Dynamics within a tunable harmonic/quartic waveguide
aa r X i v : . [ phy s i c s . a t o m - ph ] F e b Dynamics within a tunable harmonic/quartic waveguide
Rudolph N. Kohn, Jr. ∗ and James A. Stickney Space Dynamics Laboratory, Albuquerque, New Mexico 87106, USA
Abstract
We present an analytical solution to the dynamics of a noninteracting cloud of thermal atoms in acigar-shaped harmonic trap with a quartic perturbation along the axial direction. We calculate thefirst and second moments of position, which are sufficient to characterize the trap. The dynamicsof the thermal cloud differ notably from those of a single particle, with an offset to the oscillationfrequency that persists even as the oscillation amplitude approaches zero. We also present somenumerical results that describe the effects of time-of-flight on the behavior of the cloud in order tobetter understand the results of a hypothetical experimental realization of this system. ∗ [email protected] . INTRODUCTION Cold atom interferometers have proven themselves to be valuable tools for examining avariety of effects, leveraging the wave nature of matter and the large rest mass of atoms tomake extremely precise measurements. They have been used to measure gravity [1–3], multi-axis accelerations [4–6], rotations [7–9], electric polarizability [10], fundamental quantitiessuch as the fine structure constant [11], and to test predictions of general relativity [12].In general, these interferometers use controlled light pulses to apply coherent momentumkicks to separate clouds into subsets with different momenta and reflect them back towardeach other. Between kicks, the atoms evolve in free space or along a waveguide. In most ofthe guided atom interferometers, laser pulses reflect the atoms long before the confinementturns the atoms around [7, 8, 10], but there is at least one example of an interferometerallowing the atoms to complete a full oscillation in the confinement potential [13].Interferometers produced from trapped atoms can be smaller than their free space coun-terparts, and the trapping potentials can hold the atoms against gravity, but the existenceof a preferred axis of oscillation makes precise alignment of the excitation beams critical [7].Using the potential to reflect the clouds means that, for a given separation, the interroga-tion time can be longer. However, uncontrolled variation in the trap potential can causeunwanted effects, and interactions between atoms can also weaken the signal [13]. In general,trapping potentials will always depart somewhat from perfect harmonicity, if only becauseof errors introduced in the fabrication process or inherent in the trap design. Therefore, itis extremely valuable to be able to identify and compensate for unwanted deviations in thetrap shape, especially if it can be done without altering the hardware.To this end, we developed a method for producing atom chip traps which permit finecontrol of the potential along the axis of a magnetic waveguide. Several polynomial termscan be controlled by tuning the currents through several pairs of wires and the spacing of thewires can minimize higher order contributions to the potential [14]. These tunable atom chipwaveguides allow us, in theory, to produce carefully tailored potentials, but imperfections inthe manufacturing process call for fine-tuning in order to approach the desired potential asclosely as possible. Therefore, it is crucial to examine the dynamics of atoms in the potentialsand have a clear theoretical understanding of the effects of deviations in the potential.In this paper, we will examine the dynamics of a cold cloud of atoms in a one-dimensional2rap with harmonic and quartic components. We will assume the other two axes are well-confined, harmonic, conservative, and separable. The harmonic and quartic terms are par-ticularly interesting because the harmonic term is solvable and the quartic term is oftenthe leading unwanted contribution in trapped atom interferometers [13]. With a few minorapproximations, we will derive solutions for the behavior of single particles and ensemblesand show that there are qualitative differences between them.The purely harmonic case can be thoroughly described using Boltzmann’s kinetic theory.One of the more counterintuitive results involves the behavior of the clouds at long times.While it might be assumed that a system of atoms in a perfectly harmonic trap wouldeventually reach some kind of thermal equilibrium, certain excitation modes, such as themonopole