Quantum Information and Many-Body Physicswith 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-84084 Fisciano

(SA)

&

Quantum Physics Division – ISI Foundation for Scientific Interchange,Viale Settimio Severo, I-10133 Torino

http://www.sa.infn.it/QuantumTheory/Camerino, 25-10-2008 1

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Theory ofEntanglement and

Quantum Information

Quantum Informationand Many-Body

Physics

Atoms and Photons asQuantum Simulators:

Ion Traps, OpticalLattices, Coupled

Cavity Arrays

Quantum Informationwith Many-Body

Systems: Applicationsof QuantumSimulators

Areas of research at the crossroads:

• A) Simulating strongly correlated matter with atom-optical systems:

A1) Optical lattices, the Bose-Hubbard model, and the Mott insulator-superfluid 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.

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Research Directions (1)

• C) Quantum information tasks in models of interacting many-body systems:

Open quantum spin chains with XX interactions and modified endcouplings - Long distance entanglement and quantum teleportation inthe _ – _ model.

• D) Closing the loop – Implementation with quantum simulators:

Realizing quantum spin chains supporting long-distance, high fidelityqubit teleportation in suitably engineered arrays of coupled cavities.

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Research Directions (2)

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Methods of quantum information and theory ofentanglement in many-body physics

• Ground state of the Bose-Hubbard model optical lattices: A paradigmatic instance ofquantum entanglement vs. separability and factorization. In the correlated superfluidphase the GS is entangled (the hopping terms dominate in the Hamiltonian, and correlatethe different sites of the lattice). In the uncorrelated Mott-insulating phase the inter-sitetunneling is suppressed, the Hamiltonian is a sum of single-site contributions (the on-siterepulsive interactions) and the GS is a product of single-site wave functions: the GS isfactorized (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 phasetransitions, 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?

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Entanglement and quantum phase transitionsin interacting quantum spin systems

rxxryyrzzzXYZxijyijzijii,ji1HJSSJSSJSShS2r|ij|=++−=−∑∑

Fundamental systems of interacting qubits (spin 1/2) undergoing quantum phasetransitions at zero temperature. The XYZ Hamiltonian: It comprises the mostimportant models of quantum spins coupled by exchange interaction terms,including the Ising, Heisenberg, XY, XX, XXZ models. Moreover, most relevantmodels of quantum spin chains for quantum information tasks (more on this in thefollowing ).

Includes models with short, finite, and long-range interactions. Most cases arenon exactly solvable, with some notable exceptions, like the XY model.

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B1) Entanglement and quantum phase transitions in quantum spin chains

XY model:xxyyzXYii1ii1iiiHSSSShS++=−+Δ+∑∑

Anisotropic spin model ( 0 ≤ Δ < 1). Phase transition: A spontaneousmagnetization develops along the x axis as the external transverse field his varied. The divergence of the first derivative (with respect to h) of thevon Neumann block entanglement entropy between a single spin and therest of the system signals the onset of the quantum phase transition:

Ent

angl

emen

t en

trop

y

Δ = 0.25 (--------)Δ = 0.28 ( )

Quantum critical points

Factorization points

Magnetic field

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B2) Ground state factorization in systems of interactingqubits (quantum spin chains)

In the XY model, we have just seen an instance of a factorization point: Thequantum entanglement vanishes and the ground state becomes factorized, even if thesystem is strongly interacting! Is this an acccidental phenomenon, a coincidence, or ithides a more profound physical picture?

Exploiting methods inspired by entanglement theory, it is possible to derive a generaltheory of interaction balancing and determine the conditions for the occurrence offull 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.

Single Qubit Unitary Operations:A|BABUOI=⊗

ATrO0=

Hermitian:

Unitary:

Traceless:

†2 AAAAOOOI==

† AAOO=

B2) Factorization in quantum spin systems:Single-qubit unitary operations (SQUOs)

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Response of the system to controlled, nontrivial external perturbations.

Factorization:FABABψ≡φ⊗χ

FextrextrA|BA|BABABABABiffU:Uψ=ψ∃ψ=ψ

Factorization � Invariance under the action of SQUOs

The “extremal” SQUO is uniquely defined

S. M. Giampaolo and F. I., Phys. Rev. A 76, 042301 (2007)

extrA|BU

Theorem:

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B2) Factorization and SQUOs in quantum spin systems

Pure State:

{}()A|B2EAABABUS()mind,ρ=ψψ%

()2ABABABABd,1ψψ=−ψψ%%

S. M. Giampaolo and F. I., Phys. Rev. A 76, 042301 (2007)

Theorem: Distance � Entanglement (von Neumann entropy):

Transformed State:

A|BABABUψ=ψ%

Hilbert-space distance:

B2) Factorization in quantum spin systems:SQUOs and entanglement

ABΨ

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A|BABABUψ=ψ% ABABABABˆˆ QQQΔ=ψψ−ψψ%%

