Earliest stochastic Schr odinger equations from foundations Lajos - PowerPoint PPT Presentation

Earliest stochastic Schr¨ odinger equations from foundations Lajos Di´ osi Research Institute for Particle and Nuclear Physics (transforms in 2012 into: Wigner Physics Center) December 7, 2011 Lajos Di´ osi (Research Institute for Particle and Nuclear Physics (transforms in 2012 into: Wigner Physics Center)) Earliest stochastic Schr¨ odinger equations from foundationsDecember 7, 2011 1 / 11

Outline Early Motivations 1970-1980’s 1 1-Shot Non-Selective Measurement, Decoherence 2 Dynamical Non-Sel. Measurement, Decoherence 3 Master Equation 4 1-Shot Selective Measurement, Collapse 5 Dynamical Non-selective Measurement, Collapse 6 Dynamical Collapse: Diffusion or Jump 7 Dynamical Collapse: Diffusion or Jump - Proof 8 Revisit Early Motivations 1970-1980’s 9 Lajos Di´ osi (Research Institute for Particle and Nuclear Physics (transforms in 2012 into: Wigner Physics Center)) Earliest stochastic Schr¨ odinger equations from foundationsDecember 7, 2011 2 / 11

Early Motivations 1970-1980’s Early Motivations 1970-1980’s Interpretation of ψ is statistical. Sudden ‘one-shot’ collapse ψ → ψ n is central. If collapse takes time? Hunt for a math model (Pearle, Gisin, Diosi) New physics? Lajos Di´ osi (Research Institute for Particle and Nuclear Physics (transforms in 2012 into: Wigner Physics Center)) Earliest stochastic Schr¨ odinger equations from foundationsDecember 7, 2011 3 / 11

1-Shot Non-Selective Measurement, Decoherence 1-Shot Non-Selective Measurement, Decoherence Measurement of ˆ A , pre-measurement state ˆ ρ , post-measurement state, decoherence: ˆ n A n ˆ n ˆ P n = ˆ P n ˆ ˆ P m = δ nm ˆ A = � P n ; � I , I ˆ ρ ˆ � ρ → ˆ P n ˆ P n n Off-diagonal elements become zero: Decoherence. Example: ˆ σ Z = |↑��↑|− |↓��↓| , ˆ P ↑ = |↑��↑| , ˆ A = ˆ P ↓ = |↓��↓| , ρ = ρ ↑↑ |↑��↑| + ρ ↓↓ |↓��↓| + ρ ↑↓ |↑��↓| + ρ ↓↑ |↓��↑| ˆ → ˆ ρ ˆ P ↑ + ˆ ρ ˆ P ↑ ˆ P ↓ ˆ P ↓ = ρ ↑↑ |↑��↑| + ρ ↓↓ |↓��↓| Replace 1-shot non-selective measurement (decoherence) by dynamics! Lajos Di´ osi (Research Institute for Particle and Nuclear Physics (transforms in 2012 into: Wigner Physics Center)) Earliest stochastic Schr¨ odinger equations from foundationsDecember 7, 2011 4 / 11

Dynamical Non-Sel. Measurement, Decoherence Dynamical Non-Sel. Measurement, Decoherence Time-continuous (dynamical) measurement of ˆ k A k ˆ A = � P k : 2 [ˆ A , [ˆ d ˆ ρ/ dt = − 1 A , ˆ ρ ]] Solution: [ˆ A , [ˆ � k ˆ � ρ ˆ � A k A l ˆ ρ ˆ A 2 A 2 A , ˆ ρ ]] = P k ˆ ρ + k ˆ P k − 2 P k ˆ P l k k k , l d (ˆ ρ ˆ 2 ˆ P n [ˆ A , [ˆ ρ ]]ˆ 2 ( A m − A n ) 2 (ˆ ρ ˆ P n ˆ P m ) / dt = − 1 A , ˆ P m = − 1 P n ˆ P m ) Off-diagonals → 0, diagonals=const Example: ˆ A = ˆ σ z , d ˆ ρ/ dt = − 1 2 [ˆ σ z , [ˆ σ z , ˆ ρ ]] ρ ( t ) = ρ ↑↑ (0) |↑��↑| + ρ ↓↓ (0) |↓��↓| ˆ + e − 2 t ρ ↑↓ (0) |↑��↓| + e − 2 t ρ ↓↑ (0) |↓��↑| → ρ ↑↑ (0) |↑��↑| + ρ ↓↓ (0) |↓��↓| Lajos Di´ osi (Research Institute for Particle and Nuclear Physics (transforms in 2012 into: Wigner Physics Center)) Earliest stochastic Schr¨ odinger equations from foundationsDecember 7, 2011 5 / 11

