Most classical many-body systems are chaotic in the sense that trajectories that start off nearby end up in completely different -- essentially independent -- places. (This is more or less intuitive, consider what happens when you fuck up a pool shot.) Since a quantum trajectory is, intuitively, just a smeared-out quantum trajectory, one might expect the quantum mechanical evolution of a chaotic system to spread out a narrowly focused initial wavefunction all over trajectory space: it would appear to follow from this logic that the steady-state wavefunctions, i.e. the eigenfunctions, should be a featureless average over trajectories.
In fact this isn't true, and it turns out that closed classical trajectories, even if they're unstable to small perturbations, leave "scars" of high density on the eigenfunctions, and therefore on the steady-state and equilibrium density profiles of various systems. The scarring effect was first really explained by Eric Heller in 1984:
Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits
Phys. Rev. Lett. 53, 1515 - 1518 (1984)
Eric J. Heller
In the latest PRL, Huang et al. extend this work to relativistic quantum systems described by the Dirac equation. Among other things, the new work offers a nice example of how the electron gas in graphene, which obeys the Dirac equation, can be used to study features of relativistic quantum mechanics that would otherwise be experimentally inaccessible.
Relativistic Quantum Scars
Phys. Rev. Lett. 103, 054101 (2009)
Liang Huang et al.
Abstract. The concentrations of wave functions about classical periodic orbits, or quantum scars, are a fundamental phenomenon in physics. An open question is whether scarring can occur in relativistic quantum systems. To address this question, we investigate confinements made of graphene whose classical dynamics are chaotic and find unequivocal evidence of relativistic quantum scars. The scarred states can lead to strong conductance fluctuations in the corresponding open quantum dots via the mechanism of resonant transmission.
Monday, July 27, 2009
Tuesday, July 21, 2009
Topologically interesting superconductors
Cooper pairs with orbital angular momentum -- p-wave or d-wave -- have order parameters that are potentially more interesting than the U(1) order parameter of a BCS superconductor. The basic idea is that there are three p orbitals, five d orbitals, etc. instead of a single s orbital; therefore, e.g., p_x + i p_y and p_x - i p_y are equally good orbitals to Cooper pair in. In superconductors, the underlying crystal lattice picks out the axes and reduces the orbital symmetry group to a discrete one: e.g., in strontium ruthenate, one has domains of p_x + ip_y and of p_x - ip_y, separated by what appear to be domain walls. However, if one could make p- or d-wave pairs with cold fermions, one would presumably have Cooper pairs spontaneously breaking a large continuous symmetry, and this, in principle, could give rise to interesting sorts of topological defects. (e.g. triplet superfluids have half-quantum vortices, in which the orbital and spin parts each rotate by pi when you go around a loop, so that the total wavefunction returns to itself but the spatial part doesn't. These vortices turn out to be interesting for quantum computation.)
There are two new arxiv papers on this topic; the first is from Nigel Cooper and Gora Shlyapnikov, both quite well-known:
---
Stable Topological Superfluid Phase of Ultracold Polar Fermionic Molecules
N. R. Cooper, G. V. Shlyapnikov
http://arxiv.org/abs/0907.3080
Abstract: We show that single-component fermionic polar molecules confined to a 2D geometry and dressed by a microwave field, may acquire an attractive $1/r^3$ dipole-dipole interaction leading to superfluid p-wave pairing at sufficiently low temperatures even in the BCS regime. The emerging state is the topological $p_x+ip_y$ phase promising for topologically protected quantum information processing. The main decay channel is via collisional transitions to dressed states with lower energies and is rather slow, setting a lifetime of the order of seconds at 2D densities $\sim 10^8$ cm$^{-2}$.
---
The idea here is something like this. You trap a bunch of dipolar molecules in a 2D trap and impose an external magnetic field to line up the dipoles. Next, you use microwaves to have the make the dipole moments precess (a la NMR); if you do this right, the time-averaged dipole-dipole interaction is attractive and you get Cooper pairing. The authors don't offer an intuitive explanation of why p-wave pairing is preferred; I assume it has to do with the fact that dipolar-molecule systems are naturally ferromagnetic because of the external fields keeping the dipoles lined up, so the natural ground state tends to have all spins lined up pointing along the field, which favors spin-triplet pairing and therefore p-wave pairing.
The other paper is about d-wave pairing:
---
Textures and non-Abelian vortices in atomic d-wave paired Fermi condensates
Authors: H. M. Adachi, Y. Tsutsumi, J. A. M. Huhtamäki, K. Machida
http://arxiv.org/abs/0907.2972
Abstract: We report on fundamental properties of superfluids with d-wave pairing symmetry. We consider neutral atomic Fermi gases in a harmonic trap, the pairing being produced by a Feshbach resonance via a d-wave interaction channel. A Ginzburg-Landau (GL) functional is constructed which is symmetry constrained for five component order parameters (OP). We find OP textures in the cyclic phase and stability conditions for a non-Abelian fractional 1/3-vortex under rotation. It is proposed how to create the intriguing 1/3-vortex experimentally in atomic gases via optical means.
---
The 1/3-quantum vortex is interesting even if it isn't realizable. What seems to me most interesting about this paper is that it carefully accounts for the influence of the harmonic trap on vortex energetics -- that defects which would be energetically unstable in bulk might be stable in a trap is an interesting possibility.
