1. Introduction

The excitation of THz coherent optical phonons using femtosecond laser pulses has been demonstrated by several groups during the past decade [1]. More recently, ultrafast pulses were also used to generate a squeezed phonon field in KTaO₃ [2]. As in the electromagnetic case [3], phonon squeezing provides a way for experimental measurements to overcome the standard quantum limit for noise imposed by vacuum fluctuations. In this letter, we report on the simultaneous excitation of coherent and squeezed phonon fields in SrTiO₃. These fields are associated with, respectively, a low-frequency phonon of symmetry A _1g and a continuum of transverse acoustic (TA) modes. The nature of these fields can be understood by referring to the two-atom cell shown in Fig. 1 and the accompanying movies. Because the wavelength of visible light is commonly much larger than the lattice parameter, the relevant wavevectors of laser-induced phonon fields are considerably smaller than the size of the Brillouin zone. Hence, every unit cell behaves almost the same way.

Let r denote the averaged relative position between the atoms and u the instantaneous deviation from the equilibrium position for each atom. Consider also the variance 〈u ²〉. In the coherent state, r oscillates in time but the variances remain constant while squeezed fields show constant r and varying 〈u ²〉 (notice that the two frequencies are, generally, unrelated). Finally, both the relative position and the variances oscillate in the combined situation applying to SrTiO₃. A significant difference between coherent and squeezed lattice fields created with light pulses is that the former are monochromatic (more precisely, their frequency spectrum is discrete) while the latter are not, even in the harmonic approximation. However, squeezed fields may exhibit nearly discrete behavior in situations where the continuum spectrum is dominated by frequencies associated with peaks in the phonon density of states [2].

2. Theory of phonon field generation

The Hamiltonian relevant to our problem is H=H ₀-|E(t)|²∑_q χ(q). Here, H ₀=∑_q($P_{\mathbf{q}}^2$+${\mathrm{\Omega }}_{\mathbf{q}}^2$ $Q_{\mathbf{q}}^2$)/2 is the harmonic contribution to the lattice energy, E(t) is the magnitude of the pump electric field, and Q _q and P _q are the amplitude and the momentum of the phonon with frequency Ω_q and wave vector q. To second-order in the atom displacements, χ(q)=χ ₁δ_q,0+χ ₂(q) where [1,2]

 

Fig. 1. Schematic diagram showing a unitary cell containing two atoms. The averaged relative position between the atoms is labeled r, the instantaneous displacement with respect to the equilibrium position is indicated by u, and the dashed circles represent the square root of the variance 〈u ²〉. The following links show animations illustrating motion associated with coherent, squeezed, and combined coherent-squeezed fields. [Media 1] [Media 2] [Media 3]

Download Full Size | PDF

Here, χ is an effective electronic susceptibility, which includes factors depending on the experimental geometry. Its derivatives, ∂χ/∂Q and ∂ ² χ/∂Q ², are proportional to linear combinations of components of the polarizability tensors associated with first- and second-order Raman scattering [1,2]. Since χ ₁∝Q _q≡0 (optical modes), Eq. (1) gives a force density F acting on Q ₀ which is proportional to the electric field intensity. This interaction is associated with coherent states [1 ]. If we ignore phonon dispersion and dissipation, the quantum-mechanical equation of motion for the expectation value of the phonon amplitude, 〈Q ₀〉, is the same as the classical expression and given by

As discussed in [2], χ ₂(q) is responsible for squeezing. This term, reflecting contributions of pairs of modes at ±q, represents a change in the phonon frequency of ΔΩ_q≈ν ²/(2Ω_q) with

The varying frequency leads to a time-dependent variance, 〈$Q_{\mathbf{q}}^2$〉, but it does not affect 〈Q _q〉 unless the latter quantity is different than zero. The variance satisfies the equation

Consider now an optical pulse of width τ₀ such that τ₀≪2π/Ω_q. Then, we can approximate E ²=(4πI ₀/nc)δ(t) and the solutions to Eq. (3) and Eq. (5) are

and, to lowest order in I ₀,

Here I ₀ is the integrated intensity of the pulse, n is the refractive index and W _q=πI ₀/(ncΩ_q). Provided that the factor multiplying sin(2Ω_q t) is sufficiently large so as to overcome the thermal contribution, the variance dips below the standard quantum limit, ${\mathrm{\sigma }}_{\mathbf{q}}^2$=ħ/(2Ω_q), for some fraction of the cycle. Specifically, the conditions for quantum-squeezing at low intensities are W _q(∂ ² χ/∂Q ²)>n _q and n _q≪1; n _q is the Bose factor. Thus, a situation may arise in an experiment where the high- but not the low-frequency modes become squeezed below ħ/(2Ω_q).

