Floquet engineering-based frequency demodulation method for wireless THz short-range communications

This study introduces a novel theoretical framework for detecting and decoding wireless communication signals in the nanoscale range operating at terahertz (THz) frequencies. Initially, we investigate the Floquet states in a dressed 2D semiconductor quantum well and derive an analytical expression to determine its longitudinal conductivity. The results indicate that the longitudinal conductivity of a dressed 2D semiconductor can be tailored to specific requirements by manipulating the frequency of the external dressing field. Furthermore, carefully selecting the intensity and polarization type of the external dressing field enables fine-tuning and optimization of the conductivity. To evaluate the effectiveness of each dressing field configuration, we present a figure of Merit (FoM) assessment that determines the maximum possible change in conductivity within the considered frequency range. The proposed theory introduces a mechanism capable of identifying frequency-modulated communication signals in the THz range and performing frequency demodulation. We comprehensively analyze of the demodulator’s transfer function in the receiver. Consequently, we establish that the transfer function exhibits linear behavior over a specific frequency range, rendering it suitable for frequency demodulation. Finally, we provide a numerical illustration of a frequency demodulation scenario. The breakthrough uncovered in this study opens up possibilities for the development of high-efficiency, lightweight, and cutting-edge chip-scale wireless communication devices, circuits, and components.


Introduction
For several decades, the ever-growing need for rapid data sharing and processing across various applications and services has been a driving force behind the continuous growth in bandwidth requirements for wireless communication systems. This phenomenon, commonly known as Edholm's Law, posits that bandwidth and data rates will approximately double every 18 months [1,2]. As wireless networks approach their maximum capacity, there is an escalating need for higher data rates. Consequently, the exploration of higher frequency bands, such as millimeter wave (mmWave), terahertz (THz), or optical frequencies, has gained significant importance. These advanced technologies have been employed in state-of-the-art short-range wireless communication systems. Especially, the research community is actively investigating the potential of higher frequency bands to meet the ever-growing demands for bandwidth and data rates in chip-scale applications. Modern chip-scale systems are currently well-positioned to adopt mmWave technology within a few gigahertz (GHz) bandwidth ranges. However, considering the increasing demand for higher throughput, this choice may present limitations. The aggregate capacity of the channel at these frequencies may not be sufficient to address both current and future requirements. On the other hand, optical wireless communication (OWC) systems offer substantial bandwidth capabilities and can support high data rates. However, these systems face challenges due to safety regulations concerning skin exposure, which impose limitations on power transmission budgets. Moreover, optical receivers in OWC systems are susceptible to background noise, including shot noise generated by ambient light sources, thereby compromising their performance [2,3]. Considering the constraints posed by mmWave and OWC technologies, THz communications are being regarded as a promising candidate for achieving ultra-high data rates in chip-scale communication applications.
Multi-core chip systems are pervasive in contemporary computing, incorporating multiple processing units within a single device. In recent years, manufacturers have endeavored to enhance performance by augmenting the number of processing units and reducing the size of chip systems. Consequently, the substantial increase in the number of individual computing nodes within a unit volume has led to a notable rise in the complexity of interconnections among them. Moreover, conventional interconnects demonstrate sluggish and unwieldy performance, resulting in communication, rather than computation, becoming the primary bottleneck in the overall performance of multi-core processing systems. To address this issue, wireless communication techniques operating in the THz range offer a promising opportunity for high-speed wireless interconnections [2,4]. Furthermore, in recent years, there has been a notable surge in the advancement of compact, efficient, and integrated THz technology for generating and detecting THz signals at the chip-scale [5]. These advancements are a culmination of interdisciplinary collaborations involving multiple fields, including low-dimensional substances [6][7][8], electronic semiconductors and photonic devices [9,10], heterogeneous integration [11,12] and system packaging [13,14]. Moreover, recent research has been directed towards miniaturizing THz technology through the utilization of plasmonic nanostructures [15][16][17], quantum-cascade lasers and spasers [18][19][20][21], quantum thermal devices [22][23][24], nanowires [25,26], plasmonic waveguides [27][28][29], and novel metamaterials [6,[30][31][32].
The advancement of a diverse array of THz technologies has successful integrated electronics and photonics into comprehensive system-level solutions [5,33]. The successful application of photonic techniques in generating and modulating carrier signals for transmitters has not only improved data rates but also accelerated the development of potential applications. This achievement is made possible by utilizing high-frequency telecom components such as lasers, modulators, and photomixers [34,35]. The adoption of photonic technology in these components enables the creation of compact and lightweight transmitter front-ends. Conversely, electronic methods are predominantly employed in the research studies for receiver development. Various techniques have been explored for detecting THz wireless signals, but the most commonly used method involves a waveguide-integrated detector that employs GaAs Schottky barrier diodes (SBDs) [33,36]. However, SBDs are susceptible to reverse leakage current, which can introduce inaccuracies in the measurement and control of high-frequency circuits [37,38]. The subsequent phase of the receiver mechanism involves demodulating the identified signal to separate the information signal from the carrier signal. Successful implementation of this task requires the integration of various signal processing components within the receiver system. Additionally, the receiver antenna must be aligned with the wavelength of the THz carrier signal, posing a challenge in miniaturizing the receiver end. The aforementioned factors contribute to the development of a receiver that is bulky, heavy, and unreliable, prompting researchers to focus on devising novel receiver technologies that are miniature and more power-efficient. This study presents a comprehensive theoretical framework for developing a nanoscale THz signal detector and demodulator utilizing Floquet engineering techniques. This contribution is aimed at advancing nanoscale wireless communication technology and could play a pivotal role in the design and fabrication of state-of-the-art communication components for future generations.