breathing mode, actually persist indefinitely, even in the presence of isotropic,energy-independent, elastic collisions [15]. The persistence of the monopole breathing modewas demonstrated experimentally in a system of cold atoms by Lobser et al. [16] in 2015.Some simple substitutions show that, in addition to monopole breathing, center-of-massoscillations of a small cloud along the weak axis of a cylindrically symmetric harmonic trapwill also persist indefinitely, so long as the axes are separable, and collisions are isotropic,elastic, and energy-independent [15]. We will refer to such oscillations as “sloshing” hence-forth. As we will show below, the addition of a quartic perturbation has several effects.Most importantly, the quartic perturbation causes initially close atoms with slightly differ-ent energies to gradually separate, effectively randomizing their phases and resulting in thegradual decay of sloshing and the spreading out of the cloud in a quasi-thermal state at thecenter of the trap, even in the absence of collisions. We note that this quasi-thermalization,or “dephasing” can be used to characterize the anharmonic contributions to the trappingpotential. In one of our tunable atom chips, the parameters can be adjusted to minimizedephasing and iteratively approach a perfectly harmonic potential. In addition, we will seebelow that the quartic contribution alters the frequency of the trap for clouds of atoms, evenat infinitesimally small sloshing amplitudes. Finally, the use of two independent parametersto describe our traps necessitates the measurement of two independent characteristics of thecloud. The center of mass position, h x i and the size of the cloud σ , requiring two momentsof position, are sufficient to uniquely determine the shape of such a trap.The paper is divided into several sections. In Section II, we will proceed through theanalytical solution of the one-dimensional harmonic-quartic trap. Section III describes the3esults of the theory and examines some of the finer details. In Section IV, we will usenumerical methods to calculate the behavior of the clouds after some time of flight, which isa common technique used to observe clouds of cold atoms. Finally, we will summarize ourconclusions and describe some future paths for inquiry in Section V. II. THEORY
In this paper, we will assume that the motion of the atomic gas in the x directioncan be decoupled from the pure harmonic motion in the y and z directions, i.e. H t = H ( x, p x ) + H ⊥ ( y, z, p y , p z ). The trap along the x-axis is mostly harmonic, and the mostsignificant deviation is a quartic term. Thus, the Hamiltonian H = p m + mω (cid:18) x x x (cid:19) (1)governs the dynamics, with m as the atomic mass, ω as the harmonic trap frequency, and x describing the quartic contribution. We have also used p as shorthand for p x and willcontinue to do so going forward. x can be either positive or negative, and its magnitudecorresponds to the value of x where the forces due to the harmonic and quartic terms areequal.In the limit where the oscillation amplitude A is small, i.e. A ≪ | x | , the dynamics ofa particle in the Hamiltonian given by Eq. (1) can be approximated by a sinusoid with anamplitude-dependent frequency. The position of the particle is approximately x ( t ) A = cos(Ω t − φ ) + 132 A x cos(3Ω t − φ ) (2)where A is the amplitude and φ is the initial phase of the particle. The amplitude-dependentfrequency is Ω ω = 1 + 38 A x , (3)and does not depend on the initial phase. We neglect the part of the dynamics whichoscillates at the frequency 3Ω in later calculations because its magnitude is extremely smallcompared to the Ω term.The effect of the perturbation on the dynamics of a single particle becomes dependent ononly the amplitude of oscillation. It is convenient to recast this in terms of the unperturbedenergy E = p m + 12 mω x , (4)4 M o m e n t u m ωt = 0 −5 0 5 −505 ωt = 1 −5 0 5−505 M o m e n t u m ωt = 5 −5 0 5Position −505 ωt = 20 −5 0 5Position FIG. 1. Diagrams of an atomic cloud as dephasing causes quasi-thermal equilibrium. The figure isseparated into four parts, with the top subplot showing the phase space density of the cloud at agiven time, and the bottom subplot shows the density distribution in space only. As time passes,atoms with small differences in energy lose phase coherence until the cloud eventually spreads outalong the bottom of the trap, almost as if it has thermalized, even though the model contains nointeractions at all. where x and p are the