Treating Bose-Hubbard model by means of classical ......Treating Bose-Hubbard model by means of classical mechanics or Quantum-classical transition in many-body systems Andrey R. Kolovsky

Treating Bose-Hubbard model by means of classical

mechanicsor

Quantum-classical transition in many-body systems

Andrey R. Kolovsky

Kirensky Institute of Physics, 660036 Krasnoyarsk, RussiaSiberian Federal University, 660041 Krasnoyarsk, Russia

Dresden, February 2016

Andrey R. Kolovsky (Kirensky Institute of Physics, 660036 Krasnoyarsk, Russia Siberian Federal University, 660041 Krasnoyarsk, Russia)Treating Bose-Hubbard model by means of classical mechanicsor Quantum-classical transitionDresden, February 2016 1 / 19

Quantum chaos

Main conjecture: Energy spectrum of a quantum system with underlying chaoticclassical dynamics has the same statistical properties as spectrum of the randommatrix (of the same universality class).

Distribution of normalized distances between the nearest levels

s = (Ej+1 − Ej)ρ(Ej) – normalized distances

P(s) = π

2 s exp(−π

4 s2)

– chaotic systems

P(s) = e−s – regular (integrable) systems

Integrated distribution I (s) =∫ s

0 P(s ′)ds ′

[1] A.R.Kolovsky and A.Buchleitner, Quantum chaos in Bose-Hubbard model, Europhys. Lett. 68, 632 (2004).

[2] A.R.Kolovsky, Persistent current of atoms in a ring optical lattice, New J. of Phys. 8, 197 (2006).


Bose-Hubbard model

H = E0

L∑

l=1

nl −J

2

L∑

l=1

(a†l+1al + h.c .

)+

U

2

L∑

l=1

nl (nl − 1) , nl = a†l al

−2 −1 0 1 2

0

1

2

3

4

5

x/d

V(x

)l=1 l=L

φl(x)

Figure: Cold atoms (open circles) in an optical lattice. Green line is thesingle-particle Wannier function.


Bose-Hubbard model

H = −J

2

L∑

l=1

(a†l+1al + h.c .

)+

U

2

L∑

l=1

nl (nl − 1)

Fock basis: |n〉 = |n1, n2, . . . , nL〉 where∑

l nl = N

Canonical transformation: bk = (1/√

L)∑

l exp(i2πkl/L)al

H = −J∑

k

cos

(2πk

L

)b†k bk +

U

2L

∑

k1,k2,k3,k4

b†k1

bk2 b†k3

bk4 δ(k1 − k2 + k3 − k4)


Transition to chaos

H = −J

2

L∑

l=1

(a†l+1al + h.c .

)+

U

2

L∑

l=1

nl(nl − 1) +L∑

l=1

ǫl nl

−10 0 100

0.2

f(E

)

(a)

−10 0 100

0.2

(b)

0 1 2 3 40

1

s

I(s)

(c)

a

b

Figure: Density of states (upper panels) and integrated level spacing distribution(lower panel) as compared to the Poisson and Wigner-Dyson distributions.Parameters are N = 7, L = 9, J = 1, ǫ = 0.2, U = 0.02 (a) and U = 0.2 (b).


Classical limit

H = −J

2

L∑

l=1

(a∗l+1al + c .c .) +g

2

L∑

l=1

|al |4 , g =UN

L

Coherent SU(L) states: |a〉 = 1√N!

(∑Ll=1 al a

†l

)N

|vac〉

Equation on the Husimi function f (a, t) = |〈a|Ψ(t)〉|2∂f

∂t= H , f + O

(1

N

)

If the initial distribution f (a, t = 0) is a δ-function then the above equation isequivalent to DNLSE equation:

i∂al

∂t= −J

2(al+1 + al−1) + g |al |2al

[3] F. Trimborn, D. Witthaut, and H. J. Korsch, Exact number conserving phase-space dynamics of the L-site Bose-Hubbard

model, Phys. Rev. A 77, 043631 (2008)


