The total momentum of $N$ interacting bosons or fermions in a cube equipped with periodic boundary conditions is a conserved quantity. Its eigenvalues follow a probability distribution, determined by the thermal equilibrium state. While in non-interacting systems the distribution is normal with variance $\sim N$, interaction couples the single-particle momenta, so that the distribution of their sum is unpredictable, except for some implications of Galilean invariance. First, we present these implications which are strong in 1D, moderately strong in 2D, and weak in 3D. Then, we speculate about the possible form of the distribution in fluids, crystals, and superfluids. The existence of phonons suggests that the total momentum can remain finite when $N\to\infty$. We argue that in fluids the finite momenta distribute continuously, but their integrated probability is smaller than 1, because the momentum can also tend to infinity with $N$. In the fluid-crystal transition we expect that the total momentum becomes finite with full probability and distributed over a lattice, and that in the fluid-superfluid transition a delta peak appears only at zero total momentum. Based on this picture, we discuss the superfluid flow in both the frictionless and the dissipative cases, and derive a temperature-dependent critical velocity. Finally, we show that Landau's criterion for excitations in moving superfluids is an in some cases correct result of an erroneous derivation.
The total momentum of N interacting bosons or fermions in a cube equipped with periodic boundary conditions is a conserved quantity. Its eigenvalues follow a probability distribution, determined by the thermal equilibrium state. While in non-interacting systems the distribution is normal with variance ∼N, interaction couples the single-particle momenta, so that the distribution of their sum is unpredictable, except for some implications of Galilean invariance. First, we present these implications which are strong in 1D, moderately strong in 2D, and weak in 3D. Then, we speculate about the possible form of the distribution in fluids, crystals, and superfluids. The existence of phonons suggests that the total momentum can remain finite when N→∞. We argue that in fluids the finite momenta distribute continuously, but their integrated probability is smaller than 1, because the momentum can also tend to infinity with N. In the fluid-crystal transition we expect that the total momentum becomes finite with full probability and distributed over a lattice, and that in the fluid-superfluid transition a delta peak appears only at zero total momentum. Based on this picture, we discuss the superfluid flow in both the frictionless and the dissipative cases, and derive a temperature-dependent critical velocity. Finally, we show that Landau's criterion for excitations in moving superfluids is an in some cases correct result of an erroneous derivation.
The probability distribution of the total momentum P is studied in N-particle interacting homogeneous quantum systems at positive temperatures. Using Galilean invariance we prove that in one dimension, the asymptotic distribution of P/N is normal at all temperatures and densities, and in two dimensions, the tail distribution of P/N is normal. We introduce the notion of the density matrix reduced to the center of mass and show that its eigenvalues are N times the probabilities of the different eigenvalues of P. A series of results is presented for the limit of sequences of positive definite atomic probability measures, relevant for the probability distribution of both the single-particle and the total momentum. The P = 0 ensemble is shown to be equivalent to the canonical ensemble. Through some conjectures we associate the properties of the asymptotic distribution of the total momentum with the characteristics of fluid, solid, and superfluid phases. Our main suggestion is that in interacting quantum systems above one dimension, in infinite space, the total momentum is finite with a nonzero probability at all temperatures and densities. In solids, this probability is 1, and in a crystal, it is distributed on a lattice. Since it is less than 1 in two dimensions, we conclude that a 2D system is always in a fluid phase, so if the hexatic phase existed classically, it would be destroyed by quantum fluctuations. For a superfluid, we conjecture that the total momentum is zero with a nonzero probability and otherwise its distribution is continuous. We define a macroscopic wave function based on the density matrix reduced to the center of mass. We discuss how dissipation can give rise to a critical velocity, predict the temperature dependence of the latter, and argue that Landau’s criterion cannot explain superfluidity and its breakdown at positive temperatures. We also comment on the relation between superfluidity and Bose-Einstein condensation.
