The document discusses the investigation of quantum coherence in arrays of superconducting qubits, focusing on dynamic localization effects within these systems. It presents mathematical models and results demonstrating how oscillatory behaviors of current and mean-square displacement are influenced by specific parameters like electric field modulation and dephasing rates. Furthermore, it highlights the potential of using current measurements to estimate coherence and incoherence within the superconducting qubit chain.

1. Probing quantum coherence in arrays of superconducting qubits
Alexandra M. Liguori, Susana F. Huelga, Martin B. Plenio
Institut f¨ur Theoretische Physik, Universit¨at Ulm
15th
March 2011, DPG, Dresden
A. Liguori Probing quantum coherence in arrays of superconducting qubits

2. Outilne
1 Original dynamic localisation effect
2 Dynamic localisation in superconducting qubit chains as tool to evaluate
coherence in system
A. Liguori Probing quantum coherence in arrays of superconducting qubits

3. Dynamic localisation on infinite chain
Time-dependent Hamiltonian ( = 1)
H(t) = V
m=+∞
m=−∞
(σ+
mσ−
m+1 + σ−
mσ+
m+1
) −
m=+∞
m=−∞
(E0 + E1) cos(ωt)σ+
mσ−
m ,
V coupling strength between nearest-neighbours
σ±
m = (σx
m ± iσy
m)/2 acting on m-th site (σx
and σy
Pauli matrices)
E0 energy difference between adjacent sites, ω frequency
Mean-square displacement as function of E1 oscillates sinusoidally
Interaction picture → effective Hamiltonian with JE0
ω
Bessel function
Heff
I = V
m=+∞
m=−∞
JE0
ω
(
E1
ω
)(σ+
mσ−
m+1 + σ−
mσ+
m+1) ,
RESULTS [Dunlap&Kenkre, PRB (1986); Holthaus&Hone, Phil. Mag. B (1996)]
Argument of Bessel function is oscillatory function of time, with oscillation
frequency proportional to magnitude of electric field, i.e.
if E0 = nω, n ∈ R, and E1/ω is a zero of Bessel function J
then mean-square displacement oscillates sinusoidally ⇒ initially localised
particle remains localised ⇔ DYNAMICAL LOCALISATION
A. Liguori Probing quantum coherence in arrays of superconducting qubits

4. Dynamic localisation on infinite chain
Time-dependent Hamiltonian ( = 1)
H(t) = V
m=+∞
m=−∞
(σ+
mσ−
m+1 + σ−
mσ+
m+1
) −
m=+∞
m=−∞
(E0 + E1) cos(ωt)σ+
mσ−
m ,
V coupling strength between nearest-neighbours
σ±
m = (σx
m ± iσy
m)/2 acting on m-th site (σx
and σy
Pauli matrices)
E0 energy difference between adjacent sites, ω frequency
Mean-square displacement as function of E1 oscillates sinusoidally
Interaction picture → effective Hamiltonian with JE0
ω
Bessel function
Heff
I = V
m=+∞
m=−∞
JE0
ω
(
E1
ω
)(σ+
mσ−
m+1 + σ−
mσ+
m+1) ,
RESULTS [Dunlap&Kenkre, PRB (1986); Holthaus&Hone, Phil. Mag. B (1996)]
Argument of Bessel function is oscillatory function of time, with oscillation
frequency proportional to magnitude of electric field, i.e.
if E0 = nω, n ∈ R, and E1/ω is a zero of Bessel function J
then mean-square displacement oscillates sinusoidally ⇒ initially localised
particle remains localised ⇔ DYNAMICAL LOCALISATION
A. Liguori Probing quantum coherence in arrays of superconducting qubits

5. Dynamic localisation on finite chain
Finite chain interacting with environment ( = 1)
RESULTS [Vaziri&Plenio, New J. Phys. (2010)
if E0 = nω, n ∈ R, and E1/ω is a zero of Bessel function J
⇒ mean-square displacement oscillates sinusoidally → suppression of
transport for some values of modulation E1 & initially localised particle
remains localised;
effective coupling rates from averaging over transition amplitudes ⇒
suppression of transport is coherence effect due to destructive interference;
with dephasing noise oscillations still exist but amplitude decreases with
increasing dephasing.
A. Liguori Probing quantum coherence in arrays of superconducting qubits

6. Superconducting qubits
Effective two-level systems with a controllable transition frequency between their
eigenstates → general superconducting qubit Hamiltonian
Hqubit = −Ezσz
+ Xcontrol σx
(1)
→ depending on form of Ez and Xcontrol , superconducting qubit can be of charge,
phase or flux type.
Flux or persistent current qubit
A superconducting loop interrupted by three Josephson junctions, two with
capacitance C1 and the third with C2
Josephson junctions coupling constants J1 and J2
Value of external magnetic flux Φ = 0.5Φ0 (Φ0 = h/2e superconducting flux
quantum) ⇒ either in the right-hand or in the left-hand current state
Appropriate choice of parameters J1,2 and C1,2 ⇒ tunneling between the two
classical states can occur
Xcontrol from (1) is tunneling amplitude ∆
Energy splitting Ez = 2Ip(Φ − 0.5Φ0)
proportional to detuning Φ − 0.5Φ0, with Ip circulating current.
A. Liguori Probing quantum coherence in arrays of superconducting qubits

