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In this paper, the master equation for the coupled lossy waveguides is solved using the thermofield dynamics (TFD) formalism. This formalism allows the use of the underlying symmetry algebras SU(2) and SU(1, 1), associated with the Hamiltonian of the coupled lossy waveguides, to compute entanglement and decoherence as a function of time for various input states such as NOON states and thermal states.

Recently, there has been a lot of interest in studying entanglement using coupled waveguides [

For the efficient use of these waveguides in the field of quantum information, the generated entanglement should not decohere with time [

Thermofield dynamics (TFD) [

The basic formalism of TFD is as follows: corresponding to the creation and annihilation operators

where,

where the state

In TFD, the master equation is

where,

In optical communications, coupled waveguides are used as transmission medium. The linear coupling between two wave guides is used to transfer the power one wave guide to another. To study the decoherence and entanglement properties of the two coupled waveguides, the model of Rai and Agarwal [

The Hamiltonian described by,

where mode a corresponds to first wave guide and mode b corresponds to the second waveguide. The mode a and b obey bosonic commutation relations. The evanescent coupling in terms of distance between the two wave guide is given by J. The density operator has a time evolution given by

In the presence of a damping term (system reservoir interaction), the evolution equations are governed by the Liouville equation

where,

where

In absence of loss, the Heisenberg Equations for the field operators give their time evolution as,

To study the entanglement and decoherence properties as function of time for the coupled wave guide system, we solve the exactly master Equation (7) using TFD. This allows us to study the response to the coupling of different input states such as number, NOON and thermal states, in the coupled waveguide system. In particular, we will show that in the absence of damping, an input vacuum state evolves into a two mode SU(2) coherent state. In the presence of damping, we show that the vacuum state evolves into a two mode squeezed state and a thermal state into a thermal squeezed state.

The time evolution of the density operator corresponding to the two coupled waveguides without damping consisting of fields in the modes a and b from Equations (4), (5) is determined by,

Now we apply the TFD-formalism (Equation (3)), by doubling the Hilbert space and get the Schrodinger type wave equation

here, the Hamiltonian

and can be decoupled into non-tildian and tildian parts:

where,

Then the solution of Equation (9) is given by

where,

It is clear from the above that, the two Hamiltonians H and

To calculate the decoherence and entanglement properties of the coupled lossy wave guide as a master Equation (8), (equivalently, the Schrodinger like Equation (9)), the underlying symmetries associated with the Hamiltonians (Equations (11), (12)) are used. To see this symmetries explicitly we define the following operators.

which satisfy the SU(2) algebra,

with number operator,

The Hamiltonian from Equation (11) in terms of the SU(2) generators is,

Hence, the underlying symmetry of the Schrodinger like Equation (9) is

Using the disentanglement formula [

here,

The density matrix in the number state basis is given by

where,

The entanglement properties are calculated by taking the partial transpose of

The eigenvalues are

For a bipartite system, the entropy is defined as the von-Neumann entropy of the reduced density matrix traced with respect to one of the systems as

such that

We can also quantify the entanglement of the system by studying the time evolution for the logarithmic negativity [

where

Now we consider various cases of optical input states.

Case-1: For two photon system as an input (i.e., N = 2):

The entropy of entanglement of the two photon input state (both

which is shown in dotdashed curve of

Now we consider the logarithmic negativity of the various two photon states, as this is different for various two photon states, unlike the entropy.

Case-1 (a). If we take the input state as

which is shown in thick curve of _{N} increases with time and attains a maximum value of 1.32875 for Jt = 0.42879, this is the maximally entangled state. Further, for Jt = 0.785212 we get the dips at E_{N} = 1 (the coincidence rate of the output modes of the beam splitter will drop to zero, when the identical input photons overlap perfectly in time), due to Hong-Ou-Mandel interference [_{N} vanishes. At later times, we see a periodic behavior, attributed to the inter-waveguide coupling (J).

Case-1 (b). Now we take the input state as

which is shown in dotted curve of _{N} increases and attains a maximum value of 1.32193 at Jt = 0.785212, then decreases and eventually becomes equal to zero at Jt = 1.57061. Thus the state

becomes disentangled at this point of time. At later times we see a periodic behavior and the system gets entangled and disentangled periodically. Unlike the earlier case for the

Case-1 (c). For two photon input NOON state:

Then the logarithmic negativity of this state is

which is shown in thick curve of _{N} increases with time and attains a maximum value of 1.32875 for Jt = 0.356988, which is the point of maximal entanglement. Further, for Jt = 0.785212, E_{N} vanishes (we will see later that as we increase the number of photons in the NOON state, the entanglement does not vanish). At later times, we see a periodic behavior, attributed to the inter-waveguide coupling (J).

