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Fastest Effectively Adiabatic Transitions for a Collection of Harmonic Oscillators Frank Boldt,*,† Peter Salamon,*,‡ and Karl Heinz Hoffmann*,† †

Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany Department of Mathematics and Statistics, San Diego State University, San Diego, California 92182, United States

‡

ABSTRACT: We discuss fastest effectively adiabatic transitions (FEATs) for a collection of noninteracting harmonic oscillators with shared controllable real frequencies. The construction of such transitions is presented for given initial and final equilibrium states, and the dependence of the minimum time control on the interval of achievable frequencies is discussed. While the FEAT times and associated FEAT processes are important in their own right as optimal controls, the FEAT time is an added feature which provides a measure of the quality of a shortcut to adiabaticity (STA). The FEAT time is evaluated for a previously reported experiment, wherein a cloud of Rb atoms is cooled following a STA recipe that took about twice as long as the FEAT speed limit, a time efficiency of 50%.

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macro level have been introduced.17,18 Another application area is the definition of finite-time availability19 at the quantum level.20,21 Finally, we mention that such FEATs have also been found to exist for certain spin systems.22 In the present paper, we extend our former work regarding frictionless controls for harmonic oscillators.1,23−25 While our previous presentations were correct and complete only for the case where the control is limited to a frequency range between the initial and final frequencies, Stefanatos et al.26,27 have shown that it is sometimes worthwhile to add additional branches to the optimal control. That case occurs when the allowed controls include a frequency range extending significantly beyond the range given by the initial and final frequencies of the transition. Our original paper on the optimal control of this system1 found certain three-switch-point protocols to be optimal for connecting any two equilibrium states using the control interval defined by the equilibrium values of the control frequency at the equilibrium states involved. The situation turned out to be much more complicated however once the control interval is enlarged to include frequencies lower than the final equilibrium value or higher than the initial equilibrium value. Schmiedl and Seifert considered the problem with the control frequency ranging from −∞ to ∞ and dismissed it as uninteresting28 because the control can be done in zero time for this control range. Stefanatos et al. showed that for finite, albeit significantly larger intervals, the three-switch-point solutions in refs 1 and 25

INTRODUCTION For the parametric harmonic oscillator, optimal protocols can be determined which lead to fastest effectively adiabatic processes (FEATs).1 These processes maximize the work done in a minimum time by a thermodynamic process. Such processes are only effectively adiabatic because during the process the populations of the several energy levels are not constant, but the initial populations are recovered at the final time having the new energy values. Any interrupted version of the process would however leave some energy in so-called parasitic modes which would also be excited by most ways of controlling the system. This phenomenon has been dubbed quantum friction.2−6 While the excitations in such parasitic modes is in principle recoverable, it is “marked-for-loss” in the sense that the energy in these modes becomes thermalized by any ensuing thermal contact. Besides FEATs, a second family of fast effectively adiabatic frictionless protocols, so-called shortcuts to adiabaticity (STA)7 have also been found. These STAs are governed by special (scale invariant) controls based on Ermakov invariants. Because FEATs represent the upper bound for the speed of effectively adiabatic processes, they provide a speed limit for STAs. Any process faster than a FEAT will necessarily invoke unavoidable quantum friction. Knowing this theoretical upper speed limit allows one to quantify the potential speed-up of an experiment. We provide an example of this below, wherein we compute predicted results for FEATs matching the experimental realization of a STA8 for cooling a cloud of Rb atoms trapped in a harmonic potential at 130 nK. FEATs are useful in a variety of contexts. For instance, they provide the necessary strokes for building up operating cycles for heat engines or refrigerators at the quantum level2−4,10−16 in the same sense as optimal processes for heat engines at the © 2016 American Chemical Society

Special Issue: Ronnie Kosloff Festschrift Received: November 30, 2015 Revised: January 25, 2016 Published: January 26, 2016 3218

