Raman coupling of Zeeman sublevels in an alkali-metal Bose-Einstein condensate

We investigate amplitude and phase control of the components of the spinor order parameter of a ${}^{87}\mathrm{Rb}$ Bose-Einstein condensate. By modeling the interaction of the multilevel atomic system with a pair of Raman-detuned laser pulses, we show that it is possible to construct a pulse-sequence protocol for producing a desired state change within a single Zeeman manifold. We present several successful elementary tests of this protocol in both the $F=1$ and $F=2$ Zeeman manifolds of ${}^{87}\mathrm{Rb}$ using the $D_1$ transitions. We describe specific features of the interaction, which are important for multimode, spatially varying field configurations, including the role of state-dependent, light-induced potentials.

Figure 1

Experimental geometry indicating the relative orientation of the Raman beams, the magnetic quantization axis, and gravity. The inset is a schematic indicating the linkage pattern characteristic of the ${\sigma }^{+}\text{−}{\sigma }^{-}$ coupling configuration.

Figure 2

State linkage diagram for $({\sigma }^{+},{\sigma }^{-})$ Raman coupling of the $F=1$ ground-state manifold of ${}^{87}\mathrm{Rb}$ via the $D_1$ transitions. The magnitude of the Zeeman shift of the states by the small magnetic bias field is exaggerated for clarity.

Figure 3

Representation of the fictitious magnetic field due to the application of the ${\sigma }^{+}$ laser field (left), the ${\sigma }^{-}$ field (center), and both ${\sigma }^{+}$ and ${\sigma }^{-}$ fields together (right). For these plots, the detuning is assumed to be midway between the $F^{\prime }=1$ and $F^{\prime }=2$ excited state manifolds.

Figure 4

Detuning dependence of the scalar light shift $\tilde{\alpha }$, Zeeman light shift $\tilde{\delta }$, and two-photon Rabi frequency $\tilde{\Omega }$. The scalar and Zeeman light shifts are expressed in terms of their components, which depend on intensities of the ${\sigma }^{+}$ and ${\sigma }^{-}$ polarized laser fields. ($I_{+},I_{-}=1\mathrm{mW}∕{\mathrm{cm}}^2$, $B=1.33\mathrm{G}$.)

Figure 5

Detuning dependence of the intensities required to cause a 180° $\theta$ rotation of the pseudospin vector describing the Raman-coupled $\mid F=1,m_F=-1⟩$ and $\mid F=1,m_F=1⟩$ states, subject to the condition that $\tilde{\delta }=0$. Negative intensities correspond to a sign change in the direction of rotation of $\theta$. Pulse duration $\tau =20\mu \mathrm{s}$. Magnetic bias field $B=1.33\mathrm{G}$.

Figure 6

Detuning dependence of the intensities required to cause a 180° $\varphi$ rotation of the pseudospin vector describing the Raman-coupled $\mid F=1,m_F=-1⟩$ and $\mid F=1,m_F=1⟩$ states. Negative intensities correspond to a sign change in the direction of rotation of $\varphi$. $\tau =20\mu \mathrm{s}$. $B=1.33\mathrm{G}$.

Figure 7

State linkage diagram for $({\sigma }^{+},{\sigma }^{-})$ Raman coupling of the $F=2$ ground-state manifold of ${}^{87}\mathrm{Rb}$ via the $D_1$ transitions.

Figure 8

Detuning dependence of the scalar light shift ${\tilde{\alpha }}_{23}$, Zeeman light shift ${\tilde{\delta }}_{23}$, and two-photon Rabi frequency ${\tilde{\Omega }}_{23}$ associated with the $2\leftrightarrow 3$ subspace of the $F=2$ manifold. The scalar and Zeeman light shifts are expressed in terms of their components, which depend on the intensities of the ${\sigma }^{+}$ and ${\sigma }^{-}$ polarized laser fields. ($I_{+}$, $I_{-}=1\mathrm{mW}∕{\mathrm{cm}}^2$; $B=17\mathrm{G}$.)