There are two new arxiv papers on this topic; the first is from Nigel Cooper and Gora Shlyapnikov, both quite well-known:
---
Stable Topological Superfluid Phase of Ultracold Polar Fermionic Molecules
N. R. Cooper, G. V. Shlyapnikov
http://arxiv.org/abs/0907.3080
Abstract: We show that single-component fermionic polar molecules confined to a 2D geometry and dressed by a microwave field, may acquire an attractive $1/r^3$ dipole-dipole interaction leading to superfluid p-wave pairing at sufficiently low temperatures even in the BCS regime. The emerging state is the topological $p_x+ip_y$ phase promising for topologically protected quantum information processing. The main decay channel is via collisional transitions to dressed states with lower energies and is rather slow, setting a lifetime of the order of seconds at 2D densities $\sim 10^8$ cm$^{-2}$.
---
The idea here is something like this. You trap a bunch of dipolar molecules in a 2D trap and impose an external magnetic field to line up the dipoles. Next, you use microwaves to have the make the dipole moments precess (a la NMR); if you do this right, the time-averaged dipole-dipole interaction is attractive and you get Cooper pairing. The authors don't offer an intuitive explanation of why p-wave pairing is preferred; I assume it has to do with the fact that dipolar-molecule systems are naturally ferromagnetic because of the external fields keeping the dipoles lined up, so the natural ground state tends to have all spins lined up pointing along the field, which favors spin-triplet pairing and therefore p-wave pairing.
The other paper is about d-wave pairing:
---
Textures and non-Abelian vortices in atomic d-wave paired Fermi condensates
Authors: H. M. Adachi, Y. Tsutsumi, J. A. M. Huhtamäki, K. Machida
http://arxiv.org/abs/0907.2972
Abstract: We report on fundamental properties of superfluids with d-wave pairing symmetry. We consider neutral atomic Fermi gases in a harmonic trap, the pairing being produced by a Feshbach resonance via a d-wave interaction channel. A Ginzburg-Landau (GL) functional is constructed which is symmetry constrained for five component order parameters (OP). We find OP textures in the cyclic phase and stability conditions for a non-Abelian fractional 1/3-vortex under rotation. It is proposed how to create the intriguing 1/3-vortex experimentally in atomic gases via optical means.
---
The 1/3-quantum vortex is interesting even if it isn't realizable. What seems to me most interesting about this paper is that it carefully accounts for the influence of the harmonic trap on vortex energetics -- that defects which would be energetically unstable in bulk might be stable in a trap is an interesting possibility.
Saturday, July 18, 2009
Kondo polarons
A few days ago I had what I thought was a good idea: to look at the behavior of moving magnetic impurities, and see if they interpolate somehow between the Kondo effect and the polaron as you tune the mass of the impurity down from infinity. (There are some obvious ways to realize the interpolation experimentally using cold Fermi gases.) I was disappointed to find that someone else got there first, and tantalizingly recently (May 2008); Lamacraft's result is, however, really interesting, and would've been beyond my powers to derive:
Kondo Polarons in a One-Dimensional Fermi Gas
Austen Lamacraft
PRL 101, 225301 (2008)
Abstract: We consider the motion of a spin-1/2 impurity in a one-dimensional gas of spin-1/2 fermions. For antiferromagnetic interaction between the impurity and the fermions, the low temperature behavior of the system is governed by the two-channel Kondo effect, leading to the impurity becoming completely opaque to the spin excitations of the gas. As well as the known spectral signatures of the two-channel Kondo effect, we find that the low temperature mobility of the resulting `Kondo polaron' takes the universal form $\mu\to \frac{3\hbar v_F^2}{2\pi k_B^2T^2}$, in sharp contrast to the spinless case where $\mu\propto T^{-4}$.
The really interesting thing here is that the case of the mobile impurity differs drastically from that of the fixed (infinite-mass) impurity because certain processes that are permitted with an infinite-mass impurity are forbidden, because of the impossibility of simultaneously conserving energy and momentum, in a finite-mass impurity. One consequence of this is that, whereas the infinite-mass impurity couples to a single scattering channel (even combinations of left- and right-moving traveling waves) the finite-mass impurity couples independently to the left- and right-moving channels. This means that the 1D Kondo problem with recoil is not the one-channel Kondo problem but the far more exotic two-channel Kondo problem.
Kondo Polarons in a One-Dimensional Fermi Gas
Austen Lamacraft
PRL 101, 225301 (2008)
Abstract: We consider the motion of a spin-1/2 impurity in a one-dimensional gas of spin-1/2 fermions. For antiferromagnetic interaction between the impurity and the fermions, the low temperature behavior of the system is governed by the two-channel Kondo effect, leading to the impurity becoming completely opaque to the spin excitations of the gas. As well as the known spectral signatures of the two-channel Kondo effect, we find that the low temperature mobility of the resulting `Kondo polaron' takes the universal form $\mu\to \frac{3\hbar v_F^2}{2\pi k_B^2T^2}$, in sharp contrast to the spinless case where $\mu\propto T^{-4}$.
The really interesting thing here is that the case of the mobile impurity differs drastically from that of the fixed (infinite-mass) impurity because certain processes that are permitted with an infinite-mass impurity are forbidden, because of the impossibility of simultaneously conserving energy and momentum, in a finite-mass impurity. One consequence of this is that, whereas the infinite-mass impurity couples to a single scattering channel (even combinations of left- and right-moving traveling waves) the finite-mass impurity couples independently to the left- and right-moving channels. This means that the 1D Kondo problem with recoil is not the one-channel Kondo problem but the far more exotic two-channel Kondo problem.
