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W. Yi, S. Diehl, A. J. Daley, P. Zoller Driven-dissipative many-body pairing states for cold fermionic atoms in an optical lattice,
New J. Phys. 14 055002, (2012-05-01),
URL doi:10.1088/1367-2630/14/5/055002 (ID: 717858)
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We discuss the preparation of many-body states of cold fermionic atoms in an optical lattice via controlled dissipative processes induced by coupling the system to a reservoir. Based on a mechanism combining Pauli blocking and phase locking between adjacent sites, we construct complete sets of jump operators describing coupling to a reservoir that leads to dissipative preparation of pairing states for fermions with various symmetries in the absence of direct inter-particle interactions. We discuss the uniqueness of these states, and demonstrate it with small-scale numerical simulations. In the late time dissipative dynamics, we identify a "dissipative gap" that persists in the thermodynamic limit. This gap implies exponential convergence of all many-body observables to their steady state values. We then investigate how these pairing states can be used as a starting point for the preparation of the ground state of Fermi-Hubbard Hamiltonian via an adiabatic state preparation process also involving the parent Hamiltonian of the pairing state. We also provide a proof-of-principle example for implementing these dissipative processes and the parent Hamiltonians of the pairing states, based on Yb171 atoms in optical lattice potentials.
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D. Marcos, A. Tomadin, S. Diehl, P. Rabl Photon condensation in circuit QED by engineered dissipation,
New J. Phys. 14 055005, (2012-05-01),
URL doi:10.1088/1367-2630/14/5/055005 (ID: 717877)
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We study photon condensation phenomena in a driven and dissipative array of superconducting microwave resonators. Specifically, we show that by using an appropriately designed coupling of microwave photons to superconducting qubits, an effective dissipative mechanism can be engineered, which scatters photons towards low-momentum states while conserving their number. This mimics a tunable coupling of bosons to a low temperature bath, and leads to the formation of a stationary photon condensate in the presence of losses and under continuous-driving conditions. Here we propose a realistic experimental setup to observe this effect in two or multiple coupled cavities, and study the characteristics of such an out-of-equilibrium condensate, which arise from the competition between pumping and dissipation processes.
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C. Bardyn, M. Baranov, E. Rico Ortega, A. Imamoglu, P. Zoller, S. Diehl Majorana Modes in Driven-Dissipative Atomic Superfluids With Zero Chern Number,
Phys. Rev. Lett. 109 130402, (2012-09-25),
URL doi:10.1103/PhysRevLett.109.130402 (ID: 717881)
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We investigate dissipation-induced p-wave paired states of fermions in two dimensions and show that dissipation can break the bulk-edge correspondence present in Hamiltonian systems in a way that leads to the appearance of spatially separated Majorana zero modes in a phase with vanishing Chern number. We construct an explicit model of a dissipative vortex that traps a single of these modes and establish its topological origin by mapping it to a one-dimensional wire where we observe a non-equilibrium topological phase transition characterized by an abrupt change of a topological invariant (winding number). Engineered dissipation opens up possibilities for experimentally realizing such states with no Hamiltonian counterpart.
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C. Kraus, S. Diehl, P. Zoller, M. Baranov Preparing and probing atomic Majorana fermions and topological order in optical lattices,
New J. Phys. 14 113036, (2012-11-27),
URL doi:10.1088/1367-2630/14/11/113036 (ID: 717882)
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We introduce a one-dimensional system of fermionic atoms in an optical lattice whose phase diagram includes topological states of different symmetry classes. These states can be identified by their zero-energy edge modes which are Majorana fermions. We propose several universal methods of detecting the Majorana edge states, based on their genuine features: zero-energy, localized character of the wave functions, and induced non-local fermionic correlations.
