Quantum entanglement in coupled lossy waveguides using SU(2) and SU(1,1) Thermo-algebras

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.


Introduction
Recently, there has been a lot of interest in studying entanglement using coupled waveguides [1,2].
Specially designed photonic waveguides have provided a laboratory tool for analyzing coherent quantum phenomena and have a possible application in quantum computation [3]. The entanglement between waveguide modes is at the heart of many of these experiments and has been widely studied [4]. Using coupled silica waveguides Politi et.al. [5] have reported control of multiphoton entanglement directly on-chip and demonstrated integrated quantum metrology opening the way for new quantum technologies. They have been able to generated two and four photon NOON states on the chip and observed quantum interference, which further enhances the capabilities for quantum interference and quantum computing. Among various types of entangled states, NOON states are special with two orthogonal states in maximal super position thus enhancing their use in quantum information processing [6].
For the efficient use of these waveguides in the field of quantum information, the generated entanglement should not decohere with time [7]. It is well known that losses have a substantial effect on the wave guides. Therefore, the time evolution of the entanglement is of interest in the context of quantum information processing using lossy waveguides. The entanglement between waveguide modes and how loss affects this entanglement has recently been studied by Rai et.al. [4], by using a quantum Liouville equation. In this paper, we approach this problem from the viewpoint of thermofield dynamics [8]. This formalism has the advantage of solving the master equation exactly for both pure and mixed states, converting thermal averages into quantum mechanical expectation values, by doubling the Hilbert space. The formalism thus makes it simpler to calculate the effects of noise and decoherence in the coupled two mode waveguide system. We look at the effect of different type of input states and show the efficiency of the states for quantum information theory.

Thermofield dynamics
Thermofield dynamics(TFD) [8,9,10,11,12] is a finite temperature field theory. It has been applied to many branches of high energy physics and many-body systems. TFD has been used to solve the master equation, which helps in studying the temporal evolution of entanglement. In particular, it has been used to study entanglement in the presence of a Kerr medium using an SU(1,1) disentanglement theorem for arbitrary initial conditions by reducing it to a Schrodingerlike equation [13,14]. Thus, all the techniques available to solve the Schrodinger equation can be used to solve the master equation. We derive the effect of losses giving rise to decoherence by solving the master equation exactly, using TFD, and then compute the decoherence and entanglement properties of some two mode waveguide systems.
The basic formalism of TFD is as follows: corresponding to the the creation and annihilation operators a † and a, which act on the physical space H, we introduce "thermal or tildian" operatorsã † andã, which act on an augmented(fictitious) spaceH [15,16,17]. The operators a and a † commute withã andã † and the sets of basis vectors {|n >} and {|ñ >} span the Hilbert spaces H andH, respectively, and {|n > ×|m >} = |n,m > are the basis operators of the doubled Hilbert space. The Identity operator is |I >= |n > ⊗|ñ >= |n,ñ >. We define a temperature dependent " thermal vacuum state", where, tanh θ(β) = e −βω/2 and the corresponding thermal number distribution isn(β) = 1/(e βω − 1). The statistical average of any operator is equal to the vacuum expectation value of the operator with respect to the thermal vacuum. In particular, the density operator ρ acting on a Hilbert space H is a state vector |ρ α >, 1 2 ≤ α ≤ 1 in the extended Hilbert space, so that averages of operators with respect to ρ are expressed as scalar products, where ,the state |ρ α >=ρ α |I > andρ α = ρ α ⊗ I. Although, in some applications of TFD the α = 1 2 formalism is used, in this paper we work with the α = 1 formalism such that < A >< I|A|ρ >=< A|ρ > In TFD, the master equation is

Coupled waveguide system
In optical communications, coupled waveguides are used as a 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 [4] is used.
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 In the presence of a damping term(system reservoir interaction), the evolution equations are governed by the Liouville equation where, where γ is dissipation in the material of the waveguide.
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, in the absence of damping, an input vacuum state evolves into a two mode SU(2) coherent state. In the presence of damping, the vacuum state evolves into a two mode squeezed state and a thermal state into a thermal squeezed state.

