Open quantum systems and ultracold atoms

In several branches of physics it is common to have a system that can be thought as made by two (or more) parts: only one of these parts is of interest, while it would be desirable to theoretically describe all the remaining parts in the lightest possible way. This situation is typical of complex systems and therefore it is not possible in general to de ne a common strategy for the description of the systems, but each case has to be considered by its own. Indeed, if attention is not paid, something relevant such as emergent phenomena can be missed. When quantum systems are involved composite systems are known as Open Quantum Systems (OQS) and in the pas decades they have been studied extensively. In the OQS language, the part of the total system we are interested in is called the subsystem, while the other parts are called the environment or the bath. It is not possible in general to de ne a unique way to describe OQS due to their complexity. Although, in the majority of the situations the bath is much larger than the subsystems in terms of number of degrees of freedom. A common strategy consists in tracing out the environment’s degrees of freedom in order to study, is some approximation, only the subsystem. The key point here is the approximation made on the trace operation: a balance has to be found between the loss of information on the environment and the simplification of the description for the subsystem. The final effect of this procedure are some equations for the subsystem’s dynamics, where the influence of the environment is encoded in some parameters. These parameters are subjected to memory effects: de-pending on the applied approximations they can evolve or not in time. In general, a time dependency points at a non negligible effect of the subsystem on the environment in the considered timescale. The advantages of this procedure are clear: indeed reducing the degrees of freedom, without losing completely their influence on the subsystem, allows to simplify the theoretical description and heavily lowers the computational complexity of the numerical treatment. Disadvantages are clear too: an excessive simplification will determine an impossibility in describing also subsystem’s main properties. The great advantage of the OQS description is that it can be applied to different systems without losing validity: in the present work it is applied to different systems of ultracold atoms. This will allow us to predict some properties and compare theoretical predictions with experimental findings. In the eld of OQS it is quite common to describe them with master equation such as the Lindblad equation, that describe the density matrix of the subsystem when the environment is not subjected to any memory effect, i.e. is Markovian. We instead use a different approach build on Quantum Field Theory (QFT) for systems out of equilibrium, developed by Keldysh and others (and therefore known as Keldysh formalism). In this formalism, semiclassical equation of motion for the subsystem are derived, where semiclassical means that equations are classical in their form while the signature of the quantum nature of the whole system is embedded in the coefficients of the equations themselves. The first system we study is composed of an arbitrary number of non-interacting heavy impurities in contact with a free fermionic bath of atoms. The only interaction present is the one between impurities and the atoms of the bath, modeled as a contact interaction; a situation that is similar to what is observed in polaron physics. This system of heavy impurities has analogies with quark-gluon plasma systems and, after the fermionic degrees of freedom are traced out, a mediated interaction between impurities is present as a result of an exchange of fermions. A key property of the mediated interaction is that it is always attractive, irregardless of the original bare interaction. Now we perform a well controlled chain of approximations to obtain an induced potential and the semiclassical equations of motion for the impurities position in time. These equations are the so called generalized Langevin equation (GLE) and the quantum nature of the system is now encoded in three terms: force, friction and noise. The induced potential has a real and an imaginary part and it strongly depends on the bath: in this case semianalytical expressions for both parts can be derived because the fermion of the bath are non-interacting. We found that real part of the complex potential presents divergencies that has to be treated properly in a renormalization procedure, while divergencies are absent in the imaginary part. Real part generates the force term of the GLE via its gradient, while the imaginary part generates friction and noise. Regarding the friction, it is made of two parts: the first one is a constant term while the second depends on the distance between impurities. The distance-dependent term of the friction is present when there is more than one impurity in the bath and reflects the polarization of the bath induced by each impurity. Indeed, friction is related to the collisions between impurities and particles of the bath: if we look at a single impurity we see that these collisions can be modified by the presence of other impurities and so friction has to show some dependence on the distance between impurities. Moreover, we are able to prove that friction is present in this system also in the zero-temperature limit as a consequence of the energy spectrum of the fermions of the bath. The force term in the GLE is always attractive at short distances, while it disappears as the distance increases enough.The GLE is derived for an arbitrary number of impurities, but we consider two different scenarios to better understand the behaviour of the whole system: a single impurity and two impurities. The single impurity scenario is an example of quantum