The Josephson effect in superfluid ^{3}He
was studied in the 1980's by Avenel and Varoquaux [1].
More recently, the flow of superfluid ^{3}He-B through a 65x65
array of nanometer size apertures has
been measured by Backhaus et al at
Berkeley [2,3]. They find in the
current-phase relation a new branch, so-called
π state (pi state).
The π state has subsequently been observed also
in a single narrow slit [4].

In order to explain the π state,
we have made calculations in two limiting
cases. The first case is one large aperture, where the 18-component
order parameter is solved numerically in
and around the aperture using the Ginzburg-Landau theory of ^{3}He.
We find that the calculations have to be made in three dimensions because the
π state has broken symmetry
(m'm2') even in a circularly symmetric orifice.
The current-phase relationship (figure) shows the existence of the
π state for sufficiently large apertures.
The physical interpretation is that the
π state corresponds to a phase slip
by a half-quantum vortex.
The figure correnspons to the case with equal spin-orbit rotation matrices
on the two sides of the weak link, where the current-phase relation is
strongly hysteretic. The hysteresis is not usually present when more
general
boundary conditions are used, and the π
states should therefore be experimentally observable.

As the second case we have made calculations for an aperture that is small
compared to the superfluid coherence length.
The presence of the π state in such a
"pinhole" was first demonstrated by Yip using a simplified model [5].
We have done the pinhole calculation selfconsistently and find that
the π state occurs in a single pinhole only
at very low temperatures, approximately 0.2 T_{c}.
The situation is different in an array of pinholes.
There the π state can appear at higher
temperatures because of *anisotextural* Josephson effect:
the texture changes as a function of the phase difference.
We find that this mechanism gives a good quantitative explanation for the
π states and bistability observed in
Berkeley [2,3].
For alternative theories trying to explain the
π state see Refs. [5-8].

The next step is to understand
the dynamics of ^{3}He Josephson junctions.

- O. Avenel and E. Varoquaux, Phys. Rev. Lett.
**60**, 416 (1988). - S. Backhaus, S. Pereverzev, R.W. Simmonds, A. Loshak, J.C. Davis,
and R.E. Packard, Nature
**392**, 687 (1998). - A. Marchenkov, R.W. Simmonds, S. Backhaus, A. Loshak,
J.C. Davis, and R.E. Packard, Phys. Rev. Lett.
**83**, 3860 (1999). - O. Avenel, Yu. Mukharsky, and E. Varoquaux,
Physica
**280**, 130 (2000). - S.-K. Yip, Phys. Rev. Lett.
**83**, 3864 (1999), cond-mat 9907096. - N. Hatakenaka, Phys. Rev. Lett.
**81**, 3753 (1998); J. Phys. Soc. Japan**67**, 3672 (1998). - O. Avenel, Y. Mukharsky, and E. Varoquaux, Nature
**397**, 484 (1999). - A. Smerzi, S. Raghavan, S. Fantoni, and S.R. Shenoy, cond-mat 0011298.

- J.K. Viljas and E.V. Thuneberg:
*Theory of the π state in*, Phys. Rev. Lett.^{3}He Josephson junctions**83**, 3868 (1999), cond-mat 9907181. - J.K. Viljas and E.V. Thuneberg:
*Pinhole calculations of the Josephson effect in*, Phys. Rev. B^{3}He-B**65**, 64530 (2002), cond-mat 0107052. - J.K. Viljas and E.V. Thuneberg:
*Equilibrium simulations of 2D weak links in p-wave superfluids*, J. Low Temp. Phys.**129**, 423 (2002), cond-mat/0210052. - J. Viljas and E. Thuneberg,
*Stability of A-B phase boundary in a constriction*, Physica B**329-333**, 86-87 (2003), preprint. - E. V. Thuneberg,
*Theory of Josephson Phenomena in Superfluid*, AIP Conference Proceedings^{3}He**850**, 103 (2006), cond-mat/0509504. - Transparencies of a talk:
π states and textural effects
in superfluid
^{3}He weak links (Sept. 2002, J. Viljas, pdf file, 684 kb).

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30.8.2005, Erkki Thuneberg, Email