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Selection of stimulated Raman scattering signal by entangled photons
Optics Communications 383 (2017) 581–585
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Selection of stimulated Raman scattering signal by entangled photons Purevdorj Munkhbaatar
a,b
b,c,⁎
, Kim Myung-Whun
crossmark
a
Department of Physics and Electronics, National University of Mongolia, Ulaanbaatar 210646, Mongolia b Institute of Photonics and Information Technology, Chonbuk National University, Jeon-ju 54896, Republic of Korea c Department of Physics, Chonbuk National University, Jeon-ju 54896, Republic of Korea
A R T I C L E I N F O
A BS T RAC T
Keywords: Excitation-probe measurement Entangled photon pulses Stimulated Raman scattering
We propose an excitation-probe measurement method utilizing entangled photon pulses. The excitation-probe signal is dominated by stimulated Raman scattering as well as two-photon absorption when the time delay between the excitation pulse and the probe pulse is shorter than the pulse duration. We demonstrate that the two-photon-absorption signal can be suppressed when the photons of the pulses are entangled. The stimulated Raman scattering signal can be composed of many peaks distributed over broad photon energies owing to the transitions between numerous quantum states in complex materials. We show that the desired peaks among the many peaks can be selected by controlling the thickness of the nonlinear crystal, the pump pulse center frequency, and the polarization of the excitation pulse and probe pulse.
1. Introduction Excitation-probe (or pump-probe) technique is useful to investigate the optical excitations between quantum states of matter [1,2]. Light pulse can excite electric charges, thereby creating an excited state in the sample material. The excited state decays spontaneously to the ground state or to a quasi-ground state by emitting photon fields. In the presence of strong radiation such as in the excitation-probe measurement, stimulated emission becomes more dominant while spontaneous emission becomes negligible. During the excitation-probe measurement, a part of probe pulse interferes with the emitted photon field, producing measurable change in the intensity of probe pulse itself, which often appears as sinusoidal function of pulse delay time [3]. The sinusoidal modulation is the signature of a stimulated Raman scattering (SRS) process in the sample material [4–6]. In complex materials, various interactions participate in the formation of Raman active modes. Simple harmonic lattice interaction is most prevailing, but spin-exchange or orbital-exchange interaction between neighboring charges also constitute Raman active modes [1]. The interactions are often strongly coupled to each other, so independent modes can be merged into one composite mode soon after the excitation. Early time-delay signal of the excitation-probe measurement is essential to understand the intrinsic properties of the Raman modes. However, the conventional excitation-probe signal suffers from the strong nonlinear background when excitation pulse and probe pulse are overlapped. Fortunately, stimulated Raman scattering (SRS) and two-photon absorption (TPA) are dominant in simple excitation-
⁎
probe processes [3]. If it is possible to separate the TPA signal from the SRS signal in typical excitation-probe measurement geometry, the early time-delay SRS signal can be investigated by the excitation-probe measurement. Some techniques have been invented to discriminate the SRS and TPA signals. Some of the techniques utilized the combinations of classical optical devices including electro-optical or acousto-optical devices [7,8]. Recently, quantum mechanically entangled light becomes used to improve the performance of nonlinear spectroscopic techniques. A theoretical approach showed that the intensity of TPA can be manipulated in the excitation-probe measurement if entangled photons were used [9]. A quantum SRS technique was proposed theoretically, which suggested the combination of entangled light source produced via type-II parametric down conversion and Hanburry−Brown−Twiss type interferometry to enhance the resolution and selectivity of Raman signals [10]. The study theoretically demonstrated that SRS signals obtained by the sixth-order perturbation of entangled photon pulse fields can eliminate the off-resonant background and can select Raman gain and Raman loss processes. However, so far it has been rarely shown the way to suppress the intensity of TPA in the excitation-probe measurement by utilizing the entanglement of photon pulse excitationprobe measurement. Conventional excitation-probe measurements in many laboratories are performed with two optical modes with wave vectors k1 and k2 and frequencies ω1 and ω2 in the simplest description. We have thought about a way of suppressing TPA signal in two-mode geometry. Suppose (k1, ω1) is the excitation mode and (k2 , ω2 ) is the probe mode. Typical
Corresponding author at: Institute of Photonics and Information Technology, Chonbuk National University, Jeon-ju 54896, Republic of Korea. E-mail address: [email protected] (K. Myung-Whun).
http://dx.doi.org/10.1016/j.optcom.2016.09.061 Received 2 April 2016; Received in revised form 9 September 2016; Accepted 29 September 2016 Available online 14 October 2016 0030-4018/ © 2016 Elsevier B.V. All rights reserved.
