# Minimum Total Noise in Wave-Mixing Processes

Authors: Savita Gill; Sunil Rani; Nafa Singh
Journal: International Journal of Optics
 (2012)

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## 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 (Tmin) of a higher-order squeezed state always increases with the increase in nonclassicality associated with higher-order squeezing. Thus, from Tmin, one can conclude that highly nonclassical states have large amplitude fluctuations.


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## Body

## 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 thatTmin 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 as(1)Y1=12(A2+A†2),Y2=i2(A2-A†2),
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=aeiωt and A†=a†e-iωt.The operatorsY1 and Y2 obey the commutation relation(2)[Y1,Y2]=i(4N+2).
This leads to the uncertainty relation(3)ΔY1ΔY2≥〈(2N+1)〉,
where N is the usual number operator.Amplitude-squared squeezing is said to exist inY1 variable if
(4)(ΔY1)2<〈2N+1〉.
or the squeezing f is
(5)f=(ΔY1)2-〈2N+1〉<0.
Total noise of a quantum state of a single mode, having density matrix ρ, is defined as(6)T(ρ)=(ΔX1)2+(ΔX2)2.
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:(7)(ΔX1)2(ΔY1)2≥12〈X2〉2,(ΔX2)2(ΔY1)2≥12〈X1〉2.
We can combine the above relations with the identity(8)〈2N+1〉=〈X1〉2+〈X2〉2
to obtain(9)12(〈2N+1〉)=12(〈X1〉2+〈X2〉2)≤((ΔX1)2+(ΔX2)2)(ΔY1)2,
or(10)T=(ΔX1)2+(ΔX2)2≥12(〈2N+1〉)(ΔY1)2.
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:
(11)Tmin⁡=12(〈2N+1〉)(ΔY1)2.
Here again we see that, for fixed 〈N〉 as ΔY1 decreases, the total noise must increase. Thus, Tmin 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 frequencyω1 and emission of two probe photons of frequency ω2, and three signal photons of frequency ω3 where(12)2ω1=2ω2+3ω3.
The Hamiltonian for this process is given as follows (ℏ = 1):(13)H=ω1a†a+ω2b†b+ω3c†c+g(a2b†2c†3+a†2b2c3),
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):(14)dAdt=∂A∂t+i[H,A].
By using the short-time approximation technique, we expand A(t) by using Taylors series expansion and retaining the terms up to g2t2 as(15)A(t)=A-2igtA†B2C3+g2t2×[2ANB2NC3-9A†A2NB2NC2-18A†A2NB2NC-6A†A2NB2-4A†A2NBNC3-36A†A2NBNC2-72A†A2NBNC-24A†A2NB-2A†A2NC3-18A†A2NC2-36A†A2NC-12A†A2],
where  NA=A†A,NB=B†B,and NC=C†C.Initially, we consider the quantum state of the field amplitude as a product of coherent state for the fundamental modeA and the vacuum state for modes B and C, that is,
(16)|ψ〉=|α〉A|0〉B|0〉C.
Using (1), (15), and (16), a straightforward but strenuous calculation yields(17)[ΔY1A(t)]2=〈ψ|Y1A2(t)|ψ〉-〈ψ|Y1A(t)|ψ〉2=2|α|2+1-12g2t2[2|α|4(cos⁡4θ+3)+10|α|2+2].
Using (15) and (16), number of photons in mode A may be expressed as
(18)N(t)=A†(t)A(t)=|α|2-24g2t2|α|4.
Now, we can substitute (17) and (18) in (5) to obtain a closed form analytic expression for f as(19)f=(ΔY1)2-〈2N+1〉=-12g2t2[2|α|4(cos⁡4θ+1)+10|α|2+2].
And similarly substituting (17) and (18) in (11), we have(20)Tmin⁡=12(〈2N+1〉)(ΔY1)2=122|α|2+1-48g2t2|α|42|α|2+1-12g2t2[2|α|4(cos⁡4θ+3)+10|α|2+2],
where θ is the phase angle, with α=|α|eiθ. 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 frequencyω1 and emission of two probe photons of frequency ω2 and signal photon of frequency ω3, where(21)2ω1=2ω2+ω3.
The Hamiltonian for this process is given as follows (ℏ = 1):(22)H=ω1a†a+ω2b†b+ω3c†c+g(a2b†2c†+a†2b2c).
Short-time second-order solution of this Hamiltonian is(23)A(t)=A-2igtA†B2C+g2t2×[2ANB2NC-A†A2NB2-4A†A2NBNC-4A†A2NB-2A†A2NC-2A†A2],
where NA=A†A, NB=B†B, and NC=C†C.Using (23) and (16), a straightforward but strenuous calculation yields(24)[ΔY1A(t)]2=〈ψ|Y1A2(t)|ψ〉-〈ψ|Y1A(t)|ψ〉2=2|α|2+1-4g2t2[2|α|4(cos⁡4θ+3)+8|α|2+1].
Using (23) and (16), number of photons in mode A may be expressed as
(25)N1A(t)=A†(t)A(t)=|α|2-4g2t2|α|4.
The respective values of f and Tmin can similarly be calculated as we have done for seven-wave mixing and that yields(26)f=(ΔY1)2-〈2N+1〉=-4g2t2[2|α|4(cos⁡4θ+2)+8|α|2+1],(27)Tmin⁡=122|α|2+1-48g2t2|α|42|α|2+1-12g2t2[2|α|4(cos⁡4θ+3)+10|α|2+2].
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 (Tmin) in seven-wave mixing and five-wave mixing processes are shown in Figures 1 and 2, respectively.(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)(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.
(a)(b)The degree of higher-order squeezing varies with the phase of the input coherent lightθ, 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 Tmin.


## 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 Figures1 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).



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