It was suggested more than thirty years ago by Fulde and Ferrell[@ff], and Larkin and Ovchinnikov[@lo], that an inhomogeneous superconductor with an order parameter that oscillates spatially may be stabilized by a large external magnetic or internal exchange field. Such a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state has never been observed in conventional low-$T_c$ superconductors. Recently it has attracted renewed interest in the context of organic, heavy-fermion, and high-$T_c$ cuprate superconductors[@gloos; @yin; @norman; @rainer; @shimahara; @murthy; @dupuis; @modler; @tachiki; @geg; @sh2; @maki; @samokhin; @buzdin; @yang; @symington; @pickett]. These new classes of superconductors are believed to provide conditions that are favorable to the formation of FFLO state, because many of them are i) strongly type II superconductors so that the upper critical field $H_{c2}$ can easily approach the Pauli paramagnetic limit; and (ii) layered compounds so that when a magnetic field is applied parallel to the conducting plane, the orbital effect is minimal, and the Zeeman effect (which is the driving force for the formation of FFLO state) dominates the physics. Indeed, some experimental indications of the existence of the FFLO state have been reported[@gloos; @modler; @geg; @symington].

The main difficulty in the experimental search for the FFLO state is that just like the BCS state, the FFLO state is a superconducting state. The distinction between the two is a subtle difference in the structure of the superconducting order parameter, which is difficult to detect using ordinary experimental techniques. Previous experiments have focused on thermodynamic signatures of possible phase transitions from the BCS to FFLO state, which is believed to be first order (see, however, Ref. ). But such signatures can be caused by other phase transitions that have nothing to do with the superconducting order parameter. Thus it is very difficult to establish the presence of an FFLO state this way without any ambiguity.

In this paper we propose using the Josephson effect to detect the existence of FFLO states. Our proposal has some similarity in spirit to the basic ideas behind the “phase sensitive" experiments[@phase] that established the predominant $d$-wave symmetry of the order parameter in cuprate superconductors. Specifically, we predict: (i) The Josephson current in a Josephson junction between a conventional BCS superconductor and an FFLO superconductor is suppressed, due to the difference in momenta of the order parameters. (ii) The Josephson current may be recovered by applying a properly chosen magnetic field in the junction, with field strength and direction depending on the momentum of the order parameter of the FFLO superconductor; it thus provides a way to measure the momentum of the order parameter directly.

In the rest of the paper, we demonstrate the above effects first by using the Ginsburg-Landau theory, and then by presenting a microscopic derivation. We discuss the experimental implication and feasibility of our proposal toward the end of the paper.

[*Ginsburg-Landau Theory*]{}. The effects we predict are most easily demonstrated using a Ginsburg-Landau theory. For simplicity we consider a two-dimensional BCS superconductor, described by a spatially dependent superconducting order parameter $\Psi_{BCS}({\bf r})$, which is coupled to a two-dimensional FFLO superconductor, described by an order parameter $\Psi_{FFLO}({\bf r})$[@note]. We consider the two Josephson junction geometries shown in Figure 1. Since the physics for the two geometries is similar we focus our discussion on geometry a) and simply state the results for geometry b). For geometry a) the Josephson coupling term in the free energy takes the form (in the absence of any magnetic field) $$H_J=-t\int{d^2{\bf r}}[\Psi_{FFLO}^*({\bf r})\Psi_{BCS}({\bf r})+ c.c.].
\label{eq1}$$ In the ground state of a BCS superconductor, $\Psi_{BCS}({\bf r})= \psi_0$ is a constant. However, in an FFLO superconductor the order parameter is a superposition of components carrying finite momenta: $$\Psi_{FFLO}({\bf r})=\sum_{m}\psi_m e^{i{\bf k}_m\cdot{\bf r}},$$ and is oscillatory in space. In the absence of magnetic flux inside the junction, the total Josephson current is $$\begin{aligned}
I_J&=& {\rm Im}\left[t
\int{d^2{\bf r}}\Psi^*_{BCS}({\bf r})\Psi_{FFLO}({\bf r})\right]
\nonumber\\
&=&\sum_m {\rm Im}
\left[t\psi_0^*\psi_m\int{d^2{\bf r}}e^{i{\bf k}_m\cdot{\bf r}}\right].\end{aligned}$$ Clearly, due to the oscillatory nature of the integrand, the Josephson current is suppressed in such a junction. The same result is clearly true for geometry b).

