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Super-Adiabatic Particle Number:
Vacuum Particle Production in Real Time
Gerald Dunne
University of Connecticut
& Helmholtz Institut Jena,
Theoretisch-Physikalisches Institut
Friedrich-Schiller-Universität Jena
KITP Program Frontiers of Intense Laser Physics: August 5, 2014
R. Dabrowski & GD: 1405.0302, PhysRevD90(2014)
Schwinger Effect: Pair Production from Vacuum
8
Gerald V. Dunne
⇥
E
e
e+
~
Schwinger critical field: 2eE mc
∼ 2mc2 ⇒
2 3
m c
Figure
electric, field
can
apart2 a virtual
1016 V/cm
Ic ≈
4 ×tear
1029 W/cm
Ec ≈ 2. A≈ static
e~
an asymptotic electron and positron, as shown on the lef
field does not
break this
virtual dipole apart, as shown o
experimentally
inaccessible
scales
out of the page.
but, estimate based on constant-field approximation ...
Importance of Pulse Shape: Quantum Interference
I
realistic short laser pulses have temporal structure
[envelope, carrier-phase, chirp, focussing, ...]
I
particle production is extremely sensitive to these, due to
quantum interference
I
we can learn to take advantage of this, to enhance the
particle production effect
I
critical intensity can be lowered several orders of magnitude
Importance of Pulse Shape: Quantum Interference
constant E field
monochromatic
or single pulse
pulse with
sub-cycle
structure;
carrier phase
effect
chirped pulse,
Gaussian beam, ...
Particle Number in Quantum Field Theory
I
the Schwinger effect: non-perturbative production of
electron-positron pairs when an external electric field is
applied to the quantum electrodynamical (QED) vacuum
I
Parker-Zeldovich effect: cosmological particle production
due to expanding cosmologies
I
Hawking radiation: particle production due to black holes
and gravitational horizon effects
I
Unruh radiation: particle number as seen by an
accelerating observer
Particle Number in Quantum Field Theory
QED effective action:
(Heisenberg/Euler, Feynman, Schwinger, Nambu, ...)
h0out | 0in i ≡ exp
i
Γ[A]
~
vacuum persistence probability:
2
2
2
|h0out | 0in i| = exp − Im Γ[A] ≈ 1 − Im Γ[A]
~
~
Particle Number in Quantum Field Theory
QED effective action:
(Heisenberg/Euler, Feynman, Schwinger, Nambu, ...)
h0out | 0in i ≡ exp
i
Γ[A]
~
vacuum persistence probability:
2
2
2
|h0out | 0in i| = exp − Im Γ[A] ≈ 1 − Im Γ[A]
~
~
Fredholm determinant:
(Feynman, Schwinger, Matthews/Salam, ...)
Γ[A] = ln det (i D
/ + m+i)
,
simple computation for constant E field
Dµ ≡ ∂µ −
ie
Aµ
~c
Particle Number in Quantum Field Theory
QED effective action:
(Heisenberg/Euler, Feynman, Schwinger, Nambu, ...)
h0out | 0in i ≡ exp
i
Γ[A]
~
vacuum persistence probability:
2
2
2
|h0out | 0in i| = exp − Im Γ[A] ≈ 1 − Im Γ[A]
~
~
Fredholm determinant:
(Feynman, Schwinger, Matthews/Salam, ...)
Γ[A] = ln det (i D
/ + m+i)
,
Dµ ≡ ∂µ −
simple computation for constant E field
for realistic laser pulses we want more information:
I
momentum distributions, spectra, ...
I
optimization and quantum control
I
backreaction
ie
Aµ
~c
Particle Number in Quantum Field Theory
|0iin
t = −∞
|0iout
intermediate time
t = +∞
Particle Number in Quantum Field Theory
local information?
E 2 m2 π
Im Γ = V4 3 e− E
4π
? →?
Z
d4 x
m2 π
E 2 (x) − E(x)
e
4π 3
???
locally constant field approximation misses a lot of important
physics: quantum interference
how to formulate this in general ?
perturbative effective field theory:
Z
Γpert [A] = d4 x Lpert F, ∂F, ∂ 2 F, ....
non-perturbative effective field theory:
Im Lnon−pert (x) = ?
Puzzles with Particle Production
in addition to experimental challenges, vacuum particle
production presents fundamental conceptual and computational
problems
I
can it be seen ?
