Quantum simulation of the Anderson Hamiltonian with an array of coupled nanoresonators: delocalization and thermalization effects
- John Lozada-Vera^{1},
- Alejandro Carrillo^{1},
- Olimpio P de Sá Neto^{2},
- Jalil K Moqadam^{1},
- Matthew D LaHaye^{3} and
- Marcos C de Oliveira^{1}Email author
https://doi.org/10.1140/epjqt/s40507-016-0047-3
© Lozada-Vera et al. 2016
Received: 16 December 2015
Accepted: 30 May 2016
Published: 8 June 2016
Abstract
The possibility of using nanoelectromechanical systems as a simulation tool for quantum many-body effects is explored. It is demonstrated that an array of electrostatically coupled nanoresonators can effectively simulate the Bose-Hubbard model without interactions, corresponding in the single-phonon regime to the Anderson tight-binding model. Employing a density matrix formalism for the system coupled to a bosonic thermal bath, we study the interplay between disorder and thermalization, focusing on the delocalization process. It is found that the phonon population remains localized for a long time at low enough temperatures; with increasing temperatures the localization is rapidly lost due to thermal pumping of excitations into the array, producing in the equilibrium a fully thermalized system. Finally, we consider a possible experimental design to measure the phonon population in the array by means of a superconducting transmon qubit coupled to individual nanoresonators. We also consider the possibility of using the proposed quantum simulator for realizing continuous-time quantum walks.
Keywords
1 Introduction
The achievement of the ground state of mechanical motion using nanoelectromechanical systems (NEMS), as demonstrated recently in remarkable experiments [1–6], opens up a new path for studying quantum behavior in macroscopic systems. Having been able also to coherently control and cleverly measure the state of the mechanical resonator [1, 7–11], an immediate possibility to explore is the use of the NEMS as building blocks for fabricating analog quantum simulators to reproduce many-body quantum physics [12, 13]. Analog quantum simulators are dedicated and controllable devices, which can imitate (within some accuracy) the evolution of certain types of Hamiltonians. Various quantum systems have already been investigated for quantum simulation. Previously, quantum simulators were experimentally implemented using ultracold quantum gases [14], trapped ions [15], photonic quantum systems [16] and superconducting circuits [17, 18]. Recently, an array of optomechanical resonators has been suggested for simulating many-body nonlinear driven dissipative quantum dynamics [19].
One particularly important phenomena that emerges due to the wave-like nature of matter at the quantum regime is Anderson localization [20–22], a phenomena in which waves fail to propagate in disordered media due to interference. Anderson localization has been observed with many experimental setups, including microwaves [23–25], light waves [26–32], ultrasound [33, 34] and matter waves [35–40]. Given the ability to achieve the quantum regime in mechanical resonators, it would be extremely appealing to observe mechanical localization in an array of NEMS, where thermal effects become relevant. Furthermore, arrays of NEMS could be used to simulate continuous-time quantum walk (CTQW) dynamics [41]. Quantum walks are the quantum version of random walks that have a crucial role in designing efficient quantum algorithms that outperform classical algorithms [42]. Implementation of CTQW has been already realized in a four-site circle using a two-qubit nuclear magnetic resonance quantum computer [43] and in a waveguide array [44]. A NEMS-based quantum simulator would provide a means to implement CTQW with phonons, opening up new opportunities for quantum algorithms and universal quantum computation [45].
In this paper, we propose a 1D array of coupled nanomechanical resonators for simulating the Anderson Hamiltonian, namely, a discrete tight-binding model without on-site interactions. The coupling between resonators is electrostatic (capacitive coupling), but could also be mechanic (elastic coupling) in order to be improved. The disorder can be induced in a controlled and predetermined manner through the appropriate variation in the design and nanofabrication of the geometrical dimensions of the nanoresonators in the array [46]. A qubit coupled to the chain can be used as a device for both, initializing and measuring the occupation probabilities of excitations. We also discuss the physical implementations of CTQWs using the proposed quantum simulator.
2 Model
The Hamiltonian (4) is the Bose-Hubbard Hamiltonian without inter-mode interactions, and a diagonal disorder in the on-site energies \(\omega_{j}\). When the total number of phonons is restricted to 1, this Hamiltonian reduces to the Anderson tight-binding Hamiltonian (see below).
3 Closed system: Anderson localization
Localization, from a general point of view, is a mesoscopic phenomena displayed by waves as they propagate through a disordered medium. It is built upon the interference of the many randomly scattered waves, which at sufficiently large distances, collude to produce a suppression in the amplitude of propagation. This behavior is characterized by the exponential decay of the wave-function, with a decay length ξ, known as the localization length. The celebrated scaling theory of localization [47] describes how the transition between the different diffusion regimes depend upon the size L, and dimension D, of the system, independent of the microscopic intricacies of the disorder. According to the theory, depending on L, three different transport regimes can be recognized: ballistic (\(\xi\gg L\)), where the size of the system is too small and scattering events are rare; diffusive (\(\xi\lesssim L\)), where some weak localization effects may take place; and strong localization (\(\xi\ll L\)), for large systems [48]. Of course, the detailed form of the localization depends on the type of disorder potential considered and its energy spectrum.
