Broken scale-invariance in time-dependent trapping potentials
The response of a cold atom gas with contact interactions to a smoothly varying external harmonic confinement in the non-adiabatic regime is studied. The time variation of the angular frequency is varied such that the system is, for vanishing or infinitely strong contact interactions, scale invariant. The time evolution of the system with broken scale invariance (i.e., the time evolution of the system with finite interaction strength), is contrasted with that for a scale invariant system, which exhibits Efimovian-like expansion dynamics that is characterized by log-periodic oscillations with unique period and amplitude. It is found that the breaking of the scale invariance by the finiteness of the interactions leads to a time dependence of the oscillation period and amplitude. It is argued, based on analytical considerations for atomic gases of arbitrary size and numerical results for two one-dimensional particles, that the oscillation period approaches that of the scale-invariant system at large times. The role of the time-dependent contact in the expansion dynamics is analyzed.
The response of a system to a perturbation lies at the heart of important concepts such as linear response theory Fetter and Walecka (2003), characterizing whether a system is chaotic or not Stöckmann (2006), state engineering Nikolopoulos and Jex (2013); Menchon-Enrich et al. (2016), and adiabatic transport Polo et al. (2016). This paper considers the quantum mechanical response of an initial state to a time-dependent variation of the system Hamiltonian. In the context of cold atom systems Polkovnikov et al. (2011); Yan et al. (2013); Aoki et al. (2014); Makotyn et al. (2014); Langen et al. (2015); Will et al. (2015); Cetina et al. (2016); Blakie and Porto (2004); Gericke et al. (2007); Rosi et al. (2013); Masuda et al. (2014), two limiting cases have received considerable attention, the regime where the system Hamiltonian is changed adiabatically and the regime where the system Hamiltonian is quenched, i.e., changed significantly over a time scale that is short compared to the intrinsic time scales of the system (essentially instantaneously). The time dynamics of small cold atom systems has specifically attracted a great deal of attention recently, either because the few-body dynamics is interesting in its own right or as a model for understanding intricate many-body dynamics Lode et al. (2012); Sykes et al. (2014); Zürn et al. (2012, 2013); Gharashi and Blume (2015); Ebert et al. (2016); Corson and Bohn (2016). Here, we consider an “in between” case in which the system Hamiltonian is changed continuously on a time scale that is comparable to the intrinsic time scale of the system Hamiltonian at time , where is the time at which the time variation of the Hamiltonian is turned on. Starting with an eigenstate at , we focus on the regime where the long time dynamics exhibits oscillations.
Reference Deng et al. (2016) discussed an intriguing analogy between the dynamics of an -atom system with vanishing or infinitely large two-body zero-range interactions under time-dependent external harmonic confinement and the static three-body Efimov solution Efimov (1970); Braaten and Hammer (2006). The former exhibits, for a properly chosen time variation of the trap, log-periodic expansion dynamics Deng et al. (2016) while the latter is, for infinitely large -wave scattering length, characterized by log-periodic energy spacings Efimov (1970). It was shown Deng et al. (2016) that the log-periodic time evolution can be traced back to the scale invariance of the -body Hamiltonian. Specifically, by changing the angular trap frequency with time, the harmonic oscillator length can be effectively removed from the problem, leaving a scale-invariant space-time Hamiltonian that is governed by intriguing long-term dynamics. The present paper investigates how the dynamics changes when the two-body interactions define a length scale. How does the time evolution change when the interaction strength increases from zero to infinity? Do the long time oscillations survive for finite interaction strengths? Can one still define an oscillation period?
The remainder of this paper is organized as follows. Section II introduces the system Hamiltonian and theoretical background, with Secs. II.2 and II.3 discussing the cases where the two-body interactions, respectively, do not and do define a meaningful length scale. Section III presents our numerical results for a two-particle system and discusses the findings. Last, Sec. IV summarizes.
