Cluster formation in two-component Fermi gases

TL;DR

This paper investigates the conditions under which cluster states form in two-component Fermi gases, revealing that a sufficiently large effective range of interaction leads to clustering, with implications for experimental observation.

Contribution

It identifies a universal threshold for cluster formation based on the effective range of interactions, advancing understanding of few-body physics in Fermi gases.

Findings

Cluster states form when the effective range exceeds ~0.46 times the scattering length.

Structural differences between cluster and gas-like states are characterized.

The study provides conditions for experimental detection of cluster states.

Abstract

Two-component fermions are known to behave like a gas of molecules in the limit of Bose-Einstein condensation of diatomic pairs tightly bound with zero-range interactions. We discover that the formation of cluster states occurs when the effective range of two-body interaction exceeds roughly $0.46$ times the scattering length, regardless of the details of the short-range interaction. Using explicitly correlated Gaussian basis set expansion approach, we calculate the binding energy of cluster states in trapped few-body systems and show the difference of structural properties between cluster states and gas-like states. We identify the condition for cluster formation and discuss potential observation of cluster states in experiments.

Figures

Equations

$$H=\sum _{i_{\uparrow }=1}^N(-\frac{{\hslash }^2}{2m}{\nabla }_{i_{\uparrow }}^2+\frac{1}{2}m{\omega }^2{\vec{r}}_{i_{\uparrow }}^2)+\sum _{i_{\downarrow }=1}^N(-\frac{{\hslash }^2}{2m}{\nabla }_{i_{\downarrow }}^2$$

$$+\frac{1}{2}m{\omega }^2{\vec{r}}_{i_{\downarrow }}^2)+\sum _{i_{\uparrow }=1}^N\sum _{i_{\downarrow }=1}^N{V}_{2b}(r_{i_{\uparrow }{i}_{\downarrow }}),$$

Full text

Introduction: A fermion is a particle that follows Fermi-Dirac statistics, which gives rise to Pauli exclusion principle Fermi (1934). The Pauli exclusion principle states that two or more identical fermions cannot occupy the same quantum state within a quantum system. The result is the emergence of Fermi pressure that prevents white dwarfs and neutron stars from gravitational collapse Koester and Chanmugam (1990).

Fermi pressure is also responsible for stabilizing dilute two-component Fermi gases Petrov et al. (2004); Astrakharchik et al. (2004); Fregoso and Baym (2006); Blume (2012). As the strength of two-body interaction changes, the two-component Fermi gases experience a crossover from weakly correlated Bardeen-Cooper-Schrieffer (BCS) pairing to a Bose-Einstein condensate (BEC) of tightly bound pairs Regal et al. (2004a); Zwierlein et al. (2004). In the BEC limit of the BCS-BEC crossover, two unlike fermions form a molecule, which is a composite boson Astrakharchik et al. (2004); Petrov et al. (2004). The system is then governed by dimer-dimer interactions between molecules. Structureless bosons with two-body interactions are known to form cluster states, such as three-body Efimov states Greene et al. (2017). However, such cluster states have not yet been found in two-component Fermi gases in the BEC limit. Previous numerical calculations in few-body systems Blume and Daily (2009, 2011) and larger systems Blume et al. (2007); Astrakharchik et al. (2004) focused on the zero-range limit and no cluster states were found. The reason is that the Fermi pressure between identical fermions prevents such cluster states to form. However, cluster states could in principle exist with finite-range interactions. In this Letter, we show that the formation of cluster states occurs when the effective range of the two-body interaction exceeds roughly $0.46$ times the scattering length, regardless of the details of the short-range interaction.

The identification of such cluster states are crucial in three ways. First, the formation of clusters in small two-component Fermi gases in three dimension (3D) corresponds to a phase transition from a droplet-like phase to a gas-like phase in many-body systems. This is reminiscent of the Luttinger liquid and gas of molecule transition in one dimension Law and Feldman (2008). Second, the condition for cluster formation is important for preparing such systems in the lab. The magnitude of effective range can approach that of the scattering length in the vicinity of a Feshbach resonance Dyke et al. (2013). Current technology allows preparing Fermi gases interacting through large effective range Chin et al. (2010); Hazlett et al. (2012). The stability of the system and the atom loss rate will be strongly affected by the cluster formation. Third, two-component Fermi gases share similarities with the low-density regions of a neutron star interior where the scaled interaction strength varies and the effective range is, in general, not negligible van Wyk et al. (2018). Therefore, the parameter regime in certain parts of the neutron star can overlap with our study. Our results can provide insights in stability of local regions inside a neutron star.

