Pacific Institute for Theoretical Physics

1. DECOHERENCE at the CROSSROADS Pacific Institute for Theoretical Physics FEB20-22, 2005 VANCOUVER

2. MISSION of PITP • PITP is an international institute, funded internationally, with an international mission- • to bring together groups of high-quality researchers, from around the world, and foster • path-breaking new research in all branches of theoretical physics. • Recognizing that theoretical physics is central to the whole of science, PITP fosters • links to other subjects, including chemistry & biology, and tries to ‘seed’ new • developments in these areas. • Recognizing the decisive influence that physics and other sciences have in the modern • world, PITP will advise and assist in areas within its competence, and provide • information to the general public about theoretical physics and related topics. Advisory Board -Prof P.W. Anderson (Princeton) <at large> http://pitp.physics.ubc.ca/index.html

3. TOKYO PiTP EUROPE KOREA (APCTP, SeaQUest, etc) Alberta, Perimeter CITA, CQIQC Sherbrooke + other Canadian centres AUSTRALIA (ARC, AAS, NSW, Queensland) STANFORD PITP: Main activities 1. Meetings (conferences, schools, workshops); 30 since April 2003 2. Support of research networks 3. Support of PITP visitor centre + nodes RESEARCH NETWORKS • Quantum Condensed Matter • 2. Complex Systems • 3. Strings and Particles • 4. Cosmology & Astrophysics

4. see http://pitp.physics.ubc.ca/

5. NOW for the TALK…….

6. PCE STAMP “COHERENCE WINDOWS in SOLID-STATE SYSTEMS” PITP/CQIQC CONFERENCE on “DECOHERENCE at the CROSSROADS” (Feb 20-22, 2006 Physics & Astronomy UBC Vancouver Pacific Institute for Theoretical Physics

7. PART 1: DECOHERENCE MECHANISMS in SOLID-STATE QUBIT SYSTEMS WORK DONE in COLLABORATION with: I TUPITSYN M SCHECHTER

8. How do REAL Solids (%99.9999) behave at low Energy? In almost all real solids, a combination of frustrating interactions, residual long-range interactions, and boundaries leads to a very complex hierarchy of states. These often have great difficulty communicating with each other, so that the long-time relaxation properties and memory/aging effects are quite interesting- for the system to relax, a large number of objects (atoms, spins, etc.) must simultaneously reorganise themselves . This happens even in pure systems Results for Capacitance (Above) & Sound velocity and dielectric absorption (Below) for pure SiO2 , at very low T A model commonly used to describe the low-energy excitations (which is certainly appropriate for many of them) is the ‘interacting TLS model’, with effective Hamiltonian: ABOVE: structure of low-energy eigenstates for interacting TLS model, before relaxation

9. SOLID-STATE QUBITS: Theoretical Designs & Experiments Here are a few: (1) Superconducting SQUID qubits (where qubit states are flux states); all parameters can be controlled. (2) Magnetic molecule qubits (where an easy axis anisotropy gives 2 low energy spin states, which communicate via tunneling, and couple via exchange or dipolar interactions. Control of individual qubit fields is easy in principle- interspin couplings less so... (3) Spins in semiconductors (or in Q Dots). Local fields can be partially controlled, & the exchange coupling is also controllable.

10. 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.

11. 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. Our central Qubit is thus coupling to a set of effective spins; ie., to a “SPIN BATH”. Unlike for the oscillators, we cannot assume these couplings are weak. For simplicity assume here the bath spins are a set{sk} of 2-level systems, which interact with each other only very weakly (because they are localised). We then get the following low-energy effective Hamiltonian (compare previous slide): H (Wo) = { [Dt+exp(-i Sk ak.sk) + H.c.] + eotz (qubit) + tz wk.sk + hk.sk (bath spins) + inter-spin interactions Nowthecouplingswk , hktothebathspins (the 1stbetweenbathspin & qubit, the 2nd to external fields) are often very STRONG (much larger than the inter-bath spin interactions or even than D).

