Geometric Phase and Superconducting Flux Quantization
Geometric Phase and Superconducting Flux Quantization 
Walter A. Simmons & Sandip S. Pakvasa 
Department of Physics and Astronomy 
University of Hawaii at Manoa 
Honolulu, Hi  96822 
Abstract 
In a ring of s-wave superconducting material the magnetic flux is 
quantized in units of 0 2
Φ = .  It is well known from the theory of 
Josephson junctions that if the ring is interrupted with a piece of d-
wave material, then the flux is quantized in one-half of those units 
due to a additional phase shift of π  .  We reinterpret this 
phenomenon in terms of geometric phase. 
We consider an idealized hetero-junction superconductor with pure s-
wave and pure d-wave electron pairs. We find, for this idealized 
configuration, that the phase shift of π  follows from the discontinuity 
in the geometric phase and is thus a fundamental consequence of 
quantum mechanics. 
Geometric phase has been contained in quantum mechanics since 
the foundations of the field were set down in the early twentieth 
century; however, the phase and its importance were not recognized 
for some time.  Pancharatnam1 discovered the classical geometric 
phase in optics in 1956 and Berry’s important 1987 quantum 
mechanics paper2 stimulated the rapid development of the field.  By 
1992, Anandan3, in a review article in Nature, was able to conclude 
that the phase had been convincingly demonstrated.  The first 
application of geometric phase to Josephson Junctions was carried 
out by Anandan and Pati in 1997.  They showed that the zero voltage 
tunneling supercurrent is geometric in nature and that it is 
proportional to the speed of the state vector in projective Hilbert 
space.4
The appearance of a phase discontinuity of π±  arising from 
geometric phase, under certain circumstances, was shown5 to be a 
general feature of quantum mechanics in 2003, but has so far found 
only limited application6.   Here we show that a well known 
phenomenon7,8 in superconductivity, the quantization of magnetic 
flux in one half of the usual unit9, which is 0 2
Φ = , can be interpreted 
as an effect of the discontinuity in geometric phase.  This phase shift 
in superconductors has been understood in terms of the physics of 
the Josephson junctions and the result has been applied to high 
temperature superconductors10 in order to test the idea that they 
involve d-wave electron pairs.   Our application considers an 
idealized hetero-junction superconductor with pure s-wave and pure 
d-wave electron pairs.  We find, for this idealized configuration, that 
the phase shift of π  follows from the discontinuity in the geometric 
phase4 and is thus a fundamental consequence of quantum 
mechanics. 
Applications of quantum geometric phase have been made in nearly 
every branch of physics, from fundamental material science11,12 to 
quantum computing with superconducting nanocircuits13, as well as 
in chemistry14, and it has been suggested that phase may become 
important in biology15.   
Since the phase has been long present, but not fully recognized, 
some applications entail reinterpreting known phenomena in terms of 
the phase.  An important and illustrative example is the 
reinterpretation of the so called Guoy effect in optics as a geometrical 
phase16, which we will summarize below.   
An idealized superconducting ring, which consists of a composite of 
s-wave material and d-wave material, will, in the absence of external 
electromagnetic fields, exhibit quantization of the magnetic flux in 
units of one half of the usual unit, 0 2
Φ = .  This half-unit quantization 
of the magnetic flux will occur whenever there is an odd number of 
phase shifts of magnitude π  in the circuit.  The theoretical argument6 
for the π  phase shift in a composite ring was based upon the 
dynamics of the Josephson junction and on thermodynamics, and 
has experimental support.   
We next explain the quantum mechanics of the phase shift and then 
we shall proceed to reinterpret the s-wave/d-wave superconductor 
hetro-junction. 
It has long been known17 that when a light beam converges to a 
focus, then diverges again, the light experiences a phase shift whose 
magnitude depends upon the details.  For example, for a beam with a 
Gaussian profile and a very small waistline at the focus, an abrupt 
phase shift of 
π  occurs for each of the two transverse directions, for 
a total phase shift of π .  This result, which follows from standard 
classical electrodynamics, has been reinterpreted in terms of 
geometric phase15.  For one transverse direction, the complex 
curvature of the wave reverses at the focus; the geometric phase, 
which is directly related to the curvature, changes by 
π .   
