Regimes of atomic diffraction: Raman versus Bragg diffraction in retroreflective geometries

We provide a comprehensive study of atomic Raman and Bragg diffraction when coupling to a pair of counterpropagating light gratings (double diffraction) or to a single one (single diffraction) and discuss the transition from one case to the other in a retroreflective geometry as the Doppler detuning changes. In contrast to single diffraction, double Raman loses its advantage of high diffraction efficiency for short pulses and has to be performed in a Bragg-type regime. Moreover, the structure of double diffraction leads to further limitations for broad momentum distributions on the efficiency of mirror pulses, making the use of (ultra)cold ensembles essential for high diffraction efficiency.

Figure 1

Schematic setup of single diffraction in a retroreflective geometry, realized by a mirror and a $\lambda /4$ wave plate. The atom with initial momentum $p_0$ interacts with only one of the two counterpropagating optical gratings since the other one is Doppler detuned. Each grating consists of two light fields with frequencies ${\omega }_{\mathrm{b}}$ and ${\omega }_{\mathrm{r}}$, whose corresponding wave numbers are $k_{\mathrm{b}}$ and $k_{\mathrm{r}}$, respectively.

Figure 2

Single Raman versus single Bragg diffraction. The schematic setup (a) shows an atom interacting with two counterpropagating light fields of frequencies ${\omega }_{\mathrm{b}}$ and ${\omega }_{\mathrm{r}}$. A photon from a plane wave with frequency ${\omega }_{\mathrm{b}}$ is absorbed, while a subsequent photon with frequency ${\omega }_{\mathrm{r}}$ is emitted in the opposite direction causing a total recoil of $\hslash K\equiv \hslash (k_{\mathrm{b}}+k_{\mathrm{r}})$. During the process, the atom gains kinetic energy $\hslash {\omega }_K$, the recoil energy. The energy-momentum diagram for single Raman diffraction (b) shows that this process is resonant if the energy difference $\hslash ({\omega }_{\mathrm{b}}-{\omega }_{\mathrm{r}})$ between the light fields is equal to the recoil energy in addition to the energy difference $\hslash {\omega }_{\mathrm{eg}}$ between the internal ground state $\left|\mathrm{g}\right.〉$ and excited state $\left|\mathrm{e}\right.〉$. For single Bragg diffraction (c), the process is resonant if $\hslash ({\omega }_{\mathrm{b}}-{\omega }_{\mathrm{r}})$ is equal to the recoil energy. The dashed lines denote off-resonant higher-order transitions.

Figure 3

Energy-momentum diagrams for double Raman (a) and double Bragg diffraction (b). The atom interacts with two pairs of counterpropagating light fields and gains during this process the momentum $\hslash K$ which can be associated with the recoil frequency ${\omega }_K$. The two optical gratings allow transitions in opposite directions. Higher-order diffraction is possible for both mechanisms, as shown by dashed lines. They are off-resonant by integer multiples of the recoil energy $\hslash {\omega }_K$. Here, some of the transitions in Raman diffraction have been omitted, as they are off-resonant by the energy difference $\hslash {\omega }_{\mathrm{eg}}$ between ground state $|\mathrm{g}\rangle$ and excited state $|\mathrm{e}\rangle$.

Figure 4

Transition function $|G_{\mathrm{\Delta }\tau }^{\left(\mathrm{M}\right)}(p_f,p_i){|}^2$ for a single-Bragg mirror pulse of a Gaussian width $\mathrm{\Delta }\tau =12.5\mu \mathrm{s}$. The diagonals denote the momentum transfer between $p_i$ and $p_f$ which is always an integer multiple of $\hslash K$. The process $\left|0\right.〉\to \left|\hslash K\right.〉$ (green rectangles) is resonant. The parameter regime is such that also second-order processes $\left|-0.5\hslash K\right.〉\to \left|1.5\hslash K\right.〉$ (blue circles) have a nonvanishing probability. The resonance width ${\mathcal{W}}_{\mathrm{S}}$ is defined through the FWHM in Eq. (12a).