12. DYNAMICS of DECOHERENCE 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

13. The COHERENCE WINDOW In solid-state qubit systems, the coherence window arises because of the large separation of energy scales typically existing between spin and oscillator baths. This coherence window exists in ALL solid-state systems- we look here at magnetic systems M Dube, PCE Stamp, Chem Phys 268, 257 (2001) PCE Stamp, J Q Comp & Computing 4, 20 (2003) PCE Stamp, IS Tupitsyn, Phys Rev B69, 014401 (2004) ENERGY (K) 104 If we now fix the operating frequency D of the qubits to lie well below the high phonon frequencies, but well above the characteristic nuclear spin frequencies (given by hyperfine couplings, then the phonons are too fast to cause decoherence, & the nuclear spins too slow. ELECTRONS (in conductors) 102 PHONONS Log (td-1) Phonon Decoherence 1 Nuclear spin Decoherence Aij 10-2 NUCLEAR SPINS Vkk’ 10-4 Log D

14. NUCLEAR SPIN BATH in MAGNETIC SYSTEMS: The LiHoxY1-xF4 system This system is usually treated as the archetypal Quantum Ising system: However the Ho nuclear spin actually plays a profound role in the physics: (1) It blocks transitions until we get to very high fields (see left) (2) The only way to understand the quantum spin glass phase is by incorporating the nuclear spins (and also the transverse dipolar terms); see below right (3) The decoherence is completely governed by the nuclear spins down to the lowest temperatures (phonon effects disappear below roughly 250 mK Stamp, P.C.E., Tupitsyn, I.S., Phys Rev B69, 014401 (2004) M Schechter, PCE Stamp, PRL 95, 267208 (2005)

15. DECOHERENCE in the Fe-8 Molecule At low applied transverse Fields, decoherence switches on very fast- expect incoherent spin relaxation: Stamp, P.C.E., Tupitsyn, I.S., Phys Rev B69, 014401 (2004) However, at high fields, system can be in coherence window, in which qubit dynamics is too fast for nuclear spins to follow, but still much slower than phonons This frequency window we call the coherence window- note that typically

16. SOME EXPTS RIGHT: Expts on Tunneling magn. molecules & Ho ions Wernsdorfer et al, PRL 82, 3903 (1999); and PRL 84, 2965 (2000); and Science 284, 133 (1999) R. Giraud et al., PRL 87, 057203 (2001) LEFT: ESR on Mn dimer system Expts on the quantum phase transition in LiHoF4 S Hill et al, Science 302, 1015 (2003) RIGHT: NMR on Mn-12 tunneling molecules H.M. Ronnow et al., Science 308, 389 (2005) A. Morello et al., PRL 93, 197202 (2004)

17. DECOHERENCE in Superconducting Qubits • The oscillator bath (electrons, photons, phonons) decoherence • rate: tf-1 ~ Dog(D,T) coth (D/2kT) (Caldeira-Leggett). This is often many orders of magnitude smaller than the experimental decoherence rates. • The spin bath decoherence will be caused by a combination • of charge & spin (nuclear & paramagnetic) defects- in • junction, SQUID, and substrate. 1/tf = Do (Eo/8D0)2 PCE Stamp, PRL 61, 2905 (1988) NV Prokof’ev, PCE Stamp, Rep Prog Phys 63, 669 (2000) Q. Comp & Comp 4, 20 (2003) The basic problem with any theory-experiment comparison here is that most of the 2-level systems are basically just junk (coming from impurities and defects), whose characteristics are hard to quantify. Currently ~10 groups have seen coherent oscillations in superconducting qubits, and several have seen entanglement between qubit pairs. I Chiorescu et al., Science 299, 1869 (2003) RW Simmonds et al., PRL 93, 077003 (2004)

18. PART 2: SOME INTERESTING NEW RESULTS for the DYNAMICS of DECOHERENCE WORK DONE in COLLABORATION with: M HASSELFIELD NV PROKOF’EV T LEE A HINES G SEMENOFF G MILBURN YC CHEN

19. New Kinds of Order: Topological Q Fluids The archetype for all topological Quantum fluids discussed so far is the quantum Hall fluid. It has a remarkable RG flow, predicted in 1984 by Laughlin, which leads to a complex ‘nested’ phase diagram of MI transitions (see Zhang et al). The underlying symmetry in the 2d parameter space is SL(2,Z), the same as that of an interacting set of vortices and charges. This is the same symmetry as that possessed by a large class of string theories. There is a v interesting model encapsulating all these features- the ‘dissipative Hofstadter model’, (cf. Callan et al). It describes open string theories, but also flux phases, Josephson junction arrays, and even interacting 3-wire Q wire junctions (Affleck et al). Kitaev has shown that lattice anyon systems should be able to do ‘topological quantum computing’, almost immune from decoherence, & implementable on JJ arrays (Doucot & Ioffe), or on ‘Kagome lattice’ systems (Kitaev & Freedman).

