Quantum entanglement based on surface phonon polaritons in condensed matter systems

Condensed matter systems are potential candidates to realize the integration of quantum information circuits. Surface phonon polariton (SPhP) is a special propagation mode in condensed matter systems. We present an investigation on the entanglement of SPhP modes. The entangled pairs are generated from entangled photons injected to the system. Quantum performances of entangled SPhPs are investigated by using the interaction Hamiltonian and the perturbation theory. The wave mechanics approach is taken to describe the coupling process as a comparison. Finally, the correlation of system is examined. A whole set of descriptions of SPhP entanglement thus are presented.

entanglement of SPhP modes. The entangled pairs are generated from entangled photons injected to the system. Quantum performances of entangled SPhPs are investigated by using the interaction Hamiltonian and the perturbation theory. The wave mechanics approach is taken to describe the coupling process as a comparison.
Finally, the correlation of system is examined. A whole set of descriptions of SPhP entanglement thus are presented.
Quantum entanglement has become one of the most attractive topics during the past two decades. The quantum entanglement systems are described by entangled states that could not be decomposed into products of single particle states 1 . Due to the development of experimental technologies in recent years, quantum entanglement has been used in varieties of domains, such as quantum cryptography 2 , quantum imaging 3 , and quantum computation 4 . Usually, the entanglement property of the multiparticle system is demonstrated by entangled photon states 5 , for the reason that coupling between photon and environment is relatively small, the decoherence time of the entangled system thus is longer than that of the electrons based systems. However, there are still many rooms to improve in photon systems. For example, bulk optical elements are normally used, which confines the flexibility and scalability of the photon system in practical applications.
To seek for a more practical entanglement system, kinds of improving schemes have been proposed. For example, the usually used spontaneous parametric down-conversion (SPDC) source of entangled photon pairs could be substituted by a domain-engineered nonlinear photonic crystal which integrates the function of some optical elements together 6 . Entanglement systems based on metallic microstructure devices which could support surface plasmon polariton (SPP) mode are also investigated for effective miniaturization 7,8 , as the SPP has a unique capability to confine the electromagnetic field in the perpendicular direction. Recently, owing to meaningful distinct characters, entanglement of phonon modes in condensed matter systems also attracts plenty of attention, and some relative interesting results have been reported 9,10 .
In this letter, we present an investigation of quantum entanglement in condensed matter systems. Unlike the traditional ones 9, 10 in which the entangled phonon states are generated by Raman scattering, the entangled states which we investigate in this work are surface phonon polariton (SPhP) pairs. The SPhP mode is a transverse magnetic (TM) mode vibration resulting from the coupling of an infrared photon (TM mode) with a transverse-optic (TO) phonon 11,12 . For classical phonon modes in condensed matter systems, thermal fluctuation usually causes serious interference in measurement and destroys the quantum coherence 10 where β is the propagation constant of the surface mode. ε d and ε c are dielectric constants of the two materials located at both sides of the interface, respectively. ε c represents the relative permittivity of the lossy polar dielectric material that could be given by the formulation 15 22 22 ( ) ( ) 1 ω LO and ω TO are the frequencies of the longitudinal optical phonon and the transverse optical phonon, respectively. And the term iγω represents the damping. Usually, γ is much smaller than ω; so for frequencies falling between the TO and LO frequencies, the dielectric constant have a negative real part and a positive imaginary part. Thus the propagation constant also shows a similar form, just like that of the surface plasmon polariton. For a detailed illustration, we take SiC system as an example.
Corresponding dielectric permittivity and dispersion relation of SiC are shown in Fig.   1(a) and Fig. 1(b), respectively. Other parameters are given as ω L =969cm -1 , As a derivation, when the thickness of the condensed matter becomes thinner enough, the SPhPs supported by the top and bottom surfaces will couple with each other to form a long range surface phonon polariton (LRSPhP) mode. The LRSPhP mode experiences lower propagation attenuation, which is desired in practical applications. As a consequence, our following discussions are mainly focused on the LRSPhP mode.
