Dynamics of Sound Waves
in Interacting Bose Gases
Dirk - André Deckert
Department of Mathematics
University of California Davis
March 19, 2014
Joint work with J. Fröhlich (ETH), P. Pickl (LMU), and A. Pizzo (UCD).
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
1 / 20
Dynamics of Sound Waves
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
2 / 20
Model of the Bose Gas
i∂t Ψt (x1 , . . . , xN ) = HΨt (x1 , . . . , xN )


N
X
X
∆xj
−
+α
U(xj − xk )
H=
2
j=1
k<j
with U ∈ Cc∞ , and initially the gas particles are quite regularly arranged:
Ψ0 is “close” to a product state
Ψ0 (x1 , . . . , xN ) =
N
Y
Λ−1/2 ϕ0 (xj ),
kΨ0 k2 = 1
j=1
for a smooth one-particle wave function ϕ0 with:
kϕ0 k∞ = 1 and supported in a box of volume Λ ⊂ R3 .
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
3 / 20
Model of the Bose Gas
i∂t Ψt (x1 , . . . , xN ) = HΨt (x1 , . . . , xN )


N
X
X
∆xj
−
+α
U(xj − xk )
H=
2
j=1
k<j
with U ∈ Cc∞ , and initially the gas particles are quite regularly arranged:
Ψ0 is “close” to a product state
Ψ0 (x1 , . . . , xN ) =
N
Y
Λ−1/2 ϕ0 (xj ),
kΨ0 k2 = 1
j=1
for a smooth one-particle wave function ϕ0 with:
kϕ0 k∞ = 1 and supported in a box of volume Λ ⊂ R3 .
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
3 / 20
A physically relevant scaling
Gas density: ρ =
N
Λ.
For the product state Ψ0 one finds
*
+
N
ϕ 2
X
0 Ψ0 , α
U(x − xk )Ψ0 = N α U ∗ 1/2 (x) = ρ α U ∗ |ϕ0 |2 (x).
Λ
k=1
For U ∈ Cc∞ (R3 ) one formally has:
−
X
∆xj
+α
U(xj − xk ) = O(1) + O(αρ)
2
k<j
Hence, for
α∼
1
,
ρ
one can expect non-trivial dynamics for ρ 1.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
4 / 20
A physically relevant scaling
Gas density: ρ =
N
Λ.
For the product state Ψ0 one finds
*
+
N
ϕ 2
X
0 Ψ0 , α
U(x − xk )Ψ0 = N α U ∗ 1/2 (x) = ρ α U ∗ |ϕ0 |2 (x).
Λ
k=1
For U ∈ Cc∞ (R3 ) one formally has:
−
X
∆xj
+α
U(xj − xk ) = O(1) + O(αρ)
2
k<j
Hence, for
α∼
1
,
ρ
one can expect non-trivial dynamics for ρ 1.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
4 / 20
A physically relevant scaling
Gas density: ρ =
N
Λ.
For the product state Ψ0 one finds
*
+
N
ϕ 2
X
0 Ψ0 , α
U(x − xk )Ψ0 = N α U ∗ 1/2 (x) = ρ α U ∗ |ϕ0 |2 (x).
Λ
k=1
For U ∈ Cc∞ (R3 ) one formally has:
−
X
∆xj
+α
U(xj − xk ) = O(1) + O(αρ)
2
k<j
Hence, for
α∼
1
,
ρ
one can expect non-trivial dynamics for ρ 1.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
4 / 20
Microscopic dynamics:
i∂t Ψ(x1 , . . . , xN ) = HΨ(x1 , . . . , xN ),
H=
N
X
j=1

