Final Report Summary - INTERPOL (Interactions between polarons in polarized Fermi gases)
(1) INTRODUCTION
An ideal atomic Fermi gas consists of fermionic atoms having the same spin state that do not interact with each other. When a certain amount of atoms with opposite spin is added to such an ideal Fermi gas, then the atoms with opposite spin can interact in the s-wave channel and one obtains a gas that is commonly called interacting Fermi gas. Such an interacting system presents a crossover between the BEC state and the BCS regime. On the BEC side a strong coupling of a spin-up atom with a spin-down atom takes place and fermionic atoms give rise to bosonic molecules, which can subsequently condense and form a Bose Einstein Condensate (BEC). Instead on the BCS side, fermionic atoms of opposite spin and momenta interact over a large distance resulting in a weak coupling responsible for Cooper pairs formation. The BEC-BCS crossover is then driven by the amplitude and the sign of a single parameter, namely the s-wave scattering length between fermions with opposite spins.
Of particular interest is the case of spin imbalanced Fermi gases, that is with a larger number of spin up particles (Nup > Ndown). Recent cold atom experiments have reached an unprecedented accuracy in measuring the equation of state of an interacting Fermi gas [1]. This has allowed to confirm that in strongly spin-polarized configurations the minority atoms dressed by the Fermi sea of the majority atoms form a normal gas of quasiparticles called Fermi polarons [2].
Whereas the basic one-body properties of these polarons seem to be quite well understood (such as their binding energy with the Fermi sea, and their effective mass), the interaction between polarons is still a debated issue and is the focus of the INTERPOL project.
(2) OBJECTIVES AND RESULTS
Within the INTERPOL project we addressed the scattering problem of two finite momentum polarons, which may be realized experimentally by producing quasi-monochromatic beams of polarons and studying their scattering properties. Specifically, we have tackled this problem using different theoretical approaches and in what follows I shall describe in detail our methods and results.
(2.1) SPATIAL EXTENSION OF A POLARON
Since the ansatz [3,4] has proven to be very solid to describe both the energy and the state of a single polaron, we used it to evaluate the spatial extension of a single polaron [5], which is an observable of interest in the scattering process of two polarons and was not yet known.
From our numerical calculations we found that the presence of the impurity in a Fermi gas modifies the sermonic density around the impurity in a spherically symmetric way with a long-range spatial extension. More specifically, we analytically found that the damping of the fermionic density around the impurity scales with the inverse-fourth-power of the distance from the impurity. This implies that the polaron is a spatially extended quasi-particle with infinite root-mean-square radius, which suggests that the interaction between polarons may be of long-range type.
(2.2) VARIATIONAL ANSATZ
We have tackled the scattering problem of two finite momentum polarons using a generalization of the variational ansatz [3,4] to the case of two minority atoms sharing one particle-hole excitation of the Fermi sea. We have derived an integral equation for the energy of the two scattering impurities and we have solved it numerically. After a careful analysis of the solution of the scattering event resulting from the ansatz, we concluded that the ansatz is not appropriate to describe such a scattering process.
We found that the main reason for the failure of the ansatz is the lack of self consistency, since, under the usual scattering boundary conditions, the ansatz predicts an energy different from the sum of the energies of the incoming polarons, as it should be.
(2.3) A MOVING POLARON
Since the variational approach based on a generalization of the ansatz [3,4] has failed to address the scattering problem of two finite momentum polarons we decided to use perturbation theory.
In the weakly attractive limit, where the interaction between the impurity and the fermions is small and negative (-1 << g < 0), we considered the problem of a single impurity moving with wave vector K in a fermi gas; we solved the problem exactly up to order two in the small parameter gand we obtained an analytic expression both for the real and the imaginary part of the complex energy of the moving polaron [6]. Remarkably, we found that the effective mass of the polaron is
not a smooth function of K and for a unit impurity-to-fermion mass ratio we found a divergent behaviour of the second derivative when K is at the Fermi surface. The solution of such a problem constitutes a building block for the two polarons problem since one can generalize this approach to the case of two colliding impurities.
(3) CONCLUSION
In our work we first addressed an issue of the single polaron problem which was not yet considered:
The deformation of the density profile around the impurity. We showed that the perturbation induced by the impurity has a long-range tail and presents Friedel-like oscillations. In an indirect way our work gives a useful information on the two polaron problem since it shows that the polaron is a spatially extended object, which suggests a long-range interaction between polarons.
Second, we obtained an important result but negative: The ansatz [3,4], which has proven to work extremely well both for the energy and the state of the single polaron problem, cannot be generalized to the case of two colliding polarons without introducing inconsistencies. This result then disqualifies the use of such an ansatz for the two impurity problem.
Third, using perturbation theory in a systematic way we solved the problem of a moving polaron in a Fermi gas and we obtained an analytic expression for its complex energy. We considered also the effect of a small finite temperature and we concluded that our results can be measured by current experiments using radio frequency spectroscopy and constitutes a building block for the scattering of two polarons.
REFERENCES
[1] N. Navon et. al., Science 328, 729 (2010).
[2] C. Lobo et. al., Phys. Rev. Lett. 97, 200403 (2006).
[3] F. Chevy, Phys. Rev. A 74, 063628 (2006).
[4] R. Combescot, S. Giraud, Phys. Rev. Lett. 101, 050404 (2008).
[5] C. Trefzger, Y. Castin, Europhysics Letters 101, 30006 (2013).
