Double Bragg diffraction: A tool for atom optics

The use of retroreflection in light-pulse atom interferometry under microgravity conditions naturally leads to a double-diffraction scheme. The two pairs of counterpropagating beams induce simultaneously transitions with opposite momentum transfer that, when acting on atoms initially at rest, give rise to symmetric interferometer configurations where the total momentum transfer is automatically doubled and where a number of noise sources and systematic effects cancel out. Here we extend earlier implementations for Raman transitions to the case of Bragg diffraction. In contrast with the single-diffraction case, the existence of additional off-resonant transitions between resonantly connected states precludes the use of the adiabatic elimination technique. Nevertheless, we have been able to obtain analytic results even beyond the deep Bragg regime by employing the so-called “method of averaging,” which can be applied to more general situations of this kind. Our results have been validated by comparison to numerical solutions of the basic equations describing the double-diffraction process.

Figure 1

Schematic setup of Bragg diffraction. A two-level atom is interacting with two counterpropagating light fields of frequencies ${\omega }_a$ and ${\omega }_b$, which are far detuned from the atomic transition.

Figure 2

Momentum transfer and resonances in Bragg diffraction. The absorption of a photon from one field and the emission of a photon into the field traveling in the opposite direction results in a recoil of the value $\hslash K\equiv \hslash (k_b+k_a)$. This process is resonant, if the difference between the two lasers $\Delta \omega ={\omega }_b-{\omega }_a$ is equal to the change of kinetic energy, which is proportional to $p^2$. The dashed transitions are off resonant and suppressed; the gray line denotes the sum of potential and kinetic energy of the virtual excited state. ${\omega }_{\mathrm{r}}=\hslash K^2/\left(2M\right)$ denotes the recoil frequency.

Figure 3

Populations $|g_n^{\left(1\right)}{|}^2$ of momentum states for single Bragg diffraction in the quasi-Bragg regime. The initial condition ${\mathit{\gamma }}^{\left(1\right)}\left(0\right)={(0,1,0,0)}^{\mathrm{T}}$ corresponds to the dressed-state formulation, and the adiabaticity parameter is $\varepsilon =0.1$. The two bottom lines are magnified by a factor of 50 and show the small populations in the off-resonant states $n=-1,2$.

Figure 4

Populations $|g_n^{\left(1\right)}{|}^2$ of momentum states for single Bragg diffraction in the quasi-Bragg regime. In contrast to Fig. 3 now the initial condition is ${\mathit{g}}^{\left(1\right)}\left(0\right)={(0,1,0,0)}^{\mathrm{T}}$, which corresponds to the bare-state formulation. The adiabaticity parameter is again $\varepsilon =0.1$. The two bottom lines are magnified by a factor of 50 and show the small populations in the off-resonant states $n=-1,2$. The dotted lines represent numerical results.

Figure 5

Schematic setup of double Bragg diffraction. Two pairs of counterpropagating light waves (pair 1 and pair 2) induce a Bragg scattering process in opposite directions. In order to distinguish both pairs, they have orthogonal polarizations ${\mathit{\sigma }}_1$ and ${\mathit{\sigma }}_2$. This orthogonality can be achieved by a retroreflecting geometry using a mirror and a $\lambda /4$ wave plate.

Figure 6

Transitions between magnetic sublevels due to circular polarizations. Laser pair 1 with polarization ${\mathit{\sigma }}_{-}$ drives the transitions between $|g\rangle$ and $|e_{-}\rangle$, which have the energy difference $\hslash {\omega }_{-}$, with a Rabi frequency ${\Omega }_{-}$; laser pair 2 with polarization ${\mathit{\sigma }}_{+}$ causes the transitions between $|g\rangle$ and $|e_{+}\rangle$, which have the energy difference $\hslash {\omega }_{+}$, with a Rabi frequency ${\Omega }_{+}$. The kinetic energy is neglected in this figure and $m_z$ denotes the magnetic quantum number.

Figure 7

Momentum transfer and resonances in double Bragg diffraction. The Bragg scattering processes induced by each pair of lasers correspond to the one of Fig. 2, but are of opposite directions. The frequencies of the excited states are ${\omega }_{+}$ and ${\omega }_{-}$. The dashed transitions are off resonant and get suppressed.

Figure 8

Rabi oscillations to realize a beam splitter (a) and mirror (b) in the deep Bragg regime. For the adiabaticity parameter $\varepsilon \ll 1$ the quantity $|{\gamma }_n^{\left(1\right)}{|}^2$ corresponds to the population of a momentum state $|n\hslash K\rangle$. For the demonstration of a beam splitter (a) the initial condition is ${(0,1,0)}^{\mathrm{T}}$. Starting in $|0\rangle$, a superposition $\left(\right|-\hslash K\rangle +|\hslash K\rangle )/\sqrt{2}$ is generated for $\sqrt{2}\Omega t=\pi /2$. For the demonstration of a mirror (b) the initial condition is ${(1,0,0)}^{\mathrm{T}}$. Starting in $|\hslash K\rangle$, the whole population is transferred to $|-\hslash K\rangle$ at the time $\sqrt{2}\Omega t=\pi$.