initial coordinate and momentum.For the remainder of the paper, we will describe the dynamics of a single particle to be x ( t ) = r Emω cos(Ω t − φ ) , (5)where tan φ = p /mωx and Ω ω = 1 + 38 2 Emω x . (6)5rom these expressions, we see that any particle can be placed on a constant energy ellipsein phase space and traces out that ellipse with a frequency that depends only on energy,independent of the initial phase of the oscillation. Fig. 1 illustrates this effect by plottingthe phase space distribution of a cloud in a harmonic-quartic trap at several different times.A classical cloud of atoms can be described by its phase space density, f ( x, p, t ), wherethe phase space density describes the probability density to find an atom with the position x and momentum p at time t . The n th moment of position of a cloud of atoms is h x n i = Z ∞−∞ Z ∞−∞ dxdp x n f ( x, p, t ) . (7)In general, the phase space density f ( x, p, t ) evolves in time, but in a non-interacting systemeach element of phase space moves independently from all others and can be tracked as itevolves. By backtracking each point and its phase space density back to t = 0, one canremove the dynamics from f ( x, p, t ) and incorporate them into x n such that h x n i = Z ∞−∞ Z ∞−∞ dxdp x nR ( x, p, t ) f ( x, p ) , (8)where x R ( x, p, t ) is the backtracked position at t = 0 that the position at ( x, p, t ) nowrepresents.To simplify the integration, it is convenient to transform Eq.(8) into a polar coordinatesystem such that x = p ξσ x cos φ and p = p ξσ p sin φ, (9)where ξ = E/k B T is the ratio between phase space energy, Eq. (4), and thermal energy,where T is the initial temperature of the cloud. Both coordinate and momentum are scaledby the thermal standard deviations, σ p = p mk B T and σ x = p k B T /mω , (10)to make them unitless. The angle tan( φ ) = σ x p/σ p x is the initial polar angle of some point( x, p ) at t = 0. In this new coordinate system x R = p ξσ x cos(Ω t + φ ) , (11)and Eq. (8) becomes h x n i ( √ σ x ) n = Z ∞ Z π dξdφσ x σ p ξ n/ cos n (Ω t + φ ) f ( ξ, φ ) , (12)6here f ( ξ, φ ) is simply the initial phase space density distribution converted to the newcoordinate basis.First consider the case where the atomic cloud is prepared in thermodynamic equilibriumat temperature T , in the trapping potential with frequency ω . At time t = 0 the cloud isgiven a kick, resulting in initial center of mass conditions p D = p ξ D σ p sin φ D and x D = p ξ D σ x cos φ D . (13) ξ D and φ D are defined with respect to x D and p D in the same way that ξ and φ aredefined in terms of x and p , such that ξ D = x D σ x + p D σ p and tan( φ D ) = σ x p D /σ p x D . (14)Starting from a thermal distribution in the harmonic trap, we displace it by x D and p D , andconvert to the polar coordinate system, leading to f = 12 πσ x σ p exp (cid:16) − ξ − ξ D + 2 p ξξ D cos( φ − φ D ) (cid:17) . (15)Substituting Eq. (15) into Eq. (12) and converting the integral to the new variables yields h x n i ( √ σ x ) n = e − ξ D π Z ∞ dξξ n/ e − ξ Z π dφ cos n (Ω t + φ D + φ ) exp (cid:16) p ξξ D cos φ (cid:17) . (16)The trigonometric power law reduction formula, cos n θ = P m c nm cos mθ , permits furthersimplification. Analytic expressions for the coefficients c nm can be found in Gradshteyn andRyzhik [17], on page 31. In this case, the three elements of interest are c = 1, c = 1 / c = 1 / h x n i ( √ σ x ) n = 12 X m c nm (Υ nm + c.c ) , (17)with the Υ nm given asΥ nm = e − ξ D + im ( ωt + φ D ) Z ∞ dξξ n/ exp [ − (1 − im Λ t ) ξ ] I m (2 p ξξ D ) , (18)where Λ ≡ ωσ x / x and I m is the modified Bessel function of the first kind. The integralsin Υ , Υ and Υ must be solved to calculate closed-form expressions for h x i and h x i .7he integrals in Υ and Υ can be solved in terms of Gradshteyn and Ryzhik [17], section6.631, equation 4. Υ can be solved in terms of equation 1 in the same section. The resultsare Υ = 1 + ξ D , (19)Υ = √ ξ D (1 − i Λ t ) exp (cid:20) − ξ D + ξ D (1 − i Λ t ) + i ( ωt + φ D ) (cid:21) , (20)and Υ = ξ D (1 − i Λ t ) exp (cid:20) − ξ D + ξ D (1 − i Λ t ) + i (2 ωt + 2 φ D ) (cid:21) . (21)With these solutions, it is possible to calculate the center-of-mass position and size of thecloud as a function of time for any chosen set of parameters that satisfies the requirementthat the initial kick produce a maximum displacement in position much less than | x | . III. RESULTS
