Nonclassical effects in two coupled oscillators at non resonant region

We consider two coupled quantum harmonic oscillators with different free frequencies. Here the interaction between the two modes does not involve the Rotating Wave Approximation (RWA). The Heisenberg equations of motion are solved analytically using an approximate technique and the solutions are used to measure nonclassicalities associated with the system. The nonclassicality criteria chosen in this study are experimentally measurable. The analytical solutions are matched with numerical simulations with the help of a numerical toolbox. It is evident from the time developments of the operators that nonclasities, namely squeezing and quantum entanglement are present in the system. The counting statistics for the oscillator with lower free energy and the coupled mode are shown to be sub-Poissonian for a particular set of parameters. Consequently, the system may exhibit quantum mechanical antibunching.

The ubiquitous nature of nonclassical states is apparent in the rapid developments of the fields like quantum computation and communication. In particular, nonclassical states are essential in the studies of quantum cryptography [19,20], dense coding [21,22], quantum teleportation [23,24]. In this paper, the nonclassical properties, namely squeezing, antibunching and quantum entanglement are studied in case of two coupled harmonic oscillators. For a nonclassical state, the Glauber-Sudarshan p-function is more singular than δ-function. Although a single photon state is the most nonclassiacal of the quantum states, it may involve a number of photons [25]. The negativity of Wigner function, Mandel's Q parameter, etc., serve as the criteria for the measurement of nonclassicality. The present study involves some practically measurable nonclassicality criteria.
For light field and harmonic oscillator, theoretical studies on squeezed states are widespread [26][27][28]. They are accompanied by experimental works [29,30]. Squeezed states are relevant for the preparation of Fock states of harmonic oscillator strongly coupled to a single two level atom [31]. These states attract wide interests and fundamental connections between the squeezed state and entanglement are studied [8,[32][33][34][35]. For producing antibunched photons, the single photon sources remain the essential component. Or reversibly, presence of antibunching confirms the single photon emitter in the system. The process like parametric down conversion are used to generate antibunched photons [36][37][38]. Quantum entanglement is the one of the where ω 1 and ω 2 are the natural frequencies of the first and the second oscillator respectively. Here without any loss of generality, the energy of the first one is taken greater than the energy of the second one, i.e., ω 1 >ω 2 . Expressing the transformations of position (x) and momentum (p) in terms of dimensionless bosonic creation and annihilation operators as x a a p i a a , Neglecting the vacuum energy terms and taking ÿ=1 from here through the rest of the study, the Hamiltonian now becomes

Solution
The Heisenberg equations of motion can be derived from the Hamiltonian (4) The above Heisenberg equations of motion were solved at resonant condition [16] and comparisons of dynamical behaviours of various observables have been made between the solutions with or without RWA. The disparity between two results increases with the dimensionless time as expected. In this study the general solutions of equation (5) are achieved from an approximate perturbative approach resembling that in the references [41,42]. The solution of the first of equation ( and b(t) follows in the same manner as a(t). The commutators in equation (6) are evaluated and the terms comprising of a, a † , b, b † or any combination of them are taken up to the desired order of the coupling constant along with their time dependent coefficients. The trial solutions of equation (6) assumes the following form taking up to are the functions of λ 2 whereas f 6 (h 6 ) and f 7 (h 7 ) depend upon λ 3 . This is evident from the sequence of the terms in equation (7) that next two terms would contain only a and a † but are not considered since the coefficients would carry λ 4 . The same reasoning goes with equation (8) also. The initial conditions for the functions f i s and h i s are f 1 (0)=h 1 (0)=1 whereas f i (0)=h i (0)=0 for i=2, 3, 4, 5, 6, 7 and the reason behind this choice is that if there is no coupling constant (λ=0), all the terms other than the first one would vanish. This is equally evident from equation (6) also. The corresponding solutions for f i s and h i s are , Δ=ω 1 −ω 2 and Σ=ω 1 +ω 2 . The validity of the solution is given by the limit λt<1. Therefore the dynamics of various observables are explored throughout the study with λ=1. This is worth noting that the trial solutions (7) and (8)

Particle number dynamics
The initial states of both the oscillator modes are considered coherent. If añ | and bñ | are the eigenkets of the field operators a and b respectively, the eigenvalue equations are a a a a ñ = ñ The initial composite coherent state is given as which gives rise to the following eigenvalue equations for composite system at t=0 The quantities where θ and f are the phases of α and β respectively.
We first consider the mean particle number for the first oscillator mode. The average particle number for the first one is given by The O(λ 2 ) calculation for the above mean particle number is The solution (15) is a better approximation as expected. This is evident from figure 1(a). But both the 3  l ( ) and the 2  l ( )solutions agree well with the numerical simulation in the region of dimensionless time λt<0.1. The simulations are done by the quantum optical toolbox QuTip 3.1.0 [43]. To investigate the unitary time evolution of closed system (4), the toolbox function qutip.essolve is used. Here the exponential-series technique is considered for the time evolution of the initial state of the system. Figure 1(c)) shows that the average particle number is not conserved throughout the time evolution.

Nonclassicalities
As a matter of fact, all the states of light are nonclassical or quantum states. Among all, the single photon state is the most nonclassical one. For coherent states, the Glauber -Sudarshan P function is a δ function, and for all other pure nonclassical states, P function becomes negative for some regions in phase space [44]. Among different nonclassical states, this study considers quadrature squeezing, sub-Poissonian quantum statistics with antibunching and quantum entanglement. Various experimentally realizable criteria for different types of nonclassicality are used in the present study as described below.

Quadrature squeezing
The non vanishing commutator a a , is responsible for the vacuum fluctuation for the electromagnetic field. The fluctuation over one quadrature can be lower than the vacuum fluctuation at the cost of the other one. This phenomenon is known as squeezing. The quadrature defined for squeezing is as follows  18 Squeezing occurs if fluctuation in any of the quadrature in equations (18) and (19) assumes the value lower than the vacuum fluctuation, i.e., X a 2 ) . Using (7), (18) and (19), the simplified formula for the quadrature fluctuations in the first mode is given by The quadrature for the coupled mode squeezing are written as X t a t a t b t b t 1 2 2 22 The coupled mode squeezing is given by the following expression The Time evolution of quadrature variances are shown in figure 2.

Quantum statistics
The second order correlation function for the first mode corresponding to no time delay is given by g a t a t a t a t a t a t 0 . implies the sub-Poissonian distribution of oscillator-mode number and is associated with the nonclassical effect, quantum antibunching [25,45,46]. As is well-known, the distribution Similarly, the condition for b mode is as follows,