Quantum Chaos in Composite Systems Karol ˙ Zyczkowski in collaboration with Lukasz Pawela and Zbigniew Pucha� � la (Gliwice) Smoluchowski Institute of Physics, Jagiellonian University, Cracow and Center for Theoretical Physics, Polish Academy of Sciences, Warsaw University of Hradec Kr´ alov´ e , May 10, 2017 K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 1 / 28
The collaboration with Peter K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 2 / 28
The collaboration with Peter started in 1990 in Germany... K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 2 / 28
joint papers on wave chaos K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 3 / 28
Closed systems, Unitary Dynamics & Quantum Chaos ’Quantum chaology’: analogues of classically chaotic systems Quantum analogues of classically chaotic dynamical systems can be described by random matrices a). autonomous systems – Hamiltonians: Gaussian ensembles of random Hermitian matrices, (GOE, GUE, GSE) b). periodic systems – evolution operators: Dyson circular ensembles of random unitary matrices, (COE, CUE, CSE) Universality classes ( for unitary dynamics ) Depending on the symmetry properties of the system one uses ensembles form orthogonal ( β = 1); unitary ( β = 2) and symplectic ( β = 4) ensembles. The exponent β determines the level repulsion, P ( s ) ∼ s β for s → 0, where s stands for the (normalised) level spacing, s i = φ i +1 − φ i . see e.g. Fritz Haake , Quantum Signatures of Chaos K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 4 / 28
Classical kicked top model - Haake, Ku´ s, Sharf 1987 Discrete dynamics on a sphere: X 2 + Y 2 + Z 2 = 1 X ′ = Re ( X cos p + Z sin p + iY ) e ikZ cos p − X sin p , Y ′ = Im ( X cos p + Z sin p + iY ) e ikZ cos p − X sin p , Z ′ = − X sin p + Z cos p . linear rotation parameter: p = π/ 2, kicking strength k k = 2 . 0 k = 2 . 5 k = 3 . 0 k = 6 . 0 transition to chaos K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 5 / 28
Quantum kicked top model - Haake, Ku´ s, Sharf 1987 Discrete dynamics in Hilbert space of dimension N = 2 j + 1 � + ∞ i) Hamiltonian H ( t ) = pJ y + k 2 j J 2 j = −∞ δ ( t − n ) z ii) Unitary evolution operator U = exp[ − i ( k / 2 j ) J 2 z ] exp[ − ipJ y ] Level spacing distribution P ( s ) (where s = ( φ i +1 − φ i ) N / 2 π ) a) k ∈ [0 . 1 , 0 . 3] (regular dynamics) b) k ∈ [10 . 0 , 10 . 5] (chaotic dynamics) N = 201 K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 6 / 28
Quantum unitary kicked top - conclusions A) In the case of classically regular motion the level spacing distribution P ( s ) displays level clustering, ( Poisson ) B) In the case of classically chaotic motion the level spacing distribution P ( s ) displays level repulsion, ( Wigner ) Unitary evolution matrices U display statistical properties of circular unitary ensemble and their eigenvectors are generic and delocalized . K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 7 / 28
Open Systems and spectral properties of nonunitary evolution operators K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 8 / 28
joint papers on chaotic scattering K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 9 / 28
Interacting Systems & Nonunitary Dynamics Due to interaction of an open system with an environment one needs to work with mixed states, which arise by averaging ( partial trace ) over the environment . ρ ′ = Φ( ρ ) (in the space of density matrices!) Quantum maps: Enviromental form (interacting quantum system !) ρ ′ = Φ( ρ ) = Tr E [ V ( ρ ⊗ ω E ) V † ] . where ω E is an initial state of the environment, while V is non–local unitary; VV † = ✶ . Kraus form ρ ′ = Φ( ρ ) = � i A i ρ A † i , where the Kraus operators satisfy relation i A † � i A i = ✶ , which implies that the trace is preserved. K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 10 / 28
