Research Article | Open Access
Higher-Order Amplitude Squeezing in Six-Wave Mixing Process
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.
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  found squeezed state of the second harmonic when a beam of light propagates through a nonlinear crystal. Later, Hillery  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  proposed the generation of amplitude-cubed squeezing in the fundamental mode in second and third harmonic generation. Jawahar and Jaiswal  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 , in nanodisplacement measurement , and in optical storage , 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 , higher-order harmonic generation [22–25], parametric amplification , Raman  and hyper-Raman processes , 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  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 where and are the real and imaginary parts of the field amplitude, respectively. and are slowly varying operators defined by The operators and obey the commutation relation which leads to the uncertainty relation () A quantum state is squeezed in variable if Amplitude-squared squeezing is defined in terms of operators and as The operators and obey the commutation relation , where is the usual number operator which leads to the uncertainty relation Amplitude-squared squeezing is said to exist in variable if Amplitude-cubed squeezing is defined in terms of the operators The operators and obey the commutation relation Relation (10) leads to the uncertainty relation Amplitude-cubed squeezing exists when Real and imaginary parts of fourth-order amplitude are given as The operators and obey the commutation relation and satisfy the uncertainty relation Fourth-order squeezing exists when
3. Squeezing in Fundamental Mode in Six-Wave Mixing Process
The model considers the process involving absorption of three pump photons of frequency each, going from state to state and emission of two probe photons from state to state with frequency each. The atomic system returns to its original state by emitting one signal photon of frequency from state to . The process is shown in Figure 1.
The Hamiltonian for this process is as follows () in which is a coupling constant. , , and , respectively, are the slowly varying operators for the three modes at ,, and . are the usual annihilation (creation) operators associated with the relation .
The Heisenberg equation of motion for mode is Using (17) in (18), we obtain Similarly, we obtained the relations for and as Expanding using Taylor’s series expansion by assuming the short-time interaction of waves with the medium and retaining the terms up to , we obtain The real quadrature component for squeezing of field amplitude in fundamental mode is given as For spontaneous interaction, we consider the quantum state as a product of coherent state for the fundamental mode and the vacuum state for the modes and , that is, where is the complex field amplitude of the fundamental mode. Using (21)–(23), we obtain the expectation values as Therefore, where is the phase angle, with and .
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 > 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, Retaining the terms up to we obtain Therefore, which is negative, indicating that squeezing will occur for those values of for which , in the fundamental mode in stimulated interaction under short-time approximation. The effect of the stimulated interaction is represented by the factor .
Using (21) and (23), the second-order amplitude is expressed as For second-order squeezing, the real quadrature component for the fundamental mode is expressed as Using (23) and (31) in (32), we get the expectation values in spontaneous six-wave mixing process as Therefore, The number of photons in mode may be expressed as Thus, using condition (23), we get Subtracting (37) from (35), we get Using initial condition (28), we obtain squeezing for the stimulated process as The right-hand sides of (38) and (39) are negative for all values of for which 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 as and the real quadrature component for third-order squeezing in the fundamental mode is expressed as Using (23) and (41), we get the expectation values for spontaneous interaction as Subtracting (42) from (43), we get Using (23) and (36), we have Subtracting (45) from (44), we get Using (28), we obtain the stimulated process as The right-hand sides of (46) and (47) are negative, for all values of for which , 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 as The real quadrature component for fourth-order squeezing in fundamental mode is given as Using (23) and (48) in (49), we get the expectation values as Therefore, subtracting (50) from (51), we obtain Using (23) and (36), we have Subtracting (53) from (52), we get Using (28), we obtain the stimulated process as
The right-hand sides of (54) and (55) are negative, for all values of for which , indicating the existence of squeezing in fourth-order field amplitude in the fundamental mode in the spontaneous and stimulated processes, respectively.
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 and 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 as Using (33) and (35), SNR in amplitude-squared squeezing is given as Using (42) and (44), SNR in amplitude-cubed squeezing is expressed as Using (50) and (52), SNR in fourth-order squeezing is expressed as
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 , , , and for spontaneous and right-hand sides of relations (30), (39), (47), and (55) by , , and for stimulated interaction for field amplitude, amplitude-squared, amplitude-cubed, and for fourth-order field amplitude, respectively. Taking and = 0, the variations of , , , and with photon number for spontaneous interaction and of , , and with and for stimulated interaction are shown from Figures 2, 3, 4, and 5.
A comparison between results of spontaneous and stimulated processes shows the occurrence of multiplication factor . 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 . The signal-to-noise ratio is found to be higher in lower orders as shown in Figure 6.
Figures 2, 3, 4, and 5 show that squeezing increases nonlinearly with , 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 . 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 .
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