Sensing Coherent Phonons with Two-photon Interference

Detecting coherent phonons pose different challenges compared to coherent photons due to the much stronger interaction between phonons and matter. This is especially true for high frequency heat carrying phonons, which are intrinsic lattice vibrations experiencing many decoherence events with the environment, and are thus generally assumed to be incoherent. Two photon interference techniques, especially coherent population trapping (CPT) and electromagnetically induced transparency (EIT), have led to extremely sensitive detection, spectroscopy and metrology. Here, we propose the use of two photon interference in a three level system to sense coherent phonons. Unlike prior works which have treated phonon coupling as damping, we account for coherent phonon coupling using a full quantum-mechanical treatment. We observe strong asymmetry in absorption spectrum in CPT and negative dispersion in EIT susceptibility in the presence of coherent phonon coupling which cannot be accounted for if only pure phonon damping is considered. Our proposal has application in sensing heat carrying coherent phonons effects and understanding coherent bosonic multi-pathway interference effects in three coupled oscillator systems.

Phonons are packets of vibrational energy that shares many similarity with its bosonic cousin photons. Advances in nanofabrication has enabled many parallels between the development of photon and phonon control. Parallel developmets in passive control techniques include photonic 1 versus phonoic crystals 2 , optical 3 versus acoustic metamaterials 4 etc. Development in active manipulation of electromagnetic waves through light-matter interaction have led to creation of nanoscale optical emitters 5 and gates 6 and similar progress have been made in controlling phonons using their interaction with matter especially in the realms of optomechanics 7 and phononic devices 8,9 . Phonons span a vast frequency range and while techniques to control and sense lower frequency coherent phonons have been well-developed [10][11][12][13][14][15][16][17][18][19] , heat carrying coherent terahertz acoustic phonons have been harder to measure directly due to the small wavelength and numerous scattering mechanism at these small wavelengths 20 . In the past, THz crystal phonons have been generated and detected in low temperature experiments with defect doped crystals [21][22][23][24] , with experimental evidence of coherent phonon generation [25][26][27] . At the same time, interpretation of non-equilibrium phonon transport, with the advancement of nanoscale electrical heating and ultrafast optical pump-probe techniques, have allowed us to infer phonon coherence from broadband thermal conductivity measurements [28][29][30][31][32] . There have been also interest of using defectbased techniques as a thermal probe using perturbation to energy levels due to changes in temperature 33 . Furthermore, surface deflection techniques with ultrafast optics have also been used to generate phonons close to THz frequencies in materials [34][35][36][37] . Defect-based techniques are attractive compared to both thermal conductivity measurement and deflection techniques due to its ability to directly access atomic length scales where THz phonon wavelength resides. Also, the energy levels in the excited state electron manifold of these defects can match the phonon energy precisely 21,38,39 , resulting in a narrow band phonon detector. In light of the success of defect-based optical absorption techniques in coupling directly to high frequency phonons, we propose the use of two photon interference to measure the coherence properties of these phonons. Two photon interference techniques, with the most famous being coherent population trapping (CPT) 40 and electromagnetically induced transparency (EIT) 41 , have been widely adopted in spectroscopy and metrology in atomic 42,43 and defect-based systems [44][45][46][47][48] . However, CPT and EIT usually excludes the possibility of a ground state coupling 49 or merely treating the ground state coupling as thermal bath 47 .
In this paper, we propose the possibility of using the presence of coherent coupling of two ground states in a Λ system by THz acoustic phonons of the host material as a coherent phonon sensor. We show two experimentally observable effects, namely an asymmetric excited state population lineshape in CPT and an anomalous dispersion profiles in EIT measurements, which only occurs in the presence of coherent phonon coupling to a lattice phonon mode. Our proposal has the potential for direct implementation in defect-based phonon detection experiments mentioned earlier 21,38,39 and extends traditional two couple oscillator models in two photon interference to a three-coupled-oscillator models 50,51 . Our result will also be applicable for three-way coupled system such as microwave driven quantumbeat lasers 52,53 , designed opto/electro-mechanical schemes 54,55 or phonon-based quantum memories 14,56,57 .