Ii System Hamiltonian and theoretical background
ii.1 General considerations
We consider atoms with mass and position vectors () under external isotropic harmonic confinement with time-dependent angular trapping frequency and two-body zero-range interactions between each pair of particles and . The coupling constant , which has units of , can be tuned in cold atom experiments via Feshbach resonance techniques Chin et al. (2010). In the non-interacting limit () and the infinitely strongly-interacting limit (), the two-body interactions do not define a meaningful length scale Giorgini et al. (2008); Zwerger (2011) and the hyperradial and hyperangular degrees of freedom decouple Jonsell et al. (2002); Werner and Castin (2006); footnote7-1 (). In this case, the one-dimensional Schrödinger-like equation for the hyperradius ,
takes a particularly simple form,
where is a constant that is determined by the eigenvalue of the, in general, -dimensional differential equation for the hyperangular degrees of freedom (at this point, we do not separate off the center of mass degrees of freedom). While determining for is, in general (for ), a highly non-trivial task Werner and Castin (2006); von Stecher et al. (2008); von Stecher and Greene (2009), a crucial point is that the -body dynamics is, for and , fully governed by the one-dimensional Schrödinger-like equation for the hyperradius . The solutions to Eq. (2) and hence of the -body system with zero-range interactions of vanishing or infinite strength have been discussed extensively in the literature since the 60s Lewis (1967, 1968); Lewis and Riesenfeld (1969); Popov and Perelomov (1969a, b); Camiz et al. (1971); Castin (2004); Moroz (2012); Ebert et al. (2016).
In what follows we assume for , where is a (real) constant. The solution to Eq. (2) can then be reduced to solving the differential equation
It should be noted that the constant does not enter into the differential equation for the scaling function ; it enters into the wave packet for solely through the functional form of the eigenstate at .
Importantly, the hyperangular degrees of freedom are not affected by the time variation of . This implies that the hyperangular portion of the wave function is stationary and that the full wave packet , i.e., the wave packet that depends on the hyperradius and hyperangles , where collectively denotes all the position vectors, i.e., , can be readily constructed from Eq. (4), provided the hyperangular portion of the wave function is known at .
Historically Lewis (1967, 1968); Lewis and Riesenfeld (1969), the wave packet dynamics of Eq. (2) has been introduced and analyzed in connection with the corresponding classical harmonic oscillator equation for the generalized coordinate ,
where, as in the quantum case, for . Seeking complex solutions of the form , one obtains coupled differential equations for and . The differential equation for is, for the initial conditions and , independent of and identical to that given in Eq. (3). This correspondence between the absolute value of the classical generalized coordinate and the quantum mechanical scaling function is used in Sec. II.2 to elucidate some characteristics of the quantum mechanical -body problem in which the two-body interactions do not define a meaningful length scale.
If the coupling constant is finite, the hyperradial and hyperangular degrees of freedom are, in general, coupled. This implies that the wave packet dynamics depends, in general, on all degrees of freedom; it cannot be reduced to a one-dimensional problem. We note, however, that the center of mass degrees of freedom can be separated off (see Sec. III for more details). The coupling between the hyperangular and hyperradial degrees of freedom also implies that the quantum-classical correspondence breaks down.
In the remainder of this paper we consider a time variation of for which the time-dependent harmonic oscillator length , where , decreases as for , i.e., we consider
If the angular frequency changes little when increases from to , where is the characteristic time scale of the non-interacting system for (), the resulting dynamics is adiabatic. This situation is realized when is much larger than 1. In what follows, we primarily look at time variations characterized by “medium” speeds, i.e., situations for which is not large compared to (our numerical calculations in Sec. III, e.g., use ).
ii.2 Scale invariant interactions
This section focuses on the and cases. Using Eq. (4), the expectation values can be calculated analytically for any positive integer . We find that these expectation values are fully determined by the scaling function and the initial expectation value ,
The expression was used in Ref. Deng et al. (2016) to analyze the experimental expansion images. Equation (8) implies that the quantum mechanical hyperradial motion can be interpreted using the correspondence with the time-dependent classical harmonic oscillator, i.e., the expectation value and its time variation can be visualized in “phase space” by plotting as a function of . More generally, the expectation value and its time variation can be visualized in phase space by plotting as a function of .
where , and that for reads Lewis (1968)
where and . Reference Deng et al. (2016) made the nice observation that the solution exhibits log-periodic behavior reminiscent of and formally equivalent to the solutions to the static three-body Efimov problem. In fact, the symbol is introduced to make the connection to the static three-body Efimov scenario more explicit Deng et al. (2016).
