Common-mode noise rejection using fringe-locking method in WEP test by simultaneous dual-species atom interferometers
1. IntroductionBenefitting from the highly developed atom interferometry technology, cold atoms, possessing both internal and external degrees of freedom, become ideal probes in many precision measurements. They have been successfully used in measuring gravity acceleration,[1–10] gravity gradient,[7,8,11–16] rotation,[17–23] magnetic field gradient,[24–27] etc. Atom interferometers also play an important role in fundamental physics, such as the measurement of fine structure constant, the determination of gravitational constant G,[28–30] and the test of the weak equivalence principle (WEP).[2,10,31–37]
The weak equivalence principle, as one of the cornerstones of Einstein’s general relativity, states that all masses fall in the same way in a gravitational field regardless of their internal structure and composition. Verifications of WEP using macroscopic masses have achieved a level of 10
,[38,39] while the best level for testing WEP on quantum basis is at the level of 10
.[2,10] Testing WEP using microscopic particles still stimulated wide interest since the development of the neutron interferometer.[40] The advantage of quantum-basis tests is that, quantum objects offer more possibilities to break WEP and meanwhile they also afford potentially higher precision and well defined properties.[34,41,42] Up to now, WEP tests on quantum basis have been performed between atoms and macroscopic bodies,[2,10] between atoms of different species,[34] and different permutation symmetries, e.g., bosons and fermions,[35] as well as the same atoms of different hyperfine levels,[31] or in different spin orientations.[37] In these tests, the corresponding gravity accelerations of different objects are usually independently measured and then compared, in which situation the vibration noise cannot be common-mode rejected. To overcome this difficulty, WEP test using simultaneous dual-species atom interferometers is of particular interest for its intrinsic capability of the common-mode vibration noise rejection (CMVNR).[32,36] WEP tests of this kind have already been performed by several groups, achieving a level of 10
,[36] and tests with higher precision have been proposed,[43,44] or even under development.[45,46]
However, CMVNR in WEP tests of dual-species kind, especially using non-isotope species, is not so direct as that, in atom gradiometers. In the latter case, it is well developed to extract the differential phase shift
in a common-mode noises immune way by the ellipse fit method[47] or Bayesian estimation.[48–50] In WEP tests of dual-species kind, however, the scale factors
are usually different for the two interferometers. The difference originates from the different effective Raman wave vectors used for the atom species
and 2 when the pulses separation times Tj are the same. The different scale factors cause two aspects of complexity in the signal extraction of WEP tests. Firstly, the interested signal
is not proportional to the differential phase shift, which excludes the direct extraction of
using the usual ellipse fit method or Bayesian estimation. Secondly, the induced phase fluctuations by vibration noise are also different for the two interferometers, which increases the difficulty to the CMVNR. The scale factors can be made the same by using different pulses separation times Tj, which is particularly favorable in the case of the ratio
close to unity.[49] However, in this situation, the vibration noise is not exactly the same as experienced by the two interferometers, which thus excludes the possibility of perfect common-mode rejection. Alternatively, it is proposed to simultaneously measure the vibration noise by an auxiliary sensor and then reconstruct the fringes.[51,52] The effectiveness of this method depends on the quality of the correlation between real and measured vibrations, which is hard to ensure when the aimed precision of the WEP test is beyond the intrinsic noise of state-of-the-art vibrations sensors. This problem may also be mathematically resolved by an improved ellipse fit method, Bayesian estimation or direct phase extraction.[33,50,52] However, these solutions either require complex computation (sometimes even causing a bias result) or suffer from the trouble of separating the WEP violation signal from total differential phase shift.
It is already clear that the gravity acceleration or the vibration induced phase shift is linear to
, and we further note it is the conventional non-linear phase extraction process from the interference fringe that complicates the WEP violation signal separation and the CMVNR. Actually, the fringe-locking method (FLM) has already been adopted formerly for single interferometers[53–56] and recently for coupled interferometers,[16] by which the signal extraction of the interferometer can be linearized. In this work, we propose to operate the dual-species atom interferometers in the fringe-locking manner to linearize the signal extraction, which promises a good CMVNR, especially in the case of low level vibration noise. We note that, in the WEP test of dual-species kind, it only needs a change in the control of the Raman lasers effective frequencies to perform the FLM, and the corresponding signal extraction is direct. Moreover, this FLM allows for an unbiased determination of
.
