z=−2J ~ p1−z2sinϕ • z=|Ψ1|2− |Ψ2|2: population imbalance • ϕ=ϑ2−ϑ1: relative phase • pendulum with driving and length depending on momentum (4)Mean-Field (2

Driven Dimer Model

Michael Hartmann

22. April 2015

Model

• Hamiltonian H=−J

b^†₁b2+b^†₂b1

+U

2 X

j=1,2

nj(nj−1) +(t) (n2−n1)

=H₀+(t)H₁

• bj,b^†_j bosonic creation and anhilation operators

• nj ≡b^†_jbj number operator

• J hopping amplitude

• U onsite energy

• (t)function periodic in time:

(t) =µ +µ sin (ωt)

Mean-Field

• valid forN → ∞,U N = const

• dynamics may be described by two wavefunctionsΨj =|Ψj|e^iϑj

• differential equations

˙ ϕ= 2J

~

√ z

1−z² cosϕ+N U z

~ − 2(t)

~

˙

z=−2J

~

p1−z²sinϕ

• z=|Ψ1|²− |Ψ2|²: population imbalance

• ϕ=ϑ₂−ϑ₁: relative phase

• pendulum with driving and length depending on momentum

Mean-Field (2)

• differential equations

˙ ϕ= 2J

~

√ z

1−z² cosϕ+N U z

~ − 2(t)

~

˙

z=−2J

~

p1−z²sinϕ

• steady states for(t)≡0:

• z= 0,ϕ= 0, π

• z=±q

1−N^4J2U²²,ϕ= 0 ifJ/U <0and _N^4J2U²² ≤1

• z=±q

1−N^4J2U²²,ϕ=π ifJ/U >0and _N^4J2U²² ≤1

• z=±1,ϕ=±π/2

Poincar ´e plots

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

z

ϕ/π

J/ =−0.5,N U/ = 2,(t) = 0

Poincar ´e plots

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

z

ϕ/π

J/ = +0.5,N U/ = 2,(t) = 0

Poincar ´e plots

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

z

ϕ/π

order 1 order 3 order 5

J/ =−1,N U/ =−2,µ = 2.7,µ = 2.5,ω= 3

Dissipation

Heisenberg equation:

d

dt%=−i

~[H(t), %] + γ

~

2c%c^†−c^†c%−%c^†c

⇒ d

dt%~=L(t)~% where

• L(t) =L0+(t)L1+LD

• Diehl et al.:

D(%) =

2c%c^†−c^†c%−%c^†c

, c=

b^†₁+b^†₂

(b₁−b₂)

• Poletti et al.:

D(%) =− X 1

2[nj,[nj, %]]

Integration

Zassenhaus formula:

e^t(A+B)=e^tAe^tBe^C2t2e^C3t3. . . , C₂ =−[A, B]/2, . . . Propagator:

U(t, t₀) = T exp Z t

t0

dτL(t)

= T exp





N−1

X

j=0

Z δ 0

dτL(t_j+τ)





t_j =t₀+j(t−t₀)/N

Integration

Zassenhaus formula:

e^t(A+B)=e^tAe^tBe^C2t2e^C3t3. . . , C2 =−[A, B]/2, . . . Propagator:

U(t, t0) = T exp Z t

t0

dτL(t)

= T exp





N−1

X

j=0

Z δ 0

dτL(tj+τ)





≈T

N−1

Y

j=0

exp Z δ

0

dτ (L0+LD +(t_j+τ)L1)

+O(δ)

t =t +j(t−t )/N

Integration

Zassenhaus formula:

e^t(A+B)=e^tAe^tBe^C2t2e^C3t3. . . , C2 =−[A, B]/2, . . . Propagator:

U(t, t₀) = T exp Z t

t0

dτL(t)

= T exp





N−1

X

j=0

Z δ 0

dτL(t_j+τ)





≈T

N−1

Y

j=0

exp Z δ

0

dτ (L⁰+L^D+(tj+τ)L¹)

+O(δ)

≈T

N−1

Y

j=0

exp [(L0+LD)δ] exp

L1

Z δ 0

dτ (tj+τ)

+O(δ)

t =t +j(t−t )/N

Floquet-Propagator

• assumeU(T,0)is diagonalizable

• Floquet

|%_i(T)i=λ_i|%_i(0)i where

• λieigenvalues ofU(T,0)(characteristic multipliers)

• %ieigenmatrices ofU(T,0)

• λ_i= 1: asymptotic state

• |λ_i|<1:|%_i(0)i^t→∞−→ 0

Spectrum of Floquet-Propagator

−0.5

−0.25 0 0.25 0.5

−0.5 −0.25 0 0.25 0.5 0.75 1

=(λ)

<(λ)

N = 10,J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Spectrum of Floquet-Propagator

−0.5

−0.25 0 0.25 0.5

−0.5 −0.25 0 0.25 0.5 0.75 1

=(λ)

<(λ)

N = 20,J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Spectrum of Floquet-Propagator