The approximation E ²∝δ(t) leads to a simple expression for the lattice wavefunction. Let Ψ^- be the wavefunction at t=0^- immediately before the pulse strikes. Integration of the Schrödinger equation gives the wavefunction at t=0⁺

For Ψ^-=|0〉 and χ ₂≡0 (|0〉 is the harmonic oscillator ground state), this wavefunction describes a Glauber coherent state while, for χ ₁≡0, it shows squeezing similar to that obtained for the electromagnetic field in two-photon coherent states [3]. We should emphasize the fact that the spectrum of coherent phonon fields generated impulsively is discrete, containing a finite set of δ-function peaks [1], whereas squeezed fields involve a continuum of modes throughout the Brillouin zone. The continua in KTaO₃ [2] and SrTiO₃ (see later) are quasi-monochromatic for they are dominated by frequencies associated with van Hove singularities in the phonon density of states [2]

3. SrTiO₃

Strontium titanate undergoes an antiferro-distortive phase transition at T_C ≈110 K from a cubic perovskite, point group O _h, to a low-temperature tetragonal structure of symmetry D _4h [4]. With decreasing T, the dielectric constant shows a dramatic increase, reaching a plateau of ~10⁴ below 3 K. This plateau reflects quantum fluctuations which suppress the transition into the ferroelectric state [5] (for electric-field- and stress-induced ferroelectricity, see [7–9]). This behavior, referred to as quantum paraelectricity [5], has been extensively discussed in the literature [6]. Here, we are interested primarily in the change of symmetry associated with the transformation at 110 K. Second-order scattering is observed in SrTiO₃, i.e., χ ₂(q)≠0 for both T>T_C and T<T_C [10]. However, there are no first-order Raman-allowed modes [4] or, alternatively, χ ₁≡0 for T>T_C (note that, in KTaO₃, χ ₁≡0 at all temperatures). In the tetragonal phase, T<T_C , group theory predicts χ ₁≠0 for phonons of various symmetries [4]. This applies in particular to the A _1g-mode at 48 cm^-1 and the E _g-mode at 15 cm^-1 which are the soft phonons associated with the phase transition [4]. It follows that below T_C , SrTiO₃ meets the conditions required for the impulsive excitation of a combined coherent-squeezed field.

4. Experiments

The details of the experiment are described in the earlier work on KTaO₃ [2]. Fig. 2(a) shows time-domain results using a standard pump-probe configuration in the transmission geometry. The data were obtained at 7 K from a ~5×5×0.5 mm³ single crystal of SrTiO₃ oriented with the cubic [001] axis perpendicular to the large face. We used a mode-locked Ti-sapphire laser providing pulses of full width ≈75 fs centered at 810.0 nm. The oscillator had a repetition rate of 80 MHz giving an average power of ~80 mW for the pump and ~30 mW for the probe pulse which were focused to a 70-μm-diameter spot. The polarizations of the pump and probe beam were along the cubic [010] and [100] directions, respectively.

The Fourier transform in Fig. 2(b) shows peaks at ~1.3 and ~6.9 THz. Based on a comparison with spontaneous Raman spectra [10] we ascribe them to the soft A _1g-phonon and the 2TA overtone. Consistent with the spontaneous results [10], the second-order feature is dominated by a sharp peak very close to twice the frequency of TA modes at the X and M points of the Brillouin zone [11]. The observation of the A _1g-mode and the 2TA continuum conclusively proves that terms of both χ ₁ [Eq. (1)] and χ ₂(q) [Eq. (2)] character participate in the excitation process and, therefore, that the overall coherence is that of a combined coherent-squeezed field. It should be noted that these results reveal no evidence of interaction between the coherent and the squeezed modes, i.e., the two fields are excited independently by the pump.

 

Fig. 2. (a) Normalized transmitted intensity of the probe pulse as a function of the delay for the A _1g-symmetry configuration. (b) Fourier transform of the time-domain data.

Download Full Size | PDF

In closing, we note that, because of domain structure [4], the tetragonal axis of the crystal, ĉ, has no unique direction in the laboratory and, therefore, the selection rules are not known a priori below T_C . However, symmetry considerations indicate that E _g-modes are only allowed if the polarizations have components parallel to ĉ. Since the data in Fig. 2 show a single first-order feature of symmetry A _1g, we conclude that the tetragonal axis in this case is along the cubic [001]. Other results (not shown) exhibit additional oscillations associated with the E _g-phonon at 4.3 THz [10] which, accordingly, correspond to domains for which ĉ is perpendicular to [001].