Dynamics of Cu₂O Rydberg Excitons -Kerr-Effect and Quantum Beats

We investigate the nonlinear refraction and the nonlinear Kerr phase shift which are due to Rydberg excitons induced in Cu₂O crystal by short-time pulses. We observe the phenomenon of quantum beats, analogous to that observed in exciton emission spectra, obtained in the same conditions. The calculated temporal evolution show dependence on the exciton state number, the applied laser power, and quantities related to dissipative processes.

Since the first observation of Rydberg Excitons (REs) in Cuprous Oxide (Cu₂O) [1], the physics of REs come a long way, ranging from linear optical properties in bulk crystals, to strong Kerr nonlinearities [2,3]. When the first period of investigation was concentrated on effects resulting from stationary excitation, recently, the dynamics of excitons has attracted large attention. In this paper we present the results of the time dependent behavior of nonlinear refraction index and Kerr phase shift, focusing the attention on the relation between the mentioned phenomena and quantum beats. Our calculation is based on the so-called Real Density Matrix Approach (RDMA), adapted to the case of short-pulse excitations, and supplemented by the time dependent functions, resulting from interference between neighboring exciton states. In Section II we recall the basic RDMA equations, determining the linear and nonlinear susceptibility. We also present the definitions of quantities which will be described in the following. In Section III we transform the presented equations to the form, which will be then applied for calculation of time dependent nonlinear optical functions. Section IV is devoted to analysis of dissipative processes and calculation of quantities related to them. In Sections V and VI we calculate the nonlinear refraction index and the Kerr phase shift in the given exciton state, showing the dependence on time, the state number, and the applied laser power. The results of calculations are presented in Section VII. We present our conclusions in Section VIII (Figures 1,2). The appendix contains supplementary information.

Recently experiments with Cu₂O crystals were performed, when two waves with energies close to E_g/2 and finite pulse duration, excited a population of excitons in various states E_n[4], n stands for the set of quantum numbers n,l m. The population decays then with a lifetime τ_n. The emission line shapes show deviations from the decay line, which are due to interference effects between neighboring exciton states and are termed quantum beats [4,5]. The lifetimes can be read off from the shape of time-dependent susceptibility, which consists of linear susceptibility χ⁽¹⁾ , and nonlinear ones χ ⁽³⁾ [2,6].

χ⁽¹⁾ (χ,t) = (2/ ε₀) ) ∑_nψ_osc,n (t)(G_n(t,t’)F(t’ )) × (|c_n|² (E_n+E_g) )((E_g+E_n) ² –ħ² [(ω₁+ω₂) +i/ω_n]²)^-1   ,                (1)

${\chi }^{\left(\text{3}\right)}(\chi ,\text{t})=-({\text{2M}}_0{}^{\text{2}}/{\epsilon }_0)\text{ x}{\sum }_n{\text{c}}_n\left({\text{A}}_{\text{n}}+{\text{B}}_{\text{n}}\right)\left({\text{E}}_n+{\text{E}}_{\text{g}}\right){\{{[{\text{E}}_{\text{g}}+{\text{E}}_n-(h{\omega }_{\text{1}}+h{\omega }_{\text{2}})]}^{\text{2}}+{(h/{\tau }_n)}^{\text{2}}\}}^{-\text{1}}$

$\begin{array}{l}\times {\sum }_n‘({\text{T}}_{\text{1}}/{\tau }_n‘){\text{ c}}_n’{\varphi }_n’\left(0\right){\left({\text{E}}_{\text{g}}+{\text{E}}_n‘\right)}^{\text{2}}-h^{\text{2}}({\omega }_{\text{1}}+{\omega }_{\text{2}}+\text{i}/{\tau }_n{‘}^{)}\\ \times {\text{ G}}_n\left(\text{t},\text{t}‘‘\right)\text{F}\left(\text{t}‘‘\right){\text{G}}_n‘\left(\text{t}‘‘,\text{t}‘\right)\text{F}\left(\text{t}‘\right){\psi }^{\text{2}}{}_{\text{osc},n}\left(\text{t}\right).\end{array}$

