Random matrix ensembles for quantum spins and decoherence François David IPhT Saclay & CNRS J. Stat. Mech. (2011) P01001 + work in progress In honor of T om Spencer Susy & Random Matrices, IHP 3-5 avril 2012 1 mercredi 11 avril 12
Plan 1. The model: quantum spin + random matrices 2. The evolution functional: exact solution 3. Evolution of coherent and incoherent states 4. Quantum diffusion regime & initial conditions: to be or not to be Markovian 5. Extensions: spin clusters 2 mercredi 11 avril 12
Plan 1. The model: quantum spin + random matrices 2. The evolution functional: exact solution 3. Evolution of coherent and incoherent states 4. Quantum diffusion regime & initial conditions: to be or not to be Markovian 5. Extensions: spin clusters Apologies No disorder .... No SUSY ... 2 mercredi 11 avril 12
Decoherence “large” external “small” quantum ⊗ system system | ψ � | φ � E S • Decoherence = disappearance - or rather inobservability - of the quantum correlations between • some states of a system s, through its (weak) coupling with an external system E (heat bath, environment, etc.) • or more generally a few “individualized” degrees of freedom (pointer states, semi-classical variables, collective coordinates, etc.) of a large isolated macroscopic system ( a 1 | ψ 1 ⇤ + a 2 | ψ 2 ⇤ ) � | φ ⇤ ⇥ a 1 | ψ 0 1 ⇤ � | φ 0 1 ⇤ + a 2 | ψ 0 2 ⇤ � | φ 0 2 ⇤ 3 mercredi 11 avril 12
• I shall present a simple toy model • based on very standard ideas: • spin and coherent states (Takahashi & Shibata, 1975) • random matrix hamiltonians (Mello, Pereyra & Kumar, 1988) • which have been much applied for the spin 1/2 case ( j = 1/2 , Q-bit, 2 level system) • but some (relatively) novel aspects • general spin j (from quantum to classical spin) • generic interaction (novel random matrix ensembles) • It allows to study analytically several aspects decoherence • in particular the crossover between unitary quantum dynamics and stochastic diffusion in classical phase space 4 mercredi 11 avril 12
I - The model A quantum SU(2) spin S + an external system E spin = j dim( H S ) = 2 j + 1 dim( H E ) = N � j Single spin: For large spin j → ∞ the spin becomes a classical object Classical phase space is the 2-sphere The coherent states behave as quasi classical states n · � | � n � ( � S ) | � n � = j | � n � , Dynamics of the coupled spin: H = H S ⊗ 1 E + H SE + 1 S ⊗ H E The Hamiltonians: • Slow spin dynamics H S = 0 (no dissipative & thermalisation effects) • Dynamic of the external system generic H E → H SE 5 mercredi 11 avril 12
The interaction Hamiltonian The interaction hamiltonian is given by a Gaussian random matrix ensemble, with the only constraint that the ensemble in invariant under SU (2) × U ( N ) spin external system For this, go to Wigner representation of spin operators αβ � W ( lm ) ⇥ r α | H | s β ⇤ = H rs j ⊗ j = 0 ⊕ 1 ⊕ · · · ⊕ 2j αβ ⌃ j � ⇥ j ⇤ 2 l + 1 l j W ( l,m ) ⌅ � A rs = � r | A | s ⇥ = A rs � A s r m 2 j + 1 � r,s = − j W ( lm ) It is enough to take for the independent gaussian random αβ variables with zero mean and variance depending only on l and with the Hermiticity constraint. ⇣ ⌘ ( l, − m ) W ( lm ) W ( l,m ) = ( − 1) m W = ∆ ( l ) Var βα αβ αβ 6 mercredi 11 avril 12
We thus get a matrix ensemble characterized by the variances ∆ = { ∆ ( l ) , l = 0 , 1 , · · · 2 j } NB: The l=m=0 term represents the Hamiltonian H E With this GU(2)xU( N ) ensemble, the 2-points correlator is α δ r u H rs αβ H tu γδ = δ αδ δ βγ D rs,tu s t γ β 2 j ⇤ ⇤ ⌅ j ⇧ ⌅ j ⇧ ∆ ( l ) 2 l + 1 l j l j ⇤ ⇤ ] = δ s − r,t − u D rs,tu = ⇤ ⇤ t u − t u s r − s r 2 j + 1 ⇤ ⇤ l =0 This representation allows to use diagrammatic rules to resum perturbative expansions in the interaction. Standard ribbon propagator for the N indices, more complicated structure for the spin indices, but still planar. 7 mercredi 11 avril 12
II - The evolution functional separable state → entangled state → mixed state for S | ψ 0 ⌅ � | φ 0 ⌅ ⇥ | Φ ( t ) ⌅ , ρ S ( t ) = tr E ( | Φ ( t ) ⌅⇤ Φ ( t ) | Evolution functional e − itH ( · ⊗ ρ E (0)) e itH � � ρ S ( t ) = M ( t ) · ρ S (0) , M ( t ) = tr E For simplicity, start from a random state | ψ E � Then the evolution functional is I I dx dy 2 i π e it ( x − y ) G ( x, y ) M ( t ) = 2 i π � G ( x, y ) = 1 1 1 N tr E x − H ⊗ S y − H 8 mercredi 11 avril 12
