1. Dynamics of a superconducting qubit coupled to quantum two-level systems in its environment
Robert Johansson
(RIKEN, The Institute of Physical and Chemical Research, Japan)
In collaboration with:
Sahel Ashhab (RIKEN)
Alexandre Zagoskin (UBC, RIKEN, Loughborough U.)
Franco Nori (The University of Michigan, RIKEN) 1

2. Overview
Decoherence of qubit coupled to two-level systems (TLS) Physica C 444,
Driven dynamics of qubit coupled to a TLS New J. Phys. 8, 103.
Using TLSs with long coherence times as qubits.
PRL 97,
Robert Johansson
11/29/2019

3. Overview
Decoherence of qubit coupled to two-level systems (TLS) Physica C 444,
Driven dynamics of qubit coupled to a TLS New J. Phys. 8, 103.
Using TLSs with long coherence times as qubits.
PRL 97,
Robert Johansson
11/29/2019

4. Superconducting phase qubit
Energy
Phase across
Josephson junction
Robert Johansson
Physica C 444, 45-52
11/29/2019 4

5. Evidence of TLSs in the environment
Strong coupling
Long coherence times
Bias current
Simmonds et al., PRL (2004)
Cooper et al., PRL (2004)
Robert Johansson
Physica C 444, 45-52
11/29/2019 5

6. What if there are no visible splittings, but many invisibly small splittings?
Small splittings should still contribute to relaxation.
Each TLS contributes a small relaxation peak at its natural frequency.
Total noise spectrum could still look smooth.
Also weakly coupled TLSs could have long coherence times
Bias current
Simmonds et al., PRL (2004)
Robert Johansson
Physica C 444, 45-52
11/29/2019 6

7. Quantum degrees of freedom in the environment
If an environmental degree of freedom has a longer decoherence time than its coupling strength to the qubit, it must be treated as a quantum object, i.e., the noise power spectrum is not enough to describe it.
Model of a qubit coupled to a single TLS:
We must treat this combined system quantum mechanically.
Robert Johansson
Physica C 444, 45-52
11/29/2019 7

8. Model of a qubit coupled to a TLS
The qubit is in resonance with the TLS:
We use a Bloch-Redfield master equation without the secular approximation to model the background decoherence rates and for the qubit and TLS:
Solving for and tracing out the TLS gives the qubit dynamics:
and the corrections to the qubit decoherence rates can be extracted.
Robert Johansson
Physica C 444, 45-52
11/29/2019 8

9. Correction to qubit relaxation rate due to the TLS
Perturbation theory in quantum picture
Traditional weak-coupling approximation
(noise power spectrum approach)
Shnirman et al., PRL (2005)
The two approaches differ when the condition:
is not satisfied.
(Similar results for correction to the dephasing rate)
Robert Johansson
Physica C 444, 45-52
11/29/2019

10. Relative corrections to relaxation rate as function of time
Transient:
Longer Þ More memory Þ Non-markovian behavior
(Similar results for dephasing rates)
Robert Johansson
Physica C 444, 45-52
11/29/2019 10

11. Overview
Decoherence of qubit coupled to two-level systems (TLS) Physica C 444,
Driven dynamics of qubit coupled to a TLS New J. Phys. 8, 103.
Using TLSs with long coherence times as qubits.
PRL 97,
Robert Johansson
11/29/2019

12. Characterizing the TLSs
Distribution of:
resonance frequencies
energy splittings
can be measured in experiments:
Martinis et al., PRL (2005)
Is it possible to find the distribution of values of n(q) and n(TLS)?
Robert Johansson
New J. Phys. 8, 103
11/29/2019 12

13. Driven Qubit+TLS system
Let’s take:
and small l
q=angle between n and z
Robert Johansson
New J. Phys. 8, 103
11/29/2019 13

14. Driven Qubit+TLS system
Let’s take:
and small l
Rabi resonance peaks at each eigenfrequency:
q=angle between n and z
Robert Johansson
New J. Phys. 8, 103
11/29/2019 14

15. Qubit+TLS energy levels: how to probe them
Time
driving frequency
driving frequency
Robert Johansson
New J. Phys. 8, 103
11/29/2019 15

16. Effects of decoherence
- Oscillations are damped in the time domain, with an exponentially decaying envelope function. Resonance peaks are widened in the frequency domain.
Solid line: no decoherence.
Dashed line: strong TLS decoherence Þ TLS becomes weakly coupled.
Dotted line: moderate decoherence on both. Þ Narrow features are suppressed.
Dash-dotted line: strong qubit decoherence. Þ Qubit cannot perform Rabi oscillations.
Robert Johansson
New J. Phys. 8, 103
11/29/2019 16

17. Overview
Decoherence of qubit coupled to two-level systems (TLS) Physica C 444,
Driven dynamics of qubit coupled to a TLS New J. Phys. 8, 103.
Using TLSs with long coherence times as qubits.
PRL 97,
Robert Johansson
11/29/2019

18. Can we use coherent TLSs as qubits?
Superconducting phase qubit
TLS impurities
in the junction
Josephson
junction
Experiments show that some environmental TLSs have coherence times comparable to, or even longer, than the qubit.
Question: Can we use those TLSs as qubits?
Warning: We have no control over the parameters of TLSs. They are formed accidentally during fabrication.
Robert Johansson
Phys. Rev. Lett. 97,
11/29/2019 18

19. Can we use coherent TLSs as qubits?
Superconducting phase qubit
TLS impurities
in the junction
Josephson
junction
Experiments show that some environmental TLSs have coherence times comparable to, or even longer, than the qubit.
Question: Can we use those TLSs as qubits? Yes, in principle.
We do not need to control the TLS parameters, we only need to be able to measure them.
Robert Johansson
Phys. Rev. Lett. 97,
11/29/2019 19

20. TLS parameters
Need to know: resonance frequencies and coupling strengths for the different TLSs.
Simmonds et al., PRL (2004)
Cooper et al., PRL (2004)
Robert Johansson
Phys. Rev. Lett. 97,
11/29/2019 20

21. Single qubit gate on the TLSs
Qubit in ground state at t=0
Time
Manipulate
qubit
Or: If you manipulate the qubit very fast, you do not need to go out of resonance in the middle.
Robert Johansson
Phys. Rev. Lett. 97,
11/29/2019 21

22. i-SWAP operation on TLSs
Time
Qubit in ground state at t=0
Single qubit operations + i-SWAP = universal set of gates
Robert Johansson
Phys. Rev. Lett. 97,
11/29/2019 22

23. Conclusions
We find corrections to relaxation and dephasing rates of a qubit weakly coupled to a TLS with long coherence rates.
We identify features in the dynamics of a driven qubit coupled to a TLS that can be used to characterize the TLS.
We suggest that TLSs with long coherence times can be used as qubits, by using the Josephson junction qubit as a bus.
Robert Johansson
11/29/2019