In the above equation M₀ is the interband-transition integrated dipole strengths, ε₀ the vacuum electric susceptibility constant, w₁,w₂ the frequencies of the impinging waves, ϕ_n are the excitonic eigenfunctions, E_lm the related eigenvalues, t_n their lifetimes, and |c_n|² the corresponding oscillator strengths. The quantity T₁ denotes the inter-band recombination time, and A_n, B_n represent the intra-band dissipation [7]. The function F(t) determines the impulse shape, and G(t,t′) is the Green function appropriate for the exciton decay equation. The functions ψ_osc,n describe the quantum beats. The exciton emission reported in [4] is described by the imaginary part of χ⁽³⁾ . Having χ⁽¹⁾ and χ⁽³⁾ , we are also able to determine the nonlinear index of refraction n₂, defined as [2].

$n^2=\mathrm{Re}{\chi }^{\left(3\right)}{(c{\epsilon }_0{n}_0{}^2)}^{-1},n_0{}^2={\epsilon }_b(1+\mathrm{Re}{\chi }^{\left(1\right)}/{\epsilon }_b) ,$            (2)

where c the vacuum light velocity, and ε_b the bulk crystal dielectric constant. The total index of refraction is

n² = ε_b + χ⁽¹⁾ + χ⁽³⁾|E_prop|²  ,

n(I) = n₀ +n₂I,             n₀ ²  = ε_b + χ⁽¹⁾ ,

I = (c/2) n₀ε₀|E_prop|² .                                                   (3)

E_prop being the amplitude of the electromagnetic wave. This amplitude is related to the intensity I of the incoming wave by the relation

|E_prop|² = 2ζ I                                                                          (4)

where z ≈ 377Ω is the impedance of free space. Note that we have to distinguish the refraction index n from the state number n. The phase shift is calculated from

∆f =( wL/c) [n(I)−n(0)],                                                       (5)

where n(I) is the total refraction index obtained for average intensity I inside the crystal, and L is the crystal length. As we will show in the following, the nonlinear optical effects depend, in the case of short-pulse excitation, on time t, and applied laser power P. The time t = 0 corresponds to the maximum of the incoming pulse. We find it opportune to use in the formulas the nominal laser power p. The dependence τ_n(p) of the exciton life time was observed experimentally [4], and can be described by the formulas [6]

τ_n = τ₀ exp( nb₁ −n2 b₂ −Pb₃ ).                                         (6)

The coefficients τ₀,b₁,b₂,b₃ (τ₀ in ps, b₃ in 1/mW)

b₁ = 0.98, b₂ = 0.04, b₃ = 6.6×10⁻³ , τ₀ = 0.197,                  (7)

were obtained from fitting experimental results from [4].

Below we discuss the dependence n(P). Both susceptibilities, linear and nonlinear, depend on W through the dependencies τ_n(P) and A_n(P), B_n(P).  As observed in experiments, the values of lifetimes for n > 7 arrive at a plateau. For higher states  we observe a slow increase, tending to a value τ∞. The scaling in this interval can be described by the formula

τ_n00 = τ_nS = τ∞S[1−exp (b₃p −n³ λ_S ]  , S excitons,

τ_∞S = 22.89 ps, λ_S = 5.36×10⁻³ .                                     (8)

With the above statements, we calculate the dependence χ⁽¹⁾ (p), χ⁽³⁾ (p).  Since we aim to connect the Kerr effect with quantum beats, we refer to experiments [4], and consider the two-photon absorption resonant with various Rydberg states, focusing the attention on the resulting one-photon emission at twice the pump energy

hω₁+hω₂ = E_g .                                                               (9)

  In what follows we consider the lowest even states of the so-called yellow series excitons in Cu₂O. For these states

Eg ≈ 2 eV, E_n µ 10⁻² eV, 0.1 ≤ Γ_n ≤ 1 meV,                   (10)

and, in consequence, the following approximations can be made

Reχ⁽¹⁾ (w,t) ≈ (2/ε₀) ∑_n ψ_osc,n(t)

