This document discusses the justification of canonical quantization of the Josephson effect and modifications due to large capacitance energy. It presents the commonly used canonical quantization approach and derives the Josephson junction Hamiltonian in second quantization. It also discusses corrections to the Cooper pair box model that arise when the capacitance energy is comparable to the superconducting gap.

1. Justification of canonical quantization of Josephson
effect
(and its modifications due to large capacitance energy)
Krzysztof Pomorski and Adam Bednorz
Uniwersytet Warszawski
Wydział Fizyki
Instytut Fizyki Teoretycznej
E-mail:pomorski@fuw.edu.pl
27 października 2015
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 1 / 29

2. Content
1 Introduction
Simplest model of tunneling JJ
RCSJ model of JJ and concept of washboard potential with
quasiparticles
Phase sc qubit
Charge sc qubit
2 Research results
Commonly used canonical quantization
Second quantization JJ Hamiltonians
JJ non-linear capacitance (combination of I and II quantization)
Literature confirmation
Corrections to Cooper pair box in I quantization
Conclusions
Future persepctives
Literature
Technical issues
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 2 / 29

3. Tunneling Josephson junction:simplest model
Rysunek : Tunneling Josephson junction (JJ)[from F.Nori, Nature 2005] :
H = HL + HR + HT with
HL = EL|L >< L|, HR = ER |R >< R|, HT = ET (|R >< L| + |L >< R|) with
|ψ >= ψL|L > +ψR |R >, ER , EL >> ET .
H|ψ >=
EL ET
ET ER
ψL
ψR
=
i
d
dt
ψL
ψR
(1)
We obtain DC and AC Josephson relations having ∆φ = φR − φL = 2φ
with I = I0 sin(2φ) and 2e
d
dt 2φ(t) = V (t).
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 3 / 29

4. RCSJ model and concept of washboard potential.
I(t) = I0 sin(2φ) +
2e
1
R
2dφ
dt
+
2e
C
d22φ
dt2
(2)
0 =
d
2dφ
U(2φ) +
2e
1
R
2dφ
dt
+
2e
C
d22φ
dt2
, where (3)
U(2φ, t) = I0 cos(2φ) − 2φI(t) is washboard-potential.
Obvious analogies with anharmonic mechanical oscillator! For small
Josephson junction R → ∞.
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 4 / 29

5. Concept of washboard potential+quasiparticles:phase qubit
We set R → ∞ and ∆φ = 2e x2
x1
Ax (x)dx quite much equivalent to
London relation j = const × A.
H|2φ >= ((
i
d
2dφ
)2
−I ×2φ+EJ cos(2φ))|2φ >= (H0+U(2φ))|2φ > (4)
Operators 2φ, i
d
2dφ play role x and p. The commutation relation
[x, p] = i is analogical to [ i
d
2dφ , 2φ] = i .
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 5 / 29

6. Superconducting Cooper pair box qubit
Rysunek : Schematics of single Cooper pair box, from Physica Scripta 77,
V.Bouchiat, 1997
N|n >= n|n >, Hint = −EJ (|i >< i+1|+|i+1 >< i|), Hint = −EJcos(2φ),
(5)
[n, 2φ] = i , n =
i
d
2dφ
(6)
H = EC (N − n0)2
+ Hint (7)
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 6 / 29

7. 3 basic superconducting qubit architectures
Rysunek : From ’Physics Today’ F.Nori 2005
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 7 / 29

8. Structures considered-Josephson junction biased by
different circuits
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 8 / 29

9. Commonly used canonical quantization of Josephson
effect, which was never fully justified!
In first quantization we have:
HC = Q2
/2C = CV 2
/2 (8)
where Q is the charge and V is the voltage between 1 and 2.
HL = LI2
/2 = ( φ/e)2
/2L (9)
HJ = −e I1 cos(2φ)/2 (10)
The canonical quantization means that we take the Hamiltonian
H(φ, Q) = HC (Q) + HL/I (φ) + HJ(φ) (11)
as a function of conjugate variables φ and Q and replace everything by
their quantum counterparts , including conversion of Poisson bracket
(φ, Q) = e/ into commutator between [φ, Q] = ie. It essentially means
Q → e∂/i∂φ.
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 9 / 29

10. Hamiltonian of the point-like tunneling JJ in second
quantization (and 2 parameters Θ and φ) .
Having Hamiltonian parts responsible for bulk superconductor hBCS and
non-superconducting region heT (in second quantization using simplified
Bogoliubov-de Gennes formalism) of the following form
hBCS =
−i∂x ∆
∆ i∂x
eh
, heT = −τδ(x)
0 eiφ
e−iφ 0
12
. (12)
Since HT transfers between 1 and 2 the states θ are in different
superconductors. In momentum space we have k = ∆ sinh θ and
|θ± = (2 cosh θ)−1/2 e±θ±/2
±e θ±/2 (13)
in the eh basis, with ±(θ) = ±∆ cosh θ and Ee = +(θ+) − −(θ−).
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 10 / 29