Recently, applying light-matter interaction to manipulate solid-state systems has become a central focus of research. This approach has gained significant interest due to its potential to induce novel quantum phases that are not achievable in equilibrium [39][40][41]. By employing powerful periodic drives, such as ultra-fast optical pulses, one can modify the quantum state of electronic or atomic degrees of freedom and influence the underlying microscopic interactions. As a result, it becomes possible to stabilize non-equilibrium states with tailored macroscopic properties that surpass what can be achieved with static systems. The concept of Floquet engineering has emerged from these advancements, which involves customizing the Floquet electronic band structure through a periodic drive to alter material properties [42]. This encompasses the discovery of novel non-equilibrium topological states of matter [43], the engineering of correlated quantum phases [44], and the manipulation of quantum many-body systems [42,45,46]. In the realm of Floquet physics, one can predict the behavior of driven systems without relying on perturbation methods by treating the quantum system and the electromagnetic field as a single composite quantum system, known as the dressed system. The external electromagnetic radiation applied in this context is called the dressing field. In a recent study by Wackerl et al [42], they derived a closed analytical expression for the direct current (DC) conductivity of a driven quantum system. The authors conducted a comprehensive investigation that involved both theoretical and numerical analyses. Their study challenged the conventional perturbation approach and revealed that the previous findings [47,48] had overemphasized the effect of the driving field on transport properties. The analysis employed the effective Hamiltonian generated by the driven-induced gauge fields, which provided an accurate depiction of the distinctive features of the driven quantum system. By utilizing the time-dependent effective Hamiltonian approach, precise solutions for the time-dependent Schrödinger equation can be obtained. The utilization of Floquet theoretical techniques enables the dynamic generation of intriguing and exotic quantum properties in target materials by selecting a compatible dressing field. Theoretical investigations on the application of Floquet physics can be found in various subfields of physics and engineering [42,49,50]. However, a comprehensive exploration of its implementation in nanoscale wireless communication for data demodulation techniques remains a subject of further inquiry. Thus, this study employs the Floquet-Drude conductivity expression and the Floquet-Fermi golden rule [40,42] to investigate the correlation between the frequency of the dressing field and the longitudinal conductivity of a two-dimensional (2D) semiconductor under illumination. Subsequently, we conduct an in-depth examination of utilizing the Floquet formalism in information processing techniques focused on chip-scale wireless communication.
The current study investigates the electrical conductivity of a dressed 2D semiconductor quantum well and presents a methodology for identifying and decoding THz-range wireless communication signals. To begin with, the analysis focuses on exploring the Floquet states in a dressed 2D semiconductor quantum well. In this context, it is assumed that the 2D quantum well behaves as a 2D electron gas (2DEG) with a parabolic dispersion relation. Despite its simplicity, this simplified model effectively incorporates various prevalent semiconductor materials, such as gallium arsenide (GaAs). Subsequently, a theoretical expression for the longitudinal conductivity is derived, fully utilizing the Floquet-Drude conductivity formula. Our findings demonstrate that the longitudinal conductivity of a 2D semiconductor quantum well can be tailored based on the frequency, intensity, and polarization type of the external dressing field. Special attention is given to the influence of frequency on the conductivity variations. In a quantum Floquet system, the radiation frequency impacts the Floquet state, thereby offering the opportunity to customize the charge transport properties under external radiation. Detailed numerical computations are carried out to investigate the association between the conductivity of the dressed 2D semiconductor quantum well and the frequency of the dressing field, considering diverse aspects such as the intensity and polarization type of the dressing field. Building upon this understanding, our study proposes a theoretical approach to effectively detect modulated communication signals in the THz frequency range and perform frequency demodulation. We examine a wireless communication system consisting of a straightforward photonic-inspired transmitter and a 2D semiconductorbased receiver. The transmitter is assumed to employ the continuous-phase frequency-shift keying (CPFSK) technique to modulate the digital message signal into the transmitting signal. Furthermore, it is assumed that the wireless signal propagation occurs along a clear, direct path between the transmitter and receiver. The transmitted signal undergoes a series of focusing lenses before being concentrated on the 2D semiconductor quantum well in the receiver. In the receiver, the 2D semiconductor quantum well experiences the transmitted modulated signal, which acts as a dressing field and affects the transport properties of electrons in the lowdimensional semiconductor system. By utilizing our knowledge of the longitudinal conductivity of the dressed 2D semiconductor quantum well, we can distinguish frequency changes in the received signal. Subsequently, the digital message signals can be decoded using a basic voltage divider and comparator. We present a comprehensive analysis of the received modulated signal and the transfer function of our proposed receiver system in the presence of a dressing field. Our analysis demonstrates that employing a GaAs-based 2D quantum well can establish a linear correlation between the output voltage and the received signal frequency within a limited slot of the THz frequency spectrum. Additionally, we predict the detection voltage output for each digital message signal within this frequency window. As a final step, we provide numerical evidence showcasing the ability of a proposed Floquet engineering-based 2D semiconductor quantum well receiver to perform FSK demodulation and effectively recover digital information signals. The findings of our study introduce an innovative methodology for receiving and interpreting digital modulated signals within the THz range by employing a semiconductor substance with a thickness of only a few nanometers. The results of our study unlock a whole new realm of possibilities for developing cutting-edge chip-scale wireless communication technology. With the potential for high-efficiency and lightweight designs, our findings will open up exciting new directions for creating next-generation wireless devices, circuits, and components. The implications of this breakthrough will undoubtedly pave the way for innovative advancements in the field of chip-scale wireless communications.

