## Ultrafast optical excitation of a combined coherent-squeezed phonon
field in SrTiO_{3}

Optics Express, Vol. 1, Issue 12, pp. 385-389 (1997)

http://dx.doi.org/10.1364/OE.1.000385

Acrobat PDF (123 KB)

### 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

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

_{3}[2

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]

*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.

## 2. Theory of phonon field generation

*H*=

*H*

_{0}-|

*E*(

*t*)|

^{2}∑

_{q}

*χ*(

**q**). Here,

*H*

_{0}=∑

_{q}(

*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

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]

*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

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]

*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

**102**, 207–220 (1997). [CrossRef]

*Q*

_{0}〉, is the same as the classical expression and given by

**275**, 1638–1640 (1997). [CrossRef] [PubMed]

*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

*Q*

_{q}〉 unless the latter quantity is different than zero. The variance satisfies the equation

_{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

*I*

_{0},

*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,

*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}).

*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

^{+}

^{-}=|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

**102**, 207–220 (1997). [CrossRef]

*continuum*of modes throughout the Brillouin zone. The continua in KTaO

_{3}[2

**275**, 1638–1640 (1997). [CrossRef] [PubMed]

_{3}(see later) are quasi-monochromatic for they are dominated by frequencies associated with van Hove singularities in the phonon density of states [2

**275**, 1638–1640 (1997). [CrossRef] [PubMed]

## 3. SrTiO_{3}

*T*

_{C}≈110 K from a cubic perovskite, point group

*O*

_{h}, to a low-temperature tetragonal structure of symmetry

*D*

_{4h}[4

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]

*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

5. K. A. Müller and H. Burkard, “SrTiO_{3}: an intrinsic quantum paraelectric below 4 K,” Phys. Rev. B **19**, 3593–3602 (1979). [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]

*quantum paraelectricity*[5

5. K. A. Müller and H. Burkard, “SrTiO_{3}: an intrinsic quantum paraelectric below 4 K,” Phys. Rev. B **19**, 3593–3602 (1979). [CrossRef]

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]

_{3},

*i.e*.,

*χ*

_{2}(

**q**)≠0 for both

*T*>

*T*

_{C}and

*T*<

*T*

_{C}[10

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

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]

*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

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]

*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

_{3},” Phys. Rev. Lett. **21**, 16–19 (1968). [CrossRef]

*T*

_{C}, SrTiO

_{3}meets the conditions required for the impulsive excitation of a combined coherent-squeezed field.

## 4. Experiments

_{3}[2

**275**, 1638–1640 (1997). [CrossRef] [PubMed]

^{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.

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

*A*

_{1g}-phonon and the 2TA overtone. Consistent with the spontaneous results [10

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]

*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.

_{3},” Phys. Rev. Lett. **21**, 16–19 (1968). [CrossRef]

*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

**48**, 2240–2248 (1968). [CrossRef]

*ĉ*is perpendicular to [001].

## Acknowledgments

*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. |

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 |

3. | See, e. g., D. F. Walls and G. J. Wilburn, |

4. | P. A. Fleury, J. F. Scott, and J. M. Worlock, “Soft phonon modes and the 110°K phase transition in SrTiO |

5. | K. A. Müller and H. Burkard, “SrTiO |

6. | See: W. Zhong and D. Vanderbilt, “Effect of quantum fluctuations on structural phase transitions in SrTiO |

7. | P. A. Fleury and J. M. Worlock, “Electric-Field-Induced Raman scattering in SrTiO |

8. | D. E. Grupp and A. M. Goldman, “Giant piezoelectric effect in strontium titanate at cryogenic temperatures,” Science |

9. | H. Uwe and T. Sakudo, “Stress-induced ferroelectricity and soft modes in SrTiO |

10. | W. G. Nielsen and J. G. Skinner, “Raman spectrum of strontium titanate,” J. Chem. Phys. |

11. | W. G. Stirling, “Neutron inelastic scattering study of the lattice dynamics of strontium titanate: harmonic models,” J. Phys. C |

**OCIS Codes**

(270.1670) Quantum optics : Coherent optical effects

(290.5910) Scattering : Scattering, stimulated Raman

(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors

**ToC Category:**

Focus Issue: Coherent Phenomena in Solids

**History**

Original Manuscript: September 30, 1997

Published: December 8, 1997

**Citation**

Gregory Garrett, John Whitaker, Ajay Sood, and Roberto Merlin, "Ultrafast Optical Excitation of a Combined Coherent-Squeezed Phonon field in SrTiO3," Opt. Express **1**, 385-389 (1997)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-12-385

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### References

- R. Merlin, Generating coherent THz phonons with light pulses, Solid State Commun. 102, 207-220 (1997). [CrossRef]
- 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]
- See, e. g., D. F. Walls and G. J. Wilburn, Quantum Optics (Springer, Berlin, 1994), chap. 2.
- P. A. Fleury, J. F. Scott and J. M. Worlock, Soft phonon modes and the 110 0 K phase transition in SrTiO3, Phys. Rev. Lett. 21, 16-19 (1968). [CrossRef]
- K. A. Mller and H. Burkard, SrTiO3: an intrinsic quantum paraelectric below 4 K, Phys. Rev. B 19, 3593-3602 (1979). [CrossRef]
- See: W. Zhong and D. Vanderbilt, Effect of quantum fluctuations on structural phase transitions in SrTiO3 and BaTiO3, Phys. Rev. B 53, 5047-5050 (1996), and references therein. [CrossRef]
- P. A. Fleury and J. M. Worlock, Electric-Field-Induced Raman scattering in SrTiO3 and BaTiO3, Phys. Rev. 174, 613-623 (1968) [CrossRef]
- D. E. Grupp and A. M. Goldman, Giant piezoelectric effect in strontium titanate at cryogenic temperatures, Science 276, 392-394 (1997). [CrossRef] [PubMed]
- H. Uwe and T. Sakudo, Stress-induced ferroelectricity and soft modes in SrTiO3, Phys. Rev. B 13, 271-286 (1976). [CrossRef]
- W. G. Nielsen and J. G. Skinner, Raman spectrum of strontium titanate, J. Chem. Phys. 48, 2240-2248 (1968). [CrossRef]
- W. G. Stirling, Neutron inelastic scattering study of the lattice dynamics of strontium titanate: harmonic models, J. Phys. C 5, 2711-2730 (1972). [CrossRef]

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