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Abstract
An atom interferometer based on Doppler-insensitive Raman transition is proposed, which has sharply peaked interference fringes for multi-wave interference. We show that two sets of counter-propagating Doppler-insensitive Raman beam pairs can be used to construct a new type of multi-wave beam splitter, which can be used to construct an atom interferometer. Although the spacing between adjacent diffraction orders of the interferometer is small, they can be distinguished by the internal state of the atom. Our analysis shows that the width of the fringes of this atom interferometer is inversely proportional to the width (duration) of the beam splitter and the Rabi frequency of the Raman beams, that is, the interferometer can achieve high resolution at high light intensity and long pulse width.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nowadays, atom interferometers have become powerful tools for precision measurements, such as those used to measure the gravity [1,2], the gravity gradient [3,4], the rotation [5,6], and the Newtonian gravitational constant [7,8]. In addition to these applications, atom interferometers are also used in fundamental physics research, such as in testing the equivalence principle [9–11], probing the dark energy [12], and testing the local Lorentz invariance [13]. The phase resolution of an atom interferometer is one of the main limiting factors for these applications. One way to improve the resolution of an atom interferometer is to suppress technical noise. However, this method does not help to overcome the basic limitations determined by the structure of the atom interferometer, such as the quantum projection limit [14]. Another way to improve the resolution of an atom interferometer is to explore new interference structure. Although a new atom interferometer architecture is unlikely to be suitable for all current applications, it may have advantages for specific studies.
Since the cold atom interferometer was first used to measure the free-fall acceleration [15], atom interferometers based on Raman transitions have been at the core of atomic interferometry. When using Raman transitions, atoms can be transferred from one ground state to another, and the phase of the stimulated atoms can be controlled by the phase of the Raman beams. For the case where the Raman beams are far from resonating with the excited state transitions, the Raman transition of a three-level atom can be equivalent to the process of a two-level atom stimulated by a monochromatic electromagnetic wave [16]. In experiments, the standing waves formed by two electromagnetic waves propagating in opposite directions are usually used to diffract atoms. This suggests that we can use two sets of counter-propagating Raman beam pairs to form a standing-wave-like beam splitter, which is called “Raman standing waves” in this paper.
The effect of the atoms diffracted by the laser standing waves is one of the Kapitza-Dirac effects. The Kapitza-Dirac effect shows particles can be diffracted by waves, which was predicted by Kapitza and Dirac in 1933 [17]. This effect has been demonstrated with atom beam [18], cold atoms [19], and electrons [20]. It is a useful tool in high-resolution spectroscopy [21], metrology [22–24], and fundamental physics [25,26]. In the reported atomic experiments, the standing waves are usually set to be off-resonant to state transitions to avoid the spontaneous emission of the excited states [27]. This work proposes that the Raman standing waves can be used to construct a double ground state diffraction to avoid spontaneous emission.
We show that an atom interferometer based on Raman standing wave pulses can be built, which is a new type of multi-wave interferometer. By using two sets of counter-propagating Doppler-insensitive Raman beam pairs, the atomic Kapitza-Dirac effect in the Raman-Nath regime can be performed. A multi-wave atom interferometer can be constructed using two Raman standing wave pulses with a certain time interval to split and overlap the atomic wave packets. The interference fringes of this interferometer can be obtained from the probability of atomic internal state transition. Our analysis shows that this interferometer can achieve a sharp interference fringes, which can be used to measure the phase of laser-atom interaction with high resolution.
In this paper, section 2 introduces a method for constructing Raman standing waves using Doppler-insensitive Raman beams. The evolution equations and solutions of atomic states in Raman standing waves are described in detail in section 3. The interference of atoms under the action of two Raman standing wave pulses is described in section 4. The performance of the atom interferometer based on this interference is also discussed in this section. In section 5, we describe the advantages of our scheme and possible applications. The summary and discussion of the article is displayed in section 6.