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I. Boettcher, J. M. Pawlowski, S. Diehl Ultracold atoms and the Functional Renormalization Group,
Nucl. Phys. B 228 63, (2012-07-02),
URL doi:10.1016/j.nuclphysbps.2012.06.004 (ID: 718053)
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We give a self-contained introduction to the physics of ultracold atoms using functional integral techniques. Based on a consideration of the relevant length scales, we derive the universal effective low energy Hamiltonian describing ultracold alkali atoms. We then introduce the concept of the effective action, which generalizes the classical action principle to full quantum status and provides an intuitive and versatile tool for practical calculations. This framework is applied to weakly interacting degenerate bosons and fermions in the spatial continuum. In particular, we discuss the related BEC and BCS quantum condensation mechanisms. We then turn to the BCS-BEC crossover, which interpolates between both phenomena, and which is realized experimentally in the vicinity of a Feshbach resonance. For its description, we introduce the Functional Renormalization Group approach. After a general discussion of the method in the cold atoms context, we present a detailed and pedagogical application to the crossover problem. This not only provides the physical mechanism underlying this phenomenon. More generally, it also reveals how the renormalization group can be used as a tool to capture physics at all scales, from few-body scattering on microscopic scales, through the finite temperature phase diagram governed by many-body length scales, up to critical phenomena dictating long distance physics at the phase transition. The presentation aims to equip students at the beginning PhD level with knowledge on key physical phenomena and flexible tools for their description, and should enable to embark upon practical calculations in this field.
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A. Tomadin, S. Diehl, M. Lukin, P. Rabl, P. Zoller Reservoir engineering and dynamical phase transitions in optomechanical arrays,
Phys. Rev. A 86 033821, (2012-09-14),
URL doi:10.1103/PhysRevA.86.033821 (ID: 718117)
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We study the driven-dissipative dynamics of photons interacting with an array of micromechanical membranes in an optical cavity. Periodic membrane driving and phonon creation result in an effective photon-number conserving non-unitary dynamics, which features a steady state with long-range photonic coherence. If the leakage of photons out of the cavity is counteracted by incoherent driving of the photonic modes, we show that the system undergoes a dynamical phase transition to the state with long-range coherence. A minimal system, composed of two micromechanical membranes in a cavity, is studied in detail, and it is shown to be a realistic setup where the key processes of the driven-dissipative dynamics can be seen.
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M. Cordin, P. Amann, A. Menzel, E. Bertel, M. Baranov, S. Diehl, J. Redinger, C. Franchini Cleavage surface of the BaFe2−xCoxAs2 and FeySe1−xTex superconductors: A combined STM plus LEED study,
Phys. Rev. B 86 167401, (2012-10-19),
URL doi:10.1103/PhysRevB.86.167401 (ID: 718263)
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Massee et al. [ Phys. Rev. B 80 140507 (2009)] found on the cleavage planes of BaFe2−xCoxAs2 two different long-range ordered structures, i.e., a (2×1) phase present only after cleavage at low temperature and a √2×√2 phase observed after cleavage at room temperature. These results apply generally to 122 Fe-based superconductors, but have been discussed controversially [for a summary of the conflicting views, see Hoffman Rep. Prog. Phys. 74 124513 (2011)]. Here we support the interpretation of Massee et al. In addition, we argue that the existence of different long-range ordered structures corresponding to the same coverage in different temperature regimes is associated with the melting of a charge density wave and removal of an associated periodic lattice distortion (CDW/PLD) in the substrate as T is increased. At sufficiently low temperature the fluctuating CDW/PLD order parameter is stabilized by the adsorbate in a lock-in type mechanism. Accordingly, we interpret the surface structures observed on the 122 Fe pnictide surfaces as evidence for the presence of CDW fluctuations at low temperature, but with a wave vector differing from that of the antiferromagnetic spin-density fluctuations.
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A. Privitera, I. Titvinidze, S. Chang, S. Diehl, A. J. Daley, W. Hofstetter Loss-induced phase separation and pairing for 3-species atomic lattice fermions,
Phys. Rev. A 84 021601, (2011-08-03),
URL doi:10.1103/PhysRevA.84.021601 (ID: 717349)
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We study the physics of a three-component Fermi gas in an optical lattice, in the presence of a strong three-body constraint arising due to three-body loss. Using analytical and numerical techniques, we show that an atomic color superfluid phase is formed in this system and undergoes phase separation between unpaired fermions and superfluid pairs. This phase separation survives well above the critical temperature, giving a clear experimental signature of the three-body constraint.
