A Wigner Function Approach to Coherence in a Talbot-Lau Interferometer
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A Wigner Function Approach to Coherence in aTalbot-Lau Interferometer
Eric Imhof *, James Stickney and Matthew Squires Space Dynamics Laboratory, Utah State University Research Foundation, North Logan, UT 84341, USA;[email protected] (J.S.) Kirtland Air Force Base, Albuquerque, NM 87117, USA U.S. Air Force Research Laboratory, Kirtland Air Force Base, Albuquerque, NM 87117, USA;[email protected] (M.S.) * Correspondence: [email protected]; Tel.: +1-505-846-7260Academic Editors: A. Kumarakrishnan and Dallin S. DurfeeReceived: 3 May 2016; Accepted: 16 June 2016; Published:
Abstract:
Using a thermal gas, we model the signal of a trapped interferometer. This interferometeruses two short laser pulses, separated by time T , which act as a phase grating for the matter waves.Near time 2 T , there is an echo in the cloud’s density due to the Talbot-Lau effect. Our model usesthe Wigner function approach and includes a weak residual harmonic trap. The analysis shows thatthe residual potential limits the interferometer’s visibility, shifts the echo time of the interferometer,and alters its time dependence. Loss of visibility can be mitigated by optimizing the initial trapfrequency just before the interferometer cycle begins. Keywords: trapped atom interferometry; Wigner function; Talbot-Lau interferometer;coherence time
1. Introduction
Cold atom interferometry has been investigated for precision measurement applications [1,2],particularly inertial navigation [3–6]. Atom interferometers have demonstrated orders of magnitudeimprovement in bias stability over commercial navigation grade ring laser gyroscopes [7] and similargains are expected for accelerometers, gravimeters, magnetometers, and more.Transitioning the technology to a real-world device has proven difficult. The most sensitiveatom interferometers use a 10-meter long apparatus [8]. These measurements rely on a Raman pulsetechnique which changes the internal state of the interrogated atoms. Because of the difficulty inconfining multiple states with a magnetic field, atoms are allowed to propagate freely, necessitating alarge system.Single internal state splitting has allowed atoms to be trapped for the duration of theinterferometer cycle, reducing the apparatus length to a few millimeters [9]. Techniques for confinedsplitting include double-well potentials [10], optical lattices [11], and standing wave pulses [12,13].However, these interferometers have used Bose-Einstein condensates, which require cooling stagesthat increase power consumption, decrease possible repetition rates, and lower atom numbers.One single state technique has been shown to work at thermal ( i.e. , non-condensed)temperatures [14–16]. These interferometers, in the “Talbot-Lau” configuration, confine the atomicsample in two directions and allow free propagation in the third. In an ideal situation, thepotential along the third direction would vanish. However, due to the finite size of the device anduncontrollable external fields, there is residual potential along the waveguide.
Unfortunately, the residual potential and other field imperfections reduce coherence times [13,17,18].
Recent research has demonstrated a high degree of control over the residual field [19]. Here, we
Atoms , toms , , x 2 of 11 analyze the effect of a controlled residual potential in a Talbot-Lau interferometer with a gas of cold,thermal atoms using a Wigner function approach.
2. Interferometer Operation
To prepare the atomic gas for the interferometer cycle, a laser cooled sample is loaded into amagnetic trap with frequencies ω ( e ) i , where i = ( x , y , z ) . The collision rate is directly proportional tothe geometric average of these trap frequencies ¯ ω ( e ) = ( ω ( e ) x ω ( e ) y ω ( e ) z ) , so ¯ ω ( e ) should be madeas large as possible to maximize the efficiency of the evaporative cooling. In typical atom chipexperiments, the gas is evaporatively cooled in a trap with frequency ¯ ω ( e ) ∼ π ×
200 Hz.Once the atoms are cooled to a temperature on the order of
T ∼ µ K, the potential isadiabatically transformed into a trap that tightly confines the atoms in the radial direction, withfrequencies ω y = ω z = ω ⊥ ∼ π ×
200 Hz; and in the axial direction, with frequency ω x = ω . Just before the interferometer cycle starts, the potential is non-adiabatically transformed intoa waveguide potential, while holding the radial trap frequency constant to reduce the effects oftransverse excitations. In a realistic device, there remains a residual potential along the waveguidewith frequency ω .Once the atoms are loaded in the waveguide, the interferometer cycle begins. In this analysis, weconsidered the case of the trapped atom Talbo t-Lau interferometer schematically shown in Figure 1.The figure traces the different paths that an initially stationary atom could experience when movingthrough the device. Time moves from left to right, and the displacement of the atom along thewaveguide is shown in the vertical direction. d i s p l a ce m e n t Figure 1.
