Sound waves and modulational instabilities on continuous-wave solutions in spinor Bose-Einstein condensates

We analyze sound waves (phonons, i.e. Bogoliubov excitations) propagating on continuous-wave (cw) solutions of repulsive $F=1$ spinor Bose-Einstein condensates (BECs) such as ${}^{23}\mathrm{Na}$ (which is antiferromagnetic or polar) and ${}^{87}\mathrm{Rb}$ (which is ferromagnetic). Zeeman splitting by a uniform magnetic field is included. All cw solutions to ferromagnetic BECs with vanishing $M_F=0$ particle density and nonzero components in both $M_F=\pm 1$ fields are subject to modulational instability (MI). Modulational instability increases with increasing particle density. Modulational instability also increases with differences in the components' wave numbers; this effect is larger at lower densities but becomes insignificant at higher particle densities. Continuous-wave solutions to antiferromagnetic (polar) BECs with vanishing $M_F=0$ particle density and nonzero components in both $M_F=\pm 1$ fields do not suffer MI if the wave numbers of the components are the same. If there is a wave-number difference, MI initially increases with increasing particle density and then peaks before dropping to zero beyond a given particle density. The cw solutions with particles in both $M_F=\pm 1$ components and nonvanishing $M_F=0$ components do not have MI if the wave numbers of the components are the same, but do exhibit MI when the wave numbers are different. Direct numerical simulations of a continuous wave with weak white noise confirm that weak noise grows fastest at wave numbers with the largest MI and show some of the results beyond small-amplitude perturbations. Phonon dispersion curves are computed numerically; we find analytic solutions for the phonon dispersion in a variety of limiting cases.

Figure 1

Band diagram (frequency as a function of wave vector) for phonons propagating on top of a CNLS-type cw solution (vanishing $M_F=0$ field) of a ${}^{87}\mathrm{Rb}$ BEC where the $M_F=1$ and $-1$ spin components have the same wave vectors and amplitudes 2 and $2.5$, respectively. The energies (chemical potentials) come from the real parts of the frequencies $\left(E=\hslash \mathrm{Re}\left[\omega \right]\right)$, which are shown as solid lines, and the MI comes from the imaginary parts of the frequencies, which are dotted lines. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 2

Band diagram (frequency as a function of wave vector) for phonons propagating on top of a CNLS-type cw solution (vanishing $M_F=0$ field) of a ${}^{87}\mathrm{Rb}$ BEC where the $M_F=1$ and $-1$ components have wave vectors that differ by $k_1-k_{-1}=1$ and amplitudes 2 and $2.5$, respectively. The energies (chemical potentials) come from the real parts of the frequencies $\left(E=\hslash \mathrm{Re}\left[\omega \right]\right)$, which are shown as solid lines, and the MI comes from the imaginary parts of the frequencies, which are dotted lines. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 3

Band diagram (frequency as a function of wave vector) for phonons propagating on top of a CNLS-type cw solution (vanishing $M_F=0$ field) of a ${}^{23}\mathrm{Na}$ BEC where the $M_F=1$ and $-1$ spin components have the same wave vectors and amplitudes 2 and $2.5$, respectively. The energies (chemical potentials) come from the real parts of the frequencies $\left(E=\hslash \mathrm{Re}\left[\omega \right]\right)$, which are shown as solid lines, and the MI comes from the imaginary parts of the frequencies, which are dotted lines. In this case, the frequencies are real valued, so there is no MI. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 4

Band diagram (frequency as a function of wave vector) for phonons propagating on top of a CNLS-type cw solution (vanishing $M_F=0$ field) of a ${}^{23}\mathrm{Na}$ BEC where the $M_F=1$ and $-1$ components have wave vectors that differ by $k_1-k_{-1}=1$ and amplitudes 2 and $2.5$, respectively. The energies (chemical potentials) come from the real parts of the frequencies $\left(E=\hslash \mathrm{Re}\left[\omega \right]\right)$, which are shown as solid lines, and the MI comes from the imaginary parts of the frequencies, which are dotted lines. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 5

Band diagram (frequency as a function of wave vector) for phonons propagating on top of an $\left(n=0\right)$-type cw solution of a ${}^{87}\mathrm{Rb}$ BEC where the $M_F=1$ and $-1$ spin components have the same wave vectors and amplitudes 2 and $2.5$, respectively. The energies (chemical potentials) come from the real parts of the frequencies $\left(E=\hslash \mathrm{Re}\left[\omega \right]\right)$, which are shown as solid lines. The imaginary parts of the frequencies, shown as dotted lines, are zero, so there is no MI.