Theoretical formalism
This section presents a theoretical foundation for the behavior of 2D semiconductor quantum well when subjected to a high-intensity dressing field. Firstly, we establish the wave function solutions for Floquet states in the dressed 2D quantum system. We then apply this solution to the general Floquet-Drude conductivity formula to explore the connection between the longitudinal conductivity and the parameters of the dressing field. Lastly, we derive the transfer function for the proposed receiver.

Floquet states in dressed 2D semiconductor quantum well
The system of interest is a 2D semiconductor quantum well exposed to a dressing field. Here, we assume that the electrons present in the 2D semiconductor exhibit free movement and function as a 2D electron gas (2DEG). Additionally, we employ a free electron model with a parabolic dispersion to simulate the behavior of the 2DEG system. While the model does not explicitly incorporate the crystal lattice potential and electron-electron interactions, we can make modifications to address these limitations to some extent. In the present modification, the electron mass m transform into an effective electron mass denoted as m e [51,52]. Further, we assume that the 2D semiconductor is an isotropic medium and m e becomes simply a scalar. Now, consider an isolated 2DEG subjected to an external high-intensity electromagnetic field as illustrated, in figure 1. It is crucial to note that selecting the driving field frequency in the off-resonant regime is imperative to prevent photon absorption and system heating [53]. Following this selection, the driving field demonstrates characteristics of a pure (unabsorbable) dressing field within the context of our analysis. Considering the significant difference in scale between the wavelength of the dressing field and the thickness of the 2D semiconductor quantum well, it is reasonable to presume that the dressing field exhibits uniform behavior within the 2DEG at a particular point in time. To understand the actions of an individual electron within the dressed 2DEG, it is necessary to analyze its wave function. The time-dependent Schrödinger equation must be satisfied by the wave function solution ψ(r, t) of a single electron [45,54] Here, ÿ is the reduced Planck's constant, e is the magnitude of the elementary charge, is the position vector in the 2D coordinate space, and A(t) is a vector potential representation of the applied dressing field. It is essential to acknowledge the significance of adopting the Coulomb gauge for the vector potential in deriving this expression.
Initially, we contemplate that the dressing field being applied is polarized linearly and exhibits an electric field that is polarized in the x-direction. Thus, we can represent the electric field as Here, E is the amplitude of the electric field, ω is the angular frequency of the dressing field, and d  is the unit vector that is pointed to the subscript direction d = x, y. For this scenario, we can model the linearly polarized dressing field using the following vector potential with Coulomb gauge assumptions The Gordon-Volkov method [45,55] enables the identification of precise solutions to the time-dependent Schrödinger equation. By using the principles of Floquet theory [56] and considering the finite dimensions of the 2DEG, it is possible to represent these wave function solutions, as Floquet states in the momentum space in a comprehensible manner Figure 1. The 2D semiconductor quantum well is positioned in the xy-plane and exposed to a dressing field oriented perpendicular to the same plane. Low-resistance ohmic contacts connect the P 1 and P 2 points to the 2D semiconductor quantum well.
where ò k are the quasienergies, u(k, t) are the time-periodic Floquet modes for each quantized wave vector k values. As a consequence of the time-periodicity inherent in Floquet modes, it is possible to express them as a Fourier expansion . Readers interested in the details of the derivation can find them in the supplementary information file, section A.

Floquet-Drude conductivity
In recent research by Wackerl et al [42], a novel theoretical framework has been proposed that integrates linearresponse theory and the Floquet formalism to investigate the Drude conductivity.
This framework presents a comprehensive methodology for predicting the Drude conductivity in the presence of a high-intensity electromagnetic field. As depicted in figure 1, our system comprises a 2D semiconductor quantum well subjected to a high-intensity electromagnetic field, known as the dressing field. In a subsequent discussion, we leverage the effects of this dressing field to propose a nanoscale frequencydemodulator, utilizing the modulated transmitted signal as the dressing field. Prior to delving into that aspect, it is necessary to examine the relationship between the Drude conductivity and the characteristics of the dressing field. By assuming that the electrons in the 2DEG system follow the free electron model with a single energy band, we can anticipate the x-directional DC-limit of the conductivity when the probe bias is applied across the x-direction. Using figure 1 as a reference for our system, we can calculate the longitudinal DC conductivity between P 1 and P 2 points as Here, the particle distribution function is denoted as f, while k  represents the area of the momentum space. The quantity corresponds to the n-th diagonal element of the inverse scattering time matrix [42]. The derivation of this expression involves the utilization of the Floquet-Fermi golden rule and the t-t¢ formalism [42,45,49]. According to the Floquet-Fermi golden rule, the inverse scattering time 1/τ(ò, k) of electrons in a dressed system can be expanded twice using the Fourier series expansion. Consequently, the resulting representation is the inverse scattering time matrix, with signifying the (n,n)-th element of the Fourier expansions. In order to calculate the cumulative impact of the scattering time, it becomes necessary to account for the contribution of each diagonal element of this matrix to the expression for conductivity, as demonstrated in equation (12).