In the Anderson model (which is the one of interest in this paper), disorder is modeled by a δ-correlated potential V, consisting of a series of spatially uncorrelated barriers,^{1} with a finite maximum amplitude intensity Δ, called disorder strength. One can as well picture a periodic lattice with randomly shuffled on-site energies. Strictly speaking, a proper phase transition from extended to localized states only exists for \(D=3\), at some critical disorder intensity or, interchangeably, some critical energy known as the mobility edge. For \(D<2\) it can be shown that all states are localized, but the localization length for \(D=2\) can easily exceed the size of the system for weak disorder or high enough particle energies, thus admitting the existence of a diffusive transport regime. In contrast, for \(D=1\), (almost^{2}) all single-particle eigenstates are localized even for a vanishingly small Δ and there is no phase transition.^{3}
It is interesting to explore the physics that is difficult to be included in the Anderson model, and which is nonetheless present in a real implementation of the system. Specifically, the influence of thermal effects due to a surrounding reservoir for the resonators is considered in the following.
4 Open system
The effect of the reservoir on the system include loss of excitations due to imperfections in the medium. It simply means that the single phonons in the chain, eventually would leak into the medium. This phenomena can be understood like an incoherent spontaneous emission. The expected behavior of a system subject to an amplitude damping channel is the escape of all excitations in the chain. However, while such decoherence process is taking effect, we can argue the existence of enough quantum correlations as to assure the existence of localization phenomena. Working in a weak resonator-bath interaction regime (something like three orders of magnitude less than the resonator-resonator interaction), the correlations remain in the system until the leaking process has taken away all measurable probability of excitations. We can clearly see this behavior in Figure 3(b) for \(\gamma=1\) MHz, \(\bar{n}=10^{-2}\), and normalized disorder \(\Delta/J=15\). Similarly to Figure 3(a) a single excitation in the central resonator is taken as the initial condition. We see after equilibration a behavior which similar to he one characterising localization in Figure 3(a) but for a increasing uniform thermal excitation base. In Figure 3(c) the same initial condition and dynamics is employed but for a smaller disorder \(\Delta/J=2\). Now we see a typical thermal behavior of a non-localized density profile. Remark that for \(\gamma=1\) MHz there is enough time for equilibration.
Figures 5 and 6 show that the behavior of the entanglement is consistent with that of the population probability distribution from Figure 3. The concurrence and the population both spread through the available sites or become localized due to the disorder. The presence of an environment does not introduce new dynamics. The only effect of the environment, as expected, is to destroy quantum coherences with a rate proportional to \(\exp(-\gamma t )\), regardless of the presence of localization. The figures also show the sudden death of the entanglement [50] in which the concurrence between the central nanomechanical resonator and all the others destroys periodically.
The inclusion of thermal effects does not immediately change the main feature of the localization (Figure 6 (bottom)). The system is quickly localized as expected from the Anderson-like Hamiltonian with disordered diagonal terms. With dissipation, localization is still a quantum effect in the sense that correlations between the resonators remain until the dissipation takes away all possible dynamics. As time goes on, depending on the temperature, all the states become thermalized. Even for very low temperatures, before all excitations leave the chain, the equilibrium state will be a thermalized state. However, the thermal relaxation rate is slow enough that localized phonon populations could still be measured prior to decaying.
5 Measurement
The preparation and measurement protocol (Figure 8(b)) would rely upon utilizing the resonant limit of this model. To proceed, the transmon would initially be detuned in energy from all of the modes in the array and prepared in its first excited state with a microwave pi-pulse applied through a coplanar waveguide (CPW) cavity [55]. Through the application of a flux pulse, the transmon would then be brought into resonance with one of the nanoresonators and allowed to interact for one-half a Rabi cycle (\(t_{\mathrm{Rabi}}=\pi/2\lambda_{j}\)), thus transferring its excitation to the nanoresonator mode. Next, after a predetermined delay, the location of the phonon in the nanoresonator array would be measured by scanning the transmon’s transition energy \(\hbar\nu_{a}\) (via a flux ramp) through the range of nanoresonator energies. Upon achieving resonance with the populated nanoresonator mode, the transmon would transition through a Rabi transfer back to its first excited state, which could be measured through dispersive read-out of the transmon via the CPW cavity [55, 56]. As me mentioned the precise location of the nanoresonator could be determined from a map of frequencies using electron beam imaging and finite element simulations. Also, it should be noted that the coherent exchange of excitations through the resonant interaction between a piezoelectric disk resonator and superconducting phase qubit has been demonstrated previously [1]; thus it is expected that a similar technique could be adapted to the case considered here.
