Phase shift of double-diffraction Raman interference due to high-order diffraction states

Double-diffraction Raman interference is a novel interference scheme developed in recent years. It has a potentially high measurement precision due to the symmetric interference loop with identical internal states. In this paper, we analyze the high-order diffraction process in double-diffraction Raman interference and derive state evolution equations that include multiple diffraction processes. The population evolution of the states and the Mach-Zehnder interference fringe are calculated under the action of a Raman laser. The thermal momentum distribution averaging and state grouping methods are employed to solve the problem of decoherence at different interference exit ports. We compare the phase shifts calculated by integrating the sensitivity function and the two-photon frequency shift and by numerically solving the state evolution equations with high-order diffraction states. The good agreement between the results of the two methods shows that the two-photon frequency shift is the main reason for the phase shift induced by the higher-order diffraction states. Finally, the test error of an equivalent principle test experiment due to this effect is analyzed. This paper may be helpful for the analysis and suppression of the systemic error in such experiments.

Figure 1

Schematic diagrams of different Raman processes: (a) single-diffraction Raman process, (b) single-diffraction Raman process with two incident lasers and a retroreflective mirror, (c) double-diffraction Raman process with three lasers and a retroreflective mirror, and (d) double-diffraction Raman process with two incident lasers and a retroreflective mirror when the velocity of the atom is zero. The dashed lines represent the copropagating Raman and Bragg transitions.

Figure 2

Atomic states and Raman lasers used for the derivation of the evolution equations.

Figure 3

Evolution of the state populations over time. The red, blue, green, and cyan colors represent the $|a,0\rangle , |b,1\rangle , |b,-1\rangle$, and $|a,-2\rangle$ states, respectively. (a) ${\delta }_{1,0}={\delta }_{2,0}={\delta }_a={\delta }_b=0, {\mathrm{\Omega }}_{\text{eff1}}={\mathrm{\Omega }}_{\text{eff2}}=1$. (b) ${\delta }_{1,0}={\delta }_{2,0}={\delta }_a={\delta }_b=0, {\mathrm{\Omega }}_{\text{eff1}}={\mathrm{\Omega }}_{\text{eff2}}=5$. (c) ${\delta }_{1,0}={\delta }_{2,0}={\delta }_a={\delta }_b=0, {\mathrm{\Omega }}_{\text{eff1}}=1, {\mathrm{\Omega }}_{\text{eff2}}=0.8$. (d) ${\delta }_{1,0}=-2, {\delta }_{2,0}=2, {\delta }_a={\delta }_b=0, {\mathrm{\Omega }}_{\text{eff1}}={\mathrm{\Omega }}_{\text{eff2}}=0.3$.

Figure 4

(a) Schematic diagram of M-Z interference and exit ports of the counterpropagating double-diffraction Raman process at $n=1$. The red solid elliptic curve 1 denotes the interference loop induced by the double-diffraction Raman process, and the blue dashed elliptic curves 2 and 3 denote the interference loops induced by the single-diffraction Raman process. (b) Schematic diagram of M-Z interference with the blow-away process; the single-diffraction Raman interference loops are eliminated.

Figure 5

M-Z interference fringes of the double Raman process. The blue solid and red dashed lines indicate the fringes with and without the state elimination processes, respectively. (a) ${\delta }_{1,0}={\delta }_{2,0}={\delta }_a={\delta }_b=0$ and ${\mathrm{\Omega }}_{\text{eff1}}={\mathrm{\Omega }}_{\text{eff2}}=1$. The Raman pulse durations are $t_{\pi /2}=\frac{\sqrt{2}\pi }{2}$ and $t_{\pi }=\sqrt{2}\pi$. (b) ${\delta }_{1,0}={\delta }_{2,0}={\delta }_a={\delta }_b=0$ and ${\mathrm{\Omega }}_{\text{eff1}}={\mathrm{\Omega }}_{\text{eff2}}=1.2$. The Raman pulse durations are $t_{\pi /2}=\frac{\sqrt{2}\pi }{2}$ and $t_{\pi }=\sqrt{2}\pi$.

Figure 6

The phase shift of the double-diffraction Raman interference fringes obtained by scanning ${\delta }_{1,0}$. The parameters used in the calculation are as follows: ${\delta }_{2,0}={\delta }_a={\delta }_b=0$; the effective Rabi frequencies of the three Raman pulses are 1, 0.9, and 0.8 with ${\mathrm{\Omega }}_{\text{eff1}}={\mathrm{\Omega }}_{\text{eff2}}$; and the pulse lengths are $\frac{\sqrt{2}\pi }{2}, \sqrt{2}\pi$, and $\frac{\sqrt{2}\pi }{2}$, respectively. The navy blue solid and blue dashed lines represent the results calculated using Eqs. (11) and (8), respectively. The red dashed line represents the phase shift calculated using Eq. (8) without high-order diffraction, and the green short dashed line represents the differential phase with and without high-order diffraction. The inset shows an enlarged view of the navy blue solid line and the green short dashed line.

Figure 7

Interference fringes calculated using the two methods. The parameters used in the calculation are as follows: (a) ${\delta }_{1,0}={\delta }_{2,0}={\delta }_a={\delta }_b=0$ and ${\mathrm{\Omega }}_{\text{eff1}}={\mathrm{\Omega }}_{\text{eff2}}=1$. The Raman pulse durations are $t_{\pi /2}=\frac{\sqrt{2}\pi }{2}$ and $t_{\pi }=\sqrt{2}\pi$. (b) ${\delta }_{1,0}={\delta }_{2,0}={\delta }_a={\delta }_b=0$ and ${\mathrm{\Omega }}_{\text{eff1}}={\mathrm{\Omega }}_{\text{eff2}}=1.2$. The Raman pulse durations are $t_{\pi /2}=\frac{\sqrt{2}\pi }{2}$ and $t_{\pi }=\sqrt{2}\pi$. The blue solid line and red dashed line are the fringes calculated using the average and the state grouping methods, respectively. Here, the state elimination process is considered.

Figure 8

Differential phases calculated using the two methods. The parameters used in the calculation are as follows: ${\delta }_{2,0}={\delta }_a={\delta }_b=0$; the effective Rabi frequencies of the three Raman pulses are 1, 0.9, and 0.8 with ${\mathrm{\Omega }}_{\text{eff1}}={\mathrm{\Omega }}_{\text{eff2}}$; and the pulse lengths are $\frac{\sqrt{2}\pi }{2}, \sqrt{2}\pi$, and $\frac{\sqrt{2}\pi }{2}$, respectively. The blue solid line and red dashed line are the results calculated using the average and the state grouping methods, respectively.