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  1. SPINS, CHARGES, LATTICES, & TOPOLOGY IN LOW d Meeting of “QUANTUM CONDENSED MATTER” network of PITP (Fri., Jan 30- Sunday, Feb 1, 2004; Vancouver, Canada) For all current information on this workshop go to All presentations will go online in the next week on PITP archive page:


  3. PROBLEM #1 The theoretical problem is to calculate the dynamics of the “M-qubit” reduced density matrix for the following Hamiltonian, describing a set of N interacting qubits (with N > M typically): H = Sj( Djtjx + ejtjz ) + Sij Vij tiz tjz + Hspin({sk}) + Hosc({xq}) + int. The problem is to integrate out the 2 different environments coupling to the qubit system- this gives the N-qubit reduced density matrix. We may then average over other qubits if necessary to get the M-qubit density matrix operator: rNM({tj}; t) The N-qubit density matrix contains all information about the dynamics of this QUIP (QUantum Information Processing) system- & all the quantum information is encoded in it. A question of some theoretical interest is- how do decoherence rates in this quantity vary with N and M ?

  4. Feynman & Vernon, Ann. Phys. 24, 118 (1963) PW Anderson et al, PR B1, 1522, 4464 (1970) Caldeira & Leggett, Ann. Phys. 149, 374 (1983) AJ Leggett et al, Rev Mod Phys 59, 1 (1987) U. Weiss, “Quantum Dissipative Systems” (World Scientific, 1999) A qubit coupled to a bath of delocalised excitations: the SPIN-BOSONModel Suppose we have a system whose low-energy dynamics truncates to that of a 2-level system t. In general it will also couple to DELOCALISED modes around (or even in) it. A central feature of many-body theory (and indeed quantum field theory in general) is that (i) under normal circumstances the coupling to each mode is WEAK (in fact ~ O (1/N1/2)), where N is the number of relevant modes, just BECAUSE the modes are delocalised; and (ii) that then we map these low energy “environmental modes” to a set of non-interacting Oscillators, with canonical coordinates {xq,pq} and frequencies {wq}. It then follows that we can write the effective Hamiltonian for this coupled system in the ‘SPIN-BOSON’ form: H (Wo) = {[Dotx + eotz] qubit + 1/2 Sq (pq2/mq + mqwq2xq2) oscillator + Sq [ cqtz + (lqt+ + H.c.)] xq }interaction Where Wois a UVcutoff, and the {cq, lq} ~ N-1/2.

  5. UCL 16 A qubit coupled to a bath of localised excitations: the CENTRALSPIN Model P.C.E. Stamp, PRL 61, 2905 (1988) AO Caldeira et al., PR B48, 13974 (1993) NV Prokof’ev, PCE Stamp, J Phys CM5, L663 (1993) NV Prokof’ev, PCE Stamp, Rep Prog Phys 63, 669 (2000) Now consider the coupling of our 2-level system to LOCALIZED modes. These have a Hilbert space of finite dimension, in the energy range of interest- in fact, often each localised excitation has a Hilbert space dimension 2. From this we see that our central Qubit is coupling to a set of effective spins; ie., to a “SPIN BATH”. Unlike the case of the oscillators, we cannot assume these couplings are weak. For simplicity assume here that the bath spins are a set {sk} of 2-level systems. Now actually these interact with each other very weakly (because they are localised), but we cannot drop these interactions. What we then get is the following low-energy effective Hamiltonian (recall previous slide): H (Wo) = { [Dt+exp(-i + H.c.] + eotz (qubit) + + (bath spins) + inter-spin interactions Thecrucialthinghereisthatnowthecouplings wk , hktothebathspins- thefirstbetweenbath spinandqubit,thesecondtoexternalfields- are oftenverystrong (muchlargerthaneitherthe inter-spininteractionsoreventhanD).