Optics Communications 383 (2017) 581–585
P. Munkhbaatar, K. Myung-Whun
Hamiltonian can be found in Ref. [15]. Fig. 1b schematically illustrates the structure of the material system’s energy eigenstates. | g represents a set of eigenstates, i.e. {| g1 , | g2 , …}; | g1 is the ground state. The energy difference between | gn and | gn+1 arises from the phonon-orbital coupling energy difference for the different octahedral distortion and orbital ordering configuration. The second set, | e ={| e1 , | e2 , …}, represents the set of photoexcited eigenstates. The excited state corresponds to one hole and one double-occupied state. The energy difference between | en and | en+1 arises from phonon-orbital coupling between the sites, similar as in {| gn }. The energy difference between | gn and | en arises mainly from Coulomb energy (U), since U is about 10 times the phonon-orbital coupling energy. The third set of higher-energy (~2U) photoexcited states, | f (not shown in Fig. 1a), is also considered in the interaction process. The values of U , J , Q3, q , and ωph used in the calculations were5t , 0.06t , 0. 27t , 0. 07t , and 0.2t , respectively. t is the hopping integral parameter representing the kinetic energy gain of the Mn ion electrons [14]. In the proposed excitation-probe setup, the probe signal is determined solely by the (vi) and (viii) pathways; loop diagrams for these pathways are given in Fig. 1c. In the (vi) pathway, the system evolves from the ground state (| g ) to an excited state (| e ) by absorbing a photon, then to a low-energy state (| g′ ) by emitting a photon. Then it assumes another excited state (| e′ ) by absorbing a photon, then finally returns to the ground state (| g ) by emitting a photon. In the (viii) pathway, the system undergoes evolution from the ground state (| g ) to an excited state (| e ) by absorbing a photon, then to a low-energy state (| g′ ) by emitting a photon; then, the same process is repeated once in the bra-vectors. The frequency-resolved signal, S (ω1, ω2 ), is given by the following equation [3]:
measurement detects the intensity change of the (k2 , ω2 ) mode resulting from the cooperation of the eight quantum pathways invoked by the (k1, ω1) mode. The pathways are divided into the TPA group and the SRS group [9]. All the pathways in the TPA group contain optical correlation functions of the form a†a†aa ; here, a (a†) is the annihilation (creation) operator of the optical modes. The possible optical correlation functions for the TPA pathways are a†2a†1a1a 2 , a†2a†1a 2a1 , a†1a†2a 2a1 , and a†1a†2a1a 2 . The optical correlation functions of the SRS group are a†2a 2a†1a1 , a†2a1a†1a 2 , a†1a1a†2a 2 , and a†1a 2a†2a1 . For the classical optical modes, the intensity change due to all of the eight correlation functions become identical: a†2 a†1 a 2 a1 ~ E1 2 E2 2 [9,11]. This is the reason why the conventional excitation-probe method can hardly discriminate the TPA and SRS signals. However, if the quantum mechanical nature of the optical modes is harnessed, the intensity change due to TPA and SRS can be distinguished. In this paper, we propose a method to suppress the contribution of TPA to the excitation-probe signal by controlling the quantum states of the excitation and probe pulses. We have applied the proposed method to simulate the excitation-probe signal of a model material system and demonstrate herein that the proposed method can suppress the TPA signal and select the SRS signal specifically. 2. Theory Fig. 1a shows the proposed measurement setup. Here the photon pairs can be produced in pulsed mode by means of spontaneous parametric down conversion (SPDC). A type-I nonlinear crystal is pumped by a Gaussian pulse with temporal duration T1 and central frequency ωp . The pulsed photon pairs can be described by a nonseparable wave function Ψ [12]:
Ψ =
∑i,j ϕijb1i†b2j†|0
,
ϕij = C exp[−T12[ (ωp − ω1i − ω2j )]2 ] sinc[T2(ω2j − ω1i )]
S (ω1, ω2 )~Im[χ (vi) (ω1, ω2 )] a 2†a1a1†a2 +Im[χ (viii) (ω1, ω2 )] a1†a 2a 2†a1 .
(1)
(5) (2)
The third-order material susceptibility functions are as follows:
where T2 is the entanglement time, defined by the difference between the inverse group velocities of the two photons. T2 can be controlled by changing the thickness of the SPDC crystal. Fock state b1i†b2j † | 0 has one photon in each mode k1 and k2 . C is proportional to the nonlinearity of the SPDC crystal and the pump electric field amplitude ℇex [13].. Inspired by the work of Roslyak et al., we included an interferometer to control the quantum states of pulsed photon pairs [9]. The two photon modes (k1, ω1) and (k2 , ω2 ) produced in the nonlinear crystal are directed into the input port of the broadband 50/50 beam splitter, which mixes the photon fields. The photons leaving the beam splitter can be described by a1 for the excitation photon field and a 2 for the probe photon field [14]:
a1 =
1 1 (−b1 + Ib2 ), a 2 = (Ib1 − b2 ). 2 2
χ (vi) (ω1, ω2 ) = χ (viii) (ω1, ω2 ) =
−1 3!
∑e ∑g ∑e
−1 3!
′
′
∑e ∑g ∑e ′
μgβe μeα g μgα e μegβ Neg (ω2 )Ng g (ω2 − ω1)Ne g (ω2 ) 1
′
′ ′ ′ ′
1
′
1
1
′
1
α μgα e μeβ g μgβ e μeg Neg (ω1)N*g g (ω1 − ω2 )N*e g (ω1) 1
′ ′ ′ ′
1
1
′
1
′
1
(6) (7)
μegα, β
is the dipole matrix element between the two energy here, eigenstates | e and | g of the full Hamiltonian (H ), namely, e r ⃗ g ; r ⃗
(3)
We assumed that the excitation-probe signal is composed of eight pathway contributions [3,9]. When the proposed setup is used, the following six photon correlation functions among the eight functions are all equal to zero:
a1†a 2†a 2a1 = a1†a 2†a1a 2 = a 2†a1†a2a1 = a 2†a1†a1a2 = a1†a1a 2†a 2 = a 2†a 2a1†a1 =0.
(4)
Calculations of these functions are given in Appendix A. Nonzero contributions thus arise only from the other two pathways; a 2†a1a1†a2 of the (vi) pathway and a1†a 2a 2†a1 of the (viii) pathway. We considered crystalline LaMnO3 as the model material system. To mimic one lattice layer of the orbital- and spin-ordered LaMnO3 crystal, we built a Hamiltonian for an ideal Mn cluster composed of four Mn3+ ions. Details of the model material system and the
Fig. 1. (a) Proposed excitation-probe setup, (b) schematic structure of energy eigenstates of the model material system, and (c) closed-time path-loop diagrams for (vi) and (viii) pathways.
