In this paper, we investigate the distribution statistics of photons in a single mode radiation field subjected to two-photon absorption (TPA) and the factors that contribute to squeezing and antibunching of photons, leading to the generation of nonclassical light. TPA is a nonlinear optical phenomenon in which the atoms interact with the light field by absorbing two photons simultaneously. The motivation to study TPA is the recent intense activity on nanocrystallites/quantum dots. Further, it is the only nonlinear optical phenomenon that can be analytically studied. The simultaneous occurrence of squeezing and antibunching is studied with small initial photon numbers by solving the master equation for TPA of a single mode radiation directly by numerical integration, without going through analytical procedure. The results are compared with those of analytical/numerical procedures available. Further, the discussion on the parameters of squeezing and antibunching for short-time (ST) as well as long-time is done comprehensively in the present work by taking up the ST approximation and summation of ST (SST) procedure along with the exact numerical method.
A consequence of the quantization of radiation is the fluctuations associated with the zero point energy called vacuum fluctuations [
The coherent states of light exhibit minimum quantum noise and affect the quadratures, for example, amplitude and phase, equally. The correlations that can be introduced between them may reduce the noise in amplitude at the cost of increasing it in the phase. This is known as amplitude squeezed light [
Another type of squeezed light involves reducing the fluctuations in one of the two standard orthogonal quadratures in such a way that the variance in that quadrature becomes less than the quantum noise limit of 1/4. This is known as quadrature or ordinary squeezed light. The interaction Hamiltonian in this case is of the form a † 2 + h . c .
The intensity fluctuations of the optical field are described by the second-order correlation function [
To generate nonclassical light, normally, a coherent light is allowed to interact in a nonlinear fashion with a medium. The phase space contour line which is observed as a circle for the initial coherent state would then become an ellipse. A review of experiments, main achievements and progress made in technology used in the production and detection of quadrature squeezed light, from the first successful production in 1985 to 2015 is presented in [
Although squeezing and antibunching of photons are observed in light generated by a variety of nonlinear optical processes, it is useful to study the process that admits analytical solution and one such process is two-photon absorption (TPA). Moreover, TPA attracts further interest as it may allow squeezing and antibunching to occur simultaneously.
Normally, the master equation for TPA involving the reduced density matrix operator of the single mode light field is solved by the generating function method [
The study on photon statistics of the internal and the external fields of a microcavity [
In this paper, we consider Hamiltonian of the form a † 2 + h . c . for the generation of quadrature or ordinary squeezed light using TPA. An attempt has been made to solve the master equation for TPA of a single mode radiation field numerically without going through analytical procedure and obtain the required factorial moments as a function of the dimensionless time parameter for different initial photon numbers. In §2, the theory behind the photon statistics is developed by defining the parameters required to describe antibunching and squeezing of photons. To discuss the parameters of that describe squeezing and antibunching for short-time (ST) as well as long-time comprehensively, we take up the ST approximation, summation of ST (SST) procedure and exact numerical integration method. §3 gives the methods of solving the master equation. Finally, the results are discussed in §4.
The intensity fluctuations of the optical field are described by the correlation function G(2). The degree of second order coherence is a measure of the correlation of the light intensities at two space-time points. It is defined in terms of the positive and negative frequency parts of the light field.
For a single mode radiation field, the normalized form of G(2) in terms of the creation a † and annihilation a operators of the field is given by
g ( 2 ) = 〈 a † a † a a 〉 / 〈 a † a 〉 2 (1)
or
g ( 2 ) − 1 = Q / 〈 a † a 〉 (2)
where
Q = ( 〈 ( a † a ) 2 〉 − 〈 a † a 〉 2 − 〈 a † a 〉 ) / 〈 a † a 〉 (3)
is called the Mandel parameter. In terms of number operator n = a † a , we have 〈 a † a 〉 = 〈 n 〉 and variance 〈 ( Δ n ) 2 〉 = 〈 n 2 〉 − 〈 n 〉 2 . For Poisson statistics, 〈 ( Δ n ) 2 〉 = 〈 n 〉 and hence g ( 2 ) = 1 ( Q = 0 ) . This is true for coherent light where the uncertainties in the two quadratures are equal and photons are randomly distributed. If g ( 2 ) > 1 ( Q > 0 ) then the photon number fluctuation follows super-Poisson statistics and photons are bunched. Finally, if g ( 2 ) < 1 ( Q < 0 ) sub-Poisson statistics is obeyed and this corresponds to antibunched light where photons lose correlations. Thus the sign of Q becomes an important factor for the confirmation of both sub-Poissonian and antibunching in a single mode light field.