The schematic of a Talbot-Lau interferometer. An atomic cloud is split in space (verticalaxis) by a laser pulse at time t =
0. The resulting diffracted orders separate, and are further diffractedat t = T . At the recombination time t = T , the various orders overlap, allowing a probe laser toproduce a back scattered signal from the periodic atomic distribution. We only show two diffractionorders because for typical laser pulses, higher orders are suppressed. At time t =
0, the atomic cloud is illuminated with a short, standing wave laser pulse that actsas a diffraction grating. The pulse is sufficiently short that it is in the Kapitza-Dirac regime, i.e. , theatoms do not move for the duration of the laser pulse. The pulse splits the wave function for eachatom into several momentum states separated by the two photon recoil momentum δ P = hk l , where k l is the wave number of the laser beams.After the laser pulse, the atomic cloud propagates in the waveguide for a time T , at which pointit is illuminated with a second laser pulse. The paths of the different momentum states are shownas blue lines between 0 and T . Ideally, the momentum of each mode should be constant in time.However, the residual curvature along the waveguide will cause the paths to become curved (notshown in the figure), giving rise to decoherence.For simplicity, it is assumed that the laser pulse at time T has the same strength and affects theatomic wave function in the same manner. Each of the momentum states that were populated after toms , , x 3 of 11 the first laser pulse are split into several modes. After the second laser pulse, the number of possiblepaths increases dramatically. However, near time 2 T , the different paths come together to form adensity modulation that has the same period as the standing wave.An extraordinary feature of a Talbot-Lau interferometer is that the location of the density echo isindependent of the initial velocity of the atom. For example, if the initial atom in Figure 1 had somemomentum, each of the diffracted orders would gain this additional momentum. After tracing outall possible paths, it is easy to show that the density modulation appears in exactly the same locationas for the initially stationary atom. As a result, the density echo is still visible even when the initialatomic gas is relatively hot.In the absence of external forces, the density echo will have the same relative phase as thestanding wave laser pulse. However, if there is a force on the cloud, the echo will move in responseto the force. By detecting the shift in the echo, it is possible to measure the force on the cloud.This phase shift can be measured by reflecting a traveling wave off the density modulation.Due to the Bragg effect, there will be a strong backscattered signal for the duration of the echo.By heterodyning the back-reflected light with a reference beam, the phase of the density echo canbe determined.In this paper, we present a theoretical model of a trapped Talbot-Lau interferometer that includesthe decoherence due to the residual potential curvature. We use the Wigner function approach tomodel the dynamics of a thermal gas, which can be extended to include more complex laser pulsesequences [18]. For brevity, only the simple case of a two-pulse interferometer is discussed. Ourmodel predicts the amplitude of backscattered light for an arbitrary initial Wigner function and isthen specialized to the case of an initial thermal distribution. Decoherence due to finite temperatureand initial axial trap frequency are discussed. Finally the model is used to determine the ideal axialfrequency for a given initial phase space density and residual potential.