I
optimization and quantum control ?
I
particle concept in time-dependent background fields ?
I
backreaction ?
I
cascades ?
I
maximum electric field ?
I
correlations and quantum noise ?
I
entanglement entropy and mutual information ?
Mode Decomposition of Fields
spatial homogeneity
∗ (t)
⇒ Φk (t) = ak fk (t) + b†−k f−k
mode Klein-Gordon equation:
2
d
2
+ ωk (t) fk (t) = 0
dt2
Schwinger effect:
2
2
ωk2 (t) = m2 + k⊥
+ kk − Ak (t)
Parker-Zeldovich effect: metric ds2 = −dt2 + a2 (t) dΣ2
2k + d − 1
2k + d − 3
2
2
2
2
sech (Ht)
ωk (t) = H γ +
2
2
Adiabatic Particle Number
In a time-dependent background field there is no unique
separation into positive and negative energy states with which
to identify particles and anti-particles [Dirac, DeWitt,
Parker].
For a slowly varying time-dependent background we can define
an adiabatic particle number with respect to a reference basis
that reduces to ordinary positive and negative energy plane
waves at intial and final times when the background is
constant
Particle Number in Quantum Field Theory
|0iin
t = −∞
|0iout
intermediate time
t = +∞
Bogoliubov Transformation
Bogoliubov transformation to time-dependent
creation/annihilation operators:
ãk (t)
b̃†−k (t)
=
ak
αk (t) βk∗ (t)
βk (t) αk∗ (t)
b†−k
unitarity: |αk (t)|2 − |βk (t)|2 = 1
∗
Φk (t) = ak fk (t) + b†−k f−k
(t)
∗
= ãk (t)f˜k (t) + b̃†−k (t)f˜−k
(t)
f˜k (t): reference mode functions
∗
fk (t) = αk (t) f˜k (t) + βk (t) f˜−k
(t)
Bogoliubov Transformation & Adiabatic Particle
Number
time-dependent adiabatic particle number:
Ñk (t) ≡ h0| ã†k (t) ãk (t) |0i = |βk (t)|2
vacuum: ak |0i = 0 = b−k |0i
total number of particles produced in mode k:
Ñk ≡ Ñk (t = +∞) = |βk (t = +∞)|2
in this talk, consider the full time evolution of the adiabatic
particle number Ñk (t), as it evolves from an initial value of zero
to some final asymptotic value Ñk ≡ Ñk (t = +∞).
Bogoliubov Transformation & Adiabatic Particle
Number
immediate problem:
time-dependent adiabatic particle number, Ñk (t), depends on
the reference mode functions and so is not unique, at
intermediate times
even though the final value at t = +∞ is unique
Basis Dependence of Adiabatic Particle Number
0.0006
0.0005
0.0004
0.0003
0.0002
0.0001
0.0000
-15
-10
-5
0
5
10
15
20
t
Schwinger effect: E(t) = E sech2 (at). Pulse parameters:
E = 0.25, a = 0.1, k⊥ = 0, kk = 0
blue and red defined w.r.t. different reference mode functions
Basis Dependence of Adiabatic Particle Number
close-up at late times
0.00002
0.000015
0.00001
5. ´ 10-6
0
15
20
25
30
35
40
45
t
blue & red defined w.r.t. different reference mode functions
same final value, but very different at intermediate times
Basis Dependence of Adiabatic Particle Number
∗ (t)
reference functions: fk (t) = αk (t) f˜k (t) + βk (t) f˜−k
Rt
1
f˜k (t) ≡ p
e−i Wk
2 Wk (t)
f˜k (t) solves time-dependent oscillator equation if

!2 
Ẅk
3 Ẇk 
Wk2 = ωk2 − 
−
2Wk
4 Wk
leading order adiabatic expn : Wk (t) = ωk (t), cf. WKB
also need momentum field operators Πk (t) = Φ̇†k (t)
1
1
∗
˙
˜
fk (t) = −i Wk (t) + Vk (t) αk (t) fk (t)+ i Wk (t) + Vk (t) βk (t) f˜−k
(t)
2
2
Basis Dependence of Adiabatic Particle Number
∗ (t)
reference functions: fk (t) = αk (t) f˜k (t) + βk (t) f˜−k
Rt
1
f˜k (t) ≡ p
e−i Wk
2 Wk (t)
f˜k (t) solves time-dependent oscillator equation if

!2 
Ẅk
3 Ẇk 
Wk2 = ωk2 − 
−
2Wk
4 Wk
leading order adiabatic expn : Wk (t) = ωk (t), cf. WKB
also need momentum field operators Πk (t) = Φ̇†k (t)
1
1
∗
˙
˜
fk (t) = −i Wk (t) + Vk (t) αk (t) fk (t)+ i Wk (t) + Vk (t) βk (t) f˜−k
(t)
2
2
freedom in Wk (t) & Vk (t) encodes arbitrariness in defining
positive and negative energy states at intermediate times
Basis Dependence of Adiabatic Particle Number
Conventional choices:
1. Wk (t) = ωk (t) and Vk (t) = 0 → time evolution of
Bogoliubov coefficients:
!