2. Setup of Raman beams
A schematic diagram of a typical Raman transition in a $\Lambda$ configuration is shown on the left side of Fig. 1. A three-level atom with two ground states (denoted as $\left |a\right \rangle$ and $\left |b\right \rangle$) and an excited state (denoted as $\left |c\right \rangle$) interacts with two Raman lasers. The frequencies and the wave vectors of the two lasers are $\omega _1,\vec {k}_1$ and $\omega _2,\vec {k}_2$, respectively. The one-photon detuning ($\Delta$) is set to be much larger than the natural width of the excited state to suppress spontaneous emission. The two-photon detuning ($\delta _{\pm }$) is determined by the atomic momentum and the frequencies of the lasers with the relation
Fig. 1. (Left) The $\Lambda$ configuration of the Doppler-insensitive Raman transition. The one-photon detuning of the Raman beams ($\Delta$) is large to suppress the spontaneous emission. $\delta _{\pm k}$ represents the corresponding two-photon detuning when the atom gains (for $\delta _{+k}$) or loses (for $\delta _{-k}$) an effective photon momentum of $\hbar k$, where the modulus of effective wave vector is $k=k_1-k_2$ for Doppler-insensitive transition. (Right) The setting of Doppler-insensitive Raman standing waves, where $\omega _{1}$ and $\omega _{2}$ are the frequencies of the two lasers, respectively. A pair of Doppler-insensitive Raman beams is consist of two co-propagating lasers ($\omega _1$ and $\omega _2$) with a same polarization of $\sigma _{+}$ (or $\sigma _{+}$). Two sets of Doppler-insensitive Raman beams with opposite directions and polarizations perpendicular to each other will form the Raman standing waves.
When the two Raman beams are parallel to each other, there are two Raman transition configurations. The first one is the Doppler-sensitive Raman transition, where the two Raman beams with the same polarization of $\sigma _+$ (or $\sigma _-$) are counter-propagating and have a large modulus of effective wave vector, which is $k_1+k_2$. In this configuration, Eq. (1) is a strict limit for atomic momentum transitions. Therefore, the Raman beams can only stimulate atoms in a narrow momentum range, that means that this transition is sensitive to the atomic velocity. Atom interferometers based on such Raman transitions are often used as inertial sensors, for example as gravimeters.
The second type of Raman transition is the Doppler-insensitive Raman transition, where the two Raman beams with the same polarization of $\sigma _+$ (or $\sigma _-$) propagate in the same direction, and the modulus of effective wave vector is $k=k_1-k_2$. In this setup, the detuning induced by the Doppler effect $\vec {k}\cdot \vec {v}$ and the photon recoil $\hbar k^2/2m$ are much smaller than the Fourier width of the Raman beams. Therefore, almost all atomic momentum states resonate simultaneously with the same pair of Raman beams. Although the momentum spacing between adjacent diffraction orders is small in this transition, adjacent diffraction can be distinguished by the internal state transition.
When the detuning $\Delta$ is large, the three-level model of the atomic Raman transition can be equivalent to the interaction of a two-level atom with a monochromatic electromagnetic wave. Based on this idea, we can use two sets of Doppler-insensitive Raman beam pairs whose polarizations are perpendicular to each other and propagate in opposite directions, as shown on the right side of Fig. 1, to form a standing-wave-like system. In this system, the atom can be transformed between the two ground states when diffracted by the Raman beams, accompanied by momentum changes of $+\hbar k$ or $-\hbar k$ in a single round of Raman transition. It is worth noting that, strictly speaking, the Raman standing waves are not true standing waves because their polarizations are perpendicular to each other. Due to this perpendicularity, the total intensity of the Raman standing waves is not as position dependent as the usual standing waves.
3. Evolution of atomic states
To simplify the model, we ignore the light shift induced by the Raman beams and use a simplified two-level equation of motion when $\Delta$ is large [16]. For the Raman standing waves shown in Fig. 1, we refer to the case where the Raman beams’ direction is upward (downward) as the $+k$ ($-k$) transition. Considering that both the $+k$ transition and the $-k$ transition of the atoms exist simultaneously, the evolution of the atomic states can be written as
To solve the evolution of atoms in different states, we first express the atomic wave function $\psi _a(x)$ and $\psi _b(x)$ with the Fourier expansion
When an atom is stimulated by a Raman standing wave pulse, its internal state and momentum state will change with the duration time, which result in an oscillation of the state transition probability. The transition probability from the atom in momentum $p$ to the momentum $p+n\hbar k$ is $J_n^2(\Omega \tau )$ according to Eq. (6). This result is in agreement with the results of using ordinary laser standing waves. It can be expected that when the initial state of the atom is in the $\left |a\right \rangle$ state, the atom will be in the $\left |b\right \rangle$ state if and only if the atom has gone through an odd number of Raman transitions. Therefore, when the atom is initially at the state $\psi _{a,0}$, i.e. $C_{a,0}=1$, the probability of the internal state transition to $\left |b\right \rangle$ state is
Fig. 2. The transition probabilities of the atomic internal states (thick solid line) and atomic momentum states (thin lines) as a function of $\Omega \tau$. The transition probabilities of the atom from the initial state of $\left |a,0\right \rangle$ to the final states of $\left |b,1\right \rangle$, $\left |b,3\right \rangle$ and $\left |b,5\right \rangle$ are shown by the blue dashed line, the red solid line, and the green dotted line, respectively. The internal state transition probability is the sum of all the odd-order momentum transition probabilities. In this work, the corresponding momentum states in the range of $n=\pm 20$ are considered.