Figure 5

Momentum distribution after a single Bragg diffraction mirror pulse of duration $\mathrm{\Delta }\tau =12.5\mu \mathrm{s}$. The initial distribution $|{\psi }_i{|}^2$ of width $\mathrm{\Delta }\wp =0.05\hslash K$ is diffracted to the final distribution $|{\psi }_{\mathrm{f}}{|}^2$. In this process not only the resonant momentum state $\left|\hslash K\right.〉$ but also spurious (i.e., nonresonant) momentum states such as $\left|-\hslash K\right.〉$ and $\left|2\hslash K\right.〉$ are populated. Moreover, some population remains in the initial momentum state $\left|0\right.〉$.

Figure 6

Resonance width $\mathcal{W}$, defined by Eq. (12), of single Raman (SRD), double Raman (DRD), single Bragg (SBD) and double Bragg (DBD) diffraction for Gaussian mirror pulses with a varying pulse duration $\mathrm{\Delta }\tau$. In general, single diffraction shows always a larger width than double diffraction. For short times, Raman and Bragg differ: Bragg diffraction has a richer structure in this regime, especially double Bragg diffraction. For short pulse durations the initially distinct peaks of the quasiresonances present in double Bragg start to merge with the central peak causing a large variation of $\mathcal{W}$. For $\mathcal{W}/\left(\hslash K\right)$ larger than unity, the width of the resonance exceeds the separation of the resonant momentum states.

Figure 7

Diffraction efficiency $\mathcal{E}$ defined by Eq. (13) for single (top) and double (bottom) Raman (left) and Bragg (right) mirror pulses. In all four figures we vary the pulse duration $\mathrm{\Delta }\tau$ and the momentum width $\mathrm{\Delta }\wp$ of the initial state. The efficiency decreases with increasing width and increasing pulse duration. One can see that single Raman diffraction (a) exhibits a high efficiency for the widest parameter range. For single Bragg (b) and double Raman (c) off-resonant higher orders lead to a decrease of efficiency for small $\mathrm{\Delta }\tau$. Additional off-resonant couplings lead to a further decrease for double Bragg diffraction (d), while quasiresonances improve the efficiency for particular durations. The vertical and horizontal dotted lines denote cuts through the three-dimensional distributions and the corresponding curves are presented in Fig. 8.

Figure 8

Diffraction efficiency $\mathcal{E}$ defined by Eq. (13) for single and double Raman (SRD and DRD) as well as single and double Bragg (SBD and DBD) mirror pulses. In panel (a) we use the pulse duration $\mathrm{\Delta }\tau =37.5\mu \mathrm{s}$, vary the momentum width $\mathrm{\Delta }\wp$ and note that the diffraction geometry governs the behavior. In panel (b) we chose $\mathrm{\Delta }\tau =25\mu \mathrm{s}$, where the efficiency of double Bragg spikes, see Fig. 7, due to quasiresonances. Thus, its efficiency approaches the single-diffraction efficiency for broad distributions. In panel (c) the pulse duration $\mathrm{\Delta }\tau$ is varied, while the momentum width of the initial state is kept at $\mathrm{\Delta }\wp =0.1\hslash K$. Except for single Raman, off-resonant higher orders decrease the efficiency for small $\mathrm{\Delta }\tau$.

Figure 9

Diffraction losses $\mathcal{L}$ defined by Eq. (14) as a function of the pulse duration $\mathrm{\Delta }\tau$ displayed for fixed momentum width $\mathrm{\Delta }\wp =0.1\hslash K$ and for the four diffraction processes [single Raman: SRD (light orange dashed), single Bragg: SBD (light green solid), double Raman: DRD (dark red dashed), and double Bragg: DBD (dark blue solid)] discussed in this article. For beam splitters (a), losses are only relevant for $\mathrm{\Delta }\tau <20\mu \mathrm{s}$ except for single Raman, which experiences no losses at all. For mirror pulses (b), diffraction losses appear in the case of single Bragg diffraction for small $\mathrm{\Delta }\tau$. However, in double diffraction we find additional losses for $\mathrm{\Delta }\tau >20\mu \mathrm{s}$.

