Quantum Information and Many-Body Physics with Atomic and Quantum Optical Systems

Transcription

1 Quantum Information and Many-Body Physics with Atomic and Quantum Optical Systems Fabrizio Illuminati Quantum Theory Group - Dipartimento di Matematica e Informatica, Università degli Studi di Salerno, Via Ponte don Melillo, I Fisciano (SA) & Quantum Physics Division ISI Foundation for Scientific Interchange, Viale Settimio Severo, I Torino Camerino,

2 Areas of research at the crossroads: Theory of Entanglement and Quantum Information Quantum Information and Many-Body Physics Atoms and Photons as Quantum Simulators: Ion Traps, Optical Lattices, Coupled Cavity Arrays Quantum Information with Many-Body Systems: Applications of Quantum Simulators Camerino,

3 Research Directions (1) A) Simulating strongly correlated matter with atom-optical systems: A1) Optical lattices, the Bose-Hubbard model, and the Mott insulatorsuperfluid transition. A2) Extensions: Fermi systems, atomic mixtures, and high-t_c atomic superfluidity. Metastability and disorder: Quantum emulsions. B) Investigating many-body physics by methods of entanglement theory and quantum information: B1) Quantum phase transitions and entanglement measures at quantum criticality. B2) Balancing of interactions and ground state factorization in quantum spin systems. Camerino,

4 Research Directions (2) C) Quantum information tasks in models of interacting manybody systems: Open quantum spin chains with XX interactions and modified end couplings - Long distance entanglement and quantum teleportation in the _ _ model. D) Closing the loop Implementation with quantum simulators: Realizing quantum spin chains supporting long-distance, high fidelity qubit teleportation in suitably engineered arrays of coupled cavities. Camerino,

5 Methods of quantum information and theory of entanglement in many-body physics Ground state of the Bose-Hubbard model optical lattices: A paradigmatic instance of quantum entanglement vs. separability and factorization. In the correlated superfluid phase the GS is entangled (the hopping terms dominate in the Hamiltonian, and correlate the different sites of the lattice). In the uncorrelated Mott-insulating phase the inter-site tunneling is suppressed, the Hamiltonian is a sum of single-site contributions (the on-site repulsive interactions) and the GS is a product of single-site wave functions: the GS is factorized (unentangled). It is a product of Fock states (number states). Some relevant questions: I)How to quantify ground-state entanglement? How is it related to quantum phase transitions, and how does it behave at the approach of quantum critical points? II) Is the ground state of interacting many-body systems always entangled (correlated)? And if not, why, how, and when is it factorized? Camerino,

6 Entanglement and quantum phase transitions in interacting quantum spin systems Fundamental systems of interacting qubits (spin 1/2) undergoing quantum phase transitions at zero temperature. The XYZ Hamiltonian: It comprises the most important models of quantum spins coupled by exchange interaction terms, including the Ising, Heisenberg, XY, XX, XXZ models. Moreover, most relevant models of quantum spin chains for quantum information tasks (more on this in the following ). rxxryyrzzzxyzxijyijzijii,ji1hjssjssjsshs2r ij =++ = Includes models with short, finite, and long-range interactions. Most cases are non exactly solvable, with some notable exceptions, like the XY model. Camerino,

7 XY model: B1) Entanglement and quantum phase transitions in quantum spin chains xxyyzxyii1ii1iiihsssshs++= +Δ+ Anisotropic spin model ( 0 Δ < 1). Phase transition: A spontaneous magnetization develops along the x axis as the external transverse field h is varied. The divergence of the first derivative (with respect to h) of the von Neumann block entanglement entropy between a single spin and the rest of the system signals the onset of the quantum phase transition: Δ = 0.25 ( ) Δ = 0.28 ( ) Entanglement entropy Quantum critical points Factorization points Magnetic field Camerino,

