Decoherence by controlled spin baths D. Rossini, T. Calarco, V. - PowerPoint PPT Presentation

Decoherence by controlled spin baths D. Rossini, T. Calarco, V. Giovannetti, S. Montangero and R. Fazio SNS - Pisa SISSA - Trieste BEC - Trento ITAMPT - Harvard cond-mat/0605051

Motivation Decoherence Paradigm models Engineered Quantum Baths - Harmonic oscillators - Spin baths

Spin baths N.V. Prokof'ev and P.C.E. Stamp, Rep. Prog. Phys. 63 , 669 (2000) indipendent spins W.H. Zurek (1982) F.M. Cucchietti, J.P Paz, and W.H. Zurek, (2005) L. Tessieri and J. Wilkie, (2003) Highly symmetric interactions and baths interacting spins D.V. Khveshchenko, (2003) C.M. Dawson et al. , (2005) S. Paganelli, F. de Pasquale, and S.M. Giampaolo, (2002) H.T. Quan et al (2006). F.M. Cucchietti, S. Fernandez-Vidal, and J.P. Paz, (2006)

The setup System Bath ε J J J A two-level system(the quantum system) is coupled to a single spin of a one-dimensional spin-1/2 chain (the environment).

Physical realizations - JJAs D. Haviland’s group

Physical realizations - optical lattices Optical lattices as simulators of interacting spin systems E. Janè et al (2003) D. Jaksch et al (1999) O. Mandel et al (2003)

The Model H = H T L + H E + H IN H T L = ω 1 | 1 �� 1 | H T L = − � | 1 �� 1 | σ z 1

Pure dephasing Unruh (1995) Palma, Suominen, and Ekert, (1996) � ρ 00 (0) � ρ 01 ( t ) ρ ( t ) = ρ 10 ( t ) ρ 11 (0) ρ 10 ( t ) = ρ 10 (0) D ( t ) D ( t ) = � e i H t e − i ( H T L + H E ) t �

Loschmidt echo L ( t ) = | D ( t ) | 2

MODELS for the environment H E

Ising chain in a transverse field N N H = − J � i +1 + (1 + γ ) σ y i σ y � (1 − γ ) σ x i σ x σ z i +1 − h i 2 i =1 i =1 � σ x � � = 0 Ising universality class γ � = 0 λ = h XY universality class 1 γ = 0 J Exact solution - free fermions Pfeuty ‘70

Heisenberg Model � i +1 + σ y i σ y � σ x i σ x i +1 + ∆ σ z i σ z � H = J i i +1 i

Heisenberg Model Constant couplings Ferromagnetic Critical region Antiferromagnetic ∆ -1 1

Ising chain environment 1 − r + r e i C t � � L ( t ) = det � � A B r i,j = � Ψ † i Ψ j � C = − B − A B j,k = − Jγ ( δ k,j +1 − δ j,k +1 ) A j,k = − J ( δ k,j +1 + δ j,k +1 ) − 2( λ + � j ) δ j,k

Ising chain environment 1 L 1 II I 0.995 0.99 0.99 0 0.2 0.4 0.6 L I ( t ) ∼ e − αt 2 a) 0.985 0 25 50 7 J t REGION I REGION II " 1 1 L $ # 2 0.8 oscillations λ < 1 0.995 0.6 0.99 0.4 constant λ > 1 0.2 b) c) 0.985 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 ! !

λ = λ c Ising chain environment 1 c L 0.995 c 0 L c ( t ) = 0.99 (1 + c 1 ln t ) c L ! 0.985 0.986 0.984 l ∞ 0.98 L c ∞ = 0.982 1 + β ln N 0.98 0.975 0.978 500 1000 2000 N 100 50 -1 0 1 2 10 10 10 10 Jt J

Ising chain - correlations in the bath If the chain is not at the critical point, the decay of the coherences is independent of the number of spins in the bath Configuration 1 Configuration 2

Configuration 1 Configuration 2 Ising chain - correlations in the bath 1 L 0.99 0.98 0.97 λ � = λ c 0 10 20 30 40 50 60 t 0.0005 δ L 0 -0.0005 -0.001 -0.0015 0 10 20 30 40 50 60 t

Ising chain environment - γ � = 0 Universality 1 L 0.99 0.98 0.97 0.96 0.95 0 50 γ = 0 . 5 100 150 200 J t 0 1 d λ α L ∞ ε -1 0.99 -2 0.98 -3 0.97 -4 0.96 -5 0.95 0 1 2 0.5 1.5 2,5 0 0.5 1 1.5 2 λ λ

XY chain environment 1 L 0.75 0.5 The chain is critical 0 200 400 600 800 1000 J t L ∞ L ∞ L ∞ 0 d λ α 1 λ =0.25 λ =0.9 0.84 0.94 0.9 ε -0.2 0.8 0.82 0.935 -0.4 0.7 0.8 -0.6 0.6 0.93 0.5 0.78 -0.8 0.925 0.4 0 0.25 0.5 0.75 1 1.25 1.5 0 0.25 0.5 0.75 1 1.25 1.5 0.76 λ λ 0.92 l 0 =0.990088 l 0 =1.18854 β =0.0128334 β =0.0952356 0.74 0.915 50 100 200 300 500 50 100 200 300 500 N N

Decoherence vs interaction & entanglement in the bath see the suggestion of

How to measure entanglement? - Entanglement between two spins in the network - Multipartite entanglement - Block entropy - Localizable entanglement - ...

Bipartite entanglement The state of the two selected spins is mixed

Measure of mixed state entanglement for two spin-1/2 states - Separable state -Entangled state state 1 √ | α � = | 00 � | α � = 2( | 00 � + | 11 � ) - “NOT” - “NOT” 1 √ | β � = | 11 � | β � = 2( | 00 � + | 11 � ) � β | α � = 0 � β | α � = 1

- construct R = ρ ( i, j )˜ ρ ( i, j ) where ρ . = σ y ⊗ σ y ρ ∗ σ y ⊗ σ y ˜ - the concurrence is defined as C ( i, j ) = max { 0 , λ 1 ( i, j ) − λ 2 ( i, j ) − λ ( i, j ) − λ 4 ( i, j ) } where s are the eigenvalues of in ascending order λ R

Decoherence vs entanglement in the bath 0,06 α λ =0 0,05 λ =0.8 0,04 λ =1 λ =1.2 0,03 C(1) 0,2 λ m 0,02 λ =2 0,1 0,01 0 -1 -0,5 0 0,5 1 Δλ 0 0 0,1 0,2 0,3 C(1)

Heisenberg chain environment - tDMRG 1 0.01 α L 0.008 0.9975 0.006 0.004 0.995 0.002 a) b) 0 0.9925 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 J t Δ critical region

Coupling to m spins in the environment Configuration A Configuration B

Configuration A Configuration B 0.06 0.045 α /m α /m Text 0.05 0.04 Weakly 0.035 0.04 dependent on m 0.03 0.03 0.94 0.96 0.98 1 1.02 1.04 1.06 λ 0.02 0.01 0 0 0.5 1 1.5 2 2.5 λ

Configuration A Configuration B Strong dependence on the number of couplings 1 L ! 0.8 0.6 m increases 0.4 0.2 0 λ 0 0.5 1 1.5 2

Conclusions ✔ Optical lattices to simulate quantum baths ✔ Exact solution for an interacting baths ✔ Numerical t-DMRG for Heisenberg bath