The New Concept of Nano-Device Spectroscopy Based on Rabi–Bloch Oscillations for Thz-Frequency Range

applied sciences

Article The New Concept of Nano-Device Spectroscopy Based on Rabi–Bloch Oscillations for THz-Frequency Range

Ilay Levie and Gregory Slepyan *

Department from Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv 39040, Israel; [email protected] * Correspondence: [email protected]; Tel.: +972-54-737-89-17

Academic Editor: Zaiquan Xu Received: 21 June 2017; Accepted: 10 July 2017; Published: 14 July 2017

Abstract: We considered one-dimensional quantum chains of two-level Fermi particles coupled via the tunneling driven both by ac and dc fields in the regimes of strong and ultrastrong coupling. The frequency of ac field is matched with the frequency of the quantum transition. Based on the fundamental principles of electrodynamics and quantum theory, we developed a general model of quantum dynamics for such interactions. We showed that the joint action of ac and dc fields leads to the strong mutual influence of Rabi- and Bloch oscillations, one to another. We focused on the regime of ultrastrong coupling, for which Bloch- and Rabi-frequencies are significant values of the frequency of interband transition. The Hamiltonian was solved numerically, with account of anti-resonant terms. It manifests by the appearance of a great number of narrow high-amplitude resonant lines in the spectra of tunneling current and dipole moment. We proposed the new concept of terahertz (THz) spectroscopy, which is promising for different applications in future nanoelectronics and nano-photonics.

Keywords: Rabi-oscillations; Bloch-oscillations; ultrastrong coupling; THz-spectroscopy

1. Introduction Since the early days of physics, spectroscopy has been a platform for many fundamental investigations. One of the central concepts in spectroscopy is a resonance, with its corresponding resonant frequency. Types of spectroscopy can also be distinguished by the nature of the interaction between the electromagnetic field (EM-field) and the condensed matter (as examples, one may mention absorption and emission of light, Raman and Compton scattering, nuclear magnetic resonance, etc.). The first period in the development of spectroscopy is associated with atomic and molecular spectroscopy. Atoms of different elements have distinct spectra, and therefore, atomic spectroscopy allows for the identification of a sample’s elemental composition. The spectral lines obtained by experiment and theory can be mapped to the Mendeleev Periodic Table of the Elements. Spectroscopic studies at this time, were central to the development of quantum mechanics. The second period in the making of spectroscopy is associated with a wide range of applications to different types of chemical substances: gases, liquids, crystals, polymers, etc. The combination of atoms or molecules into macroscopic samples leads to the creation of many-particle quantum states, whereby different materials have distinct spectra. Therefore, spectroscopy observations in a wide frequency range (from microwave until gamma-ray) became an irreplaceable tool for the identification of a sample’s structure, chemical composition, quality, etc. in material engineering, physical chemistry, and biophysics. Recent progress in electronics and informatics is associated with the development of nanotechnologies and synthesis of different types of nano-objects. The distinct

Appl. Sci. 2017, 7, 721; doi:10.3390/app7070721 www.mdpi.com/journal/applsci Appl. Sci. 2017, 7, 721 2 of 24 spectra became the attribute of nano-objects of various spatial configurations, but the same chemical composition. In this context, the future of spectroscopy development would probably be associated with the spectroscopy of a whole range of electronic devices, or their rather large components. This trend will provide tools of the tunable spectroscopy adapted for such types of tasks. The promising examples from this point of view are based on the strong and ultrastrong interactions of condensed matter with EM-field. The strong-coupling field–matter interaction is defined as a regime, in which the EM-field does not correspond to the small perturbation [1]. It leads to the periodical transitions of a two-level quantum system between its stationary states under the action of ac driving field (Rabi-oscillations—RO) [1]. The phenomenon was theoretically predicted by Rabi on nuclear spins in radio-frequency magnetic field [2], and afterwards, discovered in various physical systems, such as electromagnetically driven atoms [3], semiconductor quantum dots [4], and different types of solid-state qubits [5–9]. The simplest physical picture of the Rabi effect is given by the model of a single atom [1]. It can be essentially modified by a set of additional features, such as broken inversion symmetry [10], spatial RO propagation in the arrays of coupled Rabi-oscillators (Rabi-waves), and local field depolarization effects [11–16]. The early quantum theory of electrical conductivity in crystal lattices by Bloch, Zener and Wannier [17–20], led to the prediction that a homogeneous dc field induces an oscillatory, rather than uniform, motion of the electrons. These so-called Bloch-oscillations (BO) have never been observed in natural crystals because the scattering time of the electrons by the lattice defects is much shorter than the Bloch period [21]. In semiconductor superlattices, the larger spatial period leads to a much shorter Bloch period, and BO have been observed through the THz radiation of the electrons [22]. BO in dc biased lattices is due to wave interference, and have been observed in a number of quite different physical systems: a few interacting atoms in optical lattices [23,24], ultracold atoms [25–30], light intensity oscillations in waveguide arrays [31–37], acoustic waves in layered and elastic structures [38], atomic oscillations in Bose-Einstein condensates [39], among others. Several recent studies have investigated the dynamics of cold atoms in optical lattices subject to ac forcing; the theoretically predicted renormalization of the tunneling amplitudes has been verified experimentally. Recent observations include global motion of the atom cloud, such as giant “super-Bloch oscillations” [40]. As a result, BO was transformed from a partial physical effect, to the universal phenomenon of oscillatory motion of wave packets placed in a periodic potential, when driven by a constant force [24,41]. Recently, there have been several theoretical studies of the ultrastrong light–matter interaction regime [42–45], where the Rabi frequency becomes comparable with the frequency of the inter-zone quantum transition [43]. This regime has been realized experimentally with intersubband transitions coupled with plasmon waveguides in the infrared frequency range [46], and metallic microcavities in the THz [47–49], magnetoplasmons of two-dimensional electron gas [50], superconducting qubits [51], and molecular transitions [52]. It was shown that the system behaves like a multi-level quantum bit with non-monochromatic energy spacing, owing to the inter-particle interactions. One of the general tendencies in modern physics consists of the synthesis of different physical mechanisms in the network of one physical process, with their strong mutual influence. One of such examples have been demonstrated in [53], where the quantum chain of the coupled two-level fermion systems driven simultaneously by ac and dc fields was considered. It was shown that in the case of resonant interaction with an ac-component, the particle dynamics exhibits itself in the oscillatory regime, which may be interpreted as a combination of RO and BO, with their strong mutual influence. This type of quantum dynamics was named in [53] as Rabi–Bloch oscillations (RBO). Such a scenario dramatically differs from the individual picture of both types of oscillations due to the interactions. This novel effect is counterintuitive because of the strongly different frequency ranges for two such types of oscillation existences. In this paper, we consider the RBO in the regimes of strong and ultrastrong coupling. We present the approximate analytical solution, and results of the numerical modelling. We study both Appl. Sci. 2017, 7, 721 3 of 25

Appl.In Sci. this2017 ,paper,7, 721 we consider the RBO in the regimes of strong and ultrastrong coupling.3 ofWe 24 present the approximate analytical solution, and results of the numerical modelling. We study both the temporal dynamics and spectra, identify the spectral lines, and compare the spectra behavior in thethe strong temporal and dynamics ultrastrong and regimes. spectra, Conclusively, identify the spectral we discuss lines, the and promising compare theapplications spectra behavior of our resultsin the strongin the novel and ultrastrong types of spectroscopy. regimes. Conclusively, we discuss the promising applications of our results in the novel types of spectroscopy. 2. Structures and Models 2. Structures and Models The model system for RBO observation in the strong coupling regime is shown in Figure 1. It correspondsThe model to the system periodic for two-level RBO observation atomic chain, in the excited strong with coupling an obliquely regime incident is shown ac infield Figure in the1. strongIt corresponds coupling toregime, the periodic and driven two-level with dc atomic voltage chain, applied excited at the with ends. an The obliquely neighboring incident atoms ac ﬁeld are coupledin the strong via interatomic coupling regime,tunneling, and with driven different with dcva voltagelues of penetration applied at theat the ends. ground The neighboringand excited states.atoms are coupled via interatomic tunneling, with different values of penetration at the ground and excited states.