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P. Munkhbaatar, K. Myung-Whun
is the dipole displacement operator. The superscripts α and β indicate the polarization directions of the a1 (excitation) and a 2 (probe) fields, respectively. Neg(ω) is the auxiliary function representing the Lorentzian response [9]:
Neg(ω)=
1 . ω − (Ee − Eg )+Iγ
(8)
here, Eg (Ee ) is the energy eigenvalue of | g (| e ), and γ is the spectral broadening parameter. A phenomenological value (0.05t ) was used for γ to obtain the results shown in Figs. 2, 3, 5, and 6... 3. Results and discussion First we calculated the conventional excitation-probe signal. For parallel polarizations of the excitation pulse and probe pulse, numerous peaks were observed, arising from the superposition of the TPA and SRS processes contributed by all eight pathways (Fig. 2a). Fig. 2b shows the perpendicular polarization case. Next we applied the proposed method. Fig. 2c shows the excitationprobe signal for the parallel polarization case. Spectral structures arising from TPA contributions and some of the SRS contributions shown in Fig. 2a do not appear in Fig. 2c. For example, the structures between ω1=4.5t and ω1=4.9t and between ω2=3.6t and ω2=3.8t which arise from the TPA contributions, are not present in Fig. 2c. Fig. 2c shows two peaks along the diagonal line at ω1 = ω2=4.45t and at ω1 = ω2=4.65t . The ω1=4.45t peak arose from the g1 → e1 → g1 → e1 → g1 transition process, and the ω1=4.65t peak arose from the g1 → e2 → g1 → e2 → g1 process. In addition to these peaks, more peaks appeared along the vertical line at ω2=4.45t and at ω2=4.65t , owing to the contribution of the (vi) pathway. Peaks appeared along the horizontal line at ω1=4.45t and at ω1=4.65t owing to the contribution of the (viii) pathway. Fig. 2d shows the perpendicular polarization case. Note that the g1 → e1 → g4 → e1 → g1 transition peak is clearer than the other peaks. In an experiment, Fig. 2c and d can be obtained by measuring the spectral transmittance of the probe signal, because S (ω1, ω2 ) is proportional to the transmittance intensity change. We found an additional advantage of the proposed method. The excitation-probe signals shown in Fig. 3e and f are the products of the susceptibility function (Fig. 3a and b) and the photon correlation
Fig. 3. (a, b) Susceptibility functions for (a) the (vi) pathway (Im[χ(vi)(ω1, ω2)]) and (b) the (viii) pathway (Im[χ (viii) (ω1, ω2 )]). (c, d) Photon correlation functions for (c) the (vi) pathway ( a2†a1a1†a2 ) and (d) the (viii) pathway ( a1†a 2a 2†a1 ). (e, f) Excitation-probe signals for (e) the (vi) pathway (S (vi)(ω1, ω2 ) ) and (f) the (viii) pathway (S (viii)(ω1, ω2 ) ).
function (Fig. 3c and d) as described in Eq. (5). The proposed method allows tuning of the spectral widths and positions of the photon correlation functions. This tunability can help to enhance the excitation-probe signal of some specific group of transitions and to suppress the other signals selectively. We controlled the spectral distribution of the photon correlation function by changing a parameter of the joint distribution function, | ϕij |2 . Fig. 4 shows the two cases of | ϕij |2 ; the | ϕij |2 shown in Fig. 4a corresponds to the photons of Fig. 3c and d. Here, the parameters ωp , T1, and T2 were 8.94t , 31.0/t , and 0. 1/t . The | ϕij |2 shown in Fig. 4b
Fig. 2. (a, b) Conventional excitation-probe signals for (a) the parallel polarization and (b) the perpendicular polarization. (c, d) Excitation probe signals obtained with the proposed method for (c) the parallel polarization and (d) the perpendicular polarization.
Fig. 4. Joint spectral density functions for different photon pair entanglement times (T2 ): (a) T2 = 0. 1/t , (b) T2=0.3/t .
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P. Munkhbaatar, K. Myung-Whun
corresponds to the photons of Fig. 5a and b. Here, the parameters ωp and T1 were same; only T2 was different (T2=0. 3/t ). T2 can be controlled by changing the thickness of the nonlinear crystal [12]. Fig. 5a and b show the photon correlation functions with longer T2 (i.e. with narrower bandwidth) than those in Fig. 3c and d; Fig. 5a shows the photon correlation function for the (vi) pathway and Fig. 5b shows the function for the (viii) pathway. Fig. 5c and d show the excitation-probe signal produced by the narrow-bandwidth photon correlation functions; compared with Fig. 3e and f, the SRS peaks arising from the g1 → e1 → g4 → e1 → g1 transition remain clear whereas the other peaks are diminished. We found that the polarization states of pulses reveal the rotational symmetry of the excitations. Fig. 6a and b present the SRS signal due to the quasiparticle (phonon and phonon-coupled orbiton) excitations in the model material system, LaMnO3. Fig. 6a shows the signal of the parallel polarization state; the peaks arise from the g1 → e1 → gn → e1 → g1 transition pathways. Here we only focus on the peaks above the diagonal (ω1 = ω2 ) line. Peaks at lower parts can be understood similarly. When n = 1, the pathway corresponds to Rayleigh scattering, which results in the peak at ω1 = ω2=4.45t . When n = 2 , the pathway corresponds to Jahn–Teller phonon excitation with Ag symmetry, which results in the peak at (ω1=4.45t , ω2=4.3t ). When n = 3 and 4, the pathways involve the orbital excitation. In particular, the n = 4 transition pathway results in the so-called phonon-coupled orbiton, which results in the peak at (ω1=4.45t , ω2=3.85t ) [14]. The strength of the peaks depends on the dipole matrix elements: μgβe μeαg μgα e μeβg , as described in Eq. (6). Fig. 6b shows the perpendicular 11 1n n 1 11 polarization case; i.e, α ⊥β . Because of the broken rotational symmetry of the excited orbitals in | g3 and | g4 , μgβe μeαg μgα e μeβg remains finite for 11 1n n 1 11 the pathways involving the orbital excitation (i.e. g1 → e1→(g3, g4 )→e1 → g1), whereas the matrix elements involving | g1 and | g2 become zero. Note that the peaks at (ω1=4.45t , ω2=3.85t ) remain whereas the other peaks are suppressed in Fig. 6b. The broken rotational symmetry of the