−0.5

−0.25 0 0.25 0.5

−0.5 −0.25 0 0.25 0.5 0.75 1

=(λ)

<(λ)

N = 30,J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Spectrum of Floquet-Propagator

−0.5

−0.25 0 0.25 0.5

−0.5 −0.25 0 0.25 0.5 0.75 1

=(λ)

<(λ)

N = 40,J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Spectrum of Floquet-Propagator

−0.5

−0.25 0 0.25 0.5

−0.5 −0.25 0 0.25 0.5 0.75 1

=(λ)

<(λ)

N = 50,J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Spectrum of Floquet-Propagator (Mean-Field)

−0.5

−0.25 0 0.25 0.5

−0.5 −0.25 0 0.25 0.5 0.75 1

=(λ)

<(λ)

N = 10,J/~=−1,U/~=−2/N,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Spectrum of Floquet-Propagator (Mean-Field)

−0.5

−0.25 0 0.25 0.5

−0.5 −0.25 0 0.25 0.5 0.75 1

=(λ)

<(λ)

N = 20,J/~=−1,U/~=−2/N,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Spectrum of Floquet-Propagator (Mean-Field)

−0.5

−0.25 0 0.25 0.5

−0.5 −0.25 0 0.25 0.5 0.75 1

=(λ)

<(λ)

N = 30,J/~=−1,U/~=−2/N,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Spectrum of Floquet-Propagator (Mean-Field)

−0.5

−0.25 0 0.25 0.5

−0.5 −0.25 0 0.25 0.5 0.75 1

=(λ)

<(λ)

N = 40,J/~=−1,U/~=−2/N,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Spectrum of Floquet-Propagator (Mean-Field)

−0.5

−0.25 0 0.25 0.5

−0.5 −0.25 0 0.25 0.5 0.75 1

=(λ)

<(λ)

N = 50,J/~=−1,U/~=−2/N,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Husimi

• Husimi distribution

Q(θ, φ) =hθ, φ|%|θ, φi

• fully condensed states

|θ, φi=

N

X

n=0

v u u t

N n

! cosθ

2 n

sinθ 2e^iφ

N−n

|ni

• projected onto

q =φ/2

Evolution

• Basis

|m, ni=|m, N−mi ≡ |mi

• mparticles in left well

• nparticles in right well

• initial density matrix

%(t= 0) =|Ni hN|

• all particles att= 0in left well

Evolution

−0.5

−0.25 0 0.25 0.5 0.75 1

0 10 20 30 40 50

z=<2n1−N>/N

t/T

N=10

J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Evolution

−0.5

−0.25 0 0.25 0.5 0.75 1

0 10 20 30 40 50

z=<2n1−N>/N

t/T

N=20

J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Evolution

−0.5

−0.25 0 0.25 0.5 0.75 1

0 10 20 30 40 50

z=<2n1−N>/N

t/T

N=30

J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Evolution

−0.5

−0.25 0 0.25 0.5 0.75 1

0 10 20 30 40 50

z=<2n1−N>/N

t/T

N=40

J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Evolution

−0.5

−0.25 0 0.25 0.5 0.75 1

0 10 20 30 40 50

z=<2n1−N>/N

t/T

N=50

J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Evolution

−0.5

−0.25 0 0.25 0.5 0.75 1

0 10 20 30 40 50

z=<2n1−N>/N

t/T

N=10 N=50

J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Evolution (Mean Field)

−1

−0.5 0 0.5 1

0 10 20 30 40 50

z=<2n1−N>/N

t/T

N=10

J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Evolution (Mean Field)

−1

−0.5 0 0.5 1

0 10 20 30 40 50

z=<2n1−N>/N

t/T

N=20

J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Evolution (Mean Field)

−1

−0.5 0 0.5 1

0 10 20 30 40 50

z=<2n1−N>/N

t/T

N=30

J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Evolution (Mean Field)

−1

−0.5 0 0.5 1

0 10 20 30 40 50

z=<2n1−N>/N

t/T

N=40

J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Evolution (Mean Field)

−1

−0.5 0 0.5 1

0 10 20 30 40 50

z=<2n1−N>/N

t/T

N=50

J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Evolution (Mean Field)

−1

−0.5 0 0.5 1

0 10 20 30 40 50

z=<2n1−N>/N

t/T

N=10 N=50

J/~=−1,U/~=−2,γ/~= 0.01, µ / = 2.7,µ / = 2.5,ω= 3

Spectrum (non-driven, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 10,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 20,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 30,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 40,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 50,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 60,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 70,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 10,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 20,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 30,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 40,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 50,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 60,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Poletti et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 70,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 10,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 20,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 30,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 40,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 50,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 60,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 70,J/~=−1,U/~=−2,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 10,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 20,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 30,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 40,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 50,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 60,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0

Spectrum (non-driven, Mean-Field, Diehl et al.)

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1

=(λ)

<(λ)

N = 70,J/~=−1,U/~=−2/N,γ/~= 0.01,(t)≡0