χ(G_n(t,t′)F(t′))|c_n|² /2E_n ,                                (11)

 Re χ⁽³⁾ (w,t)≈ − (2M₀²/ε₀)∑_n,_n′ c_n c_n′ 

(A_n +B_n) j_n′ (0) T₁ (2E_n E_n′ 2t _n′)^-1

x(G_n(t,t")F(t"))(Gn(t",t‘)F(t‘))ψ²_osc,_n(t).                  (12)

The effect of the pump power on the dynamics and the exciton lifetime was studied for each state (separately) [4,6]. We perform the calculations in analogous situation, therefore we drop the summation in Eqs. (1,12) obtaining

$\text{Re}{\chi }^{\left(\text{1}\right)}(\omega ;\text{t})\approx (\text{2}/{\epsilon }_0){\psi }_{\text{osc},\text{n}}\left(\text{t}\right)\left({\text{G}}_n\left(\text{t},\text{t}‘\right)\text{F}\left(\text{t}‘\right)\right){\left|{\text{c}}_n\right|}^{\text{2}}/{\text{2E}}_n,$

$\text{Re}{\chi }^{(\text{3}})(\omega ,\text{t})\approx -({\text{2M}}_0{}^{\text{2}}/{\epsilon }_0){\left|{\text{c}}_n\right|}^{\text{2}}\left({\text{A}}_n+{\text{B}}_n\right){\varphi }_n\left(0\right){\text{ T}}_{\text{1}}({\text{2E}}_n{\text{E}}_n{‘}^{\text{2}}{\tau }_n{‘)}^{-\text{1}}$

$\text{x }\left({\text{G}}_n\left(\text{t},\text{t}‘‘\right)\text{F}\left(\text{t}‘‘\right)\right)\left({\text{G}}_n\left(\text{t}‘‘,\text{t}‘\right)\text{F}\left(\text{t}‘\right)\right){\psi }^{\text{2}}{}_{\text{osc},n}\left(\text{t}\right).$   (13)

For the further calculations we must define all the terms arriving in the above equations. We recall the definitions [6]. The coefficients c_nlm and eigenfunctions ϕ_n = ϕ_nlm are defined as follows

${\text{c}}_n={\text{ c}}_{\text{nlm}}=ò{\text{d}}^{\text{3}}\text{r}M\left(r\right){\varphi }_{\text{nlm}}\left(r\right),$

${\varphi }_n={\varphi }_{\text{nlm}} ={\text{R}}_{\text{nl}}\left(\text{r}\right){\text{ Y}}_{\text{lm}}(\theta ,\phi ),$ (14)

Y_lm(θ,φ) are the spherical harmonics, R_nl(r) are the radial functions of a corresponding hydrogen atom-like Schrödinger equation. The function M(r) is the so-called transition dipole density [8], and describes the quantum coherence between the macroscopic electromagnetic field (the wave propagating in the crystal), and the inter-band transitions. For the S type transitions it has the form

$M\left(r\right)=M_0{\text{Y}}_{00}{}^{\text{2}}\left(\text{1}/{\text{rr}}_0{}^{\text{2}}\right)\text{ exp}\left(-\text{r}/{\text{r}}_0\right),$ (16)

where r₀ is the so-called coherence radius, defined as

${\text{r}}_0={(\text{2}\mu \text{Eg}{/}^{\text{2}})}^{-\text{1}/\text{2}},$   (17)

and µ being the reduced electron-hole effective mass. The exciton oscillator strength (here for S states) reads

${\text{f}}_{\text{n}00}=n{\left(n-{\text{r}}_0/\text{a}*\right)}^{\text{2}(\text{n}-\text{1}}){\left(n+{\text{r}}_0/\text{a}*\right)}^{-\text{2}(\text{n}+\text{1})},$    (18)

where a* is the exciton effective Bohr radius. As can be seen from figure 3, for large on the approximation f_n00 ≈ 1/n³    can be used (the red curve calculated with r0 = 0.2 a*). From the formulas (14-16) we get