Having calculated the values of Υ nm , we can combine them into expressions for physicalproperties that can be easily measured in an experimental realization of this system. Thecenter-of-mass position and size of the cloud are, in general, easily observed and calculatedfrom absorption images.Υ leads to the center-of-mass position of the atomic cloud as a function of time. h x i = σ x √ ξ D [1 + (Λ t ) ] exp (cid:20) − (Λ t ) ξ D t ) (cid:21) (cid:0) cos(Φ ) − t sin(Φ ) − (Λ t ) cos(Φ ) (cid:1) , (22)where Φ = ωt + φ D + Λ tξ D / (1 + (Λ t ) ). The first moment resembles a decaying sinusoid.The initial decay is Gaussian, but at later times the denominator of the first term takesover. Υ and Υ combine to give the second moment h x i = σ x (1 + ξ D )+ σ x ξ D [1 + (2Λ t ) ] exp (cid:20) − (2Λ t ) ξ D t ) (cid:21) × (cid:0) cos(Φ ) − t sin(Φ ) − t ) cos(Φ ) + 8(Λ t ) sin(Φ ) (cid:1) , (23)where Φ = 2 ωt + 2 φ D + 2Λ tξ D / (1 + (2Λ t ) ). This has a similar form to h x i , but it oscillatesabout twice as rapidly, and decays faster as well, as seen in Figure 3. In fact, the secondmoment is less illustrative of the dynamics than the size of the cloud as a function of timebecause the position and size of the cloud are more directly measurable. The standarddeviation of the cloud’s position distribution, hereafter referred to as the “cloud size” is8 .0 0.2 0.4 0.6 0.8 1.0 t −4−3−2−101234 ⟨ x ⟩ FIG. 2. h x i as a function of time. Most of the decay to quasi-thermal equilibrium is shown. Theplot uses parameters ξ D = 8, φ D = 0, ω = 100, σ x ( t = 0) = 1, and x = 8. For these parameters,Λ ∼ = 0 . σ ( t ) = p h x i − h x i and an example is shown in Figure 4. The decay rates are importantindicators of the harmonicity of the trap, and in an experimental realization, adjusting thetrap parameters to minimize the decay rate leads toward a maximally harmonic trap.There is an alternative formulation for the first two moments that works for Λ t ≪ h x i = √ ξ D σ x (1 + (Λ t ) ) exp (cid:20) − (Λ t ) ξ D t ) (cid:21) cos (cid:18) ωt + φ D + 2Λ t + Λ tξ D t ) (cid:19) . (24)Of particular note is the argument of the cosine. The usual ωt + φ D is present, and thereis a term proportional to ξ D , representing the frequency dependence on the strength of theinitial sloshing, but the 2Λ t term is independent of the kick strength. In other words, thefrequency of oscillation for a cloud is different from that of a single particle, with a frequencydifference that tends to a nonzero value even as the oscillation amplitude approaches zero.Contrast this with Equation 5, where the value of Ω for a single particle approaches ω asthe amplitude tends to zero.Figure 2 shows an example of this solution for a set of parameters chosen to showcasethe behavior of the cloud as it approaches quasi-thermal equilibrium. The center-of-massoscillations gradually decay, and eventually no center-of-mass oscillations will be observed.9 .0 0.2 0.4 0.6 0.8 1.0 t ⟨ x ⟩ FIG. 3. h x i as a function of time for the same parameters as used in Figure 2. Note the morerapid oscillations as well as the more rapid decay of the oscillations. As discussed previously, this model does not take collisions into account, so the apparentthermalization is purely a result of the relative dephasing of parts of the cloud with differentenergies.Next, the same approximation applied to Υ leads to an approximate expression for thesecond moment of position. h x i = σ x (1 + ξ D )+ σ x ξ D (1 + (2Λ t ) ) exp (cid:20) − (2Λ t ) ξ D t ) (cid:21) × cos (cid:20) ωt + 2 φ D + 6Λ t + 2Λ tξ D t ) (cid:21) . (25)These two moments can similarly be combined to approximate the cloud size as a functionof time. IV. TIME OF FLIGHT
In an experimental setting, it is likely that the atomic clouds will be imaged after a periodof free-fall, in order to increase the size of the cloud and its observed oscillation amplitude.Limited analytical understanding of the behavior of the clouds after time-of-flight is possible,but numerical calculations can more easily illuminate these effects.The effect on the first moment is a change in amplitude and a phase shift that dependson the time of flight. Looking at the form of the first moment, Equation (22), its general10 .0 0.2 0.4 0.6 0.8 1.0 t σ x FIG. 4. The size of the cloud as a function of time, using the same parameters as Figures 2 and 3. behavior is that of a sinusoid with a slowly varying amplitude. This functional form isapproximated by h x i ≈ M ( t ) cos(Ω t + φ ) , (26)where M ( t ) is the decay function, Ω represents the real oscillation frequency of the cloud,and φ represents some initial phase.Taking the derivative of this