We take the large N limit (large external system) and make the average over H , assuming self averaging as usual. G ( x, y ) is given by a sum of planar diagrams of the standard form (rainbow diagrams) r s t u It is useful to start from the single resolvent � H ( x ) = 1 1 N tr E x − H H ( x ) is given by a sum of planar rainbow diagrams D + .... r s r s r s r s 9 mercredi 11 avril 12
These resolvents obey recursion relations � = + u v v u s r s r s r t t + = + r s t u r s t u r v w x s t u r v s t w x u Thanks to the SU(2) invariance, the solution of these equations takes a simple diagonal form in the Wigner representation H rs ( x ) = δ rs b H ( x ) ( l 1 ,m 1 ) , ( l 2 ,m 2 ) ( x, y ) = δ l 1 l 2 δ m 1 + m 2 , 0 ( − 1) m 1 b G ( l ) ( x, y ) G rs,tu ( x, y ) → W G with ✓ ◆ q 1 Resolvent for a single Wigner b x 2 − 4 b H ( x ) = ∆ (0) x − 2 b matrix (semi circle law) ∆ (0) 2 j X 2 l + 1 b ∆ (0) = N 2 j + 1 ∆ ( l ) l =0 10 mercredi 11 avril 12
Factorization The evolution functional for the density matrix of the spin ρ S ( t ) takes a simple diagonal form in the Wigner representation basis ρ S rs ( t ) → W ( l,m ) ( t ) = c M ( l ) ( t ) · W ( l,m ) (0) S S with the kernel given by a universal decoherence function ⌥ ⌥ M ( l ) ( t ) = M ( t/ τ 0 , Z ( l )) . depending on a rescaled time and a factor Z(l) t 0 = t/ τ 0 ⌃ ∆ ( l ) ⌃ Z ( l ) = τ 0 = 1 / ∆ (0) ⌃ ∆ (0) is the dynamical time scale of the system (more later) τ 0 The parameter Z(l) depends on the spin sector considered. 11 mercredi 11 avril 12
The Z(l) function The l dependence of the factor Z(l) depends on the initial variances of the GU(2) ensemble for the Hamiltonian. ⇥ j ⇤ 2 j ⌃ l ⇥ j ∆ ( l ⇥ )(2 l ⇥ + 1)( − 1) 2 j + l � + l ⌥ ∆ ( l ) = N 6-j symbol j j l l � =0 Z ( l ) = b ∆ ( l ) / b ∆ (0) Z ( l ) ∈ [ − 1 , 1] Z(l) is maximal for l=0 Z(l) takes a scaling form in the large spin limit Z ( l ) = b ∆ ( l ) / b ∆ (0) → Y ( x ) with x = l/ 2 j � l 0 Its small l behavior is quadratic in l ∆ ( l � ) (2 l � + 1) l � ( l � + 1) ¯ Z ( l ) = 1 � l ( l + 1) 1 D 0 l � =1 j ( j + 1) + · · · , D 0 = � 4 l 0 ∆ ( l � )(2 l � + 1) ¯ 12 l � =0 mercredi 11 avril 12
Example 1: l= 0 and 1 channels only coupling distribution D H l L = 8 1, 1 < total spin j = 8 1, 2, 4, 8, 16, 32, 64, 128 < from blue to red Z H l L 1.0 0.5 l ê 2j 0.2 0.4 0.6 0.8 1.0 - 0.5 - 1.0 13 mercredi 11 avril 12
Example 2: l= 0 to 12 channels coupling distribution D H l L = 8 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 < total spin j = 8 24, 48, 96, 192, 384, 768 < from blue to red Z H l L 1.0 0.5 l ê 2j 0.2 0.4 0.6 0.8 1.0 - 0.5 - 1.0 14 mercredi 11 avril 12
Example 3: l= 0 to 12 but even only channels coupling distribution D H l L = 8 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 < coupling distribution D H l L = 8 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 < total spin j = 8 24, 48, 96, 192, 384, 768 < from blue to red total spin j = 8 24, 48, 96, 192, 384, 768 < from blue to red Z H l L Z H l L 1.0 1.0 0.5 0.5 l ê 2j l ê 2j 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1.0 1.0 - 0.5 - 0.5 - 1.0 - 1.0 15 mercredi 11 avril 12
Example 4: l= 0 to 10 channels, random variances coupling distribution D H l L = 8 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 < coupling distribution D H l L = 8 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 < coupling distribution D H l L = 8 16., 0.99, 0.94, 0.44, 0.3, 0.94, 0.65, 0.96, 0.64, 0.82 < total spin j = 8 24, 48, 96, 192, 384, 768 < from blue to red total spin j = 8 24, 48, 96, 192, 384, 768 < from blue to red total spin j = 8 9, 18, 36, 72, 144, 288, 576 < from blue to red Z H l L Z H l L Z H l L 1.0 1.0 1.0 0.5 0.5 0.5 l ê 2j l ê 2j l ê 2j 0.2 0.4 0.6 0.8 1.0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1.0 1.0 - 0.5 - 0.5 - 0.5 - 1.0 - 1.0 - 1.0 16 mercredi 11 avril 12
The decoherence function is a generalized hypergeometric function � � √ dx dy H ( x ) H ( y ) H ( x ) = 1 2i π e − i t ( x − y ) x 2 − 4) M ( t, Z ) = , 2( x − 2i π 1 − Z H ( x ) H ( y ) m ⇥ ⇥ ∞ 2(2 m + 1)( n + 1) 2 (2 m )! t 2 m z n ( − 1) m + n ) = m !( m + 1)!( m − n )!( m + n + 2)! m =0 n =0 large time limit: fast algebraic decay with t except for Z close to unity Z t 17 mercredi 11 avril 12
Recommend
More recommend
