Waiting for Unruh Jorma Louko School of Mathematical Sciences, University of Nottingham Quantum Field Theory, University of York, 4–7 April 2017 Christopher J Fewster, Benito A Ju´ arez-Aubry, JL CQG 33 (2016) 165003 [arXiv:1605.01316] t x
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Plan 1. Unruh effect ◮ Long time limit: adiabatic scaling versus plateau scaling 2. Detector ◮ Unruh-DeWitt 3. Results ◮ Thermalisation time at large E gap 4. Summary
1. Unruh effect Well established ◮ Uniformly linearly accelerated observer sees Minkowki vacuum a as thermal, T = Unruh 1976 2 π ◮ Weak coupling, long time, negligible switching effects ◮ Thermal: Detector records detailed balance: P ↓ = e E gap / T P ↑
1. Unruh effect Well established ◮ Uniformly linearly accelerated observer sees Minkowki vacuum a as thermal, T = Unruh 1976 2 π ◮ Weak coupling, long time, negligible switching effects ◮ Thermal: Detector records detailed balance: P ↓ = e E gap / T P ↑ Beyond: non-stationary ◮ Non-uniform acceleration ◮ Curved spacetime: Hawking effect E.g. detector falling into a black hole “Time-dependent temperature” ?
Our aim How long does a detector need to operate to record (approximate) detailed balance, P ↓ = e E gap / T ? P ↑
Our aim How long does a detector need to operate to record (approximate) detailed balance, P ↓ = e E gap / T ? P ↑ Novel setting ◮ How long in terms of E gap , at large E gap − → experiment? ◮ Switching: smooth and compact support ◮ Mathematically precise (nothing hidden in i ǫ )
Our aim How long does a detector need to operate to record (approximate) detailed balance, P ↓ = e E gap / T ? P ↑ Novel setting ◮ How long in terms of E gap , at large E gap − → experiment? ◮ Switching: smooth and compact support ◮ Mathematically precise (nothing hidden in i ǫ ) Limitations ◮ Weak coupling − → first-order perturbation theory ◮ (3 + 1) Minkowski, massless scalar field (for core results)
How long? Adiabatic switching χ (τ) = χ (τ/λ) λ 1 τ λτ 0 Plateau switching χ (τ) λ τ τ s τ s λτ p Long time: λ → ∞
2. Detector (Unruh-DeWitt) Quantum field Two-state detector (atom) (3 + 1) spacetime dimension � 0 � � state with energy 0 real scalar field, m = 0 � 1 � � state with energy E φ | 0 � Minkowski vacuum x( τ ) detector worldline, τ proper time
2. Detector (Unruh-DeWitt) Quantum field Two-state detector (atom) (3 + 1) spacetime dimension � 0 � � state with energy 0 real scalar field, m = 0 � 1 � � state with energy E φ | 0 � Minkowski vacuum x( τ ) detector worldline, τ proper time Interaction � � H int ( τ ) = c χ ( τ ) µ ( τ ) φ x( τ ) c coupling constant switching function, C ∞ 0 , real-valued χ detector’s monopole moment operator µ
Probability of transition � 0 � � ⊗ | 0 � − → � 1 � � ⊗ | anything � in first-order perturbation theory: P ( E ) = c 2 � � � 2 � � � 0 � µ (0) � 1 � � × F ( E ) � �� � � �� � detector internals only: trajectory and | 0 � : drop! response function � ∞ � ∞ d τ ′′ e − iE ( τ ′ − τ ′′ ) χ ( τ ′ ) χ ( τ ′′ ) W ( τ ′ , τ ′′ ) d τ ′ F ( E ) = −∞ −∞ � � � � W ( τ ′ , τ ′′ ) = � 0 | φ x( τ ′ ) x( τ ′′ ) φ | 0 � Wightman function (distribution)
Stationary W ( τ ′ , τ ′′ ) = W ( τ ′ − τ ′′ ) � ∞ F ( E ) = 1 χ ( ω ) | 2 � d ω | � W ( E + ω ) 2 π −∞ Unruh ω � � � W ( ω ) = a > 0: proper acceleration e 2 πω/ a − 1 2 π � W ( − ω ) T = a = e 2 πω/ a ⇒ Unruh temperature � 2 π W ( ω )
3. Results Theorem 0. With either switching, for any fixed E , F λ ( E ) (const) × � − − − → W ( E ) λ λ →∞ ⇒ Detailed balance at λ → ∞ (as expected)
3. Results Theorem 0. With either switching, for any fixed E , F λ ( E ) (const) × � − − − → W ( E ) λ λ →∞ ⇒ Detailed balance at λ → ∞ (as expected) Theorem 1. For fixed λ , F λ ( E ) is not exponentially suppressed as E → ∞ . ⇒ Detailed balance at λ → ∞ cannot be uniform in E .
3. Results Theorem 0. With either switching, for any fixed E , F λ ( E ) (const) × � − − − → W ( E ) λ λ →∞ ⇒ Detailed balance at λ → ∞ (as expected) Theorem 1. For fixed λ , F λ ( E ) is not exponentially suppressed as E → ∞ . ⇒ Detailed balance at λ → ∞ cannot be uniform in E . Theorem 2. For either switching, F λ ( − E ) e 2 π E / a − − − − → F λ ( E ) E →∞ with exponentially growing λ ( E ) ⇒ Detailed balance at large E gap in exponentially long waiting time
Theorem 3. For adiabatic switching, F λ ( − E ) e 2 π E / a − − − − → ( ∗ ) F λ ( E ) E →∞ � � �� � has sufficiently with polynomially growing λ ( E ), provided χ ( ω ) strong falloff (Cf. Fewster and Ford 2015) ⇒ Detailed balance at large E gap in polynomially long waiting time
Theorem 3. For adiabatic switching, F λ ( − E ) e 2 π E / a − − − − → ( ∗ ) F λ ( E ) E →∞ � � �� � has sufficiently with polynomially growing λ ( E ), provided χ ( ω ) strong falloff (Cf. Fewster and Ford 2015) ⇒ Detailed balance at large E gap in polynomially long waiting time Theorem 4. For plateau switching, no polynomially growing λ ( E ) gives ( ∗ ) ⇒ Detailed balance at large E gap requires longer than polynomial waiting time.
4. Summary Detailed balance in the Unruh effect at E gap → ∞ : ◮ (3 + 1) massless scalar ◮ Polynomial waiting time suffices for adiabatically scaled switching with sufficiently strong Fourier decay ◮ No polynomial waiting time suffices for plateau scaled switching Upshots: ◮ Large E gap regime has limited relevance for defining a “time dependent temperature” ◮ Interest for (analogue) experiments?
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