For realization of the protocol, and to insure low thermal occupation (\(\overline{n}<10^{-1}\)) of the nanoresonator modes at dilution refrigerator temperatures, it would be necessary for the nanoresonator array to be composed of ultra-high frequency (UHF) modes in the GHz regime [57]. With proper design of the nanostructures’ geometries, the third-order, in-plane flexural modes could be engineered to have frequencies varying over the range of 2 to 4 GHz - note that varying levels of disorder Δ could be programmed into the array in a controlled manner by deliberating varying the dimensions of the nanostructures. We remark that the larger the disorder the smaller will the required array for the observation of localization. However that cannot be freely varied, since a large disorder could mean that the resonators are far from resonance to each other. That would imply the necessity to consider the counterrotating terms in Hamiltonian (3). Therefore is good to keep the limit \(J\ll\Delta\ll\overline{\omega_{j}}\) for the envisaged platform.
Estimates of ω assuming the standard expression from thin-beam theory for the third in-plane flexural frequency with clamped-clamped boundary conditions
ω /2 π (GHz) | Δ V (V) | λ /2 π (MHz) | J /2 π (MHz) |
---|---|---|---|
2.5 | 10 | 1.2 | 0.7 |
2.5 | 20 | 2.3 | 2.7 |
3.5 | 10 | 1.2 | 0.6 |
3.5 | 20 | 2.5 | 2.3 |
6 Conclusions and perspectives
In this paper, we have devised a quantum simulator based on nanoelectromechanical systems, for analyzing the many-body effects in quantum systems. Actually, a one-dimensional array of electrostatically coupled nanoresonators is suggested to simulate the Anderson Hamiltonian. A method is present for coupling the nanoresonator electrostatically, but other means could also be explored to establish larger coupling (such as mechanical links). By introducing a controlled source of disorder to the system, we studied the localization phenomena in an array of 50 resonators. Accordingly, the population of the first excited state of a given resonator was analyzed, however, due to the discrete nature of our system, one could not expect a smooth Gaussian-to-exponential transition in the population profile.
We also studied the influence of thermal effects due to a surrounding environment, arising in real implementation of the system. By coupling the system to bosonic thermal baths, we studied the interplay between disorder and thermalization. For sufficiently low temperatures, so that the localization is experimentally detectable with the proposed simulator, the loss of phonons due to the dissipation does not immediately destroy the phonon population localization. For higher temperatures the localization is affected by the thermal pumping of excitations into the array, which generates a fully thermalized state.
Initializing the system in a known state and measuring the evolved system, effectively, are important steps in realizing a quantum simulator. Having coupled the chain of resonators to a superconducting transmon qubit, we have suggested detailed protocols to initialize and measure the system. By applying a flux pulse, the transmon qubit is brought into resonance with the desired nanoresonator which allows to interact with the resonator hence transferring its excitation to the nanoresonator. After system evolution, the location of the phonon in the nanoresonator array can be measured by scanning the transmon’s transition energy through the range of nanoresonator energies. Having achieved resonance with the populated nanoresonator, the transmon would transfer back to its first excited state. The transmon can then be measured through dispersive read-out via the CPW cavity.
Beside simulating Anderson localization, we have also discussed the possibility of using the proposed quantum simulator for implementing the continuous-time quantum walk dynamics. The system allows to realize localization and decoherence in continuous-time quantum walks. The initialization and measurement protocols permit to inject several phonons in the chain to investigate the multi-particle quantum walks. A (non-trivial) two-dimensional version of the suggested simulator can be used to implement two-dimensional quantum walks. In such case, quantum algorithms can be implemented. It is worth to mention that to some extent the present discussion applies as well to optomechanical systems, such as in Refs. [59, 60], whose architectures are worth to be explored for quantum simulation purposes. Moreover, given the ability to control individual resonator excitations by scanning the transmon frequency, it is possible to extend the present investigation to the situation including on-site interactions. When several realizations are taken into account the effective Hamiltonian averages out to (4) plus on-site interactions on all sites, in a similar way to the Bose-Hubbard model, allowing investigation of the allowed phases and respective quantum phase transitions when the parameters are varied [61, 62]. In that way this would allow for investigation nonequilibrium steady states of quantum many-body models in similar fashion to other proposed simulators [63, 64].
For instance in 1D, this means that the correlation function of the potential \(g(z)=\langle V(x)V(x+z) \rangle= \Delta^{2} \delta(z)\).
Almost here means for almost every realization of the potential and for almost every energy, except perhaps for some pathological potentials. See [65] for example for more rigorous definitions of localization.
It is interesting to mention anyway that for some long-range correlated disorders an effective mobility edge can appear even in the 1D case [66].
Discrete models bring essentially the same qualitative results as continuous ones regarding localization effects, and they are solvable for many disorder distributions.
Declarations
Acknowledgements
JKM acknowledges financial support from Brazilian National Council for Scientific and Technological Development (CNPq), grant PDJ 165941/2014-6. MCO acknowledges support by FAPESP and CNPq through the National Institute for Science and Technology on Quantum Information and the Research Center in Optics and Photonics (CePOF). MDL acknowledges support for this work provided by the National Science Foundation under Grant DMR-1056423 and Grant DMR-1312421.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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