3. The Model
Following the prescription of [19], we assume that the potential is separable, i.e. , V ( rrr ) = V ( x ) + V ⊥ ( r ⊥ ) , and the k -vectors of the laser beams point in the x -direction. Collisions are neglected aswe have previously analyzed the effects of collisions in a similar interferometer and do not expectatom-atom collisions to have a significant impact on the results [20]. We also ignore the mean fieldinteraction, as it is mainly relevant for strongly interacting condensates, which we do not considerhere. Inclusion of these terms may be possible, but are omitted to keep the discussion concise. TheHamiltonian that governs the axial dynamics of the interferometer is one-dimensional and can bewritten as H = P M + M β X + ¯ h Ω cos ( k l X ) , (1)where X and P are the canonical operators with commutation relation [ X , P ] = i ¯ h , k l is the wavenumber of the laser, M is the atomic mass, and β is the curvature of the residual potential. Theparameter Ω is the frequency of the AC-stark shift due to the standing wave laser pulse, whichdepends on the intensity and detuning of the beam and is, in general, a function of time.The Hamiltonian can be recast in the dimensionless form H ′ = P ′ + β ′ X ′ + Ω ′ cos X ′ , (2)where P ′ = P / P , X ′ = X / X , and t ′ = t / t where P = hk l , X = k l , and t = M /4¯ hk l . Theother parameters in Equation (1) become β ′ = β t and, Ω ′ = Ω t . The other important dimensionlessparameter is the cloud temperature T ′ = T / T , where T = h k l / Mk B , where k B is the Boltzmannconstant. For Rb where the standing wave laser is near the D2 transition, t = µ s, and T = µ K. For the rest of this paper, primes will be dropped for clarity, and unless otherwise stated, allintroduced variables will be dimensionless. toms , , x 4 of 11 Since the interferometer uses an incoherent gas, the state of the system cannot be written as awave function. Instead, the system is described by the density operator ρ . The equation of motion forthe density operator, in dimensionless form, is i ˙ ρ = [ H , ρ ] , (3)where the dot denotes the time derivative and the brackets are the usual commutation operator. Thedensity operator can be recast in terms of the Wigner function, which is defined as f ( x , p ) = π Z d ξ h x + ξ | ρ | x − ξ i e − ip ξ (4)where | x i are the eigenvectors of the coordinate operator, i.e. , X | x i = x | x i . The Wigner function f ( x , p ) can be interpreted as the probability density, however for non-classical states the Wignerfunction may be negative. As a result, R dx f = P ( p ) is the momentum density of the cloud and R dp f = ρ ( x ) is the spatial density. Even when the Wigner function is negative, the densities, P and ρ are always positive.It is worth noting that the Wigner approach works for pure states as well. In this case, it isdefined as f pure ( x , p ) = π Z d ξψ ∗ ( x + ξ ) ψ ( x − ξ ) e − ip ξ . (5)We will find that the results of the incoherent process are easily extended to include the results of apure state (BEC) interferometer.Substituting Equation (4) into Equations (2) and (3) it can be shown that the equation of motionfor the Wigner function is (cid:18) ∂∂ t + p ∂∂ x − β ∂∂ p (cid:19) f ( x , p , t ) = Ω sin x (cid:20) f (cid:18) x , p − (cid:19) − f (cid:18) x , p + (cid:19)(cid:21) , (6)where the left side of the equation describes the motion of the distribution in the potential while theright side describes the interaction with the standing wave laser field.Since the duration of the laser pulses τ p is much shorter than the interferometer time T ( T ≫ τ p ),the evolution of the distribution can be separated into relatively slow dynamics when the distributionis not being illuminated and fast dynamics when it is. Additionally, since each laser pulse is short τ p ≫ ω and strong Ω ≫ ω , the pulses are in the Kapitza-Dirac regime, which occurs in theRaman-Nath limit. As a result, the coordinate and momentum derivatives in Equation (6) may beneglected during the pulse.The dynamics of the distribution for the periods when the laser is off, Ω =