DOI: 10.1021/acs.jpca.5b11698 J. Phys. Chem. A 2016, 120, 3218−3224

Article

The Journal of Physical Chemistry A are no longer optimal. If the largest allowed frequency is sufficiently large, the optimum shifts to protocols with more switch points; it saves time to go to higher kinetic energies where the oscillations are faster. While Stefanatos et al. allowed repulsive potentials, we consider only attractive potentials. Stefanatos et al.’s derivations are impressive and solve the problem completely. Our goal here is to translate Stefanatos’s solutions into the simple graphical picture of Tsirlin and co-workers25 in terms of natural physically interpretable coordinates that enable us to understand the solutions. We discuss this case of enlarged controllability using the simple graphical picture25 afforded by a dimension reduction based on the Casimir companion29 which gives a physical picture of when and how extra switch points in the trajectory can further reduce the time required for the transitions. An additional advantage of this graphical presentation is that it determines optimal trajectories based on simple geometrical arguments that lead directly to physical understanding. We begin by introducing the system, the invariant of motion, and the dimension reduction that the Casimir companion makes possible. Next, the construction of the FEAT corresponding to given parameter values is discussed, introducing different classes of controls defined by the number of switch points used in the control.

z 3̇ = −2k(t )z1 + 2z 2

Any invariant of motion allows one to reduce this set of equations. Here, the recently found Casimir companion, X,29 is utilized. In this case, the Casimir companion is given by X = z1z 2 −

1,eq

keq 2 keq

X z 2,eq

−ı

Ȧ 3 = −2k(t )A1 + 2A2

(3c)

(4a)

z 2̇ = −k(t )z 3

(4b)

(6c)

X

(7)

⎛ ωℏ ⎞ m ωℏ coth⎜ ⎟ 2 ⎝ 2kBT ⎠

(8)

For the further discussion it is helpful to introduce an ordering relation for the equilibrium states. We define this ordering relation for particular equilibrium states Pa = (z1,a,z2,a) at temperature Ta and Pb = (z1,b,z2,b) at temperature Tb via Ta > Tb ⇔ z1, a < z1, b ⇔ Pa < Pb

(9)

From eqs 6c−8 we find X (ω , T ) =

Eeq 2 keq

=

⎛ ωℏ ⎞ ℏ2 coth2⎜ ⎟ 4 ⎝ 2kBT ⎠

(10)

We will exploit our invariant X to scale all versions of the problem having different values of X into the equivalent problem with X = 1 z bi = i (11) X

which are linear in Ai . The equations of motion for expectation values zi = ⟨Ai⟩ = Tr(ρ0 Ai) with respect to the initial density operator, ρ0, read

z1̇ = z 3

2 z 2,eq

Eeq (T ) =

Together with a third operator A3 = 2 ℏ [A1, A2] = pq̂ ̂ + qp̂ ,̂ the three operators A1, A2 , and A3 form a closed Lie algebra, called the dynamical algebra. Here, [·,·] denotes the commutator, which is the particular Lie bracket for our Lie algebra. The Hamiltonian is a linear combination of A1 and A2 and hence an element of the dynamical algebra. Therefore, the Heisenberg equations of motion become an algebraic set of coupled differential equations. In particular, one finds29

(3b)

(6b)

Given a canonical ensemble of quantum mechanical parametric harmonic oscillators, the rescaled energy Eeq at temperature T and circular frequency ω satisfies

(2)

Ȧ 2 = −k(t )A3

= z 2 , eq

2 1 z 2,eq X 1 Eeq = + z 2,eq = z 2,eq 2 X z 2,eq 2

with p̂ and q̂ being the momentum and position operators, respectively. This Hamiltonian contains two parameters: particle mass, m, and the time-dependent circular frequency, ω(t), which serves as the control parameter. Introducing the operators A1 = q 2̂ , A2 = p2̂ , and k(t) = (mω(t))2, we can rewrite the scaled Hamiltonian m/ = /̃

(3a)

(6a)

from which we can calculate the expectation value of the Hamiltonian E = ⟨/⟩ at equilibrium

(1)

Ȧ 1 = A3

1 z 2,eq 2

z1,eq =

keq =

2

1 1 /̃ = k(t )A1 + A2 2 2

(5)

with k = keq fixed, it follows that an equilibrium state will be established only if the control parameter k(t) fits with the split of the total energy into potential and kinetic parts at the particular state. Applying the equipartition theorem, we can calculate keq according to

THE QUANTUM OSCILLATOR The Hamiltonian / of the harmonic oscillator reads p̂ m + ω(t )2 q 2̂ 2m 2

1 2 z3 4

The invariant of motion defines the manifold of all state trajectories z⃗(t) = (z1(t), z2(t), z3(t)) which must remain on the surface X = constant. Additionally, equilibrium states can be identified by looking for fixed points of eq 4. For such states, X one finds that z3,eq = 0 and therefore z 2,eq = z . From eq 4c