Phase space and semiclassical quantization

H = −J

2

L∑

l=1

(a∗l+1al + c .c .) +g

2

L∑

l=1

|al |4 ,

L∑

l=1

|al |2 = 1

ρqu(E ) ≈ N (N)

Nρcl

(E

N

),

∫ρcl(E )dE = 1


Density of states

−20 0 20 40 600

100

200

300

(a)

−20 0 20 40 600

100

200

300

(b)

−20 0 20 40 600

100

200

300

E−Eint

ρ

(c)

−1 0 1 2 30

0.15

(d)

−1 0 1 2 30

0.15

(e)

−1 0 1 2 30

0.15

E−Eint

ρ

(f)

Figure: Density of states of the 5-site BH model for N = 20, panels (a-c), ascompared to the classical ‘density of states’, panels (d-f). The energy is measuredwith respect to the mean interaction energy Eint = gN .


Low-energy stability islands

Relative volumes of the regular and chaotic components?

vreg = vreg (E ) → 1 if E → Emin ≈ −J

Effective Hamiltonians:

Heff = (δk + g)I + g√

I 2 − 4M2 cos(2θ) , |M | ≤ I/2 , δk = J[1 − cos(2πk/L)]

Quantizing these effective Hamiltonians we have

E (k) = E(k)0 + ~Ω(k)(n + 1/2) , Ω(k) ∼ √

gk

which is nothing else as the Bogoliubov spectrum for low-energy excitations of aBEC.

[4] A.R.Kolovsky, Semiclassical quantization of the Bogoliubov spectrum, Phys. Rev. Lett. 99, 020401 (2007); Phys. Rev. E

76, 026207 (2007)


Bogoliubov spectrum

0 1 2 3 40

4

8

12

16

20

g

(E−

E0)/

Ω

0 1 2 3 40

8

16

24

32

40

g

(E−

E0)/

Ω

Figure: Energy spectrum of the 3-site BH model for N = 20 (left) and N = 40(right) as the function of macroscopic interaction constant g = UN/L. Theenergy is measured with respect to the ground energy and is scaled according tothe Bogoliubov frequency Ω(g).


Bloch oscillations

H = −J

2

∑

l

(a†l+1al + h.c .

)+

U

2

∑

l

nl (nl − 1)+dF∑

l

l nl

H(t) = −J

2

∑

l

(a†l+1ale

−iωBt + h.c .)

+U

2

∑

l

nl(nl − 1) , ωB =dF

~


Semiclassical approach

i al = −J

2

(al+1e

iωBt + al−1e−iωBt

)+ g |al |2al

Solution for the uniform initial condition

al(t) =1√L

exp

(iJ

Fsin(ωBt) − igt

), p(t) = p0 sin(ωBt)

−1

0

1

−1

0

1−1

0

1

a1

a2

a 3

−1

0

1

−1

0

1−1

0

1

a1

a2

a 3


Stability analysis

Fcr ≈

3g , F < 2J√10gJ , F > 2J

.

log(F/g)

log(

J/g)

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

log(F/g)

log(

J/g)

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure: Increment of the dynamical instability (sum of the positive Lyapunovexponents) for L = 3 (left) and L = 15 (right).

[6] A.R.Kolovsky, e-print: cond-mat/0412195 (2004).

[7] Yi Zheng, M. Kostrun, and J. Javanainen, Phys. Rev. Lett. 93 230401 (2004).


Strong vs. weak field regime

p(t) =

exp(−γt) sin(ωBt) , F ≪ Fcr

exp (−2n[1 − cos(Ut/~)]) sin(ωBt) , F ≫ Fcr

0 40 80 120−1

0

1

(b)

0 40 80 120−1

0

1

t/TB

p

(d)

Figure: Numerical simulations of Bloch oscillations of interacting atoms forF < Fcr (upper panel) and F > Fcr (lower panel).


Strong field – quasiperiodic Bloch oscillations

p(t) = exp (−2n[1 − cos(Ut/~)]) sin(ωBt)

Figure: Dynamics of the mean momentum for F > Fcr according to Ref. [11].

