the two-particle system consisting of an impurity atom interacting with only one majority atom (*9*) is shown as a blue line in Fig. 3. In the many-body limit where a single impurity is immersed in a Fermi sea of an infinite number of majority atoms, an explicit expression for the interaction energy in a homogeneous one-dimensional system has been analytically calculated (*15*) to be (2)This interaction energy for the many-body case is shown in Fig. 3 as an orange line.

The two theories coincide for the limits of vanishing and diverging interactions. For γ = 0, the interaction energy is zero. For γ → ∞, one reaches the limit of fermionization where the energy of the impurity atom interacting with *N* majority fermions exactly matches the energy of *N* + 1 noninteracting identical fermions, and therefore the gain in interaction energy is equal to the Fermi energy *E*_{F} (*10*, *15*, *20*–*25*). Between these limits, the many-body solution consistently has a slightly larger normalized interaction energy than the two-body solution.

To compare our data to these strictly one-dimensional theories, we must consider the fact that our quasi–one-dimensional trapping potential has a finite aspect ratio of about 10:1. We estimate this effect on the interaction energy using numerical calculations (*26*–*28*) and find that the largest corrections occur for intermediate and strong interactions, and is on the order of 2% of the interaction energy (*14*). The corrected data are plotted in Fig. 3A as a function of the dimensionless interaction parameter γ; the different colors indicate the number of majority atoms. The measured interaction energy quickly approaches the many-body limit as the number of majority particles increases. To show this in more detail, we subtract the interaction energy given by the two-particle theory ε_{2} from the total interaction energy (Fig. 3B).

For the two-particle system, where the impurity interacts with only one majority atom (*N* = 1), we find excellent agreement between the experimental data (blue dots) and its analytical prediction (blue line). For two or more majority atoms (*N* ≥ 2), there is no analytic prediction for the interaction energy, but recent state-of-the-art numerical calculations exist (*18*, *26*, *29*). We compare our experimental results for *N* = 2 (green dots) to the calculations of (*26*) (green line) and find good agreement. Note that adding only one atom to the two-particle system leads to an increase of the normalized interaction energy, which is already about half of the energy gain expected for the *N* → ∞ limit.

For more than two majority particles, one observes a quick convergence of the experimentally determined interaction energy toward the many-body limit described by the analytical solution in (*15*). For strong interactions, the data points follow the shape of the many-body theory but have a slight offset. Possible reasons for this deviation might be the peak density approximation and the uncertainties in our corrections for the anharmonicity and the finite aspect ratio, which are largest for strong interactions. For weak and intermediate interactions, the determined interaction energy agrees with the many-body theory within the experimental errors for as few as four majority atoms. This fast convergence of the normalized interaction energy shows that even for very few particles, the many-body theory describes essential aspects of our system.

Further insight into the one-dimensional quantum impurity problem can be gained by considering our findings in terms of polaron-like behavior. In general, a polaron is a quasiparticle that consists of an impurity coherently dressed by particle-hole excitations of the Fermi sea (*30*–*32*). In a homogeneous one-dimensional system, polaronic quasiparticles are not expected to exist, but polaron-like behavior for weak interactions can still be observed (*33*). To understand how this behavior emerges for growing *N* requires further measurements of the effective mass, which can be obtained by investigating the excitation spectrum of the system.

For strong interactions, the polaron-like description breaks down in a one-dimensional system, and for *g*_{1D} → ∞ one approaches the limit of fermionization. Then the system’s energy reaches that of a system of (*N* + 1) noninteracting identical fermions (*34*). By crossing this point of fermionization, one can reach a metastable state where the interaction energy is larger than the Fermi energy (*10*), which might lead to the appearance of nontrivial spin correlations (*25*, *29*, *35*). For spin-balanced systems with attractive interactions, our setup could be used to study the emergence of pairing (*36*) and superfluidity in a fermionic few-particle system with tunable interactions and full control over the system’s quantum state.

## Supplementary Materials

www.sciencemag.org/content/342/6157/457/suppl/DC1

Materials and Methods

Figs. S1 to S6

Tables S1 to S3

## References and Notes

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- ↵ The dimensionless interaction parameter is also known as the Lieb Liniger parameter.
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- ↵ Because the Fermi energy is also the only remaining energy scale in a 3D system, we expect a similar convergence to a universal value determined by the Fermi energy times a numerical constant.
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Note that in (
*10*) the harmonic oscillator length*a*_{||}was defined differently and therefore the numerical value of*g*_{1D}differs by a factor of . **Acknowledgments:**We thank D. Blume and S. E. Gharashi for inspiring discussions and for providing the results of their numerical calculations, and R. Schmidt, X. Guan, and J. McGuire for very helpful discussions. I.B. thanks G. E. Astrakharchik for helpful discussions. Supported by the International Max Planck Research School for Quantum Dynamics (A.N.W. and G.Z.), the European Commission–funded Future and Emerging Technologies project Quantum Interferometry with Bose-Einstein Condensates (I.B.), European Research Council starting grant 279697, the Helmholtz Alliance (HA216-EMMI), and the Heidelberg Center for Quantum Dynamics.