Quantum Tunneling and Decoherence in Coherently Driven Double-Well - - PDF document

Quantum Tunneling and Decoherence in Coherently Driven Double-Well Potential

Akira Igarashi1 and Hiroaki S. Yamada2

1 Graduate School of Science and Technology, Niigata University, ikarashi 2-nochou 8050, Niigata 950-2181, Japan email: f99j806b@mail.cc.niigata-u.ac.jp 2 YPRL, 5-7-14, Aoyama, Niigata 950-2002, Japan email: hyamada@uranus.dti.ne.jp

Contents

• 1. Introduction
• 2. Model
• 3. Numerical Results
• 4. Summary and Discussion

1

1 Introduction

We numerically investigate influence of a polychromatic perturbation on wave packet dynamics in one-dimensional double-well potential. Some interesting results are known in monochromatic perturbed double-well system, such as enhancement or suppression of tunneling between the wells, however tunneling dynamics in polychromatically perturbed double well does not seem to be clear. In order to make it clear, we consider tunneling dynamics the one-dimensional double-well driven by coherent external field. Especially we consider the following problem.

• Does coherent tunneling occur under the perturba-

tion ?

• Does one remain against strong perturbation ?
• If the coherence of the tunneling dynamics is de-

stroyed, what kind of motion becomes dominant ? 2

2 Model

The model system is coherently driven one-dimensional double-well whose Hamiltonian is, H(p, q, t) = p2 2 + q4 4 − A(t)q2 2 , (1) A(t) = a − 1 √ M

M

• j=1

ǫi sin(Ωit + θi). (2) Here A(t) describes time-dependent perturbation and,

• q, p: Canonically conjugate position and momentum.
• a: The parameter determining the distance between

the potential well.

• M: The number of frequency component.
• {ǫi}: The i’s perturbation strength.
• {Ωi}: Mutually incommensurate and oder of unity

frequencies of the external perturbation.

• {θi}: The initial phases of the external driving force.

To make some energy doublets and for simplicity we set a = 5, ǫi = ǫ = 0.1 ∼ 1.0, and θi’s are random numbers. We take off-resonant frequencies which are far from resonance in the corresponding unperturbed problem in order to avoid energy absorption due to resonance. 3

Time dependence of the potential, the initial wave packet, and some lowest unperturbed eigen energy levels are shown

• Fig. 1.
• 8
• 6
• 4
• 2

2 4 6 8

• 3
• 2
• 1

1 2 3 V(q,t) q t=0 t=π/2Ω1 t=3π/2Ω1

• Fig. 1: The sketch of the potential at some mo-

ments, the initial packet and some eigen energy.

The classical phase space structure of this system are shown below.

5 4 3 2 1

• 1
• 2
• 3
• 4
• 5
• 4
• 3
• 2
• 1

1 2 3 4

p q (c) M=10, =0.6

8 6 4 2

• 2
• 4
• 6
• 8
• 4
• 3
• 2
• 1

1 2 3 4

p q (d) M=10, =1.0

• 1

0.8 0.6 0.4 0.2

• 0.2
• 0.4
• 0.6
• 0.8
• 1

1.9 2 2.1 2.2 2.3 2.4 2.5 2.6

p q (a) M=1, =0.1

• 1

0.8 0.6 0.4 0.2

• 0.2
• 0.4
• 0.6
• 0.8
• 1

1.9 2 2.1 2.2 2.3 2.4 2.5 2.6

p q (b) M=2, =0.1

• Fig. 2: Points are plotted at t = 2πn/Ω1 in each cases.

4

3 Numerical Results

3.1 Tunneling

To calculate wave packet tunneling, we use the following Gaussian localized as the initial state, which is localized in the right well of the potential, ψ(q, t = 0) = exp{−(q − q0)2 2σ }. (3)

• q0: the right bottom of the potential well.
• σ: the initial spread of the packet.