Previous studies discussed the instability of trapped fermionic gases with attractive interactions within the mean-field frame work and the range of interaction was identified as an important factor Fregoso and Baym (2006). Here, we pursue an $ab$ $initial$ calculation using explicitly correlated Gaussian (ECG) basis set expansion approach to study small systems consisting up to 6 particles. The microscopic approach has been proven successful in understanding physics in larger systems Blume (2012); Levinsen et al. (2017).

System Hamiltonian: We consider equal mass two-component Fermi gases in 3D consisting of $N_{\uparrow}$ spin-up and $N_{\downarrow}$ spin-down atoms ($N_{\uparrow}=N_{\downarrow}=N$) under external spherically symmetric harmonic confinement with angular trapping frequency $\omega$. The system Hamiltonian $H$ reads

[TABLE]

where $m$ denotes the mass of a single atom, $\vec{r}_{i_{\uparrow}}$ and $\vec{r}_{i_{\downarrow}}$ denote the position vector of the $i$th spin-up and down atom with respect to the trap center, respectively. We define the harmonic oscillator length $a_{\text{ho}}=\sqrt{\hbar/(m\omega)}$ and harmonic oscillator energy $E_{\text{ho}}=\hbar\omega$. Interspecies two-body interaction potential $V_{2b}$ depends on the interparticle distance $r_{i_{\uparrow}i_{\downarrow}}$, $r_{i_{\uparrow}i_{\downarrow}}=|\vec{r}_{i_{\uparrow}}-\vec{r}_{i_{\downarrow}}|$.

We consider three different short-range potentials: (i) an attractive Gaussian potential with a repulsive core, $V_{2b}^{(i)}(r)=V_{0}\exp(-r^{2}/4r_{0}^{2})-2V_{0}\exp(-r^{2}/2r_{0}^{2})$; (ii) an attractive Gaussian potential, $V_{2b}^{(ii)}(r)=V_{0}\exp(-r^{2}/2r_{0}^{2})$; and (iii) a modified attractive Gaussian potential, $V_{2b}^{(iii)}(r)=rV_{0}\exp(-r^{2}/2r_{0}^{2})$. The scattering phase shift of two particles at low energy can be expanded as $-k\cot[\delta(k)]=1/a_{s}-r_{\text{eff}}k^{2}/2$, where $k$ is the wave vector, $a_{s}$ the $s$-wave scattering length, and $r_{\text{eff}}$ the effective range. For a fixed $r_{0}$, we adjust $V_{0}$ such that $V_{2b}(r)$ has a certain $a_{s}$. For all three types considered, shallow attractive potentials do not support two-body $s$-wave bound state in free-space. As the depth increases, the potential supports successively more two-body bound states Blume (2012). In this work, we only consider attractive potentials that support one $s$-wave bound state in free space (see Supplemental Material sup ). We then calculate $r_{\text{eff}}$ for such potential. Solid, dashed, and dotted lines in Fig. 1 show potentials (i), (ii), and (iii) that produce the same $a_{s}=0.2a_{\text{ho}}$ and $r_{\text{eff}}=0.09a_{\text{ho}}$, respectively. We will show later that the condition for cluster formation does not strongly depend on the type of potential. This indicates that the physics is well described by the two parameters $a_{s}$ and $r_{\text{eff}}$ alone. Although all three types are short-range, potential (i) simulates the repulsive core in realistic atom-atom interactions and can produce a larger range of $r_{\text{eff}}$ for a fixed $a_{s}$.

We solve the time-independent Schrödinger equation for the Hamiltonian given in Eq. (Cluster formation in two-component Fermi gases) using explicitly correlated Gaussian (ECG) basis set expansion approach Mitroy et al. (2013). After separating off the center-of-mass degrees of freedom, we expand the eigenstates of the relative Hamiltonian in terms of ECG basis functions, which depend on a number of nonlinear variational parameters that are optimized through energy minimization Mitroy et al. (2013); Blume and Daily (2009, 2011); Yin and Blume (2015).

