Persistent Currents in Superconducting Quantum Interference Devices
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PERSISTENT CURRENTS IN SUPERCONDUCTING QUANTUM INTERFERENCE 
DEVICES 
F. Romeo 
Dipartimento di Fisica “E. R. Caianiello”, Università degli Studi di Salerno 
 I-84081 Baronissi (SA), Italy 
R. De Luca 
CNR-INFM and DIIMA, Università degli Studi di Salerno 
 I-84084 Fisciano (SA), Italy 
ABSTRACT 
Starting from the reduced dynamical model of a two-junction quantum interference device, a 
quantum analog of the system has been exhibited, in order to extend the well known properties of 
this device to the quantum regime. By finding eigenvalues of the corresponding Hamiltonian 
operator, the persistent currents flowing in the ring have been obtained. The resulting quantum 
analog of the overdamped two-junction quantum interference device can be seen as a supercurrent 
qubit operating in the limit of negligible capacitance and finite inductance.  
PACS: 74.50.+r, 85.25.Dq 
Keywords: Josephson junctions, d. c. SQUID, Qubit 
I   INTRODUCTION 
The d. c. SQUID (Superconducting QUantum Interference Device) is a well known system, 
widely investigated in the literature [1-3]. This system, though not confined to atomic scale in its 
dimensions, has been proposed as the basic unit for quantum computing (qubit) by resorting to a 
characteristic feature of superconductivity: macroscopic quantum coherence [4]. In general, a qubit 
can be realized by means of a two-level quantum mechanical system [5]. Therefore, the quantum 
states of a qubit can be a linear combinations of the orthogonal basis 0  and 1 , so that the Hilbert 
space generated by this basis is two-dimensional. Alternatively, a qubit state can be represented by 
elements of an infinite-dimensional Hilbert space. In this case, however, the effective potential of 
the system must show a double-well potential, in such a way that one of the two stationary states can 
be defined as state 0  and the other as state 1 . 
The electrodynamic properties of d. c. SQUIDs can be analyzed by means of two-junction 
quantum interferometer models, where each Josephson junction is assumed to be in the overdamped 
regime. The simplest possible analysis of these systems is done by assuming negligible values of the 
inductance L of a single branch of the device, so that 0
β , where 0Φ  is the elementary flux 
quantum, and 
21 JJ
=  is the mean value of the maximum Josephson currents of the junctions. 
In this case, the dynamical equation for the superconducting phase differences 1φ  and 2φ  across the 
two junctions can be written as a single equation for the average phase variable 
21 φφϕ
= . This 
equation is similar to the nonlinear differential equation governing the time evolution of a single 
overdamped junction, so that it can be defined as an equivalent single junction model, and is written 
as follows: 
sincos Bex
=+ ϕπψ
,       (1) 
where t
τ , with 
21 RRR
= , 1R  and 2R  being the resistive junction parameters, exψ  is 
the externally applied flux normalized to 0Φ  and Bi  is the bias current normalized to JI . 
Following the same type of approach, by means of a perturbation analysis, taking β  as the 
perturbation parameter, it can be shown that, to first order in β , the equivalent single junction 
model can be written as follows for a symmetric SQUID with identical junctions [6]:  
( ) ( ) ( )
2sinsinsincos1 2 Bexex
=Ψ+Ψ−+ ϕππβϕπ
,    (2) 
where n is an integer. This model allows, at least for small values of the parameter, to calculate in 
closed form some electrodynamic quantities, such as, for example, the amplitude of the half-integer 
Shapiro steps appearing in these systems [7]. It has also been shown that, by extending this model to 
SQUIDs with non-identical junction, one can obtain an effective classical double-well potential in 
which the transition from one state to the other can be enhanced by applying an opportune external 
magnetic flux [8]. However, this classical analysis by itself does not allow to define the quantum 
states of the system. Nonetheless, the equation of the motion (2) could be assumed to be a classical 
version of the time evolution of a quantum phase state. Therefore, the aim of the present work is to 
obtain, starting from the time evolution of the superconducting phase difference ϕ , the quantum 
mechanical Hamiltonian and to compute, by means of this quantum mechanical system, which in the 
classical limit reduces to Eq. (2), the persistent currents in the SQUID. It is interesting to notice that 
the resulting “Hamiltonian” quantum model derives from the overdamped limit of a classical 
dissipative system in the presence of a double well potential. The present analysis can be seen as an 
alternative approach to the study of the quantum properties of supercurrent qubits: It allows to study 
the response of the quantum system in the limit of negligible capacitance and finite values of the 
inductance, as opposed to the case usually considered in the literature, where negligible inductance 
and finite capacitance is assumed [9 - 10].   
