Derivation of 1d and 2d GrossPitaevskii equations for strongly - - PowerPoint PPT Presentation

Derivation of 1d and 2d Gross–Pitaevskii equations for strongly confined 3d bosons

Lea Boßmann

University of T¨ ubingen

Venice, 20 August 2019

Problem Results Limiting sequences Strategy of Proof

In a nutshell

Consider N interacting bosons in a BEC which are in two or one spatial directions confined by a trap of diameter ε.

Problem Results Limiting sequences Strategy of Proof

In a nutshell

Consider N interacting bosons in a BEC which are in two or one spatial directions confined by a trap of diameter ε. Let simultaneously N → ∞ and ε → 0.

Problem Results Limiting sequences Strategy of Proof

In a nutshell

Consider N interacting bosons in a BEC which are in two or one spatial directions confined by a trap of diameter ε. Let simultaneously N → ∞ and ε → 0. We show that the dynamics of this system are effectively described by a one-/two-dimensional nonlinear equation.

Problem Results Limiting sequences Strategy of Proof

In a nutshell

Consider N interacting bosons in a BEC which are in two or one spatial directions confined by a trap of diameter ε. Let simultaneously N → ∞ and ε → 0. We show that the dynamics of this system are effectively described by a one-/two-dimensional nonlinear equation. Joint work with Stefan Teufel.

References: 1d: J. Math. Phys. 60:031902; Ann. Henri Poincar´ e 20(3):1003 2d: arXiv:1907.04547

Problem Results Limiting sequences Strategy of Proof

Microscopic model

Coordinates: z = (x, y) ∈ R3 x ∈ Rd y ∈ R3−d

N-body Hamiltonian

H =

N

• j=1
• − ∆j + 1

ε2 V ⊥( yj ε )

• +

i<j

wN,ε(zi − zj)

• V ⊥: confining potential; rescaled by ε

Problem Results Limiting sequences Strategy of Proof

Microscopic model

Coordinates: z = (x, y) ∈ R3 x ∈ Rd y ∈ R3−d

N-body Hamiltonian

H =

N

• j=1
• − ∆j + 1

ε2 V ⊥( yj ε )

• +

i<j

wN,ε(zi − zj)

• V ⊥: confining potential; rescaled by ε

Pair interaction: wN,ε(z) := µ1−3βw

• µ−βz
• β ∈ (0, 1]
• w ≥ 0 spherically symmetric, bounded, suppw ⊆ B1(0)
• µ =
• N

ε3−d

−1 → µβ: effective range of the interaction

Problem Results Limiting sequences Strategy of Proof

Microscopic model

Coordinates: z = (x, y) ∈ R3 x ∈ Rd y ∈ R3−d

N-body Hamiltonian

H =

N

• j=1
• − ∆j + 1

ε2 V ⊥( yj ε )

• +

i<j

wN,ε(zi − zj)

• V ⊥: confining potential; rescaled by ε

Pair interaction: wN,ε(z) := µ1−3βw

• µ−βz
• β ∈ (0, 1]
• w ≥ 0 spherically symmetric, bounded, suppw ⊆ B1(0)
• µ =
• N

ε3−d

−1 → µβ: effective range of the interaction

Limit: (N, ε) → (∞, 0) with suitable restrictions

Problem Results Limiting sequences Strategy of Proof

Assumptions on the initial data

1 BEC:

lim

(N,ε)→(∞,0) Tr

• γ(1)

− |ϕε

0 ϕε 0|

• = 0
• γ(1)

0 : one-particle reduced density matrix of ψN,ε

Problem Results Limiting sequences Strategy of Proof

Assumptions on the initial data

1 BEC:

lim

(N,ε)→(∞,0) Tr

• γ(1)

− |ϕε

0 ϕε 0|

• = 0
• γ(1)

0 : one-particle reduced density matrix of ψN,ε

• Condensate wave function: ϕε

0(z) = Φ0(x)χε(y)

• transverse GS:
• −∆y + 1

ε2 V ⊥( y ε)

• χε(y) = E0

ε2 χε(y)

Problem Results Limiting sequences Strategy of Proof

Assumptions on the initial data

1 BEC:

lim

(N,ε)→(∞,0) Tr

• γ(1)

− |ϕε

0 ϕε 0|

• = 0
• γ(1)

0 : one-particle reduced density matrix of ψN,ε

• Condensate wave function: ϕε

0(z) = Φ0(x)χε(y)

