1.
Measurement-induced long-distance entanglement of superconducting
qubits using optomechanical transducers
Ondřej Černotík and Klemens Hammerer
Institute for Theoretical Physics, Institute for Gravitational Physics (Albert Einstein Institute), Leibniz University Hannover, Germany
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Circuit QED
Optomechanical transduction
Entanglement by measurement
Adiabatic elimination
Force sensing
Results
Max Planck Institute
for Gravitaitonal Physics
(Albert Einstein Institute)
Acknowledgements
We thank Pertti Hakonen for useful discussions. This work was funded by
the European Commission (FP7-Programme) through iQUOEMS (Grant
Agreement No. 323924). We acknowledge support by DFG through QUEST
and by the cluster system team at the Leibniz University Hannover.
Email: Ondrej.Cernotik@itp.uni-hannover.de References
[1] R. Andrews, et al., Nature Physics 10, 321 (2014).
[2] T. Bagci, et al., Nature 507, 81 (2014).
[3] K. Stannigel et al., PRL 105, 220501 (2010).
[4] N. Roch et al., PRL 112, 170501 (2014).
[5] O. Černotík, D. V. Vasilyev, and K. Hammerer, PRA 92, 012124 (2015).
[6] O. Černotík and K. Hammerer, arXiv:1512.00768.
Superconducting structures offer a promising platform for
quantum computation. Due to transition frequencies in
the microwave range, up-conversion to optical
frequencies is needed to enable long-distance signal
transmission.
Mechanical oscillators couple to both microwave and
optical fields. They are thus suitable for converting signal
from one spectral range to the other [1, 2] and can be
used to connect superconducting systems via a room-
temperature environment [3].
With the coupling , the qubit exerts a force on the
mechanical oscillator. This force can be measured with an optomechanical system;
the qubit states can be distinguished with measurement time
At the same time, the mechanical bath disturbs the qubit, leading to its dephasing at a rate
The measurement needs be faster than this dephasing which is achieved for strong
optomechanical cooperativity,
The system is too complicated to be
treated analytically or numerically, so
we adiabatically eliminate the
transducer dynamics. We use the fact
that the dynamics is Gaussian and can
be described using the first and second
statistical moments of its canonical
operators.
0 1 2 3 4 5
Time (µs)
0.0
0.4
0.8
Concurrence
Psucc=0.1
Psucc=0.5
0.0 0.2 0.4 0.6 0.8 1.0
Optical transmission τ
0.0
0.2
0.4
0.6
0.8
Concurrence
η=1
η=0.6
η=0.2
0.0 0.2 0.4 0.6 0.8 1.0
Optical transmission τ
Psucc=0.1
Psucc=0.5
Dispersive interaction of light with a qubit results in a phase
shift on the field. After a sequential interaction with two
qubits, the measurement of phase reveals the total spin,
. Starting from the initial two-qubit state
with , the entangled state can
be prepared conditionally, as demonstrated by Roch et al. [4].
Equation of motion
Formally, the system can be described using the stochastic master equation
with the Hamiltonian
describes decoherence of the qubits,
is a Lindblad term, and accounts for the
effect of the measurement.
The scheme requires only moderately large optomechanical cooperativity and can
be implemented with present technology. Remarkably, the protocol tolerates
significant amount of optical loss. See [6] for more details and other experimental
implementations.
Effective dynamics
After adiabatically eliminating the transducer degrees of freedom, the
effective equation of motion is
with
Optical loss can also be included in the model. Imperfect transmission between
the systems introduces additional dephasing of the first qubit while losses
after the second system decrease the detection efficiency , see [6] for details.
The covariance matrix of the conditional transducer
state then obeys a deterministic Riccati equation which can be
solved efficiently [5].