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2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
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Quantum thermodynamics for a model of an
expanding universe
David Edward Bruschi
Formerly: Racah Institute of Physics and Quantum Information Science Centre
the Hebrew University of Jerusalem
the Holy Land
Currently: York Centre for Quantum Technologies, Department of Physics
University of York
the United (for the moment) Kingdom
April 24, 2015
arXiv:1409.5283
In collaboration with: N. Liu, J. Goold, I. Fuentes, V. Vedral, K. Modi
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Introduction
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Relativistic and Quantum Physics
Relativity + Quantum Information = Relativistic Quantum Information
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Introduction
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(Classical) Thermodynamics and general relativity
Important achievements in general relativistic scenarios
- Three laws of black holes;
- Cosmology and big bang;
- Firewall issues.
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Introduction
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Outlook, aims and motivations
Outlook
- (Classical) Thermodynamics useful in the study of the Universe.
- Quantum processes in the universe do not necessary involve large
numbers of constituents.
- Important: Von N. Entropy of the Universe cannot change.
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Introduction
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Outlook, aims and motivations
Outlook
- (Classical) Thermodynamics useful in the study of the Universe.
- Quantum processes in the universe do not necessary involve large
numbers of constituents.
- Important: Von N. Entropy of the Universe cannot change.
Aim
Use quantum thermodynamics to understand work, entropy and energy
flows in relativistic and cosmological setups.
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Introduction
4/11
Outlook, aims and motivations
Outlook
- (Classical) Thermodynamics useful in the study of the Universe.
- Quantum processes in the universe do not necessary involve large
numbers of constituents.
- Important: Von N. Entropy of the Universe cannot change.
Aim
Use quantum thermodynamics to understand work, entropy and energy
flows in relativistic and cosmological setups.
Motivations
- The Universe is relativistic and quantum system and processes can
involve small numbers of constituents.
- Mainly: We cannot compute energy, entropy and work flows in
relativistic quantum systems.
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Models for cosmology
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Simple cosmology model
Expanding universe line/metric element
ds 2 = Ω2 (τ ) [−dτ 2 + dx 2 + . . .] ,
gµν = Ω2 (τ ) (−1, 1, . . .)
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Models for cosmology
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Simple cosmology model
Expanding universe line/metric element
ds 2 = Ω2 (τ ) [−dτ 2 + dx 2 + . . .] ,
gµν = Ω2 (τ ) (−1, 1, . . .)
Scalar quantum field
φ(t, x) = ⨋ [uk ak + uk∗ ak† ] ,
k
[ak , ak† ′ ] = δ d (k − k ′ )
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Models for cosmology
5/11
Simple cosmology model
Expanding universe line/metric element
ds 2 = Ω2 (τ ) [−dτ 2 + dx 2 + . . .] ,
gµν = Ω2 (τ ) (−1, 1, . . .)
Scalar quantum field
φ(t, x) = ⨋ [uk ak + uk∗ ak† ] ,
k
[ak , ak† ′ ] = δ d (k − k ′ )
Frequency shift and choice of conformal factor
√
√
ωin/out = k 2 + m Ω2 (τ∓∞ ),
Ω(τ ) = 1 + (1 + tanh(σ τ ))
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Models for cosmology
5/11
Simple cosmology model
Expanding universe line/metric element
ds 2 = Ω2 (τ ) [−dτ 2 + dx 2 + . . .] ,
gµν = Ω2 (τ ) (−1, 1, . . .)
Scalar quantum field
φ(t, x) = ⨋ [uk ak + uk∗ ak† ] ,
k
[ak , ak† ′ ] = δ d (k − k ′ )
Frequency shift and choice of conformal factor
√
√
ωin/out = k 2 + m Ω2 (τ∓∞ ),
Ω(τ ) = 1 + (1 + tanh(σ τ ))
Bogoliubov transformations and squeezing
†
aout,k = cosh rk ain,k + e i θk sinh rk ain,−k
,
tanh rk =
sinh (π ωout2σ−ωin )
sinh (π ωout2σ+ωin )
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Energy in this system
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Work and energy of two mode squeezing
All couple of modes (k, −k) decouple. We can focus on a single couple and
redefine ain,k ≡ ain and ain,−k ≡ bin .
Initial Hamiltonian
1
†
†
bin + ]
ain + bin
Hin = ωin [ain
2
Final Hamiltonian
1
†
†
Hout = ωout [aout
aout + bout
bout + ]
2
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Energy in this system
6/11
Work and energy of two mode squeezing
All couple of modes (k, −k) decouple. We can focus on a single couple and
redefine ain,k ≡ ain and ain,−k ≡ bin .