intermediate state allows the intensity of the SRS signal to remain finite even in the perpendicular polarization geometry of the excitation-probe pulses. Comparison of the SRS signals measured with different polarization configurations of the pulses can test the rotational symmetry of the excited states. In experiment, Fig. 6(a) and (b) can be obtained by measuring the spectral transmittance change of probe signal, ∆Τ (ω2 ) for a LaMnO3 crystalline thin film sample. The second harmonic pulse of a femtosecond pulse generated from a Ti: Sapphire laser oscillator can be used as the pump source for the SPDC process and a type-I β-BaB2O4 crystal can be used for generating the SPDC photon pulses. In this case, center wavelength of the SPDC pulse is around 800 nm. The peak at ω1 = ω2=4.5t in Fig. 6(a) is usually observed at about 650 nm in the linear absorption experiment [16]. By using a band-pass filter or a grating, the spectral components of the photon pulse in 900 – 600 nm range can be selected from the SPDC rainbow. In addition to the conventional Hong-Ou-Mandel interferometer shown in Fig. 1(a), gratings, pinholes, and a half-wave plate can be included to select the specific wavelength components of the pulses irradiated to the sample surface. For the gratings and pinholes, optical paths of pump and probe pulse should be correctly arranged to meet the phase matching condition for different wavelength components. ∆Τ (ω2 ) can be estimated by summing the number of photons with a single photon counter in a given measurement time. First, transmitted probe pulse is measured by blocking pump pulse illumination (Τ (ω2 )without ). Then the transmittance of probe pulse is measured with pump pulse illumination (Τ (ω2 )with ). By subtracting the second signal from the first one, ∆Τ (ω2 ) can be obtained. This procedure can be repeatedly done to improve the signal to noise ratio, if an optical chopper and a lock-in amplifier were combined. In present study, we considered spontaneous parametric down conversion process for the pump and probe photon pulses preparation. In the process, the intensity of generated pulses may be weak, so it may be difficult to get clear signals. Intense pulse
Fig. 5. (a, b) Photon correlation functions for (a) the (vi) pathway and (b) the (viii) pathway. (c, d) SRS signals for (c) the (vi) pathway and (d) the (viii) pathway.
Fig. 6. Excitation probe signals simulated with narrower-bandwidth photon correlation functions (T2 = 0.3/t case) for (a) parallel polarization and (b) perpendicular polarization of the excitation and probe pulses.
can be produced by parametric amplification process. Although the detailed nature of the quantum correlations can be different for parametric down conversion and for parametric amplification, quantum state signatures of light can remain also for intense pulse fields generated by the parametric amplification process as Nagasako et al. observed [14].
4. Conclusions We have proposed an excitation-probe method using entangled photon pulses. Numerical simulation demonstrated that the twophoton absorption signal can be selectively filtered in the excitationprobe signal. By controlling the crystal thickness and the pump pulse frequency for the parametric down-conversion, we can tune the bandwidth and the central frequency of the photon correlation function distribution used in the proposed method. We found that this tuning can be used to select specific transition peaks from the group of numerous stimulated Raman scattering peaks. We demonstrated that the proposed method can be used to examine the stimulated Raman scattering signal arising from orbital wave excitation in a model material, LaMnO3. 584
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P. Munkhbaatar, K. Myung-Whun
Acknowledgements This research was supported by the National Research Foundation of Korea (Grants Nos. 2012R1A1A401010025 and 2015R1D1A1A02061984).
Appendix A. Photon correlation function The photon correlation functions for some pathways are zero. Following is the proof for one of those pathways:
a1i†a 2j †a 2ja1i =
1 (−Ib2i† − b1i†)(−Ib1j † − b2j †)(Ib1j − b2j )(Ib2i − b1i ) ; 4
(A1)
for the ket side,
(Ib1j − b2j )(Ib2i − b1i ) Ψ =
∑k,l ϕkl(b2jb1i − b2ib1j )b1k †b2l†|0
= ϕij |0 − ϕji|0 = 0
(A2)
Note here that ϕij = ϕji for a type-I nonlinear crystal. Therefore, †
a1i a 2j †a 2ja1i =0.
(A3)
The photon correlation functions for the (vi) and (viii) pathways shown in Fig. 1c can be expressed as follows: †
a1i a 2ja 2j †a
1i
= a1i†a1i + a1i†a 2j †a 2ja1i .
(A4)
The second term on the right side is zero. The first term is
* ϕ b1mb2na1i†a1ib1k †b2l †|0 Ψ |a1i†a1i Ψ = ∑m, n ∑k , l 0|ϕmn kl 1
* ϕ b1mb2n(−Ib2i† − b1i†)(Ib2i − b1i )b1k †b2l †|0 = = 2 ∑m, n ∑k , l 0|ϕmn kl †
†
†
1 2
⎞ ⎛ ∑j ⎜|ϕij |2 + |ϕji|2 ⎟ ⎠ ⎝
(A5)
†
Similarly, a 2i a1ja1j a 2i = a 2i a 2i = a1i a1i .. Appendix B. Two-photon absorption signal with frequency-correlated photons One may wonder whether the quantum mechanical entanglement between the photons is necessary for the suppression of the two-photon absorption signal or whether similar results could be achieved by using frequency-correlated photons that are not entangled. To investigate this, we explore the case in which the two photons are frequency-correlated but nonetheless unentangled. To compute the photon correlation function, we consider a classically correlated two-photon state described by a density operator of the following form [17]:
⎡ ⎤2 ρ = C ∑ ⎢ϕij ⎥ b1i†b2j †|0 0|b1ib2j i, j ⎣ ⎦
(B1)
⎡ ⎤ where ⎢ϕij ⎥ is the joined distribution function of the photons and C is a normalization constant. Annihilation (creation) operators of these fields are ⎣ ⎦ represented by b1(b1†) and b2(b2†). Using this density operator, we can calculate two-photon correlation functions in group A. For example, 2
⎡ ⎤2 b†2k b†1lb2k b1l = Tr[b2k b1lρ b†2k b†1l ] = C ∑ ⎢ϕij ⎥ Tr[b2k b1l b1i†b2j †|0 0|b1ib2jb†2k b†1l ] = C [ϕlk ]2 i, j ⎣ ⎦
(B2)
All of the correlation functions are identical and nonzero, which means that we cannot suppress the TPA signal using the frequency-correlated, unentangled photons.