8 B2) Ground state factorization in systems of interacting qubits (quantum spin chains) In the XY model, we have just seen an instance of a factorization point: The quantum entanglement vanishes and the ground state becomes factorized, even if the system is strongly interacting! Is this an acccidental phenomenon, a coincidence, or it hides a more profound physical picture? Exploiting methods inspired by entanglement theory, it is possible to derive a general theory of interaction balancing and determine the conditions for the occurrence of full factorization of the many-body states in interacting quantum systems. Main results of the analysis for quantum spin models of the XYZ type: I)Factorization of the ground state is not a rare phenomenon. II)It is due to a delicate balancing between spin-spin interactions and external fields. III)It occurs irrespective of spatial dimension and interaction range. Camerino,

9 B2) Factorization in quantum spin systems: Single-qubit unitary operations (SQUOs) Single Qubit Unitary Operations: A BABUOI= Hermitian: Unitary: Traceless: AAOO= 2 AAAAOOOI== ATrO0= Response of the system to controlled, nontrivial external perturbations. Camerino,

10 B2) Factorization and SQUOs in quantum spin systems Factorization: FABABψ φ χ Theorem: FextrextrA BA BABABABABiffU:Uψ=ψ ψ=ψ Factorization Invariance under the action of SQUOs extra BU The extremal SQUO is uniquely defined S. M. Giampaolo and F. I., Phys. Rev. A 76, (2007) Camerino,

11 Pure State: Transformed State: Hilbert-space distance: B2) Factorization in quantum spin systems: SQUOs and entanglement ABΨ A BABABUψ=ψ% ()2ABABABABd,1ψψ= ψψ%% Theorem: Distance Entanglement (von Neumann entropy): {}()A B2EAABABUS()mind, ρ=ψψ % S. M. Giampaolo and F. I., Phys. Rev. A 76, (2007) Camerino,

12 Observable estimators Q of separability under the action of SQUOs [ Bipartite system. Spin i = party A. Party B = all remaining spins N/i ] A BABABUψ=ψ% B2) Quantifying factorization in quantum spin systems with SQUO-related observables ABABABABˆˆ QQQΔ=ψψ ψψ%% If: Then: A Bˆ 1)Q0,2)U,Q0 Δ i (N/i)i(N/i)Q0Δ= ψ=φ χ This in turn implies total (full) factorization in translational-invariant systems: iiq0δ= ψ= φ Camerino,

13 B2) Determining ground-state factorization in quantum spin systems with entanglement excitation energies (EXEs) Hamiltonian Structure: Entanglement excitation energies (EXEs) associated to extremal SQUOs ˆˆQIdentification:GroundGH Entanglement Excitation Energy: E0Δ= E0Δ> ˆˆ EGHGGHGΔ= %% Factorized ground state Entangled ground state S. M. Giampaolo et al., Phys. Rev. A 77, (2008) Camerino,

14 B2) Applying the general theory to the determination of the factorization points of interacting quantum spin systems Phase diagram for factorization in the XYZ models Previously known factorization rryxjj All the factorized ground states Forbidden region rrzxjj Excited states? (Factorized) G. Adesso, S. M. Giampaolo, and F. Illuminati, Phys. Rev. Lett. 100, (2008) Camerino,

15 B2) Summary on factorization in many-body systems 1) Formalism of local unitary operations. Operational- geometric approach to the characterization of separability and entanglement. 2) Theory of ground state factorization for (generally non exactly-solvable) quantum spin systems. Arbitrary lattice dimension and range of interaction. Classes of exact solutions, exploiting concepts and techniques of quantum information. 3) Factorization: A highly nontrivial balancing in strongly interacting many-body quantum systems. Detailed exposition and further results in: S. M. Giampaolo, G. Adesso, and F. Illuminati, arxiv:0811.xxxx, to appear Camerino,

16 B2) Outlook: Significance of factorization in condensed matter Extension to mixed states and multi-qubit operations. Role of geometry and interactions: Frustration, chirality, and topology. Inhomogeneous and random systems - Models on graphs: Towards quantum complexity. Beyond spins and qubits: Systems of strongly correlated matter. Physics close to factorization points. Novel approximation schemes, starting from sets of exact solutions. Excitations and other physical properties around factorization. Understanding Entanglement phase transitions [Amico et al., PRA 74, (2006)] by hierarchical measures of entanglement [Dell Anno et al., PRA 77, (2008)]. Factorization at finite at temperature (and in classical systems?). Camerino,