Bound state energy: Trapped $(1,1)$ system, consisting one spin-up and one spin-down particle has been solved analytically. The exact energy spectrum for zero-range interaction is given in Ref. Busch et al. (1998) while the contribution from the effective range is discussed in Ref. Werner and Castin (2012). When $a_{s}$ is positive and much smaller than $a_{\text{ho}}$, the binding energy of the two-particle pair with short-range interaction can be approximated by $E_{s}=-\hbar^{2}/(ma_{s}^{2})$. For $a_{s}=0.2a_{\text{ho}}$, the binding energy is approximately $-25E_{\text{ho}}$ and decreases with increasing $r_{\text{eff}}$.

For trapped two-component Fermi gases with more particles, i.e., $(N,N)$ systems, two particles with opposite spin form a molecule just like in $(1,1)$ system. Such molecules were often treated as composite bosons and an effective model with effective dimer-dimer interactions are shown to be very accurate in the zero-range limit Petrov et al. (2004); Blume and Daily (2009, 2011). In this effective model, the dimer-dimer interaction has a scattering length $a_{dd}=0.608a_{s}$, a small positive value compared to $a_{\text{ho}}$, and does not support any bound state of two dimers Petrov et al. (2004). This indicates that a dilute two-component Fermi gas behaves like a Bose gas with hard-core repulsion in the BEC and zero-range limit. However, dimer-dimer interaction is not $a$ $priori$ repulsive and cluster states can in principle exist. An important question is what is the condition for clusters to form. In the following, we will show that the effective range $r_{\text{eff}}$ plays an important role.

We calculate the relative ground state energies of $(1,1)$, $(2,2)$, and $(3,3)$ systems, $E_{11}$, $E_{22}$, and $E_{33}$, respectively, for fixed $a_{s}=0.2a_{\text{ho}}$ and a wide range of $r_{\text{eff}}$. The center-of-mass energy is simply $3\hbar\omega/2$ for all systems and is not included in our results. To identify the cluster formation between dimers, we follow Ref. Blume and Daily (2009) and subtract the energy of molecules from the four- and six-body systems, i.e., we plot $\Delta E_{22}=E_{22}-2E_{11}$ and $\Delta E_{33}=E_{33}-3E_{11}$ as a function of $r_{\text{eff}}$ in Fig. 2. Circles, squares, and diamonds are for potential (i), (ii), and (iii), respectively. For small $r_{\text{eff}}$, our results agree with previous studies: $\Delta E_{22}$ and $\Delta E_{33}$ are positive, and agree with the zero-range ground state energy for two and three weakly repulsive bosons with dimer-dimer scattering length $a_{dd}=0.608a_{s}$, which are marked as dotted lines in the insets. This indicates that the ground state is indeed a gas-like state. Energies $\Delta E_{22}$ and $\Delta E_{33}$ remain largely unchanged as $r_{\text{eff}}$ increases. As $r_{\text{eff}}$ approaches $0.09a_{\text{ho}}$, $\Delta E_{22}$ and $\Delta E_{33}$ decrease suddenly and turn negative for all three types of interactions, signalling the formation of bound states between dimers. Note that the magnitude of $\Delta E_{33}$ is several times larger than $\Delta E_{22}$, indicating that the bound state in $(3,3)$ system is indeed a three-dimer bound state, instead of a two-dimer bound state plus a single dimer. Our calculations show that $r_{\text{eff}}$ at which cluster starts to form is largely independent of the details of the two-body interactions and is roughly the same for (2,2) and (3,3) systems. This shows that $r_{\text{eff}}$ is the deciding factor for cluster formation in two-component Fermi gases.

