Terahertz modulated optical sideband generation in graphene
Exploration of optical non-linear response of graphene predominantly relies on ultra-short time domain measurements. Here we propose an alternate technique that uses frequency modulated continuous wavefront optical fields, thereby probing graphene’s steady state non-linear response. We predict frequency sideband generation in the reflected field that originates from coherent electron dynamics of the photo-excited carriers. The corresponding threshold in input intensity for optimal sideband generation provides a direct measure of the third order optical non-linearity in graphene. Our formulation yields analytic forms for the generated sideband intensity, is applicable to generic two-band systems and suggests a range of applications that include switching of frequency sidebands using non-linear phase shifts and generation of frequency combs.
Owing to its gate tunable electronic, optical and opto-electronic properties, the exploration of non-linear optical effects in graphene has attracted significant interest in experiments Hafez et al. (2018); Yoshikawa et al. (2017); Prechtel et al. (2012); Jiang et al. (2018); Gu et al. (2012); Hendry et al. (2010); Yang et al. (2011); Zhang et al. (2012); Kumar et al. (2013); Hong et al. (2013) as well as in theory Mikhailov and Ziegler (2008); Glazov and Ganichev (2014); Cheng et al. (2014); Al-Naib et al. (2014); Tamaya et al. (2016a); Cheng et al. (2015); Mikhailov (2016); Rostami and Polini (2016); Gullans et al. (2013); Yao et al. (2014). Experimentally, several non-linear optical effects such as higher harmonic generation Yoshikawa et al. (2017); Hafez et al. (2018), third order non-linearity Jiang et al. (2018); Prechtel et al. (2012); Hendry et al. (2010); Yang et al. (2011); Zhang et al. (2012); Kumar et al. (2013); Hong et al. (2013) and four wave mixing Gu et al. (2012) have been demonstrated in graphene. There are also predictions of ultra broad-band wave mixing at low powers, with generation of several side-bands at terahertz (THz) frequencies in bilayer graphene Crosse et al. (2014). Such measurements and estimates offer fundamental insights in optical nonlinear interactions and relaxation mechanisms in lower dimensional systems Yoshikawa et al. (2017); Crosse et al. (2014); Prechtel et al. (2012); Cox et al. (2017); Tamaya et al. (2016a) along with a promise of applications including compact and useful THz sources and gate tunable opto-electronic devices Bonaccorso et al. (2010); Bao and Loh (2012); Tamaya et al. (2016b). However, experiments till date have been primarily limited to ultra-short time-domain spectroscopy Dawlaty et al. (2008); Johannsen et al. (2013) which are technologically involved and intricate, and physical interpretations generally rely on large scale computation.
Here we propose an alternate technique that uses frequency modulated continuous wavefront (CW) optical fields, probing optical non-linearity in the ‘steady state’. In particular we focus on the non-linear optical sideband generation in graphene due to inter-band polarization combined with optical Bloch oscillations Boyd (2008); Schubert et al. (2014). In presence of a CW pump (frequency ) and a frequency modulated probe beam () the optically pumped population inversion and the inter-band coherence oscillate at the modulation frequency. Such coherent ‘slushing’ of the inter-band quasiparticles excitations leads to a new sideband generation at frequencies , as shown in Fig. 1. This results in distinct signatures in reflectivity along with non-linear polarization rotation at the new sideband frequency. Our formulation based on the dynamics of density matrix for a generic two band systems, can be easily applied to other materials as well.
The predicted sideband generation is a direct consequence of non-degenerate four-wave mixing due to third-order non-linearity in graphene Boyd (2008); Zhang and Voss (2011). Estimation of the corresponding intensity threshold and polarization rotation offers an alternative technique for probing non-linear optical effects and relaxation rates with CW fields in graphene Zhang and Voss (2011). Furthermore, the formulation is applicable from THz to optical domain with applications including switching with controlled non-linear phase shifts approaching with reasonable incident CW power and cascaded generation of frequency combs Burghoff et al. (2014).