11. We want to obtain/ compare it with Josephson junction Hamiltonian in
first quantization
H = HL + HJ + HC . (14)
We define
H = H − HL = +HJ + HC (15)
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 11 / 29

12. The full eigenproblem reads
E|ψ = H|ψ (16)
where the zero order state is |g with energy 0. Since HT is small, all
states |gφ will also reduce to |g in zero order. Therefore we can write an
ansatz for |ψ in the form
|ψ = dφ(ψ(φ)|gφ +
e
ψe(φ)|eφ ), (17)
where the latter sum is of higher order. Assuming explicit knowledge of
EC = − e2
2C ∂2
φ with E = E − HL/I = EJ − e2
2C ∂2
φ and plugging this form
into the eigenproblem we get
H |ψ >= E ψdφ|gφ +
e
E ψe|eφ =
dφ Egφ −
e2
2C
∂2
φ ψ|gφ = +
e
Eeφ −
e2
2C
∂2
φ ψe|eφ (18)
. Sandwiching it with gφ| we get
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 12 / 29

13. Non-linear capacitance in tunneling JJ
E ψ = Egφψ −
e2
2C
∂2
φψ − (∂φψ)(e2
/C) gφ|∂φ|gφ
−ψ(e2
/2C) gφ|∂2
φ|gφ −
e2
2C e
gφ|∂2
φψe|eφ . (19)
The factor gφ|∂φ|gφ is an analogue to the differential Berry phase, which
can be arbitrarily chosen. For our purpose, it is convenient to assume that
it is zero. Only relative Berry phase for excited states would matter but
only at high perturbation order. In the lowest order, we obtain
E ψ Egφψ −
e2
2C
∂2
φψ − ψ(e2
/2C) gφ|∂2
φ|gφ . (20)
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 13 / 29

14. Simple identities
The term gφ|∂2
φ|gφ can be evaluated inserting identity between
derivatives,
gφ|∂2
φ|gφ = gφ|∂φ|gφ gφ∂φ|gφ +
e
gφ|∂φ|eφ eφ|∂φ|gφ (21)
since we assume zero differential ground Berry phase, and from
orthogonality between g and e we get
gφ|∂2
φ|gφ = −
e
| gφ|∂φ|eφ |2
= −
e
| eφ|∂φ|gφ |2
(22)
.
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 14 / 29

15. Definition of c modifying factor
One can write down c(φ) in terms of first-quantized (oc-cupied and
em-pty) single particles states
c(φ) =
oc,em
| ψem|∂φ|ψoc |2
. (23)
One can also use the adiabatic identity
gφ|∂φ|eφ =
gφ|(∂φHT )|eφ
Eeφ − Egφ
(24)
because only HT depends on φ.
Here time is replaced with phase difference and adiabatic approximation is
taken!
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 15 / 29

16. Reference to Berry phase and scattering matrix
For the nondegenerate eigenvalues the differential Berry phase is defined by
ψ|∂φ|ψ = iγ (25)
where γ is real because 0 = ∂φ ψ|ψ = i(γ − γ∗). For degenerate case we
collect all the states of the same energy.
We construct matrix γ with
ψm|∂φ|ψn = iγmn (26)
Note that 0 = ∂φ ψm|ψn = i(γmn − γ∗
nm) which shows that γ is
Hermitian.
Therefore for scattering matrices S
iγ = S†
∂φS/2 (27)
because the outgoing waves are halves of full waves.
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 16 / 29

17. Numerical evaluation of c(φ)
0 0.2 0.4 0.6 0.8 1
t
0
0.2
0.4
0.6
0.8
1
1.2
1.4
φ
0
0.5
1
1.5
2
2.5
3
c(φ)
Rysunek : The exact dependence of c(φ) on φ ∈ [0, π/2] and t referenced at
φ = 0, c is even in φ and has the period π.
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 17 / 29

18. Literature confirmation
The next order term renormalizes capacitance
e2
/2C → e2
/2C − t2
(3 − cos 2φ)e4
/32C2
(28)
Note that higher order terms will also contain boundary deviations of ∆
near the tunneling point (also in Egφ). However, our correction remains
dominant in the case of many independent channels. Anyway, one can
calculate exact c(φ) analytically for all t (not only small) assuming
constant (bulk) ∆,
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 18 / 29

19. Large capacitance energy and Cooper pair box qubit
Rysunek : The relevant states of occupation for the Cooper pair box.
H|n > EC (N − n0)2
− EJ
i
(|i >< +2| + |i + 2 >< i|). (29)
By diagonalization, the difference of the energy levels of the qubit reads
2 4N2
g E2
C + E2
J .
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 19 / 29

20. Corrections to Cooper pair box energy
We work in the base |N > explicitly assume that Josephson energy is of
the form
HJ =
N
EJ(|N + 2 N| + |N N + 2|) (30)
In the vicinity of Ng = 0 the relevant states are |N = ±1 = |± and the
total effective Hamiltonian can be written as
H = (N2
g + 1)EC + 2Ng EC (|− −| − |+ +|) + EJ(|+ −| + |− −|) (31)
The qubit operation range is achievable if Ng EC ∼ EJ even if EC is large
(for sufficiently small Ng ). By diagonalization, the difference of the energy
levels of the qubit reads 2 4N2
g E2
C + E2
J .
However, if EC ∼ ∆, the above approach is incorrect because the
Josephson term is different.
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 20 / 29