Friday, July 17, 2009
Exciton-polariton-BEC-excitation-mediated superconductivity
New on the arxiv:
Exciton-polariton mediated superconductivity
Authors: Fabrice P. Laussy, Alexey Kavokin, Ivan Shelykh
http://arxiv.org/abs/0907.2374
Abstract: We revisit the exciton mechanism of superconductivity in the framework of microcavity physics, replacing virtual excitons as a binding agent of Cooper pairs by excitations of a real exciton-polariton Bose-Einstein condensate. We consider a model microcavity where quantum wells with a two dimensional electron gas sandwich an undoped quantum well, where an exciton-polariton condensate is formed. We show that the critical temperature for superconductivity increases with the condensate population--opening a new route towards high temperature superconductivity.
---
The usual binding mechanism for a Cooper pair is some kind of attractive electron-boson-electron interaction. In the BCS theory the bosons are taken to be phonons, but in principle they could be anything. In particular, they could be excitations of a Bose condensate. There was some old work on this by Bardeen, Baym, and Pines when they were trying to make Cooper pairs of helium-3 through interactions mediated by helium-4:
Effective Interaction of He3 Atoms in Dilute Solutions of He3 in He4 at Low Temperatures
J. Bardeen, G. Baym, and D. Pines
Phys. Rev. 156, 207 - 221 (1967)
The new work suggests using exciton-polaritons, which are electron-hole pairs hybridized with laser photons, to serve a similar purpose. The basic appeal of exciton-polariton BECs is that they have a high transition temperature compared with other BECs so in principle an exciton polariton BEC could stabilize a relatively high-temperature superconductor via the electron--exciton-BEC-excitation--electron interaction.
Exciton-polariton mediated superconductivity
Authors: Fabrice P. Laussy, Alexey Kavokin, Ivan Shelykh
http://arxiv.org/abs/0907.2374
Abstract: We revisit the exciton mechanism of superconductivity in the framework of microcavity physics, replacing virtual excitons as a binding agent of Cooper pairs by excitations of a real exciton-polariton Bose-Einstein condensate. We consider a model microcavity where quantum wells with a two dimensional electron gas sandwich an undoped quantum well, where an exciton-polariton condensate is formed. We show that the critical temperature for superconductivity increases with the condensate population--opening a new route towards high temperature superconductivity.
---
The usual binding mechanism for a Cooper pair is some kind of attractive electron-boson-electron interaction. In the BCS theory the bosons are taken to be phonons, but in principle they could be anything. In particular, they could be excitations of a Bose condensate. There was some old work on this by Bardeen, Baym, and Pines when they were trying to make Cooper pairs of helium-3 through interactions mediated by helium-4:
Effective Interaction of He3 Atoms in Dilute Solutions of He3 in He4 at Low Temperatures
J. Bardeen, G. Baym, and D. Pines
Phys. Rev. 156, 207 - 221 (1967)
The new work suggests using exciton-polaritons, which are electron-hole pairs hybridized with laser photons, to serve a similar purpose. The basic appeal of exciton-polariton BECs is that they have a high transition temperature compared with other BECs so in principle an exciton polariton BEC could stabilize a relatively high-temperature superconductor via the electron--exciton-BEC-excitation--electron interaction.
Thursday, July 16, 2009
Non-abelian FQHE with cold atoms
New in PRL:
Non-Abelian Optical Lattices: Anomalous Quantum Hall Effect and Dirac Fermions
Phys. Rev. Lett. 103, 035301 (2009)
N. Goldman,1 A. Kubasiak,2,3 A. Bermudez,4 P. Gaspard,1 M. Lewenstein,2,5 and M. A. Martin-Delgado4
Abstract: We study the properties of an ultracold Fermi gas loaded in an optical square lattice and subjected to an external and classical non-Abelian gauge field. We show that this system can be exploited as an optical analogue of relativistic quantum electrodynamics, offering a remarkable route to access the exotic properties of massless Dirac fermions with cold atoms experiments. In particular, we show that the underlying Minkowski space-time can also be modified, reaching anisotropic regimes where a remarkable anomalous quantum Hall effect and a squeezed Landau vacuum could be observed.
---
This is an interesting proposal but I'm a little skeptical about its _actual_ realizability. The basic idea behind imposing the gauge fields is to drive rather involved sequences of transitions in multilevel atoms, as outlined, e.g., here:
Non-Abelian Gauge Potentials for Ultracold Atoms with Degenerate Dark States
Phys. Rev. Lett. 95, 010404 (2005)
J. Ruseckas1,2, G. Juzeliūnas1, P. Öhberg3, and M. Fleischhauer2
Abstract: We show that the adiabatic motion of ultracold, multilevel atoms in spatially varying laser fields can give rise to effective non-Abelian gauge fields if degenerate adiabatic eigenstates of the atom-laser interaction exist. A pair of such degenerate dark states emerges, e.g., if laser fields couple three internal states of an atom to a fourth common one under pairwise two-photon-resonance conditions. For this so-called tripod scheme we derive general conditions for truly non-Abelian gauge potentials and discuss special examples. In particular we show that using orthogonal laser beams with orbital angular momentum an effective magnetic field can be generated that has a monopole component.
It feels like it'd be pretty hard to separate out the cond-mat effect one is looking for, i.e. the FQHE, from all the atomic physics details.
Non-Abelian Optical Lattices: Anomalous Quantum Hall Effect and Dirac Fermions
Phys. Rev. Lett. 103, 035301 (2009)
N. Goldman,1 A. Kubasiak,2,3 A. Bermudez,4 P. Gaspard,1 M. Lewenstein,2,5 and M. A. Martin-Delgado4
Abstract: We study the properties of an ultracold Fermi gas loaded in an optical square lattice and subjected to an external and classical non-Abelian gauge field. We show that this system can be exploited as an optical analogue of relativistic quantum electrodynamics, offering a remarkable route to access the exotic properties of massless Dirac fermions with cold atoms experiments. In particular, we show that the underlying Minkowski space-time can also be modified, reaching anisotropic regimes where a remarkable anomalous quantum Hall effect and a squeezed Landau vacuum could be observed.