II   FROM CLASSICAL TO QUANTUM MECHANICS 
Let us consider the classical dynamical equation ( )xfx =&  for the state variable x, where the dot 
notation indicates the derivative with respect to the normalized time τ. Making use of the previous 
equation, taking the time derivative of both sides, we can obtain the equation of the motion of the 
quantity x&  as follows: 
( ) ( ) ( )xfxfxxfx xx == &&& ,      (3) 
where the notation ( )xf x  stands for the partial derivative of ( )xf  with respect to x. Given the 
above equation and following the procedure described by Huang and Lin [11], the Lagrangian 
associated to this problem is obtained in the form: 
( )( )[ ]22
xfxL += & .      (4) 
Starting from the Lagrangian L, the Legendre transformation allows us to get the following classical 
Hamiltonian: 
( )( )[ ]22
xfH −= π ,      (5) 
where 
=π  is the canonical momentum conjugated to the variable x, while ( ) ( )( )2
xfxU −=  
could be considered as an effective potential. We are now interested in the quantization of the 
classical model described so far. According to the standard procedure, the recipe to transform the 
classical Hamiltonian in a quantum operator is implemented by making the substitution 
xi∂−=→ππ ˆ , xxx =→ ˆ  (in dimensionless units). From the previous definitions, the 
commutation rule [ ] ix =π̂,ˆ  for the conjugated variables follows directly. Furthermore, the 
Hamiltonian operator can be written as   ( )( )[ ]22
1ˆ xfH x +∂−= . 
The general procedure described above can be adopted to obtain the quantum model of an 
overdamped d.c. SQUID in the limit in which the reduced two-junctions interferometer model [6] 
can be applied. In the framework of this model, the phase dynamics can be written (in the 
homogeneous case) as in Eq. (2), so that the function ( )ϕff =  takes the following form: 
( ) ( ) ( )ϕϕγϕ 2sinsin baf −−= ,    (6) 
where ( )exa πψcos= , ( )exb πψπβ 2sin= , 2
Bi=γ , having chosen 0=n  for simplicity. We notice 
that this analysis cannot be extended to the similar case, considered by Grønbech-Jensen et al. [12], 
of junctions with finite capacitance. Therefore, by setting ϕ=x  and ϕπ &=  in the above general 
analysis, we notice that the phase and the voltage across the two-junction quantum interferometer 
are conjugate variables of the system. In the present case, therefore, proceeding as we said, by 
squaring ( )ϕf  and exhibiting the final result of the calculation in terms of the higher harmonics of 
the phase variable instead of powers of trigonometric functions, the  following Hamiltonian 
operator is obtained: 
                           ( ) ( ) ( ) ( )
( ) ( ) ( )γϕγϕγ
ϕϕϕϕϕ
,,2sinsin
babaab
= ,                     (7) 
where ( )
222 ba
γ  is a flux dependent energy shift which will be important in the 
following discussion. 
In order to calculate the relevant physical quantities of the system, we introduce the orthonormal  
complete basis 
n =  of the infinite-dimensional Hilbert space with the inner product 
== −∫ , where mn,δ  is the Kronecker delta. In this representation the matrix 
elements of the Hamiltonian operator can be written as follows: 
( ) ( ) ( )
( ) ( ) ( ) ( )2,2,1,1,4,4,
1,1,,
+−+−+−+−
−+−+++++
++++−
mnmnmnmnmnmnmnmn
mnmnmnmnmnmn
aabban
δδδδδ
, (8) 
where the following useful relations have been used: 
( ) ( )lmnlmninlm +− −= ,,2
sin δδϕ ,     (9a) 
( ) ( )lmnlmnnlm +− += ,,2
cos δδϕ .     (9b) 
Once the matrix elements of the Hamiltonian operator are known, we can diagonalize a reduced 
version of the complete infinite-dimensional matrix by introducing an energy cut-off. Such 
procedure can be safely carried out when we need to characterize low energy states which are 
located very far from the cut and when the number of the vectors in the basis of the reduced Hilbert 
space is able to capture the essential features of the low energy states. For instance, the Hilbert 
space spanned by the first 20 basis functions can be a very effective choice, if we need to study only 
the lowest energy states close to the ground state of our system. In fact, in our case we have noted 
that, by halving the number of the basis elements, no evident difference is present in the lowest 
energy eigenvalues.  