Rt
ω̇k (t)
α̇k (t)
0R
e2i ωk
αk (t)
=
t
βk (t)
β̇k (t)
2ωk (t) e−2i ωk
0
Ẇk (t)
2. Wk (t) = ωk (t) and Vk (t) = − W
→ time evolution of
k (t)
Bogoliubov coefficients:
!
Rt
1 3 ω̇k2 (t) ω̈k (t)
α̇k (t)
αk (t)
1R
e2i ωk
=
−
t
βk (t)
β̇k (t)
4i 2 ωk3 (t) ωk2 (t)
−e−2i ωk
−1
Computation for Time Dependent Electric Fields
I
in a chosen basis, evolve Bogoliubov coefficients αk (t) and
βk (t) from t = −∞ to t = +∞, with βk (−∞) = 0
αk (−∞)
1
=
βk (−∞)
0
−→
αk (+∞)
βk (+∞)
I
total asymptotic particle number: Ñk = |βk (t = +∞)|2
I
do for each momentum mode k, momentum of the
produced particles along the electric field direction
I
2
|βk (t=+∞)|
useful analogy: |α
2 is the reflection probability for
k (t=+∞)|
over-the-barrier
scattering
for the "Schrödinger equation"
2
d
2
2 ) f (t)
− dt2 − (kk − Ak (t)) fk (t) = (m2 + k⊥
k
Importance of Quantum Interference
I
realistic short laser pulses have temporal structure
[envelope, carrier-phase, chirp, ...]
I
particle production is extremely sensitive to these, due to
quantum interference
I
we can learn to take advantage of this, to enhance the
particle production effect
Particle Number in Quantum Field Theory
|0iin
t = −∞
|0iout
intermediate time
t = +∞
Dynamical Schwinger Mechanism
[Schützhold, GD, Gies, PRL 2008; Di Piazza, Lötstedt, Milstein, Keitel, PRL 2009]
superimpose strong slow field (optical laser) with fast weak
pulse (X-ray laser): leads to exponential enhancement
lowers Schwinger critical field by several orders of magnitude
subcycle structure of the laser.
er. We
To complete the physical picture, we consider the overmakes
all envelope of the longitudinal momentum distribution,
mentum
again for " ¼ 0, averaging over the rapid oscillations.
own in
When there are more than three cycles per pulse (! *
unction
3), the peak of the momentum distribution is located near
hat, for carrier
phase effect
[Hebenstreit, Alkofer, GD, Gies, PRL 2009]
pk ð1Þ ¼ 0, whereas for ! & 3 the peak is shifted to a
llatory
nonzero
value.
Furthermore,
the
Gaussian
width
set in of the
2
employed
, howtthe
pffiffiffiffiffiffiffiffi WKB approximation Eq. (4), which scales
~
=
$,
is
obviously
somewhat
broader
than
with
eE
0
E(t) = E0 exp − 2 cos (ω t + φ)
field is
true distribution, as is shown in Fig. 5. We can quantify
τ
distrithis discrepancy in the width, by extending the WKB result
e oscilbeyond the Gaussian approximation inherent in Eq. (4). We
direct
PHYSICAL REVIEW LETTERS
mmetric PRL 102, 150404 (2009)
14
14
o is the
5 10
5 10
14
perfect 5 10
menta,
roduc14
14
3 10
3 10
ts from 3 10 14
r-phase
esting a
14
14
rch for 1 1101014
1 10
Quantum Interference: Carrier Phase
=0
k MeV
⇥=
week ending
17 APRIL 2009
2
k MeV
0.6
0.4
0.2
0
0.2
0.4
0.6 k MeV
od in a
0.4
0.2
0
0.2
0.4
0.6
0.4
0.2
0
0.2
0.4
0.6
as first
FIG. 5. Comparison of the asymptotic distribution
~ 1Þ for k~? ¼ 0
~ 1Þ for k~?function
FIG. 4. Asymptotic distribution function fðk;
distribution function fðk;
¼0
ioniza- FIG.