We set σ = 0.3. This initial packet approximates equal- weight linear combination of the unperturbed lowest dou-

• blet. We then define PL(t),

PL(t) ≡

−∞

|ψ(q, t)|2dq (4) which can be interpreted transition probability that the ini- tially localized wave packet goes through the central energy barrier and reach the opposite well. 5

1 0.8 0.6 0.4 0.2

PL(t)

(a)

M=0 M=1 M=5 M=7 M=10 0.8 0.6 0.4 0.2 8000 6000 4000 2000

PL(t) t

(b)

M=1 M=10

• Fig. 3: Tunneling rate PL(t)

6

1 0.8 0.6 0.4 0.2

PL(t)

(a)

=0.1 =0.5 =1.0 0.8 0.6 0.4 0.2

PL(t)

(b)

=0.1 =0.5 =0.7 0.8 0.6 0.4 0.2 8000 6000 4000 2000

PL(t) t

(c)

=0.1 =0.2 =1.0

• Fig. 4: Tunneling rate PL(t)

We see coherent motion, irregular fluctuation and quasi- irregular motion as a intermediate motion. 7

3.1.1 Fourier Transform To evaluate what frequency dominates the time- dependence of PL(t), we calculate and show the Fourier transform I(ω). I(ω) =

• T

PL(t) exp(−iωt)dt

• 2

(5) where T = 9.4 × 103.

0.0e+00 1.0e+06 2.0e+06 3.0e+06 4.0e+06 0.005 0.01 0.015 0.02

I(ω) ω (c) M=5, =0.2

0.0e+00 5.0e+04 1.0e+05 1.5e+05 2.0e+05 0.1 0.2 0.3 0.4 0.5

I(ω) ω (d) M=5, =1.0

0.0e+00 5.0e+04 1.0e+05 1.5e+05 2.0e+05 1 2 3 4 5

I(ω) ω (a) M=2, =0.5

0.0e+00 5.0e+03 1.0e+04 1.5e+04 1 2 3 4 5

I(ω) ω (b) M=10, =1.0

• Fig. 5: Fourier transform of tunneling rate PL(t)

As ǫ and/or M increase, some peaks appear and frequen- cies are distributed with finite-width around the peaks. 8

3.1.2 Quantitative description of perturbation dependence of tunneling rate and the corresponding classical dynam- ics

60 50 40 30 20 10 1 0.8 0.6 0.4 0.2

count PL (c) M=7, =0.6

60 50 40 30 20 10 1 0.8 0.6 0.4 0.2

count PL M=10, =1.0 (d)

20 40 60 80 100 120 1 0.8 0.6 0.4 0.2

count PL (a) M=2, =0.1 (d)

60 50 40 30 20 10 1 0.8 0.6 0.4 0.2

count PL (b) M=3, =0.7 (d)

• Fig. 6: Histograms of PL for some combinations ǫ and M.

From the figures the variance of the value of PL(t) can be a criterion which enables to describe the difference between the coherent and incoherent tunneling dynamics. 9

In order to estimate quantitatively the difference between coherent and incoherent motion, we use the variance of PL(t) as a degree of coherence ∆PL ≡ {(PL(t) − PLT )2T }

1 2 ,

(6) where ... denotes time average for the period T. We show the degree of coherence and the maximum Lya- punov exponent of the corresponding classical system.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

∆ PL

• (a)

2 3 5 6 7 8 10 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

λ

max cl

• (a)

M=1 M=2 M=3 M=4 M=7 M=10

• Fig. 7: Degree of coherence and Lyapunov expo-

nents of the corresponding classical system.

10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10

• M

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10

• M
• Fig. 8: Classification of types of motion in quan-

tum mechanics and corresponding classical Lya- punov exponent.

The classification of the motion suggests the existence of the critical perturbation strength ǫc(M), over which the motion decoheres in quantum mechanics. 11

3.2 Wave Packet Dynamics

Next we consider the expectation value H(t) of the Hamiltonian and the deviation ∆H(t).

• 2
• 4
• 6

〈H(t)〉

(a) =0.4

M=5 M=10 4 2

• 2
• 4
• 6

8000 6000 4000 2000

〈H(t)〉 t

(b) =0.8

M=1 M=10

• Fig. 9: Expectation value of the Hamiltonian

12

• 2
• 3
• 4
• 5
• 6

〈H(t)〉

(a) M=2

=0.1 =0.5 =1.0

• 3
• 4
• 5
• 6

〈H(t)〉

(b) M=5

=0.1 =0.5 8 6 4 2

• 2
• 4
• 6

8000 6000 4000 2000

〈H(t)〉 t

(c) M=10

=0.1 =0.7 =1.0

• Fig. 10: Expectation value of the Hamiltonian

13

1 2 3 4 5

∆H(t)

(a) =0.4

M=5 M=7 M=10 5 4 3 2 1 2000 4000 6000 8000

∆H(t) t

(b) =0.8

M=1 M=3 M=10

• Fig. 11: Standard deviation of the Hamiltonian

14

1.3 1.1 0.9 0.7 0.5

∆H(t)

(a) M=2

=0.1 =1.0 4 3 2 1

∆H(t)

(b) M=5

=0.1 =0.5 =1.0 7 6 5 4 3 2 1 8000 6000 4000 2000

∆H(t) t

(c) M=10

=0.1 =0.7 =1.0

• Fig. 12: Standard deviation of the Hamiltonian

When the perturbation strength ǫ exceeds ǫc(M), activa- tion becomes more dominant than tunneling and the mo- tion changes incoherent one. 15

Next we show the plots (q H), (q p) for some com- bination ǫ and M.