III   PERSISTENT CURRENTS 
Following the procedure described above, in the present section we shall derive the behavior of 
the persistent currents associated to each eigenstate of the Hamiltonian as a consequence of the time 
reversal symmetry breaking provided by the magnetic flux.  Such a current, in units of the 
Josephson current divided by π2  (i. e., in units of 
JE , where JE  is the Josephson energy), can be 
defined as follows: 
−= ,      (10) 
where nε  and exψ  are the eigenvalues of the Hamiltonian and the normalized external magnetic 
flux, respectively. According to the above relation, the persistent current nI  can be computed once 
the pertinent eigenvalue nε  is known. Furthermore, it should be noticed that, in the absence of the 
off-diagonal terms in the Hamiltonian given in Eq. (8), the state independent persistent current 
computed by means of Eq. (10) would be given by: 
                                    ( ) ( )[ ]exexI πψβππψπ 222 sin212sin4 −−= .                                                  (11) 
The solution of the full problem can thus be seen as the state dependent correction to the above 
relation induced by the off-diagonal terms. Last point can be well understood by analyzing Figs. 1a-
b. In these figures, even tough we are in the presence of finite off-diagonal terms, the relation given 
in Eq. (11) is able to describe quite accurately the behavior of the persistent currents which appears 
insensitive to the state index due to the small value of β . When the value of β  is raised (see Fig. 
2a), the persistent currents starts to become weakly state sensitive and some deformation of the 
original shape occurs. Furthermore, the states of higher energy (see Fig. 2b) induce a behavior of 
the persistent current which is quite insensitive to the state index.  A further raising of β  (see Fig. 
3a) induces a suppression of the persistent current carried by the first excited state in the vicinity of 
half integer values of the normalized applied magnetic flux. This implies that, in the low energy 
regime (i. e., when the quantum state can be written as SSS 1
+= , where 0
 and 
 represent the ground state and the first excited state, respectively), the average persistent 
current 1
ISISI +=  close to an half integer flux is mainly related to the ground 
state properties of the system, since 10 II >>  in the vicinity of 2
=exψ  (for 2
≠exψ ). In Fig. 4a, 
raising once again β ,  it can be noticed that, in the vicinity of half integer values of the normalized 
flux, the ground state and the first excited state carries currents of opposite sign, inducing a 
competing magnetic behavior. Therefore, the average persistent current, and its magnetic behavior,  
depend, on both coefficients of the decomposition (i. e., on S0
 and S1
).  This last point 
implies that, by measuring the magnetic momentum of the system in a particular magnetic field 
configuration, we can obtain constraints on the nature of the quantum superposition. For instance, 
under these conditions, we could prepare the quantum state in such a way that the average persistent 
current is negligibly small in the vicinity of half integer values of the normalized applied magnetic 
flux. 
Furthermore, we point out that a double well potential can be obtained setting the model 
parameters as done in Fig.5 ( 2.0=β  and 7.0=exψ ), where the potential ( )ϕU  is shown. Indeed, 
we notice that for 0=γ  two low-energy degenerate states are present, the degeneracy being 
removed by means of a small current bias. Such bias can drive the response of the system toward 
one of the two minima of the potential allowing a complete control of the quantum state which can 
be exploited for technological applications. Finally, we notice that, even thought the chosen β  
values in Figs. 2 – 5 are close to the validity limits of the first order approximation of the reduced 
model in ref. [6], the above characteristic response of the system remain qualitatively valid, since 
we are here considering the leading order in the value of β .  
IV   CONCLUSION 
Starting from the reduced dynamical model of the two-junction quantum interference device, 
the applied flux dependence of persistent currents in this system has been studied in the quantum 
regime. The extension of the dissipative overdamped classical system, from classical to quantum 
mechanics, allows to consider the electrodynamical response of a supercurrent quantum bit in the 
limit of negligible capacitance and finite inductance. For null bias current and for opportune values 
of the externally applied magnetic flux, the quantum analog of the two-junction interferometer 
shows effective potential with a degenerate ground state; degeneracy can be removed by applying a 
control non-null bias current. 