In the schematic of our proposal in Fig. 1(a), a two-photon interference is created in a localized region of a medium that carries an ensemble of identical emitters with electronic energy level resembling a typical Λ system used in CPT or EIT. The optical fields driving the |2 − |1 and |3 − |2 transitions have detunings δ a and δ b with respect to the electronic energy levels of the emitters. The total Hamiltonian of the system can be written as where the electronic part satisfies the eigenvalue Eq. 1b of electronic eigenstate |m , the field part (Eq. 1c) is the usual expression that now comprises the sum of the photon modes indexed as λ with raising and lowering operators c † λ , c λ and the phonon modes indexed as k with raising and lowering operators b † k , b k . The interaction Hamiltonian in Eq. 1d has two parts, the first part being the original two photon interference Hamiltonian which realizes effects of CPT and EIT, and the other portion responsible for phonon interaction.
Using the procedure outlined in Supplementary Information (SI) similar to the method by Whitney and Stroud 49 , one arrives at the set of Eqs. S13 which specifies the equations of motion for elements of the density matrix. Note that in Eqs. S13, we are able to obtain spontaneous Γ (Eqs. S11,S12) and stimulated rates G i , W Eq. S15 directly from the equations of motion Eq. S4 without having to add damping terms unlike semi-classical approaches and this is the merit of the approach by Whitley and Stroud 49 . The spontaneous damping terms are defined as sum over all mode contributions in both optical (Eq. S11) and phonon cases (Eq. S12) while the coherent optical coupling terms G a,b are defined for coupling to a specific mode α, β (Eqs. S15a and S15b) and W for the specific phonon mode γ. A very important feature of our system is that we have now included the possibility for a coherent phonon coupling of strength W that couples to the |3 − |1 transition instead of a pure phonon damping term, and examining this feature will be the main theme of subsequent results and discussions. We would especially like to bring your attention to the definition of W in Eq. S15c where ensemble average of the phonon annihilation operator will only yield a non-zero value if the detected phonons are coherent 49 . This is because an incoherent or thermal ensemble will yield a zero ensemble average 58 . Thus, our proposed technique offer a rigorous detection of phonons rather than indirect evidence using thermal conductivity measurements.
The diagonal terms ρ 11 , ρ 22 and ρ 33 are the population of each energy level. We first solve for the steady state solution to Eq. S13 which allows us to obtain ρ 11 , ρ 22 and ρ 33 in the longtime limit. We first consider CPT where the optical field for |2 − |1 transition is tunable while transition |3 − |2 is fixed, and that both fields are of equal strength G a = G b = G.
Under the condition of no phonon damping Γ p = 0, unity optical damping Γ a = Γ b = Γ 0 and coupling W = 0, we can obtain the expression of ρ 11 , ρ 22 and ρ 33 as The dashed lines in Fig. 1 (b) plots the population of level |1 (Eq. 2a) and level |3 (Eq. 2c) which are in the ground state manifold. There is a broad resonance that peaks at zero detuning where almost half of the population is in each of the ground state. The excited state population of level |2 in Eq. 2b in Fig. 1 (c) is small for all detuning, where the dashed line also shows a broad resonance peak. However, there exist a sudden dip at δ a = 0 to zero population, a feature of complete two photon resonance in CPT 40,42,59 . Now, let us add some phonon damping Γ p = 0.1Γ 0 but assume no phonon coupling i.e. W = 0. The solid lines in Fig. 1(b) shows the population of level |1 and level |3 again where adding phonon damping reduces the population transfer between |1 and |3 at δ a = 0, leaving only 10% of population in level |3 on resonance. Fig. 1(c) show that two photon interference effect in the excited state |2 with (solid line) phonon damping is reduced on resonance.
This is physically expected as Γ p is a source of decoherence which reduces the ideal result in CPT or EIT.