FigureFigure 1. 1. ((aa)) General General illustration illustration of of the the periodic periodic two-leve two-levell atomic atomic chain chain used used as as a a model, model, indicating indicating Rabi–BlochRabi–Bloch oscillations (RBO)(RBO) inin strongstrong coupling coupling regime. regime. It isIt excitedis excited with with obliquely obliquely incident incident ac ﬁeld, ac field,anddriven and driven with dc with voltage dc voltage applied applied at the ends. at the The ends. neighboring The neighboring atoms are coupledatoms are via coupled interatomic via interatomictunneling, with tunneling, different with values different of penetration values atof thepenetration ground and at excitedthe ground states; and (b) Groundexcited andstates; excited (b)

Groundenergy levelsand excited of single energy two-level levels atom of single separated two-level by the atom transition separated energy by the}ω transition0. energy 0 .

TheThe model model system system for for RBO RBO analysis analysis in in the the ultrastrong ultrastrong coupling coupling regime regime is is shown shown in in Figure Figure 2.2. AnalysisAnalysis of of RO RO in in the the regime regime of of ultrastrong co couplingupling was was given given in in [43], [43], performing performing on on a a model model systemsystem depicted depicted in in Figure Figure 22a,a, where a mode of microcavitymicrocavity of THz frequency range is coupled with thethe electronic excitationexcitation of of a a highly highly doped doped quantum quantum well well (QW) (QW) inserted inserted inside inside the capacitor.the capacitor. There There are at areleast at two least conﬁned two subbandsconfined in subbands the QW, and in the th fundamentale QW, and subband the fundamental hosts a two-dimensional subband hosts electron a two-dimensionalgas with an arbitrary electron total numbergas with of an electrons. arbitrary The to energytal number difference of electrons. between The the ﬁrstenergy and thedifference second betweensubband the is matched first and with the the second resonance subband of the is LC matched circuit. Thewith reduction the resonance of the capacitorof the LC armatures circuit. The also reductionimplies a reductionof the capacitor of the capacitance armatures C,also which implies in turn a reduction changes the of resonantthe capacitance frequency C, of which the LC in circuit. turn changesIn order the to keep resonant the resonator frequency frequency of the LC always circuit. matched In order with to the keep frequency the resonator of the quantum frequency transition, always matchedone has towith increase the frequency the inductance, of the L.quantum The strong transiti reductionon, one of has the to capacitor increase volumethe inductance, in the quantum L. The strongLC resonator reduction allows of the for capacitor the realization volume of in the the ultrastrong quantum couplingLC resonator regime allows with for a few the electronsrealization only, of theeven ultrastrong with a single coupling electron. regime It waswith focused a few inelectr [43ons] on only, the caseeven of with an electrona single gas electron. coupled It withwas focusedthe circuit in photons[43] on the of thecase capacitor. of an electron One of gas the coupled peculiar with features the ofcircuit such photons a quantum of the mechanical capacitor. system One ofis thethe existence peculiar of features entangled of light–mattersuch a quantum excitations mechanical in analogy system with the is super-radiantthe existence and of sub-radiantentangled Dicke states in atomic clouds [43]. Appl. Sci. 2017, 7, 721 4 of 25

Appl.light–matter Sci. 2017, 7 ,excitations 721 in analogy with the super-radiant and sub-radiant Dicke states in atomic4 of 24 clouds [43].

FigureFigure 2.2. GeneralGeneral illustrationillustration ofof thethe periodicperiodic chainchain usedused asas aa model,model, indicatingindicating RBORBO inin ultrastrongultrastrong couplingcoupling regime.regime. ( (aa)) The The single single element element of of the the chain. chain. It co Itrresponds corresponds to a to highly a highly doped doped quantum quantum well well(QW) (QW) inserted inserted inside inside the thecapacitor capacitor of ofmicrocavity microcavity ofof THz THz frequency frequency range range [43]. [43]. It isis resonantlyresonantly coupledcoupled withwith a modea mode of microcavity.of microcavity. The frequencyThe frequency matching matching is reached is viareac thehed choice via ofthe inductance choice of value;inductance (b) A value; highly ( dopedb) A highly QW inserted doped QW inside inserted the capacitors inside the of thecapacitors chain of of coupled the chain microcavities. of coupled Themicrocavities. dc voltage appliedThe dc voltage to the system applied at theto the ends system of the at chain. the ends The THzof the oscillating chain. The voltage THz isoscillating applied throughvoltage is a nano-antenna;applied through (c )a Thenano-antenna; periodic potential (c) The inducedperiodic bypotential additional induced doping by additional for transforming doping the homogeneous QW to the effective superlattice. The coupling is governed by the inter-barrier electron tunneling. Appl. Sci. 2017, 7, 721 5 of 24

The system under consideration in our case, is in essence, different. We consider the periodic chain of elements [43], every one of which plays the role of the single atom in Figure1. The chain is driven by both dc and classical THz frequency field (Figure2b). The THz oscillating voltage is applied to the system through a nano-antenna [54–63] (Figure2b). The QW is assumed to be periodically doped, thus the additional periodic potential has been induced (Figure2c). We assume only one electron in the system. The coupling is governed by the inter-barrier electron tunneling. In our case, QW represents the effective super-lattice, which is able to support the electron’s BO [21] and was the first system in which BO was experimentally observed [21,41]. Of course, these two models are identical from a physical point of view. The difference is only in the reachable value of the coupling factor, which makes the contribution of anti-resonant terms significant. These models relate to the different frequency ranges (optical—Figure1, THz—Figure2). As it leads from analysis [43], the ultrastrong coupling becomes implementable in THz, while remains open in optics. Thus, the optical model will be simplified via RWA, while the THz one is based on the total Hamiltonian. We use for analysis, the method of probability amplitudes [1]. The single-electron motion is described by the wave function |Ψ(t)i and satisfies the Schrodinger equation, i}∂t|ψi = Hˆ |ψi, where Hˆ is the total Hamiltonian of the system. The Hamiltonian may be decomposed as Hˆ = Hˆ 0 + Hˆ I, where Hˆ 0 is the component of free electron movement in the chain associated with the tunneling, and described by the periodic superlattice potential V(r) (Figure2c). The second component Hˆ I, corresponds to the interaction of electron with total (dc and ac) EM-field. The electron interaction with EM-field is described in the dipole approximation [1], thus Hˆ I = −eE(r, t) · ˆr, where ˆr is the operator of electron position in the chain, and E(r, t) = EAc(r, t) + EDc(r) is the superposition of dc and ac components. Our analysis will based on the Wannier basis [17–20]; the presentation of Hamiltonian in the Wannier basis is given in [53].

3. Strong Coupling Regime: Analytical Approximation The wave function for the single-particle state of Hamiltonian is given by: |Ψ(t)i = ∑ ap(t) ap + bp(t) bp , (1) p where ap , bp are Wannier functions of the atom with number p in the excited and ground states, respectively [53], and ap(t), bp(t) the unknown variables, which are satisfy the system differential equations:

∂ap ω0 i = − pΩ a + t a + + a − − Ω cos ωt · b , (2) ∂t 2 B p a p 1 p 1 R p

∂bp ω0 i = − + pΩ b + t b + + b − − Ω cos ωt · a . (3) ∂t 2 B p b p 1 p 1 R p In this section, we will consider Equations (2) and (3) analytically, assuming for simplicity that the exact resonance condition is fulﬁlled (ω = ω0).

3.1. Travelling Rabi-Waves

Assuming the driven dc ﬁeld to be absent (ΩB = 0), we have from (2) and (3):

∂ap ω i = a + t a + + a − − Ω cos ωt · b , (4) ∂t 2 p a p 1 p 1 R p

∂bp ω i = − b + t b + + b − − Ω cos ωt · a . (5) ∂t 2 p b p 1 p 1 R p Appl. Sci. 2017, 7, 721 6 of 24

The idea of simplifying Equations (4) and (5) is based on the so-called rotating-wave approximation (RWA), which is frequently encountered in quantum optics [1]. It takes into account only the co-rotating ﬁeld with the system, and neglects the counter-rotating part, which is proportional to exp(2iω) [1]. This part oscillates very rapidly, and its average over the time −1 larger than ω is zero, if ΩR << ω (strong coupling regime). It corresponds to the exchanges bp cos ωt → (1/2)bp exp(−iωt) , ap cos ωt → (1/2)ap exp(iωt) , and leads to the equations:

