Effects of Noise in Quantum Entanglement and Coherence Alif KM Akbar Wibowo, Jusak Sali Kosasih
Bandung Institute of Technology
Abstract
In quantum information, quantum entanglement and quantum coherence holds an important role. In preserving the entanglement or the coherence, noise will be taken into account. It happens when a measurement is applied into a quantum system or simply due to the environment surrounding a quantum system. It is important to note that this note will affect the system, namely the entanglement or the coherence. As we know when the system is isolated, the entanglement or coherence would break. However, it is recently known that in a noisy environment, a measurement would instead protect the system. We consider two-level system coupled to an external environemnt. The Hamiltonian of the system is described by the Jaynes-Cummings model coupled to a bosonic bath,
H=H_S+H_B+H_I
where in the rotating-wave approximation, we have
H_S=1/2 ℏ-ω-(t) σ-_Z, H_B=ℏ-∑-_k^ ▒-〖-ω-_k b_k^** b_k 〗-, H_I= ℏ-∑-_k^ ▒-(g_k σ-_+ b_k+g^* σ-_- b_k^** )
We notice that the noise ω-(t) evolves with time. We can further describe it in ω-(t)=ω-_0+β-(t) with β-(t) is the stochastic function representing the noise. To solve the model above, we use a linear non-Markovian stochastic Schrodinger equation which gives the reduced dynamics
d/dt ψ-_t=-iHψ-_t+σ-_- ψ-_t z_t-σ-_+ ∫-_0^t▒-〖-α-(t,s) (δ-ψ-_t)/(δ-z_z ) ds〗-
where α-(t,s)=∑-_k^ ▒-〖-g_k^2 e^(-iω-_k (t-s)) 〗- is the response function and z_t is Gaussian noise of zero mean and correlations M[z_t^* z_s ]=α-(t,s),M[z_t z_s ]=0. Meanwhile, β-(t) is characterized by the statistical mean and correlation 〈-β-(t)〉-=0,〈-β-(t)β-(s)〉-=α-_1 (t,s). The master equation is as follows
d/dt ρ-=-i[H_0,ρ-]+[a,ρ-O ̅-^** ]+[σ-_z,ρ-D ̅-^** ]+h.c.
where H_0=H[β-(t)=0],D ̅-=i∫-_0^t▒-〖-ds α-_1 (t,s) δ-/(δ-β-(s))〗-, and O ̅-=i∫-_0^t▒-〖- α-_1 (t,s) δ-/(δ-β-(s)) ψ-_t 〗-. From this, the mechanism of entanglement/coherence can be studied.
Keywords: Noise, Quantum Coherence, Quantum Entanglement, Quantum Information
