Coherent states in microwave-induced resistance oscillations and zero resistance states

We investigate irradiated high-mobility two-dimensional electron systems (2DES) under low or moderated magnetic fields. These systems present microwave-induced magnetoresistance oscillations (MIRO) which, as we demonstrate, reveal the presence of coherent states of the quantum harmonic oscillator. We also show that the principle of minimum uncertainty of coherent states is at the heart of MIRO and zero resistance states (ZRS). Accordingly, we are able to explain, based on coherent states, important experimental evidence of these photo-oscillations. Such as their physical origin, their periodicity with the inverse of the magnetic field and their peculiar oscillations minima and maxima positions in regards of the magnetic field. Thus, remarkably enough, we come to the conclusion that 2DES, under low magnetic fields, become a system of quasiclassical states or coherent states and MIRO would be the smoking gun of the existence of these peculiar states is these systems.

Introduction.The first idea of coherent states or quasiclassical states was introduced by Schrödinger 1 describing minimum uncertainty constant-shape Gaussian wave packets of the quantum harmonic oscillator.They were constructed by the quantum superposition of the stationary states of the harmonic oscillator.These wave packets displaced harmonically oscillating similarly as their classical counterpart 1 .Later on, Glauber 2 applied the concept of coherent states to the electromagnetic field being described by a sum of quantum field oscillators for each field frequency or mode.These coherent states of electromagnetic radiation introduced by Glauber are extensively used nowadays in quantum optics.Coherent states [3][4][5][6] are also an essential and powerful tool in con- densed matter when describing the dynamics of quantum systems that are very close to a classical behaviour.One remarkable example of this consists of one electron under the influence of a moderate and constant magnetic field (B).The quantum mechanical solution of this problem leads us to Landau states which are mere stationary states of the quantum harmonic oscillator.Under low or moderate values of B, this system can be described by an infinite superposition of Landau states, i.e., a coherent state.The resulting wave packet oscillates classically at the cyclotron frequency (w c ) inside the quadratic potential keeping constant the Gaussian shape (see Fig. 1) and complying with the minimum uncertainty condition.
The discovery of MIRO two decades ago led to a great deal of theoretical works back then as the displacement model 10 , the inelastic model 11 and the microwave-driven electron orbits model [12][13][14][15] .According to the latter, Landau states, under radiation, spatially and harmonically oscillate with the guiding center at the radiation frequency (w) performing classical trajectories.In this swinging motion electrons are scattered by charged impurities giving rise to oscillations in the irradiated magnetoresistance, i.e., MIRO.
In this letter we demonstrate that the electron dynamics and magnetotransport in high-mobility 2DES is governed by the coherent states of the quantum harmonic oscillator.In fact, we conclude that 2DES under low or moderate B become a systems of coherent states and when irradiated, MIRO [7][8][9] bring to light the peculiar nature of these states.In other words, irradiated coherent states of the quantum harmonic oscillator are at the heart of MIRO.Accordingly, we incorporate the concept of coherent states to the microwave-driven electron orbit model [12][13][14][15] .Thus, a remarkable obtained result is that the time, τ (evolution time 16 ), it takes a scattered electron to jump between coherent states to give significant contributions to the current has to be equal to the cy-FIG.2: Schematic diagram of scattering process between coherent states Ψα and Ψ α .The scattering is quasi-elastic.The probability density for both coherent states is a constantshaped Gaussian wave packet.The process evolution time, τ is the cyclotron period.i.e., τ = 2π/wc = Tc.∆E is the energy difference between the coherent states.
clotron period, T c = 2π/w c .For different values of τ , the contribution turns out negligible.This result holds in the dark and under radiation where τ will play an essential role.Thus, MIRO is mainly dependent on τ along with w. τ also determines the peculiar B-dependent MIRO extrema position and explain the periodicity of MIRO with the inverse of B. Thus, MIRO finally reveals that coherent states of the quantum harmonic oscillator are present in high-mobility 2DES when under low B playing a lead role in magnetotransport both in the dark and under radiation.On the other hand, coherent states minimize the Heisenberg uncertainty principle and then, in our model this would establish which states can be reached by scattering.