---
This is an interesting proposal but I'm a little skeptical about its _actual_ realizability. The basic idea behind imposing the gauge fields is to drive rather involved sequences of transitions in multilevel atoms, as outlined, e.g., here:
Non-Abelian Gauge Potentials for Ultracold Atoms with Degenerate Dark States
Phys. Rev. Lett. 95, 010404 (2005)
J. Ruseckas1,2, G. Juzeliūnas1, P. Öhberg3, and M. Fleischhauer2
Abstract: We show that the adiabatic motion of ultracold, multilevel atoms in spatially varying laser fields can give rise to effective non-Abelian gauge fields if degenerate adiabatic eigenstates of the atom-laser interaction exist. A pair of such degenerate dark states emerges, e.g., if laser fields couple three internal states of an atom to a fourth common one under pairwise two-photon-resonance conditions. For this so-called tripod scheme we derive general conditions for truly non-Abelian gauge potentials and discuss special examples. In particular we show that using orthogonal laser beams with orbital angular momentum an effective magnetic field can be generated that has a monopole component.
It feels like it'd be pretty hard to separate out the cond-mat effect one is looking for, i.e. the FQHE, from all the atomic physics details.
Wednesday, July 15, 2009
Friedel sum rule and Levinson's Theorem
Having had a somewhat haphazard education, I hadn't heard of the Friedel sum rule until I needed it for my research. As usual, Google yielded nothing remotely pedagogical, just research articles, and neither Ashcroft nor AGD had anything to say about the rule. It turns out, however, that Ziman's book Principles of Solid-State Physics has a nice discussion. It also turns out that, if I'm not being a complete idiot, this rule is a case of a general scattering theory result -- again underrepresented in textbooks -- known as Levinson's theorem.
Levinson's theorem relates the zero-energy scattering phase shift of a short-range potential to the number of bound states in the potential. The Friedel sum rule relates the number of states below the Fermi energy to the scattering phase shift at the Fermi energy. The relation in both cases is that the zero-energy (Fermi-energy) phase shift is pi times the number of states. The idea behind Levinson's theorem goes something like this -- as you tune the energy of your incoming scatterer down from infinity (where there's no phase shift), the scattering phase shift goes through pi iff you hit a scattering resonance associated with a bound state in your potential. Add up all the pi's and you get the number of bound states.
The usual argument for the Friedel sum rule goes like this -- put the system in a box, the scattering phase shift feeds into the quantization condition at the edges of the box, and depending on what the phase shifts are you can change the number of states that are below the Fermi energy. I would assume, though, that you could alternatively prove it by putting the system in a box that extends up to the Fermi energy, then using Levinson's theorem to count the bound states.
I should mention that the Friedel sum rule is true for Fermi liquids as well as for the free Fermi gas, which is not terribly surprising as the states of the two are in 1-to-1 correspondence. The equivalence is proved in the following paper:
Friedel Sum Rule for a System of Interacting Electrons
J.S. Langer and V. Ambegaokar
Phys. Rev. 121, 1090 (1961)
Levinson's theorem relates the zero-energy scattering phase shift of a short-range potential to the number of bound states in the potential. The Friedel sum rule relates the number of states below the Fermi energy to the scattering phase shift at the Fermi energy. The relation in both cases is that the zero-energy (Fermi-energy) phase shift is pi times the number of states. The idea behind Levinson's theorem goes something like this -- as you tune the energy of your incoming scatterer down from infinity (where there's no phase shift), the scattering phase shift goes through pi iff you hit a scattering resonance associated with a bound state in your potential. Add up all the pi's and you get the number of bound states.
The usual argument for the Friedel sum rule goes like this -- put the system in a box, the scattering phase shift feeds into the quantization condition at the edges of the box, and depending on what the phase shifts are you can change the number of states that are below the Fermi energy. I would assume, though, that you could alternatively prove it by putting the system in a box that extends up to the Fermi energy, then using Levinson's theorem to count the bound states.
I should mention that the Friedel sum rule is true for Fermi liquids as well as for the free Fermi gas, which is not terribly surprising as the states of the two are in 1-to-1 correspondence. The equivalence is proved in the following paper:
Friedel Sum Rule for a System of Interacting Electrons
J.S. Langer and V. Ambegaokar
Phys. Rev. 121, 1090 (1961)
Tuesday, July 14, 2009
Integrability in disconnecting air bubbles
This is a rather neat bit of theory. It's motivated by the following paradox: integrable systems remember everything about their initial conditions, whereas the behavior of systems at "finite-time singularities" (the dynamical-systems equivalent of phase transitions; cf. Chapter 10 of Nigel Goldenfeld's book) exhibits "universality," which implies insensitivity to initial conditions. What happens to integrable systems with finite-time singularities, e.g. a disconnecting air bubble? It turns out that, while some properties of the bubble exhibit universal behavior, others are sensitive to the initial conditions.