Master Equation Master Equations General non-unitary (but linear!) quantum dynamics: d ˆ ρ/ dt = L ˆ ρ Lindblad form — necessary and sufficient for consistency: � L † − 1 � ρ/ dt = − i [ˆ ˆ ρ ˆ 2 ˆ L † ˆ ρ ˆ L † ˆ d ˆ H , ˆ ρ ] + L ˆ L ˆ ρ − 1 2 ˆ + . . . L L † = ˆ If ˆ L = ˆ A : ρ/ dt = − i [ˆ 2 [ˆ A , [ˆ d ˆ H , ˆ ρ ] − 1 A , ˆ ρ ]] Decoherence (non-selectiv measurement) of ˆ A competes with ˆ H . General case ˆ H � = 0 , ˆ L � = ˆ L † : untitary, decohering, dissipative, pump mechanisms compete. Lajos Di´ osi (Research Institute for Particle and Nuclear Physics (transforms in 2012 into: Wigner Physics Center)) Earliest stochastic Schr¨ odinger equations from foundationsDecember 7, 2011 6 / 11

1-Shot Selective Measurement, Collapse 1-Shot Selective Measurement, Collapse Measurement of ˆ n A n ˆ n ˆ P n = ˆ P n ˆ ˆ P m = δ nm ˆ A = � P n ; � I , I General (mixed state) and the special case (pure state), resp. pure state, ˆ mixed state: P n = | n � � n | : ˆ ρ ˆ P n ˆ P n ρ → ˆ ≡ ˆ ρ n | ψ � → | n � ≡ | ψ n � p n with - prob . p n = tr(ˆ P n ˆ ρ ) with - prob . p n = | � n | ψ �| 2 Selective measurement is refinement of non-selective. Mean of conditional states = Non-selective post-measurement state: � M ˆ ρ n = p n ˆ ρ n = n n ˆ ρ ˆ n ˆ P n | ψ � � ψ | ˆ = � P n ˆ = � P n P n Replace 1-shot selective measurement (collapse) by dynamics! Lajos Di´ osi (Research Institute for Particle and Nuclear Physics (transforms in 2012 into: Wigner Physics Center)) Earliest stochastic Schr¨ odinger equations from foundationsDecember 7, 2011 7 / 11

Dynamical Non-selective Measurement, Collapse Dynamical Non-selective Measurement, Collapse Take pure state 1-shot measurement of ˆ A = � n A n | n � � n | and expand it for asymptotic long times: | ψ (0) � evolves into | ψ ( t ) � → | n � Construct a (stationary) stochastic process | ψ ( t ) � for t > 0 such that for any initial state | ψ (0) � the solution walks randomly into one of the orthogonal states | n � with probability p n = | � n | ψ (0) �| 2 ! There are ∞ many such stochastic processes | ψ ( t ) � . Luckily, for ρ ( t ) = M | ψ ( t ) � � ψ ( t ) | ˆ we have already constructed a possible non-selective dynamics, recall: 2 [ˆ A , [ˆ d ˆ ρ/ dt = − 1 A , ˆ ρ ]] This is a major constraint for the process | ψ ( t ) � . Infinite many choices still remain. Lajos Di´ osi (Research Institute for Particle and Nuclear Physics (transforms in 2012 into: Wigner Physics Center)) Earliest stochastic Schr¨ odinger equations from foundationsDecember 7, 2011 8 / 11