If:

Observable estimators Q of separability under the action of SQUOs[ Bipartite system. Spin i = party A. Party B = all remaining spins N/i ]

A|Bˆ1)Q0,2)U,Q0Δ≥≠

i|(N/i)i(N/i)Q0Δ=⇒ψ=φ⊗χ

This in turn implies total (full) factorization in translational-invariant systems:iiQ0Δ=⇒ψ=⊗φ

B2) Quantifying factorization in quantum spin systemswith SQUO-related observables

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Then:

Entanglement Excitation Energy:

Hamiltonian Structure: Entanglement excitationenergies (EXEs) associated to extremal SQUOs

ˆˆ EGHGGHGΔ=−%%

ˆ ˆQIdentification:GroundGHState:=ψ=

E0Δ=⇔Factorized ground state

Entangled ground stateE0Δ>⇔

S. M. Giampaolo et al., Phys. Rev. A 77, 012319 (2008)

B2) Determining ground-state factorization in quantum spinsystems with entanglement excitation energies (EXEs)

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Phase diagram for factorization in the XYZ models

rryxJJ

rrzxJJExcited states? (Factorized)

Previously known factorization

All the factorized ground states

Forbidden region

B2) Applying the general theory to the determination of thefactorization points of interacting quantum spin systems

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G. Adesso, S. M. Giampaolo, and F. Illuminati, Phys. Rev. Lett. 100, 197201 (2008)

1) Formalism of local unitary operations. Operational- geometric approach tothe 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 quantuminformation.

Detailed exposition and further results in: S. M. Giampaolo, G. Adesso, and F. Illuminati,arXiv:0811.XXXX, to appear

3) Factorization: A highly nontrivial balancing in strongly interactingmany-body quantum systems.

B2) Summary on factorization in many-body systems

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• 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 quantumcomplexity.

• Beyond spins and qubits: Systems of strongly correlated matter.

• Physics close to factorization points. Novel approximation schemes, startingfrom sets of exact solutions. Excitations and other physical properties aroundfactorization. Understanding “Entanglement phase transitions” [Amico et al.,PRA 74, 022322 (2006)] by hierarchical measures of entanglement [Dell’Annoet al., PRA 77, 062304 (2008)]. Factorization at finite at temperature (and inclassical systems?).

B2) Outlook: Significance of factorization in condensed matter

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• Producing arbitrarily large numbers of identical copies of single-qubit states (Preparing the register).

• Partial factorization and qubit encoding in systems with higherlocal dimension (qudits, continuous variables).

• Identifying the working points (close or away from factorization).

• One-way quantum computation – many-qubit cluster states at oncefrom dual-rail factorization.

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B2) Significance of factorization in quantum information

Possible realizations with quantum simulators

Systems with fundamental control of spin-spin couplings and spin-chain 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.)

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B2) Investigating physics at factorization:Possible experimental realizations

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Possible solutions? Suitably Engineered quantum spin chains: Open XXspin chains with alternating couplings, or uniform bulk interactions withmodified, “weak” end bonds. At exactly T = 0, these systems can supportmaximal entanglement and perfect qubit teleportation (i.e. with unit fidelity)between the ends of the chain (“Long-Distance-Entanglement” - LDE):

C) Implementing quantum information tasks with engineeredmany-body systems: Long-distance entanglement

Qubit teleportation. Fundamental ingredients for working schemes of quantumteleportation:I)A good quantum channel (as long as possible)II)Highest possible end-to-end entanglement, and, therefore, the highest possibleend-to-end teleportation fidelity.Formidable tasks both with condensed matter- and optical-based devices.

N1xxyyXXkkk1kk1i1HJ(SSSS)−++==−+∑

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C) Different realizations of long-distance entanglementwith engineered spin chains: The ideal case

I) Infinite-distance entanglement and perfect teleportation (zero temperature):

• • • • • • • •

21/2(|||)1inputoutputℑ=ΨΦ=

A λ

1 λ

1λ

1 λ

B

Dimerized 1-D XX spin chain with fully alternated couplings ( λ << 1).Maximal Alice-Bob entanglement. Unit fidelity for teleporting anunknown input state from Alice to Bob at zero temperature. Any finitesize, arbitrary length. Very serious drawback: Extremely fragile atfinite temperature, even at very low T.

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C) Different realizations of long-distance entanglementwith engineered spin chains: The weak-end bond trick

II) Long-distance entanglement and high-fidelity teleportation ( T = 0 ):

• • • • • • • •A λ

1 1

11

1 λ

B

“Weak-end-bond” XX spin chain with uniform Bulk interactions andweak couplings at the end points of the chain ( λ << 1). Large Alice-Bobentanglement, and large teleportation fidelity at zero temperature. Slowlydecreasing with the length of the chain (distance A-B). Drawback: Fragileat finite temperature, but still robust at low enough T.

