Minimum Total Noise in Wave-Mixing Processes

Abstract

Higher-order squeezing in different optical processes such as seven-wave mixing and five-wave mixing has been studied. The total noise of a field state is a measure of the fluctuations of the field amplitude. It is shown that the minimum total noise (𝑇_min) of a higher-order squeezed state always increases with the increase in nonclassicality associated with higher-order squeezing. Thus, from 𝑇_min, one can conclude that highly nonclassical states have large amplitude fluctuations.

1. Introduction

The concept of the total noise of a quantum state was introduced by Schumaker [1]. As was pointed by Schumaker, the total noise is always greater than or equal to a half and reaches this value only for coherent states. The total noise of a field state increases as the depth of nonclassicality associated with a state increases [2]. A nonclassical state of electromagnetic field is one for which the Glauber-Sudarshan P-function either goes negative or contains derivatives of delta function [3]. Standard deviation of an observable is considered to be the most natural measure of quantum fluctuations associated with an observable [4]. Reduction of quantum fluctuation below the coherent state level corresponds to a nonclassical state. Optical fields in states with purely quantum mechanical properties are the key ingredients of quantum optics. Nonclassical properties of a radiation field such as photon antibunching and squeezing are currently of great interest and have attracted considerable attention owing to its low noise property [5–9]. Higher-order squeezing has drawn the greater attention of the community due to the rapid development of techniques for making higher-order correlation measurements in quantum optics [10–15].

In the present work, we have reported that the generation of higher-order squeezed state is possible by using seven-wave mixing and five-wave mixing processes, respectively. Further, we have also shown that T _min can be used as an indirect measure of nonclassicality of a system associated with higher-order squeezing.

2. Higher-Order Squeezing and Total Noise

Higher-order squeezing is defined in various ways. Hong and Mandel [10] and Hillery [12] have introduced the notion of higher-order squeezing of quantized electromagnetic field as generalization of normal squeezing. Amplitude-squared squeezing is defined in terms of operators 𝑌₁ and 𝑌₂ as𝑌₁=1√2𝐴²+𝐴^†2,𝑌₂=𝑖√2𝐴²−𝐴^†2, (1) where 𝑌₁ and 𝑌₂ are the real and imaginary parts of the square of field amplitude, respectively. 𝐴 and 𝐴^† are slowly varying operators defined by 𝐴=𝑎𝑒^𝑖𝜔𝑡 and 𝐴^†=𝑎^†𝑒^{−𝑖𝜔𝑡}.

The operators 𝑌₁ and 𝑌₂ obey the commutation relation𝑌₁,𝑌₂=𝑖(4𝑁+2). (2) This leads to the uncertainty relationΔ𝑌₁Δ𝑌₂≥⟨(2𝑁+1)⟩, (3) where N is the usual number operator.

Amplitude-squared squeezing is said to exist in 𝑌₁ variable if Δ𝑌₁²<⟨2𝑁+1⟩. (4) or the squeezing 𝑓 is 𝑓=Δ𝑌₁²−⟨2𝑁+1⟩<0. (5) Total noise of a quantum state of a single mode, having density matrix 𝜌, is defined as𝑇𝜌=Δ𝑋₁²+Δ𝑋₂². (6) Increased nonclassicality gives rise to increase in the total noise. This fact can be verified by associating total noise with higher-order squeezing. The uncertainty relations for the quadrature variables using Hillery’s approach [2] may be written as follows:Δ𝑋₁²Δ𝑌₁²≥12𝑋₂²,Δ𝑋₂²Δ𝑌₁²≥12𝑋₁². (7) We can combine the above relations with the identity⟨2𝑁+1⟩=𝑋₁²+𝑋₂² (8) to obtain12=1(⟨2𝑁+1⟩)2𝑋₁²+𝑋₂²≤Δ𝑋₁²+Δ𝑋₂²Δ𝑌₁², (9) or𝑇=Δ𝑋₁²+Δ𝑋₂²≥12(⟨2𝑁+1⟩)Δ𝑌₁². (10) From condition (4), T will be greater than 1/2. The minimum total noise of a state is greater as a state becomes more nonclassical: 𝑇_min=12(⟨2𝑁+1⟩)Δ𝑌₁². (11) Here again we see that, for fixed ⟨𝑁⟩ as Δ𝑌₁ decreases, the total noise must increase. Thus, 𝑇_𝑚𝑖𝑛 increases as a state becomes more squeezed and may be considered as a measure of depth of nonclassicality.