Brownian motion with constant friction. The case of two impurities is so the simplest one that can be used to study the role of force and distance dependent friction on the dynamics. In this scenario, impurities tends to come closer under the effect of the attractive force, while noise provides random thermal fluctuations. It is possible to demonstrate that the formation of a bound state, i.e. a situation where the distance between impurities is limited, is possible. The bound state formation relies on the interplay between the strength of the impurity-bath interaction and the temperature. Indeed, increasing the temperature will also increase the noise making the bound state formation more di cult if not impossible. The bound state is characterized by radius and lifetime, respectively the distance between impurities and the average time necessary to random fluctuations to break the bound state. Lifetime obtained via numerical simulations is found to to be in agreement with theoretical predictions based on Kramers rate theory. As a consequence of the renormalization procedure used for the real part of the complex potential, results for bound state and lifetime can be considered as qualitative, while results for the friction are quantitative. The second system we focus on has been studied experimentally and it is made of atoms of 6Li. The atoms are initially prepared in two different Zeeman states and the mixture is imbalanced, thus one of the Zeeman states acts as a bath and one as minority. These two states are labelled respectively as |1> and |2>. The interaction between these states is weak, therefore some of the minority atoms are moved to a third Zeeman state, labelled as |3>, that is resonantly interacting with |1>. When an atom is in j3i the formation of polarons is observed. A polaron is a quasiparticle that is formed when an impurity attracts or repels the particle of the bath: in the first case the polaron is defined attractive, in the second repulsive. Interaction between |3> and |1> can be tuned to obtain attractive or repulsive polarons in order to study some of their properties. We are interested in particular in the description of the dynamics of the polarons and therefore we derive a theory that can describe experimental data. To do that, in the experiment a Rabi coupling is added between |2> and |3> and the populations in the two levels, n2 and n3, are measured. A complete many body description of the full dynamics is extremely challenging, but we can use Keldysh formalism and some results for Zeno effect in Fermi polarons to trace out the degrees of freedom of |1> (in the approximation where the interaction between |1> and |2> is neglected) and obtain an effective theory. Some approximations are then made, such as neglecting the polaron formation process and considering the bath as Markovian, to obtain four coupled quantum Boltzmann equations for the populations n2, n3 and for the two coherence between the levels. A relevant element in these equations is the collisional integral, that estimates for processes like conversion between attractive and repulsive polaron and momentum exchange with the bath for polarons of the same type. Moreover, collisional integral is also responsible for polaron dissociation process. Interestingly, collisional integrals do not play a major role in describing the dynamics of attractive polarons, that is found to be in good agreement with experimental data also in the collisionless approximation, i.e. when polaronic collisional integrals are neglected. An explanation to this observation is that predicted lifetime for attractive polarons is much longer than the experimental timescale, therefore dissociation of attractive polarons is not relevant in our systems and we can neglect collisional integrals without losing too much precision. Indeed, we only miss some momentum exchange process that slightly modifies the observed dynamics but that are expected to be less relevant than other approximations made before such as the ladder approximation for the imaginary part of polaron self-energy. The situation is different for the repulsive polaron, because disregarding collisional integral will result in missing the decay of the population n3 that is observed experimentally. Indeed, both conversion processes and momentum exchange are now expected to be more important, but collisionless approximations is still useful to obtain some hints. When the repulsive polaron is weakly interacting, the decay of n3 is again slow compared to our timescales and the situation is similar to the attractive polaron. Therefore, collisionless approximation is not brutal and a good agreement with experiment is observed, although adding collisional integral will now generate a sensible better agreement between predictions and experiment. When the repulsive polaron is strongly interacting the timescale for the decay of n3 matches the experimental one and the collisional integral is necessary. In the strongly interacting limit experimental results can not be fully reproduced because of some major approximations made in the derivation of the quantum Boltzmann equations. Anyway, we derived a description that is able to describe the dynamics of both attractive and repulsive polarons and needs non t parameters to reproduce experimental data, differently to what has been found before in literature. In conclusion this work shows that OQS formalism, in particular the Keldysh QFT, is a powerful tool also for ultracold atomic systems. It is also worth noting that the influence of the made approximations on the final equations of motion is clear and controllable. Therefore, the precision is in principle improvable but the balance between the gain obtained with better approximations and the required e ort has to be carefully evaluated. In any case, even this semiclassical description is able to give important hints on relevant physical properties.