• 7
• 6
• 5
• 4
• 3
• 2
• 1
• 3
• 2
• 1

1 2 3

〈H〉 〈q〉 (c)

• 8
• 6
• 4
• 2

2 4 6 8 10

• 3
• 2
• 1

1 2 3

〈H〉 〈q〉 (d)

• 7
• 6
• 5
• 4
• 3
• 2
• 1
• 3
• 2
• 1

1 2 3

〈H〉 〈q〉 (a)

• 9
• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1
• 3
• 2
• 1

1 2 3

〈H〉 〈q〉 (b)

• 3
• 2
• 1

1 2 3 2 1

• 2
• 1

〈p〉 〈q〉 (c)

• 3
• 2
• 1

1 2 3 2 1

• 2
• 1

〈p〉 〈q〉 (d)

1

• 1

2 1

• 2
• 1

〈p〉 〈q〉 (a)

1

• 1

2 1

• 2
• 1

〈p〉 〈q〉 (b)

16

Next we consider the uncertainty product ∆q∆p to eval- uate how wave packet spreads in the phase space. ∆q∆p ≡

• (q − q)2
• (p − p)2.

(7)

5 3 1 8000 6000 4000 2000

∆q∆p t

(c) M=10, =0.4

8 6 4 2 8000 6000 4000 2000

∆q∆p t

(d) M=10, =1.0

3 2 1

∆q∆p

(a) M=2, =0.1

3 2 1

∆q∆p

(b) M=2, =1.0

• Fig. 13: Uncertainty product.

In the panel (a) and (b) coherent motion occur and the corresponding wave packets coherently tunnels between the

• wells. Coherent oscillations remain even for large perturba-

tion strength. On the other In the incoherent motions (the panel (c) and (d) ), the wave packets spread over the phase space and never return to the initial ones within T. 17

Finally we show contour plots of Hushimi representation ρ(x, p) for some typical cases.

• 8
• 6
• 4
• 2

2 4 6 8

• 4
• 2

2 4

p x

(a) M=1, =1.0, t=0.0

0.1 0.05 0.01

• 8
• 6
• 4
• 2

2 4 6 8

• 4
• 2

2 4

p x

(b) M=1, =1.0, t=9.4×10

3

0.1 0.05 0.01

• 8
• 6
• 4
• 2

2 4 6 8

• 4
• 2

2 4

p x

(c) M=5, =0.7, t=4.0×10

3

0.04 0.03 0.02 0.01

• 8
• 6
• 4
• 2

2 4 6 8

• 4
• 2

2 4

p x

(d) M=5, =0.7, t=7.2×10

3

0.1 0.08 0.06 0.04 0.02

• 8
• 6
• 4
• 2

2 4 6 8

• 4
• 2

2 4

p x

(e) M=10, =1.0, t=1.3×10

2

0.04 0.03 0.02 0.01

• 8
• 6
• 4
• 2

2 4 6 8

• 4
• 2

2 4

p x

(f) M=10, =1.0, t=5.5×10

3

0.04 0.03 0.02 0.01

• Fig. 14: Hushimi representation.

In incoherent motion wave packet spreads over phase space with out symmetry. 18

4

• 4. Summary and Discussion

Summary

• Existence of some types of motion and critical value
• f perturbation strength.
• Perturbation can destroy the coherent tunneling for

a perturbation strength above the critical value.

• Perturbation makes Recurrence time for incoherent

motion much larger. Discussion

• Long time behavior of quasi-irregular motion.
• What is the mechanism of change of wave packet

traveling from tunneling to activation ?

• Can it control somehow (e.g. OCT) ?
• What and how does classical chaos affect to the quan-

tum dynamics (tunneling, coherence) ?

• Semiclassical and semiquantal description of the in-

coherent tunneling

• Relation to stochastic perturbation which corre-

sponds to the limit M → ∞ and the situation without coherence.

• Resonance structure under polychromatic perturba-

tion. 19