The internal state transition probability oscillates with duration and decays as $\Omega \tau$ increases. After the Raman standing wave pulse is applied for a long time, this probability eventually ends at $1/2$. We can prove this with the formula [30]
Let the parameter $\alpha =\pi$, this equation can be simplified to the following form4. Atom interferometry
Knowing that Doppler-insensitive Raman standing waves can stimulate an atom to different momentum states, we can build an atom interferometer based on the Raman standing wave pulses. A schematic of the interferometer is shown in Fig. 3, in which the odd and even diffracted waves are distinguished by the color and form of the lines. The atomic wave packet initially in the $\left |a,p\right \rangle$ state is split by the first Raman standing wave pulse at time $t_1$. After an interrogation time of $T$, the second Raman standing wave pulse is performed at time $t_2$ to overlap the diffracted wave packets. The two Raman pulses are set to have the same width of $\tau$ to simplify the model.
Fig. 3. The atom in the initial state $\left |a\right \rangle$ with a momentum of $p$ is diffracted by two Raman standing wave , which forms a Raman-Nath atom interferometer. The blue thick solid lines indicate the atoms in state $\left |a,p\right \rangle$, and the red dashed lines indicate the atoms in state $\left |b,p\pm \hbar k\right \rangle$, and the blue thin solid lines denote the atom in state $\left |a,p\pm 2\hbar k\right \rangle$. We only briefly sketched parts of the diffracted waves.
When an atom absorbs $m$ photons at the first Raman pulse, it needs to absorb another $n-m$ photons at the second Raman pulse to have a momentum transition of $p\rightarrow p+n\hbar k$. In addition, because the diffraction is in the Raman-Nath regime, only the usual case where the interval between every two diffracted waves is much smaller than the coherence length of the atoms is considered herein. Therefore, after the second Raman pulse, the amplitudes of different paths in the same final momentum state can be directly added, which gives
Note that after the interference, all states of odd momentum are in the $\left | b \right \rangle$ state, so the internal state transition probability of the atom isThe interference fringe in this work is the relation between the phase difference $\Delta \phi$ and the internal state transition probability $P_b$, which is shown in Fig. 4. For the case of $\Omega \tau =0.5$, where only the $0$th and the first diffraction waves ($C_{a,0}$, $C_{b,1}$, and $C_{b,-1}$) are dominant, the interference is a simple sinusoidal function as shown by the dashed line in Fig. 4. When the higher diffraction orders are not negligible, such as for the case of $\Omega \tau =1.8$ or $\Omega \tau =4.2$, the interference fringes become complex. The complexity is manifested on the one hand when atoms are “multi-reflected” by Raman standing waves, the average effect of different diffraction orders has multiple extreme points. On the other hand, the peak width of the stripe narrows with increasing $\Omega \tau$.
Fig. 4. The interference fringes of the atom interferometer when change the phase of the Raman beams. When $\Omega \tau \approx 1.8$, $J_1(\Omega \tau )$ is with its maximum value and the first order diffraction is dominant. When $\Omega \tau \approx 4.2$, $J_3(\Omega \tau )$ reaches its maximum value. With the increase of the $\Omega \tau$, the higher order diffraction become more and more important. Here, $\gamma _{1/4}$ is the quarter-width, which correspond to the difference of $\Delta \phi$ when the transition probability $P_b=1/4$. The sharpness of the fringes is given by $\gamma _{1/4}$ in this work.