[9] [10] [11] [12] [13]
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Selection of stimulated Raman scattering signal by entangled photons Purevdorj Munkhbaatar
a,b
b,c,⁎
, Kim Myung-Whun
crossmark
a
Department of Physics and Electronics, National University of Mongolia, Ulaanbaatar 210646, Mongolia b Institute of Photonics and Information Technology, Chonbuk National University, Jeon-ju 54896, Republic of Korea c Department of Physics, Chonbuk National University, Jeon-ju 54896, Republic of Korea
A R T I C L E I N F O
A BS T RAC T
Keywords: Excitation-probe measurement Entangled photon pulses Stimulated Raman scattering
We propose an excitation-probe measurement method utilizing entangled photon pulses. The excitation-probe signal is dominated by stimulated Raman scattering as well as two-photon absorption when the time delay between the excitation pulse and the probe pulse is shorter than the pulse duration. We demonstrate that the two-photon-absorption signal can be suppressed when the photons of the pulses are entangled. The stimulated Raman scattering signal can be composed of many peaks distributed over broad photon energies owing to the transitions between numerous quantum states in complex materials. We show that the desired peaks among the many peaks can be selected by controlling the thickness of the nonlinear crystal, the pump pulse center frequency, and the polarization of the excitation pulse and probe pulse.
1. Introduction Excitation-probe (or pump-probe) technique is useful to investigate the optical excitations between quantum states of matter [1,2]. Light pulse can excite electric charges, thereby creating an excited state in the sample material. The excited state decays spontaneously to the ground state or to a quasi-ground state by emitting photon fields. In the presence of strong radiation such as in the excitation-probe measurement, stimulated emission becomes more dominant while spontaneous emission becomes negligible. During the excitation-probe measurement, a part of probe pulse interferes with the emitted photon field, producing measurable change in the intensity of probe pulse itself, which often appears as sinusoidal function of pulse delay time [3]. The sinusoidal modulation is the signature of a stimulated Raman scattering (SRS) process in the sample material [4–6]. In complex materials, various interactions participate in the formation of Raman active modes. Simple harmonic lattice interaction is most prevailing, but spin-exchange or orbital-exchange interaction between neighboring charges also constitute Raman active modes [1]. The interactions are often strongly coupled to each other, so independent modes can be merged into one composite mode soon after the excitation. Early time-delay signal of the excitation-probe measurement is essential to understand the intrinsic properties of the Raman modes. However, the conventional excitation-probe signal suffers from the strong nonlinear background when excitation pulse and probe pulse are overlapped. Fortunately, stimulated Raman scattering (SRS) and two-photon absorption (TPA) are dominant in simple excitation-
⁎
probe processes [3]. If it is possible to separate the TPA signal from the SRS signal in typical excitation-probe measurement geometry, the early time-delay SRS signal can be investigated by the excitation-probe measurement. Some techniques have been invented to discriminate the SRS and TPA signals. Some of the techniques utilized the combinations of classical optical devices including electro-optical or acousto-optical devices [7,8]. Recently, quantum mechanically entangled light becomes used to improve the performance of nonlinear spectroscopic techniques. A theoretical approach showed that the intensity of TPA can be manipulated in the excitation-probe measurement if entangled photons were used [9]. A quantum SRS technique was proposed theoretically, which suggested the combination of entangled light source produced via type-II parametric down conversion and Hanburry−Brown−Twiss type interferometry to enhance the resolution and selectivity of Raman signals [10]. The study theoretically demonstrated that SRS signals obtained by the sixth-order perturbation of entangled photon pulse fields can eliminate the off-resonant background and can select Raman gain and Raman loss processes. However, so far it has been rarely shown the way to suppress the intensity of TPA in the excitation-probe measurement by utilizing the entanglement of photon pulse excitationprobe measurement. Conventional excitation-probe measurements in many laboratories are performed with two optical modes with wave vectors k1 and k2 and frequencies ω1 and ω2 in the simplest description. We have thought about a way of suppressing TPA signal in two-mode geometry. Suppose (k1, ω1) is the excitation mode and (k2 , ω2 ) is the probe mode. Typical
Corresponding author at: Institute of Photonics and Information Technology, Chonbuk National University, Jeon-ju 54896, Republic of Korea. E-mail address: [email protected] (K. Myung-Whun).
http://dx.doi.org/10.1016/j.optcom.2016.09.061 Received 2 April 2016; Received in revised form 9 September 2016; Accepted 29 September 2016 Available online 14 October 2016 0030-4018/ © 2016 Elsevier B.V. All rights reserved.