In the investigation of actual substances, the occurrence of irregularities that hinder the motion of conduction electrons is a common issue, leading to reduced conductivity. The inverse scattering time is a crucial parameter for evaluating the effect of these irregularities on electron transport. The existence of such disorders can be simulated by introducing a time-invariant scattering potential perturbation V(r) in the semiconductor, represented as an assemblage of randomly dispersed defects with Gaussian white noise characteristics [42,51]. Essentially, these assumptions enable us to express the total scattering potential in the 2DEG as an aggregate of independent single-impurity potentials υ(r). Within the framework of disorder potential assumptions, the inverse scattering time matrix in a Floquet quantum system can be evaluated utilizing the Floquet-Fermi golden rule [42,45] which states that , and |k〉 is a bare electron state with k momentum. imp · á ñ presents the statistical average over realizations of the impurity disorder. Furthermore, we make the assumption that the system under consideration is a fermion system with a single energy band. Our focus is on the high-frequency regime, satisfying the condition of ωτ 0 ? 1. Here, τ 0 represents the scattering time of a non-driven system. With this assumption, we observe that the nonzero elements of the inverse scattering time matrix do not significantly contribute to conductivity. Therefore, the conductivity can be approximated by solely considering the contribution of the central element in the scattering time matrix [42]. This approximation can be evaluated by substituting n = 0 into the inverse scattering time matrix, as defined in equation (13). Moreover, the Fermi-Dirac distribution can be chosen as the particle distribution function for the dressed 2DEG where k B is the Boltzmann constant, T is the absolute temperature, and ò F is the Fermi energy of the system. It is essential to highlight that the Fermi energy is a constant quantity that is defined within a truncated momentum range, which is confined to the central Floquet state energy zone [42]. Moreover, at low temperatures conditions, i.e., k B T = ò F , it is possible to approximate the derivative of this distribution by a delta function, which peaks near the Fermi energy Substituting this finding back into the conductivity formula given in equation (12) for the central Floquet zone yields Here, we present the central element of the inverse scattering time matrix using . In this study, we aim to examine the influence of impurity-induced scattering on the energy bands and observe a noticeable phenomenon of narrow broadening in comparison to the energy levels of the electrons. As a result, the scattering time of electrons due to impurities increases while the reciprocal scattering time of conduction electrons decreases. Given these circumstances, we can employ a delta-distribution approximation technique to streamline our calculations (see appendix B for detailed derivation).
where we introduced new parameter solely for the purpose of simplifying notation representation Finally, we can substitute this derivation back into equation (17), and this leads to As a subsequent phase in our study, we aim to determine the inverse scattering times of various polarization types utilizing Floquet mode expressions. Our initial focus involves calculating the inverse scattering time of a 2DEG subjected to a linearly polarized dressing field. Accordingly, we assess the following parameter for the linearly polarized dressing field Applying the generalized Neumann's addition theorem of generalized Bessel functions[57] Substituting this expression back into the equation (13), we can find that By employing the continuum limit of 2D momentum space in polar coordinates, we are able to derive Subsequently, we may apply analogous procedures to the dressing field with circular polarization, leading to the discovery that Substitute this back into the equation (13), and simply to get Then, we can express the central element of the inverse time matrix as The Floquet-Drude conductivity expressions for each polarization type can be derived by utilizing the identified inverse scattering time values. In the case of a linearly polarized driving field, we hereby present the longitudinal conductivity We have discovered that the conductivity of the dressed 2DEG in the x-direction is influenced by both the intensity and frequency of the dressing field. Specifically, our research has primarily centered on examining how alterations in frequency impact the Floquet-Drude conductivity. Similar to the linearly polarized dressing field, in the context of the circularly polarized dressing field, it can be discerned that The Floquet-Drude conductivity, when exposed to a circularly polarized field, is subject to alterations that are dependent on both the field's frequency and intensity, as is observed in the case of linear polarization. Additionally, we can observe that the polarization type of the dressing field can also impact the conductivity of the dressed system. In the next section, we conduct a numerical analysis to quantify and gain a deeper comprehension of these effects.

Ramifications of the Floquet-Drude conductivity morphology
In this part of the study, we report the quantitative assessment of our theoretical models regarding the correlation between the longitudinal conductivity of the dressed 2DEG and the frequency of the dressing field.
Our investigation also involves examining the impact of variations in the intensity and polarization techniques of the dressing field to optimize the system configuration. Additionally, we conduct a thorough analysis of our proposed receiver system to determine the ideal frequency range for transmitting the modulation signal. Finally, we present a comprehensive overview of the chip scale wireless communication system and demonstrate its full operation. The full MATHEMATICA code for the numerical calculations is available under the supplementary materials.

The correlations of conductivity
First, we assume that the 2D quantum well is comprised of GaAs and as such, we have incorporated its specific material parameters into our computations [58]. Unless specified otherwise, the following parameters are used in the numerical calculations: the effective mass of GaAs m e = 0.071m, the Fermi energy of GaAs ò F = 0.01 eV, the reference intensity of the dressing field I 0 = 100 mW cm −2 , and reference frequency of the dressing field f c = 0.1 THz. In order to simplify the numerical analysis, we will define several normalized parameters. Firstly, we shall introduce the normalized longitudinal conductivity in the x-direction as Here, ω c = 2πf c , I = cε 0 E 2 /2 is the intensity of the dressing field, ε 0 is the vacuum permittivity, and c is the speed of light. In our study, we have demonstrated the experimental viability of our findings under specific system conditions. Our results indicate that in order to attain a driving frequency in the tens or hundreds of terahertz range, a large Fermi energy is required, surpassing the limit of the parabolic approximation. If the Fermi energy is on the meV scale, a driving frequency of several hundred terahertz only induces a significant change in conductivity when the intensity of the driving field reaches the GW cm −2 level, which is impractical to achieve in experimental settings.