X
∆xj
1
−
+
U(xj − xk )
2
ρ

k<j
Macroscopic dynamics:
i∂t ϕt (x) = h[ϕt ]ϕt (x),
hx [ϕt ] = −
∆x
+ 1 · U ∗ |ϕt |2 (x).
2
For ρ 1 one can hope to control the micro- with the macro-dynamics in a
sufficiently strong sense, e.g.,
ϕt Trx ,...,x |Ψt ihΨt | − ϕt
≤ C (t)ρ−γ ,
for γ > 0, ρ 1.
2
N
kϕt k2
kϕt k2 For fixed volume Λ, i.e., ρ = O(N), many results are available, e.g.,
Hepp ’74, Spohn ’80, Rodniaski & Schlein ’09, Fröhlich & Knowles & Schwarz ’09,
Pickl ’10, Erdõs & Schlein & Yau ’10, . . .
Spectrum for large volume: Dereziński & Napiórkowski ’13
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
5 / 20
Microscopic dynamics:
i∂t Ψ(x1 , . . . , xN ) = HΨ(x1 , . . . , xN ),
H=
N
X
j=1

X
∆xj
1
−
+
U(xj − xk )
2
ρ

k<j
Macroscopic dynamics:
i∂t ϕt (x) = h[ϕt ]ϕt (x),
hx [ϕt ] = −
∆x
+ 1 · U ∗ |ϕt |2 (x).
2
For ρ 1 one can hope to control the micro- with the macro-dynamics in a
sufficiently strong sense, e.g.,
ϕt Trx ,...,x |Ψt ihΨt | − ϕt
≤ C (t)ρ−γ ,
for γ > 0, ρ 1.
2
N
kϕt k2
kϕt k2 For fixed volume Λ, i.e., ρ = O(N), many results are available, e.g.,
Hepp ’74, Spohn ’80, Rodniaski & Schlein ’09, Fröhlich & Knowles & Schwarz ’09,
Pickl ’10, Erdõs & Schlein & Yau ’10, . . .
Spectrum for large volume: Dereziński & Napiórkowski ’13
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
5 / 20
Microscopic dynamics:
i∂t Ψ(x1 , . . . , xN ) = HΨ(x1 , . . . , xN ),
H=
N
X
j=1