Figure 1(a) shows as a function of for various .
This phase space representation illustrates that the cloud size increases continuously with increasing time. The expansion speed is greater or equal to zero for all . For [solid, dashed, dotted, and dash-dotted lines in Fig. 1(a)], the phase space trajectories for fixed resemble those of a “bouncing ball”. The value of , assuming , does not affect the overall shape of the trajectory; it merely changes the spacing between and amplitude of the “bounces” (both the spacing and amplitude decrease with decreasing ). For [dash-dash-dotted and dot-dot-dashed lines in Fig. 1(a)], the phase space trajectories show a finite positive expansion speed for all . In these cases, the expansion speed first increases, then reaches a maximum, and eventually decreases monotonically for all later times. In the limit , i.e., in the limit of the sudden removal of the trap, the expansion speed increases to a maximum that is set by the initial energy of the systems and then remains constant after that. Figure 1(b) shows as a function of for various . These phase space trajectories characterize, according to Eq. (8), the time dynamics of the expectation value . Qualitatively, Figs. 1(a) and 1(b) display the same characteristics.
To prepare for the discussion presented in the next section, it is instructive to explicitly write down, as was done in Ref. Deng et al. (2016), the differential equation for the expectation value of the square of the hyperradius,
This equation can be derived by applying Heisenberg’s equation of motion to the -particle Schrödinger equation and taking advantage of the fact that the two-body interactions do not define a meaningful length scale Deng et al. (2016). The next section considers how the differential equation for , Eq. (11), is modified if the two-body interactions define a length scale.
ii.3 Interactions with finite
As already aluded to, the hyperradial and hyperangular degrees do, in general, not decouple if the interaction strength of the two-body interactions is finite, making it challenging to tackle the full -body dynamics. To gain insight into the dynamics for finite , we consider the simplest possible scenario, namely two interacting particles in one spatial dimension. In this case, the wave packet dynamics can be determined numerically utilizing the techniques introduced in Ref. Gharashi and Blume (2015). Moreover, analytical results for limiting cases can be used to interpret the numerical results.
For the two-particle system, the square of the hyperradius can be rewritten in terms of the relative coordinate and the center of coordinate , with and (from now on we use instead of to emphasize the one-dimensional nature of the system under study). For the time variation defined in Eq. (7), the relative and center of mass degrees of freedom decouple, i.e., the full wave packet can be written as a product of the relative part and the center of mass part , . Correspondingly, can be written as . This implies that the time evolution of the relative and center of mass parts can be treated separately. This also means that the two-body system considered does not allow us to study the coupling between the hyperradial and hyperangular degrees of freedom. Nevertheless, it does allow us to investigate how the finite interaction strength enters into the dynamics of .
The differential equation for the center of mass part is given by Eq. (11) with replaced by . Hence, the time evolution of is the same as that for a single (non-interacting) particle of mass . The differential equation for the relative part reads
where the Tan contact is defined through Barth and Zwerger (2011)
Noticing that the left hand sides of Eqs. (11) and (12) have the same functional forms, we conclude that the finiteness of the interaction strength enters into the differential equation for via the time variation of the Tan contact. In fact, the right hand side of Eq. (12) vanishes for (in this case, the contact vanishes) and (in this case, vanishes) and Eq. (12) reduces to Eq. (11) in these cases. Equation (12) was first introduced for the general case in Refs. Qi (April 13, 2016); Shi (April 16, 2016).