That optical example of geometric phase is closely analogous to a 
well-known18 phase flip of π± , which occurs in optics when the 
polarization of a light beam is rotated from some initial state, through 
and beyond, another polarization state that is orthogonal to the initial 
state. 
The latter example of the geometric phase discontinuity has been 
shown to occur rather generally in quantum mechanics4.  This can be 
understood by considering the behavior of a complex quantum state 
vector as it is impelled through a series of states in Hilbert space by 
an external force.  Suppose the initial state is iΨ , the final state is 
fΨ , and some intermediate state 0Ψ  is orthogonal to the initial 
state, 0 0iΨ Ψ = . In the complex plane, the trajectory is a sequence 
of projections of each state upon its subsequent state.  The trajectory 
from the initial to the final state passes through the origin (with a 
positive or negative infinitesimal imaginary part); the phase goes 
through an inverse-tangent singularity and changes by π± .   
Finally, turning to superconductivity, we adopt the theoretical 
framework19, in which superconductivity is viewed as a consequence 
of the breaking of gauge invariance entailing the formation of Cooper 
pairs.  We consider a ring of material in which the supercurrent is 
carried by s-wave Cooper pairs.  The ring is interrupted by a section 
of material in which d-wave Cooper pairs carry the supercurrent.  The 
supercurrent passing through the idealized hetro-junction 
experiences a shift of π±  in the geometric phase due to the 
orthogonality of d-wave and s-wave.  Since the orientation of the d-
wave relative to the s-wave is not meaningful, we have no sum over 
dimensions as in the optical beam analogy; a shift of π±  is the result 
of a single inserted section of material and the half-unit magnetic flux 
quantization follows.   
Since the 1997 work of Anandan and Pati, it has been known that the 
zero-voltage current in a tunneling supercurrent arises from the 
geometry of Hilbert space and is independent of the specific 
Hamiltonian, (which is a general feature of geometric phase20).   
Recently, experiments on hetero-junctions have supported the idea 
that high  materials are d-wave superconductors.  While we are not 
discussing here realistic models of high materials here, our results 
show that a phase shift of 
π±  in s-wave/d-wave hetero-junctions 
arises from the fundamentals of quantum mechanics.   
                                      
1 Pancharatnam, S., The Proceedings of the Indian Academy of Sciences, Vol XLIV, No. 5, Sec. 
A, 247 (1956) in Collected Works of S. Pancharatnam, Oxford University Press, London (175). 
                                                                                                                
2 Berry, M.V. “Quantal Phase Factors Accompanying Adiabatic Changes”, Proc. R. Soc. Lond. 
A392, 45, (1984). 
3 Anandan, J. “The geometric phase”, Nature 360, 307 (1992). 
4  Anandan, J. & Pati, A.K., “Geometry of the Josephson effect” Physics Letters A 231, 29 (1997). 
5 Mukunda, et al “Bargmann invariants, null phase curves, and a theory of the geometric phase”, 
Phys. Rev. A 67, 042114 (2003). 
6 Simon, R. and Mukunda, N., “Bargmann Invariant and the Geometry of the Guoy Effect”, Phys. 
Rev. Letters 70, 880 (1993). 
7 Bulaevskii, L. N. , Kuzii, V. V. & Sobyanin, A. A. Superconducting system with weak coupling to 
the current in the ground state. JETP Lett. 25, 290–294 (1977). 
8 Tsuei, C.C., and Kirtley, J.R. “Paring symmetry in cuprate superconductors” Rev. Mod. Phys 72, 
969 (2000).  For more recent results, see Kirtley, et al, Nature 373, 225 (2005) 
9 Ashcroft, N.W. & Mermin, N.D., Solid State Physics, Holt, Rinehart and Winston (1976). 
10 Hilgenkamp, H., Ariando, Smilde, H.-J. H., Blank, D. H. A., Rijnders, G., Rogalla, H., Kirtley, J. 
R., and Tsuei, C. C., “Ordering and manipulation of the magnetic moments in large-scale 
superconducting pi-loop arrays”, Nature 422, 50 (2003). 