The hierarchy of equations for reduced density matrices that describes a thermodynamically equilibrium quantum system obtained earlier by the author is investigated in the momentum representation. In the paper it is shown that the use of the momentum representation opens up new opportunities in studies of macroscopic quantum systems both nonsuperfluid and superfluid. It is found that the distribution over momenta in a quantum fluid is not a Bose or Fermi distribution even in the limit of practically noninteracting particles. The distribution looks like a Maxwellian one although, strictly speaking, it is not Maxwellian. The momentum distribution in a quantum crystal depends upon the interaction potential and the crystalline structure. The momentum distribution in a superfluid contains a delta function. The momentum distribution for the condensate in a superfluid crystal consists of delta peaks that are arranged periodically in momentum space. The periodical structure remains if the condensate crystal is not superfluid.
In this work, we study the linear stability of superfluid phases of matter irrespective of the nature of microscopic degrees of freedom and the strength of interactions between them. Famously, assuming invariance under Galilean boosts and a phonon-roton single-particle dispersion relation, Landau predicted superfluid helium 4 would become unstable for large enough superfluid velocities. Here, we demonstrate that such instabilities generically follow from a change of sign of one of the eigenvalues of the matrix of second derivatives of the free energy. Our only assumption is the existence of static thermodynamic equilibrium, irrespective of any invariance under boosts or microscopic statistics. Turning on dissipation, we show that a linear dynamical instability also develops, leading to exponential growth in time of perturbations around equilibrium. Specializing to Galilean superfluids and assuming the existence of bosonic quasiparticles, our criterion reproduces Landau's critical velocity for Bose-Einstein condensates. Our criterion also reduces to the well-known maximal supercurrent in weakly coupled superconductors described either by Landau-Ginzburg or Bardeen-Cooper-Schrieffer theory. Further, it correctly reproduces the onset of the instability in relativistic, strongly coupled superfluids without quasiparticles at zero as well as finite temperature, which we construct using gauge/gravity duality. As a less trivial application of our criterion, we show that in dirty superfluids the instability manifests itself first in the thermal diffusion mode instead of the superfluid sound mode. Our work provides a simple, comprehensive, and unified description of the large superflow instability of superfluids and superconductors at any temperature independent of the microscopic details of the system and the strength of interactions.
Collective quantum phenomena are fascinating, as they repeatedly challenge our comprehension of nature and its underlying mechanisms. The qualification ``quantum'' can be attributed to a generic many-body system whenever the interference effects related to the underlying wave nature of its elementary constituents can not be neglected anymore, and a naive classical description in terms of interacting billiard balls fails to catch its most essential features. This interference phenomenon called ``quantum degeneracy'' which occurs at weak temperatures, leads to spectacular collective behaviours such as the celebrated Bose-Einstein Condensation (BEC) phase transition, where a macroscopic fraction of a bosonic system of particles collapses below a critical temperature T_c on a single-particle state. Quantum coherence, when combined with inter-particle interactions, gives rise to highly non-classical frictionless hydrodynamic behaviours such as superfluidity (SF) and superconductivity (SC). \nEven more exotic quantum phases emerge in presence of important interactions as matter reaches a ``strongly correlated regime'' dominated by quantum fluctuations, where each particle is able to affect significantly the surrounding fluid: characteristic examples are the so-called Mott-Insulator (MI) quantum phase where particles are localized on a lattice due to a strong interaction-induced blockade, along with the Tonks-Girardeau (TG) gas where impenetrable bosons in one-dimension acquire effective fermionic statistics up to a unitary transformation, and the Fractional Quantum Hall (FQH) effect which occurs in presence of a gauge field, and features a special type of elementary excitation possessing a fractional charge and obeying to fractional statistics called `anyon'. \nThese quantum many-body effects were explored in a first place in systems well isolated from the external environment such as ultra-cold atomic gases or electrons