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I. Titvinidze, A. Privitera, S. Chang, S. Diehl, M. Baranov, A. J. Daley, W. Hofstetter Magnetism and domain formation in SU(3)-symmetric multi-species Fermi mixtures,
New J. Phys. 13 035013, (2011-03-16),
URL doi:10.1088/1367-2630/13/3/035013 (ID: 717395)
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We study the phase diagram of an SU(3)-symmetric mixture of three-component ultracold fermions with attractive interactions in an optical lattice, including the additional effect on the mixture of an effective three-body constraint induced by three-body losses. We address the properties of the system in $D \\geq 2$ by using dynamical mean-field theory and variational Monte Carlo techniques. The phase diagram of the model shows a strong interplay between magnetism and superfluidity. In the absence of the three-body constraint (no losses), the system undergoes a phase transition from a color superfluid phase to a trionic phase, which shows additional particle density modulations at half-filling. Away from the particle-hole symmetric point the color superfluid phase is always spontaneously magnetized, leading to the formation of different color superfluid domains in systems where the total number of particles of each species is conserved. This can be seen as the SU(3) symmetric realization of a more general tendency to phase-separation in three-component Fermi mixtures. The three-body constraint strongly disfavors the trionic phase, stabilizing a (fully magnetized) color superfluid also at strong coupling. With increasing temperature we observe a transition to a non-magnetized SU(3) Fermi liquid phase.
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S. Diehl, E. Rico Ortega, M. Baranov, P. Zoller Topology by Dissipation in Atomic Quantum Wires,
Nature Physics 7 971, (2011-10-02),
URL doi:10.1038/nphys2106 (ID: 717686)
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Robust edge states and non-Abelian excitations are the trademark of topological states of matter, with promising applications such as "topologically protected" quantum memory and computing. While so far topological phases have been exclusively discussed in a Hamiltonian context, we show that such phases and the associated topological protection and phenomena also emerge in open quantum systems with engineered dissipation. The specific system studied here is a quantum wire of spinless atomic fermions in an optical lattice coupled to a bath. The key feature of the dissipative dynamics described by a Lindblad master equation is the existence of Majorana edge modes, representing a non-local decoherence free subspace. The isolation of the edge states is enforced by a dissipative gap in the p-wave paired bulk of the wire. We describe dissipative non-Abelian braiding operations within the Majorana subspace, and we illustrate the insensitivity to imperfections. Topological protection is granted by a nontrivial winding number of the system density matrix.
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A. Tomadin, S. Diehl, P. Zoller Nonequilibrium phase diagram of a driven and dissipative many-body system,
Phys. Rev. A 83 013611, (2011-01-18),
URL doi:10.1103/PhysRevA.83.013611 (ID: 717765)
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We study the nonequilibrium dynamics of a many-body bosonic system on a lattice, subject to driving and dissipation. The time evolution is described by a master equation, which we treat within a generalized Gutzwiller mean field approximation for density matrices. The dissipative processes are engineered such that the system, in the absence of interaction between the bosons, is driven into a homogeneous steady state with off-diagonal long-range order. We investigate how the coherent interaction affects the properties of the steady state of the system qualitatively and derive a nonequilibrium phase diagram featuring a phase transition into a steady state without long-range order. The phase diagram also exhibits an extended domain where an instability of the homogeneous steady state gives rise to a persistent density pattern with spontaneously broken translational symmetry. In the limit of low particle density, we provide a precise analytical description of the time evolution during the instability. Moreover, we investigate the transient following a quantum quench of the dissipative processes and we elucidate the prominent role played by collective topological variables in this regime.
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J. Braun, S. Diehl, M. Scherer Finite-size and Particle-number Effects in an Ultracold Fermi Gas at Unitarity,
Phys. Rev. A 84 063616, (2011-12-12),
URL doi:10.1103/PhysRevA.84.063616 (ID: 717771)
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We investigate an ultracold Fermi gas at unitarity confined in a periodic box $V=L^3$ using renormalization group (RG) techniques. Within this approach we can quantitatively assess the long range bosonic order parameter fluctuations which dominate finite-size effects. We determine the finite-size and particle-number dependence of universal quantities, such as the Bertsch parameter and the fermion gap. Moreover, we analyze how these universal observables respond to the variation of an external pairing source. Our results indicate that the Bertsch parameter saturates rather quickly to its value in the thermodynamic limit as a function of increasing box size. On the other hand, we observe that the fermion gap shows a significantly stronger dependence on the box size, in particular for small values of the pairing source. Our results may contribute to a better understanding of finite-size and particle-number effects present in Monte-Carlo simulations of ultracold Fermi gases.