Our physics picture is as follows. The existence of cluster states in two-component Fermi gases is determined by the counterbalance between the Fermi pressure between like-particles and the dimer-dimer interaction. The Fermi pressure is a short-range effect, preventing like fermions from getting close to each other through the requirement of anti-symmetrization. On the other hand, cluster formation requires all particles, like or unlike, to be at distances of the order of $r_{\text{eff}}$ Petrov et al. (2004). When $r_{\text{eff}}$ is small, the anti-symmetrization of two like fermions cannot happen within such small distance. Therefore, such interaction cannot support bound states between dimers. When $r_{\text{eff}}$ becomes larger, the dimer-dimer interaction can potentially overcome the Fermi pressure, allowing anti-symmetrization of two like fermions to happen within the range of $r_{\text{eff}}$.

Structural properties: In order to take a peak at the wave function, we calculate several structural properties for both gas-like and cluster states interacting through potential (i). First, we consider the spherically symmetric radial density $P_{1}(r)$, which tells the likelihood of finding a particle at distance $r$ from trap center, with normalization $4\pi\int_{0}^{\infty}P_{1}(r)r^{2}dr=1$. Figure 3(a) and (d) show the radial density $P_{1}(r)$ for gas-like and cluster state, respectively. For both states, $P_{1}(r)$ peaks at the trap center and decays towards the edge. The overall extent of a single particle can be measured by expectation value $\left\langle r\right\rangle=4\pi\int_{0}^{\infty}P_{1}(r)r^{3}dr$. For both $(2,2)$ and $(3,3)$ systems, the expectation values for the cluster states are about $30\%$ lower than gas-like states (see Supplemental Material sup ). We confirm that the expectation values do not differ much for different $r_{\text{eff}}$ within the gas-like state and cluster state. The drop in $\left\langle r\right\rangle$ represents a sudden change from gas-like to cluster state.

Second, we consider the scaled distribution function between unlike pair $4\pi P_{12}(r)r^{2}$, which tells the likelihood of finding two unlike particles at distance $r$ from each other, with normalization $4\pi\int_{0}^{\infty}P_{12}(r)r^{2}dr=1$. Since we consider the BEC limit, it is natural to expect a sharp peak at a short distance that is of the order of $a_{s}$ as two unlike particles form a molecule. Indeed, for both the gas-like and cluster states, a sharp peak at short distance is identified. A difference is that the distribution function for cluster state vanishes at the order of trap length $a_{\text{ho}}$ while exhibits a lower and wider second peak for the gas-like state, which corresponds to the distribution between two unlike particles that belongs to different dimers.

Third, we consider the scaled distribution function between like pair $4\pi P_{11}(r)r^{2}$, which tells the likelihood of finding two like particles at distance $r$ from each other with normalization $4\pi\int_{0}^{\infty}P_{11}(r)r^{2}dr=1$. The distribution function between like pair is a good gauge of the overall extent of the system. For gas-like state, a broad distribution centred around the trap length $a_{\text{ho}}$ is observed, confirming that the system is indeed extended to the confinement of the trap. For cluster state, we find that a peak appears at a similar distance to the unlike pair, which is an order of magnitude smaller than gas-like state. This difference clearly distinguishes the two states. Further details on the expectation values of distances between unlike and like pairs are discussed in Supplemental Material sup .

Condition for cluster formation: For a certain $a_{s}$, both the ground state energies and the structural properties exhibit a clear transition between the gas-like and cluster state that is driven by the effective range $r_{\text{eff}}$. For each $a_{s}$, boundary is determined through the sudden drop of $\Delta E_{22}$ [see, e.g., the rightmost red square in the inset of Fig. 2(a)]. We then analyze $\Delta E_{33}$ and the expectation values from structural properties sup and find no discrepancy. This allows us to map out the condition for cluster formation in Fig. 4.

We make three observations. First, the boundary between the two classes can be approximated by a linear relation $r_{\text{eff}}=0.46a_{s}$. A shorter range interaction produces a hard-core repulsive dimer-dimer interaction and results in a gas-like state. A longer range interaction, in contrast, can overcome the Fermi pressure and leads to cluster formation. Second, although we are not able to perform calculations for smaller $a_{s}$ or $r_{\text{eff}}$ due to numerical limitation, if we assume that the linear relation continues to smaller $a_{s}$, a vertical line $r_{\text{eff}}=0$ will fall entirely into the gas-like state. This suggests that cluster states are absent for the zero-range interaction and confirms the conclusion in Ref. Astrakharchik et al. (2004); Blume et al. (2007); Blume and Daily (2009, 2011). Third, as we increase $a_{s}$, the region of cluster state shrinks. This indicates that it could be more difficult to find cluster states as the system moves towards the unitarity limit, i.e., with infinitely large scattering length. This could explain the observation in Ref. Forbes et al. (2012) where no cluster state was found for a wide range of $r_{\text{eff}}$.