[6] C. Trefzger, Y. Castin, Europhysics Letters 104, 50005 (2013).
An ideal atomic Fermi gas consists of fermionic atoms having the same spin state that do not interact with each other. When a certain amount of atoms with opposite spin is added to such an ideal Fermi gas, then the atoms with opposite spin can interact in the s-wave channel and one obtains a gas that is commonly called interacting Fermi gas. Such an interacting system presents a crossover between the BEC state and the BCS regime. On the BEC side a strong coupling of a spin-up atom with a spin-down atom takes place and fermionic atoms give rise to bosonic molecules, which can subsequently condense and form a Bose Einstein Condensate (BEC). Instead on the BCS side, fermionic atoms of opposite spin and momenta interact over a large distance resulting in a weak coupling responsible for Cooper pairs formation. The BEC-BCS crossover is then driven by the amplitude and the sign of a single parameter, namely the s-wave scattering length between fermions with opposite spins.
Of particular interest is the case of spin imbalanced Fermi gases, that is with a larger number of spin up particles (Nup > Ndown). Recent cold atom experiments have reached an unprecedented accuracy in measuring the equation of state of an interacting Fermi gas [1]. This has allowed to confirm that in strongly spin-polarized configurations the minority atoms dressed by the Fermi sea of the majority atoms form a normal gas of quasiparticles called Fermi polarons [2].
Whereas the basic one-body properties of these polarons seem to be quite well understood (such as their binding energy with the Fermi sea, and their effective mass), the interaction between polarons is still a debated issue and is the focus of the INTERPOL project.
(2) OBJECTIVES AND RESULTS
Within the INTERPOL project we addressed the scattering problem of two finite momentum polarons, which may be realized experimentally by producing quasi-monochromatic beams of polarons and studying their scattering properties. Specifically, we have tackled this problem using different theoretical approaches and in what follows I shall describe in detail our methods and results.
(2.1) SPATIAL EXTENSION OF A POLARON
Since the ansatz [3,4] has proven to be very solid to describe both the energy and the state of a single polaron, we used it to evaluate the spatial extension of a single polaron [5], which is an observable of interest in the scattering process of two polarons and was not yet known.
From our numerical calculations we found that the presence of the impurity in a Fermi gas modifies the sermonic density around the impurity in a spherically symmetric way with a long-range spatial extension. More specifically, we analytically found that the damping of the fermionic density around the impurity scales with the inverse-fourth-power of the distance from the impurity. This implies that the polaron is a spatially extended quasi-particle with infinite root-mean-square radius, which suggests that the interaction between polarons may be of long-range type.
(2.2) VARIATIONAL ANSATZ
We have tackled the scattering problem of two finite momentum polarons using a generalization of the variational ansatz [3,4] to the case of two minority atoms sharing one particle-hole excitation of the Fermi sea. We have derived an integral equation for the energy of the two scattering impurities and we have solved it numerically. After a careful analysis of the solution of the scattering event resulting from the ansatz, we concluded that the ansatz is not appropriate to describe such a scattering process.
We found that the main reason for the failure of the ansatz is the lack of self consistency, since, under the usual scattering boundary conditions, the ansatz predicts an energy different from the sum of the energies of the incoming polarons, as it should be.
(2.3) A MOVING POLARON
Since the variational approach based on a generalization of the ansatz [3,4] has failed to address the scattering problem of two finite momentum polarons we decided to use perturbation theory.
In the weakly attractive limit, where the interaction between the impurity and the fermions is small and negative (-1 << g < 0), we considered the problem of a single impurity moving with wave vector K in a fermi gas; we solved the problem exactly up to order two in the small parameter gand we obtained an analytic expression both for the real and the imaginary part of the complex energy of the moving polaron [6]. Remarkably, we found that the effective mass of the polaron is
not a smooth function of K and for a unit impurity-to-fermion mass ratio we found a divergent behaviour of the second derivative when K is at the Fermi surface. The solution of such a problem constitutes a building block for the two polarons problem since one can generalize this approach to the case of two colliding impurities.
(3) CONCLUSION
In our work we first addressed an issue of the single polaron problem which was not yet considered:
The deformation of the density profile around the impurity. We showed that the perturbation induced by the impurity has a long-range tail and presents Friedel-like oscillations. In an indirect way our work gives a useful information on the two polaron problem since it shows that the polaron is a spatially extended object, which suggests a long-range interaction between polarons.
Second, we obtained an important result but negative: The ansatz [3,4], which has proven to work extremely well both for the energy and the state of the single polaron problem, cannot be generalized to the case of two colliding polarons without introducing inconsistencies. This result then disqualifies the use of such an ansatz for the two impurity problem.
Third, using perturbation theory in a systematic way we solved the problem of a moving polaron in a Fermi gas and we obtained an analytic expression for its complex energy. We considered also the effect of a small finite temperature and we concluded that our results can be measured by current experiments using radio frequency spectroscopy and constitutes a building block for the scattering of two polarons.
REFERENCES
[1] N. Navon et. al., Science 328, 729 (2010).
[2] C. Lobo et. al., Phys. Rev. Lett. 97, 200403 (2006).
[3] F. Chevy, Phys. Rev. A 74, 063628 (2006).
[4] R. Combescot, S. Giraud, Phys. Rev. Lett. 101, 050404 (2008).
[5] C. Trefzger, Y. Castin, Europhysics Letters 101, 30006 (2013).
[6] C. Trefzger, Y. Castin, Europhysics Letters 104, 50005 (2013).