0, are such that eachpart of phase space evolves classically. For simplicity, it is useful to write the classical equations ofmotion in the form ˙ xxx = Mxxx (7)where xxx = ( x , p ) is the coordinate-momentum vector, and the matrix M is M = − β ! . (8)The solution to Equation (7) can be written as xxx ( t ) = U t xxx ( ) , where U t = exp ( Mt ) . By directsubstitution it can be shown that in between the laser pulses the distribution evolves as f f ( xxx ) = f i ( U − t xxx ) . (9) toms , , x 5 of 11 The laser pulses are more involved and fundamentally quantum in nature ( i.e. , resulting innegative Wigner distributions). The effect of the laser pulse is to transform an initial Wignerdistribution f i into a final distribution f f according to f f ( Ω = ) = ∞ ∑ nk = − ∞ ( − i ) n J k ( Ξ ) J n + k ( Ξ ) e i ( n + k ) x f i (cid:16) x , p − n (cid:17) (10)for the pulse area, Ξ = R d τ Ω ( τ ) , where the functions J n are the Bessel functions of the first kind. Interms of xxx , Equation (10) can be written in the more compact form f f ( Ω = ) = ∑ nk α nk e iggg nk · xxx f i ( xxx − N n ) (11)where g nk g nk g nk = ( n + k , 0 ) , NNN n = ( n /2 ) , and α nk = ( − ) n J k J n + k .The interferometer sequence is characterized by four unique operations separated in time. Thefirst laser pulse at t = f and transforms it to f , ( f → f ).There is then a propagation period from t = T , over which the distribution transforms f → f .The second laser pulse at t = T transforms f → f . Lastly, another propagation to t = T + τ transforms the distribution to its final form f → f .Near the end of the interferometer cycle, the cloud is illuminated with a short traveling wavelaser pulse of duration τ , where τ ≪ T . To determine the time dependence of the back-scatteredlight, the Wigner function must be found for times near the echo time, i.e. , t = T + τ . By directsubstitution into Equations (9) and (11) for the interferometer cycle discussed in Figure 1, the Wignerfunction near the echo time is f = ∑ mlnk α ml α nk × exp [ i ( ggg ml · U T + ggg nk ) · U − T − τ · xxx − iggg nk · U − T · NNN m ] × f ( U − T − τ · xxx − U − T · NNN m − NNN n ) . (12)According to [17], the amplitude of the back-scattered light is proportional to S = Z d xe iggg · xxx f ( xxx ) (13)For the rest of the paper, the quantity S will be referred to as the signal of the interferometer. Changingthe integration variable from xxx to yyy , where yyy = U − T − τ xxx − U − T N m − N n , (14)the signal can be written as S = ∑ mlnk α ml α nk e i Θ mlnk Z d ye i ∆∆∆ mlnk · yyy , (15)where ∆∆∆ mlnk = ggg ml · U T + ggg nk + ggg · U T + τ , (16)and Θ mlnk = ∆∆∆ mlnk · ( U − T · NNN m + NNN n ) − ggg nk · U − T · NNN m . (17)In what follows below, it will be assumed that both the echo duration is small as compared to theinterferometer time τ ≪ T , and the residual trap curvature is β ≪ T . When these inequalities arefulfilled, only the linear contributions in both τ and β are retained. In this limit, the time propagationoperator for small values of β is U T ≈ U ( ) T + β U ( ) T , where U ( ) t = (cid:0) t (cid:1) , and U ( ) t = (cid:16) t /2 t /6 t t /2 (cid:17) , andfor small values of time τ , U τ = + M ( ) τ , where M ( ) = (cid:0) (cid:1) . toms , , x 6 of 11 Equation (16) can now be written as ∆∆∆ mlnk = ∆∆∆ ( ) mlnk + β (cid:16) ggg ml · U ( ) T + ggg · U ( ) T (cid:17) + τ (cid:16) ggg U ( ) T M ( ) (cid:17) , (18)where ∆ ( ) is given by Equation (16) where β → τ →
0. In the limit where the distributionis slowly varying, the elements of the sum in Equation (13) are vanishingly small unless ∆ ( ) = ggg ml = − ggg and ggg nk = ggg . Using the definition of ggg , these relations can be writtenas k = ( − n ) /2 and l = − ( + m ) /2. In addition, only the terms where n , ( m ) are even (odd)contribute to the signal. Equation (18) becomes independent of the indices m , l , n , k .Substituting the explicit matrix representations for ∆ and Θ , the interferometer signal is given by S = A Z dudv exp h − i β T u + i τ ′ v i f ( u , v ) , (19)where τ ′ = τ − β T and u , v are the components of the vector yyy . The parameter A in Equation (19) isthe amplitude of the signal and can be expressed as the sum A = ∑ n , even ∑ m , odd γ nm i exp (cid:20) i (cid:18) mT + m + n τ ′ + m β T (cid:19)(cid:21) , (20)where γ nm = ( − ) ( n − ) /2 + m /2 J ( − n ) /2 J ( + n ) /2 J − ( + m ) /2 J − ( − m ) /2 (21)determines proportion of the atoms scattered into each mode.Equation (19) is the primary result of this analysis, and will be used for the case of a thermalatomic cloud in Section 4.