7. Dynamic localisation in a chain of superconducting qubits
OUR MODEL: chain of interacting superconducting qubits
H(t) = E1 sin(ωt)
N
i=1
σz
i +
N
i=1
∆iσx
i +
N−1
i=1
Ji,i+1σz
i ⊗ σz
i+1 , (2)
Lsource(̺) = γ1(2σ+
1
σ−
1 ̺σ+
1
σ−
1 − {σ+
1
σ−
1 , ̺} + 2σ−
1 σ+
1
̺σ−
1 σ+
1
− {σ−
1 σ+
1
, ̺}) ,
Lsink (̺) = γN(2σ−
N̺σ+
N
− {σ+
N
σ−
N, ̺} + 2σ+
N
̺σ−
N − {σ−
Nσ+
N
, ̺}) ,
N qubits in chain
E1 field modulation
∆i tunneling amplitude for each qubit
Ji,i+1 coupling between qubits i and i + 1
γ1, γN, rates of source and sink respectively
A. Liguori Probing quantum coherence in arrays of superconducting qubits

8. Dynamic localisation in a chain of superconducting qubits
Quantum dynamical eq. for finite chain interacting with environment ( = 1)
d̺
dt
= −i[H, ̺] + Lsource(̺) + Lsink (̺) + Ldeph (̺)
with
Ldeph (̺) = γdeph
N
i=1
(2σ+
i
σ−
i ̺σ+
i
σ−
i − {σ+
i
σ−
i , ̺})
γdeph rate of dephasing noise.
Study current I as function of field modulation E1 in (2):
I = lim
t→∞
dpsink
dt
(t)
with psink (t) =
T
0
2γN̺N,N (t)dt, ̺N,N reduced density matrix of last site of chain.
A. Liguori Probing quantum coherence in arrays of superconducting qubits

9. Dynamic localisation in a chain of superconducting qubits
Quantum dynamical eq. for finite chain interacting with environment ( = 1)
d̺
dt
= −i[H, ̺] + Lsource(̺) + Lsink (̺) + Ldeph (̺)
with
Ldeph (̺) = γdeph
N
i=1
(2σ+
i
σ−
i ̺σ+
i
σ−
i − {σ+
i
σ−
i , ̺})
γdeph rate of dephasing noise.
Study current I as function of field modulation E1 in (2):
I = lim
t→∞
dpsink
dt
(t)
with psink (t) =
T
0
2γN̺N,N (t)dt, ̺N,N reduced density matrix of last site of chain.
A. Liguori Probing quantum coherence in arrays of superconducting qubits

10. Results
N = 3 superconducting qubits with experimental parameters given by group of
Prof. J. E. Mooij in Delft:
tunnelings ∆1 = 8.9 GHz, ∆2 = 18.2 GHz, ∆3 = 6.4GHz;
persistent currents I
p
1 = 411 nA, I
p
2 ≃ 350 nA, I
p
3 = 456 nA;
inter-qubit coupling J12 = J23 = J = 200 MHz.
Current I as function of modulation E1:
100 102 104 106 108 110
5
6
7
8
x 10
−10
E1
(*1010
sec−1
)
I(sec−1
)
γdeph
=0
γdeph
=0.01
γdeph
=0.02
DYNAMIC LOCALISATION
EFFECT: oscillating behaviour of
current ⇒ strongly suppressed
transport for some values of E1;
amplitudes of oscillations decrease
with increasing dephasing rates
γdeph ⇒ use current variations to
estimate coherence or incoherence
in the system.
A. Liguori Probing quantum coherence in arrays of superconducting qubits

11. 5 5.2 5.4 5.6 5.8 6
6.7
6.8
6.9
7
7.1
7.2
I (sec*10
−10
)
C(*10
−4
) Incoherence measure C [Vaziri&Plenio,
New J. Phys. (2010)]:
C =
k l
|̺k,k ̺l,l − ̺k,l̺l,k |
⇒ C as function of I for given value of E1
at which first resonance can be found
Conclusions
1 DYNAMIC LOCALISATION IN SUPERCONDUCTING QUBIT CHAIN:
oscillating behaviour of current ⇒ strongly suppressed transport for some
values of E1;
2 amplitudes of oscillations decrease with increasing dephasing ⇒ use current
variations to estimate coherence or incoherence in the system;
3 (2) ⇒ incoherence measure C can be used as effective tool to estimate
presence of coherence in superconducting qubit chain by measuring current
I at fixed E1.
A. Liguori Probing quantum coherence in arrays of superconducting qubits

12. 5 5.2 5.4 5.6 5.8 6
6.7
6.8
6.9
7
7.1
7.2
I (sec*10
−10
)
C(*10
−4
) Incoherence measure C [Vaziri&Plenio,
New J. Phys. (2010)]:
C =
k l
|̺k,k ̺l,l − ̺k,l̺l,k |
⇒ C as function of I for given value of E1
at which first resonance can be found
Conclusions
1 DYNAMIC LOCALISATION IN SUPERCONDUCTING QUBIT CHAIN:
oscillating behaviour of current ⇒ strongly suppressed transport for some
values of E1;
2 amplitudes of oscillations decrease with increasing dephasing ⇒ use current
variations to estimate coherence or incoherence in the system;
3 (2) ⇒ incoherence measure C can be used as effective tool to estimate
presence of coherence in superconducting qubit chain by measuring current
I at fixed E1.
A. Liguori Probing quantum coherence in arrays of superconducting qubits