Once again: Do accelerated detectors click more often? (Anti-Unruh - PowerPoint PPT Presentation

Once again: Do accelerated detectors click more often? (Anti-Unruh Phenomena) RQI-N Kyoto, July 4th 2017 W.G. Brenna, R. B. Mann, E. Martin-Martinez, Phys. Lett. B 757, 307 (2016) L. J. Garay, E. Martin-Martinez, J. de Ramon Phys. Rev. D 94, 104048 (2016) And Work in progress with L. J. Garay and J. de Ramon Eduardo Martín-Martínez Department of Applied Mathematics Inst. for Quantum Computing (University of Waterloo) Perimeter Institute for Theoretical Physics

Catastrophic fail

Detecting temperature We need a thermometer!

Detecting temperature We need a thermometer! To measure the temperature of a field: Stick a thermometer into the field!

Detecting temperature We need a thermometer! To measure the temperature of a field: Stick a thermometer into the field! Unruh-DeWitt detector: H I = λχ ( τ ) µ ( τ ) φ [ x ( τ ) , t ( τ )] Captures the core features of the L-M interaction

Detecting temperature Thermalization is a non-perturbative process

Detecting temperature Thermalization is a non-perturbative process Can we talk about temperature of detectors perturbatively?

Detecting temperature Thermalization is a non-perturbative process Can we talk about temperature of detectors perturbatively? Detailed Balance? Excitation-Deexcitation ratio: R ( Ω , σ ) = P + ( Ω ) P − ( Ω ) ∝ e − β Ω

Detecting temperature Thermalization is a non-perturbative process Can we talk about temperature of detectors perturbatively? Detailed Balance? Excitation-Deexcitation ratio: � � β edr ( Ω , σ , β ) = − log R ( Ω , σ , β ) R ( Ω , σ ) = P + ( Ω ) P − ( Ω ) ∝ e − β Ω Ω To what extent is this a good estimator?

The KMS condition and equilibrium states Thermality: Maximization of entropy at constant energy Well defined for systems with finite (or countably infinite) degrees of freedom.

The KMS condition and equilibrium states Thermality: Maximization of entropy at constant energy Well defined for systems with finite (or countably infinite) degrees of freedom. Warning: The Gibbs distribution is not well defined, in general, for systems of continuous variables, (e.g., quantum fields in free space)

The KMS condition and equilibrium states -(Scalar) field ˆ φ ( x )     Ingredients: -Time evolution ∂ τ    -Field state ˆ ρ

The KMS condition and equilibrium states -(Scalar) field ˆ φ ( x )     Ingredients: -Time evolution ∂ τ    -Field state ˆ ρ � � -Let be the curve generated by x ( τ ) = t ( τ ) , x ( τ ) ∂ τ

The KMS condition and equilibrium states -(Scalar) field ˆ φ ( x )     Ingredients: -Time evolution ∂ τ    -Field state ˆ ρ � � -Let be the curve generated by x ( τ ) = t ( τ ) , x ( τ ) ∂ τ Pull-back of the Wightman function on the curve: D �E ˆ � ˆ � � W ( τ , τ 0 ) := x ( τ 0 ) x ( τ ) φ φ ˆ ρ

The KMS condition and equilibrium states -(Scalar) field ˆ φ ( x )     Ingredients: -Time evolution ∂ τ    -Field state ˆ ρ � � -Let be the curve generated by x ( τ ) = t ( τ ) , x ( τ ) ∂ τ Pull-back of the Wightman function on the curve: D �E ˆ � ˆ � � W ( τ , τ 0 ) := x ( τ 0 ) x ( τ ) φ φ ˆ ρ Stationarity Condition: W ( τ , τ 0 ) = W ( τ − τ 0 ) = W ( ∆ τ )

The KMS condition and equilibrium states -(Scalar) field ˆ φ ( x )     Ingredients: -Time evolution ∂ τ    -Field state ˆ ρ � � -Let be the curve generated by x ( τ ) = t ( τ ) , x ( τ ) ∂ τ Pull-back of the Wightman function on the curve: D �E ˆ � ˆ � � W ( τ , τ 0 ) := x ( τ 0 ) x ( τ ) φ φ ˆ ρ Stationarity Condition: W ( τ , τ 0 ) = W ( τ − τ 0 ) = W ( ∆ τ ) A stationary Wightman function satisfies the KMS condition With KMS parameter if and only if β W ( ∆ τ − i β ) = W ( − ∆ τ )

The KMS condition and equilibrium states -(Scalar) field ˆ φ ( x )     Ingredients: -Time evolution ∂ τ    -Field state ˆ ρ A stationary Wightman function satisfies the KMS condition With KMS parameter if and only if β W ( ∆ τ − i β ) = W ( − ∆ τ ) (1) A state of the field is KMS with respect to the time evolution ˆ ρ ∂ τ generated by (with KMS parameter ) if (1) is satisfied β

The KMS condition and equilibrium states Connection between equilibrium and correlations Gibbs states are KMS KMS states are passive KMS is a necessary condition for thermodynamic equilibrium The parameter is called the KMS (inverse) temperature β = 1 /T kms To all effects, KMS states are thermal states

KMS Wightman functions We can play a bit with the Wightman function…

KMS Wightman functions We can play a bit with the Wightman function… One can Fourier transform the Wightman function: Z ∞ ˜ d( ∆ τ ) e − i ω ∆ τ W ( ∆ τ ) W ( ω ) = −∞

KMS Wightman functions We can play a bit with the Wightman function… One can Fourier transform the Wightman function: Z ∞ ˜ d( ∆ τ ) e − i ω ∆ τ W ( ∆ τ ) W ( ω ) = −∞ W ( ∆ τ − i β ) = W ( − ∆ τ ) Fourier Trans. the KMS condition