  6. Dynamics of Spin-Boson System The easiest way to solve for the dynamics of the spin-boson model is in a path integral formulation. The qubit density matrix propagator is written as an integral over an “influence functional” : The influence functional is defined as For an oscillator bath: with bath propagator: For a qubit the path reduces to Thence

  7. Dynamics of Central Spin model (Qubit coupled to spin bath) Consider following averages Topological phase average Orthogonality average Bias average The reduced density matrix after a spin bath is integrated out is quite generally given by: Eg., for a single qubit, we get the return probability: NB: can also deal with external noise

  8. DYNAMICS of DECOHERENCE UCL 28 At first glance a solution of this seems very forbidding. However it turns out that one can solve for the reduced density matrix of the central spin exactly, in the interesting parameter regimes. From this soltn the decoherence mechanisms are easy to identify: (i) Noise decoherence: Random phases added to different Feynman paths by the noise field. (ii) Precessional decoherence: the phase accumulated by environmental spins between qubit flips. (iii) Topological Decoherence: The phase induced in the environmental spin dynamics by the qubit flip itself USUALLY THE 2ND MECHANISM (PRECESSIONAL DECOHERENCE) is DOMINANT Precessional decoherence Noise decoherence source

  9. Pey 1.34 • The oscillator bath decoherence rate goes like • tf-1 ~ Dog(D,T) coth (D/2kT) • with the spectral function g(w,T) shown below for an Al • SQUID (contribution from electrons & phonons). All of this is • well known and leads to a decoherence rate tf-1 ~ paDoonce • kT < Do. By reducing the flux change df = (f+ - f- ) ~ 10-3, it • has been possible to make a ~ 10-7 (Delft expts), ie., a • decoherence rate for electrons ~ O(100 Hz). This is v small! Decoherence in SQUIDs A.J. Leggett et al., Rev. Mod Phys. 59, 1 (1987) AND PCE Stamp, PRL 61, 2905 (1988) Prokof’ev and Stamp Rep Prog Phys 63, 669 (2000) On the other hand paramagnetic spin impurities (particularly in the junctions), & nuclear spins have a Zeeman coupling to the SQUID flux peaking at low energies- at energies below Eo, this will cause complete incoherence. Coupling to charge fluctuations (also a spin bath of 2-level systems) is not shown here, but also peaks at very low frequencies. However when Do >> Eo, the spin bath decoherence rate is: 1/tf = Do (Eo/8D0)2as before


  11. PROBLEM #2: The DISSIPATIVE HOFSTADTER Model This problem describes a set of fermions on a periodic potential, with uniform flux threading the plaquettes. The fermions are then coupled to a background oscillator bath: We will assume a square lattice, and a simple cosine potential: There are TWO dimensionless couplings in the problem- to the external field, and to the bath: The coupling to the oscillator bath is assumed ‘Ohmic’: where

  12. The W.A.H. MODEL This famous model was first investigated in preliminary way by Peierls, Harper,, Kohn, and Wannier in the 1950’s. The fractal structure was shown by Azbel in 1964. This structure was first displayed on a computer by Hofstadter in 1976, working with Wannier. The Hamiltonian involves a set of charged fermions moving on a periodic lattice- interactions between the fermions are ignored. The charges couple to a uniform flux through the lattice plaquettes. Often one looks at a square lattice, although it turns out much depends on the lattice symmetry. One key dimensionless parameter in the problem is the FLUX per plaquette, in units of the flux quantum

  13. The HOFSTADTER BUTTERFLY The graph shows the ‘support’ of the density of states- provided ais rational

  14. The effective Hamiltonian is also written as: H = - t Sij [ci cj exp {iAij} + H.c. ] …….“WAH” lattice + SnSq lq Rn . xq + Hosc({xq}) ……coupled to oscillators (i) the the WAH (Wannier-Azbel-Hofstadter) Hamiltonian describes the motion of spinless fermions on a 2-d square lattice, with a flux f per plaquette (coming from the gauge term Aij). (ii) The particles at positions Rncouple to a set of oscillators. This can be related to many systems- from 2-d J. Junction arrays in an external field to flux phases in HTc systems, to one kind of open string theory. It is also a model for the dynamics of information propagation in a QUIP array, with simple flux carrying the info. There are also many connections with other models of interest in mathematical physics and statistical physics.

  15. EXAMPLE: S/cond arrays The bare action is: Plus coupling to Qparticles, photons, etc: Interaction kernel (shunt resistance is RN):

  16. Expt (Kravchenko, Coleridge,..)

  17. PHASE DIAGRAM Callan & Freed result (1992) Mapping of the line a=1 under z  1/(1 + inz) Proposed phase diagram (Callan & Freed, 1992) Arguments leading to this phase diagram based mainly on duality, and assumption of localisation for strong coupling to bosonic bath. The duality is now that of the generalised vector Coulomb gas, in the complex z- plane.