Mathematically, the reason that the Josephson current is suppressed here is similar to the suppression of Josephson current by an applied magnetic field in an ordinary Josephson junction between two BCS superconductors[@tinkham]. However, the physics is very different: here the suppression is due to the spatial oscillation of the [*order parameter*]{} in the FFLO state, while in the case of ordinary Josephson junction in a magnetic field, the phase of the Josephson tunneling [*matrix element*]{} is oscillatory (in a proper gauge choice). Nevertheless, the mathematical similarity allows these two effects to [*cancel*]{} each other and restore the Josephson current, as we demonstrate below.

Consider geometry a). Imagine applying a parallel magnetic field ${\bf B}\bot \hat{z}$ parallel to the planes, where $\hat{z}$ is a unit vector along the normal direction of the plane. Using the gauge ${\bf A}({\bf r})={\bf r}\times {\bf B}$ we have ${\bf A}({\bf r})=A({\bf r})\hat{z}$. The appearance of the ${\bf A}$ field affects the phase of the Josephson tunneling matrix element only; the in-plane properties in the two individual superconductors are unaffected because the ${\bf A}$ field is perpendicular to the planes. Specifically, the Josephson coupling term in the free energy becomes $$H_J=-t\int{d^2{\bf r}}[e^{{2ieA({\bf r})d\over \hbar c}}
\Psi_{FFLO}^*({\bf r})\Psi_{BCS}({\bf r})+ c.c.],$$ and the total Josephson current takes the form $$\begin{aligned}
I_J&=& {\rm Im}\left[t
\int{d^2{\bf r}}e^{{-2ieA({\bf r})d\over \hbar c}}\Psi^*_{BCS}({\bf r})
\Psi_{FFLO}({\bf r})\right]
\nonumber\\
&=&\sum_m {\rm Im}\left[t\psi_0^*\psi_m\int{d^2{\bf r}}
e^{{-2ieA({\bf r})d\over \hbar c}}e^{i{\bf k}_m\cdot{\bf r}}\right].\end{aligned}$$ Here $d$ is the distance between the two planes. It is clear from the equation above that the two oscillatory factors cancel each other when $${\bf B}={\hbar c\over 2ed}\hat{z}\times {\bf k}_m.
\label{restore}$$ At this particular ${\bf B}$, the Josephson current gets partially restored. One can determine the momenta of the order parameter in the FFLO state by searching for the ${\bf B}$’s that restore the Josephson current.

For geometry b) consider applying a perpendicular magnetic field ${\bf B}\parallel \hat{z}$. Arguments similar to those of the last paragraph imply that the Josephson current becomes restored only when the momentum of the FFLO order parameter is along the junction (note that typically this momentum will lie along high symmetry directions of the superconductor). In this case the Josephson current is partially restored when $${\bf B}=\frac{\hbar c}{2e(d+\lambda_1+\lambda_2)}\hat{z}
\label{eq2}$$ where $\lambda_1$ ($\lambda_2$) is the penetration depth of the FFLO (BCS) superconductor for the field applied normal to the superconducting plane. The appearance of the factor $d+\lambda_1+\lambda_2$ in Eq. \[eq2\] is because the magnetic field is assumed to enter each superconductor a distance equal to the penetration depth from the edge of the junction.