KMS Wightman functions We can play a bit with the Wightman function… One can Fourier transform the Wightman function: Z ∞ ˜ d( ∆ τ ) e − i ω ∆ τ W ( ∆ τ ) W ( ω ) = −∞ W ( ∆ τ − i β ) = W ( − ∆ τ ) Fourier Trans. the KMS condition W ( − ω ) = e βω ˜ ˜ W ( ω ) Detailed balance condition:

KMS Wightman functions W ( − ω ) = e βω ˜ ˜ W ( ω ) Detailed balance condition: (2) The field commutator is prop. to the imaginary part of the Wightman: [ˆ φ ( x ( τ )) , ˆ � � C ( ∆ τ ) = 2i Im W ( ∆ τ ) φ ( x ( τ 0 ))] = C ( ∆ τ ) 1 1

KMS Wightman functions W ( − ω ) = e βω ˜ ˜ W ( ω ) Detailed balance condition: (2) The field commutator is prop. to the imaginary part of the Wightman: [ˆ φ ( x ( τ )) , ˆ � � C ( ∆ τ ) = 2i Im W ( ∆ τ ) φ ( x ( τ 0 ))] = C ( ∆ τ ) 1 1 In terms of Fourier transforms: C ( ω ) = ˜ ˜ W ( ω ) − ˜ W ( − ω ) (3) For a KMS state we can combine (2) and (3)

KMS Wightman functions W ( − ω ) = e βω ˜ ˜ W ( ω ) (2) C ( ω ) = ˜ ˜ W ( ω ) − ˜ W ( − ω ) (3) For a KMS state we can combine (2) and (3) W ( ω , β ) = − ˜ ˜ C ( ω , β ) P ( ω , β ) P ( ω , β ) Where is a Planckian distribution of inverse temperature equal to the KMS parameter. 1 P ( ω , β ) = e βω − 1

Probing a KMS state: Particle detectors We couple an Unruh-DeWitt detector to the field: H I = λχ ( τ / σ ) µ ( τ )ˆ φ ( x ( τ ))

Quick aside: UDW detectors are physical ˆ H em = d · ˆ x · ˆ E = e ˆ E ,

Quick aside: UDW detectors are physical · ˆ H em = d · ˆ x · ˆ x · ˆ x · ˆ x · ˆ E | g i e i Ω t | e ih g | + h g | ˆ E | e i e − i Ω t | g ih e | . E = e ˆ ˆ E = h e | ˆ E ,

Quick aside: UDW detectors are physical · ˆ H em = d · ˆ x · ˆ x · ˆ x · ˆ x · ˆ E | g i e i Ω t | e ih g | + h g | ˆ E | e i e − i Ω t | g ih e | . E = e ˆ ˆ E = h e | ˆ E , Z h x · ˆ F ( x ) · ˆ d 3 x E ( x , t ) e i Ω t | e ih g | ˆ E ( x , t ) = i + F ∗ ( x ) · ˆ E ( x , t ) e − i Ω t | g ih e | ,

Quick aside: UDW detectors are physical · ˆ H em = d · ˆ x · ˆ x · ˆ x · ˆ x · ˆ E | g i e i Ω t | e ih g | + h g | ˆ E | e i e − i Ω t | g ih e | . E = e ˆ ˆ E = h e | ˆ E , Z h x · ˆ F ( x ) · ˆ d 3 x E ( x , t ) e i Ω t | e ih g | ˆ E ( x , t ) = i + F ∗ ( x ) · ˆ E ( x , t ) e − i Ω t | g ih e | , at ψ g ( x ) = h x | g i F ( x ) = ψ ∗ e ( x ) x ψ g ( x ) . representation of σ + + F ∗ ( x ) e − i Ω t ˆ ˆ F ( x ) e i Ω t ˆ ⇥ σ − ⇤ d ( x , t ) = e

Quick aside: UDW detectors are physical · ˆ H em = d · ˆ x · ˆ x · ˆ x · ˆ x · ˆ E | g i e i Ω t | e ih g | + h g | ˆ E | e i e − i Ω t | g ih e | . E = e ˆ ˆ E = h e | ˆ E , Z h x · ˆ F ( x ) · ˆ d 3 x E ( x , t ) e i Ω t | e ih g | ˆ E ( x , t ) = i + F ∗ ( x ) · ˆ E ( x , t ) e − i Ω t | g ih e | , at ψ g ( x ) = h x | g i F ( x ) = ψ ∗ e ( x ) x ψ g ( x ) . representation of σ + + F ∗ ( x ) e − i Ω t ˆ ˆ F ( x ) e i Ω t ˆ ⇥ σ − ⇤ d ( x , t ) = e Z d 3 x ˆ d ( x � x d , t ) · ˆ H em = χ ( t ) E ( x , t ) .

Quick aside: UDW detectors are physical Z d 3 x ˆ d ( x � x d , t ) · ˆ H em = χ ( t ) E ( x , t ) . Z µ ( t ) ˆ d 3 x F ( x − x d )ˆ H udw = e χ ( t ) φ ( x , t ) . Almost same phenomenology (except for angular momentum exchange) More details in A. Pozas and E. Martín-Martínez, Phys. Rev. D 94, 064074 (2016) And also in Richard Lopp’s talk

Probing a KMS state: Particle detectors We couple an Unruh-DeWitt detector to the field: H I = λχ ( τ / σ ) µ ( τ )ˆ φ ( x ( τ ))

Probing a KMS state: Particle detectors We couple an Unruh-DeWitt detector to the field: H I = λχ ( τ / σ ) µ ( τ )ˆ φ ( x ( τ )) χ ( η ) Where is a square integrable switching function of L 2 norm 1