~3.1Þ Asymptotic
fðk;
for k~ ¼ 0 (oscillating solid line) with the prediction
for ! ¼ 5, E0 ¼ 0:1Ecr , and " ¼ !#=2. The center of the
¼ 0:1Ecr , and " ¼ !#=4. The center of the
! ¼ 5, E0 ?
re, the for
time-domain
analogue
of multiple-slit interference, from
of Eq. (4)is(dotted
improved
ð1Þthe
% 102
keV. WKB approximation distribution is shifted to pk ð1Þ % 137 keV.
distribution
shiftedline)
to pkand
m that
based on an expansion of
Eq. (5) (dashed line) for ! ¼ 5, E0 ¼
0:1Ecr , and " ¼ 0.
ed pair vacuum
creation events. The fringes are large for " ¼ !#=2, since
sensitive dependence on the other shape parameters, such
the field strength has two peaks of equal size (though
assensitive
!.
measure of sub-cycle pulse then
structure
150404-3
opposite sign) which act as two temporally separated slits.
In fact, there is an even more distinctive dependence on
Moving the carrier phase away from " ¼ !#=2 corre-
Quantum Interference: Ramsey Effect
alternating-sign pulse sequence
[Akkermans, GD, PRL 2012]
Nk2-pulse
Nk10-pulse
3.10-13
7.10-12
5.10-12
2.10-13
3.10-12
10-13
10-12
-1.0
-0.5
0.0
0.5
1.0
k
-1.0
-0.5
0.0
N -pulse sequence ⇒ N 2 enhancement in certain modes
time-domain analogue of N-slit interference
measured in tunnel junctions (Gabelli/Reulet, 2012)
0.5
1.0
k
Quantum Interference: Complex WKB
d2
2
2
− 2 − (kk − Ak (t)) fk (t) = (m2 + k⊥
) fk (t)
dt
I
over-the-barrier scattering: turning points only in the
complex plane
I
pulses with sub-structure have multiple sets of turning
points
I
WKB integral between complex conjugate turning points
→ magnitude of creation event
I
WKB integral between neighboring turning points
→ interference phases
R Re(t )+i Im(tp )
R Re(t )
X
−2i Re(t p)
ωk (t) dt
−2i −∞ p ωk (t) dt
p
βk (∞) ≈
e
e
tp
Quantum Interference: Stokes Phenomenon
particle production requires global information to connect
t = −∞ to t = +∞
Stokes lines and anti-Stokes lines separate different regions of
asymptotic behavior
N
ImHtL
2. ´ 10-8
t1
t3
1.5 ´ 10-8
ReHtL
1. ´ 10-8
5. ´ 10-9
t2
t4
1.0
1.5
2.0
2.5
pm
particle production = Stokes phenomenon, the
appearance/disappearance of exponentially small/large terms as
Stokes lines are crossed
[Dumlu, GD, PRL 2010]
Quantum Interference: Complex WKB
2 /τ 2
E(t) = E0 e−t
cos(ω t)
2 /τ 2
E(t) = E0 e−t
4
4
2
2
0
0
-2
-2
-4
sin(ω t)
-4
-4
-2
0
2
4
-4
-2
0
2
4
Adiabatic Particle Number and Quantum
Interference
I
what effect does quantum interference have on adiabatic
particle number ?
I
what is happening near the turning points ?
tc
Stokes line
sc
Re(tc )
t⇤c
e−i
The Adiabatic Expansion: recall f˜ = √
Rt
Wk
2Wk (t)
expand Wk2 = ωk2 −
Ẅk
2Wk
−
3
4
Ẇk
Wk
2 in derivatives of ωk (t):
(0)
1. Leading order: Wk (t) = ωk (t)
(1)
2. Next-to-leading order: Wk (t) = ωk (t) −
3. Next-to-next-to-leading order:
1 ω̈k
3 ω̇k2
(2)
Wk (t) = ωk (t) −
−
4 ωk2 2 ωk3
1
−
8
1
4
ω̈k
ωk2
−
2
3 ω̇k
2 ωk3
!