20. SCHMID MODEL & DISSIPATIVE HOFSTADTER MODEL: SOME REVISIONISM Proposed phase diagram (Callan & Freed) • Two very well studied models in the • quantum dissipation community are • Schmid model (particle in • periodic lattice potential • coupled to oscillator bath) • Dissipative WAH model (now add • a uniform flux threading the 2-d • lattice plaquettes). Mapping of line a=1 under z  1/(1 + inz) However, it looks as though some very interesting features may have been missed. In the 1st place, enforcing the natural constraint of lattice periodicity on the oscillator bath changes things- and produces some remarkable new solutions. In the 2nd place, it seems as though duality actually fails in the dissipative WAH model, again, some exact results can be found. • M Hasselfield, G Semenoff, • T Lee, PCE Stamp, • hep-th/0512219 (2) PCE Stamp, YC Chen; preprint

21. Remarks on NETWORKS- the QUANTUM WALK Computer scientists have been interested in RANDOM WALKS on various mathematical GRAPHS, for many years. These allow a general analysis of decision trees, search algorithms, and indeed general computer programmes (a Turing machine can be viewed as a walk). One of the most important applications of this has been to error correction- which is central to modern software. Starting with papers by Aharonov et al (1994), & Farhi & Gutmann (1998), the same kind of analysis has been applied to QUANTUM COMPUTATION. It is easy to show that ANY quantum computation can be modeled as a QUANTUM WALK on some graph. The problem then becomes one of QUANTUM DIFFUSION on this graph, and one easily finds either power-law or exponential speed-up, depending on the graph. Great hopes have been pinned on this new development- it allows very general analyses, and offers hope of new kinds of algorithm, and new kinds of quantum error correction- and new ‘circuit designs’. It also allows a very interesting general analysis of decoherence in quantum computation (Prokof’ev & Stamp: and Hines, Milburn & Stamp, 2005), with extraordinary results. For example, for the Hamiltonian we get ‘superdiffusion’ in the long time limit- part of the density matrix still propagates diffusively (while another part propagates SUB-diffusively). Note the general implications of this result!

22. DECOHERENCE & QUANTUM WALKS – the details FREE QUANTUM BEHAVIOUR Suppose initial state is at origin: So that: Then, since One gets More generally, we can start with a wave-packet: which gives Thus, quite generally one has and that Now this is to contrasted with diffusive behaviour: and

23. DECOHERENCE DYNAMICS For the decoherent Quantum Walk Hamiltonian Or, for an initial wave-packet Now this produces a very surprising result: BUT…. In other words, the particle spends more time near the origin than classical diffusion would predict, BUT it also has a BALLISTIC part (in the long-time limit)!! More detailed evaluation of the integrals fills this picture out:

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25. MEMBERS of QUANTUM CONDENSED MATTER NETWORK British Columbia Alberta Newfoundland Ontario Quebec IK Affleck M Berciu D Bonn A Damascelli J Folk M Franz WN Hardy I Herbut GA Sawatzky PCE Stamp A Zagoskin F Zhou M Boninsegni F Marsiglio P Brumer T Devereaux S John C Kallin HY Kee YB Kim E Sorenson A Steinberg A Blais C Bourbonnais K LeHur D Senechal AM Tremblay S Curnoe United States Europe Australasia PW Anderson G Christou S Hill D Goldhaber-Gordon M Jarrell A Kitaev S Kivelson G Kotliar RB Laughlin AJ Leggett H Manoharan C Marcus G Aeppli B Barbara Y Imry S Popescu J van den Brink W Wernsdorfer RW Clark R McKenzie G Milburn Y Nakamura N Nagaosa M Nielsen M Oshikawa G Vidal J Moore DD Osheroff BL Spivak PB Wiegmann SC Zhang

26. What happens to Heff at low Energy in Solids? States in a ‘Quantum Glass’, pile up at low energy. Their structure is T- dependent in any effective Hamiltonian The main reasons for the peculiar nature of the low-energy states in most solids are (i) boundaries, and (ii) interactions which are long range and/or ‘frustrating’. Both of these are ubiquitous, even for pure systems without disorder! States pile up at low energy, but they can’t communicate with each other. Frustrating interactions At low T, the system splits into subspaces that can never communicate with each other- the effective vacuum & its structure are physically quite meaningless. A glass can only be defined by its dynamic (non-equilibrium) properties. A commonly used model effective Hamiltonian is: ‘Frustration’ means that at low energy, any local change must re-organize simultaneously a vast number of states. This forces the Hilbert space of the effective Hamiltonian to have an ‘ultrametric’ geometry. where the ‘spins’ represent 2-level systems ‘Ultrametricgeometry’ of a glass Hilbert space