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/=

(4c)

In our scaled coordinates, the hyperboloid (eq 5) becomes 1 = b1b2 − 3219

1 2 b3 4

(12) DOI: 10.1021/acs.jpca.5b11698 J. Phys. Chem. A 2016, 120, 3218−3224

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The Journal of Physical Chemistry A

Graphical Representation. Our cost functional for a given trajectory is the total time τ of the process and can be calculated by integrating the first equation of motion (eq 4)

For easy access, we summarize the correspondence between our scaled variables and their experimental counterparts in Table 1.

b1̇ = b3 ⇒ τ =

Table 1. Correspondence between Experimental and Scaled Dimensionless Variables Eexp Xk m

⟨v2⟩ X m2

b2

∫* b (1b ) db1 3

(15)

1

Eliminating the variable b3 using eq 12

⟨x2⟩

τexp

ω

T

√Xb1

mτ

k m

X(ω,T)

b3 = ±2 b1b2 − 1

(16)

and substituting into eq 15, one obtains τ=

The model introduced above can be applied to experiments in which particles are thermalized at an initial temperature T in a harmonic trap with controllable curvature ω(t) . The trap potential is able to manipulate the particles in principle without any dissipation. To make contact with such experiments the, in general, nonequilibrium states of the particles can be accessed by measuring the mean squared position ⟨x2⟩ (∼z1) and the mean squared velocity ⟨v2⟩ (∼z2), from which the potential and kinetic energies can be inferred.

OPTIMIZED CONTROL PROTOCOLS From eq 10 for the Casimir companion X, which is a constant of the motion, observe that reducing the frequency from the initial value ωi to the final value ωf (ωi > ωf) corresponds to a reduction in the temperature (cooling) by a factor of ωi/ωf. A central problem in the experimental manipulation of trapped particles is the determination of protocols, which allow a fast cooling of the system by changing the trapping potential. Here we show how such protocols can be obtained and what features they have. In technical terms we ask the question by what protocol k*(t) = (mω*(t))2 our system can be steered in minimum time from a given initial equilibrium state Pi = (b1,i, b2,i, b3,i) into a given final equilibrium state Pf = (b1,f, b2,f, b3,f). For the further discussion we restrict ourselves to cases for which the initial and final equilibrium states lie within the control range. In addition, we restrict our discussion to cooling processes. In other words, the control frequencies ki and kf satisfy

db 2 = −k(t ) db1

(17)

(18)

Thus, the problem of finding the time-optimal control k*(t) is transformed to the problem of finding the optimal twodimensional (2-D) trajectory * = (b1 , b2(b1)) minimizing the total time τ in eq 17 subject to the constraint in eq 18. This task is achieved when trajectories are chosen which maximize b2 for a given b1. In this picture, the problem becomes one of splicing together line segments in the (b1, b2) plane having only two possible slopes, kmin and kmax. Points where the slope changes represent the switch points (SPs) of the control. While projecting our 3-D problem in terms of (b1(t), b2(t), b3(t)) onto a 2-D problem (b1, b2(b1)) leads to a very intuitive graphical representation, it is achieved at some cost: we lose the sign of b3. Thus, except for equilibrium states which have b3 = 0 and lie on the hyperbola b1b2 = 1, each point (b1, b2) actually corresponds to two states: one with b3 > 0 and one with b3 < 0. For trajectories, the 2-fold nature of our visualization space means that left−right trajectories cannot change to right−left ones except by passing through some state with b3 = 0, i.e., by touching the hyperbola b1b2 = 1. We call such points reversal points (RPs). Note that points below the hyperbola are unattainable because they are not part of the image of the projection. For graphical representations of our processes, we adopt the following symbolic notation regarding particular points of a trajectory *

(13)

making the control interval I = [kmin, kmax]. From 13 we obtain b1,i ≤ b1,f; thus, in terms of our ordering relation on equilibrium states (9), we have Pi ≤ Pf

1 db1 b1b2 − 1

The ± sign in the integral always gives a positive integrand provided the sign of db1 is included, i.e., left-to-right trajectories (ḃ1 > 0) in the (b1, b2) projection of the full surface (eq 5) take place in the b3 > 0 sheet while right-to-left trajectories (b1̇ < 0) take place in the b3 < 0 sheet. Using eq 4, one finds that the control k(t) represents the negative slope of the projection of the trajectory onto the b1−b2-plane

