Since the atom interferometer was first demonstrated in 1991,^[1] this technology has been developed rapidly and highly sensitive instruments have been realized. It has been applied to many precision measurements,^[2] such as measurements of local gravity g^[3–7] and its gradients,^[8–10] the fine-structure constant α,^[11–13] the Sagnac effect,^[14–16] Newtonʼs gravitational constant G,^[17,18] the weak equivalence principle,^[19–24] and even for searching gravitational waves.^[25,26] Recently, the atom interferometer has been utilized to precisely measure the map of a magnetic field^[27,28] and the quadratic Zeeman coefficient.^[29,30] The sensitivity of an atom interferometer linearly scales with the space–time area enclosed by the matter waves.^[31] Researchers are struggling to achieve ambitious high-precision measurements through producing large-momentum-transfer atom optics^[32–36] and a long coherence time.^[37–39] For instance, Kasevichʼs group^[40] has achieved a light-pulse atom interferometer with wave packets separated by up to 54 cm on a timescale of 1 s. Precision measurements of g are of interest in a wide range of applications in geophysics,^[41] inertial navigation,^[42] and fundamental research.^[43] For high-precision gravity measurements,^[44] the atom gravimeters have achieved an accuracy of a few ( ),^[3,4] which are able to obtain the absolute gravity value directly referenced to atomic standards. However, the ultimate accuracy of the measurement still depends on the understanding of the potential systematic errors.

In the mature atom interferometer, some significant systematic errors have been considered, such as the AC stark effect,^[1,45–48] quadratic Zeeman effect,^[49–53] gravity gradient effect,^[3,54] and some other effects.^[55–62] As the atom interferometer is rapidly developed, more and more schemes are proposed to test general relativity with atom interferometry, in which a lot of minor systematic effects should be taken into account, such as the Raman-pulse-duration effect related to the gravitational field^[63–67] and the Raman-pulse-duration effect related to the inhomogeneous magnetic field. In the past few years, the Raman-pulse-duration effect related to the gravitational field has been studied extensively, while the Raman-pulse-duration effect related to the inhomogeneous magnetic field has not been considered before. In this paper, we focus on analyzing the Raman-pulse-duration effect of atom gravimeters in an inhomogeneous magnetic field. During the gravity-measurement process, there are mainly two ingredients influencing the atomic propagation: the Earthʼs gravitational field and the laser field. Here, we use the time-propagation operator to describe the dynamic behavior of a two-level-atom system in the Schrödinger picture, where the time-propagation operator is decomposed into two parts for simplicity: the free-propagation part and the interaction part with the laser field. Based on this, we can derive the atomic wave function easily, and further calculate the coupling effect of the Raman-pulse duration and the magnetic field gradient, which has not been considered before. For the typical parameters, we find that this coupling effect must be taken into account when the expected accuracy of an atomic gravimeter reaches 10⁻¹⁰ g.

The outline of this paper is as follows. In Section 2, we describe the theoretical analytical framework, showing how the time-propagation operator is used to analyze the dynamic behavior of a two-level-atom system. In Section 3, we analyze the Raman-pulse-duration effect in an inhomogeneous magnetic field, and present the calculation for the coupling effect between the Raman-pulse duration and the magnetic field gradient in detail. In Section 4, we discuss the suppression of the coupling effect between the Raman-pulse duration and the magnetic field gradient, and how to test this coupling effect experimentally. Finally, the paper is concluded in Section 5.