4. Discussion
By taking the limit where Ξ ≪
1, only the lowest order contributions to Equation (20) need to beretained. If we keep n = ± m = ± J n for the small argument, γ ≈ − γ ≈ Ξ /4, then A = sin (cid:18) τ ′ (cid:19) Ξ (cid:20) + cos (cid:18) T + β T + τ ′ (cid:19)(cid:21) . (22)Assuming that the initial distribution is a thermal cloud of temperature T that is in equilibriumwith the trap with frequency ω , the distribution f becomes f = ω π T exp − p T − ω x T ! . (23)By comparison, the initial distribution of a condensate would be well approximated by theground state of a harmonic oscillator. Using Equation (5), the pure state Wigner function is equivalentto Equation (23) when T = ω /2. During the transition from an incoherent thermal gas to a pureBEC, the distribution is a sum of f and f pure , weighted by the number of atoms in and out of theground state, where N / N = − ( T / T c ) and T c = ¯ ω ( N / ζ ( )) . N / N is the ratio of condensedatoms to the total, and ζ is the Riemann zeta function. This combined distribution can be used withEquation (19) to find the expected signal.Returning focus to the incoherent thermal gas, substituting Equations (22) and (23) intoEquation (19) and performing the integral yields S = A exp " − T (cid:18) β T ω (cid:19) − T τ ′ . (24) toms , , x 7 of 11 To quantify the signal visibility, we define the echo strength as I = Ξ R d τ ′ S , which isproportional to the total number of photons (electromagnetic energy) of the backscattered light duringthe read-out pulse. In the limit where T ≫
1, Equation (24) can be integrated, yielding I = π A T exp " −T (cid:18) β T ω (cid:19) , (25)where A = (cid:20) + cos (cid:18) T + β T (cid:19)(cid:21) . (26)Equation (25) diverges in the limit T →
0, which is clearly an unphysical result. However thenumerical integration of Equation (24) remains finite.Note that I is an oscillating function, and is well known in the β = T for β >
0. The dotted line isthe envelope of the echo strength. Figure 3 shows a schematic of Equation (25) as a function ofinterferometer time T for β < T I Figure 2.
A schematic of the echo signal strength, I , as a function of interferometer time, T , for aninterferometer in a positive residual trapping potential, i.e. , β >
0. The signal strength is proportionalto the total number of backscattered photons during the readout laser pulse. I is periodic with anincreasing frequency within an envelope defined by the dotted curve. T I Figure 3.
A schematic of the echo signal strength, I , as a function of interferometer time, T , for aninterferometer in a negative residual trapping potential, i.e. , β <
0. Like the positive potential case, thesignal strength is periodic and contained within a decaying envelope. However, the negative potentialcauses a decreasing frequency. Both positive and negative potentials have the same envelope. toms , , x 8 of 11 The oscillation frequency increases when β > β <
0, and there is a maximawhen T + β T /6 = π n . These oscillations depend only on the values of β and T . In a typicalexperiment, the oscillation frequency is much larger than depicted in Figure 2 or Figure 3. For theremainder of the paper, it will be assumed that the interferometer time is tuned to be at the peak ofan oscillation, which will be referred to as I m .In order to maximize signal strength, it is also useful to release the atomic sample into thewaveguide from the correct initial trap. Typically, the atomic gas is evaporatively cooled to atemperature T ( e ) in a trap with frequency ω ( e ) . After cooling, the trap frequencies are adiabaticallychanged to a trap with frequency ω and then released into a waveguide with residual potentialcurvature β . During the adiabatic transformation, the phase space density is constant. This conditionimplies D = T / ω is held constant, assuming the radial trap frequencies ω ⊥ are unchanged. ThenEquation (25) can be recast as I m = π ( D ( e ) ) ω exp " − ( D ( e ) ) ω ( β T ) , (27)where D ( e ) = ( T ( e ) ) / ω ( e ) is proportional to the phase space density at the end of the evaporation.For this analysis, assume the cloud is evaporatively cooled in a trap with frequency ω ( e ) = π × − and to a temperature T =
10. For Rb, these parameters correspond to a gascooled in a trap with a frequency of 20 Hz to a temperature of 14 µ K. The phase space density isproportional to D ( e ) = − /2 π . Figure 4 shows the echo strength, Equation (27), as a function ofdecompressed trap frequency ω . The remaining parameter | β | T = − , corresponds to a cycletime of 10 ms and a residual frequency of 0.3 Hz. In this case, the decompressed trap frequency isroughly half the evaporative trap frequency. ×10 I × − Figure 4.