∂ap ω 1 −iωt i = a + t a + + a − − Ω e b , (6) ∂t 2 p a p 1 p 1 2 R p

∂bp ω 1 iωt i = − a + t b + + b − − Ω e a . (7) ∂t 2 p b p 1 p 1 2 R p

For ΩR and ω have been comparable, the average of the counter-rotating part does not vanish due to the high amplitude of such rapid oscillations (ultra-strong coupling). The equations of motion, in contrast with the system (6) and (7), become analytically unsolvable and should be integrated numerically. The system (6) and (7) is a generalization of Jaynes–Cummings model for the single atom to the case of atomic chain. The ansatz for solutions of Equations (6) and (7) in the form of a travelling wave is

! −i 1 ωt ! a (t) ae 2 p = ei(pϕ−νt) e , (8) i 1 ωt bp(t) ebe 2 where ea, eb are unknown coefﬁcients, ϕ is the phase shift per unit cell, ν is the eigen frequency. Substituting (6)–(8), we obtain the system with respect to ea, eb: ! ! ν − 2t cos ϕ Ω /2 a a R · e = 0. (9) ΩR/2 ν − 2tb cos ϕ eb

Setting determinant equal to zero, we obtain the characteristic equation with respect to ν:

Ω2 (ν − 2t cos ϕ)(ν − 2t cos ϕ) − R = 0, (10) a b 4 with two solutions:  s   2  2 2 ΩR ν1,2(ϕ) = (ta + tb) cos ϕ ± (ta − tb) cos ϕ + (11)  4 

The linearly independent eigen modes correspondent to eigen frequencies (11) are:

1 i(pϕ−ν t) −i 1 ωt i 1 ωt |Ψ (t)i = e 1 ap e 2 − Λ bp e 2 (12) 1 r ∑ N 1 + Λ2 p

1 i(pϕ−ν t) −i 1 ωt i 1 ωt |Ψ (t)i = e 2 Λ ap e 2 + bp e 2 (13) 2 r ∑ N 1 + Λ2 p where N is a number of atoms in the chain and

ΩR Λ = q (14) 2 2 2 2 (ta − tb) cos ϕ + (ta − tb) cos ϕ + ΩR/4 Appl. Sci. 2017, 7, 721 7 of 24

The general solution for Rabi-oscillations propagated along the chain in the form of travelling waves is given by the linear superposition of eigen modes (12), (13) |Ψ(t)i = C1|Ψ1(t)i + C2|Ψ2(t)i, where C1,2 are arbitrary integration constants. Using Equations (12) and (13), it may be transformed to

1 −iµt |Ψ(t)i = Ap(t) ap + Bp(t) bp e (15) r ∑ N 1 + Λ2 p where µ = (ta + tb) cos ϕ, and

−i 1 Ωt i 1 Ωt −i 1 ωt ipϕ Ap(t) = C1e 2 + ΛC2e 2 e 2 e , (16)

i 1 Ωt −i 1 Ωt i 1 ωt ipϕ Bp(t) = C2e 2 − ΛC1e 2 e 2 e (17)

−1 2 2 2 The constants C1,2 satisfy the normalization condition |C1| + |C2| = 1 + Λ , (guarantee the correct normalization for the wavefunction (15)). Let us assume the system initially prepared in the excited state (Bp(0) = 0). In this case, we obtain from (16) and (17):

1 −i 1 Ωt 2 i 1 Ωt −i 1 ωt ipϕ Ap(t) = e 2 + Λ e 2 e 2 e , (18) 1 + Λ2

2iΛ Ωt i 1 ωt ipϕ Bp(t) = sin e 2 e , (19) 1 + Λ2 2 where the RO frequency is given by q 2 2 2 Ω(ϕ) = 4(ta − tb) cos ϕ + ΩR. (20)

It is important to note that it is corrected with respect to Rabi-frequency, due to the interatomic coupling via tunneling.

3.2. Tunneling Current Following the general presentation of tunneling current in terms of probability amplitudes [1], we have for RO

e h ∗ ∗i JT,p = i 2 ta Ap+1 − Ap−1 Ap + tb Bp+1 − Bp−1 Bp + c.c. 2 2 . (21) = −2e sin ϕ ta Ap + tb Bp

For the case of a completely excited initial state, we obtain using (18) and (19):

2e sin ϕ Ωt J = − t 1 + Λ4 + 2Λ2 cos Ωt + 4t Λ2 sin2 (22) T,p 2 a b 2 1 + Λ2

In this case, the tunnel current consists of two components: dc-current, and the component on the RO frequency (20). Its physical origin is similar to the so-called low-frequency RO in the two-level quantum systems with broken inversion symmetry [10]. As was predicted in [10], it leads to non-zero diagonal matrix elements of dipole moment. It is impossible for real atoms (excluding hydrogen), but implementable for artiﬁcial atoms (for example, self-organized semiconductor quantum dots). As a result, the inter-level optical transitions are accompanied by the intra-level periodical movement with Rabi-frequency, which is responsible for low-frequency radiation. It exists only in the case of asymmetry, which manifests itself in the non-identity of diagonal dipole matrix elements. In our case, Appl. Sci. 2017, 7, 721 8 of 24 the intra-band movement and correspondent asymmetry are associated with interatomic tunneling and different penetration values (ta 6= tb), respectively.

3.3. Rabi–Bloch Oscillations: Quasiclassical Model BO of the charge particles is a pure quantum effect which can be explained in a simple quasiclassical model [21,24,25]. The similar concept may be developed for quasi-particles (Rabi-oscillated electrons in our case). The periodicity of the lattice leads to a band structure of the energy spectrum and the corresponding eigen energies }ν1,2(ϕ) given by Equation (11). The eigen states |Ψ1,2(h)i given by Equations (12) and (13), are labeled by the continuous quasimomentum h; }ν1,2(h, n) and |Ψ1,2(h)i are periodic functions of h with period 2π/a, and h is conventionally restricted to the first Brillouin zone [−π/a; π/a]. Under the influence of dc field, weak enough not to induce interband transitions, a given state |Ψ1,2(h0)i evolves up to a phase factor into the state |Ψ1,2(h(t))i, with h(t) variation according to dh eE = − Dc , (23) dt } −1 or h(t) = −e} EDct + h0. Thus, this evolution is periodic with a Bloch frequency corresponding to the time required for the quasimomentum to scan a full Brillouin zone. It leads to the exchanges

ϕ → h(t)a = −ΩBt + ϕ0, (24)

ΩR Λ(ϕ) → Λ(t) = q , (25) 2 2 2 2(ta − tb) cos(ΩBt − ϕ0) + 4(ta − tb) cos (ΩBt − ϕ0) + ΩR q 2 2 2 Ω(ϕ) → Ω(t) = 4(ta − tb) cos (ΩBt − ϕ0) + ΩR, (26) where h0 = ϕ0a. Equations (24)–(26) allow the spectrum of the tunneling current in the BO regime to be analyzed. Following quasiclassical concepts, the tunneling current may be considered as RO, with modulated amplitude and frequency. As it leads from Equations (24)–(26), the spectral presentation of tunnel current is given by ∞ i(Ω−mΩB)t JT(t) ⇒ ∑ ame + c.c., (27) m=−∞ q −1R 2π 2 2 2 where Ω = (2π) 0 4(ta − tb) cos ϕ + ΩRdϕ and am are the spectral amplitudes. Below, we consider the amplitude relations in the current spectra based on the numerical solution of motion equations. Here, we note that the small detuning for which Ω − mΩB ≈ δΩB, with δ << 1, is associated with a special type of resonance motion. The above represents a superposition of harmonic oscillations, which evolves slowly in time, with a period 2π/(δΩB) >> TB,R, where TB,R are the Rabi ∼ R t and Bloch periods, respectively. Evaluating the particle position as x(t) = 0 JT(τ)dτ, we obtain ∼ −1 i(Ω−mΩ )t x(t) = (δΩB) e B + c.c. It shows the strong enhancement of spatial oscillations, which is similar to the super-Bloch oscillations recently predicted and experimentally observed in optical lattices with cold atoms [40]. In contrast with super-Bloch oscillations, in our case, there exists two tunneling channels (with respect to the different zones). Thus, the influence of ac field does not add up to renormalization of tunneling factors, but leads to their mutual influence via Rabi-oscillations.