Although we considered trapped few-body systems in this work, we expect many of our findings to apply to larger and homogeneous systems (see Supplemental Material sup ). We also note that the transition between the gas-like and cluster state happens within very small parameter range, and the changes in ground state energy and structural properties are drastic. In the many-body limit, the transition could possibly correspond to a first-order phase transition to a phase-separation phase Piatecki and Krauth (2014); Berciu (2011).

Implications for experiments: Two-component Fermi gases in the BEC limit have been prepared in different cold atom systems. For example, a mixture of two hyperfine states, $|f,m_{f}\rangle=|9/2,-9/2\rangle$ and $|9/2,-5/2\rangle$, of ${}^{40}\text{K}$ have been prepared across the BEC-BCS crossover Regal and Jin (2003). Other hyperfine states of ${}^{40}\text{K}$  Regal et al. (2004b) and other species such as ${}^{6}\text{Li}$ Bartenstein et al. (2005) have also been studied.

We consider neutral atom interactions modelled by van der Waals potential $V_{2b,\text{vdW}}(r_{ij})=-C_{6}/r_{ij}^{6}$. The relation between $r_{\text{eff}}$ and $a_{s}$ was found to be $r_{\text{eff}}=\Gamma(1/4)^{4}/(6\pi^{2})\bar{a}\left[1-2\bar{a}/a_{s}+2\left(\bar{a}/a_{s}\right)^{2}\right]$  Gao (1998) and shown to work reasonably well near broad Feshbach resonances Blackley et al. (2014). Here, $\bar{a}=[4\pi/\Gamma(1/4)^{2}]R_{\text{vdW}}$, $R_{\text{vdW}}=(mC_{6}/\hbar^{2})^{1/4}/2$, and $\Gamma(x)$ is the Gamma function. In the case of ${}^{40}\text{K}$, $R_{\text{vdW}}$ was calculated to be $64.90a_{0}$, where $a_{0}$ denotes the Bohr radius Derevianko et al. (1999). The scattering length is tunable near a Feshbach resonance between $|f,m_{f}\rangle=|9/2,-9/2\rangle$ and $|9/2,-5/2\rangle$, which is centred at $B_{\text{pk}}=224.21G$ Regal and Jin (2003). We therefore determine that the system is in gas-like state for magnetic field $200.3G<B<233.9G$ (see Supplemental Material sup ). Within this range, the two-component Fermi gas went through the BEC-BCS crossover as observed in the experiments Regal and Jin (2003). We expect cluster to form outside this range. This is consistent with the observations that two-component Fermi gases are generally stable near Feshbach resonances Petrov et al. (2004).

Cluster formation can potentially be observed through rf spectroscopy in a magnetically trapped two-component Fermi gas with thousands of atoms Regal and Jin (2003); Regal et al. (2004b). Previous experimental measurements Regal and Jin (2003) on ${}^{40}\text{K}$ are performed with $215G<B<230G$, which lies entirely within the gas-like state according to our prediction. Our results suggest that cluster formation may be observed further away from the Feshbach resonance. A more direct reproduction of the system we studied, i.e., a trapped system with only few atoms, can potentially be realized in optical tweezers Kaufman et al. (2012); Liu et al. (2018) and optical microtraps Serwane et al. (2011).

Final remarks: We studied the formation of cluster state in two-component Fermi gases by numerically calculating the ground state energy of a trapped few-body system interacting through various types of short-range potentials. Our findings corroborate the picture that the counterbalance between the Fermi pressure and the short-range interactions determines the boundary between the gas-like and cluster state. Although our calculations are performed with short-range interactions due to numerical restrictions, we expect such result to apply to realistic interactions because of the short-range nature of Fermi pressure. We also provide an estimate for the parameter regime where such cluster states could potentially be identified in the experiments.