Dynamical Collapse: Diffusion or Jump Dynamical Collapse: Diffusion or Jump Consider the dynamical measurement of ˆ A = � n A n | n � � n | , described by dynamical decoherence (master) equation: 2 [ˆ A , [ˆ d ˆ ρ/ dt = − 1 A , ˆ ρ ]] Construct stochastic process | ψ ( t ) � of dynamical collapse satisfying the master equation by ˆ ρ ( t ) = M | ψ ( t ) � � ψ ( t ) | . Gisin’s Diffusion Process (1984): A � ) 2 | ψ � + (ˆ d | ψ � / dt = − i ˆ 2 (ˆ A − � ˆ A − � ˆ H | ψ � − 1 A � ) | ψ � w t w t : standard white-noise; M w t = 0 , M w t w s = δ ( t − s ) Diosi’s Jump Process (1985/86): A � ) 2 | ψ � + 1 d | ψ � / dt = − i ˆ 2 (ˆ A − � ˆ 2 � (ˆ A − � ˆ A � ) 2 � | ψ � H | ψ � − 1 jumps | ψ ( t ) � → const. × (ˆ A − � ˆ A � ) | ψ ( t ) � at rate � (ˆ A − � ˆ A � ) 2 � Lajos Di´ osi (Research Institute for Particle and Nuclear Physics (transforms in 2012 into: Wigner Physics Center)) Earliest stochastic Schr¨ odinger equations from foundationsDecember 7, 2011 9 / 11

Dynamical Collapse: Diffusion or Jump - Proof Dynamical Collapse: Diffusion or Jump - Proof Gisin’s Diffusion Process (1984): A � ) 2 | ψ � + (ˆ d | ψ � / dt = − i ˆ 2 (ˆ A − � ˆ A − � ˆ H | ψ � − 1 A � ) | ψ � w t w t : standard white-noise; M w t = 0 , M w t w s = δ ( t − s ) Diosi’s Jump Process (1985/86): A � ) 2 | ψ � + 1 d | ψ � / dt = − i ˆ 2 (ˆ A − � ˆ 2 � (ˆ A − � ˆ A � ) 2 � | ψ � H | ψ � − 1 jumps | ψ ( t ) � → const. × (ˆ A − � ˆ A � ) | ψ ( t ) � at rate � (ˆ A − � ˆ A � ) 2 � If [ˆ H , ˆ A ] = 0, prove: 2 [ˆ A , [ˆ ρ ( t ) = M | ψ ( t ) � � ψ ( t ) | satisfies d ˆ ˆ ρ/ dt = − 1 A , ˆ ρ ]] | ψ ( t ) � → | n � | n � occurs with p n = | � n | ψ (0) �| 2 Lajos Di´ osi (Research Institute for Particle and Nuclear Physics (transforms in 2012 into: Wigner Physics Center)) Earliest stochastic Schr¨ odinger equations from foundations December 7, 2011 10 / 11

Revisit Early Motivations 1970-1980’s Revisit Early Motivations 1970-1980’s Interpretation of ψ is statistical. Sudden ‘one-shot’ collapse ψ → ψ n is central. If collapse takes time? — Why not! Hunt for a math model (Pearle, Gisin, Diosi) — Too many models! New physics? No, it’s standard physics of real time-continuous measurement (monitoring). Yes, it’s new! to add universal non-unitary modifications to QM to replace von Neumann statistical interpretation Lajos Di´ osi (Research Institute for Particle and Nuclear Physics (transforms in 2012 into: Wigner Physics Center)) Earliest stochastic Schr¨ odinger equations from foundations December 7, 2011 11 / 11