${\left|{\text{c}}_n\right|}^{\text{2} }=({\text{M}}_0{}^{\text{2}}/\text{p}){\text{ f}}_n,$ (19)

where

${\text{M}}_0{}^{\text{2}} ={\epsilon }_0{\epsilon }_{\text{b}}\text{pa}{*}^{\text{3}}{\Delta }_{\text{LT}}/{\text{2f}}_{\text{1}00 },$    (20)

with the longitudinal-transversal energy ∆_LT .

The above calculated quantities, inserted into equations (1,12) give the relations

$\text{Re}{\chi }_n{}^{\left(\text{1}\right)}(\omega ,\text{t})={\epsilon }_{\text{b}}\left({}^{}{\Delta }_{\text{LT}}/{\text{2E}}_n\right)\left({\text{f}}_n/{\text{f}}_{\text{1}00}\right)(({\text{G}}_n(\text{t},\text{t}¢)\text{F}(\text{t}¢)){\psi }^{\text{2}}{}_{\text{osc},n}\left(\text{t}\right),$

$\text{Re}{\chi }_n{}^{\left(\text{3}\right)}(\omega ,\text{t})=-(\left({}^{}{\Delta }_{\text{LT}}/{\text{2E}}_n\right)\left({\text{f}}_n/{\text{f}}_{\text{1}00}\right)$ (21)

$\text{x} {\chi }_{0n}{}^{\left(\text{3}\right)}({\text{G}}_n(\text{t},\text{t}¢¢)\text{F}(\text{t}¢¢))({\text{G}}_n(\text{t}¢¢,\text{t}¢)\text{F}(\text{t}¢)){\psi }^{\text{2}}{}_{\text{osc},n}\left(\text{t}\right),$

${\chi }_{0n}{}^{\left(\text{3}\right)}={\epsilon }_0{\epsilon }_{\text{b}}\text{p}{\Delta }_{\text{LT}}[\text{a}{*}^{\text{3}}\left({\text{A}}_n+{\text{B}}_n\right){\varphi }_n\left(0\right){\text{ T}}_{\text{1}}]{({\text{2E}}_n{}^{\text{2}}{\tau }_n)}^{-\text{1}}.$

The quantities A_n, B_n are related to nonlinear dissipative processes (for example [2,7]), and are defined as

${\text{A}}_n=<{\varphi }_n\left(r\right)|{\text{f}}_{0\text{e}}\left(r\right)>, {\text{B}}_n=<{\varphi }_n\left(r\right)|{\text{f}}_{0\text{h}}\left(r\right)>,$ (22)

where f_0e ,f_0h are the normalized Boltzmann distributions for electrons and holes, respectively. The  expressions (21) are then used in Eq. (2), giving the formulas for the refraction indices

${\text{n}}_{0n}{}^{\text{2}}={\epsilon }_{\text{b}}\left[\text{1}+{\text{C}}_n{}^{(\text{1})}{\text{F}}_n{}^{(\text{1})}\left(\text{t}\right)\right],$

${\text{n}}_{\text{2}n}={\text{ C}}_n{}^{(\text{2})}{\text{F}}_n{}^{(\text{1})}\left(\text{t}\right)],$

${\text{C}}_n{}^{\left(\text{1}\right) }=\left({}^{}{\Delta }_{\text{LT}}/{\text{E}}_n\right)\left({\text{f}}_n/{\text{f}}_{\text{1}00}\right),$    (23)

${\text{F}}_n{}^{(\text{1})}\left(\text{t}\right)=(({\text{G}}_n(\text{t},\text{t}¢)\text{F}(\text{t}¢)){\psi }^{\text{2}}{}_{\text{osc},n}\left(\text{t}\right),$