4. Ultrastrong Coupling: Results and Discussion In this section, we will consider the dynamics and spectra for the case of ultrastrong coupling. As was mentioned above, the equations of motion are analytically unsolvable, and were integrated numerically with the technique [64]. Appl. Sci. 2017, 7, 721 9 of 24

4.1. The Case of Small Bloch-Frequency In this subsection, we consider the case of Rabi-frequency comparable with the quantum transition ∼ frequency, but strongly exceeding Bloch-frequency (ΩR >> ΩB, while ΩR = ω0). The spatial–temporal behavior of the tunnel current, for different types of relative values of tunneling factors at ground and excitedAppl. Sci. 2017 states,, 7, 721 is presented in Figures3 and4. The current oscillates with Bloch frequency.10 of 25 The amplitude for the case ta = tb exceeds the case ta >> tb. The current amplitude is high frequency modulated.Appl. Sci. 2017 In, general,7, 721 the scenario looks like as in the case of strong coupling. 10 of 25

Figure 3. Space–time distribution of the tunnel current density with Rabi-oscillations (RO) and Bloch-oscillations (BO) in the separated frequency ranges. Here, the quantum transition frequency is FigureFigure 3. Space–time3. Space–time distribution distribution of the tunnel tunnel current current density density with with Rabi-oscillations Rabi-oscillations (RO) (RO)and and taken as the frequency unit, detuning is equal to zero  0 , B 0.04 , R 0.8 , Bloch-oscillationsBloch-oscillations (BO) (BO) in in the the separatedseparated frequency frequency ranges. ranges. Here, Here, the quantum the quantum transition transition frequency frequency is ta  0.4 , tb  0.04 , interatomic distance a  20 nm . The initial state of the chain is an excited is taken as the frequency unit, detuning is equal to zero (ω = ω ), Ω = 0.04, Ω = 0.8, ta = 0.4, taken as the frequency unit, detuning is equal to zero  0 , B B 0.04 , RR 0.8 , 2 22 tb = 0.04single, interatomic Gaussian wave distance packeta = 20anm0exp. The g initial state jja of the chain / is, anb excited00 single. Gaussian Gaussian t 0.4 , t 0.04 , interatomic distancej  a 20 nm . The initial state of thej chain is an excited a  b  h 0 2 2 2i   0 wave packet aj(0) = g exp −(j − j ) a /σ , bj(0) = 0. Gaussian initial position and width are j = 80, initial position and width are j  80 , g  20 , respectively.2 22Number of atoms N  128 . = single Gaussian wave packet a0exp= g jja /, b 00. Gaussian g 20, respectively. Number of atomsj N 128.     j  initial position and width are j  80 , g  20 , respectively. Number of atoms N  128 .

Figure 4. FigureSpace-time 4. Space-time distribution distribution of of the the tunnel tunnel currentcurrent density density with with RO RO and and BO BO in the in the separated separated frequency ranges. ta = t = 0.04. All other parameters are identical to Figure3. frequency ranges. ttabb 0.04 . All other parameters are identical to Figure 3. Figure 4. Space-time distribution of the tunnel current density with RO and BO in the separated

frequencyIn Figures ranges. 5 andttab 6, the0.04 corresponding. All other parameters spectra are are identical presented. to Figure Their 3. qualitative behavior dramatically changes with respect to the strong coupling regime (see Equation (27)). The main reasonIn Figures leads from 5 and the 6,strong the correspondinganalysis of the spectrsystema inare Equations presented. (2) Their and (3),qualitative which is behaviorfree from dramatically changes with respect to the strong coupling regime (see Equation (27)). The main reason leads from the strong analysis of the system in Equations (2) and (3), which is free from Appl. Sci. 2017, 7, 721 11 of 25

RWA,Appl. Sci. and2017 based, 7, 721 on Floquet theorem. Following it, the probability amplitudes of the j-th mode10 ofare 24 presented in the form In Figures5 and6, the corresponding spectra are presented. Their qualitative behavior at a  pj,, pmj im t itj  dramatically changes with respect to the strong coupling ee regime0 (see, Equation (27)). The(28) main reason leads from the strong analysisbtpj,, of them system b pmj in Equations  (2) and (3), which is free from RWA, and based on Floquet theorem. Following it, the probability amplitudes of the j-th mode are presented in the form where abpm,, j, pm j  are the !periodical functions of! the phase shift per unit cell, and j  a (t) ∞ a (ϕ) p,j = pm,j −imω0t −iνj(ϕ)t is the eigen value. Therefore, as it leads∑ from Equation (21),e thee lines , m appear at (28)the bp,j(t) m=−∞ bpm,j(ϕ) 0 frequency spectrum, and the first of them are really observed at Figures 5 and 6, in contrast with where apm,j(ϕ), bpm,j(ϕ) are the periodical functions of the phase shift per unit cell, and νj(ϕ) relation (27). Following the quasiclassical approach, we exchange Bt  0 , thus the values is the eigen value. Therefore, as it leads from Equation (21), the lines ω = mω0 appear at the frequency abspectrum,pm,, j, and pm j the first become of them time are reallydependent observed periodic at Figures functions5 and6 , inwith contrast a Bloch-period. with relation Such (27). Following the quasiclassical approach, we exchange ϕ → −Ω t + ϕ , thus the values a (ϕ), b (ϕ) amplitude modulation means the appearance of B lines0 mnpm,j , wherepm,j become time dependent periodic functions with a Bloch-period. Such amplitude0 modulationB means mn, 0,1,2,..., . The frequency-modulated factors exp[i ] , for qualitative analysis, the appearance of lines ω = mω0 + nΩB, where m, n = 0, ±1, ±2, ...,j ±∞. The frequency-modulated mayfactors be expaveraged[iνj(ϕ) ]over, for qualitativethe Bloch-period analysis, [40]. may The be do averagedminant oversupport the Bloch-periodto observable [ 40values]. The is dominant defined bysupport two first to observablemodes, with values j = 1,2. is definedThus, the by discrete two first part modes, of the with spectraj = 1,2. in the Thus, regime the discreteof ultrastrong part of couplingthe spectra corresponds in the regime to the of ultrastrongnarrow lines coupling with the corresponds frequencies to the narrow lines with the frequencies

ω =mnmω00 + nΩBB ± ppΩ (29)(29) wherewhere pp=0,0,±1, 1 and, and 1 1 Z2 2π Ω = [ν (ϕ) − ν (ϕ)]dϕ (30) 121  2 d  (30) 22π0 0

FigureFigure 5. 5. FrequencyFrequency spectrum spectrum of of the the tunneling tunneling current current density density with with RO RO and and BO BO in in the the separated separated frequencyfrequency ranges ranges at at the the atom atomicic position position with with number number jj =9090.. The The quantum quantum transition transition frequency, frequency, Rabi-frequency and Bloch-frequency are denoted by a red-colored dashed line, blue-colored Rabi-frequency and Bloch-frequency are denoted by a red-colored dashed line, blue-colored dashed–dotted line and green-colored dotted line, respectively. The quantum transition frequency dashed–dotted line and green-colored dotted line, respectively. The quantum transition frequency is is taken as the unit. ΩB = 0.04, ΩR = 0.8. All other parameters are identical to Figure3. taken as the unit. B 0.04 , R 0.8 . All other parameters are identical to Figure 3. Appl. Sci. 2017, 7, 721 11 of 24 Appl. Sci. 2017, 7, 721 12 of 25

FigureFigure 6.6.Frequency Frequency spectrum spectrum of theof the density density of the of tunneling the tunneling current current with RO with and BORO inand the BO separated in the frequency ranges. ta = tb = 0.04. All other parameters are identical to Figure5. separated frequency ranges. ttab0.04 . All other parameters are identical to Figure 5.