Although the interference fringes are complex, they also have some distinct features. For example, at the points $\Delta \phi _N=(2N+1)\pi ,\ N=0,1,2,3,\ldots$, where the interference between adjacent diffraction orders is destructive, the probability of $\left |b\right \rangle$ state always zero for all $\Omega \tau$ values. This can be proved from the expression of $C_{b,2n+1}$, which gives
Using $l=2n+1-m$ to replace the parameter $m$ in this equation, we can rewrite the form as The sum of Eq. (13) and Eq. (14) givesAnother characteristic of interference occurs when $\Delta \phi = 0$, where the interference between adjacent diffraction orders is constructive. According to the addition theorem of the Bessel function, which gives
Combined this with Eqs. (12) and (15), we can getIn addition to the phase difference, the value of $\Omega \tau$ is also one of the key parameters of the interferometer. Considering the effects of both $\Delta \phi$ and $\Omega \tau$, the probability of an atomic internal state transition is shown in Fig. 5. We can find that this atom interferometer is similar to the optical Fabry-Perot interferometer, where the light is multiply reflected by the mirrors. In optical Fabry-Perot interferometer, the amplitude-reflection coefficient is one of the main limiting factors. In this atom interferometer, the atomic internal states are reflected by the Raman standing waves, and the reflection of the atomic state is determined by $\Omega \tau$. Since the light intensity of the Raman standing waves is spatially uniform, the Rabi frequencies of the two beam splitting pulse in the interferometer are the same. Therefore, the size of $\Omega \tau$ under the same pulse width will not change with the distribution and height of the atom cloud.
Fig. 5. The probability of the atom internal state transition versus $\Omega \tau$ and $\Delta \phi$. When the phase difference between the two Raman standing waves $\Delta \phi = -\pi ,\pi$, the state transition probability is zero due to the destructive interference of the adjacent momentum states. When $\Omega \tau$ is large, the atom interferometer has a high degree of sharpness near $\Delta \phi = -\pi ,\pi$ due to a steep change in the transition probability.
When a single Raman standing wave pulse is applied, with the increase of $\Omega \tau$, the atomic internal state transition probability approaches $1/2$. A similar trend occurs when the double Raman standing wave pulses are used to form the atomic interference, except for the case near the point of $\Delta \phi =\Delta \phi _N$. So we can take the median transition probability as $1/4$, and the quarter-width $\gamma _{1/4}$, as shown in Fig. 4, is the width of the peak in atomic internal state transition. A measure of sharpness of the fringes, that is, how rapidly the transition probability rises on either side of the minimum, can be given by the quarter-width $\gamma _{1/4}$. This definition is similar to that in optical Fabry-Perot interferometers, and $\gamma _{1/4}$ can be used to characterize the phase resolution level of the interferometer.
As $\Omega \tau$ increases, the higher-order diffraction increases, and phase resolution of the interferometer increases. This is reflected in the vicinity of $\Delta \phi =\Delta \phi _N$ in Fig. 4 and Fig. 5, where the phase width corresponding to the probability changing from 0 to 1/4 is narrowed when $\Omega \tau$ is large. The relation between $\gamma _{1/4}$ and $\Omega \tau$ is shown in Fig. 6. The result shows that $\gamma _{1/4}$ is proportional to $1/\Omega \tau$, so we can fit the data using the model of $\gamma _{1/4}\approx B/\Omega \tau$. The fitting result is that the constant $B$ is approximately $0.53(1)\ \pi$. This shows that under the conditions we consider, the resolution of the interferometer increases significantly with the increase of the pulse width. However, as the pulse width increases, the Fourier width of the pulse narrows significantly. At this time, not all atomic momentum states can be excited by the laser, so the order of diffraction will eventually decrease, thereby limiting the performance of the interferometer.
Fig. 6. The relation between $\gamma _{1/4}$ and $\Omega \tau$. Each value of the quarter-width $\gamma _{1/4}$ is obtained by the difference between the two roots of the formula $P_b=0.25$ near $\Delta \phi =\pi$. By using the function of $\gamma _{1/4}=B/\Omega \tau$ to fit the data points, we find that the ratio of $\gamma _{1/4}$ to $1/\Omega \tau$ is close to constant ($B=0.53(1) \ \pi$) in the range of $\Omega \tau =0.5$ to $\Omega \tau =8.$.