To obtain a sense of how the contact changes with time, we imagine that the system changes adiabatically, i.e., we determine the contact for the two-particle system with coupling constant in a static harmonic trap with angular frequency and corresponding harmonic oscillator length . Using the expressions for the eigenenergies and eigenstates from Ref. Busch et al. (1998), the adiabatic contact for the two-body system can be calculated readily for any . In preparation for the discussion in Sec. III, we consider selected limiting cases. For the eigenstates with relative energy around and , we find, respectively,
where and (the analytical expressions for and are lengthy and not given here). The one-dimensional scattering length is inversely proportional to the one-dimensional coupling constant Olshanii (1998), . If we assume that is small and that we start in the ground state at and then change the trapping frequency adiabatically, the ratio of the adiabatic contact at time and that at time is . If, on the other hand, we assume that is large ( positive) and that we start in the ground state at and then change the trapping frequency adiabatically, the ratio of the adiabatic contact at time and that at time is . As mentioned earlier, we are primarily interested in the regime where the Hamiltonian is changed non-adiabatically. Thus, we expect that the time dependence of the contact is not described accurately by the adiabatic prescription. Instead, we expect—taking into account that displays log-periodic oscillations “on top” of an overall growth in the and limits—that the time variation of the contact for systems with finite will oscillate around the adiabatic value.
It is also interesting to consider the strongly-attractive limit ( and ), in which the two particles form a tightly bound dimer of size (). In this case, the relative wave function approaches that of two particles in free space with contact ,
Since the contact is independent of the harmonic oscillator length , we expect that the time-dependence of the trapping potential for has little impact on the initial state, provided . This is essentially saying that the trap is too weak in this limit to affect the initial state.
Iii Two-Particle Results
To solve the time-dependent Schrödinger equation in the relative coordinate for the Hamiltonian with time varying confining potential and finite , we perform numerical calculations. We use a propagator that exactly accounts for the two-body zero-range interaction Blinder (1988); Wódkiewicz (1991); Yan and Blume (2015). In brief, the relative coordinate is discretized using an equidistant grid with grid spacing . Knowing the wave packet at time , the wave packet at time is obtained by integrating the product of the propagator and the wave packet at time over the relative coordinate. This propagation step is repeated for many time steps . The accuracy of the final wave packet depends on the values of and ; implementation details can be found in Ref. Gharashi and Blume (2015). The errorbars (not shown) of the numerical results presented in this section are smaller than the symbol sizes or not visible on the chosen scales. Typical simulation parameters are and .
Figure 2 shows the expectation value as a function of time for various on a log-log scale. In all cases, the initial state at corresponds to the lowest energy eigenstate of the time-independent Hamiltonian. As a reference, the two solid lines show for the systems with scale-invariant interactions, (upper solid curve) and (lower solid curve). As discussed in Sec. II.2, the two solid curves would collapse to a single curve if the two were scaled by their corresponding initial . In the representation chosen in Fig. 2, the solid lines are offset from each other but exhibit the same oscillation amplitude and period. The dotted and short-dashed lines show the expectation value for repulsively-interacting systems with and 1, respectively. For these two finite cases, the amplitude and oscillation period “dephase” over time; this dephasing is due to the finite length scale defined by the interactions. Ignoring the oscillations, increases faster for finite than for . Using a hand waving argument, this can be understood by noticing that increases with increasing time due to the time-dependence of the trapping frequency. At large , the interactions thus effectively approach the strongly-interacting limit, explaining why the expectation value for finite is closer to that for than that for ; a more quantitative discussion is presented below.
The long-dashed and dash-dotted lines in Fig. 2 show for negative interaction strengths, i.e., for and , respectively. For these , first increases notably with increasing and then plateaus. This can be intuitively understood by realizing that the wave packet initially, oscillations aside, expands together with the trap. At later times, however, is much larger than the size of the wave packet, and the dynamics is approximately independent of the time dependence of the trap, implying that approaches , i.e., the expectation value of for a dimer with one-dimensional scattering length in free space. Indeed, this is what we observe in Fig. 2: The long-dashed and dash-dotted lines approach and , respectively, at large . We find that the small oscillations exhibited by occur on a time scale that is, roughly, set by the two-body binding energy.