17 B2) Significance of factorization in quantum information Producing arbitrarily large numbers of identical copies of singlequbit states (Preparing the register). Partial factorization and qubit encoding in systems with higher local dimension (qudits, continuous variables). Identifying the working points (close or away from factorization). One-way quantum computation many-qubit cluster states at once from dual-rail factorization. Camerino,

18 B2) Investigating physics at factorization: Possible experimental realizations Possible realizations with quantum simulators Systems with fundamental control of spin-spin couplings and spinchain engineering: - Engineered optical lattices (Sørensen & Mølmer; Kuklov & Svistunov; Duan, Demler, & Lukin) - Engineered arrays of coupled optical cavities ( Hartmann, Brandão, & Plenio; Angelakis, Santos, & Bose) - Trapped ions with external control (Porras and Cirac; Duan et al.; Wunderlich et al.; Schätz et al.) Camerino,

19 C) Implementing quantum information tasks with engineered many-body systems: Long-distance entanglement Qubit teleportation. Fundamental ingredients for working schemes of quantum teleportation: I)A good quantum channel (as long as possible) II)Highest possible end-to-end entanglement, and, therefore, the highest possible end-to-end teleportation fidelity. Formidable tasks both with condensed matter- and optical-based devices. Possible solutions? Suitably Engineered quantum spin chains: Open XX spin chains with alternating couplings, or uniform bulk interactions with modified, weak end bonds. At exactly T = 0, these systems can support maximal entanglement and perfect qubit teleportation (i.e. with unit fidelity) between the ends of the chain ( Long-Distance-Entanglement - LDE): N1xxyyXXkkk1kk1i1HJ(SSSS) ++== + Camerino,

20 C) Different realizations of long-distance entanglement with engineered spin chains: The ideal case I) Infinite-distance entanglement and perfect teleportation (zero temperature): 21/2( )1inputoutputI=ΨΦ= A λ λ λ λ B Dimerized 1-D XX spin chain with fully alternated couplings ( λ << 1). Maximal Alice-Bob entanglement. Unit fidelity for teleporting an unknown input state from Alice to Bob at zero temperature. Any finite size, arbitrary length. Very serious drawback: Extremely fragile at finite temperature, even at very low T. Camerino,

21 C) Different realizations of long-distance entanglement with engineered spin chains: The weak-end bond trick II) Long-distance entanglement and high-fidelity teleportation ( T = 0 ): λ 1 1 λ A B Weak-end-bond XX spin chain with uniform Bulk interactions and weak couplings at the end points of the chain ( λ << 1). Large Alice-Bob entanglement, and large teleportation fidelity at zero temperature. Slowly decreasing with the length of the chain (distance A-B). Drawback: Fragile at finite temperature, but still robust at low enough T. Camerino,

22 C) Different realizations of long-distance entanglement with engineered spin chains: The µ-λ model Long-distance entanglement and high-fidelity teleportation at zero and finite T : A λ µ ( λ = J 1 / J b = J N / J b ; µ = J 2 / J b = J N-1 / J b ; 1 = J b / J b ) λ B Augmented XX spin chain with uniform Bulk interactions and alternating strong/weak couplings at the end and near-end points of the chain ( λ << 1 < µ). Large Alice-Bob entanglement and teleportation fidelity at zero temperature. Slowly decreasing with the length of the chain. Robust even at moderately high T. Camerino,

23 C) End-to-end entanglement in the µ-λ spin chain Alice-Bob entanglement (normalized) Length of the chain (# of sites) λ µ T/J b Camerino,

24 C) End-to-end teleportation fidelity in the µ-λ spin chain Maximal A-B fidelity 2/3 Classical threshold Length of the chain (# of sites) λ µ T/J b Camerino,