Squeezing exists in the quadrature X 1 or X 2 of the light, defined by
X 1 = ( 1 / 2 ) ( a + a † ) (4)
and
X 2 = ( 1 / 2 i ) ( a − a † ) , (5)
if the uncertainty in X 1 or X 2 described by the parameter S 1 , 2 < 0 . i.e.
S 1 , 2 = 〈 X 1 , 2 2 〉 − 〈 X 1 , 2 〉 2 − 1 / 4 < 0 (6)
or
S 1 , 2 = ( 1 / 2 ) ( 〈 a † a 〉 − 〈 a † 〉 〈 a 〉 ± R e ( 〈 a † 2 〉 − 〈 a † 〉 2 ) ) < 0 . (7)
The maximum squeezing achievable by any process corresponds to S 1 = − 0.25 and for antibunched (sub-Poisson) light the minimum value of Mandel parameter Q is −1. For TPA, with initial coherent light α = | α | e i φ , where photon number is | α 2 | , in the stationary or steady state [
In TPA process, the atoms and the light field interact by the simultaneous absorption of two photons. It is assumed here that almost all the atoms are maintained in the ground state and hence two-photon emission can be ignored. During this interaction, the statistical properties of light field change and they depend on the initial conditions of the incident light.
We assume that the single mode radiation field of frequency ω interacts with an ensemble of N two-level atoms with a transition frequency of ω 0 , resonantly via TPA ( 2 ω = ω 0 ). A relatively small part of them are assumed to be excited during interaction and hence we call it the unsaturated TPA [
The total Hamiltonian H , describing the interaction of electromagnetic field with a nonlinear medium is
H = H F + H A + H I , (8)
where H F is the Hamiltonian operator of the field, H A Hamiltonian operator of atoms and H I is the interaction Hamiltonian. Therefore,
H = ℏ ω a † a + 1 2 ℏ ω 0 ∑ i ( c 2 i † c 2 i − c 1 i † c 1 i ) + ℏ ∑ i ( K c 2 i † c 1 i a a + K * c 1 i † c 2 i a † a † ) . (9)
Here c and c † operators refer to the ground state and excited state of the ith atom. The equation of motion for the density operator in terms of the Hamiltonian is
i ℏ d ρ / d t = [ H , ρ ] . (10)
The above Liouville equation is a particular case of a generalized equation under Markovian approximation [
d ρ / d t = L [ ρ ] , (11)
where L is a linear operator that generates a finite super operator called Lindblad operator. This operator includes all possible quantum jumps during the interaction of the field with the medium.