[*Microscopic Derivation*]{}. These effects can also be demonstrated using a microscopic approach, as we illustrate below. For simplicity and clarity, we take the zero temperature limit, and initially consider the usual Josephson effect between two BCS superconductors. When two superconductors are placed in proximity so that electrons can tunnel from one to the other, the total energy of the system contains a term that depends on the phase difference $\phi$ between the order parameters of the two superconductors: $$E_J(\phi)=E_J\cos\phi;$$ and the Josephson current is $$I_J(\phi)={2eE_J\over\hbar}\sin\phi.$$ Thus the calculation of $I_J(\phi)$ is equivalent to the calculation of the Josephson energy $E_J(\phi)$, which we perform below. We use the following tunneling Hamiltonian[@josephson; @ambegaokar]: $$H_T=\sum_{{\bf p}{\bf q}\sigma}(T_{{\bf p},{\bf q}}c_{{\bf p}\sigma}^\dagger
c_{{\bf q}\sigma}+h.c.),$$ where ${\bf p}$ and ${\bf q}$ label single electron momentum eigenstates in the two different superconductors, and $\sigma$ is the spin label (this tunneling Hamiltonian can be used for both geometries since only the matrix elements $T_{{\bf p},{\bf q}}$ will differ). The mean field Hamiltonian for a BCS superconductor ($s$-wave) takes the form: $$\begin{aligned}
H_l&=&\sum_{{\bf p}\sigma}\epsilon_{\bf p}c_{{\bf p}\sigma}^\dagger
c_{{\bf p}\sigma}+\sum_{\bf p}(\Delta_lc_{{\bf p}\uparrow}^\dagger
c_{-{\bf p}\downarrow}^\dagger+h.c.)\nonumber\\
&=&\sum_{\bf p}E_{\bf p}(\alpha_{\bf p}^\dagger\alpha_{\bf p}
+\beta_{\bf p}^\dagger\beta_{\bf p})+const.,
\label{bcs}\end{aligned}$$ where $\Delta_l$ is the order parameter of the lower superconductor, $\alpha$ and $\beta$ are Bogliubov quasiparticle operators, and $E_{\bf p}=\sqrt{\epsilon_{\bf p}^2+|\Delta_l|^2}$ is the quasiparticle energy. When both lower and upper superconductors are BCS, a second-order perturbation calculation in $H_T$ gives rise to the following expression for the Josephson energy (we have neglected terms in the second-order perturbation that are independent of the phase difference $\phi$): $$E_J(\phi)=-{1\over 2}[\sum_{{\bf p}{\bf q}}{T_{{\bf p},{\bf q}}
T_{-{\bf p},-{\bf q}}\Delta_l^*\Delta_u\over E_{\bf p}E_{\bf q}(E_{\bf p}
+E_{\bf q})}+ c.c.].
\label{ej1}$$ In general the tunneling matrix elements $T_{{\bf p},{\bf q}}$ are complex. A fact that is crucial to the existence of the Josephson effect, as originally emphasized by Josephson himself[@josephson], is that in the presence of time-reversal symmetry, $T_{-{\bf p},-{\bf q}}
=T_{{\bf p},{\bf q}}^*$; thus there is no phase oscillation in the summation of Eq. (\[ej1\]). Breaking time-reversal symmetry ([*e.g.*]{} by applying an external magnetic field) tends to suppress the Josephson effect by introducing oscillatory phases to the terms in Eq. (\[ej1\]).