in solid-state systems, within a physical context well described by ``equilibrium statistical mechanics''. Yet, over the last two decades a broad community has started investigating the possibility of stabilizing interacting quantum phases in novel nonlinear quantum optics architectures, where interacting photons have replaced their atomic and electronic counterpart. Thanks to their high level of controllability and flexibility, and the possibility of reaching the quantum degeneracy regime at exceptionally high temperatures, these platforms appear as extremely promising candidates for the ``quantum simulation'' of the most exotic many-body quantum problems: while the precursors experiments in semiconductor exciton-polariton already allow to reach the Bose-Einstein Condensation and superfluid regimes, novel platforms such as superconducting circuits, coupled cavity arrays or photons coupled to Rydberg EIT (Electromagnetically induced Transparency) atoms have entered the so-called `photon blockade' where photons behave as impenetrable particles, and open a encouraging pathway toward the future generation of strongly correlated phases with light. \nA specificity of quantum optics devices is their intrinsic ``non-equilibrium'' nature: the interplay between the practically unavoidable radiative and non-radiative losses and the external drive needed to replenish the photon gas leads the many-body system toward a steady-state presenting important non-thermal features. One one hand, an overwhelmingly large quantity of novel quantum phenomena is expected in the non-equilibrium framework, as breaking the thermal equilibrium condition releases severe constraints on the state of a quantum system and on the nature of its surrounding environment. On the other hand, we do not benefit yet of an understanding of non-equilibrium statistical mechanics comparable with its well-established equilibrium counterpart, which relies on strong historical foundations. Understanding how to tame (and possibly exploit) non-equilibrium effects in order to stabilize interesting quantum phases in a controlled manner often reveals a hard challenge. \nIn that prospect, an important conceptual issue in the non-equilibrium physics of strongly interacting photons regards the possibility of stabilizing ``incompressible quantum phases'' such as the Mott-Insulator or Fractional Quantum Hall states, and more generally to stabilize the ground-state of a given particle-number conserving Hamiltonian, in a physical context where dissipative losses can not be neglected. While being able to quantum simulate those emblematic strongly correlated quantum phases in this novel experimental context would strongly benefit to the quantum optics community, gaining such a kind of flexibility would also contribute to fill an important bridge between the equilibrium and the non-equilibrium statistical physics of open quantum systems, allowing to access in a controlled manner a whole new phenomenology at the interface between the two theories. \nIn this thesis I address those questions, which I reformulate in the following manner: \n \n-What are the conditions for the emergence of analogue equilibrium properties in open quantum systems in contact with a non-thermal environment ? \n \n-In particular, is it possible to stabilize strongly correlated quantum phases with a perfectly defined particle number in driven-dissipative photonic platforms, in spite of environment-induced losses and heating effects ? \n \nThe structure of the thesis is the following. \n \n \n[Chapter 1.] We give an overview of the physics of many-body photonic systems. As a first step we address the weakly interacting regime in the physical context of exciton-polaritons: after describing the microscopic aspects of typical experiments, we move to the discussion of non-equilibrium Bose-Einstein Condensation and the various mechanisms related to the emergence of thermal signatures at steady-state. The second part of this Chapter is dedicated to strongly interacting fluids. After drawing a quick overview of several experimental platforms presenting a good potential for the study of such physics in a near future, we discuss the relative performance of several schemes proposed in order to replenish the photonic population \n \n[Chapter 2.] We investigate the potential of a non-Markovian pump scheme with a narrow bandpass (Lorentzian shaped) emission spectrum for the generation of strongly correlated states of light in a Bose-Hubbard lattice. Our proposal can be implemented by mean of embedded inverted two-level emitters with a strong incoherent pumping toward the excited state. Our study confirms in a single cavity the possibility of stabilizing photonic Fock states in a single configuration, and strongly localized n=1 Mott-Insulator states in