11 Zak, J. “Berry’s Phase for Energy Bands in Solids”, Phys. Rev. Lett. 62, 2747 (1989). 
12 Resta, R. “Manifestations of Berry’s phase in molecules and condensed matter”, 
J.Phys.Condens. Matter 12, R107 (2000). 
13 Falci,G., Fazio, R., Palma, G.M., Siewert, J., and Vedral, V. “Detection of geometric phases in 
superconducting nanocircuits”, Nature 407, 355 (2000). 
14 Mead, C.A. “The geometric phase in molecular systems”, Rev. Mod. Phys. 64, 51 (1992). 
15 Kagan, M.L., Kepler, T. B.  & Epstein, I.R., “Geometric phase shifts in chemical oscillators”, 
Nature 349, 506 (1991). 
16 Simon, R. and Mukunda, N., “Bargmann Invariant and the Geometry of the Guoy Effect”, Phys. 
Rev. Letters 70, 880 (1993). 
17 Siegman, A.E., Lasers, University Science Books, Mill Valley, California (1986). 
18 Bhandari, R. “Polarization of light an the topological phases”, Physics Reports, 281, 1 (1997). 
19 Weinberg, S., Quantum Theory of Fields II, Cambridge University Press (1996).   
20 Aharonov, Y. & Anandan, J., “Phase Change during a Cyclic Quantum Evolution”, Phys. Rev. 
Lett 58, 1593 (1987). 

  In a ring of s-wave superconducting material the magnetic flux is quantized
in units of $\Phi_0 = \frac{h}{2e}$. It is well known from the theory of
Josephson junctions that if the ring is interrupted with a piece of d-wave
material, then the flux is quantized in one-half of those units due to a
additional phase shift of $\pi$. We reinterpret this phenomenon in terms of
geometric phase.
  We consider an idealized hetero-junction superconductor with pure s-wave and
pure d-wave electron pairs. We find, for this idealized configuration, that the
phase shift of $\pi$ follows from the discontinuity in the geometric phase and
is thus a fundamental consequence of quantum mechanics.

Geometric Phase and Superconducting Flux Quantization
Geometric Phase and Superconducting Flux Quantization 
Walter A. Simmons & Sandip S. Pakvasa 
Department of Physics and Astronomy 
University of Hawaii at Manoa 
Honolulu, Hi  96822 
Abstract 
In a ring of s-wave superconducting material the magnetic flux is 
quantized in units of 0 2
Φ = .  It is well known from the theory of 
Josephson junctions that if the ring is interrupted with a piece of d-
wave material, then the flux is quantized in one-half of those units 
due to a additional phase shift of π  .  We reinterpret this 
phenomenon in terms of geometric phase. 
We consider an idealized hetero-junction superconductor with pure s-
wave and pure d-wave electron pairs. We find, for this idealized 
configuration, that the phase shift of π  follows from the discontinuity 
in the geometric phase and is thus a fundamental consequence of 
quantum mechanics. 