We now turn the discussion to the case where the upper superconductor is in an FFLO state, while the lower one is still a BCS superconductor. We first consider zero external magnetic field, and for simplicity assume the order parameter of the FFLO state has a single momentum ${\bf k}$. The mean-field Hamiltonian in this case takes the form[@shimahara; @murthy]: $$\begin{aligned}
&H_u&=\sum_{{\bf q}\sigma}\epsilon_{\bf q}c_{{\bf q}\sigma}^\dagger
c_{{\bf q}\sigma}+E_Z\sum_{{\bf q}}
(c_{{\bf q}\uparrow}^\dagger c_{{\bf q}\uparrow}
-c_{{\bf q}\downarrow}^\dagger c_{{\bf q}\downarrow})\nonumber\\
&+&\sum_{\bf q}(\Delta_uc_{{\bf q}+{\bf k}/2\uparrow}^\dagger
c_{-{\bf q}+{\bf k}/2\downarrow}^\dagger+h.c.)\nonumber\\
&=&\sum_{\bf q}[(E_{\bf q}+Z_{\bf q})\alpha_{\bf q}^\dagger\alpha_{\bf q}
+(E_{\bf q}-Z_{\bf q})\beta_{\bf q}^\dagger\beta_{\bf q}]
+const.,
\label{fflo}\end{aligned}$$ where $E_Z$ is the Zeeman splitting between the up and down spin electrons, and $Z_{\bf q}=E_Z+v_F{\bf q}\cdot{\bf k}/(2|{\bf q}|)$. The crucial difference between (\[bcs\]) and (\[fflo\]) is that in the latter case the pairing term creates or annihilates pairs with total momentum ${\bf k}$. A second-order perturbation calculation yields the following for the Josephson energy: $$\begin{aligned}
&E_J(\phi)&=-{1\over 2}[\sum_{{\bf p}{\bf q}}{T_{{\bf p},{\bf q}+{\bf k}/2}
T_{-{\bf p},-{\bf q}+{\bf k}/2}\Delta_l^*
\Delta_u\over E_{\bf p}E_{\bf q}}\nonumber\\
&\times&({\theta(E_{\bf q}-Z_{\bf q})\over E_{\bf p}+E_{\bf q}-Z_{\bf q}}
-{\theta(-E_{\bf q}-Z_{\bf q})\over E_{\bf p}+|E_{\bf q}+Z_{\bf q}|})
+ c.c.].
\label{ej2}\end{aligned}$$ The expression (\[ej2\]) looks more complicated than (\[ej1\]) because in the FFLO superconductor i) the up and down spin quasiparticles are no longer degenerate and ii) the ground state is not a vacuum of the quasiparticles. The fundamental difference between (\[ej2\]) and (\[ej1\]), however, lies in the fact that in (\[ej2\]) the two tunneling matrix elements forming the product, $T_{{\bf p},{\bf q}+{\bf k}/2}$ and $T_{-{\bf p},-{\bf q}+{\bf k}/2}$, are no longer related by time-reversal transformation and therefore are no longer the complex conjugate of each other. This is due to the fact that the Cooper pairs in the FFLO superconductor carry a finite momentum ${\bf k}$. As a consequence of this, the terms in the summation of (\[ej2\]) have oscillatory phases, and the Josephson effect is suppressed.

Consider geometry a) again. In this case the Josephson energy may be recovered by applying a properly chosen magnetic field parallel to the planes. In the microscopic theory it is convenient to choose the following gauge: $${\bf A}=-z(\hat{z}\times {\bf B}),$$ ${\bf A}$ is then parallel to the planes. In this case the gauge field does not change the tunneling matrix elements. Let us assume the lower layer is at $z=0$, while the upper layer is at $z=d$. Thus ${\bf A}_l=0$ so there is no change in the BCS superconductor. Also ${\bf A}_u=-d(\hat{z}\times {\bf B})$ is a constant gauge field, which shifts the momenta of individual electrons in the upper plane by a constant amount $$\delta{\bf k}={ed\over \hbar c}(\hat{z}\times {\bf B}),$$ i.e., we need to replace all the ${\bf q}$’s by ${\bf q}+\delta{\bf k}$, and $-{\bf q}$’s by $-{\bf q}+\delta{\bf k}$, in Eqs. (\[fflo\]) and (\[ej2\]). We emphasize that if the upper layer were [*isolated*]{}, such a shift induced by a constant ${\bf A}$ field is a pure gauge transformation that has absolutely no physical consequence; the only reason that such a parallel magnetic field has physical effects is because electrons can tunnel from one layer to the other, and thus go through loops that enclose a finite amount of magnetic [*flux*]{}.

