^{1}, Liang Huang

^{1,2}, Ying-Cheng Lai

^{1,3,4}, Celso Grebogi

^{4}and Louis M. Pecora

^{5}

### Abstract

Chaos has long been recognized to be generally advantageous from the perspective of control. In particular, the infinite number of unstable periodic orbits embedded in a chaotic set and the intrinsically sensitive dependence on initial conditions imply that a chaotic system can be controlled to a desirable state by using small perturbations. Investigation of chaos control, however, was largely limited to nonlinear dynamical systems in the classical realm. In this paper, we show that chaos may be used to modulate or harness quantum mechanical systems. To be concrete, we focus on quantum transport through nanostructures, a problem of considerable interest in nanoscience, where a key feature is conductance fluctuations. We articulate and demonstrate that chaos, more specifically transient chaos, can be effective in modulating the conductance-fluctuation patterns. Experimentally, this can be achieved by applying an external gate voltage in a device of suitable geometry to generate classically inaccessible potential barriers. Adjusting the gate voltage allows the characteristics of the dynamical invariant set responsible for transient chaos to be varied in a desirable manner which, in turn, can induce continuous changes in the statistical characteristics of the quantum conductance-fluctuation pattern. To understand the physical mechanism of our scheme, we develop a theory based on analyzing the spectrum of the generalized non-Hermitian Hamiltonian that includes the effect of leads, or electronic waveguides, as self-energy terms. As the escape rate of the underlying non-attracting chaotic set is increased, the imaginary part of the complex eigenenergy becomes increasingly large so that pointer states are more difficult to form, making smoother the conductance-fluctuation pattern.

Controlling quantum-mechanical systems is generally a challenging problem in science and engineering. In this paper, we exploit the idea that chaos may be used to harness certain statistical features of quantum transportdynamics. While previous works elucidated the basic physics underlying the effect of chaos on quantum transport, we propose a scheme that can be implemented experimentally to systematically harness conductance fluctuations associated with quantum transport through nanostructures. Our idea can be illustrated by using a Sinai-type of open billiard quantum dot(QD), where a central circular region forbidden to classical trajectories can be generated by applying a relatively high gate voltage, and the size of the region can be controlled in a continuous manner. Since the system is open, chaos in the classical limit must be transient. We demonstrate for both non-relativistic (semiconductor two-dimensional electron gas(2DEG)) and relativistic (graphene) quantum-dot systems that, when the radius of the central potential region is varied so that the characteristics of the corresponding classical chaotic dynamics are modified, the quantum conductance-fluctuation patterns can be effectively modulated. While a semiclassical argument based on previous works on quantum chaotic scattering can be used to explain qualitatively the role of classical transient chaos of different dynamical characteristics in affecting the conductance fluctuations, we develop a formal theory based on the concept of self-energies and the complex eigenvalue spectrum of the corresponding generalized non-Hermitian Hamiltonian. The emergence of narrow resonances can be related to the magnitude of the imaginary part of the eigenvalues, and we obtain an explicit formula predicting the form of the narrow resonance. Our theory indicates that the role of continuously varying chaos in the classical limit lies in removing successively the eigenvalues with extremely small imaginary parts. Our results suggest a viable way to harness quantum behaviors of nanostructures, which can be of interest due to the feasibility of experimental implementation of our scheme.

We thank P.-P. Li for assisting in generating Fig. 4. This work was supported by AFOSR under Grant No. FA9550-12-1-0095 and by ONR under Grant No. N00014-08-1-0627. L.H. was also supported by NSFC under Grant No. 11005053.

I. INTRODUCTION

II. PROPOSED EXPERIMENTAL SCHEME AND COMPUTATIONAL METHOD

III. NUMERICAL RESULTS

IV. THEORY

V. CONCLUSIONS

### Key Topics

- Chaos
- 49.0
- Quantum dots
- 37.0
- Quantum chaos
- 25.0
- Quantum transport
- 20.0
- Quantum fluctuations
- 17.0

## Figures

Schematic illustration of a possible experimental scheme to harness transport through a semiconductor 2DEG quantum-dot system, where 2DEG is formed at the GaAs/Al_{0.3}Ga_{0.7}As hetero-interface. The heterostructure sits on a substrate (purple), covered by 300 nm SiO_{2} (blue) and contacted by Au/Cr (yellow). By applying a suitable gate voltage to generate a *circular forbidden region* (for classical orbits) at the center of the device, the resulting closed system is a Sinai billiard. Open quantum-dot system can be formed by attaching leads to the billiard system. In this paper we place the leads in the middle of the dot, as shown in the second row. Similar idea can be applied to graphene systems.