${\text{C}}_n{}^{(\text{2})}{=}^{}{\text{C}}_n{}^{(\text{1})}{\Delta }_{\text{LT}}\text{a}{*}^{\text{3}}[\text{a}{*}^{\text{3}}\left({\text{A}}_n+{\text{B}}_n\right){\varphi }_n\left(0\right){\text{ T}}_{\text{1}}]{({\text{2cE}}_n{}^{\text{2}}{\tau }_{}{}_n)}^{-\text{1}} ,$

${\text{F}}_n{}^{(\text{2})}\left(\text{t}\right)]=[\text{1}+{\text{ C}}_n{}^{(\text{1})}{\text{F}}_n{}^{(\text{1})}\left(\text{t}\right){]}^{-\text{1}}$

${\text{x}}^{ }({\text{G}}_n(\text{t},\text{t}¢)\text{F}(\text{t}¢))({\text{G}}_n(\text{t}¢¢,\text{t}¢)\text{F}(\text{t}¢)){\psi }^{\text{2}}{}_{\text{osc},n}\left(\text{t}\right),$

with the dimension [C_n⁽²⁾]=m²/W.  The equations (23) define the refraction indices for given exciton state n, and in short-pulse excitation regime. As can be seen, they reveal oscillatory behavior (quantum beats), described by the time-dependent terms   F_n⁽¹⁾ (t), F_n⁽²⁾(t). The quantities C_n⁽¹⁾ ,  C_n⁽²⁾  define  amplitudes of oscillations. Having calculated the refraction indices, one can determine the phase shift (5), which also will show oscillatory behavior at the given state [8-10].

Calculations of coefficients An, Bn

As follows from expressions (21), almost all ingredients are known, besides of the quantities An,Bn. Below we consider the exciton S states, thus the definitions (22) take the form

${\text{A}}_n={ò}^{}{\text{r}}^{\text{2}}{\text{dr R}}_{\text{n}0}\text{exp}(-{\text{r}}^{\text{2}}/\text{2}{\lambda }^{\text{2}}{}_{\text{th},\text{e}}), {\text{B}}_n={ò}^{}{\text{r}}^{\text{2}}{\text{dr R}}_{\text{n}0}\text{exp}(-{\text{r}}^{\text{2}}/\text{2}{\lambda }^{\text{2}}{}_{\text{th},\text{h}}),$

where we now denote n = (n,0,0) → n, and λ_th,e , λ_th,h are the so-called thermal lengths for electrons and holes, respectively [2,7],

${\lambda }_{\text{th},\text{e}}{=}^{\text{2}}/{\text{m}}_{\text{e}}{k}_{B}T{)}^{\text{1}/\text{2}} , {\lambda }_{\text{th},\text{h}}={{(}^{\text{2}}/{\text{m}}_{\text{h}}{k}_{B}T)}^{\text{1}/\text{2}} ,$ (25)

k_B  being the Boltzmann constant, and T is the temperature, k_B  = 8.617 x 10^-2 meV/K. The coefficients A_n, B_n depend on the temperature by definitions of l’s (25), which can be written in the form

${\lambda }_{\text{th},\text{e}}={\lambda }_{\text{th},\text{e}}/\text{a}*=\left(\text{1}/\text{a}*\right){(\text{2 }{\mu }^{\text{2}}/{\text{m}}_{\text{e}}(\text{2}\mu )k_{B}T)}^{\text{1}/\text{2}}={(\text{2 }\mu \text{R}*/{\text{m}}_{\text{e}}{k}_{B}T)}^{\text{1}/\text{2}},$

${\lambda }_{\text{th},\text{h}}={\lambda }_{\text{th},\text{h}}/\text{a}*=\left(\text{1}/\text{a}*\right){(\text{2 }{\mu }^{\text{2}}/{\text{m}}_{\text{h}}(\text{2}\mu )k_{B}T)}^{\text{1}/\text{2}}={(\text{2 }\mu \text{R}*/{\text{m}}_{\text{h}}{k}_{B}T)}^{\text{1}/\text{2}}.$

The exciton Rydberg energy R*, and the excitonic Bohr radius a*, were obtained by the formulas 