[9] A.R.Kolovsky, New Bloch period for interacting cold atoms in 1D optical lattices, Phys. Rev. Lett. 90, 213002 (2003).

[11] F.Meinert, M.J.Mark, E.Kirilov, K.Lauber, P.Weinmann, M.Grobner, and H.-C. Nagerl, Interaction-induced quantum phase

revivals and evidence for the transition to the quantum chaotic regime in 1D atomic Bloch oscillations, (2013).


Weak field – decaying Bloch oscillations

p(t) = exp(−γt) sin(ωBt) , γ ∼ n2U2

Figure: Dynamics of the mean momentum for F < Fcr according to Ref. [11].

[10] A.Buchleitner and A.R.Kolovsky, Interaction-induced decoherence of atomic Bloch oscillations, Phys. Rev. Lett. 91, 253002

(2003).

[11] F.Meinert, M.J.Mark, E.Kirilov, K.Lauber, P.Weinmann, M.Grobner, and H.-C. Nagerl, Interaction-induced quantum phase

revivals and evidence for the transition to the quantum chaotic regime in 1D atomic Bloch oscillations, (2013).Andrey R. Kolovsky (Kirensky Institute of Physics, 660036 Krasnoyarsk, Russia Siberian Federal University, 660041 Krasnoyarsk, Russia)Treating Bose-Hubbard model by means of classical mechanicsor Quantum-classical transitionDresden, February 2016 16 / 19

Quantum ensemble

∂f

∂t= H , f + O

(1

N

), f (a, t = 0) = |〈a|BEC 〉|2

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

n/nbar

P(n

)

−1 −0.5 0 0.5 10

0.05

0.1

0.15

0.2

phase/π

P(p

hase

)

0.5

1

1.5

2

30

210

60

240

90

270

120

300

150

330

180 0

0 1 2 30

0.1

0.2

0.3

0.4

n/nbar

P(n

)

−0.5 0 0.50

0.1

0.2

0.3

0.4

phase/π

P(p

hase

)

Figure: Ensemble of classical trajectories in the Bloch (amplitudes bk 6=0) andWannier (amplitudes al ) representations. Parameters are L = 5 and N = 20.

[8] A.R.K., H.J.Korsch, and E.M.Graefe, Bloch oscillations of Bose-Einstein condensates: Quantum counterpart of dynamical

instability, Phys. Rev. A 80, 023617 (2009).


Classical vs. quantum dynamics: internal decoherence

∂f

∂t= H , f + O

(1

N

), f (a, t = 0) = |〈a|BEC 〉|2

0 40 80 120−1

0

1

t/TB

p

(c)

0 40 80 120−1

0

1

(a)

0 40 80 120−1

0

1

(b)

0 40 80 120−1

0

1

t/TB

p

(d)

Figure: Dynamics of the mean momentum for dF/J = 0.1 and dF/J = 10,calculated by using the classical (left) and quantum (right) approaches.


Conclusions

We addressed the energy spectrum of the Bose-Hubbard model by using”semiclassical” approach based on classical Hamiltonian.

In particular, we obtained the Bogoliubov spectrum for low-energyexcitations of a BEC by quantizing classical tori where ~eff = 1/N .

We addressed dynamics of the Bose-Hubbard system induced by a staticfield (Bloch oscillations).

Using classical analysis we predicted two qualitatively different regimes ofBloch oscillations which have been observed in the laboratory experiment.

Remarkably, regime of decaying Bloch oscillations is perfectly reproduced bypure classical dynamics. Here we have a loop: underlying classical chaos →quantum chaos → internal decoherence → classical dynamics.

[12] A.R.Kolovsky, Bose-Hubbard Hamiltonian: Quantum Chaos approach , Int. J. of Modern Physics B 30 (2016), 1630009

(2016) [arXiv:1507.0341 (2015)]


Documents

Treating Bose-Hubbard model by means of classical ......Treating Bose-Hubbard model by means of classical mechanics or Quantum-classical transition in many-body systems Andrey R. Kolovsky