(4)
...
13 ω̈k2 99 ω̇k2 ω̈k
ω̇k ωk
1 ωk
297 ω̇k4
−
+5 5 +
+
4 ωk5
4 ωk6
2 ωk4
16 ωk7
ωk
e−i
The Adiabatic Expansion: recall f˜ = √
Rt
Wk
2Wk (t)
expand Wk2 = ωk2 −
Ẅk
2Wk
−
3
4
Ẇk
Wk
2 in derivatives of ωk (t):
(0)
1. Leading order: Wk (t) = ωk (t)
(1)
2. Next-to-leading order: Wk (t) = ωk (t) −
3. Next-to-next-to-leading order:
1 ω̈k
3 ω̇k2
(2)
Wk (t) = ωk (t) −
−
4 ωk2 2 ωk3
1
−
8
1
4
ω̈k
ωk2
−
2
3 ω̇k
2 ωk3
!
(4)
...
13 ω̈k2 99 ω̇k2 ω̈k
ω̇k ωk
1 ωk
297 ω̇k4
−
+5 5 +
+
4 ωk5
4 ωk6
2 ωk4
16 ωk7
ωk
a complicated mess !!!
...
looks hopeless
Super-Adiabatic Basis
a closer look at the adiabatic expansion:
I
adiabatic expansion is divergent (asymptotic)
I
expressions for Wk (t) in terms of ωk (t) rapidly become
more and more complicated
I
situation look uninteresting and hopeless at high orders
I
Dingle found remarkable universal large-order behavior
(j)
Super-Adiabatic Basis
Define the “singulant” variable
Z t
Fk (t) = 2i
ωk (t0 ) dt0
tc
expand
Wk (t) = ωk (t)
∞
X
(2l)
ϕk (t)
l=0
simple and universal large-order behavior:
(2l+2)
ϕk
(t) ∼ −
2 (2l + 1)!
π [Fk (t)]2l+2
,
l1
Super-Adiabatic Basis
0.001
0.000
0.000
j=2
-0.001
-0.010
-0.002
-0.015
-10
-5
0
5
10
-10
-5
0
t
5
10
t
0.0010
0.0005
0.0005
0.0000
j=4
j=3
0.0000
-0.0005
-0.0015
-0.0015
-10
-0.0005
-0.0010
-0.0010
-5
0
5
10
-10
-5
0
t
0.000
5
10
0.000
j=6
j=5
10
0.005
0.001
-0.001
-0.005
-0.002
-0.003
-10
5
t
0.002
-0.010
-5
0
5
10
-10
-5
0
t
t
0.2
0.02
0.1
0.00
j=8
j=7
(0)
Wk : 1 term
(1)
Wk : 3 terms
(2)
Wk : 8 terms
(3)
Wk : 19 terms
(4)
Wk : 41 terms
(5)
Wk : 83 terms
(6)
Wk : 160 terms
(7)
Wk : 295 terms
(8)
Wk : 526 terms
(9)
Wk : 911 terms
(10)
Wk : 1538 terms
(11)
Wk : 2540 terms
j=1
-0.005
-0.2
-0.04
-10
0.0
-0.1
-0.02
-5
0
t
5
10
-10
-5
0
t
5
10
Super-Adiabatic Basis
Berry: universal time dependence of Bogoliubov coefficient near
turning point:
βk (t) ≈
(0)
Fk
(0)
i
Erfc (−σk (t)) e−Fk
2
= Im
σk (t) ≈
Z
tc
Stokes line
t∗c
ωk (t) dt
tc
2 ωk (sc ) (t − sc )
p
2 Fk (sc )
sc
Re(tc )
t⇤c
Super-Adiabatic Particle Number
super-adiabatic particle number:
(0) 2
1 − Fk Ñk (t) ≈ Erfc (−σk (t)) e
4
recall: optimal order of truncation of an asymptotic expansion
depends on the value of the expansion parameter
optimal truncation of adiabatic expansion at order j:
1 (0) j ≈ Int
Fk − 1
2
Super-Adiabatic Particle Number: examples
0.00008
0.00006
0.00004
0.00002
0
-10
-5
0
5
10
5
10
t
5. ´ 10 - 6
4. ´ 10 - 6
3. ´ 10 - 6
2. ´ 10 - 6
1. ´ 10 - 6
0
-10
-5
0
t
Super-Adiabatic Particle Number: examples
5. ´ 10 - 6
4. ´ 10 - 6
3. ´ 10 - 6
2. ´ 10 - 6
1. ´ 10 - 6
0
-10
-5
0
5
10
5
10
t
5. ´ 10 - 6
4. ´ 10 - 6
3. ´ 10 - 6
2. ´ 10 - 6
1. ´ 10 - 6
0
-10
-5
0
t