5. Application prospects
Compared with the Doppler-sensitive Raman interferometer, there is no advantage in using the interferometer in this work to measure the inertial effect, because the phase caused by the inertial effect in our interferometer is much smaller. However, this interferometer can be used to improve the precision of experiments based on the measurement of atomic energy difference, such as the magnetic field measurement [31], and the atomic clocks [32,33]. When the ground state of the Raman transition is chosen to be sensitive to magnetic fields (eg, the $\left |F=1,m_F=1\right \rangle$ state and the $\left |F=2,m_F=1\right \rangle$ state of $^{87}$Rb atoms), the energy difference between atomic states can accumulate interference phases over time, which can be used to measure the magnetic field. Similarly, the magnetic insensitive states (eg, the $\left |F=1,m_F=0\right \rangle$ state and the $\left |F=2,m_F=0\right \rangle$ state of $^{87}$Rb atoms) can be used as a frequency standard. The laser frequency can be calibrated by comparing the deviation of the laser frequency from the atomic resonance transition. In theory, our scheme has higher potential accuracy than Ramsey interferometer-based schemes.
The scheme we propose in this work is applicable to the cold atom systems, such as the atoms in magneto-optical trap (MOT), the atoms in optical dipole trap, and the BECs. Taking $^{87}$Rb as an example, the Rabi frequency of Raman beams with a beam waist of 2 cm and a total power of 10 mW is usually several kHz, which is at least an order of magnitude larger than the Doppler shift of the atoms with a velocity of 1 m/s ($kv\approx 143$ Hz). At this time, a pulse length of a few milliseconds is sufficient to ensure the high phase sensitivity of our multi-wave interferometer. The performance of the Ramsey interferometer is comparable to that of our interferometer at $\Omega \tau = 0.5$, as shown in Fig. 4. Because the phase resolution of our interferometer is directly proportional to $1/\Omega \tau$, and the condition of $\Omega \tau\;>\;5$ is easy to achieve in experiment, we expect the best resolution of this interferometer to be an order of magnitude better than the reported works based on Ramsey interferometers.
Compared with the multi-wave interferometers in other works [34–36], the effective wavelength of the Doppler-insensitive Raman transition is much larger, and its photon recoil velocity is much smaller. For $^{87}$Rb atoms, the effective wavelength of the Raman beams is about 0.04 m. When the Raman standing waves interact with a millimeter-sized atomic cloud, the phase non-uniformity caused by atomic distribution is small. Therefore, the Raman standing waves are more suitable for experiments using interference to study the evolution of strongly coherent atomic systems (such as BEC [37]), because they have little effect on the spatial phase distribution and motion of atoms. When the Doppler-insensitive Raman beams are used, the internal state evolution and momentum change of the atom are almost the same as those under microwave. However, the wavefront and polarization characteristics of Raman standing waves are experimentally easier to manipulate than microwaves, which is also one of the advantages of our scheme.
6. Conclusion
As a new development of the Doppler-insensitive Raman transition, we find that Raman beams with this configuration can be used to construct a standing wave system. Although there have been many studies on the Kapitza-Dirac effect in the past, a common limitation in these studies is that the laser need to be off-resonance to avoid spontaneous emission of the excited state. We can avoid this problem by using Raman transition. In addition, in Raman standing waves, the polarizations of the beams traveling in opposite directions are perpendicular to each other, which means that the electric field amplitude is uniformly distributed along the propagation direction. This is an advantage not found in traditional standing waves with antinodes.
When using the Doppler-insensitive Raman standing waves as a multi-wave splitter, the diffracted atoms are marked by internal states. An atom interferometer in the Raman-Nath regime can be constructed based on this beam splitter, and interference fringe can be obtained from the atomic internal state transition probability. Moreover, this interferometer can significantly improve the phase sharpness characteristics with the increase of $\Omega \tau$, which is expected to be used in high-resolution phase measurement. Because the pulse width $\tau$ need to be short to ensure that the Fourier width of the Raman standing wave pulse can cover the frequency range of high-order momentum state transitions, the Rabi frequency of the Raman beams, $\Omega$, ultimately determines the limit of the interferometer.
In practical applications, the Doppler-insensitive Raman standing wave we proposed in this work can be applied to cold atom systems. The multi-wave atom interference technology based on this new type of beam splitter is expected to be used in atomic clock and magnetic field measurement to improve their performances. In these applications, we expect the multi-wave interferometer in this work to have better resolution than the reported works based on Ramsey interference.
Funding
National Natural Science Foundation of China (11625417, 11727809, 91636219, 91736311).
Acknowledgments
We thank Duan Xiao-chun and Deng Xiao-bing for the enlightening talk about this work.
Disclosures
The authors declare no conflicts of interest.
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