1. Introduction

For studying optical coherent phenomena, the four wave mixing (FWM) method is used. FWM technique is mostly used for observing atomic coherence [1,2] and for investigating entangled photon pairs[3,4]. FL spectra and absorption at varying probe polarization leads to significant phenomenon, which have been investigated experimentally [5,6]. By changing polarization of incident laser, leads us to transition of pathways among Zeeman sublevels. Coupling strength value is different for different transitions, which are represented by Clebsch-Gordan (CG) coefficients. By using polarization techniques, we can control the electromagnetically induced transparency (EIT) processes [7]. In multilevel EIT system rotation of polarization can be controlled by another strong laser beam [8]. Existence of two Pr³⁺ optical centers associated with impurity ions at nonequivalent cation sites have been already revealed using the absorption and luminescence spectra of a Pr^3+:Y₂SiO₅ crystal at low temperature [9].

In this paper we observe and explore the changes occur in AT-Splitting of FL signal by changing temperature, power and polarization of laser beam. We also correspond AT-Splitting of FL signal in spectral and time domain, which is controlled by multi-parameters. For this, we have confined our study to two-level system, in which polarization states of dressing fields are modulated by a quarter-wave-plate (QWP) and temperature of crystal is being changed by holding crystal into variable cryostat. The dressed state theory of modeled density matrix elements in conjunction with the phonon interaction phenomenon is used to explain the experimental results. These results applied for temperature sensors and switching devices, which have potential application in all-Optical control (controlling dressing field) and on photonic chip. Its speed is much faster which is limited by the atomic coherence time.

2. Experimental setup

Figure 1(a) shows the simplified energy levels diagram in which Pr³⁺ ions doped with YSO crystal. Y₂SiO₅ belongs to $C_{2h}^6$ space group which occupies two inequivalent crystallographic sites (I and II). Figure 1(b) shows the Energy levels of two-mode system. For this experiment (two level atomic systems), Pr³⁺: YSO crystal is held at cryostat for changing the temperature (K) to imply our experiment condition. In this experiment, the dye laser E₁ with wavelength 605.9 nm is used, where narrow scan with a 0.04 cm⁻¹ line width E₁ (ω₁, Δ₁) is pumped by an injection locked single-mode Nd:YAG laser (Continuum Powerlite DLS 9010), which repetition rate is 10 Hz, pulse width is 5 ns and have frequency detuning ${{\Delta}_i} = {\Omega}_{mn} - {\omega}_{i}$ of the pumping field. Here, ${{\Omega }_{mn}}$ is the corresponding atomic transition frequency between levels |m〉 and |n〉, ω_i (i = 1,2,3…etc.) depends on the laser frequency. When E₁ is turned on, E₁ drives the two level transition $|0 \rangle \leftrightarrow |1 \rangle$ and three narrow band correlated photons Stokes (E_S) and anti-Stokes (E_AS) along with FL are generated under SP-FWM process as shown in Fig. 1(b).

 

Fig. 1. (a) Simplified energy levels diagram of Pr³⁺ ions in YSO crystal. (b) Energy levels of two-mode system in Pr³⁺: YSO and the laser coupling configuration. (c) Experimental setup scheme, where PMT: photomultiplier tube, QWP: quarter-wave plate, PBS: polarized beam splitter, E_S is the Stokes signal and E_AS is the anti-Stokes signal. (d) Zeeman energy-levels and transition paths at different polarization states, where solid arrows shows Transitions for the linearly polarized beam and Dotted arrows shows transitions for the right circularly polarized beam. (e) Schematic image of theta angle.

Download Full Size | PDF

In our experiment, one near and two far detectors are placed at 13 cm and 118 cm, respectively from the cryostat container. Temporal outputs are obtained by digital oscilloscope and spectral signals are recorded by photomultiplier tubes which are shown in Fig. 1(c). Figure 1(d) shows transition paths and CG coefficients at different polarization states. We change polarization from linear to right-circular state of E₁ by using QWP. Although there are several transition paths for the FL signals, considering the population of each level. For FL processes, two hyperfine states ± (5/2) and ± (3/2) of δ₀ and γ₀ are involved. The optical transition from the lowest crystal-field level $|{ \pm 5/2} \rangle$ is the dominant as shown in Fig. 1(d).Polarization induced in the medium is proportional to intensity of FL signal. Polarization for second-order FL process is given as $\rho _k^{(2)} = {\varepsilon _ \circ }\mathop \Sigma \limits_{i,j} x_{ijk}^{(2)}|{{E_i}} ||{{E_j}} |$. Hence, Y₂SiO₅ crystal belongs to $C_{2h}^6$ space group, while considering all the incident beams are transverse wave, only four tensor elements are nonzero, which are denoted as $\chi _{xxy}^{(2)}$, $\chi _{xyx}^{(2)}$, $\chi _{yxx}^{(2)}$ and $\chi _{yyy}^{(2)}$. By using QWP, we change the polarization of incident laser beams from linear to right elliptical and right circular, the effective nonlinear susceptibilities can be defined as,

2.1 Density matrix element

By changing polarization from linear to right circular using QWP, we control FL signal intensity. Second-order FL signal can be expressed via Liouville pathway, $\rho _{00}^{(0)}\buildrel {{E_1}} \over \longrightarrow \rho _{10}^{(1)}\buildrel {{{E^{\prime}}_1}} \over \longrightarrow \rho _{11}^{(2)}$ and its density matrix is given as

In two level system, by opening field E₁ with phase matching condition of Stokes and anti-Stokes as k₁+k’₁=k_S+k_AS, the density matrix elements of generated signals anti-Stokes and Stokes (E_AS and E_S) channels using Liouville pathways via the perturbation chains is given respectively as,

2.2 Two-mode and three-mode correlation and squeezing

The second order correlation function $G_{ij}^{(2)}(\tau )$ between intensity fluctuations of two optical beams (i, j, i≠j) as a function of time delay and can be used to calculate the correlation between the generated outputs of SP-FWM, which is given by

3. Experimental results and discussion

In experimental results, we investigate the second-order FL signals by changing polarization, power and temperature in two-level system. Figure 2(a) shows the FL spectral signal at different polarization angles obtained by pumping field E₁. Two peak structure, AT-Splitting of each FL curve results from the dressing effect of pumping field E₁. The suppressed dip appears at resonant point ${\Delta _1} = 0$. The dressing effect is self-dressing. The change in dressing field with change in polarization from linear to right circular as shown in Fig. 2(a1) -(a3) can interpreted by ${|{{G_{{1_M}}}} |^2} = {|{Cg{G_1}} |^2}({{{\cos }^4}{\theta_1} + {{\sin }^4}{\theta_1}} )$ described by Eq. (3). The Rabi frequency $|{{G_{{1_M}}}} |$ of linearly polarized state is greater than $|{G{ +_{{1_M}}}} |$ right circularly polarized state [shown in Fig. 1(d)]. The AT-Splitting are decided by numerator term and ${|{{G_{{1_M}}}} |^2}/({\Gamma _{{1_M}{0_M}}} + i{\Delta _1})$ term in denominator of Eq. (3), which are stronger at linear polarization (Fig. 2(a1)) than at circular polarization (Fig. 2(a3)).

 

Fig. 2. (a) Intensity of second-order FL spectral signal obtained by scanning E₁ and changing E₁ polarization by using QWP in two-level system at temperature of 120 K. (b) Corresponding Time domain FL signal of (a). (c-e) are the intensities of second-order FL spectral signal by changing power of E₁ from high (7 mW) to low (1 mW) for linear, right elliptical and right circular polarizations, respectively at temperature of 120 K. (f-h) show the corresponding time domain signal intensities of (c-e) respectively.

Download Full Size | PDF

The temporal intensity signal in Fig. 2(b) is calculated in correspondence to each curve of Fig. 2(a). Figure 2(c) shows the measured FL signal for high and low power of E₁. Initially, the FL signal is observed at a high power level, as shown in Fig. 2(c1). At high power, AT-splitting is strong, caused by the dressing effect of E₁ ${|{{G_{{1_M}}}} |^2}/({\Gamma _{{1_M}{0_M}}} + i{\Delta _1})$ from Eq. (3). As the power decreases, dressing effect of E₁ reduces and AT-Splitting vanishes as seen in Fig. 2(c2). Similarly, FL intensity with elliptical polarization of dressing beam shows similar behavior as linear polarization at high and low power as shown in Fig. 2(d1) and (d2). For circular polarization in Fig. 2(e), since the dressing effect ${|{G_{{1_M}}^ + } |^2}/{\Gamma _{{1_{M + 1}}{0_M}}}$ mentioned in Eq. (3), is less as compared to dressing effect of linear and elliptical polarization, so the AT-splitting is turn out to be weak. In Fig, 2(e1) and 2(e2), one can’t see so obvious AT-Splitting in circular polarization even at high power. In addition, Fig. 2(b1), have more dressed time delay (AT splitting in time domain) as compare to 2(b3), because the dressing effect at linear polarization is stronger as compare to circular polarization. Similarly, the dressed time delay of Fig. 2(f1), 2(g1), and 2(h1) is more as compared to Fig. 2(f2), 2(g2) and 2(h2) respectively, because at high power with stronger dressing the dipole-dipole interaction increases, due to which life time decreases. The Competition between polarization dressing and phonon of intensity at temperature 120 K in Fig. 2 will be discussed in Fig. 3. From Fig. 2, this experiment provides an accurate physical mechanism for power switch due to variation in AT-splitting occurs by varying power from high (Fig.(c1,d1,e1)) to low (Fig.(c2,d2,e2)) respectively. At high power maximum AT-splitting occur, hence switch is On at high power but as we vary power from high to low, AT-splitting become minimize and state of switch changes from On-state to Off-state. The speed of power switching is about 15 ns and can be controlled by E₁ dressing field. Such, speed is limited by the atomic coherence time in nanosecond time scale and dependent on the quadrature sum of several independent contributions.

Figure 3 shows the intensity of FL spectral signal at different temperature and different polarization with pumping field E₁. Figure 3(a1)-(a4) shows AT-splitting of FL signal due to dressing effect of E₁ for linear polarization. AT-splitting is gradually decreasing from left to right as we vary the temperature from 77k to 300k. When temperature is at 77k, the effect of phonon ${{\Gamma }_{phonon}}$ is reduced due to low temperature and phonon relates with ${{\Gamma }_{i/j}}$ in Eq. (3), also reduces ${\Gamma _{{1_M}{0_M}}}$ term. Hence, dressing term ${|{{G_{{1_M}}}} |^2}/({\Gamma _{{1_M}{0_M}}} + i{\Delta _1})$ is increased in density matrix element $\rho _{{1_M}{1_M}}^{(2)}$ in Eq. (3), which results strong AT-splitting as shown in Fig. 3(a1). When temperature is increased, the effect of phonon ${{\Gamma }_{phonon}}$ increases due to high temperature and phonon relates with ${{\Gamma }_{i/j}}$ in Eq. (3), also increase ${\Gamma _{{1_M}{0_M}}}$ term. Hence, dressing term ${|{{G_{{1_M}}}} |^2}/({\Gamma _{{1_M}{0_M}}} + i{\Delta _1})$ is reduced in density matrix element $\rho _{{1_M}{1_M}}^{(2)}$ in Eq. (3) which results low AT-splitting as shown in Fig. 3(a4). Similarly, FL signal measured for elliptical and circular polarization shows similar behavior as linear polarization at different temperature in Fig. 3(b1)-(b4) and Fig. 3(c1)-(c4), respectively.

 

Fig. 3. (a-c) Intensity of FL spectral signal at different temperature for linear, right elliptical and right circular polarization, respectively. (d-g) Time domain intensity signal at different polarization for different temperature 77 K, 110 K, 160 K, and 300 K respectively. (h-j) Schematic image of dressing sensitivity versus temperature for linear, elliptical and circular polarization respectively.

Download Full Size | PDF

Figure 3(d) shows intensity of FL signal in time domain with changing polarization of input field E₁ from linear to circular at 77k. The peak intensity of each curve is determined by density matrix element, from time domain intensities $I(t) = |{\rho_{{1_M}{1_M}}^{(2)}(\theta )} |\exp ({ - \Gamma _{FL}^Lt} )$ and $I(t) = |{\rho_{{1_{M + 1}}{1_{M + 1}}}^{(2)}(\theta )} |\exp ({ - \Gamma _{FL}^Ct} )$ for linear and circular polarization, respectively. Time domain intensity is determined by $\rho _{{1_M}{1_M}}^{(2)}(\theta )$ and $\Gamma _{FL}^L = {\Gamma _{{1_M}{0_M}}} + {\Gamma _{{1_M}{1_M}}}$, which are stronger at 0^o degree (linear) than at 45^o degree (circular) shown in Fig. 3(d1) and (d3) respectively. Similarly, in Fig. 3(e)-(g), Peak intensities are stronger at linear polarization as compare to elliptical and circular polarization. The dressed time delay (Temporal AT-Splitting) in Fig. 3(d)-(f) are showing similar phenomenon to Fig. 2(b), (f) and (g). Figure 3(h)-(j) shows sensor sensitivity with respect to temperature at different polarization. The polarizer dressing field of circular polarization (Fig. 3(j)) is very weak as compare to linear (Fig. 3(h)) and elliptical polarization (Fig. 3(i)), hence sensor is not sensitive for circular polarization (Fig. 3(j)) and sensitive for linear (Fig. 3(h)) and elliptical polarization (Fig. 3(i)). At linear (Fig. 3(h)) and elliptical polarization (Fig. 3(i)), the sensor sensitivity changes by varying temperature. At low temperature range (77K-110 K), sensor is not sensitive due to the minimum effect of phonon ${{\Gamma }_{phonon}}$ at low temperature and this phononic effect is related with ${{\Gamma }_{i/j}}$ from Eq. (3), which also decreased at low temperature. At this range (110K-160 K), sensor is much sensitive and its sensitivity is increasing by increasing temperature, because the effect of phonon ${{\Gamma }_{phonon}}$ is increased and phonon relates with ${{\Gamma }_{i/j}}$ in Eq. (3) also increased. Above 160 K, the sensor sensitivity variation decreased and become static. From Fig. (3), one can see that, this experiment provides an accurate physical mechanism for temperature sensors due to the competition between phonon and polarization dressing. The variation occur in splitting area are calculated from ${S_i} = {W_i} \ast {H_i}$ (i = a and i = b for 77 K (Fig. 3(a1)) and 300 K (Fig. 3(a4) respectively). AT-splitting area results from polarization dressing, which can be controlled by phonon effect by varying temperature. Our experiment result defined the sensitivity contrast of temperature sensor as $C = ({S_{\textrm{a}}} - {S_{b)}})/({S_a} + {S_b})$. The sensor sensitivity are C = 96% from Fig. 3(a1) to Fig. 3(a4) and can be controlled by varying temperature. Such, sensitivity is limited by the competition between phonon and polarization dressing.

Figure 4 shows the FL spectral signal at different polarization angles. Two peak structure, AT-Splitting of each FL curve results from the dressing effect of pumping field E₁. The spectral dip is deeper under linear polarization (Fig. 4(a1)). Low variation occurs, as we vary polarization from linear (Fig. 4(a1)) to elliptical (Fig. 4(a3)), however the variation becomes obvious when we change the polarization from elliptical (Fig. 4(a3)) to circular (Fig. 4(a5)). Figure 4(b) shows symmetry results of polarization, when polarization is changed from linear (Fig. 4(b2)) to left circular (Fig. 4(b1)) or from linear (Fig. 4(b2)) to right circular (Fig. 4(b3)). At left circular polarization (Fig. 4(a1)) the spectral dip is not deeper, but as we vary the polarization from left circular (Fig. 4(a1)) to linear (Fig. 4(a2)), the spectral dip become deeper and obvious. Again as we vary the polarization from linear (Fig. 4(a2)) towards right circular (Fig. 4(a3)), the dip decreased accordingly.

 

Fig. 4. (a1-a5) shows intensity of FL spectral signal at various polarization angles 0⁰, 10⁰, 22.5⁰, 32⁰ and 45⁰ respectively. (b1-b3) shows intensity of FL spectral signal at polarization angle −45⁰, 0⁰ and 45⁰ respectively.

Download Full Size | PDF

Figure 5 describes the intensity noise correlation between SP-FWM and FL signals plotted using correlation function described by (Eq. (9)). The intensities for correlations are detected by photomultiplier tubes (PMT1, PMT3) with a fixed boxcar gated integrator.

 

Fig. 5. (a1-a3) and (b1-b3) Two-mode intensity noise correlation between the output signals E_S/AS and FL signal in two level by changing polarization of E₁ at 77 K and 300 K, respectively.

Download Full Size | PDF

In Fig. 5(a1-a3), when temperature is low (77 K), the effect of phonon ${{\Gamma }_{phonon}}$ and phonon relates with ${{\Gamma }_{i/j}}$ in Eq. (3) reduces, which also decrease ${\Gamma _{{1_M}{0_M}}}$ term. Hence, dressing term ${|{{G_{{1_M}}}} |^2}/({\Gamma _{{1_M}{0_M}}} + i{\Delta _1})$ is increased in density matrix element $\rho _{{1_M}{1_M}}^{(2)}$ in Eq. (3), due to which the value of intensity noise correlation is broad at low temperature. The line width of correlation signal can determined by using decoherence time of SP-FWM process. The line shape of a correlation function can be determined by ${A_{S/AS}} = {R_1}{|{{A_1}} |^2}[{e^{ - 2\Gamma _S^L|\tau |}} + {e^{ - 2\Gamma _{AS}^L|\tau |}} - 2\cos({\Omega _e}|\tau |){e^{-(\Gamma _S^L + \Gamma _{AS}^L)}}^{|\tau |}]$, which obviously shows constant parameters except decoherence rate. Figure 5(b1-b3) shows measured correlation results at 300k, hence at high temperature the effect of phonon ${{\Gamma }_{phonon}}$ and phonon relates with ${{\Gamma }_{i/j}}$ in Eq. (3) is increased, which also increased ${\Gamma _{{1_M}{0_M}}}$ term. Hence, dressing term ${|{{G_{{1_M}}}} |^2}/({\Gamma _{{1_M}{0_M}}} + i{\Delta _1})$ is decreased in density matrix element $\rho _{{1_M}{1_M}}^{(2)}$ in Eq. (3), due to which the value of intensity noise correlation is sharp at high temperature as compare to low temperature.

Figure 6 shows three mode correlation of FL and SP-FWM. Figure 6(a1) shows correlation of FL and SP-FWM at linear polarization using Eq. (12). At linear polarization the dressing effect is dominant so maximum AT splitting occurs, creating a dip in FL, which can be verified from Fig. 2(a1), hence the calculated noise correlation is maximum, because here relative nonlinear phase is induced by self-phase modulation. As the dressing effect decreased with change in polarization from linear to elliptical polarization, the correlation also decreased, which is attributed to the gain effect of FL caused by decreased dressing effect, hence in Fig. 6(a2) one can observe that, correlation calculated in right elliptical polarization is decreased in comparison with linear polarization. In Fig. 6(a3) the dressing effect is minimum at circular polarization as compare to right elliptical and linear polarization, which can be verified from the Fig. 2(a3) and Fig. 3(c).

 

Fig. 6. (a1-a3) Three-mode intensity noise correlation between signals E_{S ,} E_AS and FL signal in two level by changing polarization of E₁ at 110k. (b1-b3) shows corresponding three mode squeezing of (a1)-(a3), respectively.

Download Full Size | PDF

Hence the resultant amplitude of pure FL, pure stokes and anti-stokes decreases, which are due to the incoherent nature of FL and coherent nature of SP-FWM (Stokes and anti-Stokes). Here in Fig. 6(a3), nonlinear phase of Stoke signal ${\Delta }{\varphi _s}$ change from ${0}$ to $\pi {/4}$. Hence, the correlation is observed is minimum as compare to linear and elliptical polarization. Figure 6(b1)-(b3) shows three mode squeezing, which has been calculated using Eq. (13). The intensity difference ${\delta ^2}({\widehat I_{\textrm{FL}}} - {\widehat I_\textrm{S}} - {\widehat I_{\textrm{AS}}})$ is demonstrated by red curves in Fig. 6(b1)-(b3) and the intensity noise sum ${\delta ^2}({\widehat I_{\textrm{FL}}} + {\widehat I_\textrm{S}} + {\widehat I_{\textrm{AS}}})$ is plotted as black curves in Fig. 6(b1)-(b3). The degree of squeezing corresponds to correlation, from Fig. 6(a1)-(a3), it is very much clear that the degree of squeezing follows the amplitude of correlation. The degree of squeezing is minimum with less amplitude of correlation and vice versa. From Fig. (6), the intensity peak of correlation is varying from maximum bunching (Fig. 6(a1)) to minimum bunching (Fig. 6(a3)) by changing polarization from linear to circular respectively and also squeezing behavior changes from squeezing (Fig. 6(b1)) to anti-squeezing (Fig. 6(b3)) through nonlinear phase, showing switching phenomenon. This switching is occurred by interaction between coherent and incoherent signal by varying polarization. This can be also explained by relative nonlinear phase induced by self phase modulation (SPM) ${\Delta }{\varphi _s} = {\varphi _{\textrm{FL}}} - {\varphi _{\textrm{S}/{{\rm A}}\textrm{S}}} = 2(|{E_{\textrm{FL}}}{|^2}n_2^{\textrm{FL}} - |{E_\textrm{S}}{|^2}n_2^\textrm{S} - |{E_{\textrm{AS}}}{|^2}n_2^{{{\rm A}S}})\zeta {e^{ - {r^2}}}z/n_1^i$. The trends in correlation peak intensity are represented in Fig. 6(a1)-(a3), which demonstrates the decreasing trend with changing polarization from linear to circular. Here, the intensity noise correlation in Fig. 6(a1) and (b1) is considered as polarized switching On-state and Fig. 6(a3) and (b3) is considered as polarized switching Off-state. Figure 5 (b1)-(b3) shows the intensity different squeezing in response to correlation of Fig. 6(a1-a3). The intensity difference squeezing replicates the same pattern as the correlation, hence squeezing is switch as we change polarization from linear to circular. The trend of squeezing tends to decrease by varying polarization from linear to circular. The speed of polarization switching can be controlled by E₁ polarization dressing field, and such speed is limited by the atomic coherence time (Eq. (10)) in nanosecond time scale.

4. Conclusion

In summary, we reported the polarization and temperature dependencies of fluorescence signal and SP-FWM in two-level system. By using QWP, we observe that dressing effect changes in conjunction with phonon effect caused by varying temperature and polarization. The obtained AT-Splitting signals were analyzed in both spectral and temporal domain. Our results suggest that the dressing effects at low temperature with linearly polarized incident fields is greater than that of high temperature with elliptical and circular polarized of incident field respectively. Furthermore, our results provide an effective method to control dressing effect of temperature sensors and switching devices by varying temperature, power and polarization.