In particular, when $\delta{\bf k}=-{\bf k}/2$ (as in Eq. (\[restore\])), just as in the BCS superconductor, the Cooper pairs in the FFLO state carry [*zero*]{} momentum (but still finite gauge-invariant center-of-mass [*velocity*]{}!); in this case the Josephson energy takes the form: $$\begin{aligned}
E_J(\phi)&=&-{1\over 2}[\sum_{{\bf p}{\bf q}}{T_{{\bf p},{\bf q}}
T_{-{\bf p},-{\bf q}}\Delta_l^*
\Delta_u\over E_{\bf p}E_{{\bf q}-{{\bf k}\over 2}}}
({\theta(E_{\bf q-{{\bf k}\over 2}}-Z_{{\bf q}-{{\bf k}\over 2}})
\over E_{\bf p}+E_{{\bf q}-{{\bf k}\over 2}}-Z_{{\bf q}-{{\bf k}\over 2}}}
\nonumber\\
&-&{\theta(-E_{{\bf q}-{{\bf k}\over 2}}-Z_{{\bf q}-{\bf k}/2})
\over E_{\bf p}+|E_{{\bf q}-{{\bf k}\over 2}}+Z_{{\bf q}-{{\bf k}\over 2}}|})
+ c.c.].\end{aligned}$$ Clearly in the above summation we no longer have the oscillatory phases, and the Josephson energy and current are restored. Again, similar arguments apply for geometry b) with the conclusions that the FFLO momentum must lie along the junction to restore the Josephson effect and, if this is the case, the restoration will occur at the field given by Eq. \[eq2\].

[*Discussion.*]{} While arriving at the same basic conclusions, the microscopic and macroscopic derivations do not lead to results that are equivalent in all details. This is due to the implicit assumption made in the macroscopic derivation that the Josephson coupling is uniform throughout the junction. This assumption was not made at the microscopic level; if made, it will restrict the form of the tunneling matrix elements $T_{\bf p,q}$. To illustrate this consider an infinite junction. In this case translational invariance in the junction requires $T_{\bf p,q}=e^{i({\bf p}-
{\bf q})\cdot {\bf t}_j} T_{\bf {p,q}}$ where ${\bf t}_j$ is an arbitrary translation vector in the junction. Implementing this constraint in Eq. (\[ej1\]) gives $E_J(\phi)\propto  
\delta_{{\bf k},0}$ which is also the result of Eq. (\[eq1\]) after the spatial integration is performed.

In our discussion we have assumed that the superconductors are in the clean limit, so that we can label single electron states by their momenta, and the order parameter of the FFLO superconductor has either a single momentum or is a linear superposition of components with a discrete set of momenta. The presence of disorder will change the situation. In this case it is still true that the Josephson effect between a BCS superconductor and an FFLO superconductor is suppressed, as in the latter Cooper pairs are [*not*]{} formed between single electrons states that are related by time-reversal transformation. However, the recovery of the Josephson effect by applying a magnetic field will be decreased, since in this case the order parameter in the FFLO state is formed by superpositions of components with momentum that varies continuously; while a magnetic field can only recover the contribution to the Josephson energy of a single component with a fixed momentum. Thus experimental studies of the effects we predict should be performed in clean superconductors.

Ideally, we would like to apply this theory to FFLO superconductors that are stabilized by an internal exchange field, rather than an external magnetic field. This is because in the latter case, unless the system is truly two-dimensional, the orbital effect of the magnetic field that penetrates into the superconductor will induce vortices, which will complicate the analysis. We note that while experimentally the potential candidates for FFLO superconductors reported so far are stabilized by an external field[@gloos; @modler; @geg; @symington], recent theoretical studies suggest that an FFLO state may form in the ferromagnetic superconductor RuSr$_2$GdCu$_2$O$_8$[@pickett] without the benefit of an external magnetic field. Despite the disclaimer made above, we expect that the above results will be robust in the presence of a weak orbital effect, for example in quasi 2D systems where there is very weak interlayer coupling. In this case the orbital effects will be minimal when the magnetic field is applied in the layers.

In conclusion we have shown that while the Josephson current between a BCS and an FFLO superconductor is suppressed, it is restored by applying a magnetic field in the junction. The field strength and direction at which the Josephson current recovers depend upon the momenta of the FFLO order parameter, allowing the momenta to be measured. The observation of a field restored Josephson effect will provide unambiguous evidence for an FFLO phase.

This work was supported by NSF DMR-9971541 and the A. P. Sloan Foundation (KY), and by NSF DMR-9527035 and the State of Florida (DFA).

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Here for simplicity we assume that we are dealing with $s$-wave superconductors, whose order parameter depends only on the spatial coordinate ${\bf r}$, and has no non-trivial internal structure. However our results can be easily generalized to cases where the order parameters have non-trivial internal structure (like $d$-wave symmetry).

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