Next, we introduce coherent phonon coupling W and ignore phonon damping Γ p for the excited state level |2 given by Eq. S19. Figure  linearly decreasing trend for W < ∼ 0.1Γ 0 (shown in red solid line in Figs. 2(b,c)). However, when W is increased further, then the higher order terms in Eq. S20 starts to dominate, increasing the positive maximum value and decreasing the negative maximum, consistent with the observation of the shift in detuning as W increases in Fig. 2(b,c).
Next, we examine how phonon coupling W creates asymmetry in the peak heights in Fig.   2(b). We substitute the linear term in Eq. S20 into the steady state solution for ρ 22 (Eq. S19) to obtain difference between the positive and negative detuning as . Equation 3 is plotted as a function of W in Fig. 2(d) to show that the linear regime agrees well with the actual data from Fig. 2(a) for small values of W . Experimentally, this linearity allows direct retrieval of the value of phonon coupling W from experimental measurements of excited state population ρ 22 if optical fields couplings are much stronger than phonon The third observation is the preservation of the resonance dip to zero occupation in Fig.   2 for all W , indicating that the dark state is preserved just like in the CPT case in Fig.   1(c). The dressed state picture allows us to identify the eigenstates by diagonalizing the where the dressed states can be obtained by taking the eigenvector and eigenvalues of Eq. 4. In the absence of phonon coupling where W = 0, we obtain the familiar dressed state result of a CPT system 41 where the eigenvalues are (0, ± √ 2G) and the eigenvectors are . Equation 5a is the dark state as it does not contain any excited state |2 . Physically, this means that the ground states are mixed with no population in the excited state when the system is in a dark state.
When W is non-zero, the eigenvalues are modified to (−W, 1/2(W ± √ 8G 2 + W 2 )) and the eigenvectors become . Equation 6 shows that the dark state |a 0 is preserved even when W is non-zero. This as demonstrated in Fig. 3(a). The populations ρ 11 (t), ρ 33 (t) tend to 0.5 which is the steady state value in Fig. 1, likewise for ρ 22 (t) in after t = 300/Γ 0 . The Fourier transform of ρ 11 (t) (blue solid line in in Fig. 3(c)) shows a peak at ∼ 0.28Γ 0 . The peak matches almost the value of √ 2G where G = 0.2Γ 0 as expected in CPT 40 and from Eq. 5 41 . However, with non zero phonon term W = 0.01G a , ρ 11 (t) and ρ 33 both have a slower modulation on top of the faster optical oscillation as shown by the blue and yellow lines of population in levels |1 and |3 in Fig. 3(b). If we take the Fourier transform of ρ 11 (t) again, we obtain the red dashed spectrum in Fig. 3(c) where the first peak now shows a splitting of frequency with respect to the undisturbed case. The splitting into two frequencies at ω + ∼ 0.27Γ 0 and ω + ∼ 0.29Γ 0 resembles the splitting in eigenvalues 1/2(W ± √ 8G 2 + W 2 ) of eigenvectors in Eq. 6. Physically, phonon coupling W results in non-degenerate eigenvalue magnitudes such that |a + and |a − oscillate at different eigenfrequencies. This in turn modulates population ρ 11 (t) and ρ 33 (t), causing a splitting of the frequency compared to the case where phonon coupling W = 0.
Having looked at the CPT case, one wonders if we can use EIT technique to sense coherent phonons. In EIT, the condition for the optical fields becomes G a ≪ G b , where the |2 − |1 optical field is a now a weak probe with detuning δ a compared to a strong resonant driving field for the |3 − |1 transition. The quantity of interest in EIT is the susceptibility of the medium 41 under the incidence of the probe beam which is related to the off-diagonal steady state solution to the density matrix term χ 21 in Eq. S14. Under the condition of no phonon field and damping W = 0, Γ p = 0, we can obtain the linear susceptibility X by Taylor expansion of the steady state solution for Eq. S17 for χ 21 for small G a to obtain . Figure 4 (a,b) plots the real and imaginary susceptibility for different values of W .