25 D) Physical implementation of LD teleportation with engineered XX spin chains: Arrays of coupled cavities Motivation: Long-distance spin chain channels important alternative/complement to quantum repeaters. Building elements of quantum communication circuits. Physical Realization: Extremely challenging with strongly correlated systems of condensed matter. Solution? Use quantum simulators, with controlled engineering and implementation. Optical lattices? In principle, yes. But not very practical because the engineering of LDE requires a very high degree of control on single sites and constituents, which is hard to achieve in optical lattices. Promising alternative: Ion traps with external controlling magnetic field. Further possibility: Engineered coupled cavity arrays. Realization of effective spin Hamiltonians by purely quantum optical means Perfect control at the level of single constituents. Camerino,

26 D) Cavity quantum electrodynamics Cavity QED: A two-level atom interacts via a dipole coupling g with the photons in the cavity. Excitations are lost via spontaneous emission and cavity decay. At very high values of Q, quasi-ideal and lossless cavity. Strong coupling Photon blockade: Due to the strong interaction of the cavity mode with atoms, a single photon can modify the resonance frequency of the cavity mode in such a way that a second photon cannot enter the cavity before the first leaks out. Camerino,

27 D) Coupling cavities by quantum tunneling An array of coupled cavities: Hopping of photons between adjacent cavities occurs due to the overlap (shaded green) of the light modes (green lines). The distance between adjacent cavities can be hundreds of atomic lengths enhanced single-site addressability. Crucial difference with respect to optical lattices. Camerino,

28 D) Arrays of coupled cavities as quantum simulators Polaritons: collective atom-photon excitations that in the strong coupling regime, and in the limit of large numbers of atoms per cavity, obey the bosonic CCRs. Effective Kerr-type interactions among polaritons (Hartmann, Brandao, Plenio): Realization of Bose-Hubbard-like models of interacting and hopping polaritons. OR Effective Kerr-type photonic nonlinearities (Angelakis, Santos, Bose). Phase transitions of light: Prediction of Mott phases of polaritons and phase transition to photon superfluidity. Camerino,

29 D) Two-level systems with optical cavities Models of the µ-λ type, that support LD entanglement and LD teleportation, can be realized by adapting a scheme in which inside each cavity there sits one atom with two degenerate levels (Scheme of Angelakis, Santos, Bose) Camerino,

30 D) Realizing the µ-λ spin model in coupled cavity arrays For more details and for specific results on long-distance, high-fidelity quantum teleportation in coupled cavity arrays, see the poster by S. M. Giampaolo Camerino,

31 An overview at the crossroads A multi-facetted physical framework emerges at the confluence of condensed matter, atomic physics, quantum optics, and quantum information. I)Some interesting problems of many-body physics can be addressed by the methods of quantum information and can be tested experimentally with optical and atomic quantum simulators. They allow unprecedented degrees of control on physical parameters, flexibility in their spatial/temporal engineering, and enhanced resilience against decoherence (optical lattices, ion traps, arrays of coupled cavities): Here, we considered the instance of interaction balancing and factorization in quantum cooperative systems. II)Some models of condensed matter provide important schemes and devices for quantum information and communication. These schemes may be tested and implemented resorting to a suitable quantum simulators: Here, we discussed the instance of long-distance qubit teleportation in arrays of coupled optical cavities. Camerino,

32 A unifying final message: Light does matter Camerino,

33 This Referenzen talk was zum based Inhalt on: Giampaolo & Illuminati, PRA 76, (2008) Giampaolo & al., PRA 77, (2008) Giampaolo, Adesso, & Illuminati, PRL 100, (2008) factorization Campos Venuti & al., PRA 76, (2007) Giampaolo & Illuminati, submitted, Oct CCAs Dell Anno & al., PRA 77, (2008) entanglement Illuminati, Nature Physics 2, 803 (2006) matter more to come... Theory of SQUOs Thank you very much for your kind attention! SQUOs & QPTS Theory of LDE in XX spin chains µ λ models in Hierarchical Light does Camerino,