a lattice with n=1 density. We show that a relatively moderate hopping is responsible for a depletion of the Mott-state, which then moves toward a delocalized state reminiscent of the superfluid regime. Finally, we proceed to a mean-field analysis of the phase diagram, and unveil a Mott-to-Superfluid transition characterized by a spontaneous breaking of the U(1) symmetry and incommensurate density. The results of this Chapter are based on the following publications: \n- J. Lebreuilly, M. Wouters and I. Carusotto, ``Towards strongly correlated photons in arrays of dissipative nonlinear cavities under a frequency-dependent incoherent pumping'', C. R. Phys., 17(8), 836, 2016. \n- A. Biella, F. Storme, J. Lebreuilly, D. Rossini, R. Fazio, I. Carusotto and C. Ciuti, ``Phase diagram of incoherently driven strongly correlated photonic lattice'', Phys. Rev. A, 96, 023839, 2017. \n \n[Chapter 3.] In view of improving the performance of the scheme introduced in last chapter, and reproducing in particular the equilibrium zero temperature phenomenology in driven-dissipative photonic lattices, we develop a fully novel scheme based on the use of non-Markovian reservoirs with tailored broadband spectra which allows to mimick the effect of tunable chemical potential. Our proposal can be implemented by mean of a small number of emitters and absorbers and is accessible to current technologies. \n \nWe first analyse the case of a frequency-dependent emission with a square spectrum and confirm the possibility of stabilizing Mott insulator states with arbitrary integer density. Unlike the previous proposal the Mott state is robust against both losses and tunneling. A sharp transition toward a delocalized superfluid-like state can be induced by strong values of the tunneling or a change in the effective chemical potential. While an overall good agreement is found with the T=0 predictions, our analysis highlights small deviations from the equilibrium case in some parts of the parameters space, which are characterized by a non-vanishing entropy and the kinetic generation of doublon excitations. We finally consider an improved scheme involving additional frequency-dependent losses, and show in that case that the Hamiltonian ground-state is fully recovered for any choice of parameters. Our proposal, whose functionality relies on generic energy relaxation mechanisms and is not restricted to the Bose-Hubbard model, appears as a promising quantum simulator of zero temperature physics in photonic devices. The results of this Chapter are based on the following publication: \n- J. Lebreuilly, A. Biella, F. Storme, D. Rossini, R. Fazio, C. Ciuti and I. Carusotto, ``Stabilizing strongly correlated photon fluids with non-Markovian reservoirs'', Phys. Rev. A 96, 033828 (2017). \n \n[Chapter 4.] We adopt a broader perspective, and analyse the conditions for the emergence of analogous thermal properties in driven-dissipative quantum systems. We show that the impact of an equilibrated environment can be mimicked by several non-Markovian and non-equilibrated reservoirs. Chapter 2 already f
Introduction In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenberg spin models, classical fluids and solids, etc.) which have many unphysical low temperature properties. A case in point is a classical solid for which the specific heat follows the Dulong–Petit law, C = 3 Nk B , as T → 0, and the entropy has unphysical behavior since S → –∞. Also, thermal expansion coefficients tend to non-vanishing constants for T → 0 while the third law implies that they must be zero. While the position and momentum of a particle can be specified precisely in classical mechanics, and hence the groundstate of a solid is a perfectly rigid crystal lattice (motionless particles localized at the lattice points), in reality the Heisenberg uncertainty principle forbids such a perfect rigid crystal, even at T → 0, due to zero point motions which ‘smear out’ the particles over some region around these lattice points. This delocalization of quantum-mechanical particles increases as the atomic mass is reduced; therefore, these quantum effects are most pronounced for light atoms like hydrogen in metals, or liquid helium. Spectacular phenomena like superfluidity are a consequence of the quantum nature of the particles and have no classical counterpart at all. Even for heavier atoms, which do not show superfluidity because the fluid–solid transition intervenes before a transition from normal fluid to superfluid could occur, there are genuine effects of quantum nature. Examples include the isotope effects (remember that in classical statistical mechanics the kinetic energy part of the Boltzmann factor cancels out from all averages, and thus in thermal equilibrium no property depends explicitly on the mass of the particles).