Dynamics of entanglement in two coupled qubits

by Vivek M. Aji, Joel E. Moore

Journal club talk by Asma Al-Qasimi
Summary by So Takei



Today's discussion was led by Asma Al-Qasimi from quantum
optics who has recently joined our journal club. She
presented an article by Vivek Aji and Joel Moore which
discusses the time-evolution of entanglement between two
interacting qubits each coupled to a dissipative bath
(arXiv:cond-mat/0312145). Asma began by identifying
entanglement as the defining difference between a quantum
and a classical computer. She then stressed the difficulty
in preserving entanglement and its extreme fragility to
various sources of decoherence. She emphasized the particular
importance of considering decoherence effects in solid state
qubits because they are always subject to interactions with
a multitude of quantum degrees of freedom. Indeed, solid state
qubits generally possess much shorter decoherence times relative
to qubits implemented in, for example, cavity quantum
electrodynamics and ion traps. The speaker then underlined the
main topic of the article: the time evolution of an entangled
state of two solid state qubits in the presence of dissipation.

Asma proceeded by introducing the solid state qubit considered
in the article called the flux qubit, or the persistent current
qubit. This qubit consists of a superconducting loop with typically
three Josephson junctions that encloses a flux supplied by an
external magnetic field. By appropriately tuning the three
Josephson coupling energies and the magnetic field, the effective
low-energy physics of the system is engineered to possess two
metastable states, |0> and |1> with opposite circulating persistent
current. Tunnel splitting between the two states is determined by the
offset from (1/2)Phi_0 of the flux, where Phi_0 is the superconducting
flux quantum.

The article mutually couples two flux qubits via a Heisenberg-type
coupling of strength J. Assuming that the source of dissipation
arises from the noise in the external flux, the article couples
the z-component of the pseudospin to a bath of oscillators, which
models the environment. The primary motivation of the work is
to investigate the two qubit system with entanglement of formation
as a diagnostic. At t=0, the qubits are prepared in an entangled
state |3> in Eq.2.

The speaker then presented three main results from the work.
In the first case with J=0, the work reveals an interesting
speed up in the decay rate of entanglement relative to the
decoherence rate of individual qubits. The speed up factor
is explicitly derived to be 2/log(2). The decoherence rate is
found to be linearly proportional to temperature and the
strength of dissipation. A question was asked whether the
decay rate of entanglement is generally faster than the
decoherence rate. The speaker supported the generality of the
result by alluding to her previous studies of entanglement.

Second, she discussed the time evolution of entanglement for
a fixed temperature T and various J's. The work finds that
for larger J entanglement is completely lost in finite time.
This result seemingly implies that J encourages loss of
entanglement. However, an intriguing revival of entanglement
for large J is observed for long times. A question was asked
whether coherence also behaves in a similar oscillatory fashion.
The answer was yes. However, it was noted during the discussion
that coherence is not necessarily completely lost even at
times where entanglement is zero.

Finally, the presenter discussed the dynamics of entanglement
for a fixed J and various T's. Here, the dependence for the
second case is reproduced but for higher temperatures the long
time decay increases rapidly while the initial loss of
entanglement is hardly affected. A question was raised whether
qubits with Ising symmetry were considered. The answer was yes;
in the Ising case, no revival in entanglement is observed for
a fixed J and various T's.