Figure 6

Band diagram (frequency as a function of wave vector) for phonons propagating on top of an $\left(n=0\right)$-type cw solution of a ${}^{87}\mathrm{Rb}$ BEC where the $M_F=1$ and $-1$ components have wave vectors that differ by $k_1-k_{-1}=1$ and amplitudes 2 and $2.5$, respectively. The energies (chemical potentials) come from the real parts of the frequencies $\left(E=\hslash \mathrm{Re}\left[\omega \right]\right)$, which are shown as solid lines, and the MI comes from the imaginary parts of the frequencies, which are dotted lines. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 7

Band diagram (frequency as a function of wave vector) for phonons propagating on top of an $\left(n=0\right)$-type cw solution of a ${}^{23}\mathrm{Na}$ BEC where the $M_F=1$ and $-1$ spin components have the same wave vectors and amplitudes 2 and $2.5$, respectively. The energies (chemical potentials) come from the real parts of the frequencies $\left(E=\hslash \mathrm{Re}\left[\omega \right]\right)$, which are shown as solid lines. The imaginary parts of the frequencies, shown as dotted lines, which would create MI, are zero. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 8

Band diagram (frequency as a function of wave vector) for phonons propagating on top of an $\left(n=0\right)$-type cw solution of a ${}^{23}\mathrm{Na}$ BEC where the $M_F=1$ and $-1$ components have wave vectors that differ by $k_1-k_{-1}=1$ and amplitudes 2 and $2.5$, respectively. The energies (chemical potentials) come from the real parts of the frequencies $\left(E=\hslash \mathrm{Re}\left[\omega \right]\right)$, which are shown as solid lines, and the MI comes from the imaginary parts of the frequencies, which are dotted lines. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 9

Band diagram (frequency as a function of wave vector) for phonons propagating on top of an $\left(n=1\right)$-type cw solution of a ${}^{23}\mathrm{Na}$ BEC where the $M_F=1$ and $-1$ components have wave vectors that differ by $k_1-k_{-1}=1$ and amplitudes 2 and $2.5$, respectively. The energies (chemical potentials) come from the real parts of the frequencies $\left(E=\hslash \mathrm{Re}\left[\omega \right]\right)$, which are shown as solid lines, and the MI comes from the imaginary parts of the frequencies, which are dotted lines. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 10

Maximum MI of the CNLS-type cws (zero $M_F=0$ particle density) of a ${}^{87}\mathrm{Rb}$ BEC, as a function of the amplitudes of the $M_F=\pm 1$ fields, where the spin components have identical wave vectors $k_1=k_{-1}$ and quadratic Zeeman splitting is zero: (a) the peak MI values and (b) the wave vectors at which the maxima occur. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 11

Maximum MI of the CNLS-type cws (zero $M_F=0$ particle density) of a ${}^{87}\mathrm{Rb}$ BEC, as a function of the amplitudes of the $M_F=\pm 1$ fields, where the spin components have wave vectors that differ by $k_1-k_{-1}=1$ and the quadratic Zeeman splitting is zero: (a) the peak MI values and (b) the wave vectors at which the maxima occur. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 12

Maximum MI of CNLS-type (zero $M_F=0$ particle density) cws of a ${}^{23}\mathrm{Na}$ BEC, as a function of the amplitudes of the $M_F=\pm 1$ fields, where the spin components have wave vectors that differ by $k_1-k_{-1}=1$ and the quadratic Zeeman splitting is zero: (a) the peak MI values and (b) the wave vectors at which the maxima occur. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 13