Super-Adiabatic Particle Number: examples
5. ´ 10 - 6
4. ´ 10 - 6
3. ´ 10 - 6
2. ´ 10 - 6
1. ´ 10 - 6
0
-10
-5
0
5
10
5
10
t
0.000012
0.00001
8. ´ 10 - 6
6. ´ 10 - 6
4. ´ 10 - 6
2. ´ 10 - 6
0
-10
-5
0
t
Super-Adiabatic Particle Number: examples
5. ´ 10 - 6
0.00008
4. ´ 10 - 6
0.00006
3. ´ 10 - 6
0.00004
2. ´ 10 - 6
0.00002
1. ´ 10 - 6
0
-10
-5
0
5
10
0
-10
-5
5. ´ 10 - 6
5. ´ 10 - 6
4. ´ 10 - 6
4. ´ 10 - 6
3. ´ 10 - 6
3. ´ 10 - 6
2. ´ 10 - 6
2. ´ 10 - 6
-6
1. ´ 10 - 6
1. ´ 10
0
-10
-5
0
5
10
0
-10
-5
t
5
10
0
5
10
5
10
t
5. ´ 10 - 6
0.000012
4. ´ 10 - 6
0.00001
3. ´ 10 - 6
8. ´ 10 - 6
6. ´ 10 - 6
2. ´ 10 - 6
4. ´ 10 - 6
1. ´ 10 - 6
0
-10
0
t
t
2. ´ 10 - 6
-5
0
5
10
0
-10
-5
0
Super-Adiabatic Particle Number
realistic laser pulses have temporal sub-structure, which means
multiple pairs of complex turning points
Super-Adiabatic Particle Number
super-adiabatic particle number with multiple turning point
pairs:
interference
tc
Stokes line
sc
tc
Stokes line
Re(tc )
t⇤c
sc
Re(tc )
t⇤c
2
X
1
(p)
(0)
(p)
Ñk (t) ≈ exp 2iθk exp −Fk,tp Erfc −σk (t) 4 t
p
Super-Adiabatic Particle Number: examples
two alternating-sign pulses ⇒ constructive or destructive
interference for different k modes
Nk2-pulse
3.10-13
constructive
interference
2.10-13
destructive
interference
10-13
-1.0
-0.5
0.0
0.5
1.0
k
Super-Adiabatic Particle Number: examples
two alternating-sign pulses, with constructive interference:
0.00014
0.00012
0.00010
0.00008
0.00006
0.00004
0.00002
0.00000
- 60
- 40
- 20
0
20
40
60
20
40
60
t
0.000025
0.00002
0.000015
0.00001
5. ´ 10 - 6
0
- 60
- 40
- 20
0
t
Super-Adiabatic Particle Number: examples
two alternating-sign pulses, with constructive interference:
0.000025
0.00002
0.000015
0.00001
5. ´ 10 - 6
0
- 60
- 40
- 20
0
20
40
60
20
40
60
t
0.000025
0.00002
0.000015
0.00001
5. ´ 10 - 6
0
- 60
- 40
- 20
0
t
Super-Adiabatic Particle Number: examples
two alternating-sign pulses, with constructive interference:
0.000025
0.00002
0.000015
0.00001
5. ´ 10-6
0
-60
-40
-20
0
20
40
60
20
40
60
t
0.000035
0.00003
0.000025
0.00002
0.000015
0.00001
5. ´ 10-6
0
-60
-40
-20
0
t
Super-Adiabatic Particle Number: examples
two alternating-sign pulses, with constructive interference:
0.00014
0.000025
0.00012
0.00010
0.00002
0.00008
0.000015
0.00006
0.00001
0.00004
5. ´ 10 - 6
0.00002
0.00000
- 60
- 40
- 20
0
20
40
0
60
- 60
- 40
- 20
t
20
40
60
20
40
60
20
40
60
t
0.000025
0.000025
0.00002
0.00002
0.000015
0.000015
0.00001
0.00001
5. ´ 10 - 6
5. ´ 10 - 6
0
0
- 60
- 40
- 20
0
20
40
0
60
- 60
- 40
- 20
t
0
t
0.000035
0.000025
0.00003
0.00002
0.000025
0.000015
0.00002
0.00001
0.000015
0.00001
5. ´ 10-6
5. ´ 10-6
0
0
-60
-40
-20
0
20
40
60