Initial Hamiltonian
Final Hamiltonian
1
†
†
bin + ]
ain + bin
Hin = ωin [ain
2
1
†
†
Hout = ωout [aout
aout + bout
bout + ]
2
Start with an initial thermal state ρ with ni particles. The work W done
by spacetime is
Work performed
W =Tr ((Hout − Hin ) ρ)
=ωout nc + (ωout − ωin )(ni + 1)
with nc particles that are created.
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Energy in this system
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The “inner friction”
An adiabatic process would lead to a work cost Wad of the form
Adiabatic work
Wad = (ωout − ωin )(ni + 1)
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Energy in this system
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The “inner friction”
An adiabatic process would lead to a work cost Wad of the form
Adiabatic work
Wad = (ωout − ωin )(ni + 1)
which allows us to find the inner friction Wfric for our case as
Inner friction
Wfric =W − Wad
=ωout nc
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Energy in this system
7/11
The “inner friction”
An adiabatic process would lead to a work cost Wad of the form
Adiabatic work
Wad = (ωout − ωin )(ni + 1)
which allows us to find the inner friction Wfric for our case as
Inner friction
Wfric =W − Wad
=ωout nc
Our last step
We proceed to show that Wfric can be interpreted as an entropic quantity
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
(Created) particles are entropy
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Entropy and inner friction
Forward process
pin,out = ∣⟨nout ∣nin ⟩∣2 ⟨nin ∣ρ∣nin ⟩
Backward process
pout,in = ∣⟨nin ∣nout ⟩∣2 ⟨nout ∣ρ∣nout ⟩
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
(Created) particles are entropy
8/11
Entropy and inner friction
Forward process
Backward process
pin,out = ∣⟨nout ∣nin ⟩∣2 ⟨nin ∣ρ∣nin ⟩
pout,in = ∣⟨nin ∣nout ⟩∣2 ⟨nout ∣ρ∣nout ⟩
Fluctuation relation: sin,out ∶= − log⟨nout ∣ρ∣nout ⟩ + log⟨nin ∣ρ∣nin ⟩
pin,out = pout,in e sin,out
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
(Created) particles are entropy
8/11
Entropy and inner friction
Forward process
Backward process
pin,out = ∣⟨nout ∣nin ⟩∣2 ⟨nin ∣ρ∣nin ⟩
pout,in = ∣⟨nin ∣nout ⟩∣2 ⟨nout ∣ρ∣nout ⟩
Fluctuation relation: sin,out ∶= − log⟨nout ∣ρ∣nout ⟩ + log⟨nin ∣ρ∣nin ⟩
pin,out = pout,in e sin,out
Forward process
Pin,out (s) = ∑ pin,out δ (s − sin,out )
nin ,nout
Backward process
Pout,in (s) = ∑ pout,in δ (s − sout,in )
nin ,nout
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
(Created) particles are entropy
8/11
Entropy and inner friction
Forward process
Backward process
pin,out = ∣⟨nout ∣nin ⟩∣2 ⟨nin ∣ρ∣nin ⟩
pout,in = ∣⟨nin ∣nout ⟩∣2 ⟨nout ∣ρ∣nout ⟩
Fluctuation relation: sin,out ∶= − log⟨nout ∣ρ∣nout ⟩ + log⟨nin ∣ρ∣nin ⟩
pin,out = pout,in e sin,out
Forward process
Backward process
Pin,out (s) = ∑ pin,out δ (s − sin,out )
Pout,in (s) = ∑ pout,in δ (s − sout,in )
nin ,nout
nin ,nout
Average entropy: K (X ∣∣Y ) ∶= − ∑n px (n) [log py (n) − log px (n)] ≥ 0
s = K (Pin,out ∣∣Pout,in )
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
(Created) particles are entropy
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Results
Initial state
Entropy
ρ=
e
−βin Hin
Z
sin,out =
ωin
nc
T
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
(Created) particles are entropy
9/11
Results
Initial state
Entropy
ρ=
e
−βin Hin
sin,out =
Z
ωin
nc
T
Final result
s=
ωin
nc
T
This is our main result.
Extendable result
Number of created particles nc = sinh2 r . This result can be extended to:
̵
* Unruh effect: tanh r = exp[ kBh TωU ];
̵
* Schwarschild black hole: tanh r = exp[ kBh TωH ];
* Analogue gravity models.
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
The end
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Conclusions and outlook
Conclusions
* We have studied applications of quantum thermodynamics to setups
that appear in quantum field theory;
* Have found some entropic quantity that increases in (simple)
cosmological processes;
* The results apply to different cosmological and quantum field
theoretical scenarios.
Outlook
* Can teach us more about the physics at the overlap of relativity and
quantum mechanics;
* Drive future theoretical efforts to uncover novel physics;
* Hopefully: provide energy balance relations in quantum field
theoretical scenarios.
2nd QT conference, Medina Mayurqa, Spain - Quantum thermodynamics for a model of an expanding universe
Acknowledgments
Thank You.
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