Theoretical model.We first obtain an expression for the coherent states of a radiation-driven quantum harmonic oscillator.The starting point is the exact solution of the time-dependent Schrödinger equation of a quantum harmonic oscillator under a time-dependent force.This corresponds to the electronic wave function for a 2DES in a perpendicular B, a DC electric field, E DC , and MW radiation which is considered semi-classically.The total hamiltonian H can be written as: where the corresponding wave function solution is given by 12,17,18 : where, X(0) is the guiding center of the driven-Landau state, E 0 the MW electric field intensity, φ n is the solution for the Schrödinger equation of the unforced quantum harmonic oscillator and x 0 (t) is the classical solution of a forced harmonic oscillator: where γ is a phenomenologically-introduced damping factor for the electronic interaction with acoustic phonons and L is the classical Lagrangian.Apart from phase factors, the wave function turns out to be the same as a quantum harmonic oscillator (Landau state) where the center is driven by x 0 (t).Thus, all driven-Landau states harmonically oscillates in phase at the radiation frequency.
A coherent state denoted by |α is defined as the eigenvector of the annihilation operator â with eigenvalue α and can be expressed as a superposition of quantum harmonic oscillator states 16 , The coherent state |α can be also obtained with the displacement operator D(α) 16 acting on the quantum harmonic oscillator ground state |φ 0 , |α = D(α)|φ 0 , where the unitary operator D(α) is defined by: D(α) = e αa † −α * a .The coherent state in the position representation or wave function then reads, ψ α (x) = x|D(α)|φ 0 .We observe, according to the obtained MW-driven wave function (Eq.2), that the irradiated Landau level structure remains unchanged with respect to the undriven situation; same Landau level index and energy.Then, we conclude that the system is quantized, in the same way as the unforced quantum harmonic oscillator 18 .Thus, we can construct the driven-coherent states based on driven-Landau states similarly as if they were undriven 16 : Now applying the displacement operator, we can calculate the wave function corresponding to the coherent state of the MW-driven quantum oscillator: where, ) x (t) and p (t) are the position and momentum mean values respectively 16 where we have used that α = |α 0 |e −(iwct−ϕ) .∆x is the position uncertainty and the global phase factor, e iϑα = e α * 2 −α 2 .Then, the wave packet associated with Ψ α (x, t) is therefore given by: Thus, according to the above, the microscopic physical description of a high-mobility 2DES under low or moderate B would consist of constant-shaped Gaussian wave packets harmonically displacing with w c in the undriven case and with w c and w under radiation.
To calculate the longitudinal magnetoresistance, R xx , we first obtain the longitudinal conductivity σ xx following a semiclassical Boltzmann model [19][20][21] , being E the energy, ρ i (E) the Landau states density of the initial coherent state and W I is the electron-charged impurities scattering rate.We consider now that the scattering takes place between coherent states of quantum harmonic oscillators.Thus, ∆X 0 is the distance between the guiding centers of the scattering-involved coherent states.We first study the dark case and according to the Fermi's golden rule W I is given by, where N i is the number of charged impurities, ψ α and ψ α are the wave functions corresponding to the initial and final coherent states respectively, V s is the scattering potential for charged impurities 20 : 2S (q+q T F ) e iqxx , S being the sample surface, the dielectric constant, q T F is the Thomas-Fermi screening constant 20 and q x the x-component of − → q , the electron momentum change after the scattering event.E α and E α stand for the coherent states initial and final energies respectively.
The averaging on the impurities distribution has been considered in a very simple approach following Askerov 21 , Ando 20 et.al., and J. H.Davies 22 .Thus, if the concentration of impurities is not too high, and they are randomly distributed in the sample, the interferences caused by the impurity centers can be neglected.Then, we have ignored those interreferences and assume that the scattering due to each impurity is independent of the others.As a result the total scattering is equal to the scattering rate for one impurity center multiplied by the total number of impurities N i .