3. Seven-Wave Mixing Process

In this process, the interaction is looked upon as a process which involves the absorption of two pump photons, each having frequency ω₁ and emission of two probe photons of frequency ω₂, and three signal photons of frequency ω₃ where2𝜔₁=2𝜔₂+3𝜔₃. (12) The Hamiltonian for this process is given as follows (ℏ = 1):𝐻=𝜔₁𝑎^†𝑎+𝜔₂𝑏^†𝑏+𝜔₃𝑐^†𝑐+𝑔𝑎²𝑏^†2𝑐^†3+𝑎^†2𝑏²𝑐³, (13) in which g is a coupling constant. 𝐴=𝑎exp(iω₁t), 𝐵=𝑏exp(iω₂t), and 𝐶=𝑐exp(iω₃t) are the slowly varying operators at frequencies 𝜔₁, 𝜔₂, and 𝜔₃, a (𝑎^†), b (𝑏^†), and c (𝑐^†) are the usual annihilation (creation) operators, respectively. The Heisenberg equation of motion for fundamental mode A is given as (ℏ = 1):𝑑𝐴=𝑑𝑡𝜕𝐴[].𝜕𝑡+𝑖𝐻,𝐴 (14) By using the short-time approximation technique, we expand A(t) by using Taylors series expansion and retaining the terms up to 𝑔²𝑡² as𝐴(𝑡)=𝐴−2𝑖𝑔𝑡𝐴^†𝐵²𝐶³+𝑔²𝑡²×2𝐴𝑁²_𝐵𝑁³_𝐶−9𝐴^†𝐴²𝑁²_𝐵𝑁²_𝐶−18𝐴^†𝐴²𝑁²_𝐵𝑁_𝐶−6𝐴^†𝐴²𝑁²_𝐵−4𝐴^†𝐴²𝑁_𝐵𝑁³_𝐶−36𝐴^†𝐴²𝑁_𝐵𝑁²_𝐶−72𝐴^†𝐴²𝑁_𝐵𝑁_𝐶−24𝐴^†𝐴²𝑁_𝐵−2𝐴^†𝐴²𝑁³_𝐶−18𝐴^†𝐴²𝑁²_𝐶−36𝐴^†𝐴²𝑁_𝐶−12𝐴^†𝐴², (15) where 𝑁_𝐴=𝐴^†𝐴, 𝑁_𝐵=𝐵^†𝐵, and 𝑁_𝐶=𝐶^†𝐶.

Initially, we consider the quantum state of the field amplitude as a product of coherent state for the fundamental mode 𝐴 and the vacuum state for modes B and C, that is, 𝜓=|𝛼⟩_𝐴|0⟩_𝐵|0⟩_𝐶. (16) Using (1), (15), and (16), a straightforward but strenuous calculation yieldsΔ𝑌_1𝐴(𝑡)²=𝜓𝑌²_1𝐴(𝑡)𝜓−𝜓𝑌_1𝐴(𝑡)𝜓²=2|𝛼|²+1−12𝑔²𝑡²2|𝛼|⁴(cos4𝜃+3)+10|𝛼|²+2. (17) Using (15) and (16), number of photons in mode A may be expressed as 𝑁(𝑡)=𝐴^†(𝑡)𝐴(𝑡)=|𝛼|²−24𝑔²𝑡²|𝛼|⁴. (18) Now, we can substitute (17) and (18) in (5) to obtain a closed form analytic expression for f as𝑓=Δ𝑌₁²−⟨2𝑁+1⟩=−12𝑔²𝑡²2|𝛼|⁴(cos4𝜃+1)+10|𝛼|²+2. (19) And similarly substituting (17) and (18) in (11), we have𝑇_min=12(⟨2𝑁+1⟩)Δ𝑌₁²=122|𝛼|²+1−48𝑔²𝑡²|𝛼|⁴2|𝛼|²+1−12𝑔²𝑡²2|𝛼|⁴(cos4𝜃+3)+10|𝛼|²,+2 (20) where θ is the phase angle, with 𝛼=|𝛼|𝑒^𝑖𝜃. The right-hand side of (19) is negative and thus indicating the presence of higher-order squeezing within the short time domain of the second-order solution.

