Josephson and persistent spin currents in Bose-Einstein condensates of magnons

Using the Aharonov-Casher (A-C) phase, we present a microscopic theory of the Josephson and persistent spin currents in quasiequilibrium Bose-Einstein condensates (BECs) of magnons in ferromagnetic insulators. Starting from a microscopic spin model that we map onto a Gross-Pitaevskii Hamiltonian, we derive a two-state model for the Josephson junction between the weakly coupled magnon-BECs. We then show how to obtain the alternating-current (ac) Josephson effect with magnons as well as macroscopic quantum self-trapping in a magnon-BEC. We next propose how to control the direct-current (dc) Josephson effect electrically using the A-C phase, which is the geometric phase acquired by magnons moving in an electric field. Finally, we introduce a magnon-BEC ring and show that persistent magnon-BEC currents flow due to the A-C phase. Focusing on the feature that the persistent magnon-BEC current is a steady flow of magnetic dipoles that produces an electric field, we propose a method to directly measure it experimentally.

Figure 1

(a) Schematic representation of the quasiequilibrium magnon-BEC Josephson junction (MJJ) consisting of two coupled ferromagnetic insulators (FI) in the presence of magnetic fields ${\mathbf{B}}_{\mathrm{L}}$ and ${\mathbf{B}}_{\mathrm{R}}$. Each cloud of circles represents a single magnon-BEC. A quasiequilibrium magnon-BEC corresponds to a macroscopic coherent precession of the spins in each FI. The boundary spins ${\mathbf{S}}_{{\Gamma }_{\mathrm{L}}}$ in the left FI and ${\mathbf{S}}_{{\Gamma }_{\mathrm{R}}}$ in the right FI are relevant to the Josephson magnon current. (b) Close-up of the MJJ. The two FIs are separated by an interface of width $\Delta x$ and thereby weakly exchange-coupled with strength $J_{\mathrm{ex}}$. The variable $J$ represents the exchange interaction between neighboring spins in each FI. The applied electric field, $\mathbf{E}=E{\mathbf{e}}_y$, couples to the magnons through the A-C phase.

Figure 2

The ac Josephson effect: plots of the population imbalance $z\left(\tau \right)$ as function of the rescaled time $\tau =(2J_{\mathrm{ex}}S/\hslash )t$ obtained by numerically solving Eqs. (10a) and (10b) for the values (a) $\Delta E=0.1$, $\Lambda =0$, ${\theta }_{\mathrm{A}\text{−}\mathrm{C}}=0$, $z\left(0\right)=0.6$, and $\theta \left(0\right)=0$. As an example, for $J_{\mathrm{ex}}=0.25$ $\mu \mathrm{eV}$, $S=2$, and $g=2$, this corresponds to $B_{\mathrm{L}}-B_{\mathrm{R}}=1$ mT and the period of an oscillation is $\mathcal{T}=6$ ns (i.e., the rescaled time $\tau =1$ corresponds to $t=1$ ns). (b)–(d) show cases with vanishing $\Delta E={\theta }_{\mathrm{A}\text{−}\mathrm{C}}=0$, $z\left(0\right)=0.6$, and $\theta \left(0\right)=0$, which give ${\Lambda }_{\mathrm{c}}^{\mathrm{MJJ}}=10$. (b) $\Lambda =1$, (c) $9.99$, and (d) 11. When the value of $\Lambda$ exceeds ${\Lambda }_{\mathrm{c}}^{\mathrm{MJJ}}$, $\Lambda >{\Lambda }_{\mathrm{c}}^{\mathrm{MJJ}}$, the MQST occurs as shown in (d).

Figure 3

The dc Josephson effect through the A-C phase: plots of the population imbalance $z\left(\tau \right)$ and the relative phase $\theta \left(\tau \right)$ as function of the rescaled time $\tau =(2J_{\mathrm{ex}}S/\hslash )t$ obtained by numerically solving Eqs. (14a) and (14b) for the values (a) (b) $z_0={10}^{-6}$, $\Lambda ={10}^3(\gg 1)$, and $z\left(0\right)=\theta \left(0\right)=0$. (a) ${\theta }_{\mathrm{A}\text{−}\mathrm{C}}=0$, in which the period of the small oscillation is estimated as $\mathcal{T}\approx 0.2$ ns. (b) ${\theta }_{\mathrm{A}\text{−}\mathrm{C}}=-arcsin\left(z_0\right)=-{10}^{-6}$. (c) The breakdown of the dc Josephson effect (${\theta }_{\mathrm{A}\text{−}\mathrm{C}}=0$) due to increase of $b_0$. The transition between the dc region [(i) and (ii)] and the ac one [(iii) and (iv)] takes place for $z_0\approx 0.725$ due to the absence of the A-C phase. Under the conditions $\left(z\right(0),\theta (0\left)\right)=(0,0)$ and $\Lambda =100$ ($\gg 1$), each $z_0=b_0/\Lambda$ reads (i) 0.100, (ii) 0.724, (iii) 0.726, and (iv) 1.100. (d) Example of the recovery of the dc Josephson effect from the ac effect through the A-C phase in the region $z_0\le 1$ (i.e., $b_0\le \Lambda$). (iii) ${\theta }_{\mathrm{A}\text{−}\mathrm{C}}=0$, (v) ${\theta }_{\mathrm{A}\text{−}\mathrm{C}}=-arcsin\left(z_0\right)$.

Figure 4

(a) A schematic picture of the magnon-BEC ring. The radius of the ring reads $R$ and the applied electric field is $\mathbf{E}(\rho ,\phi )=\mathcal{E}\rho {\mathbf{e}}_{\rho }$. (b) An enlarged view of the cylindrical wire that forms the ring. The radius of the wire is ${\rho }_0$. The persistent magnon-BEC current $I_{\mathrm{BEC}}^{\mathrm{ring}}$ flows in the ring due to the A-C phase.

Figure 5

A schematic picture of the cross-section of the cylindrical wire. The electric dipole field ${\mathbf{E}}_{\mathrm{m}}$ arises from the persistent magnon-BEC current $I_{\mathrm{BEC}}^{\mathrm{ring}}$ (i.e., the magnetic dipole steady current) flowing in the magnon-BEC ring. This leads to a measurable voltage drop $V_{\mathrm{m}}$ between the points (i) and (ii). The points are (i) $(y,z)=(0,-r_0)$ and (ii) $(y,z)=(-r_0,0)$.

Figure 6

(a) A schematic picture of a magnon-BEC ring. (b) An enlarged view of the isotropic cylindrical wire that forms the magnon-BEC ring ($r_0={\rho }_0={\rho }_1={\rho }_2$). The cross-section is $\pi {\rho }_0^2$. The voltage drop due to the persistent magnon-BEC current $I_{\mathrm{BEC}}^{\mathrm{ring}}$ is $V_{\mathrm{m}}$ and the one arising directly from the applied electric field is $V$. The points are (i) $(y,z)=(0,-r_0)$ and (ii) $(y,z)=(-r_0,0)$ (see also Fig. 5). (c) An enlarged view of the anisotropic cylindrical wire that forms the ring (${\rho }_1\ll {\rho }_2$; e.g., ${\rho }_2=100{\rho }_1$). Consequently, the voltage drop $V$ directly due to the applied electric field becomes much smaller (about ${10}^{-2}$ times) than the one of (b). The cross-section remains about the same with (b). Therefore it generates about the same amount of the persistent magnon-BEC current $I_{\mathrm{BEC}}^{\mathrm{ring}}$ as for that of (b).