• transverse GS:
• −∆y + 1

ε2 V ⊥( y ε)

• χε(y) = E0

ε2 χε(y)

• Φ0 ∈ H2d(Rd)

→ evolves in time

Problem Results Limiting sequences Strategy of Proof

Assumptions on the initial data

1 BEC:

lim

(N,ε)→(∞,0) Tr

• γ(1)

− |ϕε

0 ϕε 0|

• = 0
• γ(1)

0 : one-particle reduced density matrix of ψN,ε

• Condensate wave function: ϕε

0(z) = Φ0(x)χε(y)

• transverse GS:
• −∆y + 1

ε2 V ⊥( y ε)

• χε(y) = E0

ε2 χε(y)

• Φ0 ∈ H2d(Rd)

→ evolves in time

2 Energy per particle:

lim

(N,ε)→(∞,0)

• E(ψN,ε

) − Ebβ(Φ0)

• = 0
• E(ψ) := 1

N ψ, Hψ − E0 ε2

• Ebβ(Φ) := Φ,
• − ∆x + 1

2bβ|Φ|2

Φ

Problem Results Limiting sequences Strategy of Proof

Effective d-dimensional Gross–Pitaevskii dynamics

Theorem

Under assumptions (1) and (2) and for any t ∈ R, lim

(N,ε)→(∞,0) Tr

• γ(1)(t) − |ϕε(t) ϕε(t)|
• = 0,

where ϕε(t) = Φ(t)χε and Φ(t) is the solution of i ∂

∂t Φ(t, x) =

• −∆x + bβ|Φ(t, x)|2

Φ(t, x) with Φ(0) = Φ0 and where bβ =    8πa

• |χ(y)|4 dy

β = 1 (GP) w1

• |χ(y)|4 dy

β ∈ (0, 1) (NLS)

a: scattering length of w, χ: ground state of −∆y + V ⊥(y)

Problem Results Limiting sequences Strategy of Proof

Related results

• X. Chen, J. Holmer. ARMA 2013.

d = 2 → β ∈ (0, 2

5), repulsive interactions

• X. Chen, J. Holmer. APDE 2017.

d = 1 → β ∈ (0, 3

7), attractive interactions

• J. v. Keler, S. Teufel. AHP 2016.

d = 1 → β ∈ (0, 1

3), repulsive interactions

Problem Results Limiting sequences Strategy of Proof

Simultaneous limit (N, ε) → (∞, 0)

1 1

N−1 ε

Problem Results Limiting sequences Strategy of Proof

Simultaneous limit (N, ε) → (∞, 0)

1 1

N−1 ε

• admissibility condition: ε must shrink fast enough

→ upper bound on ε

Problem Results Limiting sequences Strategy of Proof

Simultaneous limit (N, ε) → (∞, 0)

1 1

N−1 ε

• admissibility condition: ε must shrink fast enough

→ upper bound on ε

• moderate confinement: ε must not shrink too fast

→ lower bound on ε

Problem Results Limiting sequences Strategy of Proof

Parameter range for β ∈ (0, 1) d=2

β = 1

3

1

N

1

1 1

N

1

1

β = 2

3

β = 5

6

1

N

1

1 1

N

1

1

β = 11

12

Problem Results Limiting sequences Strategy of Proof

Comparison with [ChHo2013] d = 2, β ∈ (0, 2

5)

β =

3 11

1

N

1

1 1

N

1

1

β = 1

3

β = 11

30

1

N

1

1 1

N

1

1

β = 23

60

Problem Results Limiting sequences Strategy of Proof

Limiting sequences for β = 1 d=2

1

N

1

1

Problem Results Limiting sequences Strategy of Proof

Strategy of proof

• General strategy: method from [Pickl2015]

Problem Results Limiting sequences Strategy of Proof

Strategy of proof

• General strategy: method from [Pickl2015]
• Adaptation to strong confinement:
• 3d micro dynamics ↔ 1d/2d effective dynamics
• split interaction into quasi-1d/2d interaction + remainders
• remainders controllable with admissibility condition

Problem Results Limiting sequences Strategy of Proof

Strategy of proof

• General strategy: method from [Pickl2015]
• Adaptation to strong confinement:
• 3d micro dynamics ↔ 1d/2d effective dynamics
• split interaction into quasi-1d/2d interaction + remainders
• remainders controllable with admissibility condition

Thank you very much for your attention!