In the literature, the quantum behavior of the two-junction quantum interference device is 
studied by considering the charging energy of the junctions as preponderant with respect to the 
energy of the circulating currents [5, 9, 10]. In the present work it is shown that it is possible to 
obtain an Hamiltonian quantum analog of d. c. SQUIDs containing overdamped junctions in the 
limit of null capacitance and finite inductance values. In this framework, a flux qubit can be 
realized, under quite different conditions than those with high junction capacitance value [5]. 
Finally, the present analysis can also be considered as a link between classical dissipative systems 
and their corresponding quantum mechanical models.  
REFERENCES 
1. A. Barone and G. Paternò, Physics and Applications of the Josephson Effect (Wiley, NY, 
1982). 
2. K. K. Likharev, Dynamics of Josephson Junctions and circuits, Gordon and Breach, 
Amsterdam, 1986.  
3. J. Clarke and A. I. Braginski, Eds., The SQUID Handbook, Vol. I (Wiley-VCH, Weinheim, 
2004). 
4. M. F. Bocko, A. M. Herr and M. J. Feldman, IEEE Tans. Appl. Supercond. 7, 3638 (1997).  
5. J. B. Majer, F. G. Paauw, A. C. J. ter Haar, C. J. P. M. Harmans, and J. E. Mooij, Phys. Rev. 
Lett. 94, 090501 (2005). 
6. F. Romeo, R. De Luca, Phys. Lett. A 328, 330 (2004). 
7. C. Vanneste, C. C. Chi, W. J. Gallagher, A. W. Kleinsasser, S. I. Raider, and R. L. 
Sandstrom, J. Appl. Phys. 64,  242 (1988). 
8. R. De Luca, F. Romeo, Phys. Rev. B 73, 214518 (2006). 
9. G. Burkard, Phys. Rev B 71, 144511 (2005). 
10.  T. P. Orlando, J. E. Mooij, Lin Tian, Caspar H. van der Wal, L. S. Levitov, Seth Lloyd, J. J. 
Mazo, Phys. Rev B 60, 15398 (1999).  
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(2003). 
FIGURE CAPTIONS 
Fig. 1  
(a) Persistent currents 1I  (triangle) and 2I  (box) plotted as a function of the applied external flux 
exψ  and  by fixing 075.0=β  and 0=γ . (b) Persistent currents 3I  (star) and 4I  (diamond) plotted 
as a function of the applied external flux exψ  and  by fixing 075.0=β  and 0=γ . 
Fig. 2 
(a) Persistent currents 1I  (triangle) and 2I  (box) plotted as a function of the applied external flux 
exψ  and  by fixing 15.0=β  and 0=γ . (b) Persistent currents 3I  (star) and 4I  (diamond) plotted 
as a function of the applied external flux exψ  and  by fixing 15.0=β  and 0=γ . 
Fig. 3 
(a) Persistent currents 1I  (triangle) and 2I  (box) plotted as a function of the applied external flux 
exψ  and  by fixing 2.0=β  and 0=γ . (b) Persistent currents 3I  (star) and 4I  (diamond) plotted as 
a function of the applied external flux exψ  and  by fixing 2.0=β  and 0=γ . 
Fig. 4  
 (a) Persistent currents 1I  (triangle) and 2I  (box) plotted as a function of the applied external flux 
exψ  and  by fixing 25.0=β  and 0=γ . (b) Persistent currents 3I  (star) and 4I  (diamond) plotted 
as a function of the applied external flux exψ  and  by fixing 25.0=β  and 0=γ . 
Fig. 5 
 Density plot of the potential ( )ϕU  plotted as a function of the phase ϕ  and of the normalized bias 
current γ  by setting the remaining parameters as: 2.0=β  and 7.0=exψ . Lower energy states are 
represented by darker regions in the plot. 
Fig. 1  
0 0.2 0.4 0.6 0.8 1
�0.75
�0.25
0 0.2 0.4 0.6 0.8 1
Fig. 2 
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Fig. 3 
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Fig. 4 
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Fig. 5 
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