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0 < k min ≤ k f ≤ k i ≤ k max < ∞

∫* ±2

(14)

which means that in the graphical representation introduced below, the required protocol overall moves from left to right. Using the Pontryagin maximum principle,30 the time-optimal control protocol is of the bang−bang-type.1,25−27 This means that the optimal control k* takes on only the values kmin or kmax; the control k*(t) is either constant or jumps between its extremes at particular switch points with the total trajectory consisting of branches of constant k (k = kmin or k = kmax) concatenated by n switch points. The number of necessary switch points (including the initial and final switch points) determines the class of controls and will be called an n-SP protocol, i.e., a 3-SP protocol uses three switch points. Note that two of these switch points must occur at the initial and at the final time.

• initial point ( × ) • final point (+) • switch point to be optimized (△) • reversal point (□) • switch point (○) The time dependence along a trajectory is obtained by the following consideration. Along any branch of our optimal trajectory, the piecewise constancy of k gives b2 = b2,0 − k(b1 − b1,0) 3220

(19) DOI: 10.1021/acs.jpca.5b11698 J. Phys. Chem. A 2016, 120, 3218−3224

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The Journal of Physical Chemistry A for fixed k ≥ 0 and bi,0 = bi(0) . To find the explicit time dependence we can directly integrate eq 17 with eq 19 substituted in. The resultant τ(b1) can then be inverted to give b1(k , t , b0⃗ ) =

b1,0k + b2,0

+ sign(b3)

2k b1,0b2,0 − 1 k

+

b1,0k − b2,0 2k

τ1 =

⎛ 2k (k + k ) k − (k + k )(k + k ) k ⎞ min max f i min max min i f ⎟⎟ Arccos⎜⎜ (k max − k min) k f (k min − k i) ⎝ ⎠ (23)

cos(2 k t )

sin(2 k t )

1 2 k min

τ2 =

(20)

3-SP Protocols. Here we discuss the domain of possible controls which has paths with three switch points. Apart from the switch points at the initial state and the final state, a third intermediate switch point (cf. Figure 1, PS, P′S) is used.

1 2 k max

⎛ 2k (k + k ) k − (k + k )(k + k ) k ⎞ max min i f min max max f i ⎟⎟ Arccos⎜⎜ (k min − k max ) k i (k max − k f ) ⎠ ⎝ (24)

This allows one to easily determine the total process time τ = τ1 + τ2

(25)

It is convenient to introduce the function τ(ki, kf, kmin, kmax), which gives the time needed for a trajectory generated by a 3SP protocol as given by eq 22. 4-SP Protocols. For cases where kmax > ki, a reversal point PR (□) to the left of the initial state Pi can be included into the trajectory between the initial and final points. Such a trajectory corresponds to a 4-SP control. This class of protocols may be faster than 3-SP controls. Under what conditions this will be the case will be discussed later. Figure 1. For the 3-SP control, the candidates for optimal control consist of three switch points: initial state Pi, intermediate switch point PS or PS′ , and final state Pf. Because trajectories with larger b2 for a given b1 correspond to shorter process times, the trajectory (Pi →PS → Pf) is time-optimal.

In detail, following eq 15, the larger b3 (and thus b2), the faster the process. Thus, according to the shape of the surface X, in Figure 1 the control consisting of Pi connected by the solid line to PS connected by the solid line to Pf is fastest. Using simple geometry, the coordinates of the intermediate switch point PS evaluate to

Figure 2. If kmax > ki, then a reversal point PR < Pi can be added to a three-switch-point connection *1 of initial state Pi and final state Pf. This allows one to travel faster from left to right (4-SP protocol, *2 , *3). Adding another switch point PpS(r) allows one to shrink the time requirement. The optimal location of PpS(r) is not obvious; hence, a is numerical optimization of PpS(r) between PpS(0) = Pi and PpS(1) = Pmax i needed.