For a typical Raman atom interferometer, the Raman beams are formed by a pair of opposite propagation far-detuned laser beams, which are used to manipulate the atomic states. For the interaction of a three-level atom using short laser pulses, adiabatic elimination will produce an effective two-state system.^[67] Based on the approximate two-level model, we study the coupling effect between the Raman-pulse duration and the magnetic field gradient, which has not been studied before. In the Schrödinger picture, for a two-level system with two hyperfine states and , the atomic wave function can be described as 1 with U(t) the time-propagation operator,^[67,68] which can be decomposed into two parts the atomic free-propagation operator and the interaction operator as 2 This decomposition-operator method has been studied in our previous work.^[67] Here, we use a similar method to analyze the effect of the Raman-pulse duration in the interferometer. Firstly, we analyze the time-propagation operator U(t) in a typical Mach–Zehnder atom interferometer. In this configuration, the atoms are manipulated by a pulse sequence, which is used to split, reflect, and recombine the atomic wave packets, as shown in Fig. 1. The durations of the pulses are τ₁, τ₂, and τ₃, respectively, and the free-propagation time between the three pulses is T and , respectively. When considering the pulse duration, the propagation operator can be used to describe the instantaneous interaction of the atoms with the laser. is the free-propagation operator during the finite pulse duration, which can be merged into the free-propagation operator between pulses, resulting in a total free-propagation operator . Then, the time-propagation operator can be expressed as and , in which the magnetic field gradient effect has been taken into account. Based on Eq. (1), the final wave function can be obtained. Finally, we can derive the interference phase containing the coupling effect of the Raman-pulse duration and the magnetic field gradient.

Fig. 1.

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	Fig. 1. (color online) Space–time diagram for the Mach–Zehnder type atom interferometry. The pulse durations of the sequences are , and τ3, respectively. The time of free propagation between the pulses is T and , respectively. The time propagation operator U(t) for an atom interferometer can be decomposed into five parts: , , , , and .

The interaction between a two-level atomic system and an electromagnetic field can be described by Hamiltonian operator 3 where and are respectively the internal energies of the ground and excited states, and are respectively the atomic momentum and position operators, represents any external potential acting on the center of mass motion, and describes the interaction between the atomic dipole moment and the electric field , which can be written as 4 with Rabi frequency 5 where is the amplitude of the electric field. In the atom interferometer, the moving atoms suffer from the external magnetic field in the vertical direction, which can induce the shift of the atomic energy levels. The interactive Hamiltonian between the atoms and magnetic field can be expressed as 6 with and the typical parameters for ⁸⁷Rb atom. For a magnetic sensitive type interferometer, the first-order Zeeman parameter is , and the second-order Zeeman parameter is .^[69] The inhomogeneous magnetic field can be written as 7 where is the magnetic field gradient, and is the direction of the magnetic field. To simplify the calculations, can be defined as an equivalent gravitational field introduced by the magnetic field. We further define with i = I,II representing the first-order and the second-order Zeeman effects, respectively. Then the magnetic field gradient factor for the first-order Zeeman effect can be defined as 8 Similar to the gravitational effect, the corresponding potential has a direct relation with the interference path. In the typical interferometer, the atoms are prepared in magnetic-field-insensitive states to avoid the first-order Zeeman effect. However, these states still show a second-order Zeeman shift in the inhomogeneous magnetic field. When neglecting the high-order effect of , we can obtain 9 Similarly, the magnetic field gradient factor for the second-order Zeeman effect can be defined as 10 Then, equation (9) can be equivalently written as 11 For Eq. (11), the first term is independent of the interference path, not responsible for the systematic effect. Then, the interactive Hamiltonian between the atoms and magnetic field can be given by 12 The potential energy induced by the uniform gravitational field and the inhomogeneous magnetic field can be written as 13