This frequency may be identified with the frequency of RO. The spectrum defined by Equation (29) This frequency may be identified with the frequency of RO. The spectrum defined by Equation corresponds to the physical picture of RBO as a superposition of three partial motions: (i) a single (29) corresponds to the physical picture of RBO as a superposition of three partial motions: (i) a quantum transition from the ground to excited state (or vice versa); (ii) the series of periodic single quantum transition from the ground to excited state (or vice versa); (ii) the series of periodic quantum transitions between different states with frequency Ω (RO); (iii) the spatial oscillations quantum transitions between different states with frequency  (RO); (iii) the spatial oscillations as a comprehensive whole, with frequency ΩB(BO). Of course, the validity of this picture is essentially stronglyas a comprehensive limited, in view whole, of thewith fact frequency that the travellingB (BO). Of waves course, are usedthe validity as a model of this of excitation.picture is Butessentially if we go strongly to the wavepacket limited, in withview strongly of the fact defined that momentum,the travelling this waves picture are will used be as adequate a model too, of andexcitation. is supported But if we by thego to numerical the wavepacket calculations with presentedstrongly defined below. momentum, The similar generalthis picture scenario will ofbe theadequate complex too, dynamics and is supported manifests by itself the innumerical a lot of othercalculations physical presented systems. below. In particular, The similar as one general more example,scenario of the the coherent complex control dynamics of charge manifests fluxes itself in unbiasedin a lot of molecular other physical junctions systems. may In be particular, considered, as andone ismore discussed example, in [65 the]. Suchcoherent control control is induced of charge by resonances fluxes in betweenunbiased the molecular Rabi-oscillations junctions driven may bybe aconsidered, pumping laser and field, is discussed and the bridge in [65]. mediated Such tunneling control oscillationsis induced between by resonances the lowest between unoccupied the molecularRabi-oscillations orbitals driven of the donor by a andpumping acceptor. laser One field, can and see that the thebridge resonance mediated condition tunneling [65] corresponds oscillations tobetween Equation the (29) lowest for m unoccupied= 0. The frequency molecular given orbitals by Equation of the donor (30) is and associated acceptor. with One the can Rabi see frequency that the inresonance [64]. The condition frequency [65] of intra-molecularcorresponds to Equation oscillations (29) driven for m by the0 . The donor–acceptor frequency given tunneling by Equation in [65] may(30) beis identifiedassociated aswith a Bloch the frequencyRabi frequency in Equation in [64]. (29). The frequency of intra-molecular oscillations drivenWe by will the comment donor–acceptor only on tunneling the analysis in [65] and may physical be identified interpretation as a Bloch of frequency the dominative in Equation lines, which(29). will keep their resolvability in real experimental conditions. The spectra in Figures5 and6 are clearlyWe will separated comment into only low-frequency on the analysis and high-frequency and physical regions.interpretation The low-frequency of the dominative part consists lines, ofwhich the single will keep line ωtheir≈ Ω resolvabilityB, while the high-frequencyin real experimental part consistsconditions. of the The different spectra sub-harmonics in Figures 5 and (29). 6 Theare low-frequencyclearly separated part into corresponds low-frequency to the ordinary and high-frequency BO; the high frequencyregions. The spectrum low-frequency is governed part by theconsists RO. The of dipolethe single moment line and  inversionB , while densities the are high-frequency presented in Figures part7 consistsand8, and of corresponding the different spectrasub-harmonics in Figures (29).9 and The 10 ,low-frequency respectively. part corresponds to the ordinary BO; the high frequency spectrum is governed by the RO. The dipole moment and inversion densities are presented in Figures 7 and 8, and corresponding spectra in Figures 9 and 10, respectively. Appl. Sci. 2017, 7, 721 12 of 24 Appl.Appl. Sci. Sci. 2017 2017, ,7 7, ,721 721 1313 of of 25 25

FigureFigure 7. 7. Space–timeSpace–time distributiondistribution ofof thethe dipoledipole momemome momentntnt densitydensity withwith RORO andand BO BO in in the the separated separated frequencyfrequency ranges. ranges.ranges. All All parameters parameters are are identical identicalidentical to toto Figure FigureFigure 33. 3..

Figure 8. Space–time distribution of the inversion density with RO and BO in the separated frequency FigureFigure 8.8. Space–timeSpace–time distributiondistribution ofof thethe inversioninversion densitydensity withwith RORO andand BOBO inin thethe separatedseparated ranges. All parameters are identical to Figure3. frequencyfrequency ranges. ranges. All All parameters parameters are are identical identical to to Figure Figure 3. 3. Appl. Sci. 2017, 7, 721 14 of 25 Appl. Sci. 2017, 7, 721 13 of 24 Appl. Sci. 2017, 7, 721 14 of 25

FigureFigure 9.9. FrequencyFrequency spectrumspectrum ofof thethe dipoledipole momentmoment withwith RORO andand BOBO inin thethe separatedseparated frequencyfrequency Figure 9. Frequency spectrum of the dipole moment with RO and BO in the separated frequency rangesranges at at the the atom atom with with position position number numberj =j 90. 90 All. parametersAll parameters are identicalare identical to Figure to Figure3. 3. ranges at the atom with position number j  90 . All parameters are identical to Figure 3.

FigureFigure 10. 10.Frequency Frequency spectrum spectrum of inversionof inversion density density with wi ROth andRO BOand at BO the separatedat the separated frequency frequency ranges atFigureranges the atomic 10.at the Frequency positionatomic positionj spectrum= 90. Allj parametersof 90inversion. All parameters are density identical wi areth to identical FigureRO and3. toBO Figure at the 3. separated frequency ranges at the atomic position j  90 . All parameters are identical to Figure 3. AsAs oneone can can see, see, the the dipole dipole current current and inversionand inversion densities densities are modulated are modulated by the Bloch by frequency.the Bloch As one can see, the dipole current and inversion densities are modulated by the Bloch Thefrequency. reason isThe the mutualreason inﬂuenceis the mutual of BO andinfluence RO. The of lowBO frequencyand RO. componentThe low frequencyω ≈ ΩB corresponds component frequency. The reason is the mutual influence of BO and RO. The low frequency component to the mutualB corresponds inﬂuence to of the RO mutual and BO influence in the so-called of RO and RBO BO regime. in the so-called RBO regime.  B corresponds to the mutual influence of RO and BO in the so-called RBO regime. 4.2. The Case of Comparable Rabi- and Bloch Frequencies 4.2. The Case of Comparable Rabi- and Bloch Frequencies 4.2. TheIn this Case subsection, of Comparable we considerRabi- and another Bloch Frequencies case: when the Rabi- and Bloch frequencies are comparable In this subsection, we consider another case: when the Rabi- and Bloch frequencies are (Figures 11 and 12). The scenario dramatically changes; the total spectrum is not separable into comparableIn this subsection,(Figures 11 weand consider 12). The another scenario case: dramatically when the changes; Rabi- and the Blochtotal spectrumfrequencies is arenot low-frequency and high-frequency components. As one can see in Figure 12, there is the appearance of comparableseparable into (Figures low-frequency 11 and and12). high-frequencyThe scenario dramatically components. changes;As one can the see total in Figurespectrum 12, thereis not is the combined sub-harmonics of Bloch and Rabi-frequencies, according to Equation (27) (RBO regime). separablethe appearance into low-frequency of the combined and high-frequency sub-harmonics components. of Bloch andAs oneRabi-frequencies, can see in Figure according 12, there tois theEquation appearance (27) (RBO of regime).the combined sub-harmonics of Bloch and Rabi-frequencies, according to Equation (27) (RBO regime). Appl. Sci. 2017, 7, 721 15 of 25

Appl. Sci. 2017, 7, 721 14 of 24 Appl. Sci. 2017, 7, 721 15 of 25

Figure 11. The space–time distribution of the tunnel current density with RBO in the ultrastrong coupling regime. Here, the quantum transition frequency is taken as the frequency unit, zero

detuning  0 , B 0.7 , R 0.8 , ttab7.0 , and interatomic distance Figure 11. The space–time distribution of the tunnel current density with RBO in the ultrastrong Figurea  20 11. nm The. space–timeThe initial distributionstate of the of thechain tunnel is currentan excited density single with GaussianRBO in the wave ultrastrong packet coupling regime. Here, the quantum transition frequency is taken as the frequency unit, zero detuning coupling regime. Here, the quantum2 transition frequency is taken as the frequency unit, zero (ω = ω ), Ω = 0.7, Ω = 0.8, t22= t = 7.0, and interatomic distance a = 20 nm. The initial state aj 0exp0 gB  jjaR   a / b , bj 00 . Gaussian initialh position andi width are detuning  0 , B 0.7 , R 0.8 , ttab7.0 , and interatomic0 2 2 2 distance of the chain is an excited single Gaussian wave packet aj(0) = g exp −(j − j ) a /σ , bj(0) = 0. , , respectively. Number of atoms N 128 . Gaussianaj2080 nm initialg . The position20 initial and state width of are thej0 = 80,chaing = is20, an respectively. excited single Number Gaussian of atoms Nwave= 128. packet 2 22 a0exp g jja /, b 00. Gaussian initial position and width are j     j  j  80 , g  20 , respectively. Number of atoms N  128 .

FigureFigure 12.12.The The frequency frequency spectrum spectrum of tunnelof tunnel current current density density with RBOwith inRBO ultrastrong in ultrastrong coupling coupling regime atregime the atomic at the positionatomic positionj = 60. Allj parameters 60 . All parameters are identical are to identical Figure 11 to. Figure 11.