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P. Munkhbaatar, K. Myung-Whun
Hamiltonian can be found in Ref. [15]. Fig. 1b schematically illustrates the structure of the material system’s energy eigenstates. | g represents a set of eigenstates, i.e. {| g1 , | g2 , …}; | g1 is the ground state. The energy difference between | gn and | gn+1 arises from the phonon-orbital coupling energy difference for the different octahedral distortion and orbital ordering configuration. The second set, | e ={| e1 , | e2 , …}, represents the set of photoexcited eigenstates. The excited state corresponds to one hole and one double-occupied state. The energy difference between | en and | en+1 arises from phonon-orbital coupling between the sites, similar as in {| gn }. The energy difference between | gn and | en arises mainly from Coulomb energy (U), since U is about 10 times the phonon-orbital coupling energy. The third set of higher-energy (~2U) photoexcited states, | f (not shown in Fig. 1a), is also considered in the interaction process. The values of U , J , Q3, q , and ωph used in the calculations were5t , 0.06t , 0. 27t , 0. 07t , and 0.2t , respectively. t is the hopping integral parameter representing the kinetic energy gain of the Mn ion electrons [14]. In the proposed excitation-probe setup, the probe signal is determined solely by the (vi) and (viii) pathways; loop diagrams for these pathways are given in Fig. 1c. In the (vi) pathway, the system evolves from the ground state (| g ) to an excited state (| e ) by absorbing a photon, then to a low-energy state (| g′ ) by emitting a photon. Then it assumes another excited state (| e′ ) by absorbing a photon, then finally returns to the ground state (| g ) by emitting a photon. In the (viii) pathway, the system undergoes evolution from the ground state (| g ) to an excited state (| e ) by absorbing a photon, then to a low-energy state (| g′ ) by emitting a photon; then, the same process is repeated once in the bra-vectors. The frequency-resolved signal, S (ω1, ω2 ), is given by the following equation [3]:
measurement detects the intensity change of the (k2 , ω2 ) mode resulting from the cooperation of the eight quantum pathways invoked by the (k1, ω1) mode. The pathways are divided into the TPA group and the SRS group [9]. All the pathways in the TPA group contain optical correlation functions of the form a†a†aa ; here, a (a†) is the annihilation (creation) operator of the optical modes. The possible optical correlation functions for the TPA pathways are a†2a†1a1a 2 , a†2a†1a 2a1 , a†1a†2a 2a1 , and a†1a†2a1a 2 . The optical correlation functions of the SRS group are a†2a 2a†1a1 , a†2a1a†1a 2 , a†1a1a†2a 2 , and a†1a 2a†2a1 . For the classical optical modes, the intensity change due to all of the eight correlation functions become identical: a†2 a†1 a 2 a1 ~ E1 2 E2 2 [9,11]. This is the reason why the conventional excitation-probe method can hardly discriminate the TPA and SRS signals. However, if the quantum mechanical nature of the optical modes is harnessed, the intensity change due to TPA and SRS can be distinguished. In this paper, we propose a method to suppress the contribution of TPA to the excitation-probe signal by controlling the quantum states of the excitation and probe pulses. We have applied the proposed method to simulate the excitation-probe signal of a model material system and demonstrate herein that the proposed method can suppress the TPA signal and select the SRS signal specifically. 2. Theory Fig. 1a shows the proposed measurement setup. Here the photon pairs can be produced in pulsed mode by means of spontaneous parametric down conversion (SPDC). A type-I nonlinear crystal is pumped by a Gaussian pulse with temporal duration T1 and central frequency ωp . The pulsed photon pairs can be described by a nonseparable wave function Ψ [12]:
Ψ =
∑i,j ϕijb1i†b2j†|0
,
ϕij = C exp[−T12[ (ωp − ω1i − ω2j )]2 ] sinc[T2(ω2j − ω1i )]
S (ω1, ω2 )~Im[χ (vi) (ω1, ω2 )] a 2†a1a1†a2 +Im[χ (viii) (ω1, ω2 )] a1†a 2a 2†a1 .
(1)
(5) (2)
The third-order material susceptibility functions are as follows:
where T2 is the entanglement time, defined by the difference between the inverse group velocities of the two photons. T2 can be controlled by changing the thickness of the SPDC crystal. Fock state b1i†b2j † | 0 has one photon in each mode k1 and k2 . C is proportional to the nonlinearity of the SPDC crystal and the pump electric field amplitude ℇex [13].. Inspired by the work of Roslyak et al., we included an interferometer to control the quantum states of pulsed photon pairs [9]. The two photon modes (k1, ω1) and (k2 , ω2 ) produced in the nonlinear crystal are directed into the input port of the broadband 50/50 beam splitter, which mixes the photon fields. The photons leaving the beam splitter can be described by a1 for the excitation photon field and a 2 for the probe photon field [14]:
a1 =
1 1 (−b1 + Ib2 ), a 2 = (Ib1 − b2 ). 2 2
χ (vi) (ω1, ω2 ) = χ (viii) (ω1, ω2 ) =
−1 3!
∑e ∑g ∑e
−1 3!
′
′
∑e ∑g ∑e ′
μgβe μeα g μgα e μegβ Neg (ω2 )Ng g (ω2 − ω1)Ne g (ω2 ) 1
′
′ ′ ′ ′
1
′
1
1
′
1
α μgα e μeβ g μgβ e μeg Neg (ω1)N*g g (ω1 − ω2 )N*e g (ω1) 1
′ ′ ′ ′
1
1
′
1
′
1
(6) (7)
μegα, β
is the dipole matrix element between the two energy here, eigenstates | e and | g of the full Hamiltonian (H ), namely, e r ⃗ g ; r ⃗
(3)
We assumed that the excitation-probe signal is composed of eight pathway contributions [3,9]. When the proposed setup is used, the following six photon correlation functions among the eight functions are all equal to zero:
a1†a 2†a 2a1 = a1†a 2†a1a 2 = a 2†a1†a2a1 = a 2†a1†a1a2 = a1†a1a 2†a 2 = a 2†a 2a1†a1 =0.
(4)
Calculations of these functions are given in Appendix A. Nonzero contributions thus arise only from the other two pathways; a 2†a1a1†a2 of the (vi) pathway and a1†a 2a 2†a1 of the (viii) pathway. We considered crystalline LaMnO3 as the model material system. To mimic one lattice layer of the orbital- and spin-ordered LaMnO3 crystal, we built a Hamiltonian for an ideal Mn cluster composed of four Mn3+ ions. Details of the model material system and the
Fig. 1. (a) Proposed excitation-probe setup, (b) schematic structure of energy eigenstates of the model material system, and (c) closed-time path-loop diagrams for (vi) and (viii) pathways.