Schematic illustration of a possible experimental scheme to harness transport through a semiconductor 2DEG quantum-dot system, where 2DEG is formed at the GaAs/Al_{0.3}Ga_{0.7}As hetero-interface. The heterostructure sits on a substrate (purple), covered by 300 nm SiO_{2} (blue) and contacted by Au/Cr (yellow). By applying a suitable gate voltage to generate a *circular forbidden region* (for classical orbits) at the center of the device, the resulting closed system is a Sinai billiard. Open quantum-dot system can be formed by attaching leads to the billiard system. In this paper we place the leads in the middle of the dot, as shown in the second row. Similar idea can be applied to graphene systems.

(a) Conductance versus Fermi energy for four semiconductor 2DEG quantum-dot systems (bottom to top): rectangular dot, rectangular dot with a rectangular forbidden region of area , and Sinai dots of radii and . The area of the original rectangular dot is and the lead width is . (b) Conductance fluctuation patterns for graphene quantum dots of the same geometry as in (a). ^{ 20 } The curves in (a) and (b) have been shifted vertically by some arbitrary values to facilitate visualization. (c) Normalized number of *sharp resonances*, which is defined (see text) to quantify the degree of transmission fluctuations, corresponding to the four cases in (a), i.e., the top two rectangular symbols represent two rectangular quantum dots while the bottom two circles correspond to two Sinai quantum dots.

(a) Conductance versus Fermi energy for four semiconductor 2DEG quantum-dot systems (bottom to top): rectangular dot, rectangular dot with a rectangular forbidden region of area , and Sinai dots of radii and . The area of the original rectangular dot is and the lead width is . (b) Conductance fluctuation patterns for graphene quantum dots of the same geometry as in (a). ^{ 20 } The curves in (a) and (b) have been shifted vertically by some arbitrary values to facilitate visualization. (c) Normalized number of *sharp resonances*, which is defined (see text) to quantify the degree of transmission fluctuations, corresponding to the four cases in (a), i.e., the top two rectangular symbols represent two rectangular quantum dots while the bottom two circles correspond to two Sinai quantum dots.

Normalized number of sharp resonances versus radius of the central circular forbidden region [Fig. 1 ]. The area for original rectangular quantum dot is .

Normalized number of sharp resonances versus radius of the central circular forbidden region [Fig. 1 ]. The area for original rectangular quantum dot is .

Contour plot of escape rate in the parameter plane of *W* and *R*. For each parameter combination, 10^{6} random particles, each of unit velocity, are used to calculate the escape rate. For each particle, its path length in the scattering region (the Sinai billiard region) is calculated. The distribution of the path length is observed to decay exponentially, and the escape rate is the exponential rate.

Contour plot of escape rate in the parameter plane of *W* and *R*. For each parameter combination, 10^{6} random particles, each of unit velocity, are used to calculate the escape rate. For each particle, its path length in the scattering region (the Sinai billiard region) is calculated. The distribution of the path length is observed to decay exponentially, and the escape rate is the exponential rate.

(a) Conductance versus energy for one small quantum dot of area , where the number of discrete points used in the simulation is 218; (b) the corresponding imaginary parts versus real parts of eigenenergy of (cross), calculated from Eq. (13) (square) and Eq. (16) (circle). The Fermi energy is , as indicated by the arrow.

(a) Conductance versus energy for one small quantum dot of area , where the number of discrete points used in the simulation is 218; (b) the corresponding imaginary parts versus real parts of eigenenergy of (cross), calculated from Eq. (13) (square) and Eq. (16) (circle). The Fermi energy is , as indicated by the arrow.

Imaginary and real parts of the eigenenergies for (a) rectangular quantum dot, (b) rectangular dot with a rectangular forbidden region at the center, (c) Sinai quantum dot with , and (d) Sinai dot with , where for all cases. Each eigenenergy is represented by one blue circle. The red dashed lines indicate and they are just for eye guidance.

Imaginary and real parts of the eigenenergies for (a) rectangular quantum dot, (b) rectangular dot with a rectangular forbidden region at the center, (c) Sinai quantum dot with , and (d) Sinai dot with , where for all cases. Each eigenenergy is represented by one blue circle. The red dashed lines indicate and they are just for eye guidance.

Typical quantum pointer states for (a) rectangular quantum dot, (b)rectangular quantum dot with a rectangular forbidden region, (c) and (d)Sinai quantum dots of radii and , respectively. Darker regions indicate higher values of LDS. The color scale has been normalized for each panel for better visualization. The mean value of LDS in the leads are 0.0154, 0.0351, 0.0491, 0.0669 for panels (a)-(d), respectively. We observe the presence of pronounced pointer states in the two integrable cases.

Typical quantum pointer states for (a) rectangular quantum dot, (b)rectangular quantum dot with a rectangular forbidden region, (c) and (d)Sinai quantum dots of radii and , respectively. Darker regions indicate higher values of LDS. The color scale has been normalized for each panel for better visualization. The mean value of LDS in the leads are 0.0154, 0.0351, 0.0491, 0.0669 for panels (a)-(d), respectively. We observe the presence of pronounced pointer states in the two integrable cases.

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