FigureInIn FiguresFigures 12. The 1313 frequencyandand 1414,, thespectrumthe spectraspectra of tunnel ofof tunnelingtunneli currentng density currentcurrent with andand RBO dipoledipole in ultrastrong momentmoment densitiescouplingdensities forfor Ω = Ω = 0.5ω are presented. In this special case, a lot of sub-harmonics become degenerate. BBRregimeR at0.5 the atomic0 0 are positionpresented. j  In60 this. All special parameters case, are a identicallot of sub-harmonics to Figure 11. become degenerate. AsAs aa result, result, the the density density of spectral of spectral lines decreaseslines decreases and the and ability the of ability their resolution of their inresolution real experiments in real = grows,experimentsIn avoidingFigures grows, 13 a strong andavoiding 14, overlap the a strong spectra via overlap dissipative of tunneli via broadening.dissipativeng current broadening. and Appearing dipole Appearingmoment are the linesdensities are ωthe lines forω0 ω = ω ± Ω (see Figure 14), at ﬁrst, it appears they may be associated with a Mollow triplet, but  0 R0.5 are (see presented. Figure In14), this at specialfirst, it case,appears a lot they of sub-harmonicsmay be associated become with degenerate. a Mollow noBR rather0 convincing0 0 R arguments for such an interpretation exist. The reason is the degeneracy of As a result, the density of spectral lines decreases and the ability of their resolution in real the modes of different origins in our case, in contrast with Lorentz single-oscillator lines in Mollow experiments grows, avoiding a strong overlap via dissipative broadening. Appearing are the lines triplets [1].  0 0 R (see Figure 14), at first, it appears they may be associated with a Mollow Appl. Sci. 2017, 7, 721 16 of 25

Appl. Sci. 2017, 7, 721 16 of 25 triplet, but no rather convincing arguments for such an interpretation exist. The reason is the triplet,degeneracy but noof therather modes convincing of different arguments origins infor our su chcase, an in interpretation contrast with exLorentzist. The single-oscillator reason is the degeneracylines in Mollow of the triplets modes [1]. of different origins in our case, in contrast with Lorentz single-oscillator linesAppl. Sci.in Mollow2017, 7, 721 triplets [1]. 15 of 24

Figure 13. The frequency spectrum of the tunnel current density with RBO in the ultrastrong Figure 13. The frequency spectrum of the tunnel current density with RBO in the ultrastrong coupling Figurecoupling 13. regime The frequency at the atomic spectrum position of the jtunnel 60 . curreRBnt density0.5 with, tt abRBO in the5.0 ultrastrong. All other regime at the atomic position j = 60. ΩR = ΩB = 0.5, ta = tb = 5.0. All other parameters are identical parameters are identical to Figure 11. couplingto Figure 11regime. at the atomic position j  60 . RB0.5 , ttab5.0 . All other parameters are identical to Figure 11.

Figure 14. The frequency spectrum of the dipole moment density with RBO in the ultrastrong coupling Figure 14. The frequency spectrum of the dipole moment density with RBO in the ultrastrong regime at the atomic position j = 60. ΩR = ΩB = 0.5, ta = tb = 5.0. Figurecoupling 14. regime The frequency at the atomic spectrum position of thej  dipole60 .  moRBment density0.5 , withttab RBO5.0 in the. ultrastrong

4.3. Thecoupling Role ofregime the Losses at the atomic position j  60 . RB0.5 , ttab5.0 . 4.3. The Role of the Losses One of distinctive features of ultrastrong coupling is a comparable value of inter-band transition 4.3. TheOne Role of of distinctive the Losses features of ultrastrong coupling is a comparable value of inter-band with radiative and dissipative damping value. Thus, the dissipative and radiative broadening leads transition with radiative and dissipative damping value. Thus, the dissipative and radiative to theOne conﬂuence of distinctive of neighboring features spectralof ultrastrong lines. As coupling a result, theis a discrete comparable lines willvalue interchange of inter-band with broadening leads to the confluence of neighboring spectral lines. As a result, the discrete lines will transitionareas of continuous with radiative spectra. and These dissipative discrete lines damping will have value. non-Lorentz Thus, the asymmetric dissipative form. and This radiative means interchange with areas of continuous spectra. These discrete lines will have non-Lorentz broadeningthat the simple leads lossless to the modelconfluence is not of applicable neighboring for detailedspectral analysislines. As of a theresult, spectra the discrete thin structure—it lines will asymmetric form. This means that the simple lossless model is not applicable for detailed analysis interchangeallows only predictionwith areas of theof spectralcontinuous line positionsspectra. andThese their discrete relative lines amplitudes. will have For estimatingnon-Lorentz the of the spectra thin structure—it allows only prediction of the spectral line positions and their asymmetricrole of losses, form. we willThis use means the simple that the phenomenological simple lossless model model is usednot applicable by Scully etfor al., detailed for the analysis relative amplitudes. For estimating the role of losses, we will use the simple phenomenological of populationthe spectra trapping,thin structure—it lasing without allows inversion, only prediction and electromagnetically of the spectral inducedline positions transparency and their [1]. relativeThis model amplitudes. assumes thatFor estimating two levels decaythe role at aof rate losses,γ [1 ,66we]. will This use means the thatsimple the groundphenomenological state is not Appl. Sci. 2017, 7, 721 17 of 25

model used by Scully et al., for the analysis of population trapping, lasing without inversion, and electromagnetically induced transparency [1]. This model assumes that two levels decay at a rate  [1,66]. This means that the ground state is not really ground, and corresponds to the excited

Appl.stateSci. at the2017 ,lowest7, 721 energy level. The wavefunction for such model is given by 16 of 24

really ground, and corresponds to the excited state at the lowest energy level. The wavefunction for 1 it t/2 tAtaBtbeepppp  , such model is given by  2   (31) N 1 p 1 −iµt −γt/2 |Ψ(t)i = r ∑ Ap(t) ap + Bp(t) bp e e , (31) 2 p where AtBtpp,  are the sameN 1 +coefficientsΛ as in the lossless system. The tunneling current and its frequency spectrum for Q = 15 are shown in Figures 15 and 16. whereOne canAp see(t), Bthatp(t )theare oscillation the same coefficients process quickly as in the reduces lossless (Figure system. 15). The number of the resolved spectralThe lines tunneling decreases current with and decay its frequencyappearance spectrum (Figure 16), for becauseQ = 15 areof the shown large in number Figures of 15 lines and run 16. Oneinto, candue see to that the the broadening. oscillation processThe main quickly lines reduces are kept, (Figure however, 15). The their number positions of the resolvedare shifted spectral and linesamplitudes decreases are withdecreased decay with appearance respect (Figureto the lossle 16), becausess case. The of the frequency large number spectrum of lines of the run tunneling into, due tocurrent the broadening. for BR The0.5 main islines shown are in kept, Figure however, 17. The theirmechanism positions of the are spectral shifted andlines amplitudes confluence aredue decreased to the losses, with is respect especially to the well-defined lossless case. in Thethis partial frequency case spectrum via the transformation of the tunneling of currentduplets for to ΩtheirB = correspondingΩR = 0.5 is shown broadened in Figure singlets. 17. The mechanismThe non-Lore ofntz the spectralorigin of lines these confluence spectral duelines to themanifests losses, isitself especially in their well-defined asymmetry. in thisSuch partial qualitative case via thebehavior transformation agrees with of duplets general to theirprinciples corresponding of the broadenedoscillations’ singlets. theory The[67]. non-LorentzThe detailed origin description of these of spectral losses linesmust manifestsbe based itselfon the in theirconcept asymmetry. of open Suchquantum qualitative systems behavior [68], and agrees master with equation general techniqu principlese especially of the oscillations’ adopted to theory the case [67]. of The ultrastrong detailed descriptioncoupling [69]. of lossesIt goes must beyond be based the onsubject the concept of this of article, open quantumand may systems be one [68of], the and topics master for equation future techniqueresearch activity. especially adopted to the case of ultrastrong coupling [69]. It goes beyond the subject of this article, and may be one of the topics for future research activity.