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is the dipole displacement operator. The superscripts α and β indicate the polarization directions of the a1 (excitation) and a 2 (probe) fields, respectively. Neg(ω) is the auxiliary function representing the Lorentzian response [9]:
Neg(ω)=
1 . ω − (Ee − Eg )+Iγ
(8)
here, Eg (Ee ) is the energy eigenvalue of | g (| e ), and γ is the spectral broadening parameter. A phenomenological value (0.05t ) was used for γ to obtain the results shown in Figs. 2, 3, 5, and 6... 3. Results and discussion First we calculated the conventional excitation-probe signal. For parallel polarizations of the excitation pulse and probe pulse, numerous peaks were observed, arising from the superposition of the TPA and SRS processes contributed by all eight pathways (Fig. 2a). Fig. 2b shows the perpendicular polarization case. Next we applied the proposed method. Fig. 2c shows the excitationprobe signal for the parallel polarization case. Spectral structures arising from TPA contributions and some of the SRS contributions shown in Fig. 2a do not appear in Fig. 2c. For example, the structures between ω1=4.5t and ω1=4.9t and between ω2=3.6t and ω2=3.8t which arise from the TPA contributions, are not present in Fig. 2c. Fig. 2c shows two peaks along the diagonal line at ω1 = ω2=4.45t and at ω1 = ω2=4.65t . The ω1=4.45t peak arose from the g1 → e1 → g1 → e1 → g1 transition process, and the ω1=4.65t peak arose from the g1 → e2 → g1 → e2 → g1 process. In addition to these peaks, more peaks appeared along the vertical line at ω2=4.45t and at ω2=4.65t , owing to the contribution of the (vi) pathway. Peaks appeared along the horizontal line at ω1=4.45t and at ω1=4.65t owing to the contribution of the (viii) pathway. Fig. 2d shows the perpendicular polarization case. Note that the g1 → e1 → g4 → e1 → g1 transition peak is clearer than the other peaks. In an experiment, Fig. 2c and d can be obtained by measuring the spectral transmittance of the probe signal, because S (ω1, ω2 ) is proportional to the transmittance intensity change. We found an additional advantage of the proposed method. The excitation-probe signals shown in Fig. 3e and f are the products of the susceptibility function (Fig. 3a and b) and the photon correlation
Fig. 3. (a, b) Susceptibility functions for (a) the (vi) pathway (Im[χ(vi)(ω1, ω2)]) and (b) the (viii) pathway (Im[χ (viii) (ω1, ω2 )]). (c, d) Photon correlation functions for (c) the (vi) pathway ( a2†a1a1†a2 ) and (d) the (viii) pathway ( a1†a 2a 2†a1 ). (e, f) Excitation-probe signals for (e) the (vi) pathway (S (vi)(ω1, ω2 ) ) and (f) the (viii) pathway (S (viii)(ω1, ω2 ) ).
function (Fig. 3c and d) as described in Eq. (5). The proposed method allows tuning of the spectral widths and positions of the photon correlation functions. This tunability can help to enhance the excitation-probe signal of some specific group of transitions and to suppress the other signals selectively. We controlled the spectral distribution of the photon correlation function by changing a parameter of the joint distribution function, | ϕij |2 . Fig. 4 shows the two cases of | ϕij |2 ; the | ϕij |2 shown in Fig. 4a corresponds to the photons of Fig. 3c and d. Here, the parameters ωp , T1, and T2 were 8.94t , 31.0/t , and 0. 1/t . The | ϕij |2 shown in Fig. 4b
Fig. 2. (a, b) Conventional excitation-probe signals for (a) the parallel polarization and (b) the perpendicular polarization. (c, d) Excitation probe signals obtained with the proposed method for (c) the parallel polarization and (d) the perpendicular polarization.
Fig. 4. Joint spectral density functions for different photon pair entanglement times (T2 ): (a) T2 = 0. 1/t , (b) T2=0.3/t .
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corresponds to the photons of Fig. 5a and b. Here, the parameters ωp and T1 were same; only T2 was different (T2=0. 3/t ). T2 can be controlled by changing the thickness of the nonlinear crystal [12]. Fig. 5a and b show the photon correlation functions with longer T2 (i.e. with narrower bandwidth) than those in Fig. 3c and d; Fig. 5a shows the photon correlation function for the (vi) pathway and Fig. 5b shows the function for the (viii) pathway. Fig. 5c and d show the excitation-probe signal produced by the narrow-bandwidth photon correlation functions; compared with Fig. 3e and f, the SRS peaks arising from the g1 → e1 → g4 → e1 → g1 transition remain clear whereas the other peaks are diminished. We found that the polarization states of pulses reveal the rotational symmetry of the excitations. Fig. 6a and b present the SRS signal due to the quasiparticle (phonon and phonon-coupled orbiton) excitations in the model material system, LaMnO3. Fig. 6a shows the signal of the parallel polarization state; the peaks arise from the g1 → e1 → gn → e1 → g1 transition pathways. Here we only focus on the peaks above the diagonal (ω1 = ω2 ) line. Peaks at lower parts can be understood similarly. When n = 1, the pathway corresponds to Rayleigh scattering, which results in the peak at ω1 = ω2=4.45t . When n = 2 , the pathway corresponds to Jahn–Teller phonon excitation with Ag symmetry, which results in the peak at (ω1=4.45t , ω2=4.3t ). When n = 3 and 4, the pathways involve the orbital excitation. In particular, the n = 4 transition pathway results in the so-called phonon-coupled orbiton, which results in the peak at (ω1=4.45t , ω2=3.85t ) [14]. The strength of the peaks depends on the dipole matrix elements: μgβe μeαg μgα e μeβg , as described in Eq. (6). Fig. 6b shows the perpendicular 11 1n n 1 11 polarization case; i.e, α ⊥β . Because of the broken rotational symmetry of the excited orbitals in | g3 and | g4 , μgβe μeαg μgα e μeβg remains finite for 11 1n n 1 11 the pathways involving the orbital excitation (i.e. g1 → e1→(g3, g4 )→e1 → g1), whereas the matrix elements involving | g1 and | g2 become zero. Note that the peaks at (ω1=4.45t , ω2=3.85t ) remain whereas the other peaks are suppressed in Fig. 6b. The broken rotational symmetry of the intermediate state allows the intensity of the SRS signal