X
∆xj
1
−
+
U(xj − xk )
2
ρ

k<j
Macroscopic dynamics:
i∂t ϕt (x) = h[ϕt ]ϕt (x),
hx [ϕt ] = −
∆x
+ 1 · U ∗ |ϕt |2 (x).
2
For ρ 1 one can hope to control the micro- with the macro-dynamics in a
sufficiently strong sense, e.g.,
ϕt Trx ,...,x |Ψt ihΨt | − ϕt
≤ C (t)ρ−γ ,
for γ > 0, ρ 1.
2
N
kϕt k2
kϕt k2 For fixed volume Λ, i.e., ρ = O(N), many results are available, e.g.,
Hepp ’74, Spohn ’80, Rodniaski & Schlein ’09, Fröhlich & Knowles & Schwarz ’09,
Pickl ’10, Erdõs & Schlein & Yau ’10, . . .
Spectrum for large volume: Dereziński & Napiórkowski ’13
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
5 / 20
Open Key Questions
1
Can the control be maintained for large volume Λ?
2
Is the approximation good enough to be able to see OΛ,ρ (1) excitations of
the gas, e.g., sound waves, for large Λ?
3
Can the thermodynamic limit, Λ → ∞ for fixed ρ, and the mean-field limit,
ρ → ∞, be decoupled?
We demonstrate how to control the microscopic dynamics of excitations in the
following large volume regime:
Λ, ρ 1
D.-A. Deckert (UC Davis)
such that
Λ
1
ρ
Sound Waves in BECs @ Warwick
and
N = ρΛ.
March 19, 2014
6 / 20
Open Key Questions
1
Can the control be maintained for large volume Λ?
2
Is the approximation good enough to be able to see OΛ,ρ (1) excitations of
the gas, e.g., sound waves, for large Λ?
3
Can the thermodynamic limit, Λ → ∞ for fixed ρ, and the mean-field limit,
ρ → ∞, be decoupled?
We demonstrate how to control the microscopic dynamics of excitations in the
following large volume regime:
Λ, ρ 1
D.-A. Deckert (UC Davis)
such that
Λ
1
ρ
Sound Waves in BECs @ Warwick
and
N = ρΛ.
March 19, 2014
6 / 20
Tracking Excitations for large Λ
Coherent excitation of the gas:
Ψ0 (x1 , . . . , xN ) =
N
Y
Λ−1/2 ϕ0 (xj ),
for ϕ0 = Ω0 + 0 ,
given:
j=1
A smooth and flat reference state Ω0 :
supp Ω0 = O(Λ), kΩk∞ = 1 with sufficiently regular tails;
A smooth and localized coherent excitation 0 :
supp 0 = O(1).
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
7 / 20
Splitting of the dynamics: given the macroscopic dynamics ϕt use
∆x
i∂t Ωt = −
+ U ∗ |Ωt |2 − 1 Ωt .
2
(reference)
as reference state and define the excitation by
t = ϕt e ikUk1 t − Ωt
(excitation)
Control of approximation: define
(micro)
ρt
ED
1/2
Ψt Λ1/2 Ψt proj⊥
= proj⊥
Ωt Trx2 ,...,XN Λ
Ωt ,
(macro)
= |t iht |,
(micro)
(macro) and control ρt
− ρt
for large Λ, ρ.
ρt
. . . which turns out to be not so easy.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
8 / 20
Splitting of the dynamics: given the macroscopic dynamics ϕt use
∆x
i∂t Ωt = −
+ U ∗ |Ωt |2 − 1 Ωt .
2
(reference)
as reference state and define the excitation by
t = ϕt e ikUk1 t − Ωt
(excitation)
Control of approximation: define
(micro)
ρt
ED
1/2
Ψt Λ1/2 Ψt proj⊥
= proj⊥
Ωt Trx2 ,...,XN Λ
Ωt ,
(macro)
= |t iht |,
(micro)
(macro) and control ρt
− ρt
for large Λ, ρ.
ρt
. . . which turns out to be not so easy.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
8 / 20
Splitting of the dynamics: given the macroscopic dynamics ϕt use
∆x
i∂t Ωt = −
+ U ∗ |Ωt |2 − 1 Ωt .
2
(reference)
as reference state and define the excitation by
t = ϕt e ikUk1 t − Ωt
(excitation)
Control of approximation: define
(micro)
ρt
ED
1/2
Ψt Λ1/2 Ψt proj⊥
= proj⊥
Ωt Trx2 ,...,XN Λ
Ωt ,
(macro)
= |t iht |,
(micro)
(macro) and control ρt
− ρt
for large Λ, ρ.
ρt
. . . which turns out to be not so easy.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
8 / 20
New mode of approximation: Though weaker, it is strong enough to answer
most physical questions in a satisfactory way.
Theorem (Micro approximated by Macro)
e t with
Suppose kϕt k∞ is bounded on [0, T ]. There is a trajectory t 7→ Ψ
