Transfer of entanglement from a Gaussian field to remote qubits Myungshik Kim Queen’s University, Belfast UniMilano 14 December 2004

In collaboration with Mauro Paternostro Wonmin Son Helen McAneney, Ebyung Park, Pamela Wilson, Mark Tame, Jingak Jang

Why entanglement in two remote qubits? Quantum teleportation to send Quantum repeater Distributed quantum computation NON LOCAL CNOT Eisert,Jacobs, Papadopoulos,Plenio, PRA 62, (2000); Collins,Linden,Popescu, PRA64, (2000) Motivation 1

Why Gaussian? Gaussians are as natural as orange juice and sunshine Motivation 2

Contents Generation of entanglement between two quibits in a cavity Entanglement transfer through local reservoirs Remarks

Generation of entanglement Two atoms in a cavity of a single-mode field

When the cavity is in a thermal field MSK,Lee, Ahn, Knight., PRA 65, (R) (2002) Generation of entanglement Two atoms in a cavity of a single-mode field Interaction time entanglement Even when a many-mode thermal field is concerned, entanglement is generated. Braun, PRL 89, (2002)

Generation of entanglement Two atoms in a cavity of a single-mode field

Generation of entanglement Two atoms in a cavity of a single-mode field

Erasure of “which-path information” effectively entangles two qubits (Bose al,(‘99), Browne et al, (‘03)). Duan et al. Nature 414, 413 (2001) √

via an entangled system entangler

Entanglement transfer through Gaussian fields local environment a local environment b Q1Q1 Q2Q2 driving field: broadband squeezed field qubit-bosonic mode interaction: Paternostro, Son, Kim, PRL 92, (2004) Kraus & Cirac, PRL92, (2004) Rabi oscillation depends on the photon number t p Q uestions Qubits are located in respective cavities When the channel is mixed Entanglement in the steady state – related to minimum control To include spontaneous emission

Evolution of the qubits Conditions driving-field carrier frequency resonant with local modes and its bandwidth is larger than the cavity decay rate  Then we use second-order perturbation theory Markov approximation To get the master equation for the field inside the cavities Adiabatic elimination of the local field modes: Find the evolution of the qubits 4x4 Kossakowski matrix Benatti et al.,PRL(03), Ficek&Tanas, Phys.Rep.(02)

Entanglement condition Consider the Gaussian channel with its variance matrix + qubits prepared in the ground state THEN..still a bit obscure...BUT: consider the uncertainty principle for the driving field. This can be written as then it is

Entanglement condition Two remote qubits can be entangled, at some instant of time of their interaction with a correlated Gaussian channel, if and only if the channel itself is entangled. Example:  1 =  2, n=m and solve the ME. Then c=1.58 n=2.4 c=1.804 c=2.18 n=2.4 c=2.18 c=1.804 c=1.58 initial state, c=1.58 n=2.4 c=1.804 c=2.18 n=2.4 c=2.18 c=1.804 c=1.58 initial state

Channel entanglement condition Steady state entanglement The two qubits are entangled at their steady-state if and only if c > c ss For  1 =  2, n=m it is Kraus & Cirac, PRL92, (2004) Paternostro, Son, Kim, PRL 92, (2004)

Entanglement transfer through Gaussian fields local environment a local environment b Q1Q1 Q2Q2 driving field: broadband squeezed field qubit-bosonic mode interaction: Paternostro, Son, Kim, PRL 92, (2004) Kraus & Cirac, PRL92, (2004) Rabi oscillation depends on the photon number t p Q uestions Qubits are located in respective cavities  When the channel is mixed  Entanglement in the steady state  – related to minimum control  To include spontaneous emission

Summary Entanglement transfer from a Gaussian field to a qubit system Entangled remote local reservoirs suggested Entanglement condition obtained