As an initial analysis, we examine the correlation between the normalized longitudinal conductivity in the xdirection and the frequency of the dressing field. This investigation includes an evaluation of various intensities of the dressing field and two types of polarization types. Figure 2 portrays the changes in the normalized longitudinal conductivity xx s in response to the normalized angular frequency of the dressing fieldw, which can be either linearly or circularly polarized. Based on the presented graphical correlations, we can deduce that the normalized longitudinal conductivity can be tailored through the intensity and frequency of the applied dressing field. Specifically, at a constant intensity level, a decrease in frequency results in an increase in longitudinal conductivity, while higher intensity levels lead to higher conductivity values at a constant frequency. These findings hold true for both linear and circular polarized dressing fields. However, the enhancement attainable through circular polarization surpasses that of linear polarization under the same circumstances. This discovery is of great significance, as it offers potential applications in signal demodulation techniques in wireless communication. The dependence of conductivity on both the intensity and frequency of the dressing field allows for the exploitation of this relationship in both amplitude and frequency demodulation schemes. With the achieved results, we recommend utilizing circularly polarized dressing fields as the carrier signal for THz short-range wireless communication due to the greater enhancement achieved in this configuration.

Figure of Merit
The objective of this study is to present an innovative nanoscale frequency demodulation mechanism based on the obtained findings. The proposed mechanism entails keeping the intensity of the dressing field constant while adjusting its frequency according to the information signal. The system's conductivity change per unit frequency change is a crucial factor for achieving better performance. In the previous analysis, circular polarization was identified as the best method for achieving high-conductivity change. However, determining the appropriate intensity level to achieve optimal performance poses a challenge. Although a higher intensity level results in an enhanced conductivity change, it also necessitates more power and may not be the most efficient solution. Therefore, a figure of Merit (FoM) calculation is necessary to assess and compare the available options. Here, we introduce a FoM that determines the maximum conductivity change attainable when   0. 9 1.1 w , per unit of normalized intensity where I I I 0 = . Table 1 presents the outcomes of the assessment of FoM values for two distinct polarization techniques across four levels of intensity. The findings presented in this study unequivocally establish the superiority of circular-polarization method over linear-polarization method in terms of performance. Moreover, the FoM measurements indicate that the use of circular polarization with I = 1I 0 , I = 1.5I 0 , and I = 2I 0 intensity levels yield higher efficiency. Consequently, for further investigations, employing the dressing field possessing I > I 0 intensity values and circular polarization is recommended. However, the behavior of the transfer function must be scrutinized before finalizing this selection.

Receiver transfer function
Now, we investigate the receiver component of our digital wireless communication system, focusing specifically on the constituent elements and transfer function of the demodulator. Our non-coherent receiver model is based on a simple voltage divider circuit, illustrated in figure 3, where the upper resistor is represented by a 2D quantum well that is exposed to the modulated signal transmitted. We assume the unexposed resistance of the 2D quantum well to be R 0 . Additionally, a second resistor is incorporated in the lower portion of the circuit, with We use dashed lines to illustrate the relationship under the linearly polarized dressing field, while solid lines are used to represent the relationship under the circularly polarized dressing field. Furthermore, the relationship has been evaluated under four different dressing field intensity I = 0.5I 0 , I 0 , 1.5I 0 , 2I 0 levels. Here, I 0 is the reference intensity level, and ω c is the reference angular frequency of the dressing field. Here, we introduce σ 0 , which denotes the longitudinal conductivity of a 2D quantum well in an unexposed state along the x-direction. We investigate the behavior of the 2D quantum well as a THz modulated signal detector, whereby the current in the voltage divider i det is driven by the static voltage source V in and resistance of the 2D semiconductor quantum well. By measuring the current with a current meter, we are able to detect the corresponding digital signal. However, we propose a voltage measuring method for detecting the modulated information signal by measuring the output voltage V out of the voltage divider.
For a given time interval lT b < t < (k + 1)T b , we can write the transfer function of the voltage divider as Here, we make the assumption that the constant resistor in the voltage divider is independent of the frequency of the current flow. Through the substitution of the expression derived in equation (37), we are able to simplify this aspect of our analysis as follows Using the expression present in equation (30) and (32), we can find that Next, we showcase the transfer function behavior across a wide range of frequencies using the same system parameters used in the previous calculations. This is visually represented in figure 4 (a). Here, we discovered that circularly polarized field based transfer functions offer superior gains compared to linearly polarized field-based transfer functions for a given intensity level. At high frequencies, all transfer functions converge to a value of 0.5 as the normalized conductivity approaches 1, which is attributed to the behavior of Bessel's functions. When the frequency falls below the selected ω c , transfer functions exhibit non-linear behavior and converge to 1, a common occurrence for both linearly and circularly polarized fields. However, by focusing on the region outlined in red lines in figure 4 (a), we observe that most transfer functions behave linearly. As we aim to Figure 3. Signal receiver comprises a 2D semiconductor quantum well that is connected to a static voltage via a voltage divider. V in represents the static input voltage to the system, while V out is a voltage output that is correlated to the digital signal detected by the receiver.