21. The modified Josephson energy reads
EJ =
t
2t2
∆2 dθ+dθ−
(2π)2
(∆(cosh θ+ + cosh θ−) − EC )−1
(32)
.
0.5 1.0 1.5 2.0
EC
1.5
2.0
2.5
E’J EJ
Rysunek : The dependence of EJ /EJ on EC /∆ in the Cooper pair qubit. At
EC → 2∆ there is logarithmic divergence
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 21 / 29

22. Conclusions
We show that this approach is indeed correct in certain range of
parameters. We find the condition of the validity of such quantization and
the lowest corrections to the Josephson energy.
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 22 / 29

23. Future perspectives:
1 Extension results to triplet superconductor.
2 Extension results to multiband superconductor.
3 Extension results to triple superconductor Josephson junction.
4 Accounting for effects in flux-phase Josephson junction.
5 Josephson effect in superconducting quark-gluon plasma?
6 Considerations of structure Sc-qdot-Sc.
7 Considerations of Sc-Fe-Sc tunneling Josephson junction.
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 23 / 29

24. References
1. B.D.Josephson, Phys. Lett. 1, 251 (1962)
2. M.Thinkham-Introduction to superconductivity (2004)
3. K.Pomorski, A.Bednorz, http://arxiv.org/abs/1502.00511 (2015)
4. U.Eckern, G.Schon, V. Ambegaokar, Phys. Rev. B 30 6419 (1984)
5. G.Schon, and A.D.Zaikin, Phys. Rep. 198, 237 (1990)
6. A.Barone- ’Physics and applications of Josephson effect’
7. F.Nori- ’Physics today’ 2005
8. P.de-Gennes-’Superconductivity of Metals and Alloys’
9. M.V.Berry, 1984, Proc. R. Soc. London, Ser. A 392, 45 10.
’Superconducting Qubits: A Short Review’, M. H. Devoret, A. Wallraff, J.
M. Martinis
11. G.E.Blonder, M.Tinkham, T.M.Klapwijk, BTK paper, Physical Review
B, vol. 25,no. 7, pp. 45154532, 1982.
12. F.Romeo, R.Citro , Physical Review B 91, 035427 (2015) Minimal
model of point contact Andreev reĆection spectroscopy of multiband
superconductors
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 24 / 29

25. Computed scattering matrix
Normal scattering matrix is then obtained from solutions of the
eigenproblem for (12)
Se =
r iteiφ
ite−iφ r
(33)
with t = sin(τ/ ) and r =
√
1 − t2,Sh = ST
e . One can add the overall
phase eiα/ to the scattering matrix by modifying hT → hT + αδ(x)
(accounting interface properties) but it will not change any of our results
and hence we can safely disregard it.
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 25 / 29

26. Scattering matrix S
Therefore we finally get detailed version of scattering equation multiplied
by M as
4MS =
r sinh2
θ t sinh θ(i sinh θ cos φ − cosh θ sin φ)
t sinh θ(i sinh θ cos φ + cosh θ sin φ) r sinh2
θ
t2
sin φ(− cosh θ sin φ − i sinh θ cos φ) −rt sinh θ sin φ
rt sinh θ sin φ t2
sin φ(− cosh θ sin φ + i sinh θ cos φ)
t2
sin φ(− cosh θ sin φ + i sinh θ cos φ) −rt sinh θ sin φ
rt sinh θ sin φ t2
sin φ(− cosh θ sin φ − i sinh θ cos φ)
r sinh2
θ t sinh θ(−i sinh θ cos φ − cosh θ sin φ)
t sinh θ(−i sinh θ cos φ + cosh θ sin φ) r sinh2
θ
, where M = sinh2
θ + t2
sin2
φ.
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 26 / 29

27. The scattering matrix connects the modes amplitudes
Ao =
ARo
ALo
=





A1Ro
A2Ro
A1Lo
A2Lo





= S





A1Li
A2Li
A1Ri
A2Ri





= S
ALi
ARi
= SAi (34)
S1 =
Se 0
0 Sh
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 27 / 29

28. Rysunek : Energy spectrum of JJ, with the continuum parts above +∆ and below
−∆ and two phase-dependent ABSs in the gap. Possible excitations of the
ground state: (A) between ABSs, (B) between an ABS and continuum, (C)
between lower and upper continuum.
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 28 / 29

29. Analytical evaluation of c(φ)
In this approximation
c(φ) 2t2 dθ−dθ+
(4π)2
|e(θ+−θ−)/2+iφ − e(θ−−θ+)/2−iφ|2
(cosh θ+ + cosh θ−)2
. (35)
By introducing variables 2s = θ+ + θ− and 2w = θ+ − θ− we get
c(φ) t2 dsdw
(2π)2
sinh2
w + sin2
φ
cosh2
s cosh2
w
. (36)
K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 29 / 29