expression, assuming that the variation of M ( t ) is slowcompared to the sinusoid, results in ∂ h x i ∂t ≈ − Ω M ( t ) sin(Ω t + φ ) , (27)which has the same frequency of oscillation and a similar decay rate, insofar as the variationof M ( t ) is slow, to the dynamics without time of flight. Assuming the x axis is perpendicularto gravity, the horizontal velocity of the cloud is constant after the trap is released. Theresult is an expression that oscillates at the same frequency as in the trap, but with adifferent phase and amplitude that depend on the time of release and the elapsed time offlight.The behavior of the cloud size requires more careful consideration. Since the cloud isoscillating in a potential much larger than the cloud, the cloud may experience dispersionwhich can cause the cloud to expand more or less quickly after the trap is released, dependingon the phase of the oscillation at the time of release. This complication makes writing ananalytical solution, even a highly approximate one, to σ after free-fall much more difficult.11 ⟨ x ⟩ σ x FIG. 5. Pictured here are the results of a simple Monte-Carlo numerical simulation of sloshing in aharmonic/quartic trap, with trapping parameters similar to those used in earlier figures. Fourth-order Runge-Kutta integration was used to evolve the system. The black curves represent theposition and size of the cloud without taking time of flight into account. The dotted curves (redonline) the position and size of the cloud after a short time of flight (0.015). The time of flightexaggerates the oscillations in σ x , changing the shape of the curve. Figure 5 shows the results of a numerical model which shows the behavior of h x i and σ , in conditions chosen to be similar to the parameters used in Figures (2), (3), and (4).We observe that the behavior with and without time of flight are similar but the time offlight data has a larger amplitude of oscillation, a larger cloud size in general, and a slightlydifferent size profile as time goes on, with more exaggerated size oscillations in the middletimes. V. CONCLUSIONS AND OUTLOOK
The oscillations along the weak axis of a small cloud in a perfectly harmonic cigar-shapedtrap are not expected to decay significantly over time, based on a straightforward extension ofthe calculations in Gu´ery-Odelin et al. [15]. The addition of non-harmonic terms causes thecloud to gradually spread out and reach quasi-thermal equilibrium even without interactions,as atoms with different energies gradually dephase.12ur analytical model describes the dynamics of a cloud of neutral atoms in a harmonictrap with a quartic perturbation. We assume that the displacement of the cloud is lessthan the magnitude of the quartic parameter x , that mean-field effects are negligible, thatcollisions are isotropic, elastic, and energy-independent, and that the quartic contributiondoes not significantly affect the magnitude of the collisional integral.With a few minor approximations, the system is analytically solvable. The analyticalexpressions can be used as fitting functions for the observed behavior of ensembles of coldatoms in a nearly- harmonic trap. The motion of the ensemble’s center of mass and theevolution of its size permit easy determination of trap anharmonicity, and given the righttrap architecture, the anharmonic parts can be tuned to some desired value. Time-of-flight imaging makes small changes to the observed values of the variables but does notqualitatively change the behavior of the ensemble. Thus, even with time-of-flight imaging,the fit parameters for the trap will still maintain the same relative trends.The construction of a trap geometry which can control various polynomial terms whileminimizing unwanted terms is discussed in Stickney et al. [14]. The specific calculations fora trap allowing adjustment of harmonic and quartic terms, while canceling polynomial termsout to sixth order are also detailed there. Dynamic control of the shape of the trap is expectedto produce additional interesting results. For instance, because the equilibrium is not a truethermal equilibrium, it should be possible to dephase and rephase the cloud as it oscillatesas long as the collision rate is not too high, and effective “pauses” in the dephasing of thecloud might be made possible by turning off the quartic contributions. These qualities makethe dynamically controlled trap an interesting topic worthy of experimental examination. VI. ACKNOWLEDGMENTS
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