2. Review of fringe-locking methodIn light pulses atom interferometers, the interference pattern is usually manifested as the variation of the transition probability between the two ground levels of the atom. The transition probability P is typically expressed as[1,3]
which forms a cosine fringe when the controllable
φ is scanned. In Eq. (
1),
A is the fringe offset,
B is the fringe amplitude, and
indicates the phase shift induced by the physical quantity to be measured. In the conventional method, a full cosine fringe is obtained by scanning
φ step by step, as shown in Fig.
1(a), and
is then acquired by a cosine fitting. Alternatively, the FLM can be adopted,
[53–55] as shown in Fig.
1(b). In the FLM, the appropriate value of
φ, denoted as
, is found to make
(where
n is an integer). Then
φ is modulated by
respect to the center of
so that the measurement is always performed at the midfringe, alternately to the right and to the left side of the central fringe. In this case, the transition probabilities
P for every two consecutive launches can be expressed as
 | (2a) |
 | (2b) |
where
l (
) denotes the index of modulating cycles. The difference between
and
could be used as a criterion of whether
is fulfilled, which in turn enables feedback control of
φ0. According to the linear approximation of Eqs. (
2a) and (
2b) at the midfringe, the correction can be expressed as
where the fringe amplitude
B must be known in advance by scanning the full fringe. Once the corrections are made to form a closed feedback loop, the equation
is supposed to be reached, from which the value of the interested
can be deduced from
in a linear way. It is shown in Eq. (
3) that when the interferometer is operated at the midfringe, the measurement works in a linear region. This linearization character is helpful for signal extraction in the WEP test of dual-species kind, as will be discussed in detail in the following section.
3. Application in WEP testIn the above section, the FLM for single interferometer is illuminated, and the Raman laser phase is convenient to control, which thus usually plays the role of
. Generally speaking, one needs two independent controllable phases to simultaneously lock two interferometers. In Ref. [16], in addition to the Raman laser phase, the phase shift due to the magnetic field gradient is explored as the other controllable phase for an atom gravity gradiometer. In a WEP test using simultaneous dual-species atom interferometers, there are already two independent groups of Raman lasers (usually one group used for one species atom), and thus it is natural to explore the two controllable Raman laser phases for fringe locking.
For each species atom interferometer using
Raman pulses scheme, the controllable phase
, namely the Raman lasers phase, can be expressed as[33,50,52,57]
where
αj is the chirp rate of the effective Raman laser frequency used to compensate the Doppler shift due to gravity, and
is the
pulse duration. With the Raman pulses duration effect neglected,
just simplifies to
Tj2. The interested phase
, namely the phase related to the gravity acceleration, can be expressed as
[33,50,52,57]
where the gravity acceleration of
j species atoms is denoted as
gj to account for possible WEP violation. In order to clearly manifest the ability of common-mode rejection with FLM, the phase due to vibration is explicitly included in the total phase shift, which can be expressed as
[33,50,52,57]where
is the sensitivity function,
[58] and
ti is the central time of the interfering progress for the
ith shot measurement. We note that both
and
are irrelative to
.
[33,50,52,57] In the case of identical pulses separation time (namely
), identical effective Rabi frequencies (thus
), and simultaneous interferometers (thus
as well as experiencing the same vibration noise
),
and
are identical for the dual-species atom interferometers. This is exactly the situation we hope (and also are able to) manage to achieve, we then abbreviate
and
as
and
, respectively.
Once the appropriate values of Raman laser phase, denoted as φj0, are found to make
for each of the interferometers, the Raman laser phases are then modulated by
. The corresponding transition probabilities for the two interferometers for every two consecutive launches can be expressed as
 | (7a) |
 | (7b) |
In the same way as in a single interferometer, the correction is made as
where
l denotes the
lth correction (the index of
φj0 is explicitly indicated here). The
-th phase modulation center will be
. For a single interferometer, the
l-th measured value of the gravity acceleration is then expressed as
This measured value is obviously affected by the vibration noise, which can be actually explicitly deduced from the linear approximation of Eqs. (
7a) and (
7b) at the midfringe, namely
However, the measured WEP violation signal is the difference of the measured gravity accelerations, which is then expressed as
which is exactly the possible WEP violation signal one searches for. It is clearly shown from Eq. (
10) that the FLM promises a perfect CMNVR within the first order approximation for WEP tests using simultaneous dual-species atom interferometers. This common-mode rejection capability profits from the linearized signal extraction.
