Stability and stabilization of unstable condensates

Doron Cohen

Ben-Gurion University

Amichay Vardi (BGU)

Maya Chuchem (PhD, BGU)

Christine Khripkov (PhD, BGU)

Geva Arwas (PhD, BGU)

Igor Tikhonenkov (potdoc, BGU)

+ additional collaborations (see next page)

http://www.bgu.ac.il/∼dcohen

$DIP, $BSF, $ISF 0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|S|

φ=πφ=4π/5φ=3π/5φ=2π/5φ=π/5φ=0

time [Josephson periods]

BHH - dimers and trimers

Minimal models: networks (1D), billiards (2D), and coupled oscillators (BHH).

HBHH =U

2

M∑j=1

a†ja†jajaj −

K

2

M∑j=1

(a†j+1aj + a†jaj+1

)u ≡

NU

K

Dimer (M=2): minimal BHH; Bosonic Josephson junction; Pendulum physics [1].

Driven dimer: Landau-Zener dynamics [2], Kapitza effect [3], Zeno effect [4], Standard-map physics [5].

Linear trimer: minimal model for chaos; Coupled pendula physics.

Triangular trimer: minimal model with topology, Superfluidity [6], Stirring [7].

Coupled trimers: minimal model for mesoscopic thermalization [8].

[1] Chuchem, Smith-Mannschott, Hiller, Kottos, Vardi, Cohen (PRA 2010).

[2] Smith-Mannschott, Chuchem, Hiller, Kottos, Cohen (PRL 2009).

[3] Boukobza, Moore, Cohen, Vardi (PRL 2010).

[4] Khripkov, Vardi, Cohen (PRA 2012)

[5] Khripkov, Cohen, Vardi (PRE 2013).

[6] Arwas, Vardi, Cohen (arXiv 2013).

[7] Hiller, Kottos, Cohen (EPL 2008, PRA 2008).

[8] Tikhonenkov, Vardi, Anglin, Cohen, PRL (2013).

Triangular Bose-Hubbard trimer as a minimal model for a superfluid circuit

H =

M∑j=1

[U

2a†ja†jajaj −

K

2

(ei(Φ/M)a†j+1aj + e−i(Φ/M)a†jaj+1

)], M=3, u≡

NU

K

−1 −0.5 0 0.5 1−40

−20

0

20

40

I [scaled]

E

−1 −0.5 0 0.5 1−40

−20

0

20

40

60

I [scaled]

E

−1 −0.5 0 0.5 1−20

0

20

40

60

80

I [scaled]

E

−1 −0.5 0 0.5 10

50

100

150

200

I [scaled]

E

0I [scaled]

E

2NMott regime

1

2

ππ/2

(c) (a)

(b)(d)

Φ

u

self trapping

0

(d) (b)

(c) (a)

Φ

u

0.5 1 1.5 2 2.5 3

1

2

3

4

Φ

u

0.5 1 1.5 2 2.5 3

1

2

3

4

The BHH for a dimer is like a pendulum

H =U

2

∑i=1,2

ni(ni − 1) −K

2(a

†2a1 + a

†1a2), ni ≡ a†i ai, n ≡

1

2(n1 − n2) , u ≡

NU

K

N particles in a double well

is like spin j = N/2 system

H = UJ2z − KJx

Jz = occupation difference

KU

Analogous to Josephson junction

if the occupation difference N/2

H(n, ϕ) = Un2 −

NK

2cos(ϕ)

n = occupation difference

Pendulum physics - take home messages

ground-state preparation - all bosons are condensed in the lower orbital

π preparation - all bosons are condensed in the upper orbital

Edge preparation - coherent preparation with the same energy as π preparation

• The ”π” and the Edge preparations are classically unstable: irreversible decay

• Quantum mechanically: Edge preparation is quasi-irreversible trecurrances ∼√N

• Quantum mechanically: π preparation is quasi-periodic trecurrances ∼ log(N)

• You can stabilize the π preparation using high frequency periodic driving (Kapitza)

• You can stabilize the π preparation using noisy driving (Zeno)

• “Quantum Zeno effect” is in fact a classical effect

• Beyond Fermi-golden-rule: log-normal statistics of one-body decoherence

Irreversible decay or Quasi-periodic oscillations?

The initial wave-packet should overlap a continuum of eigen-states, else quasi-periodic oscillation

instead of decay. Some cases where there is no continuum contrary to semi-classical intuition:

• Anderson localization

• Small PN at the top of a barrier [with Chuchem, Khripkov, Vardi]

PN ≈

√u elliptic fixed point√u log(N/u) saddle point√N log(N/u) separatrix edge

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|S|

φ=πφ=4π/5φ=3π/5φ=2π/5φ=π/5φ=0

time [Josephson periods]

• Scars at hyperbolic points [Kaplan, Heller]

PN ≈

∞∑s=−∞

1

cosh(λs)

−1

N

2hyperbolic fixed point

How to stabilize the π preparation?

H = H0 + f(t)W

Options:

• Introduce periodic driving f(t) ∝ sin(Ωt)

• Introduce noisy driving f(t)f(t′) = 2Dδ(t− t′)

• Watch the pendulum...

successive collapses of the wavefunction...

quantum Zeno effect

To watch the pendulum is formally like introducing noise.

HBHH = UJ2z − (K + f(t))Jx

HJJ = Un2 −N

2(K + f(t)) cos(ϕ)

Kapitza physics in spherical phase-space

dρ

dt= −i[H0 + f(t)W, ρ]

f(t) = sin(Ωt)

For low Ω we get chaos, but for high frequency driving...

3 iterations and averaging over a cycle ;

dρ

dt= −i[H+ V eff, ρ]

V eff = −1

4Ω2[W, [W,H]]

[Boukobza, Moore, Cohen, Vardi (PRL 2010)]

Note: the standard Kapitza analysis assumes canonical phase space with V (ϕ) perturbation

Zeno effect

H = H0 + f(t)W, W = Jx

Noisy driving f(t)f(t′) = 2Dδ(t− t′)

2 iterations and averaging over realizations ;

dρ

dt= −i[H0, ρ]−D[W, [W,ρ]]

Noise induced radial diffusion:

Dw =w2

J

8D

|S| = exp

−

1

N[exp (8Dwt)− 1]

Should be contrasted with

|S| = exp

−

1

N8Dwt

[Khripkov, Vardi, Cohen (PRA 2012)]

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|S|

time [Josephson periods]

[Simulation: ”csr movie noise long.mpg”]

Log-normal spreading is induced

due to stretching/squeezing operations:

rt = λt ... λ2 λ1 r0

In FGR it is like setting λ = 1 + ε

rt = (1 + ε1 + ε2 + ...εt) r0

Thermalization of mesoscopic subsystems

Minimal Fokker-Planck theory for the thermalization of mesoscopic subsystems:

∂ρ

∂t=

∂

∂ε

(g(ε)Dε

∂

∂ε

(1

g(ε)ρ

))g(ε) = g1(ε) g2(E − ε)

A(ε) = ∂εDε + (β1 − β2)Dε Dε =

∫ ∞0

dω

2πω2 S(1)(ω) S(2)(ω)

Complexity of phase space might affect the thermalization.

BEC trimer: long dwell times in sticky regions are reflected in ε(t)

[Tikhonenkov, Vardi, Anglin, Cohen (PRL 2012)]