Case-2: For four photon system as an input (i.e., N = 4):

The entropy of entanglement for four photon system:

This is shown in dotted curve of

Now we consider the logarithmic negativity for each of the four photon states

Case-2 (a). For the input state as

Then the log negativity entanglement of this system is,

which is shown in thick curve of _{N} = 1.15181 and for Jt = 0.325477 dips at E_{N} = 1 due to Hong-Ou-Mandel interference [_{N} increases with time and attains a value of 1.77519 for Jt = 0.601094, this is the maximally entangled state, and for Jt = 0.785212 again we get dips at E_{N} = 1.7277 due to Hong-Ou- Mandel interference. At later times, we see a periodic behaviorr, attributed to the inter-waveguide coupling (J). Because of involvement of four photons, we can see the double the interference effect of the two photon system.

Case-2 (b). For the input state as

The Entanglement for this system is,

which is shown in dotted curve of

Case-2 (c). Now we take the input state as

The Entanglement for this system is,

which is shown in thin curve of

Case-2 (d). For four photon input NOON state:

The logarithmic negativity for this system is,

which is shown in dotdashed curve of

where,

curve of

For five photon input NOON state, the entanglement for this system is,

and

The Logarithmic entropy is shown in thin curve of

Now we consider the initial state

where,

The entanglement properties are calculated by taking the partial transpose of

The von-Neumann entropy of the reduced density matrix in terms of

Case-3 (a). For two photon system as an input (i.e., N = 2):

The entropy of entanglement for two photon is

which is shown in Figures 2(a)-(c).

Case-3 (b). For four photon system as an input (i.e., N = 4):

Then the entropy of entanglement for four photon system is

which is shown in Figures 2(d)-(f).

These figures show that for low values of

We consider now, losses in the coupled waveguides due to system-reservoir interaction with “g” as the rate of loss due to the material of the waveguide. The time evolution of the density operator equation is

with,

the following transformations,

gives

We diagonalise this Hamiltonian by applying a squeezing (Bogolubov) transformation mixing the real and tilde fields.

where,

and

where,

and

and satisfy the commutation relations

The Casimir operators are

Then the solution of Equation (45) becomes

where,

By using the SU(1, 1) disentanglement formula [

here,

with

subscript i labels A, B.

We consider an initial state

where

The entanglement properties are calculated by first taking trace of

The entropy of entanglement of the the system is

Thus, for two photon state as an input state, the entropy of entanglement of the system is

which is shown in thin curves of Figures 3(a)-(d).

For four photon state as an input state, the entropy of entanglement of the system,

which is shown in thick curves of Figures 3(a)-(d). When

Since the state is Gaussian, we can use the covariance matrix method to calculate the entanglement of the sys- tem by using Simon’s criterion [

Then, the covariance matrix is:

where,

Since the tildian fields are fictitious, we trace over them to get the covariance matrix for the physical modes,

The canonical form of covariance matrix is given by,

where,

Then the separablility condition [

The symplectic eigenvalues are defined as,

where,

and

The entanglement of the system is

For

In order to quantify the decoherence effects, we compute

The behaviour of decoherence is plotted

Taking the initial state

where,

Applying Simon’s criterion Equation (77) we see that the system is entangled iff

For

In this paper, we have shown that the formalism of thermofield dynamics is a powerful tool for exact studies of coupled waveguide systems. Indeed, we have exactly solved the master equation associated with SU(2) and

SU(1, 1) symmetries for coupled lossy waveguides with and without damping. For coupled waveguides without damping, special attention has been given to the time evolution of the NOON states as inputs and we have shown that as we increase the photon number, the entanglement of the NOON states survives with time, thus making them extremely suitable for quantum information. The solution for damped systems was obtained by transforming the master equation to a Schrodinger type equation and applying the disentanglement formulae for SU(2) and SU(1, 1). Our work extends that of Rai et al. [

M.N.K. wishes to acknowledge CSIR-UGC for a JRF fellowship. KVSSC acknowledges the Department of Science and technology, Govt of India, (fast track scheme (D. O. No: SR/FTP/PS-139/2012)) for financial support. We wish to thank Prof. C. Mukku and Prof. S. Chaturvedi for insightful comments. We also wish to thank V. Srinivasan for introducing us to the details of thermofield dynamics.

Naveen KumarMogurampally,K. V. S. ShivChaitanya,Bindu A.Bambah, (2015) Quantum Entanglement in Coupled Lossy Waveguides Using SU(2) and SU(1, 1) Thermo-Algebras. Journal of Modern Physics,06,1554-1571. doi: 10.4236/jmp.2015.611158