demodulate frequency-modulated signals through this system, we conducted a detailed analysis of transfer function behavior in this region, which we have re-illustrated in figure 4(b). If we analyze the frequency range of our proposed receiver within the range shown in figure 4 (b), we can assume that the transfer functions operate linearly with respect to frequency for the most part. However, it is crucial to determine the optimal system performance by evaluating their linearity behavior. To accomplish this, we can utilize linear regression predictions to determine the error percentages for the calculated data points of each available option and select the best one. Table 2 presents the linear regression function approximation related mean absolute percentage errors (MAPEs) for all available options in the study. Here, we consider only the frequency range   0. 9 1.1 w . Based on the analysis presented in table 2, it can be concluded that all the considered transfer functions exhibit very low errors in comparison to their linear predictions. This finding supports our assumption of linear behavior for these gain functions within the studied frequency range. Additionally, the results indicate that for both linear and circular polarization based radiation, the option with the highest intensity yields the minimum error. Thus, by considering the linearity of the transfer function in the frequency domain, we can select options with I = 2I 0 as the potential solutions. However, when comparing these findings to the results of the FoM analysis presented in table 1, we can disregard the linear-polarized based transmission. Finally, based on comparing the FoM and linearity analyzes, we select a circularly polarized field with I = 2I 0 option for our subsequent demonstration of data demodulation. The theoretical data points and linear approximation for the considering frequency range is illustrated in the figure 5.

Demonstration of system operation
The behavior of 2D quantum wells under the influence of dressing fields and the receiver's transfer function has been thoroughly studied and analyzed. Based on these findings, we propose a theoretical mechanism for demodulating frequency-modulated signals at the chip scale. This mechanism is based on a simple system comprising a signal transmitter and receiver, which is illustrated in figure 6. Here, the transmitter employs a frequency modulation technique to modulate a digital data signal onto a carrier signal. The system utilizes THz range wireless carrier signals as the medium for information transfer. The transmitter consists of a frequency modulator component, which modulates the THz signal generated by the THz range laser with the digital input data and amplifies it before transmission through a wireless signal emitter. The transmitted wireless signal is then focused on the receiver using a system of lenses. The proposed system is designed for short-range data transfers, such as chip scale wireless communications. The receiver of this proposed system is unique, as it is made of a 2D quantum well with nanoscale dimensions, making it compact and lightweight. The modulated signal is focused onto the 2D quantum well, which is known to alter its conductivity according to the applied  w against the normalized frequencyw for that frequency window. field's frequency. By providing a static voltage to the 2D quantum well, different current flows can be achieved according to the modulated frequency values, allowing for the demodulation of the carrier wireless signal and retrieval of the digital information signal. Initially, an investigation can be conducted on the signal transmission. Digital data transmission usually requires continuous waveform modulation to generate a bandpass signal suited to a transmission medium [59]. Sinusoidal carrier waves can be modulated in amplitude, frequency, or phase with a digital signal. If the modulating information signal comprises non-return-to-zero (NRZ) rectangular pulses, then the modulated parameter will switch from one discrete value to another. Three commonly used modulation schemes are amplitude-shift keying (ASK), phase-shift keying (PSK), and frequency-shift keying (FSK). This study focuses on FSK modulation-based digital information transmission and presents mathematical models to analyze it. In the general FSK scheme, a digital signal x(t) is used to control a switch that chooses the modulation frequency from a set of M oscillators. Due to the switching, the modulated signal is discontinuous at each switch, leading to relatively large sidelobes in the output spectrum if the amplitude, frequency, and phase of each oscillator are not properly adjusted. These sidelobes do not contain additional information and waste bandwidth. To overcome this issue, a variant of FSK, called continuous-phase FSK (CPFSK) can be used in which x(t) modulates the frequency of a single oscillator, resulting in a continuous phase modulation [59]. In our analysis, we assume that the transmitter system employs CPFSK modulation for signal transmission. We assume that our system takes binary NRZ rectangular pulses to represent the digital information signal, and it starts at t = 0. Thus, we can write the information signal with two digital states as  w against the normalized frequencyw for the circularly polarized field with the intensity I = 2I 0 . Here, the circular data points denote the theoretical data points obtained for the actual transfer function, while the solid line indicates the approximated linear relationship. Figure 6. The proposed system for short-range wireless communication in the THz frequency range comprises a transmitter (Tx) and a receiver (Rx). The receiver incorporates a 2D quantum well, which is subjected to a modulated dressing field.
x t a p t lT a where 1, 43 where a l represents a sequence of data digits with rate r b = 1/T b , and l is an integer. In addition, we can define the pulse shape of NRZ rectangular pulse as Here, u(t) is the Heaviside step function. After CPFSK modulation, we can identify the transmitting signal by where ω d is the modulation index. Here, A c ,ω c , and θ are the angular frequency, amplitude, and initial phase of the carrier signal, respectively. Now, we can simplify and show that x t A t a t lT p t lT cos , 46 With these findings, we can calculate the output detection voltages for each bit in the receiver. In the numerical calculations, the system utilizes the following parameters: Angular frequency of the carrier signal ω c = 0.1 THz, modulation index ω d = 0.05ω c , and data rate r b = 0.2ω d /(2π) = 1 GHz. We used a slightly lower bit rate for the purpose of demonstrating a simple scenario, but we can increase the bit rate to achieve higher performance if needed.
