De Deco coher herence ence Flavio Auer 25 th July 2017 World is - - PowerPoint PPT Presentation

De Deco coher herence ence

Flavio Auer

25th July 2017

World is Quantum!

↓

Why does it look classical? → Decoherence

Overview

• 1. Superposition and Interference
• 2. Entanglement
• 3. Density Matrices
• 4. Von Neumann Measurements
• 5. Environmental Monitoring and Damping of Interference
• 6. Einselection
• 7. Application to the Measurement Problem

Bibliography

1.1 Superpositions

Superposition Principle:

Linear combinations of states correspond to new quantum states. |ψ = 𝑑𝑜|ψ𝑜

𝑜

Example: spin-

1 2 particle with states |0 , |1

→ superposition: |ψ =

1 2 (|0 + |1 )

1.2 Interference

Double-slit experiment with electrons:

at slits: |ψ =

1 2 (|ψ1 + |ψ2 )

density at screen: 𝜍 𝑦 = 1

2 ψ1 + ψ2 2 = 1 2 ψ1 2 + 1 2 ψ2 2 + 1 2 (ψ∗ 1ψ2 + ψ1ψ∗ 2)

interference terms

wikimedia.org

• 2. Entanglement

Combine two systems A and B with Hilbert spaces ℋ

𝐵 and ℋ𝐶:

1) direct product state: |ψ 𝐵𝐶 = |0 𝐵 ⊗ |0 𝐶 ≡ |0 𝐵|0 𝐶 2) entangled state: |ψ 𝐵𝐶 =

1 2 (|0 𝐵|0 𝐶 + |1 𝐵|1 𝐶)

(cannot be factorised)

• 3. Density Matrices
• single state: 𝜍

= |ψ ψ|

• superposition |ψ = 𝑑𝑜|ψ𝑜

𝑜

: 𝜍 = |ψ ψ| = 𝑑𝑜𝑑𝑛| ψ𝑜

𝑜,𝑛

ψ𝑛|

• ff-diagonal terms: interference terms
• mixed states: 𝜍

= 𝑞𝑗| ψ𝑗

𝑗

ψ𝑗|

• reduced density matrix: 𝜍

𝐵 ≔ 𝑢𝑠

𝐶 𝜍

• 4. Von Neumann measurements
• system S with states |0 , |1
• detector D with states |𝑒0 , |𝑒1

initially: |ψ𝑗

𝑇 = 𝛽|0 + 𝛾|1 ; |ψ𝑗 𝐸 = |𝑒𝑠

→ composite system: |Φ𝑗 = |ψ𝑗

𝑇 |ψ𝑗 𝐸 = (𝛽|0 + 𝛾|1 )|𝑒𝑠

interaction: |Φ𝑔 = α|0 |d0 + β|1 |d1 (entangled)

5.1 Environmental Monitoring

Scattering of photons, air molecules etc. leads to coupling between system (ψ) and environment (E): initially: |ψ |𝐹0 =

1 2 (|ψ1 + |ψ2 ) |𝐹0

→ von Neumann evolution:

1 2 (|ψ1 |𝐹1 + |ψ2 |𝐹2 )

Schlosshauer (2008)

5.2 Damping of Interference

1 2 (|ψ1 |𝐹1 + |ψ2 |𝐹2 ) → reduced density matrix: 𝜍 =

1 2 (|ψ1 ψ1| + |ψ2 ψ2| + |ψ1 ψ2| 𝐹2|𝐹1 + |ψ2 ψ1| 𝐹1|𝐹2 )

If the environment has recorded sufficient information, |𝐹1 and |𝐹2 will be approximately orthogonal, i.e. 𝐹1|𝐹2 ≈ 0. → interference is suppressed: 𝜍 ≈

1 2 (|ψ1 ψ1| + |ψ2 ψ2|)

interference terms

• 6. Einselection

(Environment-Induced Superselection)

Problem of Preferred Basis: Basis of von Neumann measurement is arbitrary. What singles out the states |ψ1 and |ψ2 as those between which interference is suppressed? Solution: Environment-Induced Superselection The preferred states of the system emerge dynamically as those states that are most robust to interaction with the environment and thus immune to decoherence.

• 7. Application to the Measurement Problem

Aspects of the Measurement Problem (Schlosshauer 2008):

• Problem of Preferred Basis

→ einselection

• Problem of the Nonobservability of Interference

→ environmental monitoring

• Problem of outcomes

→ decoherence can explain why there are definit outcomes, ..........but not which ones

Bibliography

• H. Briegel: A perspective on possible manifestations of entanglement in biological
• systems. in: Masoud Mohseni et al. (ed.): Quantum Effects in Biology. Cambridge

2014, 277-310

• M. Schlosshauer: Decoherence and the Quantum-to-Classical Transition.

Heidelberg³ 2008

• D. Wallace: Decoherence and its role in the modern measurement problem.

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 370, 4576–4593, 2011

• W.H. Zurek: Pointer basis of quantum apparatus: Into what mixture does the wave

packet collapse? Physical Review D, 24 (6), 1516-1525, 1981

• W.H. Zurek: Environment-induced superselection rules. Physical Review D, 26 (8),

1862-1880, 1982

• W. H. Zurek: Decoherence and the transition from quantum to classical – revisited.

Los Alamos Science 27, 86-109 (2002)