Retaining only the relevant TPA absorption and emission terms, the equation of motion for the density operator of the light field [
d ρ / d t = k 1 { [ a a ρ , a † a † ] + [ a a , ρ a † a † ] } + k 2 { [ a † a † ρ , a a ] + [ a † a † , ρ a a ] } , (12)
where the two terms on right hand side represent the absorption and emission processes of TPA. As mentioned in the beginning of this section, neglecting the emission term the master equation becomes
d ρ / d τ = − ( a † 2 a 2 ρ − 2 a 2 ρ a † 2 + ρ a † 2 a 2 ) , (13)
where τ = 2 k 1 t is the dimensionless time parameter. k 1 depends on the line shape function and intensity of the radiation field and also on the number density of atoms of the medium. It is related to the TPA absorption coefficient β as
k 1 = n 0 ℏ ω c 2 β / 8 V , (14)
where
β = 3 ω χ I ( 3 ) / 2 n 0 2 ε 0 c 2 . (15)
n 0 is the refractive index and χ I ( 3 ) is imaginary part of third-order susceptibility of the medium for TPA. The diagonal and off-diagonal matrix elements of the density operator in the Fock representation are given by ρ n , m = 〈 n | ρ | m 〉 and they satisfy
d ρ n , n / d τ = ( n + 1 ) ( n + 2 ) ρ n + 2 , n + 2 − n ( n − 1 ) ρ n , n (16)
and
d ρ n , m / d τ = [ ( n + 1 ) ( n + 2 ) ( m + 1 ) ( m + 2 ) ] 1 / 2 ρ n + 2 , m + 2 − ( 1 / 2 ) n ( n − 1 ) ρ n , m − ( 1 / 2 ) m ( m − 1 ) ρ n , m (17)
where, m = n + μ , here μ denotes the degree of off-diagonality. For an initial coherent light, the diagonal elements of density matrix ( μ = 0 ) can be written as
ρ n , n ( 0 ) = 〈 n | α 〉 〈 α | n 〉 = exp ( − | α 2 | ) | α | 2 n / n ! . (18)
The off-diagonal elements of density matrix ( μ ≠ 0 ) are
ρ n , n + μ ( 0 ) = 〈 n | α 〉 〈 α | n + μ 〉 = exp ( − i μ φ ) exp ( − | α 2 | ) α n ( α * ) n + μ / n ! ( n + μ ) ! . (19)
Following [
ρ n , n + μ ( τ ) = n ! / ( n + μ ) ! Ψ n ( μ , τ ) . (20)
At τ = 0 ,
Ψ n ( μ , 0 ) = ( n + μ ) ! / n ! ρ n , n + μ ( 0 ) = exp ( − i μ φ ) exp ( − | α 2 | ) | α | 2 n ( α * ) μ / n ! . (21)
The master equation can be rewritten in terms of Ψ n as
d Ψ n ( μ , τ ) / d τ = [ n ( n − 1 ) + n μ + 1 2 μ ( μ − 1 ) ] Ψ n ( μ , τ ) + ( n + 1 ) ( n + 2 ) Ψ n + 2 ( μ , τ ) . (22)
The relevant expectation values for the evaluation of antibunching and squeezing are given by
〈 a † a 〉 = ∑ n n Ψ n ( 0 , τ ) , (23)
〈 a † 2 a 2 〉 = ∑ n n ( n − 1 ) Ψ n ( 0 , τ ) , (24)
〈 ( a † a ) 2 〉 = ∑ n n 2 Ψ n ( 0 , τ ) , (25)
〈 a † 〉 = ∑ n Ψ n ( 1 , τ ) , (26)
〈 a † 2 〉 = ∑ n Ψ n ( 2 , τ ) . (27)
Usually, the master equation is solved exactly by using a generating function that describes the change of photon statistics of an initially coherent light and Ψ n ( μ , τ ) is determined by the nth order derivative of the generating function. This leads to an expression involving an infinite sum over gamma functions with complex arguments, along with an exponential factor exp [ − n ( n − 1 ) τ ] . This factor helps in convergence of the infinite sum, allowing truncation of the series at suitable n value. Further evaluation of relevant moments is done by numerical methods [
It is of interest to check if any change in photon statistics occurs soon after the interaction of light with matter and also to avoid the labor involved in obtaining the exact solution describe above, short-time expansion procedure [
〈 a † a 〉 = | α | 2 − 2 | α | 4 τ + 2 | α | 4 ( 2 | α | 2 + 1 ) τ 2 , (28)
〈 a † 〉 = | α | e − i φ [ 1 − | α | 2 τ + ( 1 / 2 ) | α | 2 ( 3 | α | 2 + 1 ) τ 2 ] , (29)
〈 a † 2 〉 = | α | 2 e − 2 i φ [ 1 − ( 2 | α | 2 + 1 ) τ + ( 4 | α | 4 + 4 | α | 2 + 1 / 2 ) τ 2 ] , (30)
〈 a † 2 a 2 〉 = | α | 4 − 2 | α | 4 ( 2 | α | 2 + 1 ) τ + 3 | α | 4 ( 4 | α | 4 + 6 | α | 2 + 1 ) τ 2 . (31)
For studying long time behavior of photon statistics, a summation of all higher order terms is necessary [
〈 a † a 〉 = | α | 2 / ( 1 + ξ ) + ξ 2 ( 3 + ξ ) / 6 ( 1 + ξ ) 3 , (32)