Figure 10

Population in states $|-\hslash K\rangle$, $|0\rangle$, $|\hslash K\rangle$ and off-resonant higher orders $|\mathrm{other}\rangle$ after a mirror pulse for double Raman (top) and double Bragg (bottom). For a broad momentum distribution $\mathrm{\Delta }\wp =0.1\hslash K$, shown in panels (a) and (c), a significant part of the population is lost to the intermediate state $|0\rangle$ and off-resonant higher orders are only populated for short pulse durations. Whereas the latter is also true for narrow momentum distributions with $\mathrm{\Delta }\wp =0.01\hslash K$, illustrated in panels (b) and (d), we see for intermediate times an almost perfect transfer of the population and a suppression of the losses to the intermediate state.

Figure 11

Resonant (solid blue and dashed green) and spurious (dashed gray) paths in a Mach-Zehnder atom interferometer configuration for single (a) and double (b) diffraction. Through a phase scan, we are able to obtain an interference signal $I\left(\delta \varphi \right)$ in panel (c) with amplitude $A$ and contrast $C$.

Figure 12

Amplitude $A$ of a simple Mach-Zehnder interferometer as a function of the momentum width $\mathrm{\Delta }\wp$ of the initial wave packet and the pulse duration $\mathrm{\Delta }\tau$. We observe for single Raman (a), single Bragg (b), double Raman (c), and double Bragg (d) how the imperfections of the diffracting pulses influence the interference signal. In particular, the effect of the Bragg regime as well as quasiresonances are reflected in the amplitude of the signal.

Figure 13

Interferometric contrast $C$ of a double-Bragg Mach-Zehnder interferometer as a function of the momentum width $\mathrm{\Delta }\wp$ of the initial wave packet and the pulse duration $\mathrm{\Delta }\tau$. The contrast is close to unity except for parameters where quasiresonances dominate the behavior, where it drops to 0.5. However, because of the impracticality of blow-away pulses for Bragg, additional contributions from spurious paths, which are not included in this treatment, may dominate the contrast.

Figure 14

Energy-momentum diagram for double Bragg diffraction (a) for an initial momentum offset $p_0$ resonantly connected to $p_0+\hslash K$. Pulse area ${\mathcal{A}}_{\text{opt}}$ defined for single diffraction (b) leading to an optimal diffraction efficiency for an initial momentum eigenstate $\left|p_0\right.〉$ and a pulse duration $\mathrm{\Delta }\tau =37.5\mu \mathrm{s}$, comparing Raman and Bragg diffraction. We compute the efficiency for different initial momenta and Gaussian widths $\mathrm{\Delta }\wp$ and show in panel (c) the results for Bragg diffraction with a pulse duration of $\mathrm{\Delta }\tau =37.5\mu \mathrm{s}$ using the optimal pulse area from panel (b). For $p_0=0$ and small $\mathrm{\Delta }\wp$ we observe an efficiency of 0.5 corresponding to a double-diffraction beam-splitter process. For increasing $p_0$, the efficiency approaches unity and thus a single-diffraction mirror process emerges. We also display the resonance width $\mathcal{W}$ of the diffraction process as a function of $p_0$ in this density plot (orange line) and observe that it is one limiting factor of the efficiency. The straight line $p_0=\mathrm{\Delta }\wp$ (see dashed pink line) in the plot demonstrates that a significant fraction of the atoms undergo double instead of single diffraction whenever the momentum distribution has significant overlap with the degenerate momentum ($p_0=0$), i.e., $\mathrm{\Delta }\wp >p_0$, as long as the resonance is wide enough. The corresponding behavior for Raman diffraction is similar. Effects on phases, caused for example by two-photon light shifts, cannot be observed in the efficiency.