Geometric phase has been contained in quantum mechanics since 
the foundations of the field were set down in the early twentieth 
century; however, the phase and its importance were not recognized 
for some time.  Pancharatnam1 discovered the classical geometric 
phase in optics in 1956 and Berry’s important 1987 quantum 
mechanics paper2 stimulated the rapid development of the field.  By 
1992, Anandan3, in a review article in Nature, was able to conclude 
that the phase had been convincingly demonstrated.  The first 
application of geometric phase to Josephson Junctions was carried 
out by Anandan and Pati in 1997.  They showed that the zero voltage 
tunneling supercurrent is geometric in nature and that it is 
proportional to the speed of the state vector in projective Hilbert 
space.4
The appearance of a phase discontinuity of π±  arising from 
geometric phase, under certain circumstances, was shown5 to be a 
general feature of quantum mechanics in 2003, but has so far found 
only limited application6.   Here we show that a well known 
phenomenon7,8 in superconductivity, the quantization of magnetic 
flux in one half of the usual unit9, which is 0 2
Φ = , can be interpreted 
as an effect of the discontinuity in geometric phase.  This phase shift 
in superconductors has been understood in terms of the physics of 
the Josephson junctions and the result has been applied to high 
temperature superconductors10 in order to test the idea that they 
involve d-wave electron pairs.   Our application considers an 
idealized hetero-junction superconductor with pure s-wave and pure 
d-wave electron pairs.  We find, for this idealized configuration, that 
the phase shift of π  follows from the discontinuity in the geometric 
phase4 and is thus a fundamental consequence of quantum 
mechanics. 
Applications of quantum geometric phase have been made in nearly 
every branch of physics, from fundamental material science11,12 to 
quantum computing with superconducting nanocircuits13, as well as 
in chemistry14, and it has been suggested that phase may become 
important in biology15.   
Since the phase has been long present, but not fully recognized, 
some applications entail reinterpreting known phenomena in terms of 
the phase.  An important and illustrative example is the 
reinterpretation of the so called Guoy effect in optics as a geometrical 
phase16, which we will summarize below.   
An idealized superconducting ring, which consists of a composite of 
s-wave material and d-wave material, will, in the absence of external 
electromagnetic fields, exhibit quantization of the magnetic flux in 
units of one half of the usual unit, 0 2
Φ = .  This half-unit quantization 
of the magnetic flux will occur whenever there is an odd number of 
phase shifts of magnitude π  in the circuit.  The theoretical argument6 
for the π  phase shift in a composite ring was based upon the 
dynamics of the Josephson junction and on thermodynamics, and 
has experimental support.   
We next explain the quantum mechanics of the phase shift and then 
we shall proceed to reinterpret the s-wave/d-wave superconductor 
hetro-junction. 
It has long been known17 that when a light beam converges to a 
focus, then diverges again, the light experiences a phase shift whose 
magnitude depends upon the details.  For example, for a beam with a 
Gaussian profile and a very small waistline at the focus, an abrupt 
phase shift of 
π  occurs for each of the two transverse directions, for 
a total phase shift of π .  This result, which follows from standard 
classical electrodynamics, has been reinterpreted in terms of 
geometric phase15.  For one transverse direction, the complex 
curvature of the wave reverses at the focus; the geometric phase, 
which is directly related to the curvature, changes by 
π .   
That optical example of geometric phase is closely analogous to a 
well-known18 phase flip of π± , which occurs in optics when the 
polarization of a light beam is rotated from some initial state, through 
and beyond, another polarization state that is orthogonal to the initial 
state. 
The latter example of the geometric phase discontinuity has been 
shown to occur rather generally in quantum mechanics4.  This can be 
understood by considering the behavior of a complex quantum state 
vector as it is impelled through a series of states in Hilbert space by 
an external force.  Suppose the initial state is iΨ , the final state is 
fΨ , and some intermediate state 0Ψ  is orthogonal to the initial 
state, 0 0iΨ Ψ = . In the complex plane, the trajectory is a sequence 
of projections of each state upon its subsequent state.  The trajectory 
from the initial to the final state passes through the origin (with a 
positive or negative infinitesimal imaginary part); the phase goes 
through an inverse-tangent singularity and changes by π± .   
Finally, turning to superconductivity, we adopt the theoretical 
framework19, in which superconductivity is viewed as a consequence 
of the breaking of gauge invariance entailing the formation of Cooper 
pairs.  We consider a ring of material in which the supercurrent is 
carried by s-wave Cooper pairs.  The ring is interrupted by a section 
of material in which d-wave Cooper pairs carry the supercurrent.  The 
supercurrent passing through the idealized hetro-junction 
experiences a shift of π±  in the geometric phase due to the 
orthogonality of d-wave and s-wave.  Since the orientation of the d-
wave relative to the s-wave is not meaningful, we have no sum over 
dimensions as in the optical beam analogy; a shift of π±  is the result 
of a single inserted section of material and the half-unit magnetic flux 
quantization follows.   