As discussed in Sec. II.3, the length scale introduced by the two-body interaction manifests itself in the differential equation for via the time derivative of the contact [see Eq. (11)]. To analyze the role of this “new” term, thick lines in Fig. 3(a) show the contact , normalized by its initial value , for and various . For finite , displays oscillations on top of an overall decay. The overall decay is well described by the adiabatic contact , which is shown by the thin lines for each considered. As already discussed in Sec. II.3, the thin lines for the strongly-repulsive systems fall off approximately as for all (the approximation becomes better for larger ) while the thin lines for the weakly-repulsive systems transition from an initial fall-off to a fall-off for large . The non-trivial time-dependence of is responsible for the “dephasing” discussed in the context of Fig. 2.
To see this more clearly, Fig. 3(b) shows the quantity , i.e., the right hand side of Eq. (12), for various finite . While this quantity vanishes for and , it exhibits oscillations on top of an overall decrease with increasing for finite . Our analysis shows that the absolute value of each of the three terms on the left hand side of equation Eq. (12) is, for the first few oscillations, of the same order of magnitude as the absolute value of the right hand side of that equation. As time increases, the magnitude of the right hand side of Eq. (12), however, decreases faster than that of the other three terms. For positive , we find that the relative importance of the right hand side decreases faster for larger . This can, again, be intuitively understood by realizing that increases with increasing time.
To analyze the time dependence of further, we think of the dynamics displayed in Fig. 2 for finite as “close to periodic” and devise empirical measures to quantify the deviations from the “truly periodic” dynamics encountered for and . To this end, we calculate . While is zero at the beginning and end of each cycle for or (see Fig. 1), it does not go to zero for finite . For positive , we thus define cycles by looking at consecutive local minima of and by denoting the time at which the th cycle starts by and the time at which the th cycle ends by (the cycle enumeration starts with ). We define
as well as the scaling factors and ,
For and , these scaling factors are independent of and given by .
Circles, diamonds, triangles, and stars in Fig. 4 show our numerical results for Eqs. (17)-(19) for , and , respectively. For comparison, the dashed lines show the corresponding values for the scale-invariant systems. For the two largest considered (stars and triangles), and decrease monotonically with increasing . Figures 4(b) and 4(c) suggest that the large limit is given by the horizontal dashed lines, i.e., that the scale factors for finite at large approach the scale factor of the scale-invariant systems. For smaller (diamonds and circles), the scale factors first increase, then reach a maximum, and finally decrease. Although our numerics does not allow us to go beyond , Figs. 4(b) and 4(c) suggest that the scale factors for systems with these smaller values also approach the scale-invariant value in the large limit. The quantity [see Fig. 4(a)], in contrast, does not approach the value of the scale invariant systems but instead approaches a finite constant for large . The asymptotic, large value is reached faster for large than for small .
We now present analytical considerations for systems with finite positive that explain the trends displayed in Fig. 4. As mentioned already several times, increases with increasing time. Thus, we neglect the right hand side of Eq. (12) in the large limit and analyze the analytical solution, which can be found in Ref. Lewis (1968), for the differential equation assuming finite values for and , where . The reason for using a finite value for is that the wave packet at is not in an eigenstate [indeed, Fig. 4(a) shows that is finite]. We find that the analytical solution for the initial conditions applicable to an initial non-stationary state exhibits the same log-periodic oscillation period and amplitude as the analytical solution for the initial conditions applicable to an initial stationary state. This explains why the scaling factors and [see Figs. 4(b) and 4(c)] approach the dashed horizontal lines in the large limit. Moreover, since the analytical description is expected to become more accurate for larger , is expected to approach a constant in the large limit.