Memory-encoding vibrations in a disconnecting air bubble
Laura E. Schmidt, Nathan C. Keim, Wendy W. Zhang & Sidney R. Nagel
Nature Phys. 5, 343 - 346 (2009)
Many nonlinear processes, such as the propagation of waves over an ocean or the transmission of light pulses down an optical fibre1, are integrable in the sense that the dynamics has as many conserved quantities as there are independent variables. The result is a time evolution that retains a complete memory of the initial state. In contrast, the nonlinear dynamics near a finite-time singularity, in which physical quantities such as pressure or velocity diverge at a point in time, is believed to evolve towards a universal form, one independent of the initial state2. The break-up of a water drop in air3 or a viscous liquid inside an immiscible oil4, 5 are processes that conform to this second scenario. These opposing scenarios collide in the nonlinearity produced by the formation of a finite-time singularity that is also integrable. We demonstrate here that the result is a novel dynamics with a dual character.
Memory-encoding vibrations in a disconnecting air bubble
Laura E. Schmidt, Nathan C. Keim, Wendy W. Zhang & Sidney R. Nagel
Nature Phys. 5, 343 - 346 (2009)
Many nonlinear processes, such as the propagation of waves over an ocean or the transmission of light pulses down an optical fibre1, are integrable in the sense that the dynamics has as many conserved quantities as there are independent variables. The result is a time evolution that retains a complete memory of the initial state. In contrast, the nonlinear dynamics near a finite-time singularity, in which physical quantities such as pressure or velocity diverge at a point in time, is believed to evolve towards a universal form, one independent of the initial state2. The break-up of a water drop in air3 or a viscous liquid inside an immiscible oil4, 5 are processes that conform to this second scenario. These opposing scenarios collide in the nonlinearity produced by the formation of a finite-time singularity that is also integrable. We demonstrate here that the result is a novel dynamics with a dual character.
Old(-ish) stuff: quenches and coarsening
Quenches are a generic term for when you take an initial density matrix (say the thermal density matrix of Hamiltonian H) and time-evolve it with a Hamiltonian H' that's very different from H. A classical quench is typically a thermal quench -- you cool a system rapidly across a phase transition, and watch the domain walls grow. A good review of the old stuff is:
Theory of Phase-Ordering Kinetics
A.J. Bray
cond-mat/9501089 (no figures)
Adv. Phys. 43, 357 (1994)
There are three basic take-away messages -- (1) the dynamic scaling hypothesis, which states that all the time-dependence of correlation functions, response functions, etc. depends on a single scale, which is set (in some models, at least) by the average bubble size. (2) The bubble growth laws depend on whether the order parameter is locally conserved (e.g., binary fluid at separation/mixing) or not (e.g., magnets), as well as on the dimensionality of the order parameter (so, e.g., domain walls grow differently from vortices and hedgehogs). (3) Within any bubble, the system behaves like an ordered state with the equilibrium order parameter at the final temperature: therefore, response functions tend to have a two-time/two-length behavior, depending on whether they're probing the short-time respones, which is basically equilibrium, or the long-time response, which depends on the motion of the domain walls.
There are variants on this problem that involve, e.g., quenching from/to a critical point (so that the initial state has long-range correlations), or quenching across a Kosterlitz-Thouless transition, or quenching in the presence of disorder.
Zero-temperature quenches (microcanonical)
Here the initial state is a wavefunction that is very far from being an eigenfunction of the Hamiltonian it evolves under. (Usually it's the ground state of a totally different Hamiltonian but that's unnecessary.) Most of the experimental work in this field has involved quenching across the superfluid-to-Mott transition in optical lattices; it is at present unclear whether the systems ever approach a locally equilibrium state.
Here are some papers that don't necessarily agree with each other about thermalization:
Oscillating superfluidity of bosons in optical lattices
E. Altman and A. Auerbach
PRL 89, 250404 (2002)
Altman and Auerbach argue that quenching from the Mott to the superfluid phase will lead to macroscopic oscillations of the order parameter (which eventually die out because of quasiparticle collisions), and that underdamped relaxation should be observable. I'm not sure that underdamped relaxation has been seen experimentally for this particular case, but it was observed experimentally for a quench from the SF to the Mott state.
Collapse and revival of the matter wave field of a Bose–Einstein condensate
Markus Greiner, Olaf Mandel, Theodor W. Hänsch & Immanuel Bloch
Nature 419, 51 (2002)
This famous experiment demonstrated the periodic collapse and revival of Bragg peaks (indicative, usually, of phase coherence) when a BEC was quenched into the Mott state. This is an instance of underdamped relaxation -- trap-induced anharmonicities, etc., eventually kill phase coherence. Interestingly, the deeper the quench, the slower the system is to relax.
A quantum Newton's cradle
T. Kinoshita, T. Wenger, and D. Weiss
Nature 440, 900 (2006)
In this experiment, a 1D Bose gas -- bosons confined in an array of very narrow tubes -- was started off in a far-from-equilibrium condition, with half the atoms flying off to the left and the other half flying off to the right, and allowed to time-evolve. The momentum distribution didn't seem to approach a thermal distribution on the experimental timescales; this was interpreted as a consequence of the integrability of the system. (Roughly, an integrable system has a huge number of conserved quantities, and therefore retains much more information about its initial conditions than a normal system does.) This is a stronger result than underdamped relaxation; the experiment showed no tendency towards relaxation at all.
Quench dynamics and nonequilibrium phase diagram of the Bose-Hubbard model
C. Kollath, A. Lauchli, and E. Altman
PRL 98, 180601 (2007)
Kollath et al. found, rather surprisingly, that thermalization doesn't necessarily happen even in nonintegrable models. They simulated a quench from the SF to the Mott phase, using exact diagonalization techniques as well as a density-matrix renormalization group algorithm. The result is that if one quenches from the SF deep into the Mott, the system doesn't seem to thermalize. They explained their results with an argument that goes something like this -- thermalization occurs as a consequence of the decay of double occupancies; however, a double occupancy has energy U (on-site repulsion) and has to decay into quasi-particles that have energy of order J (roughly the bandwidth of the lower Mott band). If U >> J then this can only happen at very high orders in perturbation theory, so it takes forever.