The SPhP entangled pairs could be generated in the interaction process of entangled photons and the lossy condensed matter. For a clearer sense, an illustration of the model is presented in Fig. 2. When a pair of entangled photons incident on the condensed matters, they could be transferred into entangled SPhP pairs through surface gratings or the attenuated total reflection (ATR) regime. It is worth mentioning that wavelength of SPhP is usually located in mid-infrared band. Although the usually used entangled photons sources are mostly in near-infrared band, nonlinear optical processes in crystals such as ZnGeP 2 , CdSe and GaSe could be utilized to provide mid-infrared entangled photon pairs needed to generate entangled LRSPhP pairs 16 . For instance, to prepare a frequency-entangled LRSPhP pair, the corresponding entangled photons could be obtained through the traditional scheme as reported in Ref. 17.
To describe the quantum characters and relative evolution processes of LRSPhP, a canonical quantization procedure should be taken 18 with the results as, † * 2 0 where the eigenvector () k r  could be expressed as The subscript is marked only by wave number k, as the polariton state of LRSPhP is always TM mode, the usually used subscript σ for polariton could be ignored.
represents the annihilation operator of LRSPhP. And q  is given by . Γ k is the normalization coefficient which could be determined through the normalized longitudinal power flux 19 . Corresponding equation is written as Taken the quantization formulation of the LRSPhP mode, we could obtain the The coefficient 12  is given by (2) 12 The quantities marked by subscript p are relative parameters of the pump field of the SPDC process, and the perfect phase matching conditions are assumed to be satisfied.
where the λ i 's (i=1, 2, 3) are determined by the overall boundary conditions of the coupling gratings. For the entangled LRSPhPs, although the formulation of the field could not be derived through the second order nonlinearity, it is natural to determine that based on the correspondence between frequencies of the incident photons and the LRSPhP modes. The quantized field of the entangled LRSPhP modes thus are be expressed as The coefficient 12  is given by Substituting the formulations of E eph and E esp into the Hamiltonian of the electromagnetic field, the overall interaction Hamiltonian is written as  In the interaction Hamiltonian, the first and the second terms corresponds to the unperturbed portion of the Hamiltonian. The physical meaning of these two terms is quite clear. Their effective operators are both combinations of the particle number operators for the entangled states, so expectation values of these operators represent the combination detections of entangled photons and entangled LRSPhPs, respectively.
The latter two terms corresponds to the interaction of entangled states. The first one describes the generation of the entangled LRSPhPs, while the latter one represents the conjugate process. So the effective H 1 could be written as Here the coefficient 12 kk k  is given by In the derivation, we assume the conservation of energy is satisfied in the SPDC process to prepare the entangled optical fields, so the spatial integral in the interaction Hamiltonian is estimated as To simplify the form of expression, origins of z 1 and z 2 axes are set in the middle of the gratings, i.e., the coupling sections. For a more explicit comprehension of the entangled LRSPhP modes, the state vector could be derived from second order perturbation where T is the time-ordering operator. Neglecting the unrelated items,  could be written as The coefficient F k is defined by the SPDC process. We would like to take the entangled state for a detailed consideration. As we know, the SPDC-generated entangled photon state is usually entangled in frequency and wave vector 1 . In frequency space, the coupling process from photon to LRSPhP does not influence the corresponding frequency property, so the frequency entanglement derived from entangled photons could completely preserve in the entangled LRSPhPs, which is stressed by the Dirac-δ function in Eq. (19). For the case of wave vector, the situation is slightly different. The wave-vectors along z direction should be considered for the phase-matching condition of the coupling process. Generally speaking, the wave-vector entanglement has implications for the spatial correlations of the entangled pair 1 , so the quantum entanglement based on SPhP corresponds to a low dimension spatial correlation system. To investigate variations of two-entity states, the Bialynicki-Birula-Sipe equation should be generalized. The motion equation for the two-photon wave function in vacuum has been derived by Smith and Raymer 22,23 . We need to expand the equation in dealing with problems in dielectrics. The Bialynicki-Birula-Sipe equation for single photon wave function in dielectrics has been derived in Ref. 24, through a similar treatment; the governing equation for two-state evolution is obtained as (2) (2) The two-state wave function (2) d  is written as (2) (1)