Optimal Laser-Control of Two-Dimensional Nanodevices

Optimal Laser-Control of Two-Dimensional Nanodevices 1,2

E. R¨ as¨ anen

1

, A.

1,2 Castro ,

3,1,2 Werschnik ,

J.

A.

4,1,2 Rubio ,

and E. K. U.

Institut f¨ ur Theoretische Physik, Freie Universit¨ at Berlin, Germany 2 European Theoretical Spectroscopy Facility (ETSF) 3 JENOPTIK Laser, Optik, Systeme GmbH, Jena, Germany 4 Universidad del Pa´ıs Vasco, DIPC, Donostia-San Sebasti´ an, Spain

Introduction

Controlling a double quantum dot

l =0

200 0.993

100

0.978

0

0.975 0.88

initial CW

• Only transitions ij → (i ± 1)j are controllable i.e., between the red lines shown in Fig. 6. Note that in the harmonic-oscillator limits (d = 0 and d → ∞) the system is always uncontrollable.

50 iteration

1.5

2 1

0.5 1 t [ps]

21

• Considering the required pulse lengths for desired occupation accuracies, OCT is superior to the CW approach (see Fig. 3). CW OCT

d

0.5

0

(a)

x 10

x

0

1

3

pulse length [ps]

(b) l = −1

5

4.5

A [V/m] occupation

(b)

(a)

0 −5 1

4

l =0

Vext(r) t

ε (t)

Ej

j

3.5

3

3

2 1

60

1

R

0

l = −1

l =1

l =0

0.6 0.4

1

2 t [ps]

3

|L

|R L

20 ω [THz]

30

0 0

1

2

3 4 t [ps]

5

(d)

(c)

L

R t=1.16ps

t=0 (f)

(e)

t=3.49ps

t=2.33ps

l =0

5

εx

10

l =1

x10 1

εy

0.8

0 0

50

00

−5

l =2

l = −1

0.2 4

l = −2

l =1

4

with ω0 = 10 meV, V0 = 200 meV, and a = 10 nm, leading to a ring radius of r0 = 22 nm (effective-mass approximation for electrons in GaAs with m∗ = 0.067me and κ = 12.7ǫ0).

40

10

• We can flip the electron currents in multilevel transitions l ⇌ −l. The corresponding magnetic switch driven by the induced field can be used to change the spin state of a subsystem placed in the middle of the ring [2].

x10 5

30 d [nm]

(b)

0 1 2 3 4 5 t [ps]

(5)

20

occupation

0.9

A [V/m]

1 2 1 2 2 −r2/d2 ˆ H0 = − ∇ + ω0 r + V0e 2 2

10

5

occupation

where ǫ(t) = (ǫx(t), ǫy (t)) is the two-component laser field propagating in z direction, µˆ = −r, and

ij = 00

4

0.99

FIG. 3: Maximum occupation of the target state in transition l = 0 → −1 as a function of the pulse length (fixed initially). The insets show examples of the target-state densities when T = 6 ps.

• The Hamiltonian of our 2D model system confined in the xy plane is written as ˆ =H ˆ 0 − µǫ(t), ˆ H (4)

ground state controllable non−controllable

FIG. 6: Shape of the external potential of a double quantum dot, and the lowest eigenenergies as a function of the interdot distance. The right panel shows densities of six lowest eigenstates at d = 50 nm.

0

Controlling a quantum ring

11

• We are able to optimize the pulse driving the electron from the other well (L) into another (R). The picosecond charge switch is insensitive to d and to anharmonicities in the external potential.

0.999

• The control equations (1-3) can be solved iteratively, and they converge monotonically towards the optimal laser pulse ǫ(t) [4, 5]. In the numerical calculations we have employed the octopus code [6].

20

1

10

ε (ω)

ˆ is the dipole operator, where χ(t) is the Lagrange multiplier, µ and α(t) = α0f (t) restricts the pulse with a constant penalty factor α0 and with an envelope function f (t).

0.9999

Occupation

(3)

40

30

0

1.5

FIG. 2: Laser pulses (x components) during the iterative solution of the control equations (1-3) for the transition l = 0 → −1 in a quantum ring. The initial pulse (zeroth iteration) is a continuous wave, and the optimal pulse is achieved after 200 iterations. The red numbers mark the target-state occupations achieved by the pulses.