(micro)
corresponding reduced density matrix ρet
such that:
Λ
2
e t ≤ C (t) ;
Ψt − Ψ
2
ρ
1/2
(micro)
(macro) − ρt
ρt
,
≤ C (t) Λρ
e
for all times t ∈ [0, T ] and sufficiently large Λ and ρ.
This implies that the actual quantity
(micro)
(macro) − ρt
ρt
is typically small, provided Λ, ρ 1 such that
D.-A. Deckert (UC Davis)
Λ
ρ
1.
Sound Waves in BECs @ Warwick
March 19, 2014
9 / 20
Strategy of Proof
Similar to Pickl’s ’10 technique we count “bad” particles, i.e., particles that
do not follow the macroscopic dynamics given by ϕt .
For this we employ the orthogonal one-particle projectors:
p ϕt =
1
|ϕt ihϕt |,
kϕt k2
q ϕt = 1 − p ϕt ,
and define the N-particle operators:
k
Pkϕt = (q ϕt )
(N−k)
(p ϕt )
,
N
X
Pkϕt = id,
k=0
which projects onto wave functions with exactly k “bad” particles.
A quantity that was successfully used to control fixed-volume mean-field
limits is
+
*
N
X
k ϕt
ϕt
hΨt , q1 Ψt i = Ψt ,
P Ψt ,
N k
k=0
the expected ratio of “bad particles” over N.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
10 / 20
Strategy of Proof
Similar to Pickl’s ’10 technique we count “bad” particles, i.e., particles that
do not follow the macroscopic dynamics given by ϕt .
For this we employ the orthogonal one-particle projectors:
p ϕt =
1
|ϕt ihϕt |,
kϕt k2
q ϕt = 1 − p ϕt ,
and define the N-particle operators:
k
Pkϕt = (q ϕt )
(N−k)
(p ϕt )
,
N
X
Pkϕt = id,
k=0
which projects onto wave functions with exactly k “bad” particles.
A quantity that was successfully used to control fixed-volume mean-field
limits is
+
*
N
X
k ϕt
ϕt
hΨt , q1 Ψt i = Ψt ,
P Ψt ,
N k
k=0
the expected ratio of “bad particles” over N.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
10 / 20
Strategy of Proof
Similar to Pickl’s ’10 technique we count “bad” particles, i.e., particles that
do not follow the macroscopic dynamics given by ϕt .
For this we employ the orthogonal one-particle projectors:
p ϕt =
1
|ϕt ihϕt |,
kϕt k2
q ϕt = 1 − p ϕt ,
and define the N-particle operators:
k
Pkϕt = (q ϕt )
(N−k)
(p ϕt )
,
N
X
Pkϕt = id,
k=0
which projects onto wave functions with exactly k “bad” particles.
A quantity that was successfully used to control fixed-volume mean-field
limits is
+
*
N
X
k ϕt
ϕt
hΨt , q1 Ψt i = Ψt ,
P Ψt ,
N k
k=0
the expected ratio of “bad particles” over N.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
10 / 20
Strategy of Proof
However, to silhouette the O(1) excitation t against the reference state Ωt
we need much finer control on the ratio of bad particles over ρ:
Only less than ρ particles may behave badly!
Therefore, we regard
*
+
N
X
ϕt
ϕt
b it := Ψt ,
hm
m(k)Pk Ψt ,
(
m(k) =
k=0
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
k
ρ
for 0 ≤ k ≤ ρ
1 otherwise
March 19, 2014
.
11 / 20
Strategy of Proof
However, to silhouette the O(1) excitation t against the reference state Ωt
we need much finer control on the ratio of bad particles over ρ:
Only less than ρ particles may behave badly!
Therefore, we regard
*
+
N
X
ϕt
ϕt
b it := Ψt ,
hm
m(k)Pk Ψt ,
(
m(k) =
k=0
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
k
ρ
for 0 ≤ k ≤ ρ
1 otherwise
March 19, 2014
.
11 / 20
One can show:
Lemma
– same conditions – Define:
X ϕ
e t :=
Ψ
P k t Ψt
0≤k≤ρ
= Ψt projected onto the subspace with at most ρ bad particles.
Then:
et
b ϕt it ;
Ψt − Ψ
22 ≤ hm
p
(micro)
(macro) b ϕt it .
ρt
− ρt
e
≤ C (t) hm
Hence, it is sufficient to prove:
Lemma (Expected ratio of bad particles over ρ)
– same conditions –
D.-A. Deckert (UC Davis)
Λ
b ϕt it ≤ C (t) .
hm
ρ
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March 19, 2014
12 / 20
d
b ϕt it =
hm
dt