$\text{R}*=\text{ 136}00\mu /{\epsilon }_{\text{b}}{}^{\text{2}}, \text{a}*=\left({\text{m}}_0/\mu \right){\epsilon }_{\text{b}}{\text{x}}_{}0.0{\text{529 nm}}_{. }$    (26)

Using the parameter values from table 1, we obtain

Table 1: Band parameter values for Cu2O from [9], Energies in meV, masses in free electron mass m0, lengths in nm,  ∆LT  from [10].
Parameter	Value
Eg	2172.08
R*	90.88
∆LT	5x 10-2
me	0.985
mh	0.575
µ	0.363
εb	7.37
a*	1.07

${\lambda }_{\text{th},\text{e} }=\text{27}.\text{88 }{(\text{K}/T)}^{\text{1}/\text{2}}$

${(\text{2}{\lambda }_{\text{th},\text{e}})}_{}{}^{-\text{2}}=\text{6}.\text{43x1}{0}^{-\text{4}}\left(T/\text{K}\right)$

${\lambda }_{\text{th},\text{h}}=\text{36}.\text{49 }{(\text{K}/T)}^{\text{1}/\text{2}}$

${({\text{2}}^{}{\lambda }_{\text{th},\text{h}})}_{}{}^{-\text{2}}{=}^{}\text{3}.\text{75x1}{0}^{-\text{4}}\left(T/\text{K}\right)$

The above quantities are then used in the determination of A_n, B_n by equations (24). The results are given below in tables 2-4. As it can be seen from the data displayed in the tables, the coefficients A_n, B_n have alternative signs. It reflects the oscillatory behavior of the Lagrange polynomials, being constituents of the radial functions R_nl.

Table 2: Quantities A4 , A4 j4(0), B4 , B4 j4(0), and (A4+B4)0 = (A4+B4) j4(0), for 5 values of temperature.
Temp.	A4	A4 j4(0)	B4	B4 j4(0	(A4+B4)0
4.2	- 10.33	- 2.58	-26.94	-6.73	-9.31
10	1.138	0.288	-3.48	-0.87	-0.59
20	0.905	0.248	1.5	0.38	1.38
30	0.20	0.05	1.3	0.33	0.38
40	-0.237	-0.0638	0.67	0.17	0.11

Table 3: Quantities A5 , A5 j5(0), B5 , B5 j5(0), and (A5+B5)0 = (A5+B5) j5(0), for 5 values of temperature.
Temp.	A5	A5 j5(0)	B5	B5 j5(0	(A5+B4)0
4.2	- 19.36	- 2.88	-26.4	-3.933	-6.81
10	-5.57	-0.83	-13.13	-1.96	-2.79
20	-1.3	-0.194	-4.06	-0.6	-0.794
30	-0.712	-0.1	-1.8	-0.26	-0.36
40	-0.54	-0.0838	-1	-0.145	-0.225

Table 4: Quantities A6 , A6 j6(0), B6 , B6 j6(0), and (A6+B6)0 = (A6+B6) j6(0), for 5 values of temperature.
Temp.	A6	A6 j6(0)	B6	B6 j6(0)	(A6+B6)0
4.2	- 0.666	- -0.09	0.469	0.065	-0.025
10	0.225	0.03	-0.63	-1.96	-2.79
20	0.48	0.066	0.423	0.0588	0.125
30	0.187	0.026	0.55	0.076	0.1
40	-0.033	-0.004	0.372	0.052	0.048

Calculation of the nonlinear refraction index

Using Eqs. (23), we calculate the time dependence of the nonlinear refraction index for S exciton states in a Cu₂O crystal. In calculations we use a Gaussian shaped normalized pulse

$\text{F}\left(\text{t}\right)={\text{F}}_{\text{max}}{(\text{2p}{\tau }_{\text{p}}{}^{\text{2}})}^{-\text{1}/\text{2}}\text{exp}(-{\text{t}}^{\text{2}}/\text{2}{\tau }_{\text{p}}{}^{\text{2}})$   (28)    

where t_p  is the pulse temporal duration. Using this shape, we calculate the expressions including the Green function G,