The signal strength as a function of injection trap frequency, ω . After evaporation in a trapwith frequency ω ( e ) , the trap potential is adiabatically transformed to ω before the interferometercycle begins. At the start of the cycle, the trap is snapped to ω = p β , where it stays. The signalstrength peaks at for a non-zero injection frequency ω . For this case, β T = − , and D ( e ) = /2 π . For small values of ω ≪
1, the echo strength vanishes because the weak trap creates a largecloud, which experiences more de-phasing due to the residual potential. On the other hand, when ω ≫
1, the echo strength vanishes because the tight trap increases the temperature of the cloud,resulting in a shorter echo duration.The ratio of ideal starting trap frequency ω and evaporation trap frequency ω ( e ) = π × − is shown as as a function of | β | T in Figure 5. The dash-dot line is the ideal frequency if the gas iscooled to a temperature of T =
1, the solid line is the ideal frequency when T =
10, and the dashedline is when T = toms , , x 9 of 11 T0.00.51.01.52.02.5 ω ( m ) / ω ( e ) Figure 5.
The ratio of the ideal injection trap frequency, ω , to the evaporation trap frequency, ω ( e ) , as a function of | β | T . Here we use ω ( e ) = π × − , and plot for temperatures T = T =
10 (solid), and T =
20 (dash). As the ratio ω / ω ( e ) becomes greater than one, the gasshould be compressed before being released into the interferometer. This compression step raises thetemperature, but reduces the size of the cloud. For values where ω / ω e <
1, the ideal starting frequency is lower than the evaporationfrequency, i.e. , the gas should be decompressed before the beginning of the interferometer cycle. Atthe cost of increasing the cloud size, it is more advantageous the lower the temperature. For the casewhere ω / ω e >
1, the gas should be compressed, raising the temperature by reducing the size of thecloud.
5. Outlook
Tuning the interferometer time T and the injection trap frequency ω allows for maximal signalvisibility. However, these optimizations cannot overcome the exp ( − β ) dependence in Equation (25).Even a small residual potential dramatically reduces coherence times in this version of a trappedTalbot-Lau interferometer. Figure 6 shows the signal visibility, Equation (25), for several residualpotentials. The dashed curve is β = × − , the solid line is 10 − , and the dash-dot curve is 10 − .For this plot, I = π A /32 T , with A = T × − corresponds roughly to 1 ms. −2 I / I Figure 6.
The signal visibility, i.e. , the decaying envelope that limits the maximum possible signalstrength for a given interferometer time T . The decay is proportional to exp ( − β ) , causing rapidsignal loss for even small residual potentials. Here we show β = × − (dashed), 10 − (solid),and 10 − (dash-dot). T × − corresponds roughly to 1 ms. toms , , x 10 of 11 Clearly the signal visibility has a strong dependence on residual potential, which must beextremely small for coherence times compared to free space interferometers. In future work, we willexplore modifications to the trapped Talbot-Lau scheme with the potential to minimize the coherencetime’s sensitivity on residual field imperfections.The Wigner function approach allows a straightforward way to model interference in anincoherent system such as a cold atomic gas. It can be readily applied to consider different pulseschemes such as those of [17], as well as propagation in more complex confining potentials. TheTalbot-Lau interferometer’s ability to operate at thermal temperatures is a significant enough benefitto a real-world device that further study is warranted.
Acknowledgments:
This work was supported by the Air Force Research Laboratory.
Author Contributions:
Eric Imhof, James Stickney, and Matthew Squires contributed equally to this paper.
Conflicts of Interest:
The authors declare no conflict of interest.
References
1. Barrett, B.; Chan, I.; Kumarakrishnan, A. Atom-interferometric techniques for measuring uniform magneticfield gradients and gravitational acceleration.