*
Ψt , H −
N
X

+
b t  Ψt
hxj [ϕt ], m
j=1

 +
N
2
X
X
N
|ϕt |
1
b t  Ψt
U(xj − xk ) − U ∗
(xj ), m
= Ψt ,
ρ
ρ
Λ
j=1
j<k


*
+


N2
|ϕt |2
|ϕt |2


b t  Ψt
≈
Ψt , U(x1 − x2 ) − U ∗
(x1 ) − U ∗
(x2 ), m
2ρ
Λ{z
Λ
|
}
*
=:Z (x1 ,x2 )
2
N
b tϕt ] Ψt i
hΨt , [Z (x1 , x2 ), m
2ρ
N2
b ϕt ] id idΨt ,
=
Ψt , id id [Z (x1 , x2 ), m
2ρ
=
for id = p1ϕt + q1ϕt = p2ϕt + q2ϕt ,
= 16 terms.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
13 / 20
g
d
b ϕt it ≤
hm
dt
b
+
Z
g
g
b
b
+
Z
g
g
b
b
Z
g
b
+ h.c. + diagrams that conserve the particle numbers
Λ
b ϕt i
≤C (t)
0
+
hm
+
.
ρ
This yields:
d
Λ
ϕt
ϕt
b it ≤ C (t) hm
b it +
hm
dt
ρ
and, thanks to Grönwall’s Lemma, concludes the proof.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
14 / 20
The term determining the structure of the macroscopic equation is:
g
b
Z
g
g
=
N2
b ϕt ] p1ϕt q2ϕt Ψt ,
Ψt , p1ϕt p2ϕt [Z (x1 , x2 ), m
2ρ
and one finds
p1ϕt p2ϕt Z (x1 , x2 )p1ϕt q2ϕt
=
p1ϕt p2ϕt
|ϕt |2
|ϕt |2
U(x1 − x2 ) − U ∗
(x2 ) − U ∗
(x1 ) p1ϕt q2ϕt
Λ
Λ


|ϕt |2


= p1ϕt p2ϕt p1ϕt U(x1 − x2 )p1ϕt −U ∗
(x2 )p1ϕt  q2ϕt
Λ
{z
}
|
ϕt
=Λ−1 U∗|ϕt |2 (x2 )p1
− p1ϕt U ∗
|ϕt |2
(x1 )p1ϕt p2ϕt q2ϕt
Λ
| {z }
=0
=0
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
15 / 20
Effective Equation for the Excitation
∆x
2
2
∗
i∂t t = −
+ U ∗ |Ωt | − 1 + U ∗ |t | + U ∗ 2<t Ωt t
2
+ U ∗ |t |2 + U ∗ 2<∗t Ωt Ωt ,
(excitation)
If there are no blow-ups for t ∈ [0, T ] and the L2 norm is small enough, we find a
simple effective equation for the excitation:
Theorem (Small Excitations)
Let ηt solve
i∂t ηt = −
∆x
ηt + U ∗ 2<ηt ,
2
η0 = 0 .
Then:
!
−1/3
kt − ηt k2 ≤ C (t) Λ
+ sup kt k2
s∈[0,T ]
for all t ∈ [0, T ].
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
16 / 20
The evolution equation
i∂t ηt = −
∆x
ηt + U ∗ 2<ηt
2
for Fourier transform ηbt can be given by
i∂t
ηbt (k)
ηbt ∗ (−k)
ηbt (k)
= H(k)
,
ηbt ∗ (−k)
H(k) =
!
b
b
ω0 (k) + U(k)
U(k)
,
b
b
−U(k)
−ω(k) − U(k)
where ω0 (k) = k 2 /2. The eigenvalues ω(k) of H(k) fulfill
b
ω(k)2 = ω0 (k) ω0 (k) + 2U(k)
.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
17 / 20
One can explicitly solve for the dispersion relation:
r
k2 b
+ U(k).
ω(k) = |k|
4
first discovered by Bogulyubov.
For small momenta |k| we can distinguish two cases:
b
Repulsive potential, i.e., U(0)
> 0:
vsound
q
d
b
=
= U(0).
ω(k)
dk
k→0
Sound waves with arbitrary small modes travel at a strictly positive speed.
b
Attractive potential, i.e., U(0)
< 0:
b
Modes with wave vectors k such that k 2 /4 = U(k)
become static.
b
Modes such that k 2 /4 < U(k)
become dynamical unstable.
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
18 / 20
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
19 / 20
Outlook
Show that the actual ground state of the gas is “close” to a product state
much like our initial state.
Uncouple the scales Λ 1 and ρ 1.
Treat the case of non-zero temperature.
Thank you!
D.-A. Deckert (UC Davis)
Sound Waves in BECs @ Warwick
March 19, 2014
20 / 20