Let us assume that there exists a time instant, t, within the range of lT b < t < (k + 1)T b at the receiver end. Nonetheless, it is crucial to ensure that t is not a multiple of the data pulse width T b . Within this time interval, there are only two possible signal shapes can be expected. When a l = − 1, we get The received signal has a rectangular pulse shape that is modulated with sinusoidal amplitude. Here, the frequency of the sinusoidal is a bit lower than our carrier signal frequency ω c . If the received signal possesses a l =1, then we can obtain In this case, the frequency of the sinusoidal is a bit higher than our carrier signal frequency ω c . To proceed further, we can compute the Fourier transform of these two signals, which will allow us to determine their frequency domain representation. Using the convolution theorem and assuming the frequencies are in the positive range, we can derive the frequency spectrum of these two signals by The normalized frequency spectrum of these two possible receiving signals are illustrated in the figure 7. From the analysis of the figure, it can be inferred that the receiver signal is completely limited within the range of 0.9 1.1 w < < , where the transfer function of the receiver system is linear. Based on this, we can proceed to compute the average output voltage for each received signal, which is done in the following manner Then, evaluating this for each data state, we can obtain that , and 0.69240 . 54 Our nanoscale THz wireless receiver is capable of providing a 0.05293|V in | voltage difference to distinguish between the two digital data states. By using a voltage comparator and setting the reference voltage to the midpoint of the output voltage states, we can readily detect the digital information that has been modulated onto the wireless carrier signal. Figure 8 illustrates an example data signal and its corresponding modulated signal along with its frequency component. The bottom plot shows the output voltage from the receiver system, which is inverted due to the conductivity behavior of the dressed 2DEG. In order to increase data rates, our proposed system can be upgraded to an M-ary frequency shift keying (M-ary FSK) scheme, allowing for the transmission and reception of multiple bits of data within a single frequency. This proposed receive architecture can also be used to build an analog frequency modulation-based wireless communication system, given the linear behavior of the receiver's transfer function. However, the additional complexity associated with the continuous frequency range requires further analysis and will be left for future work. Our comprehensive theoretical model demonstrates the feasibility of detecting and decoding frequencymodulated signals in the THz frequency range using a 2D semiconductor quantum well. This breakthrough holds significant potential for the next generation of chip-scale wireless communication systems. The realization of a nanoscale frequency demodulation technique is pivotal to this promising advancement. Our findings are grounded in the observation that irradiating a 2DEG with a dressing field enhances its longitudinal conductivity by modifying the electron scattering probability. Electron scattering refers to the deflection of electrons from  . The top plot displays the data signal, which has two data states (a l = ± 1) that correspond to the message's bit value. The second plot illustrates the transmitted signal after the CPFSK modulation. The third plot displays the instantaneous frequency of the received signal, while the bottom plot predicts the normalized output voltage at the receiver. their original trajectory, resulting in a loss of kinetic momentum and reduced electrical conductivity. In 2DEG systems, scattering can occur through elastic processes induced by impurities and inelastic processes caused by phonons. Elastic scattering dominates in the 2D semiconductor quantum well when the temperature is low. Therefore, we assume our system operates under low-temperature conditions, where the primary contributor to damping effects in electron transport is electron scattering due to disorder impurities. The probability of scattering can be quantified by examining the overlap of wave functions between incident and scattered electrons, as described by Floquet-Fermi's golden rule [42,45]. The wave function terms depend on the intensity and frequency of the dressing field. Lower frequencies can reduce the stationary overlap of wave functions, thereby decreasing the scattering probability at low-temperature steady states. Consequently, the longitudinal conductivity experiences enhancement. The modified conductivity achieved through our approach paves the way for decoding the frequency-modulated signal in our proposed system. It is important to note that a generic interacting Floquet system generates heat when it absorbs radiation photons. To fully describe a dressed system, the interaction between photons must be taken into account. Although these interactions are complex, various strategies can be employed to mitigate the heating challenge and achieve non-equilibrium steady states [53]. For instance, operating in regimes where heating rates are strongly suppressed can lead to steady particle distribution functions in driven isolated quantum systems. In our analysis, we meticulously selected the carrier frequency and time scales to fulfill these conditions.
The presence of noise compromises the performance of our receiver mechanism by introducing fluctuations in the conductivity of the 2D semiconductor quantum well. Nevertheless, we can assume that high-frequency noise is attenuated by the receiver components according to our transfer function. Conversely, low-frequency noise affects the output voltage, but our transfer function maintains a constant gain within the low-frequency range. As a result, both digital signal states experience a similar voltage change. Given that we can achieve a larger voltage difference between the two data signal states, the effects of low-frequency noise are inconsequential to the behavior and outcomes of our proposed system. Moreover, inter-modulation distortion (IMD) arises when multiple frequency-modulated signals coexist within a system. The combination of frequencies from two simultaneous frequency-modulated signals generates new sum and difference frequencies. These newly formed frequencies can interfere with the original signals, leading to errors in information signal detection and decoding, thereby compromising communication accuracy or reducing signal quality. However, IMD can be mitigated by preventing overlap of modulated signals in the frequency or time domain. To achieve a highaccuracy receiver system with elevated data rates, we can employ a larger modulation index and advanced multiplexing techniques such as orthogonal frequency-division multiplexing (OFDM). Furthermore, as our intended application involves direct path communication over short distances within shielded environments like chips, we can disregard the impact of multi-path distortion and its associated effects.
The obtained results from this study demonstrate applicability across a diverse range of materials with compatible system parameters. Specifically, we have chosen the GaAs-based quantum well as the 2DEG under our investigation. Additionally, we have opted for an intensity in the range of 100 mW cm 2 for the dressing field, as this particular intensity level has been extensively utilized in numerous previous studies focused on Floquet engineering [42,45,60,61]. However, to achieve significant changes in conductivity within a narrow frequency range, selecting a dressing field from the THz region is imperative. Failure to do so, coupled with the properties of Bessel functions described in equation (41) and equation (42), would yield inconsequential outcomes. Thus, it is crucial to intelligently choose the frequency range of the dressing field if our objective is to effect material or intensity level modifications.