Figure 9

Beam splitter in the quasi-Bragg regime for the adiabaticity parameter $\varepsilon =0.1$ represented by the population $|g_n^{\left(1\right)}{|}^2$ of the momentum state $|n\hslash K\rangle$. The initial condition is ${\mathit{g}}^{\left(1\right)}\left(0\right)={(0,1,0)}^{\mathrm{T}}$. The dashed lines are the result of a numerical simulation, the solid lines are the approximate analytical solutions. In this regime, fast oscillations with small amplitude modify the Rabi oscillations from Fig. 8.

Figure 10

Rabi oscillations of the populations $|{\gamma }_n^{\left(1\right)}{|}^2$ of the momentum states $|(n+0.01)\hslash K\rangle$ in the deep Bragg regime ($\varepsilon =0.01$) for the off-resonant momentum $p=0.01\hslash K$. The Rabi oscillations are detuned and suppressed, which leads to velocity selectivity. The initial condition is ${(0,1,0)}^{\mathrm{T}}$ and the time is in units of the resonant frequency $\sqrt{2}\Omega$.

Figure 11

Velocity selectivity of double Bragg diffraction expressed by the Rabi oscillation of the population $|{\gamma }_{\pm 1}^{\left(1\right)}{|}^2$ in their dependence on the initial momentum (a). For $p=0$, i.e., for the resonant momenta $\pm \hslash K$, we get full Rabi oscillations, but as we leave this resonance this process is suppressed and detuned. The difference between analytical and numerical solutions (b) shows a large deviation between both solutions. This deviation is mainly due to different frequencies, not to different amplitudes. The width of the resonance is comparable in both solutions. Here we have chosen $\varepsilon =0.01$.

Figure 12

Damped Rabi oscillations in the numerically integrated population $N_{\pm 1}$ when the adiabaticity parameter is $\varepsilon =0.01$. For a broad momentum distribution, the oscillations get damped out due to the average over different frequencies and amplitudes. This effect is less pronounced for smaller values of the momentum width ${\sigma }_p$.

Figure 13

Two processes in second-order double Bragg diffraction ($\Delta \omega =2{\omega }_{\mathrm{r}}$). A four-photon transition in each direction occurs. For an initial population of the momentum states $|0\rangle$ and $|\pm 2\hslash K\rangle$, the states $|\pm \hslash K\rangle$ are just virtually occupied. Different initial conditions are sketched in Fig. 14. The dashed lines show off-resonant processes.

Figure 14

Scattering process between the states $|\pm \hslash K\rangle$ according to Eq. (37). If the atoms are initially in one of these states, a resonant Rabi oscillation between them takes place in second-order Bragg diffraction. This phenomenon can be explained by the resonant four-photon process shown above. The transitions for other initial conditions are shown in Fig. 13.

Figure 15

Rabi oscillations in second-order double Bragg diffraction, where $|{\gamma }_n^{\left(2\right)}{|}^2$ corresponds to the population of the momentum state $|n\hslash K\rangle$ for $\varepsilon \ll 1$. The initial condition ${(0,0,1,0,0)}^{\mathrm{T}}$ was chosen to focus on the diffraction process $|0\rangle \leftrightarrow |\pm 2\hslash K\rangle$, where the states $|\pm \hslash K\rangle$ are not involved. The oscillations are slightly suppressed due to an asymmetric ac Stark shift. The dashed lines show the numerical solution for $\varepsilon =0.01$.

Figure 16

Rabi oscillations in second-order double Bragg diffraction after elimination of the asymmetric ac Stark shift. The elimination was achieved by choosing $\Delta \omega =(2-32/105{\varepsilon }^2){\omega }_{\mathrm{r}}$ (solid line). The initial condition was again ${(0,0,1,0,0)}^{\mathrm{T}}$. The oscillations are not detuned anymore. The dashed line which represents the numerical solution for $\varepsilon =0.01$ and the same choice of $\Delta \omega$ is in agreement with the approximate analytical result.

Figure 17

Quasiresonances in first-order double Bragg diffraction for $\Delta \omega ={\omega }_{\mathrm{r}}$. Even though we consider first-order diffraction, a second-order process between momenta $|\pm \hslash K/2\rangle$ and $|\pm \hslash 3K/2\rangle$ is resonant.

Figure 18

Rabi oscillations between the quasiresonances $|-\hslash K/2\rangle \leftrightarrow |3\hslash K/2\rangle$ for $\Delta \omega ={\omega }_{\mathrm{r}}$ (solid lines). The oscillations are suppressed by approximately one half. The dashed lines show a numerical solution of the Rabi equations for $\varepsilon =0.01$ and confirm our analytical result.

Figure 19

Linear-linear configuration for double Bragg diffraction. The magnetic field $\mathit{B}$ is not in the $z$ direction of the pair of light fields, but in a different one denoted by $z^{\prime }$. The polarizations of the two pairs are linear and orthogonal to the $z$ direction of propagation, i.e., ${\mathit{\sigma }}_x$ and ${\mathit{\sigma }}_y$.

Figure 20

Atomic level structure. We consider three excited states $|e_{\pm }\rangle$ and $|e_z\rangle$ with the frequencies ${\omega }_{\pm }$ and ${\omega }_z$, respectively, and one ground state $|g\rangle$. The dipole moments ${\wp }_{\pm }$ and ${\wp }_z$ initiate transitions between sublevels and correspond to the polarizations ${\mathit{\sigma }}_{\pm }^{\prime }$ and ${\mathit{\sigma }}_z^{\prime }$ in the primed frame.