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S. Diehl, S. Floerchinger, H. Gies, J. M. Pawlowski, C. Wetterich Functional renormalization group approach to the BCS-BEC crossover,
Annals of Physics 615, (2010-06-02),
URL doi: 10.1002/andp.201010458 (ID: 694286)
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The phase transition to superfluidity and the BCS-BEC crossover for an ultracold gas of fermionic atoms is discussed within a functional renormalization group approach. Non-perturbative flow equations, based on an exact renormalization group equation, describe the scale dependence of the flowing or average action. They interpolate continuously from the microphysics at atomic or molecular distance scales to the macroscopic physics at much larger length scales, as given by the interparticle distance, the correlation length, or the size of the experimental probe. We discuss the phase diagram as a function of the scattering length and the temperature and compute the gap, the correlation length and the scattering length for molecules. Close to the critical temperature, we find the expected universal behavior. Our approach allows for a description of the few-body physics (scattering and molecular binding) and the many-body physics within the same formalism.
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S. Diehl, M. Baranov, A. J. Daley, P. Zoller Observability of Quantum Criticality and a Continuous Supersolid in Atomic Gases,
Phys. Rev. Lett. 104 165301, (2010-04-20),
URL doi:10.1103/PhysRevLett.104.165301 (ID: 716766)
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We analyze the Bose-Hubbard model with three-body onsite hardcore constraint, which stabilizes the system for an attractive interparticle interaction and allows, in particular, the formation of a superfluid phase of bosonic dimers. Our approach is based on an exact mapping of the constrained Hamiltonian to a theory of two coupled bosonic degrees of freedom. We demonstrate that the phase transition between atomic and dimer superfluidity is generically of the first order as a result of the Coleman-Weinberg phenomenon, while at unit filling we identify an Ising quantum critical point. At this filling, furthermore, a symmetry enhancement in the strong coupling limit leads to a continuous supersolid phase for deeply bound dimers, observable in experiments.
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S. Diehl, M. Baranov, A. J. Daley, P. Zoller Quantum Field Theory for the Three-Body Constrained Lattice Bose Gas -- Part I: Formal Developments,
Phys. Rev. B 82 064509, (2010-08-13),
URL doi:10.1103/PhysRevB.82.064509 (ID: 716839)
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We develop a quantum field theoretical framework to analytically study the three-body constrained Bose-Hubbard model beyond mean field and non-interacting spin wave approximations. It is based on an exact mapping of the constrained model to a theory with two coupled bosonic degrees of freedom with polynomial interactions, which have a natural interpretation as single particles and two-particle states. The procedure can be seen as a proper quantization of the Gutzwiller mean field theory. The theory is conveniently evaluated in the framework of the quantum effective action, for which the usual symmetry principles are now supplemented with a ``constraint principle'' operative on short distances. We test the theory via investigation of scattering properties of few particles in the limit of vanishing density, and we address the complementary problem in the limit of maximum filling, where the low lying excitations are holes and di-holes on top of the constraint induced insulator. This is the first of a sequence of two papers. The application of the formalism to the many-body problem, which can be realized with atoms in optical lattices with strong three-body loss, is performed in a related work [13].
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S. Diehl, M. Baranov, A. J. Daley, P. Zoller Quantum Field Theory for the Three-Body Constrained Lattice Bose Gas -- Part II: Application to the Many-Body Problem,
Phys. Rev. B 82 064510, (2010-08-13),
URL doi:10.1103/PhysRevB.82.064510 (ID: 716840)
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We analyze the ground state phase diagram of attractive lattice bosons, which are stabilized by a three-body onsite hardcore constraint. A salient feature of this model is an Ising type transition from a conventional atomic superfluid to a dimer superfluid with vanishing atomic condensate. The study builds on an exact mapping of the constrained model to a theory of coupled bosons with polynomial interactions, proposed in a related paper [11]. In this framework, we focus by analytical means on aspects of the phase diagram which are intimately connected to interactions, and are thus not accessible in a mean field plus spin wave approach. First, we determine shifts in the mean field phase border, which are most pronounced in the low density regime. Second, the investigation of the strong coupling limit reveals the existence of a new collective mode, which emerges as a consequence of enhanced symmetries in this regime. Third, we show that the Ising type phase transition, driven first order via the competition of long wavelength modes at generic fillings, terminates into a true Ising quantum critical point in the vicinity of half filling.