4. Five-Wave Mixing Process

In this process, the interaction is looked upon as a process which involves the absorption of two pump photons, each having frequency ω₁ and emission of two probe photons of frequency ω₂ and signal photon of frequency ω₃, where2𝜔₁=2𝜔₂+𝜔₃. (21) The Hamiltonian for this process is given as follows (ℏ = 1):𝐻=𝜔₁𝑎^†𝑎+𝜔₂𝑏^†𝑏+𝜔₃𝑐^†𝑐+𝑔𝑎²𝑏^†2𝑐^†+𝑎^†2𝑏²𝑐. (22) Short-time second-order solution of this Hamiltonian is𝐴(𝑡)=𝐴−2𝑖𝑔𝑡𝐴^†𝐵²𝐶+𝑔²𝑡²×2𝐴𝑁²_𝐵𝑁_𝐶−𝐴^†𝐴²𝑁²_𝐵−4𝐴^†𝐴²𝑁_𝐵𝑁_𝐶−4𝐴^†𝐴²𝑁_𝐵−2𝐴^†𝐴²𝑁_𝐶−2𝐴^†𝐴², (23) where 𝑁_𝐴=𝐴^†𝐴, 𝑁_𝐵=𝐵^†𝐵, and 𝑁_𝐶=𝐶^†𝐶.

Using (23) and (16), a straightforward but strenuous calculation yieldsΔ𝑌_1𝐴(𝑡)²=𝜓𝑌²_1𝐴(𝑡)𝜓−𝜓𝑌_1𝐴(𝑡)𝜓²=2|𝛼|²+1−4𝑔²𝑡²2|𝛼|⁴(cos4𝜃+3)+8|𝛼|²+1. (24) Using (23) and (16), number of photons in mode A may be expressed as 𝑁_1𝐴(𝑡)=𝐴^†(𝑡)𝐴(𝑡)=|𝛼|²−4𝑔²𝑡²|𝛼|⁴. (25) The respective values of 𝑓 and T _min can similarly be calculated as we have done for seven-wave mixing and that yields𝑓=Δ𝑌₁²−⟨2𝑁+1⟩=−4𝑔²𝑡²2|𝛼|⁴(cos4𝜃+2)+8|𝛼|²,𝑇+1_min=122|𝛼|²+1−48𝑔²𝑡²|𝛼|⁴2|𝛼|²+1−12𝑔²𝑡²2|𝛼|⁴(cos4𝜃+3)+10|𝛼|².+2 (26)(27) The right-hand side of (26) is always negative within the domain of the validity of the solution which shows the existence of higher-order squeezing.

5. Results

The presence of higher-order squeezing in seven-wave mixing and five-wave mixing processes is being shown in (19) and (26), respectively. Again, from (5), (10), and (11), it is clear that highly nonclassical states have large amplitude fluctuations. Taking 𝑔² = 10⁻⁶ and |𝛼|² = 50, the variation of squeezing (−f) and minimum total noise (𝑇_min) in seven-wave mixing and five-wave mixing processes are shown in Figures 1 and 2, respectively.

(a)

(b)

Figure 1 

(a) Variation of −f for seven-wave mixing process with respect to interaction time t and initial phase of the coherent state 𝜃. (b) Variation of 𝑇min of seven-wave mixing process with respect to interaction time t and initial phase of the coherent state𝜃.

(a)

(b)

Figure 2 

(a) Variation of −f for five-wave mixing process with respect to interaction time t and initial phase of the coherent state 𝜃. (b) Variation of 𝑇min of five-wave mixing process with respect to interaction time t and initial phase of the coherent state 𝜃.

The degree of higher-order squeezing varies with the phase of the input coherent light θ, initial photon number |𝛼|², and the interaction time t. Further, Figures 1(b) and 2(b) show that the depth of nonclassicality in higher-order squeezing can be measured in terms of 𝑇_min.

6. Conclusion

The results show the presence of higher-order squeezing in seven-wave mixing and five-wave mixing processes. The total noise which is a measure of the size of amplitude fluctuations of a state of the field always increases with the increase in nonclassicality of a system associated with higher-order squeezing. This fact is more conspicuous in Figures 1 and 2. Again, from (19), (20), (26), and (27), we can conclude that squeezing and the amount of total noise present in the system can be tuned by varying the values of initial phase of the coherent state (𝜃), number of photons present in the radiation field prior to the interaction (|𝛼|²), and the interaction time (t).