One could interpret this result as being about the observation, rather than the occurrence, of thermalization -- from a practical point of view, a system that takes very long to thermalize looks like one that doesn't thermalize.
Thermalization and its mechanism for generic isolated quantum systems
M. Rigol, V. Dunjko, and M. Olshanii
Nature 452, 854-858 (arxiv version)
This paper presents the "eigenstate thermalization" hypothesis. The hypothesis, as I understand it (which is not very well), states that the expectation values of physically interesting observables are essentially constant for eigenstates that are near each other in energy. There is some numerical reason to believe that this is true, but as far as I know there hasn't been much analytic work. Rigol et al. also suggest a generalization of this hypothesis to integrable systems; the upshot is the idea of a generalized Gibbs ensemble, which introduces "chemical potentials" (i.e. Lagrange multipliers) corresponding to all the conserved quantities.
[... to be continued]
Theory of Phase-Ordering Kinetics
A.J. Bray
cond-mat/9501089 (no figures)
Adv. Phys. 43, 357 (1994)
There are three basic take-away messages -- (1) the dynamic scaling hypothesis, which states that all the time-dependence of correlation functions, response functions, etc. depends on a single scale, which is set (in some models, at least) by the average bubble size. (2) The bubble growth laws depend on whether the order parameter is locally conserved (e.g., binary fluid at separation/mixing) or not (e.g., magnets), as well as on the dimensionality of the order parameter (so, e.g., domain walls grow differently from vortices and hedgehogs). (3) Within any bubble, the system behaves like an ordered state with the equilibrium order parameter at the final temperature: therefore, response functions tend to have a two-time/two-length behavior, depending on whether they're probing the short-time respones, which is basically equilibrium, or the long-time response, which depends on the motion of the domain walls.
There are variants on this problem that involve, e.g., quenching from/to a critical point (so that the initial state has long-range correlations), or quenching across a Kosterlitz-Thouless transition, or quenching in the presence of disorder.
Zero-temperature quenches (microcanonical)
Here the initial state is a wavefunction that is very far from being an eigenfunction of the Hamiltonian it evolves under. (Usually it's the ground state of a totally different Hamiltonian but that's unnecessary.) Most of the experimental work in this field has involved quenching across the superfluid-to-Mott transition in optical lattices; it is at present unclear whether the systems ever approach a locally equilibrium state.
Here are some papers that don't necessarily agree with each other about thermalization:
Oscillating superfluidity of bosons in optical lattices
E. Altman and A. Auerbach
PRL 89, 250404 (2002)
Altman and Auerbach argue that quenching from the Mott to the superfluid phase will lead to macroscopic oscillations of the order parameter (which eventually die out because of quasiparticle collisions), and that underdamped relaxation should be observable. I'm not sure that underdamped relaxation has been seen experimentally for this particular case, but it was observed experimentally for a quench from the SF to the Mott state.
Collapse and revival of the matter wave field of a Bose–Einstein condensate
Markus Greiner, Olaf Mandel, Theodor W. Hänsch & Immanuel Bloch
Nature 419, 51 (2002)
This famous experiment demonstrated the periodic collapse and revival of Bragg peaks (indicative, usually, of phase coherence) when a BEC was quenched into the Mott state. This is an instance of underdamped relaxation -- trap-induced anharmonicities, etc., eventually kill phase coherence. Interestingly, the deeper the quench, the slower the system is to relax.
A quantum Newton's cradle
T. Kinoshita, T. Wenger, and D. Weiss
Nature 440, 900 (2006)
In this experiment, a 1D Bose gas -- bosons confined in an array of very narrow tubes -- was started off in a far-from-equilibrium condition, with half the atoms flying off to the left and the other half flying off to the right, and allowed to time-evolve. The momentum distribution didn't seem to approach a thermal distribution on the experimental timescales; this was interpreted as a consequence of the integrability of the system. (Roughly, an integrable system has a huge number of conserved quantities, and therefore retains much more information about its initial conditions than a normal system does.) This is a stronger result than underdamped relaxation; the experiment showed no tendency towards relaxation at all.
Quench dynamics and nonequilibrium phase diagram of the Bose-Hubbard model
C. Kollath, A. Lauchli, and E. Altman
PRL 98, 180601 (2007)
Kollath et al. found, rather surprisingly, that thermalization doesn't necessarily happen even in nonintegrable models. They simulated a quench from the SF to the Mott phase, using exact diagonalization techniques as well as a density-matrix renormalization group algorithm. The result is that if one quenches from the SF deep into the Mott, the system doesn't seem to thermalize. They explained their results with an argument that goes something like this -- thermalization occurs as a consequence of the decay of double occupancies; however, a double occupancy has energy U (on-site repulsion) and has to decay into quasi-particles that have energy of order J (roughly the bandwidth of the lower Mott band). If U >> J then this can only happen at very high orders in perturbation theory, so it takes forever.
One could interpret this result as being about the observation, rather than the occurrence, of thermalization -- from a practical point of view, a system that takes very long to thermalize looks like one that doesn't thermalize.
Thermalization and its mechanism for generic isolated quantum systems
M. Rigol, V. Dunjko, and M. Olshanii
Nature 452, 854-858 (arxiv version)
This paper presents the "eigenstate thermalization" hypothesis. The hypothesis, as I understand it (which is not very well), states that the expectation values of physically interesting observables are essentially constant for eigenstates that are near each other in energy. There is some numerical reason to believe that this is true, but as far as I know there hasn't been much analytic work. Rigol et al. also suggest a generalization of this hypothesis to integrable systems; the upshot is the idea of a generalized Gibbs ensemble, which introduces "chemical potentials" (i.e. Lagrange multipliers) corresponding to all the conserved quantities.
