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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 1, Iss. 12 — Dec. 8, 1997
  • pp: 385–389
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Ultrafast optical excitation of a combined coherent-squeezed phonon field in SrTiO3

G. A. Garrett, J. F. Whitaker, A. K. Sood, and R. Merlin  »View Author Affiliations


Optics Express, Vol. 1, Issue 12, pp. 385-389 (1997)
http://dx.doi.org/10.1364/OE.1.000385


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Abstract

We have simultaneously excited a coherent and a squeezed phonon field in SrTiO3 using femtosecond laser pulses and stimulated Raman scattering. The frequency of the coherent state (~ 1.3 THz) is that of the A1g-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

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

]. More recently, ultrafast pulses were also used to generate a squeezed phonon field in KTaO3 [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]

]. As in the electromagnetic case [3

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

], 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. 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]

].

2. Theory of phonon field generation

The Hamiltonian relevant to our problem is H=H 0-|E(t)|2q χ(q). Here, H 0=∑q(Pq2+Ωq2 Qq2)/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 Ωq and wave vector q. To second-order in the atom displacements, χ(q)=χ 1δq,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

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]

]

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]
χ1=12(χQ0)Q0
(1)
χ2(q)=14(2χQq2)Qq2.
(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

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

,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]

]. Since χ 1Q q0 (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

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

]. 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

d2Q0dt2+Ω02Q0=12(χQ0)E(t)2=F.
(3)

As discussed in [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]

], 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 ΔΩqν 2/(2Ωq) with

ν2=12(2χQ0)E(t)2.
(4)

The varying frequency leads to a time-dependent variance, 〈Qq2〉, but it does not affect 〈Q q〉 unless the latter quantity is different than zero. The variance satisfies the equation

d3Qq2dt3+4[Ωq2+ν2]dQq2dt=2d(ν2)dtQq2.
(5)

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

Q0=2W0(χQ0)sin(Ω0t)
(6)

and, to lowest order in I 0,

Qq2(t)Qq2(0)[1+2Wq(2χQq2)sin(2Ωqt)].
(7)

Here I 0 is the integrated intensity of the pulse, n is the refractive index and W qI 0/(ncΩq). Provided that the factor multiplying sin(2Ωq t) is sufficiently large so as to overcome the thermal contribution, the variance dips below the standard quantum limit, σq2=ħ/(2Ωq), 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/(2Ωq).

The approximation E 2δ(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+

Ψ+=exp(q2iWqσq2[χ1δq,0+χ2(q)])Ψ.
(8)

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

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

]. 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

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

], whereas squeezed fields involve a continuum of modes throughout the Brillouin zone. The continua in KTaO3 [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]

] 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

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. SrTiO3

Strontium titanate undergoes an antiferro-distortive phase transition at TC ≈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 SrTiO3,” Phys. Rev. Lett. 21, 16–19 (1968). [CrossRef]

]. 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

5. K. A. Müller and H. Burkard, “SrTiO3: an intrinsic quantum paraelectric below 4 K,” Phys. Rev. B 19, 3593–3602 (1979). [CrossRef]

] (for electric-field- and stress-induced ferroelectricity, see [7–9

7. P. A. Fleury and J. M. Worlock, “Electric-Field-Induced Raman scattering in SrTiO3 and BaTiO3,” Phys. Rev. 174, 613–623 (1968) [CrossRef]

]). This behavior, referred to as quantum paraelectricity [5

5. K. A. Müller and H. Burkard, “SrTiO3: an intrinsic quantum paraelectric below 4 K,” Phys. Rev. B 19, 3593–3602 (1979). [CrossRef]

], has been extensively discussed in the literature [6

6. 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]

]. 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>TC and T<TC [10

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

]. However, there are no first-order Raman-allowed modes [4

4. P. A. Fleury, J. F. Scott, and J. M. Worlock, “Soft phonon modes and the 110°K phase transition in SrTiO3,” Phys. Rev. Lett. 21, 16–19 (1968). [CrossRef]

] or, alternatively, χ 1≡0 for T>TC (note that, in KTaO3, χ 1≡0 at all temperatures). In the tetragonal phase, T<TC , 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 SrTiO3,” Phys. Rev. Lett. 21, 16–19 (1968). [CrossRef]

]. 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

4. P. A. Fleury, J. F. Scott, and J. M. Worlock, “Soft phonon modes and the 110°K phase transition in SrTiO3,” Phys. Rev. Lett. 21, 16–19 (1968). [CrossRef]

]. It follows that below TC , 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

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]

]. 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

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

] we ascribe them to the soft 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]

], 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

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]

]. 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.

In closing, we note that, because of domain structure [4

4. P. A. Fleury, J. F. Scott, and J. M. Worlock, “Soft phonon modes and the 110°K phase transition in SrTiO3,” Phys. Rev. Lett. 21, 16–19 (1968). [CrossRef]

], 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 TC . 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

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

] 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 SrTiO3,” Phys. Rev. Lett. 21, 16–19 (1968). [CrossRef]

5.

K. A. Müller and H. Burkard, “SrTiO3: 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 SrTiO3 and BaTiO3,” 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 SrTiO3 and BaTiO3,” 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 SrTiO3,” 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]

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

  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 0 K phase transition in SrTiO3, Phys. Rev. Lett. 21, 16-19 (1968). [CrossRef]
  5. K. A. Mller and H. Burkard, SrTiO3: 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 SrTiO3 and BaTiO3, 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 SrTiO3 and BaTiO3, 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 SrTiO3, 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]

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