${\text{G}}_n(\text{t},\text{t}¢)=({\tau }_{\text{n}}/\text{2})\text{ exp}(-|\text{t}-\text{t}¢|/{\tau }_n{}_{}).$    (29)

In the lowest approximation we obtain

$({\text{G}}_n(\text{t},\text{t}¢)\text{F}(\text{t}¢))\approx {\text{ F}}_{\text{max} }({\tau }_n{}_{}/\text{2})\text{ exp}(-\left|\text{t}\right|/{\tau }_n).$   (30)

The nonlinear refraction index for  the state |n=n,0,0> → n has the form

${\text{n}}_{\text{2}n}\left(\text{t}\right)I={\text{ X}}_{0\text{n}}{{}^{(\text{3})}}^{ }({\text{G}}_n(\text{t},\text{t}¢¢)\text{F}(\text{t}¢¢))({\text{G}}_n(\text{t}¢,\text{t}¢¢)\text{F}(\text{t}¢)){\psi }^{\text{2}}{}_{\text{osc},n}\left(\text{t}\right),$                                                                                                                   (31)

where

${\text{X}}_{nP}{}^{(\text{3})} =-0.\text{795 x 1}{0}^{\text{3}}\text{P}\left(\text{mW}\right)({\Delta }_{\text{LT}}{\text{f}}_n/{\text{E}}_n{\text{f}}_{\text{1}00})$

$\text{x}({\Delta }_{\text{LT}}{\text{a}}_{*}{}^{\text{3}}/{\text{2cE}}_n{}^{\text{2}})\left[{\left({\text{A}}_n+{\text{B}}_n\right)}_0\right]{\text{Y}}_n{}^{\text{2}}(\text{1}{0}^{\text{3}}{\text{t}}_{\text{1}}/{\tau }_{\text{n}})$ (32)

$= -{\text{n}}^{\text{3}}\left(\text{4}.\text{25x 1}{0}^{-\text{17}}\right){\text{Y}}_n{}^{\text{2}})\left[{\left({\text{A}}_n+{\text{B}}_n\right)}_0\right]{\text{ t}}_{\text{1}}/{\tau }_n)\text{ exp}\left({\text{b}}_{\text{3}}p\right) [{\text{mm}}^{\text{2}}/\text{mW}],$

with t₁=10^-3 T1, and Ψ_n will be defined below. The matrix elements X_nP(3)  are displayed in table 5.  The total refraction index  will be calculated by Eqs. (3). First we calculate the index n₀

Table 5: Matrix elements XnP(3)  in units 10-6.
n/P	12	50	150
4	1.55	-8.31	48
5	8.0	5.52	20
6	1.39	-14.8	-40

${\text{n}}_{0n }\approx {\epsilon }_{\text{b }+}{\chi }_{0n}{}^{\left(\text{1}\right)}{\psi }_{\text{osc},n}\left(\text{t}\right)({\text{G}}_n(\text{t},\text{t}¢)\text{F}(\text{t}¢)),$

${\chi }_{0n}{}^{\left(\text{1}\right)}={\Delta }_{\text{LT}}{\Psi }_n{\text{f}}_n{({\text{2 E}}_n{\epsilon }_{\text{b}}{\text{f}}_{\text{1}00})}^{-\text{1}}=\text{1}{0}^{-\text{4}} (\text{1}/n){\Psi }_{\text{n} }.$ (33)

When using Eqs. (3), one must to insert the laser power per unit area. Having in mind the experiments in Ref. [4], where the laser with 20 µm spot diameter is focused on the sample, we obtain

$\text{mW}/(\text{p}{(\text{2}0\text{mm})}^{\text{2}})=(\text{1}0\text{9}/\text{ 4}00\text{p})(\text{W}/{\text{m}}^{\text{2}})=0.\text{795 x 1}{0}^{\text{6}})(\text{W}/{\text{m}}^{\text{2}}),$