(1) (2)

31

Vext

0.99999

ˆ i∂tΨ(t) = HΨ(t), Ψ(0) = Φi, ˆ i∂tχ(t) = Hχ(t), χ(T ) = Φf hΦf |Ψ(T )i , 1 ˆ ǫ(t) = − Im hχ(t)|µ|Ψ(t)i , α(t)

22

ij

−1

10

where ω0 = 0.5 a.u. ≈ 5.6 meV is the confinement strength and d is the distance between the two dots.

x

• Maximizing the overlap | hΨ(T )|ΦFi |2 and minimizing the fluence (time-integrated intensity) of the laser pulse leads, together with the time-dependent Schr¨odinger equation, to the control equations [2]

5

ε

target state |Φf in a finite time interval T .

#

d 2 d 2 2 2 min (x − √ ) + y , x + (y − √ ) . (6) Vext(x, y) = 2 2 2 2 2

0.996

0.986

Optimal control theory

E

"

ε [a.u.]

x 10 1

0

• OCT is a powerful ǫ(t) driving E tool to find optimal laser pulses E E the state |Ψ(t) from a given initial state |Φi = |Ψt=0 to a

ω02

0.995

x

• Here we show that using optimal control theory (OCT) [1] one can construct coherent two-state quantum switches out of quantum rings [2] and double quantum dots [3], respectively.

• The external potential describing the DQD is given by

optimal pulse

ε [V/m]

• Two-dimensional nanodevices such as semiconductor quantum dots and rings are promising candidates for these applications due to their high flexibility in size, shape, and number of confined electrons.

l =1

l = −1

[V/m]

• One of the fundamental aims in coherent quantum control is to construct tailored laser pulses that achieve logic operations for quantum computation.

(a)

1,2 Gross

(h)

(g)

0 −1 1

0.8

l = −2

0.6

t=4.66ps

2

−1 0 1

FIG. 7: Shape (inset) and spectrum (a) of the optimized pulse and the occu  pations (b) in the transport process |L → |R in a fixed time T = 5.82 ps. (c-h) Snapshots of the total electron density during the process.

0.4 0.2 0 0

1

2 t [ps]

t=5.82ps

3

4

FIG. 4: Optimized laser pulses for transitions l = −1 → 1 (a) and l = −2 → 2 (b), and the occupations of the states involved in the transitions.

• The frequencies (∼ 1012 Hz) and intensities (∼ 1012 W/cm2) are already accessible experimentally [7], and precise pulse shaping could be achieved applying quantum cascade lasers [8].

References

−4 −3 −2 −1 0 1 2 3 4

l

FIG. 1: (a) Shape of the external confining potential for a quantum ring and an example of a circularly polarized laser field. (b) Energy-level spectrum of a quantum ring. The transitions are allowed along the dashed line so that ∆l = ±1.

• The initial pulse for transition l = 0 → −1 is a circularly polarl=±1 ized continuous wave (CW) having a resonant frequency ωl=0 and amplitude A = ΩR/µ = π/µT , where ΩR is the Rabi frequency (π-pulse condition). The CW yields | hΨ(T )|ΦFi |2 ≈ 0.88. • The optimized pulses lead to occupations close to one after only a few iterations (see Fig. 2).

[1] A. P. Peirce, M. A. Dahleh, and H. Rabitz, Phys. Rev. A 37, 4950 (1988); R. Kosloff, S. A. Rice, P. Gaspard, S. Tersigni, and D. J. Tannor, Chem. Phys. 139, 201 (1989).

t=0

t=0.6 ps

t=1.2 ps

[2] E. R¨as¨anen, A. Castro, J. Werschnik, A. Rubio, and E. K. U. Gross, Phys. Rev. Lett. 98, 157404 (2007). [3] E. R¨as¨anen, A. Castro, J. Werschnik, A. Rubio, and E. K. U. Gross, submitted, http://arxiv.org/abs/0707.0179. [4] W. Zhu and H. Rabitz, J. Chem. Phys. 109, 385 (1998).

t=1.6 ps

t=2.2 ps

t=3.2 ps

FIG. 5: Time-dependent electron density |Ψ(r, t)|2 (yellow: low, black: high) and the current j(r, t) (arrows) at different times in the transition l = −1 → 1 driven by the optimized pulse [see Fig. 4(a)].

[5] J. Werschnik, Quantum Optimal Control Theory: Filter Techniques, Time-Dependent Targets, and Time-Dependent Density-Functional Theory, (Cuvillier, G¨ottingen, 2006). [6] A. Castro et al., Phys. Stat. Sol. (b) 243, 2465 (2006). [7] M. Tonouchi, Nature Photonics 1, 97 (2007). [8] F. Eickemeyer et al., Phys. Rev. Lett. 89, 047402 (2002).