Nonunitary Dynamics & mixed quantum states Due to interaction with environment the image of a pure state becomes ρ ′ = Φ( | ψ �� ψ | ) � = ( ρ ′ ) 2 mixed , Set M N of all mixed states of size N M N := { ρ : H N → H N ; ρ = ρ † , ρ ≥ 0 , Tr ρ = 1 } example: M 2 = B 3 ⊂ ❘ 3 - Bloch ball with all pure states at the boundary The set M N is compact and convex : ρ = � i a i | ψ i �� ψ i | where a i ≥ 0 and � i a i = 1. The set of mixed quantum states has N 2 − 1 real dimensions , *) **) For N ≥ 3 the set M N of mixed states is neither a polytope nor an ellipsoid with a smooth surphace. The set of pure states forms only a small (measure zero!) part of the boundary of the set M N of mixed states. K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 11 / 28
Two coupled quantum kicked tops Miller and Sarkar (1998), Demkowicz-Dobrza´ nski and Ku´ s (2008), Lakshminarayan (2010) Two spins j 1 and j 2 described in space of dimension D = (2 j 1 + 1)(2 j 2 + 1) Unitary evolution operator V = U 12 ( U 1 ⊗ U 2 ) � − i k � − i π � � 2 j J 2 where U i = exp exp , i = 1 , 2 2 J y i z i � − i ǫ � U 12 = exp j J z 1 ⊗ J z 2 . Global dynamics of two coupled spins is unitary , but the evolution of the reduced state is not ! | φ 0 � ∈ H 1 ⊗ H 2 , � V t | φ 0 �� φ 0 | ( V † ) t � σ t = Tr B We analyze mixed states σ t obtained by partial trace over the other spin... K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 12 / 28
Coupled kicked tops - level density of reductions Two spins j 1 and j 2 described each in dimension N i = 2 j i + 1. Let c = (2 j 1 + 1) / (2 j 2 + 1) denote the ratio of both dimensions 0 . 30 0 . 50 P ( x ) P ( x ) 0 . 15 0 . 25 0 . 00 0 . 00 0 1 2 0 . 0 1 . 5 3 . 0 x x 3 . 0 0 . 30 P ( x ) P ( x ) 1 . 5 0 . 15 0 . 0 0 . 00 0 2 4 0 4 8 x x Level density P ( x ) of the rescaled eigenvalue x = λ 1 N 1 for rectangularity c = 0 . 2 , 0 . 5 , 1 . 0 and 4 . 0 compared with Marchenko-Pastur distribution. K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 13 / 28
K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 14 / 28
Hradec Kr´ alov´ e K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 14 / 28
Random Matrices and Marchenko-Pastur distribution (1967) Let G be a nonhermitian random matrix of size N 1 × N 2 with i.i.d. Gaussian complex variables. Then the spectrum of a positive matrix W = GG † / Tr GG † has density given by P c ( x ) = 1 � ( x − x − ) ( x + − x ) , 2 π x 1 ± √ c � 2 . � where x = N 1 λ , rectangularity c = N 1 / N 2 and support x ± = For square matrices c = 1 this expression reduces to the standard Marchenko – Pastur distribution √ 1 − x / 4 P 1 ( x ) = , x ∈ [0 , 4], π √ x equivalent to setting x = y 2 with y distributed according to Wigner semicircle Vladimir Marchenko & Leonid Pastur (2000) K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 15 / 28
Quantum coupled kicked top - first observations In the case of classically chaotic motion the level density P ( x ) of � � partially reduced states, σ = Tr 2 | ψ 12 �� ψ 12 | , is described by the Marchenko-Pastur distribution, so it conforms to predictions of random matrices . Questions concerning time evolution : – Do any initial state φ 0 leads to a ’generic’ mixed state � V t | φ 0 �� φ 0 | ( V † ) t � σ t = Tr B after a sufficiently long evolution time t ? – Are density matrices σ t distributed uniformly (with respect to the flat, Hilbert-Schmidt measure) inside the set M N of mixed quantum states ? K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 16 / 28
Symmeterized Marchenko–Pastur distribution Trace distance between two states D Tr ( ρ, σ ) = � ρ − σ � 1 = Tr | ρ − σ | is used to describe their distinguishability. To compute it we analyze distribution of eigenvalue µ of the difference ρ − σ , where both states are random. It is given by symmetrized Marchenko–Pastur distribution, f c ( x ) = MP c ( x ) ⊞ MP c ( − x ), where x = N 1 µ and ⊞ denotes free multiplicative convolution . In the case of HS measure, (rectangularity c = 1) we obtain a normalized symmetric distribution � √ �� 2 / 3 � − 1 − 3 x 2 + 3 + 33 x 2 − 3 x 4 + 6 x 1 + 3 x f 1 ( x ) = . √ � √ �� 1 / 3 � 3 + 33 x 2 − 3 x 4 + 6 x 2 3 π x 1 + 3 x and analogous formulae for f c ( x ) with an arbitrary parameter c = N 1 / N 2 > 0. K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 17 / 28
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