## Abstract

We have simultaneously excited a coherent and a squeezed phonon field in
SrTiO_{3} using femtosecond laser pulses and stimulated Raman
scattering. The frequency of the coherent state (~ 1.3 THz) is that
of the *A*
_{1g}-component of the soft mode responsible
for the cubic-tetragonal phase transformation at ≈ 110 K. The
squeezed field involves a continuum of transverse acoustic phonons dominated by
a narrow peak in the density of states at ~ 6.9 THz.

© Optical Society of America

## 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 KTaO_{3} [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 SrTiO_{3}. 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 SrTiO_{3}. 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*
_{1}δ_{q,0}+*χ*
_{2}(**q**)
where [1,2]

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 τ_{0} such that
τ_{0}≪2π/Ω_{q}. Then, we can approximate
*E*
^{2}=(4π*I*
_{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}=π*I*
_{0}/(*nc*Ω_{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
KTaO_{3} [2] and SrTiO_{3} (see later) are quasi-monochromatic
for they are dominated by frequencies associated with van Hove singularities in the
phonon density of states [2]

## 3. SrTiO_{3}

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 ~10^{4} 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
SrTiO_{3}, *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
KTaO_{3}, *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}
,
SrTiO_{3} 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 KTaO_{3} [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 mm^{3} single crystal of
SrTiO_{3} 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.

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].

## Acknowledgments

R. M. would like to thank his colleagues at UAM, where he was a *Profesor
Visitante Iberdrola de Ciencia y Tecnología*, for warm
hospitality. Supported by the NSF through the Center for Ultrafast Optical Science
under grant STC PHY 8920108 and by the ARO under contract DAAH04-96-1-0183.

## References and links

**1. **R. Merlin, “Generating coherent THz phonons with light pulses,” Solid State Commun. **102**, 207–220 (1997). [CrossRef]

**2. **G. A. Garrett, A. G. Rojo, A. K. Sood, J. F. Whitaker, and R. Merlin, “Vacuum squeezing of solids: macroscopic quantum states driven by light pulses,” Science **275**, 1638–1640 (1997). [CrossRef] [PubMed]

**3. **See, e. g., D. F. Walls and G. J. Wilburn, *Quantum Optics* (Springer, Berlin, 1994), chap. 2.

**4. **P. A. Fleury, J. F. Scott, and J. M. Worlock, “Soft phonon modes and the 110°K phase transition in SrTiO_{3},” Phys. Rev. Lett. **21**, 16–19 (1968). [CrossRef]

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**6. **See: W. Zhong and D. Vanderbilt, “Effect of quantum fluctuations on structural phase transitions in SrTiO_{3} and BaTiO_{3},” Phys. Rev. B **53**, 5047–5050 (1996), and references therein. [CrossRef]

**7. **P. A. Fleury and J. M. Worlock, “Electric-Field-Induced Raman scattering in SrTiO_{3} and BaTiO_{3},” Phys. Rev. **174**, 613–623 (1968) [CrossRef]

**8. **D. E. Grupp and A. M. Goldman, “Giant piezoelectric effect in strontium titanate at cryogenic temperatures,” Science **276**, 392–394 (1997). [CrossRef] [PubMed]

**9. **H. Uwe and T. Sakudo, “Stress-induced ferroelectricity and soft modes in SrTiO_{3},” Phys. Rev. B **13**, 271–286 (1976). [CrossRef]

**10. **W. G. Nielsen and J. G. Skinner, “Raman spectrum of strontium titanate,” J. Chem. Phys. **48**, 2240–2248 (1968). [CrossRef]

**11. **W. G. Stirling, “Neutron inelastic scattering study of the lattice dynamics of strontium titanate: harmonic models,” J. Phys. C **5**, 2711–2730 (1972). [CrossRef]