to remain finite even in the perpendicular polarization geometry of the excitation-probe pulses. Comparison of the SRS signals measured with different polarization configurations of the pulses can test the rotational symmetry of the excited states. In experiment, Fig. 6(a) and (b) can be obtained by measuring the spectral transmittance change of probe signal, ∆Τ (ω2 ) for a LaMnO3 crystalline thin film sample. The second harmonic pulse of a femtosecond pulse generated from a Ti: Sapphire laser oscillator can be used as the pump source for the SPDC process and a type-I β-BaB2O4 crystal can be used for generating the SPDC photon pulses. In this case, center wavelength of the SPDC pulse is around 800 nm. The peak at ω1 = ω2=4.5t in Fig. 6(a) is usually observed at about 650 nm in the linear absorption experiment [16]. By using a band-pass filter or a grating, the spectral components of the photon pulse in 900 – 600 nm range can be selected from the SPDC rainbow. In addition to the conventional Hong-Ou-Mandel interferometer shown in Fig. 1(a), gratings, pinholes, and a half-wave plate can be included to select the specific wavelength components of the pulses irradiated to the sample surface. For the gratings and pinholes, optical paths of pump and probe pulse should be correctly arranged to meet the phase matching condition for different wavelength components. ∆Τ (ω2 ) can be estimated by summing the number of photons with a single photon counter in a given measurement time. First, transmitted probe pulse is measured by blocking pump pulse illumination (Τ (ω2 )without ). Then the transmittance of probe pulse is measured with pump pulse illumination (Τ (ω2 )with ). By subtracting the second signal from the first one, ∆Τ (ω2 ) can be obtained. This procedure can be repeatedly done to improve the signal to noise ratio, if an optical chopper and a lock-in amplifier were combined. In present study, we considered spontaneous parametric down conversion process for the pump and probe photon pulses preparation. In the process, the intensity of generated pulses may be weak, so it may be difficult to get clear signals. Intense pulse
Fig. 5. (a, b) Photon correlation functions for (a) the (vi) pathway and (b) the (viii) pathway. (c, d) SRS signals for (c) the (vi) pathway and (d) the (viii) pathway.
Fig. 6. Excitation probe signals simulated with narrower-bandwidth photon correlation functions (T2 = 0.3/t case) for (a) parallel polarization and (b) perpendicular polarization of the excitation and probe pulses.
can be produced by parametric amplification process. Although the detailed nature of the quantum correlations can be different for parametric down conversion and for parametric amplification, quantum state signatures of light can remain also for intense pulse fields generated by the parametric amplification process as Nagasako et al. observed [14].
4. Conclusions We have proposed an excitation-probe method using entangled photon pulses. Numerical simulation demonstrated that the twophoton absorption signal can be selectively filtered in the excitationprobe signal. By controlling the crystal thickness and the pump pulse frequency for the parametric down-conversion, we can tune the bandwidth and the central frequency of the photon correlation function distribution used in the proposed method. We found that this tuning can be used to select specific transition peaks from the group of numerous stimulated Raman scattering peaks. We demonstrated that the proposed method can be used to examine the stimulated Raman scattering signal arising from orbital wave excitation in a model material, LaMnO3. 584
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Acknowledgements This research was supported by the National Research Foundation of Korea (Grants Nos. 2012R1A1A401010025 and 2015R1D1A1A02061984).
Appendix A. Photon correlation function The photon correlation functions for some pathways are zero. Following is the proof for one of those pathways:
a1i†a 2j †a 2ja1i =
1 (−Ib2i† − b1i†)(−Ib1j † − b2j †)(Ib1j − b2j )(Ib2i − b1i ) ; 4
(A1)
for the ket side,
(Ib1j − b2j )(Ib2i − b1i ) Ψ =
∑k,l ϕkl(b2jb1i − b2ib1j )b1k †b2l†|0
= ϕij |0 − ϕji|0 = 0
(A2)
Note here that ϕij = ϕji for a type-I nonlinear crystal. Therefore, †
a1i a 2j †a 2ja1i =0.
(A3)
The photon correlation functions for the (vi) and (viii) pathways shown in Fig. 1c can be expressed as follows: †
a1i a 2ja 2j †a
1i
= a1i†a1i + a1i†a 2j †a 2ja1i .
(A4)
The second term on the right side is zero. The first term is
* ϕ b1mb2na1i†a1ib1k †b2l †|0 Ψ |a1i†a1i Ψ = ∑m, n ∑k , l 0|ϕmn kl 1
* ϕ b1mb2n(−Ib2i† − b1i†)(Ib2i − b1i )b1k †b2l †|0 = = 2 ∑m, n ∑k , l 0|ϕmn kl †
†
†
1 2
⎞ ⎛ ∑j ⎜|ϕij |2 + |ϕji|2 ⎟ ⎠ ⎝
(A5)
†
Similarly, a 2i a1ja1j a 2i = a 2i a 2i = a1i a1i .. Appendix B. Two-photon absorption signal with frequency-correlated photons One may wonder whether the quantum mechanical entanglement between the photons is necessary for the suppression of the two-photon absorption signal or whether similar results could be achieved by using frequency-correlated photons that are not entangled. To investigate this, we explore the case in which the two photons are frequency-correlated but nonetheless unentangled. To compute the photon correlation function, we consider a classically correlated two-photon state described by a density operator of the following form [17]:
⎡ ⎤2 ρ = C ∑ ⎢ϕij ⎥ b1i†b2j †|0 0|b1ib2j i, j ⎣ ⎦
(B1)
⎡ ⎤ where ⎢ϕij ⎥ is the joined distribution function of the photons and C is a normalization constant. Annihilation (creation) operators of these fields are ⎣ ⎦ represented by b1(b1†) and b2(b2†). Using this density operator, we can calculate two-photon correlation functions in group A. For example, 2
⎡ ⎤2 b†2k b†1lb2k b1l = Tr[b2k b1lρ b†2k b†1l ] = C ∑ ⎢ϕij ⎥ Tr[b2k b1l b1i†b2j †|0 0|b1ib2jb†2k b†1l ] = C [ϕlk ]2 i, j ⎣ ⎦
(B2)
All of the correlation functions are identical and nonzero, which means that we cannot suppress the TPA signal using the frequency-correlated, unentangled photons.
[9] [10] [11] [12] [13]
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