While the discussion above shows that certain properties of the large dynamics can be predicted analytically, the full wave packet dynamics for finite may be quite intricate, i.e., the quantities and alone may not describe the full story. To corroborate this notion, we consider the time-dependent overlap between the wave packet and the adiabatic eigenstate for the same and confinement with harmonic oscillator length ,
Figure 5 shows (the absolute value is taken to eliminate arbitrary phase factors) as a function of time for various coupling strengths . For (solid line) and (dash-dotted line), is equal to 1 at the end of each cycle, reflecting the fact that the wave packet returns, provided the appropriate scaling of the coordinate is applied, to its orginal shape. Interestingly, takes on its minimal values at the “half-cycle” times, i.e., at the times where coincides with the corresponding adiabatic value. For finite , the norm of the overlap displays, as for the scale-invariant interactions, oscillatory behavior. However, the oscillations are on top of an overall decrease of the overlap. This implies that the finite length scale continually introduces a dephasing, i.e., the time-dependent wave packet is increasingly less similar to the adiabatic eigenstate. Said differently, excited adiabatic states get mixed in more with increasing time . Taking a slightly different view point, this means that a full description of the wave packet at the end of the empirically defined cycles requires knowledge not only of but of for all .
This paper investigated the expansion dynamics of a harmonically trapped cold atom system with zero-range interactions. For , the angular trap frequency is equal to the constant and the system is in an eigenstate. For , a scale-invariant trap potential is realized by varying the angular frequency smoothly according to . The scale invariance can be most readily seen from Eq. (11), where and occur with the same powers in all three terms. More formally, we can look at how the Hamiltonian changes if the position coordinates are multiplied by . In this case, the kinetic energy term picks up an extra factor while the time-independent trapping potential picks up an extra factor of ; thus, the system Hamiltonian possesses a scale (namely the harmonic oscillator length). Scaling the time by (this follows from the fact that time is, dimensionally, divided by energy and that energy scales as one over length to the power of two), the time-dependent trapping potential, which contains terms like , picks up a factor of , just as the kinetic energy term; thus, the system Hamiltonian does not possess a scale, i.e., it is scale-invariant.
If the interaction strength vanishes or is infinitely large, the entire systems Hamiltonian is scale-invariant. In this case, the expansion dynamics has been investigated experimentally and theoretically in Ref. Deng et al. (2016). It was found that the cloud size follows so-called Efimovian expansion dynamics with logarithmically spaced oscillation periods and amplitudes.
The present paper investigated how the expansion dynamics changes if the interaction strength is finite. To address this question, the simplest non-trivial system consisting of two one-dimensional particles was investigated. It was found that the expansion dynamics of systems with finite and positive still exhibits oscillatory behavior. However, the oscillation period and amplitude are no longer logarithmically spaced. Instead, a dephasing that is governed by the time derivative of the contact is observed. At large times (depending on the value of , this may imply large numbers of “close-to-periodic” oscillations), we found that the cloud size is again, at least approximately, governed by a unique oscillation period and amplitude. We note that the role of the contact in non-equilibrium situations was previously investigated in two-atom quantum quenches Corson and Bohn (2016), where a rapid change of the interaction strength (a quench) induced a ballistic component into the contact. In the scenario considered in the present paper, the oscillations of the contact are the result of the continuously changing trapping potential. The work done on the system for triggers an interplay between the harmonic oscillator parts of the Hamiltonian and the two-body interaction terms. When the cloud is extremely dilute (large times), the effective interaction strength of the one-dimensional system becomes large [ increases as ]. Thus, the system is, again, effectively scale-invariant at large times, implying—using the general solutions for the scale-invariant system with modified initial conditions—close-to-log-periodic expansion dynamics. While our analysis in Sec. III considered two one-dimensional particles with finite interaction strength, the results generalize to systems consisting of more particles and with other dimensionalities.
The design of time-dependent trapping potentials has played an important role over the past 60 years or so. Here, the role of finite two-body interaction strengths on the Efimovian-like expansion dynamics was investigated. In other contexts (see, e.g., Ref. Chen et al. (2010)), time-dependent trapping potentials have been used to design frictionless non-adiabatic atom cooling trajectories. It would be interesting to include atom-atom interactions in those contexts via the contact and to quantify the energy distribution among the various modes, building on the ideas put forward in Ref. Ebert et al. (2016).
Support by the National Science Foundation (NSF) through Grant No. PHY-1415112 is gratefully acknowledged. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by NSF Grant No. OCI-1053575, and the WSU HPC.
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