Dynamics of a Quantum Phase Transition

Zurek, Wojciech H.; Dorner, U.; Zoller, P.

doi:10.1103/physrevlett.95.105701

Cited by 799 publications

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“…The second transition "unfreezes" the Mott state into the HH superfluid. Technically, this approach cannot be fully adiabatic due to the exact degeneracy of the ground states in the HH spectrum; a Kibble-Zurek model implies that this will lead to the spontaneous formation of domains [34]. However, this will not create entropy beyond the randomization of domains.…”

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confidence: 99%

Observation of Bose–Einstein condensation in a strong synthetic magnetic field

et al. 2015

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Extensions of Berry's phase and the quantum Hall effect have led to the discovery of new states of matter with topological properties. Traditionally, this has been achieved using gauge fields created by magnetic fields or spin orbit interactions which couple only to charged particles. For neutral ultracold atoms, synthetic magnetic fields have been created which are strong enough to realize the Harper-Hofstadter model. Despite many proposals and major experimental efforts, so far it has not been possible to prepare the ground state of this system. Here we report the observation of BoseEinstein condensation for the Harper-Hofstadter Hamiltonian with one-half flux quantum per lattice unit cell. The diffraction pattern of the superfluid state directly shows the momentum distribution on the wavefuction, which is gauge-dependent. It reveals both the reduced symmetry of the vector potential and the twofold degeneracy of the ground state. We explore an adiabatic many-body state preparation protocol via the Mott insulating phase and observe the superfluid ground state in a three-dimensional lattice with strong interactions.Topological states of matter are an active new frontier in physics. Topological properties at the single particle level are well understood; however, there are many open questions when strong interactions and correlations are introduced [1, 2] as in the ν = 5/2 state of the fractional quantum Hall effect [3] and in Majorana fermions [4][5][6]. For neutral ultracold atoms, new methods have been developed to create synthetic gauge fields. Forces analogous to the Lorentz force on electons are engineered either through the Coriolis force in rotating systems [7][8][9], by phase imprinting via photon recoil [10][11][12][13][14], or lattice shaking [15,16]. Much of the research with ultracold atoms has focused on the paradigmatic HarperHofstadter (HH) Hamiltonian, which describes particles in a crystal lattice subject to a strong homogeneous magnetic field [17][18][19]. For magnetic fluxes on the order of one flux quantum per lattice unit cell, the radius of the smallest possible cyclotron orbit and the lattice constant are comparable, and their competition gives rise to the celebrated fractal spectrum of Hofstadfer's butterfly whose sub-bands have non-zero Chern numbers [20] and Dirac points.So far, it has not been possible to observe the ground state of the HH Hamiltonian, which for bosonic atoms is a superfluid Bose-Einstein condensate. It is unknown whether this is due to heating associated with technical noise, non-adiabatic state preparation, or inelastic collisions. These issues are complicated, since all schemes for realizing the HH Hamiltonian use some form of temporal lattice modulation and therefore are described by a time-dependent Floquet formalism. The HH model arises after time averaging the Floquet Hamiltonian, but it is an open question to what extent finite interactions and micromotion lead to transitions between Floquet modes and therefore heating [21][22][23][24]. Bose-Einstein condensat...

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Observation of Bose–Einstein condensation in a strong synthetic magnetic field

et al. 2015

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“…The quantum Ising model quenches a state according to Hamiltonian (1) so, when g is changed in time, superpositions of different locations of a kink (e.g., α| · · · ↑↑↓↓↓↓ · · · +β| · · · ↑↑↑↓↓↓ · · · +γ| · · · ↑↑↑↑↓↓ · · · ) are allowed, and, indeed, inevitable 10,11 . Spreading of a localized kink will come about as a consequence of the kinetic term, gσ x n , in equation (1).…”

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confidence: 99%

Non-local quantum superpositions of topological defects

2011

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Topological defects (such as monopoles, vortex lines, or domain walls) mark locations where disparate choices of a broken symmetry vacuum elsewhere in the system lead to irreconcilable differences 1,2 . They are energetically costly (the energy density in their core reaches that of the prior symmetric vacuum) but topologically stable (the whole manifold would have to be rearranged to get rid of the defect). We show how, in a paradigmatic model of a quantum phase transition, a topological defect can be put in a nonlocal superposition, so that -in a region large compared to the size of its core -the order parameter of the system is "undecided" by being in a quantum superposition of conflicting choices of the broken symmetry. We dub such a topological Schrödinger cat state a 'Schrödinger kink', and devise a version of a double-slit experiment suitable for topological defects to describe one possible manifestation of the phenomenon. Coherence detectable in such experiments will be suppressed as a consequence of interaction with the environment. We analyze the environment-induced decoherence and discuss its role in symmetry breaking.Topological defects are the epitome of locality. An example of a defect occurs in the quantum Ising model where a lattice of spins interact ferromagnetically with their nearest neighbors, i.e., the Hamiltonian contains an interaction of the form −σ z n σ z n+1 for the n th spin on the lattice. The entire collection of spins achieves its lowest energy when they are all aligned. However, there are two choices for this lowest energy state, | · · · ↑↑↑↑ · · · or | · · · ↓↓↓↓ · · · . Both are energetically identical but each of them breaks the symmetry of the Hamiltonian, which has no preference between "up" and "down".A configuration that includes topological defects could arise, for example, in a driven phase transition, i.e., "a quench". In the Ising model, a quench can be induced when the transverse field strength g is changed in the HamiltonianWhen the external field is strong enough (g 1), it "wins" and all spins align with σ x . A decrease in g, however, will lead to a phase transition at g = 1 with the symmetry breaking term −σ z n σ z n+1 trying to align all the spins in their z-direction. The choice of whether the spins will point up or down is made locally. As a result of causality and the finite speed at which signals propagate, the size of these domains will be determined by rate of change of g 3-8 . This will lead to configurations, such as | · · · ↑↑↓↓ · · · , where the topological defect marks the location of a "switch" from one broken symmetry ground state to the other. In this one-dimensional example, the defect is a kink (a domain wall), but, in a sense, all the topological defects (monopoles, vortex lines, and so on) that exist in higher dimensions "look the same". A discussion of the consequences of this process for the post-transition state is beyond the scope of this work (but see, e.g., Refs. 9-11). We only note that the density of the resulting kinks will be finite.The...