Figure 15. The space–time distribution of tunneling current density with RBO in the ultrastrong Figure 15. The space–time distribution of tunneling current density with RBO in the ultrastrong coupling regime with dissipation. Insert: the individual energy spectrum of the single atom in the chain. coupling regime with dissipation. Insert: the individual energy spectrum of the single atom in the The model assumes that two levels decay at a rate γ, due to spontaneous emission. Thus, the ground chain. The model assumes that two levels decay at a rate  , due to spontaneous emission. Thus, state is not really ground; it corresponds to the excited state at the lowest energy level. The dissipation the ground state is not really ground; it corresponds to the excited state at the lowest energy level. property is characterized by a Q-factor, deﬁned as Q = ω0/2γ. Realistic values for ultrastrong coupling casesThe dissipation are Q = 15–30 property [43]. Here, is charac the quantumterized transitionby a Q-factor, frequency defined is taken as Q as the  frequency0 /2 . Realistic unit, detuning values

isfor zero ultrastrong(ω = ω 0coupling), ΩB = cases0.7, Ω areR = Q 0.8= 15–30, ta = [43].tb = Here7.0,, interatomicthe quantum distance transitiona = frequency20 nm, and is takenQ = 15.as h 0 2 2 2i The initial state of the chain is an excited single Gaussian wave packet aj(0) = g exp −(j − j ) a /σ the frequency unit, detuning is zero  0 , B 0.7 , R 0.8 , ttab7.0 , 0 bj(0) = 0. Gaussian initial position and width are j = 80, g = 20, respectively. Number of atoms interatomic distance a  20 nm , and Q = 15. The initial state of the chain is an excited single N = 128. Appl. Sci. 2017, 7, 721 18 of 25

2 22 Appl.Gaussian Sci. 2017, 7 , wave721 packet ag0exp jja / b 00. Gaussian initial18 of 25 j     j 

position and width are j  80 , g  20 , respectively. 2Number22 of atoms N  128 . Gaussian wave packet ag0exp jja / b 00. Gaussian initial j     j  Appl. Sci.position2017, 7 and, 721 width are j  80 , g  20 , respectively. Number of atoms N  128 . 17 of 24

Figure 16. The frequency spectrum of tunneling current density with RBO in the ultrastrong

coupling regime with dissipation in the atomic position j  60 . All other parameters are identical Figure 16. The frequency spectrum of tunneling current density with RBO in the ultrastrong coupling toFigure Figure 16. 15. The frequency spectrum of tunneling current density with RBO in the ultrastrong regime with dissipation in the atomic position j = 60. All other parameters are identical to Figure 15. coupling regime with dissipation in the atomic position j  60 . All other parameters are identical to Figure 15.

Figure 17. The frequency spectrum of tunneling current density with RBO in the ultrastrong coupling Figure 17. The frequency spectrum of tunneling current density with RBO in the ultrastrong regime with dissipation in the atomic position j = 60. ΩB = ΩR = 0.5, ta = tb = 5.0. All other coupling regime with dissipation in the atomic position j 60 . 0.5 , parameters are identical to Figure 15.  BR Figure 17. The frequency spectrum of tunneling current density with RBO in the ultrastrong ttab5.0 . All other parameters are identical to Figure 15. coupling regime with dissipation in the atomic position j 60 . 0.5 , Another topic for future activity is the implementation of the scenarioBR considered before.

The subjectAnotherttab of topic this5.0 article.for All futureother is limited parameters activity by theis are the guidance identical implem to ofentation Figure some candidates15. of the scenario for such considered potential experiments.before. The subjectAs one of example, this article the multi-excitonis limited by the dynamics guidance in theof some linear candidates chain of identical for such molecules potential with experiments. transition Asdipole oneAnother example, moment topic the perpendicular formulti-exciton future activity to dynamics the chainis the in axisimplem the (H-aggregate linearentation chain of of conﬁguration)the identical scenario molecules considered [70], may with bebefore. transition marked. The ∼ ∼ dipoleThesubject number moment of this of article perpendicular emitters is limited in [70 to ]by isthe theN chain guidance= 10 axis÷ 25 (H-aggregateof, thesome distance candidates configuration)a = for1.2 suchnm, [70],potentialta,b may= 2.6 experiments.be÷ marked.23 meV, ∼ 7 8 TheandAs onenumberτ =example,2/ γof =emitters the1.0 multi-excitonns. in The [70] reachable is dynamicsN 10 values in 25 the, of the linear the distance ac chain ﬁeld ofa10 identical 1.2÷ 10 nm, V/mmolecules tab, for2.6 the with frequencies 23transition meV, ∼ ωdipole≈ ω moment0 = 2.6 perpendiculareV noted in [ 70to] the correspond chain axis to (H-aggregate the strong coupling configuration) regime. [70], The may agreed be marked. values are implementable for the chain of 1D semiconductor quantum dots [14], which makes it another The number of emitters in [70] is N 10 25, the distance a  1.2 nm, tab, 2.6 23 meV, candidate for RBO experiments in the strong coupling regime. The suitable structure for RBO ultrastrong coupling was implemented via plasmon polaritons in parabolic semiconductor quantum wells [71]. The electronic feedback microcavity was composed of circular capacitor elements connected Appl. Sci. 2017, 7, 721 19 of 25

78 and 2/ 1.0 ns. The reachable values of the ac field 10 10 V/m for the frequencies

0 2.6 eV noted in [70] correspond to the strong coupling regime. The agreed values are implementable for the chain of 1D semiconductor quantum dots [14], which makes it another candidate for RBO experiments in the strong coupling regime. The suitable structure for RBO ultrastrong coupling was implemented via plasmon polaritons in parabolic semiconductor quantum wells [71]. The electronic feedback microcavity was composed of circular capacitor elements connected via the wires of finite inductance. The cavity has been tuned by changing the electrical Appl. Sci. 2017, 7, 721 18 of 24 size of the components. A parabolic potential was approximated by digitally alloying

GaAs/Al0.15 Ga 0.85 As layers. The well has been surrounded by Al0.3 Ga 0.7 As barriers, containing via the wires of finite inductance. The cavity has been tuned by changing the electrical size of Si- doping. The use of parabolic quantum wells enabled very large values of the ratio of R /0 the components. A parabolic potential was approximated by digitally alloying GaAs/Al0.15Ga0.85As for THz frequency range (  / = 0.205 for the lower doped sample and  / = 0.27 for the layers. The well has been surroundedR 0 by Al0.3Ga0.7As barriers, containing Si-δRdoping.0 The use of heavierparabolic doped quantum one) to wells be achieved enabled veryin [71]. large values of the ratio of ΩR/ω0 for THz frequency range (ΩR/ω0 = 0.205 for the lower doped sample and ΩR/ω0 = 0.27 for the heavier doped one) to be 4.4.achieved Stark Effect in [71 in]. One-Dimensional Atomic Chains In this section, we consider the 1D chain of two-level atoms driven by dc field only. For a 4.4. Stark Effect in One-Dimensional Atomic Chains single atom, the interaction with a dc field leads to the well-known Stark effect, which consists of the shiftingIn this section,of energy we levels consider [48]. the For 1D reachable chain of two-levelfield values, atoms the drivenshift is by small, dc field and only. considered For a single as a perturbationatom, the interaction of the energy with a dcspectrum field leads (weak to the coupli well-knownng regime) Stark [48]. effect, The which dc field consists couples of the all shifting eigen statesof energy of the levels atom, [48 ].thus For reachablethe multi-level field values, model thewith shift summation is small, and over considered the complete as a perturbation set of atomic of termsthe energy is applicable spectrum to (weakreal atoms. coupling Here, regime) we show [48]. that The the dc Stark field coupleseffect manifests all eigen itself states in of 1D the chains atom, too,thus thebut multi-levelits qualitative model picture with summation differs from over the the completesingle-atomic set of atomiccase. The terms main is applicable reason is to realthe performanceatoms. Here, weability show of that ultrastrong the Stark effectcoupling, manifests on itselfwhich in 1Dwe chainsfocus too,below. but itsThe qualitative system pictureunder considerationdiffers from the is single-atomicshown in Figure case. 18. The The main dipole reason moments is the performance of the atoms ability is assumed of ultrastrong to be coupling,oriented orthogonallyon which we focusto the below. chain Theaxis, system and the under dc field consideration is directed is under shown the in Figureangle 18., Thewith dipole respect moments to the axis.of the atoms is assumed to be oriented orthogonally to the chain axis, and the dc field is directe d under the angle α, with respect to the axis.