Two Coupled waveguides without damping(γ = 0)
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 eqs. (4,5) is determined by, Now we apply the TFD-formalism(eq.(3)), by doubling the Hilbert space and get the Schrodinger type wave equation here, the HamiltonianĤ is given bŷ and can be decoupled into non-tildian and tildian parts: Then the solution of eq.(9) is given by where, |ρ(0) > is an intial state in H ⊗H.
It is clear from the above that, the two Hamiltonians H andH are independent. Hence, we can work with one of the Hamiltonians but since we are only interested in the physical states, we trace over the tilde states.
To calculate the decoherence and entanglement properties the master equation (8),is solved using the underlying symmetries associated with the Hamiltonians(eqns (11,12)) . To see these symmetries explicitly we define the following operators, which satisfy the SU (2) algebra, with number operator, N = a † a + b † b.

|α(t)|
The density matrix in the number state basis is The entanglement properties are calculated by taking the partial transpose of ρ and computing the eigenvalues of the resulting matrix [20] : C na,na C * ma,ma |n a , N − m a >< m a , N − n a | + C ma,ma C * na,na |m a , N − n a >< n a , N − m a |. (21) The eigenvalues are such that ρ a = T r b (ρ) and {λ i } correspond to the set of eigenvalues of the reduced density operator. For the general number state, with the eigenvalues given by eqns (22,23) the entropy is We can also quantify the entanglement of the system by studying the time evolution for the logarithmic negativity [21], which is an easily computable measure of distillable entanglement. For a bipartite system described by the density matrix, ρ the log negativity is, where ρ T is the partial transpose of ρ and the symbol denotes the trace norm. Also N (ρ) is the absolute value of the sum of all the negative eigenvalues of the partial transpose of ρ. The log negativity is a non-negative quantity and a non-zero value of E N would mean that the state is entangled.

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 |1, 1 > and |2, 0 > ) is The logarithmic negativity of the various two photon states is different for various two photon states, unlike the entropy.
Case-1(a.) If we take the input state as |ψ in >= |1, 1 >, the output state is Then we can write the log negativity entanglement of this system(from eq.(26)) as, which is shown in thick curve of Fig Case-1(b.) Now we take the input state as |ψ in >= |2, 0 >, the output state is The log negativity for this system is , and is shown in dotted curve of Fig.1(a). In this case, E 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 |1, 1 > input state, we do not see any interference effects. Clearly the entanglement dynamics of the states |1, 1 > and |2, 0 > are different.
Case-1(c.) For two photon input NOON state: The logarithmic negativity of this state is shown in thick curve of Fig.1(c). At time t = 0, we begin with an entangled |2002 > state as input and thus the value of log negativity is one. We see periodic dips due to the Hong-Ou-Mandel interference, which is characteristic of NOON states. E 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 Then the log negativity entanglement of this system , E N = log 2 [1 + 2(a 1 a 2 + a 1 a 3 + a 1 a 4 + a 1 a 5 + a 2 a 3 + a 2 a 4 + a 2 a 5 + a 3 a 4 + a 3 a 5 + a 4 a 5 ] (32) is shown in thick curve of Fig.1(b). The system starts at t = 0, with a separable input state and  The Entanglement for this system is, and is shown in dotted curve of Fig.1(b). We see the small difference in interference pattern with the |2, 2 > state. The Hong-Ou-Mandel effect and the maximally entangled state occur at lower values.
Case-2(c.) Now we take the input state as |ψ in >= |4, 0 >, theoutput state is The Entanglement for this system is shown in thin curve of Fig.1(b).
For five photon input NOON state, the entanglement for this system is, E N = log 2 [1+2(e 1 e 2 +e 1 e 3 +e 1 e 4 +e 1 e 5 +e 1 e 6 +e 2 e 3 +e 2 e 4 +e 2 e 5 +e 2 e 6 +e 3 e 4 +e 3 e 5 +e 3 e 6 +e 4 e 5 +e 4 e 6 +e 5 e 6 )]  Fig.1(c). We see that as the photon number increases the NOON state gets more and more robust and shows that "high-noon" states can be used for more

Entanglement properties for Input Thermal States
Now we consider the initial state |ρ(0) > to be the two mode thermal state. Then, in the TFD formalism the time evolved state ρ(t) is The corresponding von-Neumann entropy is: Case-3(a.) Two photon system input(i.e., N = 2): The entropy of entanglement for the two photon thermal input state is and is shown in Figs.2(a,b,c). These figures show that for low values ofn a andn b , the system sustains entanglement, but as the system gets more thermalized, it decoheres and the entropy of entanglement tends to zero. The thermal effect mimics a damping effect in the system.