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S. Diehl, A. Tomadin, A. Micheli, R. Fazio, P. Zoller Dynamical Phase Transitions and Instabilities in Open Atomic Many-Body Systems,
Phys. Rev. Lett. 105 015702, (2010-07-01),
URL doi:10.1103/PhysRevLett.105.015702 (ID: 717159)
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We discuss an open driven-dissipative many-body system, in which the competition of unitary Hamiltonian and dissipative Liouvillian dynamics leads to a nonequilibrium phase transition. It shares features of a quantum phase transition in that it is interaction driven, and of a classical phase transition, in that the ordered phase is continuously connected to a thermal state. Within a generalized Gutzwiller approach which includes the description of mixed state density matrices, we characterize the complete phase diagram and the critical behavior at the phase transition approached as a function of time. We find a novel fluctuation induced dynamical instability, which occurs at long wavelength as a consequence of a subtle dissipative renormalization effect on the speed of sound.
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P. Amann, M. Cordin, C. Braun, B. A. Lechner, A. Menzel, E. Bertel, C. Franchini, R. Zucca, J. Redinger, M. Baranov, S. Diehl Electronically driven phase transitions in a quasi-one-dimensional adsorbate system,
Eur. Phys. J. B (2010-02-02),
URL doi:10.1140/epjb/e2010-00026-5 (ID: 717176)
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A quasi-1D system is prepared using the Pt(110) surface as a template. The electronic surface resonance structure is studied by angle-resolved photoemission spectroscopy for the clean surface as well as for different Bromine coverages. A Fermi surface mapping reveals saddle points at the Fermi level in the interior of the surface Brillouin zone. Correspondingly, a maximum in the static response function χ(q, 0) at the connecting vector q is expected. With 1/2Gx < q < 2/3Gx one observes indeed a 3-fold periodicity around defects and a 2-fold periodicity at low temperature for ΘBr = 0.5 ML. Cooling of a defect-free c(2×2)-Br/Pt(110) preparation counter-intuitively results in a loss of long-range order. Motivated by DFT calculations this is attributed to an anomalous order-order phase transition into the (2×1) phase accompanied by intense, strongly anisotropic fluctuations within a temperature range of ~200 K. The peculiar behaviour is rationalised in terms of a competition between inter-adsorbate repulsion and an adsorbate triggered 2kF interaction in the substrate.
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S. Diehl, W. Yi, A. J. Daley, P. Zoller Dissipation-Induced d-Wave Pairing of Fermionic Atoms in an Optical Lattice,
Phys. Rev. Lett. 105 227001, (2010-11-22),
URL doi:10.1103/PhysRevLett.105.227001 (ID: 717276)
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We show how dissipative dynamics can give rise to pairing for two-component fermions on a lattice. In particular, we construct a \\\\\\\"parent\\\\\\\" Liouvillian operator so that a BCS-type state of a given symmetry, e.g. a d-wave state, is reached for arbitrary initial states in the absence of conservative forces. The system-bath couplings describe single-particle, number conserving and quasi-local processes. The pairing mechanism crucially relies on Fermi statistics. We show how such Liouvillians can be realized via reservoir engineering with cold atoms representing a driven dissipative dynamics.
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M. Cordin, B. A. Lechner, P. Amann, A. Menzel, E. Bertel, C. Franchini, R. Zucca, J. Redinger, M. Baranov, S. Diehl Phase transitions driven by competing interactions in low-dimensional systems,
Europhysics Letters 92 26004, (2010-11-16),
URL doi: 10.1209/0295-5075/92/26004 (ID: 717348)
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Variable-temperature scanning tunnelling microscopy is used to study an order-order phase transition in a virtually defect-free quasi–one-dimensional surface system. The phase transition is driven by competing electronic interactions. The phase diagram is captured by a modified Landau formalism containing a coupling term between two different subsystems. The extra term has the effect of a spontaneously generated field which drives the phase transition. The proposed formalism applies to a variety of problems, where competing interactions produce sometimes counter-intuitive ordering phenomena.