[... to be continued]
Monday, July 13, 2009
Old stuff: Mott Insulators, Disorder, 1D gases
(Many of these links are cribbed from Brian DeMarco's now-defunct blog.)
Phase Coherence and Superfluid-Insulator Transition in a Disordered Bose-Einstein Condensate
by: Yong P. Chen, J. Hitchcock, D. Dries, M. Junker, C. Welford, R. G. Hulet
Abstract: We have studied the effects of a disordered optical potential on the transport and phase coherence of a Bose-Einstein condensate (BEC) of 7Li atoms. At moderate disorder strengths (V_D), we observe inhibited transport and damping of dipole excitations, while in time-of-flight images, random but reproducible interference patterns are observed. The interference reflects phase coherence in the disordered BEC and is interpreted as speckle for matter waves. At higher V_D, the interference contrast diminishes as the BEC fragments into multiple pieces with little phase coherence.
arXiv:0710.5187v1 [cond-mat.other]
Non-equilibrium coherence dynamics in one-dimensional Bose gases
Authors: S. Hofferberth1,2, I. Lesanovsky3, B. Fischer1, T. Schumm2 & J. Schmiedmayer1,2
Abstract: Low-dimensional systems provide beautiful examples of many-body quantum physics1. For one-dimensional (1D) systems2, the Luttinger liquid approach3 provides insight into universal properties. Much is known of the equilibrium state, both in the weakly4, 5, 6, 7 and strongly8, 9 interacting regimes. However, it remains a challenge to probe the dynamics by which this equilibrium state is reached10. Here we present a direct experimental study of the coherence dynamics in both isolated and coupled degenerate 1D Bose gases. Dynamic splitting is used to create two 1D systems in a phase coherent state11. The time evolution of the coherence is revealed through local phase shifts of the subsequently observed interference patterns. Completely isolated 1D Bose gases are observed to exhibit universal sub-exponential coherence decay, in excellent agreement with recent predictions12. For two coupled 1D Bose gases, the coherence factor is observed to approach a non-zero equilibrium value, as predicted by a Bogoliubov approach13. This coupled-system decay to finite coherence is the matter wave equivalent of phase-locking two lasers by injection. The non-equilibrium dynamics of superfluids has an important role in a wide range of physical systems, such as superconductors, quantum Hall systems, superfluid helium and spin systems14, 15, 16. Our experiments studying coherence dynamics show that 1D Bose gases are ideally suited for investigating this class of phenomena.
http://www.nature.com/nature/journal/v449/n7160/full/nature06149.html
Nature 449, 324-327 (20 September 2007)
Tonks–Girardeau gas of ultracold atoms in an optical lattice
Authors: Belén Paredes, Artur Widera, Valentin Murg, Olaf Mandel, Simon Fölling, Ignacio Cirac, Gora V. Shlyapnikov, Theodor W. Hänsch and Immanuel Bloch
Abstract: Strongly correlated quantum systems are among the most intriguing and fundamental systems in physics. One such example is the Tonks–Girardeau gas1, 2, proposed about 40 years ago, but until now lacking experimental realization; in such a gas, the repulsive interactions between bosonic particles confined to one dimension dominate the physics of the system. In order to minimize their mutual repulsion, the bosons are prevented from occupying the same position in space. This mimics the Pauli exclusion principle for fermions, causing the bosonic particles to exhibit fermionic properties1, 2. However, such bosons do not exhibit completely ideal fermionic (or bosonic) quantum behaviour; for example, this is reflected in their characteristic momentum distribution3. Here we report the preparation of a Tonks–Girardeau gas of ultracold rubidium atoms held in a two-dimensional optical lattice formed by two orthogonal standing waves. The addition of a third, shallower lattice potential along the long axis of the quantum gases allows us to enter the Tonks–Girardeau regime by increasing the atoms’ effective mass and thereby enhancing the role of interactions. We make a theoretical prediction of the momentum distribution based on an approach in which trapped bosons acquire fermionic properties, finding that it agrees closely with the measured distribution.
Nature 429, 277-281 (20 May 2004)
http://www.nature.com/nature/journal/v429/n6989/abs/nature02530.html
Formation of Spatial Shell Structure in the Superfluid to Mott Insulator Transition
Authors: Simon Fölling, Artur Widera, Torben Müller, Fabrice Gerbier, and Immanuel Bloch
Abstract: We report on the direct observation of the transition from a compressible superfluid to an incompressible Mott insulator by recording the in-trap density distribution of a Bosonic quantum gas in an optical lattice. Using spatially selective microwave transitions and spin-changing collisions, we are able to locally modify the spin state of the trapped quantum gas and record the spatial distribution of lattice sites with different filling factors. As the system evolves from a superfluid to a Mott insulator, we observe the formation of a distinct shell structure, in good agreement with theory.
http://link.aps.org/abstract/PRL/v97/e060403
Phys. Rev. Lett. 97, 060403 (2006)
Phase Coherence and Superfluid-Insulator Transition in a Disordered Bose-Einstein Condensate
by: Yong P. Chen, J. Hitchcock, D. Dries, M. Junker, C. Welford, R. G. Hulet
Abstract: We have studied the effects of a disordered optical potential on the transport and phase coherence of a Bose-Einstein condensate (BEC) of 7Li atoms. At moderate disorder strengths (V_D), we observe inhibited transport and damping of dipole excitations, while in time-of-flight images, random but reproducible interference patterns are observed. The interference reflects phase coherence in the disordered BEC and is interpreted as speckle for matter waves. At higher V_D, the interference contrast diminishes as the BEC fragments into multiple pieces with little phase coherence.