Two coupled waveguides with damping(γ = 0)
We consider now, losses in the coupled waveguides due to system-reservoir interaction with 'γ' as the rate of loss due to the material of the waveguide. The time evolution of the density operator equation is the following transformations, give the Hamiltonian We diagonalise this Hamiltonian by applying a squeezing (Bogolubov) transformation mixing the real and tilde fields. where, and r 1 and r 2 are the squeezing parameters and |µ 1 | 2 −|ν 1 | 2 = 1 and |µ 2 | 2 −|ν 2 | 2 = 1. The final Hamiltonian is written as,Ĥ where, and The generators of the SU(1,1) algebra in terms of the modes A and B are given by and satisfy the commutation relations The Casimir operators are Then the solution of eq.(45) becomes where, By using the SU (1, 1) disentanglement formula [18], one can write eq.(61) as, here, subscript i labels A, B.
We consider an initial state |ρ(0) >= N m,n ρ m,n (0)|m,m, n,ñ >, in TFD notation. This gives us an exact solution of the density matrix : where C(t)is an overall phase factor due to . This is the exact solution for the density matrix of the coupled lossy system of waveguides.

Calculation of Entanglement of the System for γ = 0
The entanglement properties are calculated by first taking trace of ρ(t) over the tilde space and then taking the partial transpose of ρ and calculating the eigenvalues of the resulting matrix similar to the case without damping. The eigenvalues are 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 and is shown in thin curves of Figs.3(a,b,c,d).
For a four photon state as an input state, the entropy of entanglement of the system, is shown in thick curves of figs3(a,b,c,d). When γ goes to zero, we get the same entropy in fig1(d) (without damping). As we increase the value of γ, we can see the damping effect, for four photon system there is more damping than the two photon system. So, one can say that as we increase the input photons, the system will decohere more. This can also be seen by calculating the decoherence parameter.
Since the state is Gaussian, we can use the the covariance matrix method to calculate the entanglement of the system by using Simon's criterion [24]. The density matrix can be written as where S(r) is the squeezing matrix and R(φ) is the rotation matrix mixing real and tilde fields. In our case,θ = 45 o (see in eqs(47, 48)) and 'r' is squeezing parameter(see in eqs(52, 53)), and ρ(0) is the initial state of two mode system. Now we take the initial state ρ(0) to be the two mode vacuum state, |ρ(0) >= |0, 0,0,0 >. To calculate the entanglement of the time evolved state ρ (t) we go over to phase space description by following transformations, Then, the covariance matrix is: where, p = e 2iJt e −2γt cosh2r 1 , q = e 2iJt e −2γt cosh2r 2 , s = e −2γt sinh2r 1 , t = e −2γt sinh2r 2 .
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 [24] for any two mode state is The symplectic eigenvalues are defined as, where,∆ = Detα + Detβ − 2Detγ = 2(p + q)(p * + q * ) + 2(s + t) 2 The entanglement of the system is For r 1 = r 2 = r, the entanglement for two mode vacuum states without damping(i.e., γ = 0) is shown in fig4(a) and with damping(i.e., γ = 0) is shown in fig4(b). We see that as damping increases the entanglement decreases, but, for low damping, the system seems to sustain entanglement to a large extent, so that it is quite robust for applications.
In order to quantify the decoherence effects, we compute ρ 2 as The behaviour of decoherence is plotted figs5. We have considered two cases fig5(a), shows the variation of decoherence with time for strong coupling for various values of γ and fig5(b), shows the evolution of decoherence with weak coupling. For strong coupling, the system decoheres in an oscillatory fashion and saturates to a non-zero value, while for weak coupling, one that for even short times, as the value of damping coefficient increases the system decoheres, to a very low value, very fast.
Applying Simon's criterion eq.(77) we see that the system is entangled iff (n 1 + n 2 ) 4 e −4γt [cosh 2 2r − sinh 2 2r) 2 + 1 16 ≥ (n 1 + n 2 ) 2 2 e −2γt [cosh 2 2r + sinh 2 2r) For r 1 = r 2 = r, and n 1 = n 2 = n , this condition is satisfied for the values of r given in the figure 5, in which we plot the logarithmic negativity as a function ofn, for different values of r. We see that as the system not only gets less entangled for high values of γ (quantified by r), but also for large n(external heat bath). So that in the presence of a heat bath the effect of damping increases and both have to be considered when generating entanglement in the lab by using coupled cavities.

Conclusion
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 al [4], as it gives the exact solution for the master equation, and, in addition shows how the entanglement behaves for input thermal states. Our results have also shown that the entanglement of the system can withstand a certain amount of damping, suggesting that it can be used for applications such as quantum computation, even if the waveguides are lossy. Furthermore we have shown the effect of an external heat bath on the system, by applying our methods to thermal input states. Our method shows the usefulness of thermofield dynamics in quantum entanglement problems, quite orthogonal to the approach given in ref. [25], and allows us to handle damping in entanglement generation properties. We propose to apply this formalism to coupled light-atom systems, to shed further light on the effect of damping on the generation of entanglement.