Based on Eqs. (3), (4), and (13), the Hamiltonian for a two-level atomic system in a uniform gravitational field and an inhomogeneous magnetic field can be written as 14 Through some simple calculations (see Appendix A), we can derive the unitary time-propagation operator U(t) as 15 with 16 Here, is the Hamiltonian of the center of mass, and α denotes the sweeping rate of lights’ frequency, which is used to compensate the frequency shift induced by the Doppler effect for the moving atoms. More details can be found in Appendix A. To further simplify the calculations, U(t) can be decomposed into two parts: the free propagation operator 17 which describes the free propagation of the atoms, and the laser-propagation operator 18 which describes the instantaneous interaction between the atoms and lasers. For a typical Mach–Zehnder atom interferometer, and π pulses are used to split and reflect the atomic wave packets, respectively, where the laser-propagation operator can be respectively written as 19 20 In this paper, we consider the plane wave approximation for the wave-packets of the atoms and assume that the atoms are initially in the state , which can be expressed by the wave function 21 with the starting time . According to the timing diagram of the Raman-pulse sequences (see Fig. 1), the final wave function can be written as 22 This transforming matrix between the initial and final wave functions is just like the optical propagation matrix. In this treatment, to obtain the final wave function, we just need the initial wave function and the propagation operator. With the specific free propagation operator and the laser propagation operator, the final wave function can be obtained as 23 Finally, the interference phase can be obtained from the probability of atom in state , based on the relation . When the atoms in state are measured finally, the related interference phases are 24 While the related phases are 25 with 26 when the atoms in the state are finally measured. In the atom interferometer, the interference phase is composed of three parts, atomic free-propagation phase shift , atomic instantaneous interactions with laser phase shift , and the observation position-dependent phase shift . By assuming that the atoms in the state are measured, the total interference phase can be written as 27 Δ φ total = ( Z B 1 − Z B 2 ) ︸ Δ φ prop − ( Z A 2 + Z E 2 ) ︸ Δ φ laser + k 1 · x − k 2 · x ︸ Δ φ obs . Strictly, the evolution of atoms is the evolution of a Gauss wave packet. Arbitrary position can be written as , where is the central position of the atoms. For the interference measurement we are concerned with, can be rewritten as 28 with the classical position of the atoms in the upper (lower) path, and the observation position, which can be chosen as . Usually, the first term in the above equation can be merged into the atomic free-propagation phase shift, and the last term is the separation phase shift. Here, the total interference phase contains the phase shift induced by the uniform gravitational field and the inhomogeneous magnetic field. We can decompose the total phase into two parts: arising from the uniform gravitational field and arising from the inhomogeneous magnetic field. According to Eqs. (24)–(27), the former can be written as 29 Assuming , , and neglecting the quadratic term of pulse duration τ, we can derive 30 which is consistent with the previous works.^[63–65,67] For different definitions for the time T, the final phase shift can be retrieved consistently. For example, when is defined as the time in between the centers of two consecutive Raman pulses, the consistent result can be derived through a replacement of . In this paper, we focus on the interference phase shift . For convenience, we decompose into three parts: the atomic free propagation phase shit , the instantaneous interaction phase shift between the atoms and laser, and the phase shift related to the position of observation. From Eqs. (24)–(27), these phase shifts can be written as 31 Usually, we can ignore the phase shift , since it is the high-order effect of τ. Finally, we can derive the total phase as 32 From Eq. (32), we know that is proportional to the magnetic field gradient . To minimize the effect of the magnetic field inhomogeneity and the first-order Zeeman effect, it is necessary to prepare the atoms in the magnetic-field-insensitive states of . However, the second-order Zeeman effect caused by the weak magnetic field gradient cannot be avoided. To calculate the second-order Zeeman effect, the magnetic field factor should be replaced by Eq. (10), and the related phase shift is 33 Apparently, the phase shift related to the magnetic field gradient can be divided into two parts: the pulse-duration-independence part and the pulse-duration-dependence part. The former is the main term and wholly agreed with the previous analysis,^[27] which shows that the phase shift related to the magnetic field gradient can be further reduced with a symmetrical interferometer. The latter we are concerned with is tiny, and can be written as 34 which has not been studied before. From this equation, the coupling effect of the pulse-duration τ and the magnetic field gradient is also related to the initial atomic velocity. As the pulse-duration-dependence part contributes a smaller systematic error than the pulse-duration-independence part, it is very urgent to minimize the effect of the magnetic field gradient, which usually requires the well-controlled magnetic fields and extensive magnetic shielding. Based on the typical experimental parameters 35 we can reasonably evaluate these effects, the pulse-duration-dependence effect is about 5.5 ×10⁻¹¹ g, which is about thirty times smaller than the pulse-duration-independence effect. When further increasing the magnetic field gradient and pulse duration, the pulse-duration-dependence effect becomes more prominent. We believe that the pulse-duration effect related to the magnetic field gradient plays an important role when one estimates a high-precision atom interferometer, such as the accuracy of 10⁻¹⁰ g. This effect may also be paid more attention to in the high-precision measurements of gravity gradient,^[8–10] the fine structure constant,^[11–13] the Sagnac effect,^[14–16] and the Newtonian gravitational constant.^[17,18]