  18. DIRECT CALCULATION of m (Chen & Stamp) We wish to calculate directly the time evolution of the reduced density matrix of the particle. It satisfies the eqtn of motion: The propagator on the Keldysh contour g is: The influence functional is written in the form:

  19. Influence of the periodic potential We do a weak potential expansion, using the standard trick Without the lattice potential, the path integral contains paths obeying the simple Q Langevin eqtn: The potential then adds a set of ‘delta-fn. kicks’:

  20. One can calculate the dynamics now in a quite direct way, not by calculating an autocorrelation function but rather by evaluating the long-time behaviour of the density matrix. If one evaluates the long-time behaviour of the Wigner function one then finds the following, after expanding in the potential: We now go to some rather detailed exact results for this velocity, in the next few slides ….



  23. DIAGONAL & CROSS-CORRELATORS: It turns out from these exact results that not all of the conclusions which come from a simple analysis of the long-time scaling are confirmed. In particular we do not get the same phase diagram, as we now see …

  24. We find that we can get some exact results on a particular circle in the phase plane- the one for which K = 1/2 The reason is that on this circle, one finds that both the long- and short-range parts of the interaction permit a ‘dipole’ phase, in which the system form close dipoles, with the dipolar widely separated. This happens nowhere else. One then may immediately evaluate the dynamics, which is well-defined. If we write this in terms of a mobility we have the simple results shown:

  25. RESULTS on CIRCLE K = 1/2 The results can be summarized as shown in the figure. For a set of points on the circle the system is localised. At all other points on the circle, it is delocalised. The behaviour on this circle should be testable in experiments.

  26. Conclusions • In the weak-coupling limit (with dimensionless couplings ~ l ), the • disentanglement rate for a set of N coupled qubits, is actually linear • in N provided Nl < 1 • In the coherence window, this is good for quite large N • In the dissipative Hofstadter model duality apparently fails. There is • actually a whole set of ‘exact’ solutions possible on various circles. It will be interesting to explore decoherence rates for topological computation- note that the bath couplings are local but one still has to determine the couplings to the non-local information

  27. THE END

  28. The dynamics of the density matrix is calculated using path integral methods. We define the propagator for the density matrix as follows: This propagator is written a a path integral along a Keldysh contour: All effects of the bath are contained in Feynman’s influence functional, which averages over the bath dynamics, entangled with that of the particle: The ‘reactive’ part & the ‘decoherence’ part of the influence functional depend on the spectral function:

  29. UCL 19 DYNAMICS of the DIPOLAR SPIN NET NV Prokof’ev, PCE Stamp, PRL 80, 5794 (1998) JLTP 113, 1147 (1998) PCE Stamp, IS Tupitsyn Rev Mod Phys (to be publ.). The dipolar spin net is of great interest to solid-state theorists because it represents the behaviour of a large class of systems with “frustrating” interactions (spin glasses, ordinary dipolar glasses). It is also a fascinating toy model for quantum computation: H = Sj (Djtjx + ejtjz) + Sij Vijdiptiz tjz + HNN(Ik) + Hf(xq) + interactions For magnetic systems this leads to the picture at right. Almost all experiments so far are done in the region where Do is small- whether the dynamics is dipolar-dominated or single molecule, it is incoherent. However one can give a theory of this regime. The next great challenge is to understand the dynamics in the quantum coherence regime, with or without important inter-molecule interactions

  30. UCL 20 Quantum Relaxation of a single NANOMAGNET Structure of Nuclear spin Multiplet  Our Hamiltonian: When D<<Eo (linewidth of the nuclear multiplet states around each magbit level), the magbit relaxes via incoherent tunneling. The nuclear bias acts like a rapidly varying noise field, causing the magbit to move rapidly in and out of resonance, PROVIDED |gmBSHo| < Eo Tunneling now proceeds over a range Eoof bias, governed by the NUCLEAR SPINmultiplet. The relaxation rate is G ~ D2/Eo for a single qubit. Fluctuating noise field Nuclear spin diffusion paths NV Prokof’ev, PCE Stamp, J Low Temp Phys 104, 143 (1996)

  31. UCL 30 The path integral splits into contributions for each M. They have the effective action of a set of interacting instantons The effective interactions can be mapped to a set of fake charges to produce an action having the structure of a “spherical model” involving a spin S The key step is to then reduce this to a sum over Bessel functions associated with each polarisation group.