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S. Diehl, H. C. Krahl, M. Scherer Three-Body Scattering from Nonperturbative Flow Equations,
Phys. Rev. C 78 034001, (2008-09-05),
URL doi:10.1103/PhysRevC.78.034001 (ID: 547706)
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We consider fermion-dimer scattering in the presence of a large positive scattering length in the frame of functional renormalization group equations. A flow equation for the momentum dependent fermion-dimer scattering amplitude is derived from first principles in a systematic vertex expansion of the exact flow equation for the effective action. The resummation obtained from the nonperturbative flow is shown to be equivalent to the one performed by the phenomenological integral equation by Skorniakov and Ter-Martirosian (STM). However, the flow equation approach allows to integrate out fermions and bosons simultaneously, in line with the fact that the bosons are not fundamental but build up dynamically as fluctuation induced bound states of fermions. In particular, the STM result for atom-dimer scattering is obtained by choosing the relative cutoff scales of fermions and bosons such that the fermions are integrated out already at the initial stage of the RG evolution. The STM picture, based on an effective interacting theory for ``fundamental'' fermions and dimers, is then recovered.
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S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. Büchler, P. Zoller Quantum States and Phases in Driven Open Quantum Systems with Cold Atoms,
Nature Physics 4 878, (2008-09-07),
URL doi:10.1038/nphys1073 (ID: 576233)
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An open quantum system, whose time evolution is governed by a master equation, can be driven into a given pure quantum state by an appropriate design of the system-reservoir coupling. This points out a route towards preparing many body states and non-equilibrium quantum phases by quantum reservoir engineering. Here we discuss in detail the example of a \emph{driven dissipative Bose Einstein Condensate} of bosons and of paired fermions, where atoms in an optical lattice are coupled to a bath of Bogoliubov excitations via the atomic current representing \emph{local dissipation}. In the absence of interactions the lattice gas is driven into a pure state with long range order. Weak interactions lead to a weakly mixed state, which in 3D can be understood as a depletion of the condensate, and in 1D and 2D exhibits properties reminiscent of a Luttinger liquid or a Kosterlitz-Thouless critical phase at finite temperature, with the role of the ``finite temperature'' played by the interactions.
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B. Kraus, H. Büchler, S. Diehl, A. Kantian, A. Micheli, P. Zoller Preparation of Entangled States by Quantum Markov Processes,
Phys. Rev. A 78 042307, (2008-05-02),
URL doi:10.1103/PhysRevA.78.042307 (ID: 576234)
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We investigate the possibility of using a dissipative process to prepare a quantum system in a desired state. We derive for any multipartite pure state a dissipative process for which this state is the unique stationary state and solve the corresponding master equation analytically. For certain states, such as the cluster states, we use this process to show that the jump operators can be chosen quasilocally, i.e. they act nontrivially only on a few, neighboring qubits. Furthermore, the relaxation time of this dissipative process is independent of the number of subsystems. We demonstrate the general formalism by considering arbitrary matrix-product states or projected entangled pair states. In particular, we show that the ground state of the Affleck-Kennedy-Lieb-Tasaki model can be prepared employing a quasi-local dissipative process.
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S. Floerchinger, M. Scherer, S. Diehl, C. Wetterich Particle-hole fluctuations in the BCS-BEC Crossover,
Phys. Rev. B 78 174528, (2008-11-26),
URL doi:10.1103/PhysRevB.78.174528 (ID: 606176)
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The effect of particle-hole fluctuations for the BCS-BEC crossover is investigated by use of functional renormalization. We compute the Gorkov effect and the critical temperature for the whole range in the scattering length a. On the BCS side for small negative a we recover the Gorkov approximation, while on the BEC side of small positive a the particle-hole fluctuations play no important role, and we find a system of interacting bosons. In the unitarity limit of infinite scattering length our quantitative estimate yields Tc/TF=0.264. We also investigate the crossover from broad to narrow Feshbach resonances—for the latter we obtain Tc/TF=0.204 for a−1=0. A key ingredient for our treatment is the computation of the momentum dependent four-fermion vertex and its bosonization in terms of an effective bound-state exchange.
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