The V s matrix element is given by [19][20][21] : and the term I α,α [19][20][21] , 13) After lengthy algebra we obtain an expression for I α,α , x (t)2(∆x) 2 4 (14 where q x (t)is given by, On the other hand, (16) where t and t are the initial and final times for the scattering event and τ is the evolution time between coherent states.Thus, t = t + τ .We have considered also that for low values of B, |α 0 | |α 0 |.Developing the above exponential we can finally get to, For typical experimental values of B, |α 0 | 2 > 50 and thus, I α,α → 0. Accordingly, the scattering rate and conductivity would be negligible too.Nonetheless, there is an important exception when τ equals the cyclotron period T c : τ = 2π wc .In other words, the scattered electron begins and ends in the same position in the Landau orbit.Only in this case I α,α = 0. Thus, only scattering processes fulfilling the previous condition of τ will efficiently contribute to the current.The rest contributions can be neglected.Finally the expression of I α,α reads 19 , where X (0) − X(0) = [−q y 2(∆x) 2 ], 19 .This, in turn, leads us to a final expression for W I , where n i is the charged impurity density and θ is the scattering angle.The density of initial Landau states ρ i (E) can be obtained by using the Poisson sum rules to get to 23 , ρ i (E) = m * π 2 1 − 2 cos 2πE wc e −πΓ/ wc .Finally, gathering all terms and solving the energy integral, we obtain an expression for σ xx that reads, where χ s = 2π One important condition that features coherent states is that they minimize the Heisenberg uncertainty principle.Thus, for the time-energy uncertainty relation 16 , ∆t∆E = h.For our specific problem, ∆t = τ that implies ∆E = w c , ∆E being the energy difference between scattering-involved coherent states.Thus, we obtain two conditions for the scattering between coherent states to take place, first τ = 2π wc and second, the energy difference equals w c .There are also physical reasons that endorse the latter specially in high-mobility samples where the levels are very narrow in terms of states density.In these systems the only efficient contributions to scattering are the ones corresponding to aligned Landau levels (see Fig. 2), i.e., when ∆E = n × w c .The most intense of them is when n = 1 that corresponds to the closest in distance coherent states or smallest value of ∆X 0 , (see Eq. 18).This agrees with the condition that when n = 1, the Heisenberg uncertainty principle is minimized.The two conditions discussed above hold in the dark and under radiation.For the latter case, MIRO reveals, the important role played by τ in the based-on-coherent states magnetotransport processes.
When we turn on the light, the term that is going to be mainly affected in the σ xx expression is the distance between the coherent states guiding centers, i.e., ∆X 0 .This average distance now turns into ∆X M W 24,25 If we consider, on average, that the scattering jump begins when the MW-driven oscillations is at its midpoint, (wt = 2πn, n being a positive integer), and being τ = 2π/w c , we end up having, This result affects dramatically σ xx and in turn R xx .Now photo-oscillations rise according to ∆X M W and its built-in sine function.In Fig. 3 we present schematic diagrams for the different situations regarding MIRO peaks and valleys and ZRS.In the undriven scenario an electron in the initial coherent state, scatters with charged impurities and jumps to the final coherent state minimizing the Heisenberg uncertainty principle.The latter condition determines what coherent states can be connected via scattering.On average the advanced distance is ∆X 0 = X 0 − X 0 , (see Fig. 3a).When the light is on, depending on the term A sin 2π w wc , some times the minimum uncertainty final state will be further away than in the dark regarding the initial state position.Thus, on average, ∆X M W > ∆X 0 and R xx will be larger, giving rise to peaks (see Fig. 3b).On the other hand, other times the final coherent state will be closer and ∆X M W < ∆X 0 and R xx will be smaller, giving rise to valleys, (see Fig. 3c).Finally, when the driven coherent states are going backward and the radiation power is large enough, the final state, minimizing the uncertainty principle, will be behind the initial state in the dark (see Fig. 3d).However, the scattered electron can only effectively jump forward due to the DC electric field direction and the final coherent state can never be reached; in the forward direction there is no final coherent state fulfilling the minimum uncertainty condition and the scattering can not be completed.Thus, the system reaches the ZRS scenario where the electron remains in the initial coherent state.