Our formulation starts with Hamiltonian of an electronic system interacting with an electro-magnetic field. It can be described using the dipole approximation Aversa and Sipe (1995), i.e., . Here is the bare Hamiltonian, is the electronic charge, and is the electric field. For simplicity we focus on a generic two band system Singh et al. (2017, 2018a, 2018b), with its quasiparticle dispersion described by the Hamiltonian, , where is a vector composed of real scalar elements and is a vector composed of the identity and the three Pauli matrices. The eigenvalues for conduction/valence band are, , where . Accordingly, the state vectors are given by, . The frequency modulated electromagnetic field is, composed of a primary pump beam (of amplitude and frequency ) and a probe beam (of amplitude and frequency , where ). In general the pump and the probe fields can have different polarization angles, and , respectively, so that and for vertical incidence.
The dynamics of the two band system described above is obtained by analytically solving the equation of motion (EOM) for the density matrix (). The diagonal elements of comprise of the carrier distribution in the conduction () and valence () bands, while the off-diagonal elements capture the inter-band coherence. The incident optical field ‘pumps’ the carriers from the valence band to the conduction band via vertical transitions. This optical pumping of carriers is countered by damping terms originating from the vacuum fluctuations, electron-electron interactions, electron-phonon interactions, and disorder, leading to a finite population inversion () – shown in Fig. 1. Including the damping terms phenomenologically in the EOM of the density matrix leads to the following set of coupled optical Bloch equation (OBE) Chaves et al. (2016); Singh et al. (2017, 2018a, 2018b),
The inter-band Rabi frequency can be expressed in terms of the inter-band optical matrix element as: , where is the vertical transition frequency. In Eqs. (1)-(2), and are the phenomenological damping rates of the inter-band population inversion and coherence, respectively. For simplicity, we assume these rates to be constants.
For incident CW field, competition between optical pumping and decay rates, results in an eventual steady state. In this regime, analytical solutions can be obtained by making the following ansatz for the population inversion and inter-band coherence Boyd (2008):
Here, the superscript is used to denote the steady state solution of the OBEs in presence of CW pump field leading to optical response at frequency Chaves et al. (2016); Singh et al. (2017, 2018a, 2018b). Presence of a probe field leads to slowly oscillating sidebands which are denoted by the superscript and results in response at frequencies . Since the population inversion is a real quantity, one is forced to add a new term with the superscript in the ansatz for the population inversion, such that . This leads to an additional contribution in the inter-band coherence, and accordingly, these new terms (with superscript ) leads to optical response at frequency . This non-linear term in the inter-band coherence combines with the optical Bloch oscillations leading to the sideband response in the current. Its consequence in reflectivity and polarization rotation is the primary focus for this work.
Here, , and we have defined the dimensionless parameter - which uniquely characterizes the non-linearity in the system Chaves et al. (2016); Singh et al. (2017, 2018a, 2018b). See supplementary material (SM) 111Supplementary material detailing the 1) steady state calculation, 2) longitudinal and transverse conductivities for the pump, probe and new sideband, and 3) nonlinear phase shift in reflection coefficient and polarization (Kerr) rotation. for details of the calculation. Equation (5) can be systematically expanded in powers of with denoting the equilibrium distribution (Fermi function denoted by ) in absence of optical interactions, while the terms denote the correction to the modified distribution function. The limit is the saturation limit of maximum population inversion with . The sideband population inversion corresponding to the probe frequency component is Note1 () can be expressed as,
Here, we have defined,
It can easily be checked that in both the limiting cases of vanishing intensity of the pump beam () as well as in the saturation limit (), as expected. Recall that and the analytical expressions for the components of the inter-band coherence are presented in Sec. S1 of the SM Note1 (). These sideband components of the density matrix generate a new optical sideband whose amplitude and polarization depend on the amplitude and polarization of the incident pump beam.
The optical conductivity at different frequencies can now be obtained via the calculation of the charge current response: , where is the effective velocity operator. The corresponding optical conductivity matrix can be expressed as a Brillouin zone sum of the momentum resolved conductivity matrix: , with and denoting the dimensionality of the system. The momentum resolved optical conductivity matrix corresponding to the pump frequency is given by
Here, and denotes the outer product of the optical matrix element vectors. The momentum resolved optical conductivity corresponding to the probe frequency sideband is
Equation (11) clearly emphasizes the gain in the optical response at the probe frequency . As stated earlier, the response at frequencies and interfere leading to a sideband generation at the frequency . The details of the calculations are presented in Sec. S1 and S2 of the SM Note1 ().