Conclusions
The present study investigated the electrical conductivity of a dressed 2D semiconductor quantum well and proposed a methodology for identifying and decoding THz-range wireless communication signals. The analysis focused on the Floquet states in a dressed 2D semiconductor quantum well and derived a theoretical expression for the longitudinal conductivity using the Floquet-Drude conductivity formula. The scattering effects of impurities were explored using the Floquet-Fermi golden rule. The results demonstrated that the longitudinal conductivity could be controlled by the frequency, intensity, and polarization type of the external dressing field. Numerical computations were conducted to examine the relationship between the conductivity of the dressed 2D semiconductor quantum well and the frequency of the dressing field. An FoM evaluation was also introduced to determine the maximum attainable conductivity change within the considered frequency range for each available dressing field configuration. A comprehensive analysis of the transfer function of the demodulator in the receiver of our digital wireless communication system was presented. It was shown that the transfer function behaved as a linear function within a specific frequency range, enabling its utilization for frequency demodulation. The study proposed a theoretical approach for detecting modulated communication signals in the THz frequency range and performing frequency demodulation. The proposed receiver system utilized a GaAs-based 2D quantum well and successfully executed FSK demodulation, recovering digital information signals. The observed modifications were primarily caused by the effect of the dressing field on the free electron wave function in the 2D semiconductor quantum well, resulting in reduced electron scattering attributed to impurities. This breakthrough introduces an innovative methodology for receiving and interpreting digitally modulated signals within the THz range, employing a semiconductor material with a thickness of only a few nanometers. These findings open up exciting possibilities for the development of highly efficient, lightweight, and advanced wireless communication technologies that have the potential to revolutionize chip-scale wireless communication. We defer certain crucial generalizations to future investigations. Recent literature [39,62] has introduced innovative analytical concepts pertaining to the Floquet-Hamiltonian, which accurately describes heating dynamics. While our findings were derived under stable, low-temperature conditions with specific system parameter selections, relevant studies imply that dissipation resulting from inter-particle collisions might induce heating in quantum Floquet systems at large. Consequently, we intend to employ this comprehensive depiction to elucidate the characteristics of more generalized systems.
where |k| = k. It is important to notice that in the free electron model of the 2DEG, we have assumed that the considering system is isotropic. Therefore, we can identify that k k k x x 2 2 = + , and k d is the electron wave vector component in the d-direction. This can be easily solved by direct integration over time t. Here, without loss of generality we have assumed that the dressing field is switched on at t = 0, and F(0) = 0. Then the solution for F(t) can be identified as Although after a significant time period F(t) function tends to go to infinite value, our wavefunction solution ψ (r, t) always takes a finite value. Because F(t) function only contribute for an oscillation motion to the wavefunction solution ψ(r, t). Given that the semiconductor sample has a finite size, we can conclude that the wave vector of the electron under consideration must be quantized. Consequently, multiple solutions for the wavefunction exist (for each quantum number of k value) Here, ε k = ÿ 2 k 2 /2m e is the quantized energy levels for a bare electron with quantized k values where L d is length of the considering system in the d-direction. Furthermore, the wavefunction of the dressed electron can be re-write as follows To proceed, we must employ Floquet theory to identify the quasienergies and time periodic Floquet modes for these wavefunction solutions. Floquet theory provides a separation of the time evolution into a periodic component and an exponential component, with the latter featuring the quasienergy [49]. As a result, we can factorize the wavefunction into a portion that varies linearly with time and a portion that varies periodically with time. This allows us to identify the quasienergies as follows e E m 4 , which only depend on the magnitude of the wave vector k. Furthermore, we can identify the Floquet modes as This can be re-written as Therefore, finally we can present the wavefunction solution for our dressed system as Floquet states in momentum space We can introduce the left-hand circularly polarized dressing field with an electric field without losing any generality in this scenario where E is the amplitude of the electric field, ω is the frequency of the dressing field, and d  is the unit vector showing into the subscript direction. In addition, it is possible to represent the dressing field in the Coulomb gauge as a vector potential Under the free electron model [63], we can choose a wavefunction solution ansatz for the time-dependent Schrödinger equation as same as the previous subsection Here, k is the electron wave vector, V is the volume of the metal sample, and F(t) is only a function of time that we need to identify. Here this solution represents a wave traveling in the positive r direction, and a corresponding wave traveling in the opposite direction as well. Then, we can obtain a first order differential equation by substituting this ansatz into the Schrödinger equation Here, without loss of generality we have assumed that the dressing field is switched on at t = 0, and F(0) = 0. By integration over the time, the solution for F(t) can be found as Although after a significant time period F(t) function tends to go to infinite value, our wavefunction solution ψ (r, t) always takes a finite value. Because F(t) function only contribute for an oscillation motion to the wavefunction solution ψ(r, t). Since we consider a finite size metal sample, we can identify that the wave vector of the considering electron has been quantized. This leads to a number of solutions (for each single quantum number of k) for the wavefunction Here, ε k = ÿ 2 k 2 /2m e is the quantized energy levels for a bare electron with quantized k values where L d is length of the considering system in the d-direction. In addition, it is possible to express the wavefunction of the dressed electron in the following manner.