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“…The Kibble-Zurek scaling has been verified in various exactly solvable spin models and systems of interacting bosons [4,13,16,17,11,5]; it has been generalized to quenching through a multicritical point [31], across a gapless phase [20,23], and along a gapless line [26,28], and to systems with quenched disorder [14], white noise [18], infinite-range interactions [27], and edge states [30]. Studies have also been made to estimate the defect density for quenching with a non-linear form [21], an oscillatory variation of an applied magnetic field [32] or under a reversal of the magnetic field [33].…”

Section: Introductionmentioning

confidence: 99%

“…In the article in this book by Mondal, Sengupta and Sen [36], the quenching dynamics through a gapless phase and quenching with a power-law form of the change of a parameter as well as the possibility of experimental realizations have been discussed in detail. It should be mentioned that in addition to the density of defects in the final state, the degree of non-adiabaticity can also be quantified by looking at various quantities like residual energy [9,37] and fidelity [4].…”

Section: Introductionmentioning

confidence: 99%

“…The non-equilibrium dynamics of a quantum system when quenched very fast [3] or slowly across a quantum critical point [4,5] has attracted the attention of several groups recently. The possibility of experimental realizations of quantum dynamics in spin-1 Bose condensates [6] and atoms trapped in optical lattices [7,8] has led to an upsurge in studies of related theoretical models [3,4,5,9,10,11,12,13,14,15,16,17,18,19,20,21,23,24,25,26,27,28,29,30,31,32,33].In this review, we concentrate on the dynamics of quantum spin chains swept across a quantum critical or multicritical point or along a gapless line by a slow (adiabatic) variation of a parameter appearing in the Hamiltonian of the system. Our aim is to find the scaling form of the density of defects (which, in our case, is the density of wrongly oriented spins) in the final state which is reached after the system is prepared in an initial ground state and then slowly quenched through a quantum critical point.…”

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confidence: 99%

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Defect Production Due to Quenching Through a Multicritical Point and Along a Gapless Line

Divakaran

Mukherjee

Dutta

et al. 2010

Lecture Notes in Physics

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The exciting physics of quantum phase transitions has been explored extensively in the last few years [1,2]. The non-equilibrium dynamics of a quantum system when quenched very fast [3] or slowly across a quantum critical point [4,5] has attracted the attention of several groups recently. The possibility of experimental realizations of quantum dynamics in spin-1 Bose condensates [6] and atoms trapped in optical lattices [7,8] has led to an upsurge in studies of related theoretical models [3,4,5,9,10,11,12,13,14,15,16,17,18,19,20,21,23,24,25,26,27,28,29,30,31,32,33].In this review, we concentrate on the dynamics of quantum spin chains swept across a quantum critical or multicritical point or along a gapless line by a slow (adiabatic) variation of a parameter appearing in the Hamiltonian of the system. Our aim is to find the scaling form of the density of defects (which, in our case, is the density of wrongly oriented spins) in the final state which is reached after the system is prepared in an initial ground state and then slowly quenched through a quantum critical point. The dynamics in the vicinity of a quantum critical point is necessarily non-adiabatic due to the divergence of the relaxation time of the underlying quantum system which forces the system to be infinitely sluggish; thus the system fails to respond to a change in a parameter of the Hamiltonian no matter how slow that rate of change may be! We first recall the Kibble-Zurek argument [34,35] which predicts a scaling form for the defect density following a slow quench through a quantum critical point. We assume that a parameter g of the Hamiltonian is varied in a linear fashion such that g − g c ∼ t/τ, where g = g c denotes the value of g at the quantum critical point and τ is the quenching time. Our interest is in the adiabatic limit, τ → ∞. The energy gap of the quantum Hamiltonian vanishes at the critical point as (g − g c ) νz whereas the relaxation time ξ τ , which is inverse of the gap, diverges at the critical point. It is clear that non-adiabaticity becomes important at a timet when the characteristic time scale of the quantum system (i.e., the relaxation time) is of the order of the inverse of the rate of change of the Hamiltonian; this yields

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Dynamics of a Quantum Phase Transition

Cited by 799 publications

References 31 publications

Observation of Bose–Einstein condensation in a strong synthetic magnetic field

Observation of Bose–Einstein condensation in a strong synthetic magnetic field

Non-local quantum superpositions of topological defects

Defect Production Due to Quenching Through a Multicritical Point and Along a Gapless Line

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