The momentum resolved optical conductivity due to the newly generated sideband is given by,
Equation (12) highlights the optical response generated at the new sideband frequency , and is one of the significant finding of this work. This new sideband response originates from the third order non-linearity in graphene. The dependence of the longitudinal optical conductivities on the non-linearity parameter and the pump polarization angle is shown in Fig. 2. The transverse component of the optical conductivity are presented in Fig. S3 of the SM Note1 (). As expected, both and reduce to the universal optical conductivity of graphene, , in the linear response regime of . However, the new sideband contribution is finite only in the non-linear regime of , and vanishes in the linear response as well as in the saturation regime ().
The generated sideband would leave its signature in a range of optical and photo-conductivity measurements Prechtel et al. (2012). Here, we focus on its impact in optical reflectivity, In particular, we explore the pump power and polarization angle dependence of the reflectivity (amplitude and phase in ), and the Kerr angle Yoshino (2013); Singh et al. (2018b). For graphene, though small, the reflectivity is routinely measured Nair et al. (2008); Mak et al. (2008); Singh et al. (2017, 2018b), while the phase of the reflection coefficient can be measured using a generic interference setup. Thus , and can be probed as a function of the probe laser power and polarization angle (see Fig. 1). The dependence of the and components of the reflection coefficients on the respective optical conductivities in graphene are tabulated in Table 1 222The simplified expressions in the last column are obtained using , along with – which works in the case of graphene..
To compare the reflection amplitude and phase of the sideband 333 Reflectivity and polarization rotation for the newly generated sideband are defined in reference to the amplitude of backward propagating field , and its polarization relative to the polarization of the incident pump beam. with that of the pump beam, we define the following:
where for reflectance measured at the sideband frequencies . The dependence of the ratios of the reflectance and defined in Eq. 13, is shown in Fig. 3 as a function of the pump beam intensity (). Clearly the sideband response at manifests only in the non-linear regime of . In the optical regime (say ), the estimated damping constants in graphene Zhang and Voss (2011) are . Using these values, the condition in graphene corresponds to a CW laser intensity , which is reasonable Prechtel et al. (2012). Furthermore, at reasonable CW powers we also observe non-linear phase shifts in excess of for the new sideband. Such large non-linear phase shifts is of great interest in a range of switching applications in THz and optical domains. The polarization angle dependence of the reflection probability and its phase is shown in Fig. 4(a)-(b).
Non-linear optical response in graphene also generates a finite , which in turn leads to Kerr rotation (polarization rotation of the reflected beam)Yoshino (2013); Singh et al. (2018b). The Kerr rotation angle for and polarized incident pump beam is given by Yoshino (2013); Singh et al. (2018b),
where can be expressed in terms of the reflection coefficients (see Table 1). The variation of the Kerr angle for the and components for pump, probe and the new sideband beam as a function of is shown in Fig. 4 (c)-(d). The polarization rotation of the sideband seems to be significantly large and different from that corresponding to the pump and probe frequencies.
In summary, we predict generation of a new modulated optical sideband in graphene in presence of a CW frequency modulated pump-probe setup. Physically, the ‘slushing’ of the inter-band coherence due to interference of the pump and the probe results in the generated sideband that carries unique signature of the third order non-linear response in graphene. Experimentally, this manifests in the polarization, reflectivity, and in the phase of the reflection coefficient (see Fig. 3) at the sideband frequencies. In particular, the peak of the sideband gain occurs at a thereshold, characterized by a single parameter set by system decay rates and the pump power. A careful characterization of generated sideband gain can thereby provide a direct method of characterizing non-linear response of two-band systems with CW fields, in contrast to traditional, technologically involved time domain measurements. It also suggests a range of applications that include switching of frequency sidebands using non-linear phase shifts and generation of frequency combs.
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- (42) Reflectivity and polarization rotation for the newly generated sideband are defined in reference to the amplitude of backward propagating field , and its polarization relative to the polarization of the incident pump beam.