The ferroelectric photo ground state of SrTiO3: Cavity materials engineering

Significance Controlling collective phenomena in quantum materials is a promising route toward engineering material properties on demand. Strong THz lasers have been successful at inducing ferroelectricity in SrTiO3. Here we demonstrate, from atomistic calculations, that cavity quantum vacuum fluctuations induce a change in the collective phase of SrTiO3 in the strong light–matter coupling regime. Under these conditions, the ferroelectric phase is stabilized as the ground state, instead of the quantum paraelectric one. We conceptualize this light–matter hybrid state as a material photo ground state: Fundamental properties such as crystal structure, phonon frequencies, and the collective phase of a material are determined by the quantum light–matter coupling in equilibrium conditions. Cavity-coupling adds a new dimension to the phase diagram of SrTiO3.


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In order to predict the properties of SrTiO3 embedded in an optical cavity we introduce the following atomistic quantum 13 electrodynamical (QED) Hamiltonian (1): [1] 15 where ωc is the frequency of the photons in the cavity which is set by the cavity length L ⊥ , a † and a are the corresponding creation the potential energy surface shown in Fig. 1(c) of the main text, which includes the intrinsic phonon non-linearities of SrTiO3. 20 The FES and lattice modes are parameterized in terms Q f and Qc respectively (see next section for more details on Q f . The

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Born effective charges and the 2D potential energy surface expansion coefficients k are determined within DFT (density 22 functional theory) using the Perdew-Burke-Ernzerhof (PBE) functional (2) as described in Ref. (3). Furthermore, we assume 23 the Born-effective charge Z f to be not affected by the light-matter coupling. We stress that in the Hamiltonian it is essential to 24 take into account the diamagnetic as it guarantees the existence of a groundstate bound from below (4 The strength of the cavity light-phonon coupling is determined by the Born effective charge and the photon mode amplitude 31 A0. For the Γ-phonon mode coupled to the dipole component of the electromagnetic field the mode amplitude is given by (5, 6):  to be rescaled accordingly as compared to the corresponding quantities for the FES phonon mode. This is because, even if of 43 minor importance, the FES phonon mode involves the motion of all the other atoms in the unit cell. Since within density 44 functional perturbation theory with the PBE functional (DFPT@PBE) (7) the FES mode has an imaginary frequency, the 45 phonon mode is ill defined and therefore we evaluated such mode directly from the difference of the atomic position in the 46 optimized paraeletric and ferroelectric geometry. Specifically, we defined the FES eigenvector as: where I is the index running over the atoms of the unit cell and s are the atomic basis vectors of the two different geometries.

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The eigevector above and Q f are then related by: 51 with x f the actual FES mode parameter, i.e. the one that is independent of the specific atoms and to which the phonon 52 effective masses and Born effective charges are standardly referred to. To calculate the effective mass and charge for the Q f 53 parameter, the following formula can be applied: 55 the PBE functional. The full diagonalization of the Hamiltonian is performed on a simple product basis set |Q f ⊗ |Qc ⊗ |n 58 which consists of a 50 × 25 real space grid for the phononic coordinates Q f and Qc and up to n = 9 Fock number states as a 59 basis for the photons. We obtain a FES mode frequency of 0.44 THz, which reproduces well the experimental results (8-10).

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In the main text we characterized the photo-groundstate of SrTiO3 in terms of the generalized FES mode frequency, mean 62 displacement of the lattice vibration and Von Neumann entropy of the photonic sub-system. The latter quantity is commonly 63 used to describe the amount of correlation between a given sub-system and all the others, which in our case is just the phononic 64 system. The Von Neumann entropy is defined as: where ηi are the eigenvalues of the density matrix of the chosen subsystem, which for the photons is defined as: with Trpn meant as the trace over the phononic states. The resulting photonic entropy is the one shown in Fig. 2(c) of the 69 main text.

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To further characterize the photo-groundstate of SrTiO3, we report the expectation value of the squared FES mode 71 displacement, the purity and the expectation value of the photon number. These quantities are shown in Fig. S1 as a function 72 of the coupling strength. We point out that the maximum of the mean squared displacement Q 2 f is at ωc = 3 THz, which is 73 off-resonant with the FES mode frequency.

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The existence of an optimal for Q 2 f is a consequence of the trade-off between the delocalization and the dipole matrix elements 75 between the phononic states coupled by the cavity photons. Indeed the higher ωc, the higher the phononic excited states 76 that are coupled to the groundstate. In turns, this means that resulting delocalization gets larger but at the same time the 77 dipole-matrix element becomes smaller.

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Beside the Von Neumann entropy another way to characterize the correlation between light and matter is to evaluate the 79 so-called purity (11). This is defined as follows: 80 γ = Tr ρ 2 ph . [8]

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A purity value that deviates from 1 means that the groundstate cannot be factorized in a simple tensor product of a phononic

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In this section, we describe an alternative analytic simple approach to extend the theory of dynamical localization to the case 91 of the quantized light field in a cavity (12, 13).

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A. Effective Hamiltonian. The eigenvalue problem associated with the QED Hamiltonian in the first section can be rewritten in 93 the following matrix form: [11] coupling and frequency. To make this clear we write H1 = A0PH1 and factorize: Only if ω A0P the continued fractions can be neglected and the leading term reads 107 as in the main text A0 defines the photon mode volume and P is a c-number that sets the scale of the photon-phonon momentum 108 matrix. Under this condition we can use the resolvent as a Neumann series and write [14] 110 The Neumann series converges for ω > max({E 0λ }) − E, where H0ψ λ = E 0λ ψ λ , but Eq. (14) is only a valid approximation for 111 the full Hamiltonian as long as the truncation at n = 1 is possible.

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The series formulation of the effective Hamiltonian is useful because it yields to leading order in 1/ω: 118 which has to be solved separately for each eigenstate.

C. Localisation in SrTiO3.
To describe the photon induced localization we consider the system within a two-level approximation. 120 We choose as the two levels, two Gaussians which are localized in the left and right well of the 1D FES mode energy potential 121 respectively. In matrix form this translates to: 123 so that H0 = tσx and H1 = A0P σy. Here P is directly the L-R momentum matrix element. The high frequency approximation 124 from the previous subsection then reads: The second term is only shifting the eigenvalues, while the first one gives full localisation if

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In order to calculate the phase diagram that we presented in the main text, we need to include the effect of temperature in our 131 theory. To do so we apply Kubo's formula for the linear response of a thermal state to a perturbation described by: where Z f , the FES mode effective charge, is assumed to be temperature independent. Applying Kubo's formula, the resulting 134 polarizability takes the form: cavity coupling strengths. Note that the artificial broadening δ is kept constant with temperature however the increase in 140 temperature introduces a finite population in the excited states which explains the appearance of further peaks. 141 We then define a characteristic temperature dependent FES mode frequency from such a response function as: