Higher-Order Amplitude Squeezing in Six-Wave Mixing Process

Abstract

We investigate theoretically the generation of squeezed states in spontaneous and stimulated six-wave mixing process quantum mechanically. It has been found that squeezing occurs in field amplitude, amplitude-squared, amplitude-cubed, and fourth power of field amplitude of fundamental mode in the process. It is found to be dependent on coupling parameter “g” (characteristics of higher-order susceptibility tensor) and phase values of the field amplitude under short-time approximation. Six-wave mixing is a process which involves absorption of three pump photons and emission of two probe photons of the same frequency and a signal photon of different frequency. It is shown that squeezing is greater in a stimulated interaction than the corresponding squeezing in spontaneous process. The degree of squeezing depends upon the photon number in first and higher orders of field amplitude. We study the statistical behaviour of quantum field in the fundamental mode and found it to be sub-Poissonian in nature. The signal-to-noise ratio has been studied in different orders. It is found that signal-to-noise ratio is higher in lower orders. This study when supplemented with experimental observations offers possibility of improving performance of many optical devices and optical communication networks.

1. Introduction

Over the past three decades, particular attention has been focused on theoretical investigations and experimental observations in generation of squeezed light, for improving the performance of many optical devices and optical communication networks. The concept of squeezed light is concerned with reduction of quantum fluctuations in one of the quadrature, at the expense of increased fluctuations in the other quadrature. In general, the two important nonclassical effects, squeezing and antibunching (or Sub-Poissonian photon statistics), are not interrelated; that is, some states exist that exhibit the first but not the second and vice versa. However, squeezing can be detected using simple photon counting in higher-order sub-Poissonian statistics.

A lot of work has appeared in the literature on the theoretical and experimental investigations on generation of squeezed states of electromagnetic field. Mandel [1] found squeezed state of the second harmonic when a beam of light propagates through a nonlinear crystal. Later, Hillery [2] defined amplitude-squared squeezing and showed that amplitude-squared squeezed states can be of use in reducing noise in the output of certain nonlinear optical devices. Hong and Mandel [3, 4] introduced the notion of Nth-order squeezing as a generalization of the second-order squeezing. Zhan [5] proposed the generation of amplitude-cubed squeezing in the fundamental mode in second and third harmonic generation. Jawahar and Jaiswal [6] extended the results obtained by Zhan for amplitude-cubed squeezing in the fundamental mode during second and third harmonic generations to kth order. The significant experimental observations include gravity wave detection [7–10], in optical communication [11], in nanodisplacement measurement [12], and in optical storage [13], and interferometer enhancement [14, 15]. The experimental detections and applications confirm the importance of the theoretical investigations into various optical processes such as four- and six-wave mixing [16–20], eight-wave mixing [21], higher-order harmonic generation [22–25], parametric amplification [26], Raman [27] and hyper-Raman processes [28], and so forth. Higher-order sub-Poissonian statistics have been studied by a number of authors such as those in [29–31]. The conversion of higher-order squeezed light into nonclassical light with high sub-Poissonian statistics and its experimental detection has been discussed in [31–33].

Recently, Giri and Gupta [19] have investigated the squeezing effects in six-wave mixing process. In this paper, we propose a different model for the same interaction process. Also, this paper shows one of the distinguished examples of nonlinear processes when light exhibits both squeezing and sub-Poissonian photon statistics at the same time. Squeezing in field amplitude, amplitude-squared, amplitude-cubed, and in fourth-order amplitude has been studied in fundamental mode for the proposed model. The photon statistics and dependence of squeezing on photon number have also been investigated.

2. Definition of Squeezing and Higher-Order Squeezing

Squeezing is a purely quantum mechanical phenomenon which cannot be explained on the basis of classical physics. The coherent states do not exhibit nonclassical effects, but a superposition of coherent states can exhibit normal squeezing, higher-order squeezing, and sub-Poissonian photon statistics. A coherent state changes to a superposition of coherent states when it interacts with a non linear medium. Squeezed states of an electromagnetic field are the states with reduced noise below the vacuum limit in one of the canonical conjugate quadratures. Normal squeezing is defined in terms of the operators 𝑋₁=12𝐴+𝐴^†,𝑋₂=12𝑖𝐴−𝐴^†, (1) where 𝑋₁ and 𝑋₂ are the real and imaginary parts of the field amplitude, respectively. 𝐴 and 𝐴^† are slowly varying operators defined by 𝐴=𝑎𝑒^𝑖𝜔𝑡,𝐴^†=𝑎^†𝑒^{−𝑖𝜔𝑡}. (2) The operators 𝑋₁ and 𝑋₂ obey the commutation relation𝑋₁,𝑋₂=𝑖2 (3) which leads to the uncertainty relation (ℏ=1) Δ𝑋₁Δ𝑋₂≥14. (4) A quantum state is squeezed in 𝑋_𝑖 variable if Δ𝑋_𝑖<12for𝑖=1or2. (5) Amplitude-squared squeezing is defined in terms of operators 𝑌₁ and 𝑌₂ as𝑌₁=12𝐴²+𝐴^†2,𝑌₂=1𝐴2𝑖²−𝐴^†2. (6) The operators 𝑌₁ and 𝑌₂ obey the commutation relation [𝑌₁,𝑌₂]=𝑖(2𝑁+1), where 𝑁 is the usual number operator which leads to the uncertainty relation Δ𝑌₁Δ𝑌₂≥1𝑁+2. (7) Amplitude-squared squeezing is said to exist in 𝑌_𝑖 variable if Δ𝑌_𝑖²<1𝑁+2for𝑖=1or2. (8) Amplitude-cubed squeezing is defined in terms of the operators𝑍₁=12𝐴³+𝐴^†3,𝑍₂=1𝐴2𝑖³−𝐴^†3. (9) The operators 𝑍₁ and 𝑍₂ obey the commutation relation𝑍₁,𝑍₂=𝑖29𝑁²+9𝑁+6. (10) Relation (10) leads to the uncertainty relationΔ𝑍₁Δ𝑍₂≥149𝑁²+9𝑁+6. (11) Amplitude-cubed squeezing exists whenΔ𝑍_𝑖²<149𝑁²+9𝑁+6for𝑖=1or2. (12) Real and imaginary parts of fourth-order amplitude are given as𝐹₁=12𝐴⁴+𝐴^†4,𝐹₂=1𝐴2𝑖⁴−𝐴^†4. (13) The operators 𝐹₁ and 𝐹₂ obey the commutation relation𝐹₁,𝐹₂=𝑖216𝑁³+24𝑁²+56𝑁+24 (14) and satisfy the uncertainty relation (ℏ=1)Δ𝐹₁Δ𝐹₂≥1416𝑁³+24𝑁²+56𝑁+24. (15) Fourth-order squeezing exists whenΔ𝐹_𝑖²<1416𝑁³+24𝑁²+56𝑁+24for𝑖=1or2. (16)

3. Squeezing in Fundamental Mode in Six-Wave Mixing Process

The model considers the process involving absorption of three pump photons of frequency 𝜔₁ each, going from state |1⟩ to state |2⟩ and emission of two probe photons from state |2⟩ to state |3⟩ with frequency 𝜔₂ each. The atomic system returns to its original state by emitting one signal photon of frequency 𝜔₃ from state |3⟩ to |1⟩. The process is shown in Figure 1.

Figure 1 

Six-wave interaction model.

The Hamiltonian for this process is as follows (ℏ=1)𝐻=𝜔₁𝑎^†𝑎+𝜔₂𝑏^†𝑏+𝜔₃𝑐^†𝑐+𝑔𝑎³𝑏^†2𝑐^†+𝑎^†3𝑏²𝑐, (17) in which 𝑔 is a coupling constant. 𝐴=𝑎𝑒^{𝑖𝜔1𝑡}, 𝐵=𝑏𝑒^{𝑖𝜔2𝑡}, and 𝐶=𝑐𝑒^{𝑖𝜔3𝑡}, respectively, are the slowly varying operators for the three modes at 𝜔₁,𝜔₂, and 𝜔₃.𝑎(𝑎^†),𝑏(𝑏^†),𝑐(𝑐^†) are the usual annihilation (creation) operators associated with the relation 3𝜔₁=2𝜔₂+𝜔₃.

The Heisenberg equation of motion for mode 𝐴 is 𝐴̇=𝜕𝐴[]𝜕𝑡+𝑖𝐻,𝐴. (18) Using (17) in (18), we obtain𝐴̇=−3𝑖𝑔𝐴^†2𝐵²𝐶. (19) Similarly, we obtained the relations for 𝐵̇ and 𝐶̇ as𝐵̇=−2𝑖𝑔𝐴³𝐵^†𝐶^†,𝐶̇=−𝑖𝑔𝐴³𝐵^†2. (20) Expanding 𝐴(𝑡) using Taylor’s series expansion by assuming the short-time interaction of waves with the medium and retaining the terms up to |𝑔𝑡|², we obtain 𝐴(𝑡)=𝐴−3𝑖𝑔𝑡𝐴^†2𝐵²3𝐶+2𝑔²𝑡²×6𝐴^†𝐴²𝐵^†2𝐵²𝐶^†𝐶+6𝐴𝐵^†2𝐵²𝐶^†𝐶−4𝐴^†2𝐴³𝐵^†𝐵𝐶^†𝐶−2𝐴^†2𝐴³𝐶^†𝐶−𝐴^†2𝐴³𝐵^†2𝐵²𝐶^†𝐶−4𝐴^†2𝐴³𝐵^†𝐵𝐶^†𝐶−2𝐴^†2𝐴³. (21) The real quadrature component for squeezing of field amplitude in fundamental mode 𝐴 is given as𝑋_1𝐴(𝑡)=12𝐴(𝑡)+𝐴^†(𝑡). (22) For spontaneous interaction, we consider the quantum state as a product of coherent state for the fundamental mode 𝐴 and the vacuum state for the modes 𝐵 and 𝐶, that is,𝜓=|𝛼⟩_𝐴|0⟩_𝐵|0⟩_𝐶, (23) where 𝛼 is the complex field amplitude of the fundamental mode. Using (21)–(23), we obtain the expectation values as𝜓𝑋²_1𝐴(𝑡𝜓=1)4𝛼²+𝛼^∗2+2|𝛼|²+1−6𝑔²𝑡²𝛼²|𝛼|⁴+𝛼²|𝛼|²+𝛼^∗2|𝛼|⁴+𝛼^∗2|𝛼|²+2|𝛼|⁶,𝜓𝑋_1𝐴(𝑡)𝜓²=14𝛼²+𝛼^∗2+2|𝛼|²−6𝑔²𝑡²𝛼²|𝛼|⁴+𝛼^∗2|𝛼|⁴+2|𝛼|⁶. (24)(25) Therefore,Δ𝑋_1𝐴(𝑡)²=𝑋²_1𝐴(𝑡)−𝑋_1𝐴(𝑡)²=141−6𝑔²𝑡²𝛼²|𝛼|²+𝛼^∗2|𝛼|²,Δ𝑋_1𝐴(𝑡)²−14=−3𝑔²𝑡²|𝛼|⁴cos2𝜃, (26)(27) where 𝜃 is the phase angle, with 𝛼=|𝛼|𝑒^𝑖𝜃 and 𝛼^∗=|𝛼|𝑒^−𝑖𝜃.

The right-hand side of the expression (27) is negative, indicating that squeezing will occur in the first-order amplitude in the fundamental mode in six-wave mixing process for which cos 2𝜃> 0 for spontaneous interaction.

In parallel to the spontaneous interaction, the stimulated emission is caused due to the coupling of the atom to the other states of the field. Therefore, the study of squeezing in stimulated interaction in six-wave mixing process requires initial quantum state as a product of coherent states for modes 1, 2 and vacuum state for 3, that is,𝜓=|𝛼⟩_𝐴𝛽_𝐵|0⟩_𝐶. (28) Retaining the terms up to𝑔²𝑡², we obtain 𝜓𝑋_1𝐴(𝑡)𝜓²=14𝛼²+𝛼^∗2+2|𝛼|²−3𝑔²𝑡²𝛼²|𝛼|⁴+𝛼^∗2|𝛼|⁴+2|𝛼|⁶×𝛽⁴+4𝛽²,𝜓𝑋+2_21𝐴(𝑡𝜓=1)4𝛼²+𝛼^∗2+2|𝛼|²+1−3𝑔²𝑡²×𝛼²|𝛼|⁴+𝛼²|𝛼|²+𝛼^∗2|𝛼|⁴+𝛼^∗2|𝛼|²+2|𝛼|⁶×𝛽⁴+4𝛽²+2. (29) Therefore,Δ𝑋_1𝐴(𝑡)²−143=−2𝑔²𝑡²|𝛼|⁴𝛽⁴+4𝛽²+2cos2𝜃, (30) which is negative, indicating that squeezing will occur for those values of 𝜃 for which cos2𝜃>0, in the fundamental mode in stimulated interaction under short-time approximation. The effect of the stimulated interaction is represented by the factor (|𝛽|⁴+4|𝛽|²+2).

Using (21) and (23), the second-order amplitude is expressed as𝐴²(𝑡)=𝐴²−6𝑖𝑔𝑡𝐴^†2𝐴+𝐴^†𝐵²𝐶−3𝑔²𝑡²×𝐴^†2𝐴⁴+𝐴^†𝐴³𝐵^†2𝐵²+4𝐵^†𝐵+2. (31) For second-order squeezing, the real quadrature component for the fundamental mode is expressed as𝑌_1𝐴(𝑡)=12𝐴²(𝑡)+𝐴^†2(𝑡). (32) Using (23) and (31) in (32), we get the expectation values in spontaneous six-wave mixing process as𝜓𝑌_1𝐴(𝑡)𝜓²=14𝛼⁴+𝛼^∗4+2|𝛼|⁴−12𝑔²𝑡²×𝛼⁴|𝛼|⁴+𝛼⁴|𝛼|²+𝛼^∗4|𝛼|⁴+𝛼^∗4|𝛼|²+2|𝛼|⁸+2|𝛼|⁶,𝜓𝑌²_1𝐴(𝑡𝜓=1)4𝛼⁴+𝛼^∗4+2|𝛼|⁴+4|𝛼|²+2−12𝑔²𝑡²×𝛼⁴|𝛼|⁴+3𝛼⁴|𝛼|²+2𝛼⁴+𝛼^∗4|𝛼|⁴+3𝛼^∗4|𝛼|²+2𝛼^∗4+2|𝛼|⁸+4|𝛼|⁶. (33)(34) Therefore,Δ𝑌_1𝐴(𝑡)²=𝑌²_1𝐴(𝑡)−𝑌_1𝐴(𝑡)²=144|𝛼|²+2−24𝑔²𝑡²𝛼⁴|𝛼|²+𝛼⁴+𝛼^∗4|𝛼|²+𝛼^∗4+|𝛼|⁶. (35) The number of photons in mode 𝐴 may be expressed as𝑁_1𝐴(𝑡)=𝐴^†(𝑡)𝐴(𝑡)=𝐴^†𝐴+3𝑖𝑔𝑡𝐴³𝐵^†2𝐶^†−𝐴^†3𝐵²𝐶−3𝑔²𝑡²𝐴^†3𝐴³𝐵^†2𝐵²+4𝐵^†𝐵+2+9𝑔²𝑡²𝐴^†2𝐴²+4𝐴^†𝐴+2𝐵^†4𝐵⁴𝐶^†𝐶. (36) Thus, using condition (23), we get𝑁_1𝐴(𝑡)+12=|𝛼|²+12−6𝑔²𝑡²|𝛼|⁶. (37) Subtracting (37) from (35), we getΔ𝑌_1𝐴(𝑡)²−𝑁_1𝐴(𝑡)+12=−12𝑔²𝑡²|𝛼|⁶+|𝛼|⁴cos4𝜃. (38) Using initial condition (28), we obtain squeezing for the stimulated process asΔ𝑌_1𝐴(𝑡)²−𝑁_1𝐴(𝑡)+12=−6𝑔²𝑡²|𝛼|⁶+|𝛼|⁴×𝛽⁴+4𝛽²+2cos4𝜃. (39) The right-hand sides of (38) and (39) are negative for all values of 𝜃 for which cos4𝜃>0 and thus shows the existence of squeezing in the second order of the field amplitude in spontaneous and stimulated interaction under short-time approximation.

Using (21), cubed-amplitude is expressed as𝐴³(𝑡)=𝐴³−3𝑖𝑔𝑡3𝐴^†2𝐴²+6𝐴^†𝐴+2𝐵²𝐶−18𝑔²𝑡²𝐴^†4𝐴+3𝐴^†3𝐵⁴𝐶²−32𝑔²𝑡²×3𝐴^†2𝐴⁵+6𝐴^†𝐴⁴+2𝐴³𝐵^†2𝐵²+4𝐵^†𝐵+2, (40) and the real quadrature component for third-order squeezing in the fundamental mode is expressed as𝑍_1𝐴(𝑡)=12𝐴³(𝑡)+𝐴^†3(𝑡=1)2𝐴³+𝐴^†3−3𝑖𝑔𝑡𝐴^†2𝐴²+6𝐴^†𝐴+2𝐵²𝐶+3𝑖𝑔𝑡𝐴^†2𝐴²+6𝐴^†𝐴+2𝐵^†2𝐶^†−18𝑔²𝑡²𝐴^†4𝐴+3𝐴^†3𝐵⁴𝐶²−18𝑔²𝑡²𝐴^†𝐴⁴+3𝐴³𝐵^†4𝐶^†2−32𝑔²𝑡²×3𝐴^†5𝐴²+6𝐴^†4𝐴+2𝐴^†3+3𝐴^†2𝐴⁵+6𝐴^†𝐴⁴+2𝐴³. (41) Using (23) and (41), we get the expectation values for spontaneous interaction as 𝜓𝑍_1𝐴(𝑡)𝜓²=14𝛼⁶+𝛼^∗6+2|𝛼|⁶−6𝑔²𝑡²×3|𝛼|⁴+6|𝛼|²+2𝛼⁶+𝛼^∗6+2|𝛼|⁶,𝜓𝑍²_1𝐴(𝑡𝜓=1)4𝛼⁶+𝛼^∗6+2|𝛼|⁶+9|𝛼|⁴+18|𝛼|²+6−6𝑔²𝑡²3|𝛼|⁴+15|𝛼|²+20𝛼⁶+𝛼^∗6+6|𝛼|¹⁰+30|𝛼|⁸+40|𝛼|⁶. (42)(43) Subtracting (42) from (43), we getΔ𝑍_1𝐴(𝑡)²=149|𝛼|⁴+18|𝛼|²+6−54𝑔²𝑡²×|𝛼|²+2𝛼⁶+𝛼^∗6+2|𝛼|⁸+4|𝛼|⁶. (44) Using (23) and (36), we have149𝑁²_1𝐴(𝑡)+9𝑁_1𝐴(𝑡)=1+649|𝛼|⁴+18|𝛼|²+6−108𝑔²𝑡²|𝛼|⁸+2|𝛼|⁶. (45) Subtracting (45) from (44), we getΔ𝑍_1𝐴(𝑡)²−149𝑁²_1𝐴(𝑡)+9𝑁_1𝐴(𝑡)+6=−27𝑔²𝑡²|𝛼|⁸+2|𝛼|⁶cos6𝜃. (46) Using (28), we obtain the stimulated process asΔ𝑍_1𝐴(𝑡)²−149𝑁²_1𝐴(𝑡)+9𝑁_1𝐴(𝑡)+6=−272𝑔²𝑡²|𝛼|⁸+2|𝛼|⁶𝛽⁴+4𝛽²+2cos6𝜃. (47) The right-hand sides of (46) and (47) are negative, for all values of 𝜃 for which cos6𝜃>0, indicating the existence of squeezing in cubed amplitude in the fundamental mode in the spontaneous and stimulated processes.

For fourth-order squeezing, amplitude is expressed as𝐴⁴(𝑡)=𝐴⁴−12𝑖𝑔𝑡𝐴^†2𝐴³+3𝐴^†𝐴²+2𝐴𝐵²𝐶−6𝑔²𝑡²×𝐴^†2𝐴⁶+3𝐴^†𝐴⁵+2𝐴⁴𝐵^†2𝐵²+4𝐵^†𝐵+2. (48) The real quadrature component for fourth-order squeezing in fundamental mode is given as 𝐹_1𝐴(𝑡)=12𝐴⁴(𝑡)+𝐴^†4(𝑡). (49) Using (23) and (48) in (49), we get the expectation values as 𝜓𝐹_1𝐴(𝑡)𝜓²=14𝛼⁸+𝛼^∗8+2|𝛼|⁸−24𝑔²𝑡²×|𝛼|⁴+3|𝛼|²+2𝛼⁸+𝛼^∗8+2|𝛼|¹²+6|𝛼|¹⁰+4|𝛼|⁸,𝜓𝐹²_1𝐴(𝑡𝜓=1)4𝛼⁸+𝛼^∗8+2|𝛼|⁸+16|𝛼|⁶+72|𝛼|⁴+96|𝛼|²+24−24𝑔²𝑡²×|𝛼|⁴+7|𝛼|²+14𝛼⁸+𝛼^∗8+2|𝛼|¹²+18|𝛼|¹⁰+64|𝛼|⁸+68|𝛼|⁶. (50)(51) Therefore, subtracting (50) from (51), we obtainΔ𝐹_1𝐴(𝑡)²=1416|𝛼|⁶+72|𝛼|⁴+96|𝛼|²+24−24𝑔²𝑡²×4|𝛼|²+12𝛼⁸+𝛼^∗8+12|𝛼|¹⁰+60|𝛼|⁸+68|𝛼|⁶. (52) Using (23) and (36), we have1416𝑁³_1𝐴(𝑡)+24𝑁²_1𝐴(𝑡)+56𝑁_𝐴=1+24416|𝛼|⁶+72|𝛼|⁴+96|𝛼|²+24−24𝑔²𝑡²12|𝛼|¹⁰+60|𝛼|⁸+68|𝛼|⁶. (53) Subtracting (53) from (52), we getΔ𝐹_1𝐴(𝑡)²−1416𝑁³_1𝐴(𝑡)+24𝑁²_1𝐴(𝑡)+56𝑁_𝐴+24=−48𝑔²𝑡²|𝛼|¹⁰+3|𝛼|⁸cos8𝜃. (54) Using (28), we obtain the stimulated process asΔ𝐹_1𝐴(𝑡)²−1416𝑁³_1𝐴+24𝑁²_1𝐴(𝑡)+56𝑁_1𝐴(𝑡)+24=−24𝑔²𝑡²|𝛼|¹⁰+3|𝛼|⁸𝛽⁴+4𝛽²+2cos8𝜃. (55)

The right-hand sides of (54) and (55) are negative, for all values of 𝜃 for which cos8𝜃>0, indicating the existence of squeezing in fourth-order field amplitude in the fundamental mode in the spontaneous and stimulated processes, respectively.

Using (23) and (36), the statistics of fundamental mode in six-wave mixing is found to be sub-Poissonian, given asΔ𝑁_1𝐴²−𝑁_1𝐴=−12𝑔²𝑡²|𝛼|⁶. (56)

4. Signal-to-Noise Ratio

Signal-to-noise ratio is defined as ratio of the magnitude of the signal to the magnitude of the noise. With the approximations 𝜃=0 and |𝑔𝑡|²≪1,the maximum signal-to-noise ratio (in decibels) in field amplitude and higher orders is given in the following.