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C) Different realizations of long-distance entanglementwith engineered spin chains: The µ-λ model

Long-distance entanglement and high-fidelity teleportation at zeroand finite T :

A λ

µ 1

11

1 λ

B ( λ = J1 / Jb = JN / Jb ; µ = J2 / Jb = JN-1 / Jb ; 1 = Jb / Jb )

Augmented XX spin chain with uniform Bulk interactions andalternating strong/weak couplings at the end and near-end points of thechain ( λ << 1 < µ). Large Alice-Bob entanglement and teleportationfidelity at zero temperature. Slowly decreasing with the length of thechain. Robust even at moderately high T.

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C) End-to-end entanglement in the µ-λ spin chain

Alice-Bob entanglement (normalized)

Length of the chain (# of sites)

λ µ T/Jb 0.15 7 5 � 10-5 - - - - - - 0.2 5 2 � 10-4 - - - - - - 0.4 3 2 � 10-3 - - - - - - 0.2 1 5 � 10-3 - - - - - -

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C) End-to-end teleportation fidelity in the µ-λ spin chain

Maximal A-B fidelity

Length of the chain (# of sites)

λ µ T/Jb 0.15 7 5 � 10-5 - - - - - - 0.2 5 2 � 10-4 - - - - - - 0.4 3 2 � 10-3 - - - - - - 0.2 1 5 � 10-3 - - - - - -

2/3 Classical threshold

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D) Physical implementation of LD teleportation withengineered XX spin chains: Arrays of coupled cavities

Motivation: Long-distance spin chain channels important alternative/complementto quantum repeaters. Building elements of quantum communication circuits.

Physical Realization: Extremely challenging with strongly correlated systems ofcondensed matter. Solution?

Use quantum simulators, with controlled engineering and implementation.Optical lattices? In principle, yes. But not very practical because the engineeringof 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 ofeffective spin Hamiltonians by purely quantum optical means –Perfect control at the level of single constituents.

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D) Cavity quantum electrodynamics

Cavity QED: A two-level atom interacts via a dipole coupling g with thephotons in the cavity. Excitations are lost via spontaneous emission andcavity decay. At very high values of Q, quasi-ideal and lossless cavity.

Strong coupling – Photon blockade: Due to the strong interaction of thecavity mode with atoms, a single photon can modify the resonance frequencyof the cavity mode in such a way that a second photon cannotenter the cavity before the first leaks out.

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D) Coupling cavities by quantum tunneling

An array of coupled cavities: Hopping of photons betweenadjacent cavities occurs due to the overlap (shaded green) of thelight modes (green lines).

The distance between adjacent cavities can be hundreds ofatomic lengths � enhanced single-site addressability. Crucialdifference with respect to optical lattices.

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D) Arrays of coupled cavities as quantum simulators

Polaritons: collective atom-photon excitations that in the strong couplingregime, and in the limit of large numbers of atoms per cavity, obey thebosonic 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). Phasetransitions of light: Prediction of Mott phases of polaritons and phase transition tophoton superfluidity.

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D) Two-level systems with optical cavities

Models of the µ-λ type, that support LD entanglement and LDteleportation, can be realized by adapting a scheme in which insideeach cavity there sits one atom with two degenerate levels (Scheme ofAngelakis, Santos, Bose)

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D) Realizing the µ-λ spin model in coupled cavity arrays

For more details and for specific results on long-distance, high-fidelityquantum teleportation in coupled cavity arrays, see the poster by S. M.Giampaolo

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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 ofquantum information and can be tested experimentally with optical and atomic quantumsimulators. They allow unprecedented degrees of control on physical parameters,flexibility in their spatial/temporal engineering, and enhanced resilience againstdecoherence (optical lattices, ion traps, arrays of coupled cavities): Here, we considered the instance of interaction balancing and factorization inquantum cooperative systems.

II)Some models of condensed matter provide important schemes and devices for quantuminformation and communication. These schemes may be tested and implemented resortingto a suitable quantum simulators:Here, we discussed the instance of long-distance qubit teleportation in arrays ofcoupled optical cavities.

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A unifying final message:

Light does matter

Referenzen zum Inhalt

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This talk was based on:

Giampaolo & Illuminati, PRA 76, 042301 (2008) Giampaolo & al., PRA 77, 012319 (2008) SQUOs & QPTS

Giampaolo, Adesso, & Illuminati, PRL 100, 197201 (2008) Theory offactorization

Campos Venuti & al., PRA 76, 052328 (2007) Giampaolo & Illuminati, submitted, Oct. 2008 µ−λ models inCCAs

Dell’Anno & al., PRA 77, 062304 (2008) Hierarchicalentanglement

Illuminati, Nature Physics 2, 803 (2006) Light doesmatter

more to come....

Theory of SQUOs

LDE in XX spin chains

Thank you very much for your kind attention!