Maximum MI of $\left(n=0\right)$-type cws of a ${}^{87}\mathrm{Rb}$ BEC, as a function of the amplitudes of the $M_F=\pm 1$ fields, where the spin components have wave vectors that differ by $k_1-k_{-1}=1$ and the quadratic Zeeman splitting is zero: (a) the peak MI values and (b) the wave vectors at which the maxima occur. The (dimensionless) reference amplitude for this plot is $A_{\mathrm{ref}}\equiv \sqrt{[{\hslash }^2{(k_1-k_{-1})}^2/8m+q{B}^2]/\left|c_2\right|}=\left(8\right|c_2{\left|\right)}^{-1/2}\approx 5.103$. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 14

Maximum MI of $\left(n=0\right)$-type cws of a ${}^{23}\mathrm{Na}$ BEC, as a function of the amplitudes of the $M_F=\pm 1$ fields, where the spin components have wave vectors that differ by $k_1-k_{-1}=1$ and the quadratic Zeeman splitting is zero: (a) the peak MI values and (b) the wave vectors at which the maxima occur. The (dimensionless) reference amplitude for this plot is $A_{\mathrm{ref}}\equiv \sqrt{[{\hslash }^2{(k_1-k_{-1})}^2/8m+q{B}^2]/c_2}={\left(8c_2\right)}^{-1/2}\approx 2.368$. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 15

Maximum MI of $\left(n=1\right)$-type cws of a ${}^{23}\mathrm{Na}$ BEC, as a function of the amplitudes of the $M_F=\pm 1$ fields, where the spin components have wave vectors that differ by $k_1-k_{-1}=1$ and the quadratic Zeeman splitting is zero: (a) the peak MI values and (b) the wave vectors at which the maxima occur. The (dimensionless) reference amplitude for this plot is $A_{\mathrm{ref}}\equiv \sqrt{[{\hslash }^2{(k_1-k_{-1})}^2/8m+q{B}^2]/\left|c_2\right|}={\left(8c_2\right)}^{-1/2}\approx 2.368$. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 16

(a) Modulational instability band diagram for phonons in a BEC with the (dimensionless) parameters $\hslash =1,m=1,c_0=1$, and $c_2=-.4793$, on top of an $\left(n=0\right)$-type cw with amplitudes $A_1=2.5,A_0=3.1826$, and $A_{-1}=2.0$; wave numbers $k_1=0.5,k_0=0$, and $k_{-1}=-0.5$; and frequencies ${\omega }_1=10.6861,{\omega }_0=10.6722$, and ${\omega }_{-1}=10.6583$. (b) Spectral density at the start. The peak in the middle is the cw and the remainder of the spectrum is statistically flat. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 17

Amplitudes of the BEC spin fields $M_F=1,0,-1$, in dimensionless variables, at time $t=176.7.$ The magnitudes of the amplitudes are solid lines, the real parts dashed lines, and the imaginary parts dotted. The cw structure is visible and the amplified noise is also large enough to be visible to the eye. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 18

Snapshot at dimensionless time $t=176.7$. (a) The (dimensionless) magnetization components $(m_x,m_y,m_z)$ and the particle density of the $m=0$ spin components $|{\varphi }_0{|}^2$. The pattern is the spin texture. (b) Squared amplitudes of the fields in momentum space, i.e., the particle density as a function of wave number, divided by the domain length. The BEC field with spins $M_F=1,0,-1$ are represented by solid, dashed, and dotted lined, respectively. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 19

Amplitudes of the BEC spin fields $M_F=1,0,-1$ at (dimensionless) time $t=372.9$. The magnitudes of the amplitudes are solid lines, the real parts dashed lines, and the imaginary parts dotted. This is a snapshot of dynamical turbulence. The cw has been destroyed by amplified noise and phonons are no longer helpful in describing the system. All quantities in the figure are dimensionless; see Eqs. (3).

Figure 20

Snapshot at dimensionless time $t=372.9$. (a) The (dimensionless) magnetization components $(m_x,m_y,m_z)$ and the particle density of the $m=0$ spin components $|{\varphi }_0{|}^2$. The pattern is the spin texture. (b) Squared amplitudes of the fields in momentum space, i.e., the particle density as a function of wave number, divided by the domain length. The BEC fields with spins $M_F=1,0,-1$ are represented by solid, dashed, and dotted lines, respectively. All quantities in the figure are dimensionless; see Eqs. (3).