Phys. Rev. A , , 063623.2. Muntinga, H.; Ahlers, H.; Krutzik, M.; Wenzlawski, A.; Arnold, S.; Becker, D.; Bongs, K.; Dittus, H.;Duncker, H.; Gaaloul, N.; et al. Interferometry with Bose-Einstein Condensates in Microgravity.
Phys.Rev. Lett. , , 093602.3. Cahn, S.B.; Kumarakrishnan, A.; Shim, U.; Sleator, T.; Berman, P.R.; Dubetsky, B. Time-Domain de BroglieWave Interferometry. Phys. Rev. Lett. , , 784–787.4. Cronin, A.; Schmiedmayer, J.; Pritchard, D. Optics and interferometry with atoms and molecules. Rev. Mod.Phys. , , 1051–1129.5. Adams, C.; Sigel, M.; Mlynek, J. Atom optics. Phys. Rep. , , 143–210.6. Geiger, R.; Menoret, V.; Stern, G.; Zahzam, N.; Cheinet, P.; Battelier, B.; Villing, A.; Moron, F.; Lours, M.;Bidel, Y.; et al. Detecting inertial effects with airborne matter-wave interferometry.
Nat. Commun. , ,474.7. Durfee, D.; Shaham, Y.; Kasevich, M. Long-term stability of an area-reversible atom-interferometer Sagnacgyroscope. Phys. Rev. Lett. , , 240801.8. Dickerson, S.M.; Hogan, J.M.; Sugarbaker, A.; Johnson, D.M.S.; Kasevich, M.A. Multiaxis Inertial Sensingwith Long-Time Point Source Atom Interferometry. Phys. Rev. Lett. , , 083001.9. Wang, Y.; Anderson, D.; Bright, V.; Cornell, E.; Diot, Q.; Kishimoto, T.; Prentiss, M.; Saravanan, R.; Segal, S.;Wu, S. Atom Michelson interferometer on a chip using a Bose-Einstein condensate. Phys. Rev. Lett. , , 090405.10. Schumm, T.; Kruger, P.; Hofferberth, S.; Lesanovsky, I.; Wildermuth, S.; Groth, S.; Bar-Joseph, I.;Andersson, L.M.; Schmiedmayer, J. A Double Well Interferometer on an Atom Chip. Quantum Inf. Process. , , 537–558.11. Hilico, A.; Solaro, C.; Zhou, M.K.; Lopez, M.; Pereira dos Santos, F. Contrast decay in a trapped-atominterferometer. Phys. Rev. A , , 053616.12. Horikoshi, M.; Nakagawa, K. Dephasing due to atom-atom interaction in a waveguide interferometer usinga Bose-Einstein condensate. Phys. Rev. A , , 031602.13. Burke, J.; Sackett, C. Scalable Bose-Einstein-condensate Sagnac interferometer in a linear trap. Phys. Rev. A , , 061603.14. Wu, S.; Su, E.; Prentiss, M. Time domain de Broglie wave interferometry along a magnetic guide. Eur. Phys.J. D , , 111–118.15. Xiong, W.; Zhou, X.; Yue, X.; Chen, X.; Wu, B.; Xiong, H. Critical correlations in an ultra-cold Bose gasrevealed by means of a temporal Talbot-Lau interferometer. Laser Phys. Lett. , , 125502.16. Mok, C.; Barrett, B.; Carew, A.; Berthiaume, R.; Beattie, S.; Kumarakrishnan, A. Demonstration of improvedsensitivity of echo interferometers to gravitational acceleration. Phys. Rev. A , , 023614.17. Wu, S. Light Pulse Talbot-Lau Interferometry with Magnetically Guided Atoms ; Harvard University: Cambridge,MA, USA, 2007. toms , , x 11 of 11
18. Su, E.; Wu, S.; Prentiss, M. Atom interferometry using wave packets with constant spatial displacements.
Phys. Rev. A , , 043631.19. Stickney, J.; Kasch, B.; Imhof, E.; Kroese, B.; Crow, J.; Olson, S.; Squires, M. Tunable axial potentials for atomchip waveguides. 2014, arXiv:1407.6398.20. Stickney, J.; Squires, M.; Scoville, J.; Baker, P.; Miller, S. Collisional decoherence in trapped-atominterferometers that use nondegenerate sources. Phys. Rev. A , , 013618.c (cid:13)(cid:13)