-60
-40
-20
0
Super-Adiabatic Particle Number: examples
two alternating-sign pulses ⇒ constructive or destructive
interference for different k modes
Nk2-pulse
3.10-13
constructive
interference
2.10-13
destructive
interference
10-13
-1.0
-0.5
0.0
0.5
1.0
k
Super-Adiabatic Particle Number: examples
two alternating-sign pulses, with destructive interference:
0.0001
0.00008
0.00006
0.00004
0.00002
0.0000
- 60
- 40
- 20
0
20
40
60
20
40
60
t
8. ´ 10 - 6
6. ´ 10 - 6
4. ´ 10 - 6
2. ´ 10 - 6
0
- 60
- 40
- 20
0
t
Super-Adiabatic Particle Number: examples
two alternating-sign pulses, with destructive interference:
8. ´ 10 - 6
6. ´ 10 - 6
4. ´ 10 - 6
2. ´ 10 - 6
0
- 60
- 40
- 20
0
20
40
60
20
40
60
t
8. ´ 10 - 6
6. ´ 10 - 6
4. ´ 10 - 6
2. ´ 10 - 6
0
- 60
- 40
- 20
0
t
Super-Adiabatic Particle Number: examples
two alternating-sign pulses, with destructive interference:
8. ´ 10-6
6. ´ 10-6
4. ´ 10-6
2. ´ 10-6
0
-60
-40
-20
0
20
40
60
20
40
60
t
0.00003
0.000025
0.00002
0.000015
0.00001
5. ´ 10-6
0
-60
-40
-20
0
t
Super-Adiabatic Particle Number: examples
two alternating-sign pulses, with destructive interference:
0.0001
8. ´ 10 - 6
0.00008
6. ´ 10 - 6
0.00006
4. ´ 10 - 6
0.00004
2. ´ 10 - 6
0.00002
0
0.0000
- 60
- 40
- 20
0
20
40
- 60
60
- 40
- 20
8. ´ 10 - 6
8. ´ 10 - 6
6. ´ 10 - 6
6. ´ 10 - 6
4. ´ 10 - 6
4. ´ 10 - 6
-6
2. ´ 10 - 6
2. ´ 10
0
20
40
60
20
40
60
20
40
60
t
t
0
0
- 60
- 40
- 20
0
20
40
60
- 60
- 40
- 20
t
0
t
8. ´ 10-6
0.00003
0.000025
-6
6. ´ 10
0.00002
4. ´ 10-6
0.000015
0.00001
2. ´ 10-6
5. ´ 10-6
0
0
-60
-40
-20
0
t
20
40
60
-60
-40
-20
0
t
Super-Adiabatic Particle Number: examples
alternating-sign pulses with coherent constructive
interference:
1
1
0
0
� 75
� 50
� 25
0
25
t
1: 4
50
75
� 100
� 50
0
50
t
1: 4: 9
100
1
Super-Adiabatic Particle Number: examples
momentum spectrum for alternating-sign pulses:
0.025
0.025
0.020
0.020
0.015
0.015
0.015
0.015
0.010
0.010
0.010
0.010
0.010
0.010
0.005
0.005
0.005
0.005
0.005
0.005
0.025
0.025
0.025
0.020
0.025
0.020
0.020
0.015
0.000
0.000
-2 -2
-1 -1
0 0
kÈÈkÈÈ
11
22
0.020
0.015
0.000
0.000
-2
-1
-1
00
kÈÈkÈÈ
0.025
0.020
0.015
0.010
0.005
0
kÈÈ
1
2
0.000
-2
-1
0
kÈÈ
1
2
1 1
2
2
0.000
0.000
-2 -2
Super-Adiabatic Particle Number: examples
momentum spectrum for alternating-sign pulses:
Note
thatthat
as as
a function
of Note
thethelongitudinal
momentum
these
spectra
n-slitspectra
interference
Note
a function
of
longitudinal
momentum
theseparticle
particle
spectra represent
represent
the n-slit
interference
that
as a function
of thek,k,longitudinal
momentum
k, these the
particle
represent the
pattern,
here
probed
in in
thethe
time-domain
from
vacuum
[69].