Figure 9

Detuning dependence of the scalar light shift ${\tilde{\alpha }}_{12}$, Zeeman light shift ${\tilde{\delta }}_{12}$, and two-photon Rabi frequency ${\tilde{\Omega }}_{12}$ associated with the $1\leftrightarrow 2$ subspace of the $F=2$ manifold. The scalar and Zeeman light shifts are expressed in terms of their components, which depend on the intensities of the ${\sigma }^{+}$ and ${\sigma }^{-}$ polarized laser fields. ($I_{+}$, $I_{-}=1\mathrm{mW}∕{\mathrm{cm}}^2$; $B=17\mathrm{G}$.)

Figure 10

Plot of the intensities required to cause a 180° $\theta$ rotation (a) or $\varphi$ rotation (b) of the pseudospin vector describing the Raman-coupled $\mid F=2,m_F=0⟩$ and $\mid F=2,m_F=2⟩$ states, subject to the condition that ${\tilde{\delta }}_{23}=0$. Negative intensities correspond to a sign change in the direction of rotation of $\theta$. Cross hatching indicates imaginary-valued, nonphysical solutions. Pulse duration $\tau =20\mu \mathrm{s}$. Magnetic bias field $B=17\mathrm{G}$.

Figure 11

Population in the ∣1,1⟩ state after a $20\mu \mathrm{s}$ Raman laser pulse as a function of square root of the product of the beam intensities. The vertical error bars indicate the scatter range in the measurements, and the horizontal error bars indicate the uncertainty in the beam intensity due to laser power fluctuations. The gray line is the theoretically predicted intensity dependence.

Figure 12

Population in the ∣2,0⟩ state after a $20\mu \mathrm{s}$ Raman laser pulse as a function of square root of the product of the beam intensities. The vertical error bars indicate the scatter range in the measurements, and the horizontal error bars indicate the uncertainty in the beam intensity due to laser power fluctuations.

Figure 13

Applying two successive $\theta =\pi ∕2$ pulses with no phase shift results in complete transfer to the final state (a). Turning on only one of the Raman lasers causes a phase shift, rotating the pseudospin vector in $\varphi$. For a $\varphi =\pi$ rotation (b), the population returns to the initial state.

Figure 14

Experimental results of applying the $\theta =\pi ∕2$, $\varphi$, $\theta =\pi ∕2$ pulse sequence described above to the $\mid 1,-1⟩\leftrightarrow \mid 1,1⟩$ pseudospin system. Application of a $\varphi$ changing pulse is shown to cause the expected oscillation in the final state population. The gray line is a theoretical prediction taking into account a nonzero $\tilde{\delta }$ caused by lowering the ${\sigma }_{-}$ beam intensity by 5%.

Figure 15

Fractional population transferred to ∣2,0⟩ and $\mid 2,-2⟩$ from a BEC initially in the ∣2,2⟩ state by a $20\mu \mathrm{s}$ Raman laser pulse for different choices of the two-photon detuning $\delta$. The magnitude and ratio of the intensities were set for an effective $\theta =1.5\pi$ pulse area on two-photon resonance with the $\mid 2,2⟩\leftrightarrow \mid 2,0⟩$ transition, with the detuning of the beams set at ${\Delta }_{D_1}=-800\mathrm{MHz}$. The applied magnetic bias field was $17\mathrm{G}$, for which ${\delta }_{12}-{\delta }_{23}=166\mathrm{kHz}$.

Figure 16

Numerical prediction of the population transferred to ∣2,0⟩ (a) and $\mid 2,-2⟩$ (b) from a BEC initially in the ∣2,2⟩ state, as a function of two-photon detuning and effective pulse area. The ratio of the intensities was set to eliminate $\tilde{\delta }$ for the $\mid 2,2⟩\leftrightarrow \mid 2,0⟩$ transition. ${\Delta }_{D_1}=-800\mathrm{MHz}$, $B=17\mathrm{G}$, for which ${\delta }_{12}-{\delta }_{23}=166\mathrm{kHz}$. The theoretical curves shown in Fig. 15 are lineouts from these plots at the indicated location of $\theta =1.5\pi$. The highest peaks correspond to complete population transfer.