〈 a † 〉 = α * / ( 1 + ξ ) 1 / 2 + ξ 2 α * / 8 | α | 2 ( 1 + ξ ) 5 / 2 , (33)
〈 a † 2 〉 = exp ( − 2 i φ ) exp ( − τ ) 〈 a † a 〉 , (34)
〈 a † 2 a 2 〉 = | α | 4 / ( 1 + ξ ) 2 − | α | 2 ξ / ( 1 + ξ ) 4 . (35)
The master equation was solved directly using the eigenvalue method [
In the present work, Ψ n ( μ , τ ) is obtained directly by the numerical integration of the master Equation (22) by employing a standard integration procedure [
The results and discussion on the parameters of squeezing and antibunching for short-time (ST) as well as long-time is done in the present work comprehensively for comparison purpose by taking up the ST approximation, summation of ST (SST) procedure and exact numerical method. To check the reliability of our numerical results, a comparison of the variation of diagonal elements ρ n , n as a function of n for various dimensionless time τ values, for the initial photon number 9 is made with those of Simaan and Loudon [
From
τ-values, in fact it moves close to zero only, as indicated by Garcia-Fernandez et al. [
In
Finally,
takes place much earlier for large initial photon number. The analysis of the results indicates that the minimum value of Q that is achievable in TPA is −1/2, the stationary state value itself, even at a time of about 7.0 and it remains constant thereafter, with fewer than 10 initial photons. In other words, for initial photon number | α | 2 > 9 the maximum antibunching corresponding to the stationary state value is reached much sooner, whereas, SST calculation of Mandel parameter predicts only a minimum value of ?1/3 against ?1/2.
Summarizing, after the interaction of initially coherent light with a two-photon absorber, it becomes nonclassical by acquiring squeezing and antibunching of photons. Though squeezing is lost after some time, the light remains an antibunched one.
The simultaneous observation of squeezing and antibunching is made with small initial photon numbers by solving the master equation for TPA of a single mode radiation directly by numerical integration, without going through analytical procedure. Further, the discussion on the parameters of squeezing and antibunching for short-time (ST) as well as long-time is done comprehensively in the present work by taking up the ST approximation, summation of ST (SST) procedure and exact numerical method.
The results obtained by us agree well with those of already existing analytical/numerical procedures in the literature. The nonclassical parameters of light calculated by the ST approximation deviate very much from those of SST and exact numerical methods, except for exceedingly small time intervals in the beginning of interaction.
It is observed from our analysis that to know the amount of maximum squeezing and the time at which it is attained, it is only sufficient to use the expression of SST and avoid the complicated exact numerical procedure. On increasing the initial photon number, the minimum value of squeezing parameter is deeper and moves toward shorter time, but the duration of squeezing is smaller. The maximum amount of squeezing that could be attained in TPA is 33% or 1/3rd (corresponding to 1.8 dB noise reduction in the standard quadrature) of the lowest achievable value of ?0.25.
The Mandel parameter describing antibunching of photons obtained from the numerical method reaches the stationary state value of ?1/2 even at earlier times for initial photon number ≥ 9 and remains unchanged thereafter.
Thus, after acquiring the nonclassical characteristics of squeezing and antibunching on interaction with a two-photon absorber, the initial coherent light loses squeezing after some time but antibunching of photons persists.
Latha, G.M., Sripriya, M. and Ramesh, N. (2017) Generation of Nonclassical Light by Unsaturated Two-Photon Absorption. Optics and Photonics Journal, 7, 139-150. https://doi.org/10.4236/opj.2017.79014