Usually, the -vector reversal technique is adopted in the gravity measurements, since this technique can well reject some effects independent of the direction of the Raman-pulse wavevector, such as the second-order Zeeman effect and AC Stark effect. Here we just consider the second-order Zeeman effect in gravity measurements. With the -vector reversal technique, the phase shift related to the second-order Zeeman effect can be mostly suppressed, but cannot be canceled completely due to the spatial nonoverlap in the two configurations. The phase shift of an atom interferometer in two configurations of the -vector reversal technique can be expressed as 36 with and respectively the phase shifts related to the second-order Zeeman effects in the and configurations, which can be written as 37 where is the recoil velocity. Then, the differential result only related to the second-order Zeeman effect can be written as 38 It should be noted that this residual Zeeman phase shift has nothing to do with the symmetry of the atom interferometer, and the rejection results are mainly determined by the magnetic field. With the typical experimental parameters 39 we estimate the total residual second-order Zeeman effect in differential measurements as , while the coupling part of the Raman pulse duration and magnetic field gradient is about . Although this coupling effect is small, it is necessary to analyze, since it is a non-ignorable effect in the future precision atom interferometers aiming at testing general relativity.

As we know, the atom interferometer has achieved a high sensitivity of with short integration time,^[5] which is a challenging sensitivity for a direct observation. However, a light-pulse atom interferometer for ⁸⁷Rb with a duration of 2T = 2.3 s has been demonstrated,^[37] which inferred the acceleration sensitivity of . In this case, the coupling effect between the Raman-pulse duration and the magnetic field gradient may be easily detected. To achieve a possible observation for the coupling effect of the pulse duration and magnetic field gradient, one can choose the atom interferometer working in a magnetic sensitive sub-level (eg ). In this configuration, the coupling effect related to the first-order Zeeman effect is calculated as 40 From this equation, this coupling effect can be measured by modulating the Raman-pulse duration theoretically. With the typical experimental parameters 41 42 we can simply estimate this coupling effect as , which seems observable experimentally, since the magnetically sensitive interferometer shows a long-term sensitivity better than .^[21]

In the real measurement, some other systematic errors should be considered, for example, the AC stark shift. The phase shift due to the AC stark shift is^[45] 43 where are the frequency shifts of the Raman transition at the time of the first and the third pulses, and are the corresponding effective Rabi frequencies. It must be noted that both and are proportional to the intensity of the two Raman lasers, which means that there will be no AC stark phase shift contribution by changing the Raman pulse duration, i.e., the AC stark phase shift is just a constant bias. However, the changes of the intensity ratio will lead to the fluctuations of the AC stark shift, which has been estimated at the level of .^[3] Similarly, we can estimate this effect about in our setup.

Therefore, the effect of the Raman-pulse duration related to the magnetic field gradient can be detected with a magnetic sensitive atom interferometer.

With a simplified time-propagation operator model, we have calculated the Raman-pulse-duration effect in an inhomogeneous magnetic field for a high-precision atom interferometer, where the coupling effect between the Raman-pulse duration and magnetic field gradient is focused on. Finally, we find that for a typical atom gravimeter, this coupling effect is very tiny, but cannot be ignored when one estimates a higher-precision atom gravimeter, such as the accuracy of 10⁻¹⁰ g. In addition, we have a simple discussion on the possibility of testing this effect.