Using (25) and (26), signal-to-noise ratio in field amplitude is defined asSNR₁=20∗log₁₀𝑋_1𝐴(𝑡)²Δ𝑋_1𝐴(𝑡)²=20∗log₁₀2|𝛼|². (57) Using (33) and (35), SNR in amplitude-squared squeezing is given asSNR₂=20∗log₁₀2|𝛼|⁴+|𝛼|²3|𝛼|²+2. (58) Using (42) and (44), SNR in amplitude-cubed squeezing is expressed asSNR₃=20∗log₁₀3|𝛼|⁴+6|𝛼|²+29|𝛼|²+18. (59) Using (50) and (52), SNR in fourth-order squeezing is expressed asSNR₄=20∗log₁₀|𝛼|⁶+3|𝛼|⁴+2|𝛼|²5|𝛼|⁴+21|𝛼|²+17. (60)

5. Results

The results show the presence of squeezing in field amplitude, amplitude-squared, amplitude-cubed, and fourth-order field amplitude of fundamental mode in six-wave mixing process. To study squeezing, we denote the right-hand sides of relations (27), (38),(46), and (54) by 𝑆_𝑥, 𝑆_𝑦, 𝑆_𝑧, and 𝑆_𝑓 for spontaneous and right-hand sides of relations (30), (39), (47), and (55) by 𝑆^′_𝑥, 𝑆^′_𝑦, 𝑆^′_𝑧, and 𝑆^′_𝑓for stimulated interaction for field amplitude, amplitude-squared, amplitude-cubed, and for fourth-order field amplitude, respectively. Taking |𝑔𝑡|²=10⁻⁴ and 𝜃 = 0, the variations of 𝑆_𝑥, 𝑆_𝑦, 𝑆_𝑧, and 𝑆_𝑓 with photon number |𝛼|² for spontaneous interaction and of 𝑆^′_𝑥, 𝑆^′_𝑦, 𝑆^′_𝑧,and 𝑆^′_𝑓 with |𝛼|²and |𝛽|²for stimulated interaction are shown from Figures 2, 3, 4, and 5.

(a)

(b)

Figure 2 

Dependence of first-order squeezing (a) 𝑠𝑋 with |𝛼|2 in spontaneous and (b) 𝑠′𝑋 with |𝛼|2 and |𝛽|2 in stimulated six-wave mixing process.

(a)

(b)

Figure 3 

Dependence of amplitude-squared squeezing (a) 𝑠𝑦with |𝛼|2 in spontaneous and (b) 𝑠′𝑦 with |𝛼|2 and |𝛽|2 in stimulated six-wave mixing process.

(a)

(b)

Figure 4 

Dependence of amplitude-cubed squeezing (a) 𝑆𝑧 with |𝛼|2 in spontaneous and (b) 𝑆′𝑧 with |𝛼|2 and |𝛽|2 in stimulated six-wave mixing process.

(a)

(b)

Figure 5 

Dependence of fourth-order field amplitude squeezing (a) 𝑆𝑓 with |𝛼|2 in spontaneous and (b) 𝑆′𝑓 with |𝛼|2 and |𝛽|2 in stimulated six-wave mixing process.

A comparison between results of spontaneous and stimulated processes shows the occurrence of multiplication factor (|𝛽|⁴+4|𝛽|²+2). It implies that squeezing in the fundamental mode in stimulated interaction is greater than corresponding squeezing in spontaneous interaction. It is also seen that maximum squeezing occurs when 𝜃=0. The signal-to-noise ratio is found to be higher in lower orders as shown in Figure 6.

Figure 6 

Signal-to-noise ratio for different order squeezing.

6. Conclusion

Figures 2, 3, 4, and 5 show that squeezing increases nonlinearly with |𝛼|², which is directly dependent upon the number of photons. The squeezing in any order during stimulated interaction (Figures 5(b), 4(b), 3(b), and 2(b)) is higher than the squeezing in corresponding order in spontaneous (Figures 5, 4, 3, and 2) interaction by a factor (|𝛽|⁴+4|𝛽|²+2). The squeezing is higher in higher orders in both processes. Thus, the higher-order squeezing associated with higher order nonlinear optical processes makes it possible to achieve significant noise reduction.

It has also been found that the fundamental mode of field amplitude shows sub-Poissonian behavior as shown in relation (56). The signal-to-noise ratio is higher in lower orders squeezed states as reported earlier for Raman process [34].