4. Simulation and resultEquation (9) is based on the linear approximation of the measurement equations (7a) and (7b), and so is the consequent common-mode rejection. Actually, vibration noise would cause a departure of the measurement point from the midfringe, which would then affect the linear approximation. It is easily imagined that this affection will increase with the noise level. Since the vibration noise induced phases for the two interferometers are different, the relative sites of the measurement points at the respective fringe are also different, which will limit the common-mode rejection capability of this fringe-locking method. The rejection capability will be investigated here by numerical simulation. For simplicity, in the simulation the transition probabilities are re-written as
 | (11a) |
 | (11b) |
where
is equivalent to
, and
is defined as
. With this redefinition,
in the absence of WEP violation. The vibration noise is simulated by randomly drawing the values of
in a Gaussian distribution with a standard deviation of
. To simulate the fringe locking, the Raman laser phase
(
) is modulated by
with a center of
φ10 (
φ20), and the corrections for every two consecutive launches are made as Eq. (
8) (the initial values of
and
are
and
, respectively). The corresponding measured phase shifts due to gravity acceleration are then
and
, respectively, from which the interested possible WEP violation signal can be deduced as
.
In the numerical simulation, the dependence of the common-mode rejection efficiency on the vibration noise level (characterized by
) and the ratio of the effective Raman wave vector (characterized by r) is investigated. For each
and r, 104 modulation cycles are simulated for the two interferometers, in which process
pairs
are generated with
and
and 104 pairs (
) are obtained. The influence of the vibration noise on the interested signal is characterized by the Allan deviation of the
, which is calculated by the obtained pairs (
). Since the simulated vibration noise is white noise, the calculated Allan deviation of
, denoted as
, scales down by the inverse square-root of the number of measurement, namely
. Here N denotes the number of measurements, and
stands for the measurement sensitivity, which is obtained by white noise model fitting of
versus N.
The efficiency of the common-mode noise rejection is characterized by the rejection ratio
. The simulation result for the dependence of the rejection ratio on the vibration noise level
is shown in Fig. 2. According to the logarithmic fit in Fig. 2, the rejection ratio scales by
per octave, which is very close to the expected value of
per octave as a result of the third order expansion of the sinusoid function. This means that the CMVNR capability of the FLM improves quite fast as the vibration noise decreases. This simulation also indicates that even if the vibration noise is as large as 1 rad, there is still about 33 dB rejection ratio for the 39K versus 87Rb dual-species interferometers.
The simulation for different r is then performed with a fixed vibration noise level of
mrad. The result is shown in Fig. 3, which also shows a logarithmic relation. According to the logarithmic fit, the rejection ratio scales down by
per octave. It is shown that even if the deviation of the ratio from unity is as large as 1, there is still about 110 dB rejection ratio for
mrad.
In addition to the CMVNR capability, the bias of the extracted
is also highly important. We have checked that the average value of
extracted by this fringe-locking method is exactly equal to the differential value of the set
and
, whatever the vibration noise level (as long as
rad) and the ratio of the effective Raman wave vector are.
5. Discussion and conclusionThe simulation for different fringe amplitudes of the two interferometers (namely
) is also performed, and the dependence of the rejection ratio on
and r does not change. This is consistent with our knowledge. In the presence of phase noises only, the absolute value of the amplitude does not matter much for the FLM as long as it is known exactly. In an actual experiment, the fringe amplitudes of the two interferometers are pre-determined by scanning the full fringe and then performing cosine fittings. In the fringe-locking mode, the realtime information about the fringe amplitudes is lost, of which possible drift will affect the fringe-locking. This can be resolved by occasionally switching back to the full-fringe recording mode to get renewed fringe amplitudes.
In conclusion, we have shown the capability of CMNVR using FLM in WEP test by simultaneous dual-species atom interferometers. We note that it is convenient to perform this method in the dual-species interferometers, and it is also direct to extract the signal. Of importance, this signal extraction approach allows for an unbiased determination of the gravity accelerations difference, and allows a good common-mode noise rejection, especially in the low vibration noise level. This will thus alleviate the demand for the vibration noise isolation in WEP test of the dual-species kind.