In
we
theseInmomentum
momentum
pattern,
here
probed
time-domain
fromthe
thequantum
quantum
vacuumfrom
[69].the
In Figure
Figure 13
13vacuum
we show
show[69].
these
pattern,
here probed
in
the
time-domain
quantum
Figure 13 we show
spectra
(for(for
thethe
final
asymptotic
particle
number
N
=
Ñ
(t
=
+∞)),
for
the
cases
of
one,
two
and
three
pulses,
of two a
k
k
spectra
final
asymptotic
particle
number
N
=
Ñ
(t
=
+∞)),
for
the
cases
of
one,
two
and
three
pulses,
k
k
spectra (for the final asymptotic particle number Nk = Ñk (t = +∞)), for the cases of of
one,
the the
forms
(36),
(37),
andand
(39),
respectively.
forms
(36),
(37),
(39),
respectively.
the
forms
(36), (37), and (39), respectively.
Super-Adiabatic Particle Number: examples
t = −7.5
t = −7.5
FIG. 15:
t = −7.5
t = +7.5
t = +7.5
t = +7.5
t = +30
t = +30
t = +30
As in (14), but plotted over specific time ranges, ending with cross-sections at t = −7.5, t = +7.5, and t = +30,
Super-Adiabatic Particle Number
2
X
1
(p)
(0)
(p)
exp 2iθk exp −Fk,tp Erfc −σk (t) Ñk (t) ≈ 4 t
p
I
super-adiabatic particle number is smooth in time, and in
momentum
I
sensible definition because of universality (Dingle/Berry)
Super-Adiabatic Particle Number: de Sitter space
cosmological particle production: particle number in an
expanding/contracting universe
I
particle production in 4 dimensional spacetime (Mottola)
I
particle production in even dimensional dS spacetime,
but no particle production in odd dimensional dS spacetime
(Bousso et al)
question: physical reason for this difference ?
Super-Adiabatic Particle Number: 4d de Sitter space
0.000025
0.00015
0.00002
0.00010
0.000015
0.00001
0.00005
5. ´ 10-6
0.00000
-15
-10
-5
0
5
10
15
0
-15
-10
-5
t
5
10
15
5
10
15
5
10
15
t
0.000025
0.000025
0.00002
0.00002
0.000015
0.000015
0.00001
0.00001
5. ´ 10-6
5. ´ 10-6
0
-15
0
-10
-5
0
5
10
15
0
-15
-10
-5
t
0
t
0.000025
0.00004
0.00002
0.00003
0.000015
0.00001
0.00002
5. ´ 10-6
0.00001
0
-15
-10
-5
0
t
5
10
15
0
-15
-10
-5
0
t
cf. two-pulse coherent-constructive Schwinger effect
Super-Adiabatic Particle Number: 3d de Sitter space
0.00012
0.00010
6. ´ 10-6
0.00008
4. ´ 10-6
0.00006
0.00004
2. ´ 10-6
0.00002
0.00000
-15
-10
-5
0
5
10
15
0
-15
-10
-5
6. ´ 10-6
6. ´ 10-6
4. ´ 10-6
4. ´ 10-6
-6
2. ´ 10-6
2. ´ 10
0
-15
-10
-5
0
0
5
10
15
5
10
15
5
10
15
t
t
5
10
15
0
-15
-10
-5
t
0
t
0.000025
6. ´ 10-6
0.00002
0.000015
4. ´ 10-6
0.00001
2. ´ 10-6
0
-15
5. ´ 10-6
-10
-5
0
t
5
10
15
0
-15
-10
-5
0
t
cf. two-pulse coherent-destructive Schwinger effect
Conclusions
I
adiabatic particle number is basis dependent at
non-asymptotic times
I
preferred super-adiabatic basis: optimal truncation of
adiabatic expansion
I
super-adiabatic particle number is smooth across “particle
creation events”
I
super-adiabatic particle number reveals quantum
interference
I
difference between dS3 and dS4 is quantum interference
I
future: implications for back-reaction and quantum noise
I
future: analogue systems: time-dependent Bogoliubov
transformation problems