Download MISiS-02-08-2015

Динамический эффект Лэмба в
связанной системе
сверхпроводникового кубита и СВЧрезонатора
Д. С. Шапиро, А. А. Жуков, В. В. Погосов,
Ю. Е. Лозовик
Центр фундаментальных и прикладных исследований,
Всероссийский научно-исследовательский институт автоматики им. Н. Л. Духова
(ГК «Росатом»), Москва
НИТУ МИСиС, 2 июля 2015 г.
Outline
• Motivation: cavity QED nonstationary effects
• Main idea: dynamical Lamb effect via tunable
qubit-photon coupling
• Theoretical model: nonstationary Rabi
model beyond RWA
• Results: system dynamics; a method to enhance
the effect
• Summary
Motivation-1
Superconducting circuits with Josephson junctions
- Quantum computation (qubits)
- A unique platform to study cavity QED nonstationary
phenomena
First observation of the dynamical Casimir effect – tuning
boundary condition for the electric field via an additional SQUID
С. M. Wilson, G. Johansson, A. Pourkabirian, J. R. Johansson, T. Duty, F. Nori, P. Delsing,
Nature (2011).
Motivation-2
Casimir effect (1948)
Two conducting planes attract each other
due to vacuum fluctuations (zero-point energy)
Motivation-3
Dynamical Casimir effect: prediction (Moore, 1970)
- Generation of photons (always in couples)
- Difficult to observe in experiments with massive mirrors
- Indirect schemes are needed
Motivation-4
Dynamical Casimir effect: first observation (2011)
tuning an
inductance
- Superconducting circuit system: high and fast tunability!
- Statistics of photon states is in agreement with theory
С. M. Wilson, G. Johansson, A. Pourkabirian, J. R. Johansson, T. Duty, F. Nori, P. Delsing, Nature (2011).
Motivation-5
Natural atom in a nonstationary cavity
А.А. Белов, Ю.Е. Лозовик, B.JI. Покровский, Лэмбовский сдвиг ридберговских
атомов в резонаторе, ЖЭТФ (1989).
N. B. Narozhny, A. M. Fedotov, and Yu. E. Lozovik,
Dynamical Lamb effect versus dynamical Casimir effect, PRA (2001).
Atom can be parametrically excited even if cavity
sizes far exceed quantum-mechanical atom size
Motivation-6
Two channels of atom excitation
(nonadiabatical modulation):
- Absorption of Casimir photons
- Nonadiabatical modulation of atomic level Lamb shift:
dynamical Lamb effect.
Excitation probability is proportional to the square of the Lamb shift modulation.
Lamb shift: tiny shift of atom energy levels due to vacuum
fluctuations. Dominant contribution to the shift is due to the
zero-point fluctuations of photon field.
Lamb shift for artificial macroscopic “atoms” (qubits) is not something illusive.
Moreover, strong coupling regime is possible in contrast to natural atoms.
A. Fragner, M. Goppl, J. M. Fink, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, A.
Wallraff, Science (2008).
Motivation-7
How dynamical Lamb effect can be isolated from
other mechanisms of atom excitation?
Fedotov, Narozhny, Lozovik, PRA (2001)
Difficult to implement !
Our idea is to suggest a realization of the dynamical Lamb effect in
superconducting circuit systems (natural atom  qubit).
Main idea-1
Dynamically tunable qubit-resonator coupling
- In contrast to the optical system with a natural atom, it is possible
to dynamically tune also an effective photon-qubit coupling
A. J. Hoffman, S. J. Srinivasan, J. M. Gambetta, A. A. Houck, “Coherent control of
a superconducting qubit with dynamically tunable qubit-cavity coupling”, PRB (2011).
M. S. Allman, F. Altomare, J. D. Whittaker, K. Cicak, D. Li, A. Sirois, J. Strong, J. D.
Teufel, R.W. Simmonds, “rf-SQUID-Mediated Coherent Tunable Coupling between a
Superconducting Phase Qubit and a Lumped-Element Resonator”, PRL (2010).
S. Zeytino˘glu, M. Pechal, S. Berger, A. A. Abdumalikov Jr., A. Wallraff, S. Filipp
“Microwave-induced amplitude- and phase-tunable qubit-resonator coupling in circuit
quantum electrodynamics”, PRA (2015).
…
- This possibility makes superconducting qubit-resonator system a
very promising candidate for the observation of the dynamical
Lamb effect
Main idea-2
Two possible realizations
• Dynamical coupling of a qubit (at rest !) with two resonators:
nonadiabatic switching.
Similar to the scheme
with a natural atom
• A simplified scheme: dynamical coupling with a single resonator
No Casimir photons!
Theoretical model-1
Rabi model beyond the rotating wave approximation
(one mode photon field)
photons
qubit
Stationary Lamb shift (perturbation theory in V):
coupling
Theoretical model-2
• Qubit-photon coupling:
RWA, conserves excitation number.
Dressed states.
Counter RWA term, a perturbation.
Responsible for the Lamb shift
• V1 -- slow Rabi oscillations of both the photon
number and qubit state
Theoretical model-3
Rotating wave approximation
• Wave function
• Spectrum
Theoretical model-4
Nonstationary system: qualitative picture after the
coupling is turned on
(full resonance)
small parameter:
Fast and slow degrees of freedom:
V1 produces slow Rabi-like
oscillations;
V2 generates fast oscillations
Superposition of slow and fast oscillations
Theoretical model-5
Hamiltonian splitting
Slow oscillations appear automatically
Results-1: single switching
+ full resonance
Probability of a parametric
qubit excitation
Excitation probability is proportional to the square of the Lamb shift modulation
Results-2: single switching
Number of generated photons
An excellent agreement between a simple analytical treatment and numerics
The effect, however, is very weak. Strong coupling regime?
Results-3: parametric driving
-- integrands are not oscillating in sign
• Huge enhancement of excitation probability !
Results-4: parametric driving
Photon states are strongly squeezed
Results-5: parametric driving
First-order photon
correlation function
Second-order photon
correlation function
Results-6: parametric driving
Universal behavior
- Qualitatively correct result, but quantitatively not so
good.
- Universality, however, does exist
Results-7: parametric driving
Squeezing
Results-9: possible experiment
How to observe experimentally?
Photon field:
- Parametric excitation of photons
- Statistics of photon states is different from statistics
for Casimir photons (both even and odd states are
populated)
- Slow Rabi-like oscillations of various photon
characteristics  photons are coupled to a qubit.
Dynamical Lamb effect can be parasitic for a quantum computation:
uncontrollable qubit excitation
Summary
• Dynamical Lamb effect can be realized in
tunable superconducting qubit-resonator
systems
• Theoretical description of system’s evolution
upon the modulation of qubit-resonator
coupling constant
• Parametric pumping as an efficient method to
enhance the effect
D. S. Shapiro, A. A. Zhukov, W. V. Pogosov, Yu. E. Lozovik, Phys. Rev. A 91, 063814 (2015)
• Частота резонатора – 10 GHz
• g – 1-100 MHz
• Decoherence – 1-30 MHz or smaller in new
transmons
• Quality factor 10^4
• Resonator size - centimeter
Дипольное приближение