arXiv:0710.5187v1 [cond-mat.other]
Non-equilibrium coherence dynamics in one-dimensional Bose gases
Authors: S. Hofferberth1,2, I. Lesanovsky3, B. Fischer1, T. Schumm2 & J. Schmiedmayer1,2
Abstract: Low-dimensional systems provide beautiful examples of many-body quantum physics1. For one-dimensional (1D) systems2, the Luttinger liquid approach3 provides insight into universal properties. Much is known of the equilibrium state, both in the weakly4, 5, 6, 7 and strongly8, 9 interacting regimes. However, it remains a challenge to probe the dynamics by which this equilibrium state is reached10. Here we present a direct experimental study of the coherence dynamics in both isolated and coupled degenerate 1D Bose gases. Dynamic splitting is used to create two 1D systems in a phase coherent state11. The time evolution of the coherence is revealed through local phase shifts of the subsequently observed interference patterns. Completely isolated 1D Bose gases are observed to exhibit universal sub-exponential coherence decay, in excellent agreement with recent predictions12. For two coupled 1D Bose gases, the coherence factor is observed to approach a non-zero equilibrium value, as predicted by a Bogoliubov approach13. This coupled-system decay to finite coherence is the matter wave equivalent of phase-locking two lasers by injection. The non-equilibrium dynamics of superfluids has an important role in a wide range of physical systems, such as superconductors, quantum Hall systems, superfluid helium and spin systems14, 15, 16. Our experiments studying coherence dynamics show that 1D Bose gases are ideally suited for investigating this class of phenomena.
http://www.nature.com/nature/journal/v449/n7160/full/nature06149.html
Nature 449, 324-327 (20 September 2007)
Tonks–Girardeau gas of ultracold atoms in an optical lattice
Authors: Belén Paredes, Artur Widera, Valentin Murg, Olaf Mandel, Simon Fölling, Ignacio Cirac, Gora V. Shlyapnikov, Theodor W. Hänsch and Immanuel Bloch
Abstract: Strongly correlated quantum systems are among the most intriguing and fundamental systems in physics. One such example is the Tonks–Girardeau gas1, 2, proposed about 40 years ago, but until now lacking experimental realization; in such a gas, the repulsive interactions between bosonic particles confined to one dimension dominate the physics of the system. In order to minimize their mutual repulsion, the bosons are prevented from occupying the same position in space. This mimics the Pauli exclusion principle for fermions, causing the bosonic particles to exhibit fermionic properties1, 2. However, such bosons do not exhibit completely ideal fermionic (or bosonic) quantum behaviour; for example, this is reflected in their characteristic momentum distribution3. Here we report the preparation of a Tonks–Girardeau gas of ultracold rubidium atoms held in a two-dimensional optical lattice formed by two orthogonal standing waves. The addition of a third, shallower lattice potential along the long axis of the quantum gases allows us to enter the Tonks–Girardeau regime by increasing the atoms’ effective mass and thereby enhancing the role of interactions. We make a theoretical prediction of the momentum distribution based on an approach in which trapped bosons acquire fermionic properties, finding that it agrees closely with the measured distribution.
Nature 429, 277-281 (20 May 2004)
http://www.nature.com/nature/journal/v429/n6989/abs/nature02530.html
Phase Coherence of an Atomic Mott Insulator
Authors: Fabrice Gerbier, Artur Widera, Simon Fölling, Olaf Mandel, Tatjana Gericke, and Immanuel Bloch
Abstract: We investigate the phase coherence properties of ultracold Bose gases in optical lattices, with special emphasis on the Mott insulating phase. We show that phase coherence on short length scales persists even deep in the insulating phase, preserving a finite visibility of the interference pattern observed after free expansion. This behavior can be attributed to a coherent admixture of particle-hole pairs to the perfect Mott state for small but finite tunneling. In addition, small but reproducible kinks are seen in the visibility, in a broad range of atom numbers. We interpret them as signatures for density redistribution in the shell structure of the trapped Mott insulator.
Phys. Rev. Lett. 95, 050404 (2005)
http://link.aps.org/abstract/PRL/v95/e050404
Formation of Spatial Shell Structure in the Superfluid to Mott Insulator Transition
Authors: Simon Fölling, Artur Widera, Torben Müller, Fabrice Gerbier, and Immanuel Bloch
Abstract: We report on the direct observation of the transition from a compressible superfluid to an incompressible Mott insulator by recording the in-trap density distribution of a Bosonic quantum gas in an optical lattice. Using spatially selective microwave transitions and spin-changing collisions, we are able to locally modify the spin state of the trapped quantum gas and record the spatial distribution of lattice sites with different filling factors. As the system evolves from a superfluid to a Mott insulator, we observe the formation of a distinct shell structure, in good agreement with theory.
http://link.aps.org/abstract/PRL/v97/e060403
Phys. Rev. Lett. 97, 060403 (2006)
What this blog is for
This blog is supposed to be a repository of recent (and not-so-recent) physics papers I find interesting. I'm a grad student in condensed matter theory; my current research is primarily on (1) applications of condensed matter ideas in cavity QED and (2) nonequilibrium condensed matter physics, but I've been known to be interested in other things. I don't know, at this stage, whether the plan is to blog papers before or after I've read them, but if the latter, I'll try to do summaries.
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