The shape of the real and imaginary susceptibility for W = 0 in Eq. 7 are typical EIT susceptibility 41 showing a sharp inflection at zero detuning δ a = 0 for the real part and a sharp dip for the imaginary part. The dip to zero for the imaginary part (blue solid line in Fig. 4(b)) physically indicates zero absorption where the transparency window in EIT refers to.
When we have phonon coupling W > 0, we see changes in dispersion in Fig. 4(a,b). The change in the real part in Fig. 4(a) follows a decrease in the sharpness of the inflection which can also be due to effects of damping. However the negative anomalous imaginary part on resonance in Fig. 4(a) cannot be caused by damping. Damping will only reduce the size of the dip similar to the result of excited state population |2 in Fig. 1(c). Thus, the presence of anomalous imaginary susceptibility at resonance is another good measure for the strength of phonon coupling W . Physically, negative anomalous imaginary susceptibility should indicate gain rather than loss, which means that we not only have transparency, but possibly amplification. The details of this possibility will be discussed in a future study.
Experimentally, this scheme offers a rigorous way to detect coherent phonons in the THz frequency range which is responsible for heat condition. As mentioned earlier, these defectbased detection techniques have the characteristic of being narrow band and yet tunable 38,39 and has been employed successfully in understanding many aspects of phonon transport in crystals 23 and interfaces 60 . These crystals can be interfaced with other materials phonon detectors 61 , making our proposed method directly applicable to detecting coherent phonons in thermal transport.
To experimentally realize our proposal, three challenges need to be addressed. Firstly, CPT or EIT have yet been experimentally demonstrated with THz energy separation between the ground state manifold to our knowledge. However, we believe that with the advent of frequency combs, locking two laser in the THz range is certainly possible 62 and we may soon see such an experiment being performed. Secondly, phase fluctuation in any of the optical or phonon fields will affect the quality of the photon-phonon interference. Experimental demonstrations of CPT and EIT typically use the same laser source to generate two frequencies 40,41 , leading to the same phase fluctuations in both optical fields. Dalton and Knight 63 specifically addressed this issue for two photon interference where Λ will be spared of any decoherence but not in a ladder system. Here, our two-photon-phonon interference is a composite of Λ and ladder systems and the net effect will be a reduced interference effect. Lastly, due to phase fluctuation, the coherent phonon field must carry the same phase fluctuation as the optical field, so we must generate the phonons in a coherent manner with the same laser field for the |2 − |1 and |3 − |1 transitions. This is possible with the advent of coherent phonon sources in defect-based systems [25][26][27] , material systems 34-37 and nanofabricated systems 11,[13][14][15][16][17][18][19] .
Our work differs from the field optomechanics and non-linear coherent phonon control 64 .
Optomechanics primarily relies on coupling a mechanical mode to a designed optical cavity for coherent phonon control. It is remarkable that quantum coherence of phonons has been and characterizing phonon coherence in thermal transport using correlation functions 31 . It is thus evident that characterizing high frequency coherent acoustic phonons in materials using quantum mechanical description are only starting to be explored.
Lastly, we would like to mention the relevance of our work not limited to phonon sensing, but also to three-way interference problems 54,55 and coupled oscillator systems 50,51 . Our theory is not limited to just phonon coupling of the ground state manifold but any bosonic field. Thus, the predicted asymmetry in the excited state population, modulation in population time dynamics and the anomalous EIT dispersion will also be observable in any of the above systems, paving way to understanding and engineering multiple interference pathways in more complex multilevel systems.
In conclusion, we have proposed a coherent phonon sensing scheme that utilized the existing two photon interference techniques to rigorously test the presence of coherent phonons. This derivation follows closely the work of Whitley and Stround [49]. Consider a Λ system described in the main text where the frequency difference between the |2 −|1 transition is expressed as Ω a = (E 2 − E 1 )/h and the |2 − |3 transition as Ω b = (E 2 − E 3 )/h. The photon field driving the |2 − |1 transition has a detuning δ a = Ω a − ω a and the field driving the |2 − |3 transition has a detuning of δ b = Ω b − ω b . The total Hamiltonian of the system can be written as Eqs. 1a-d where the atomic part satisfies the eigenvalue Eq. 1b, the field part (Eq. 1c) is the usual expression that now comprises the sum of the photon modes indexed as λ with raising and lowering operators c † λ , c λ and the phonon modes indexed as k with raising and lowering operators b † k , b k . The interaction Hamiltonian in Eq. 1d has two parts, the first part being the original two photon interference Hamiltonian which realizes effects of CPT and EIT, and the other portion responsible for phonon coupling here. Notice that we did not use the rotating wave approximation for the phonon part.