$I=p(\text{mW})\text{ x }0.\text{795 x 1}{0}^{\text{3}} )(\text{mW}/{\text{mm}}^{\text{2}}),$ (34)

Calculation of the phase shift

Using Eqs. (3) and (5) we calculate

${\text{n}}_n\left(I\right)={\text{ n}}_{0n}+{\text{n}}_{\text{2}n}I,$  (35)

${\text{n}}_n\left(0\right)=\surd {\epsilon }_{\text{b }+}{\chi }_{0n}{}^{\left(\text{1}\right)}{\psi }_{\text{osc},n}\left(\text{t}\right)\text{ exp}(-\left|\text{t}\right|/{\tau }_{\text{n},\left(p=0\right)}).$

The above result is now used to calculate

${\text{n}}_n\left(I\right)-{\text{n}}_n\left(p=0\right)={\text{Dn}}_n+{\text{n}}_{\text{2}n}I,$ (36)

$\Delta {\text{n}}_n=[\text{ exp}(-\left|\text{t}\right|/{\tau }_n{}_{,p})-\text{ exp}(-\left|\text{t}\right|/{\tau }_n{}_{,(p=0)})]{\psi }_{\text{osc},n}\left(\text{t}\right).$

Thus the final expression for the phase shift in the exciton state n has the form

$\Delta {\varphi }_n{}_{,\text{P}}=(\omega \text{L}/\text{c})\left[{\text{n}}_n\left(I\right)-{\text{n}}_n\left(p=0\right)\right]=(\omega \text{L}/\text{c})(\Delta {\text{n}}_n+{\text{n}}_{\text{2}n}I) .$ (37)

Using the expressions (35,37), with definitions (A.1)-(A.3), we obtain the formulas for the time dependence of the refraction indices n2,n(P), and the phase shifts ∆φ₄ − ∆φ₆. The lineshapes of the nonlinear refraction indices, for three applied laser powers, are presented in figures 4-10. We observe the general tendency of increasing the value with increasing state number and applied power. Simultaneously, the quantum beats appear. Their amplitudes decrease with time and the beats disappear after about 20 ps. In two cases we observe the inversion of sign of the indices with the increasing power (the states n = 4,6) (Tables 6,7).

Table 6: Exciton beatings characteristic frequencies Ωn1n2, corresponding to (n2 D↓ / n1 S →) transitions.
n2 D/n1 S ® ↓	3	4	5	6	7
3.1	2.3	4
3.2	3.06
4.1		1.3	2.29
4.2		1.745	1.851
5.1			0.835	1.12
5.2			1.11	0.845
6			2.53	0.58	0.59
7				1.6	0.42

Table 7: Exciton beatings characteristic frequencies Ωn1n2, corresponding to (n2 D↓ / n1 S →) transitions, for n1,2 = 9-11,13-15.
n2 D/n1 S ® ↓	9	10	11	13	14	15
9	0.346	0.167	0.32
10	0.7	0.15	0
11	0.926	0.395	0.243
13				0.172	0.035	0.07
14				0.29	0.15	0.046
15				0.386	0.24	0.14

We presented the method based on the so-called real density matrix approach, to describe the time evolution of the nonlinear refraction index and the Kerr phase-shift, induced in Cu₂O crystal by short-time pulses. We analyzed the case of two-photon excitation of S excitons, for various excitation power. The calculated line shapes show quantum beats, where the frequencies are the same as those observed in emission spectra. It is due to the fact, that the frequencies depend only on involved excitonic eigen-energies. The amplitudes of the oscillations show a quite complicated dependence on the exciton state number and the applied laser power. The Kerr-effect for Cu₂O S excitons was investigated previously in the case of stationary excitation. To our best knowledge, the time evolution was not investigated. Therefore, a comparison of our theoretical results with experiment is lacking. Future experimental results, together with the presented theory, can give an insight into the role of dissipative processes, which, to a considerable extent, determine the quantum beats shape.

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