Results.In Fig. 4 we present calculated results of the irradiated R xx vs B for a radiation frequency of 103 GHz and T = 1 K.The dark case is also exhibited.In our simulations all results have been based on experimental parameters corresponding to the experiments by Mani et al. 7 .We obtain clear MIRO where the minima positions are indicated with arrows and, as in experiments 7 , correspond to, w wc = j + 1 4 , j being a positive integer.Minima positions show a clear 1/4-cycle shift, which is a universal property that features MIRO and shows up in any experiment about MIRO irrespective of the sort of carrier 29 and platform 30 .In the minima corresponding to j = 1, ZRS are found.Now with the help of our present model based on coherent states we can explain such a peculiar value for the minima position.Thus, it is straightforward to check out that if we substitute equation w wc = j + 1 4 in ∆X M W , (Eq.22), we would obtain minima values of the latter and in turn of R xx .Therefore, from the minima positions relation we can obtain the value 2π/w c which would be the "smoking gun" that would reveal the presence of coherent states of quantum harmonic oscillators sustaining the magnetorresistance of high-quality 2DES.Another evidence of the latter would be the MIRO periodicity with the inverse of B (see inset of Fig. 4) that would be explained by the presence of τ in the argument of the sine function.
Summary.Summing up, we have demonstrated that magnetoresistance in a high mobility 2DES under MW radiation can be explained in terms of the coherent states of the quantum harmonic oscillator.When irradiated these systems give rise to MIRO that reveals the presence of these quasi-classical states in high-quality samples when under low B. These MW-driven coherent states have been used to calculate irradiated magnetoresistance finding that the principle of minimum uncertainty of coherent states is crucial to understand MIRO and their properties and zero resistance states.We conclude that any experiment on irradiated magnetoresistance of 2D systems, regardless of carrier and platform 29,31 , showing MIRO reveals the existence of coherent states of the quantum harmonic oscillator.We expect that dealing with even higher mobility samples, (µ > 10 7 ), it would be possible to achieve the quantum superposition of coherent states yielding, for instance in the case of two, even and odd coherent states of the quantum harmonic oscillator.Then, when irradiated, we expect that MIRO would evolve showing striking results revealing the presence of coherent states superposition 32,33 .This work was supported by the MCYT (Spain) grant PID2020-117787GB-I00.

2 FIG. 1 :
FIG.1: Schematic diagrams of coherent states: The probability density of the coherent state is a constant-shaped Gaussian distribution, whose center oscillates in a harmonic potential similarly as its classical counterpart.The lower part exhibits the 2D approach.

FIG. 3 :
FIG. 3: Schematic diagrams for electron scattering between coherent states in the dark (undriven) and with radiation (mw-driven).a) Undriven scattering.The average distance (advanced distance) between initial |ψα and final coherent state, |ψ α , is ∆X(0).This distance mainly determines Rxx.b) MW-driven scattering giving rise to peaks.Now the average advanced distance is larger because the final state, minimizing the Heisenberg uncertainty principle, is farther than the dark position due to the swinging motion of the drivencoherent states.c) MW-driven scattering giving rise to valleys.When the final coherent state is closer we obtain MIRO valleys.b) Situation when MW power is high enough and the states go backwards.In this scenario the final state ends up behind the initial state dark position and the scattering jump can not take place.

FIG. 4 :
FIG. 4: Calculated magnetoresistance as a function of B, for a radiation frequency of 103 GHz and T = 1 K.The dark case is also exhibited.Minima positions are indicated with arrows corresponding to, w wc = j + 1 4 , j being a positive integer.Zero resistance states are obtained around B 0.2T .Inset: irradiated magnetoresistance showing periodicity vs 1/B.
2 k B T / w c , k B being the Boltzmann constant, E F the Fermi energy and Γ the Landau level width.To obtain R xx we use the relation R xx = σxx