The coupling coefficient g λ a and g λ b are stands for interaction of the photon dipole interaction for the |2 − |1 and |2 − |3 transition in Fig. 1(a) respectively and the coupling coefficient ζ k stands for electron phonon interaction. The magnitude of the coupling constants are given by In Eqs. S1 and S2,ǫ λ is the unit polarization vector of the λ mode, d a = 2|r|1 and d b = 2|r|3 are the dipole moments with 3|r|1 = 0. V l stands for the quantization volume for photons.
However, we allow for electron phonon coupling between |1 and |3 and the coupling coefficient is defined by Eq. S3 [IYS64, Toy03] where Ξ is the deformation potential, ρ is the density v k is the group velocity of mode k, V p stands for the quantization volume for phonons.
Using the equation of motion for a single time operator given byȮ(t) = (ih) −1 [O, H], and using the Hamiltonian in Eqs. 1a-1d, we find the atom-field system evolves according to the following equatioṅ Eqs. S4f and S4g can be integrated from initial time t = 0 to give Eqs. S5a and S5b can be substituted into Eqs. S4a-S4e to express electronic density matrix operators as initial conditions of field operators. We first make the harmonic approximation to Eqs. S5a and S5b to obtain The validity of this approximation is discussed in Whitley and Stroud [49] where we also assume that the time intervals are much shorter than the Rabi or natural lifetime for both the photon and phonon field.
Using the harmonic approximation in Eqs. S6, we simplify the photon field in Eq. S5a to since only the values of t ′ within a few optical periods of t are important to the integral.
Likewise, we apply the harmonic approximation to the phonon operator in Eq. S4g to obtain As we did not adopt rotating wave approximation for the phonon field, the combined operator Note that Eq. S9 is invariant with respect to its conjugate. We can thus ignore conjugation considerations on the phonon operators later in our derivation.

RELATIONSHIP BETWEEN TURNING POINTS AND DETUNING
Solving the steady state solution to Eq. S17 for χ 11 , χ 22 , χ 33 , one obtains the population in each level ρ 11 , ρ 22 and ρ 33 . Here, we examine the coherent population trapping (CPT) case where G a = G b = G and that Γ a = Γ b = Γ 0 which is the scaling factor for the entire system [49].
Thus, all units in our calculations are subjected to scaling factor Γ 0 . In the CPT calculations, we assume that δ b = 0 and δ a is varied as described in the main text. In CPT experiments, ρ 22 is usually the main indicator in experiments and we also focus on this observable here. The solution for the excited state population ρ 22 under the condition W = 0, Γ p = 0 is given in Eq. 2b in the main text. Here, we show the solution under which Γ p = 0 which gives The turning points for Eq. S19 can be obtained by taking the equating first order derivative to zero. One of the roots will be at δ a = 0 which is the same as CPT. Two roots are imaginary and two roots are real, one for the turning point for δ a > 0 and one for δ a < 0. Here, we Taylor expand the roots for small W to the fourth power to obtain δ b,max ≃ ± 2 3/4 G − 1 2 where when W → 0, we obtain the turning points at ± ± 2 3/4 G as described in Fig. 1(c) of the main text. The linear shift term in Eq. S20 in − 1 2 1 2 + 1 √ 2 W is independent of the sign of the root and is valid for small W as shown in Fig. 2.

TIME DEPENDENCE OF POPULATION
The time dependence of the population is obtained by solving the eigenvalues s m , eigenvectors ν(m) of matrix A in Eq. S18 and its reciprocal eigenvector w(m). The time dependent solution of Eq. S17 is then given by [49]