Figure 18. General illustration of the periodic two-level atomic chain used as a model, indicating Figure 18. General illustration of the periodic two-level atomic chain used as a model, indicating Stark effect in the ultrastrong coupling regime. The chain is placed obliquely between the armatures Stark effect in the ultrastrong coupling regime. The chain is placed obliquely between the armatures of plane capacitor with applied dc voltage. The dipole moments (blue arrowheads) are oriented of plane capacitor with applied dc voltage. The dipole moments (blue arrowheads) are oriented orthogonally to the chain axis. Dc ﬁeld (red arrowheads) are oriented angularly with respect to orthogonally to the chain axis. Dc field (red arrowheads) are oriented angularly with respect to the the chain axis. The choice of the angle allows for obtaining the arbitrary relative value between chain axis. The choice of the angle allows for obtaining the arbitrary relative value between Rabi- Rabi- and Bloch frequencies. and Bloch frequencies.

TheThe analytical analytical model, model, in in general, general, is issimila similarr to the to theRBO-model RBO-model developed developed and andused usedbefore. before. The waveThe wave function function is given is given by by |Ψ(t)i = ∑ ap(t) ap + bp(t) bp (32) p where the probability amplitudes satisfy the system ∂ap 1 i = ω − pΩ a + t a + + a − − Ω b , (33) ∂t 2 0 B p a p 1 p 1 R p ∂bp 1 i = − ω + pΩ b + t b + + b − − Ω a , (34) ∂t 2 0 B p b p 1 p 1 R p Appl. Sci. 2017, 7, 721 19 of 24

−1 −1 where ΩR = e} dabEDc sin α, ΩB = e} aEDc cos α, are the Rabi- and Bloch frequencies, respectively. In contrast with RBO, these two frequencies are defined by the same field (dc), but their relative values may be arbitrary, via the corresponding choice of the field direction (angleα). Let us assume that ΩR ≈ ω0 (35) and ΩR >> ΩB. (36) Equation (35) corresponds to the resonant condition. It is a counterintuitive resonance, which does not couple with any oscillating forces, and represents the attribute of a ultrastrong coupling regime. This resonance legitimates the using of a two-level model, in contrast with the single-atomic case. This inequality (36) means high amplitude of BO. It allows us to separate the inter-atomic and intra-atomic motions, and describe the last one in terms of quasiclassical approximation. The wave function of inter-atomic motion is given as a superposition of eigen states (12) and (13). The account of intra-atomic motion is made via exchange (24). As a result, the total motion is described by the wave function (15) with q 2 2 2 Ω(t) = (ta − tb) cos (ΩBt − ϕ0) + ΩR, (37)

ΩR Λ(t) = q . (38) 2 2 2 (ta − tb) cos(ΩBt − ϕ0) + (ta − tb) cos (ΩBt − ϕ0) + ΩR The application of this solution in relation to tunneling current (21) leads to the spectrum (29), in the case of Stark effect. Another approach consists in the numerical solution of Equations (33) and (34), which is simplified by the absence of oscillation coefficients. The typical behavior of space–time distribution of inversions and tunneling currents, as well as the frequency spectrum of tunneling currents, are shown in Figures 19–21. The scenario of the process may be explained in the following way: the ordinary Stark effect for the single atom consists in the perturbed energy spectrum and mixing of states due to the dc field [48]. In our case, the situation is similar, however, the energy shifting is not small. The system is initially prepared in the case of non-perturbed states, which is not related to the eigen modes of the system with the dc field applied. As a result, the new eigen modes become coupled, and we observe their mutual oscillations in inversion with Rabi-frequency (see Figure 19). These oscillations propagate along the chain with Rabi-period, due to the tunneling, and produce the intra-zone current (Figure 20). The value of this current strongly increases due to the resonant condition (35). The spectrum of this current is a set of lines ω = nΩR, n = 1,2, ... (see Figure 21). In the opposite case ΩR << ΩB (dc field directed predominately along the chain), the inter-zone coupling becomes small, which leads to the suppression of intra-atomic motion. Thus, the total motion adds up to the oscillatory inter-atomic tunneling, which corresponds to the ordinary BO. Appl. Sci. 2017, 7, 721 21 of 25 coupling becomes small, which leads to the suppression of intra-atomic motion. Thus, the total Appl. Sci. 2017, 7, 721 21 of 25 motion adds up to the oscillatory inter-atomic tunneling, which corresponds to the ordinary BO. coupling becomes small, which leads to the suppression of intra-atomic motion. Thus, the total Appl.motion Sci. 2017 adds, 7, 721up to the oscillatory inter-atomic tunneling, which corresponds to the ordinary BO.20 of 24

Figure 19. Space–time distribution of inversion density with Stark effect. The initial state of the 2 chain is an excited single Gaussian wave packet ag0exp jja 22 / , Figure 19. Space–time distribution of inversion density with Starkj  effect. The initial  state  of the chain Figure 19. Space–time distribution of inversion densith y with0 Stark2 2 2effect.i The initial state of the is an excited single Gaussian wave packet aj(0) = g exp −(j − j ) a /σ , bj(0) = 0. Gaussian initial bj 00 . Gaussian initial position and width are j  80 , g  20 , respectively,2 22 chain is an excited 0 single Gaussian wave packet ag0exp jja / , position and width are j = 80, g = 20, respectively, ΩB = 0.9jω0, ta = tb =9.0 , number  of atoms N=B 128.0.90 , ttab9.0 , number of atoms N  128 . bj 00 . Gaussian initial position and width are j  80 , g  20 , respectively,

B 0.90 , ttab9.0 , number of atoms N  128 .

Figure 20. Space–time distribution of tunneling current density with Stark effect. All parameters Figureare identical 20. Space–time to Figure 19distribution. of tunneling current density with Stark effect. All parameters are identical to Figure 19. Figure 20. Space–time distribution of tunneling current density with Stark effect. All parameters are identical to Figure 19. Appl. Sci. 2017, 7, 721 21 of 24 Appl. Sci. 2017, 7, 721 22 of 25

Figure 21. The frequency spectrum of tunneling current density with Stark effect. All parameters Figure 21. The frequency spectrum of tunneling current density with Stark effect. All parameters are are identical to Figure 15. identical to Figure 15.

5. Conclusions 5. Conclusions In summary, we developed the physical model of 1D two-level atomic chain, with the tunneling In summary, we developed the physical model of 1D two-level atomic chain, with the inter-atomic coupling simultaneously driven by dc and ac fields in the regimes of strong and tunneling inter-atomic coupling simultaneously driven by dc and ac fields in the regimes of strong ultrastrong coupling. The approximate analytical solutions and results of computer modeling of and ultrastrong coupling. The approximate analytical solutions and results of computer modeling Rabi-Bloch oscillations have been presented and discussed. There is also considered limited cases, of Rabi-Bloch oscillations have been presented and discussed. There is also considered limited such as the Stark effect, Rabi-waves, and Bloch oscillations. The temporal dynamics, as well as cases, such as the Stark effect, Rabi-waves, and Bloch oscillations. The temporal dynamics, as well the frequency spectra of tunneling currents, have been presented. It was shown that the passage to as the frequency spectra of tunneling currents, have been presented. It was shown that the passage ultrastrong coupling dramatically changes the scenario of Rabi–Bloch oscillations. This manifests to ultrastrong coupling dramatically changes the scenario of Rabi–Bloch oscillations. This manifests in the appearance of additional lines in the spectra of the tunneling current. These lines are identified in the appearance of additional lines in the spectra of the tunneling current. These lines are as a contribution of anti-resonant interactions, and remain even in the case of high losses characteristic identified as a contribution of anti-resonant interactions, and remain even in the case of high losses of ultrastrong coupling. The promising potential applications of the obtained results in novel types of characteristic of ultrastrong coupling. The promising potential applications of the obtained results THz spectroscopy for nano-electronics have been proposed. in novel types of THz spectroscopy for nano-electronics have been proposed. Acknowledgments: Gregory Slepyan acknowledges support from the project FP7-PEOPLE-2013 -IRSES-612285Acknowledgments: CANTOR. Gregory Slepyan acknowledges support from the project FP7-PEOPLE-2013-IRSES-612285 CANTOR. Author Contributions: Developments of the physical models, derivation of the basis equations, interpretation ofAuthor the physical Contributions: results Developments and righting the of the paper physical have models, been done derivati by Ilayon of Levie the basis and Gregoryequations, Slepyan interpretation jointly. Theof the MATHEMATICA physical results Numericaland righting Python the paper calculations have been and done figures by were Ilay producedLevie andby Gregory Ilay Levie. Slepyan jointly. The ConflictsMATHEMATICA of Interest: NumericalThe authors Python declare calculations no conflict and of figures interest. were produced by Ilay Levie. Conflicts of Interest: The authors declare no conflict of interest. References

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