Since the 1997 work of Anandan and Pati, it has been known that the 
zero-voltage current in a tunneling supercurrent arises from the 
geometry of Hilbert space and is independent of the specific 
Hamiltonian, (which is a general feature of geometric phase20).   
Recently, experiments on hetero-junctions have supported the idea 
that high  materials are d-wave superconductors.  While we are not 
discussing here realistic models of high materials here, our results 
show that a phase shift of 
π±  in s-wave/d-wave hetero-junctions 
arises from the fundamentals of quantum mechanics.   
                                      
1 Pancharatnam, S., The Proceedings of the Indian Academy of Sciences, Vol XLIV, No. 5, Sec. 
A, 247 (1956) in Collected Works of S. Pancharatnam, Oxford University Press, London (175). 
                                                                                                                
2 Berry, M.V. “Quantal Phase Factors Accompanying Adiabatic Changes”, Proc. R. Soc. Lond. 
A392, 45, (1984). 
3 Anandan, J. “The geometric phase”, Nature 360, 307 (1992). 
4  Anandan, J. & Pati, A.K., “Geometry of the Josephson effect” Physics Letters A 231, 29 (1997). 
5 Mukunda, et al “Bargmann invariants, null phase curves, and a theory of the geometric phase”, 
Phys. Rev. A 67, 042114 (2003). 
6 Simon, R. and Mukunda, N., “Bargmann Invariant and the Geometry of the Guoy Effect”, Phys. 
Rev. Letters 70, 880 (1993). 
7 Bulaevskii, L. N. , Kuzii, V. V. & Sobyanin, A. A. Superconducting system with weak coupling to 
the current in the ground state. JETP Lett. 25, 290–294 (1977). 
8 Tsuei, C.C., and Kirtley, J.R. “Paring symmetry in cuprate superconductors” Rev. Mod. Phys 72, 
969 (2000).  For more recent results, see Kirtley, et al, Nature 373, 225 (2005) 
9 Ashcroft, N.W. & Mermin, N.D., Solid State Physics, Holt, Rinehart and Winston (1976). 
10 Hilgenkamp, H., Ariando, Smilde, H.-J. H., Blank, D. H. A., Rijnders, G., Rogalla, H., Kirtley, J. 
R., and Tsuei, C. C., “Ordering and manipulation of the magnetic moments in large-scale 
superconducting pi-loop arrays”, Nature 422, 50 (2003). 
11 Zak, J. “Berry’s Phase for Energy Bands in Solids”, Phys. Rev. Lett. 62, 2747 (1989). 
12 Resta, R. “Manifestations of Berry’s phase in molecules and condensed matter”, 
J.Phys.Condens. Matter 12, R107 (2000). 
13 Falci,G., Fazio, R., Palma, G.M., Siewert, J., and Vedral, V. “Detection of geometric phases in 
superconducting nanocircuits”, Nature 407, 355 (2000). 
14 Mead, C.A. “The geometric phase in molecular systems”, Rev. Mod. Phys. 64, 51 (1992). 
15 Kagan, M.L., Kepler, T. B.  & Epstein, I.R., “Geometric phase shifts in chemical oscillators”, 
Nature 349, 506 (1991). 
16 Simon, R. and Mukunda, N., “Bargmann Invariant and the Geometry of the Guoy Effect”, Phys. 
Rev. Letters 70, 880 (1993). 
17 Siegman, A.E., Lasers, University Science Books, Mill Valley, California (1986). 
18 Bhandari, R. “Polarization of light an the topological phases”, Physics Reports, 281, 1 (1997). 
19 Weinberg, S., Quantum Theory of Fields II, Cambridge University Press (1996).   
20 Aharonov, Y. & Anandan, J., “Phase Change during a Cyclic Quantum Evolution”, Phys. Rev. 
Lett 58, 1593 (1987). 