⎛ 1 + k b 2 − b (b + k b ) max 1,f 1,f 2,i min 1,i PS = ⎜⎜ , − b ( k k ) ⎝ 1,f max min k maxb1,f (b2,i + k min(b1,i − b1,f )) − k min ⎞ ⎟⎟ b1,f (k max − k min) ⎠

(21)

The path of such a 3-SP trajectory is shown in Figure 1. When all three switch points, Pi, PS, and Pf, are known, the explicit moments of switching and the total process time τ for the class of 3-SP protocols can be calculated. This leads to ⎧k i , ⎪ ⎪ k min , k*(t ) = ⎨ ⎪ k max , ⎪ ⎩kf ,

Possible cases are illustrated in Figure 2. Here, three different trajectories *1, *2 , and *3 can be identified by their respective sequence of points following our symbolic notation. In detail, *1 is established following a 3-SP protocol, whereas *2 and *3 are governed by a 4-SP protocol. The difference between *2 and *3 is that for *3 the additional reversal point (□) coincides with the additional switch point (△). Because kmax > ki, the system can now evolve to even lower b1, allowing b2 to increase, which enlarges b3 and hence possibly speeds up the process. However, the particular path of lowering b1 needs extra time. This trade-off might imply that a trajectory *2 between *1 and *3 is optimal. Note that no such trajectories are possible if kmax = ki.

for t = 0 for 0 < t ≤ τ1 for τ1 < t < τ1 + τ2 for t = τ1 + τ2

(22)

For any values of the extremal frequencies kmin and kmax, the values of τ1 and τ2 given in ref 1 can be used 3221

DOI: 10.1021/acs.jpca.5b11698 J. Phys. Chem. A 2016, 120, 3218−3224

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The Journal of Physical Chemistry A The additional switch point (the second of trajectories *2 and *3) is denoted as PpS(r) with r specifying the switch point’s location as the fraction of the way the path follows the line up and to the left with k = kmax between the initial state, × (Pi = PR, r = 0), and the point where this line intersects the hyperbola, △ (Pmax = PR, r = 1). It is necessary to consider all reversal i points PR(r) between r = 0 and r = 1. The third switch point PS(r) needs to be adjusted (cf. Figure 2) simultaneously by the constraint of having to reach Pf. Optimal Trajectories. From this conceptional discussion, the time-optimal control can be found, taking into account three kinds of paths: *1:

Pi → PS(0) → Pf

*2:

Pi → PSp(r ) ⎯⎯⎯⎯→ PS(r ) → Pf

*3:

Pi → PSp(1) → PS(1) → Pf

kmax). Now notice that as the latter term is certainly positive it suffices to show that τ(kR,u , kR,l , k min , k max ) > τ(kR,u , k f , k min , k max )

(27)

to show that the trajectory *4 is time-inferior. Taking the derivative of the first term τ(kR,u, kf, kmin, kmax) with respect to kf, one can show that for values kmin < kf < kR,u < kmax the time always increases as kf is moved to smaller values, i.e., to the right in Figure 3. Thus, the trajectory *4 can be excluded from the set of possible time-optimal trajectories. We mention that by similar arguments additional lower reversal points to the right of the Pf can be excluded. Thus, it follows that additional reversal points between Pf and Pmin are always leading to longer trajectory times. A further consequence is that the branch ending in Pf is always a kmax-branch. 6-SP protocols. Using six switch points (6-SP protocols) might speed up the process further. This refinement (cf. Figure 4) is of a different kind than the one leading from a 3-SP protocol to a 4-SP protocol.

PR (r )

with notation in compliance with Figure 2. The optimal location of PR(r) can be obtained by varying r according to max b2,R (r ) = b2,i + r(b2,i − b2,i)

b1,R (r ) =

1 b2,R (r )

(26)

For the boundary value r = 1, the reversal point coincides with the additional switch point. The question of which choice of r leads to a time-optimal control is not trivial.26 Direct comparison of the total process time for the optimal parameter r with the time for the optimized 3-SP protocol allows one to decide whether the 4-SP protocol is time-optimal. Additional Lower Reversal Point between Pf and Pmin. An interesting question is whether the introduction of an additional reversal point between Pf and Pmin could be beneficial. The situation is shown in Figure 3.

Figure 4. A six-switch-point connection *6SP including a further lower reversal point PlR between Pi and Pf can be faster in comparison to a four-switch-point connection *4SP , cf. Figure 2. Such paths include an upper reversal point PuR which is left of the critical point Pmax i . This critical point does not necessarily belong to the trajectory because a previous switch and reversal point, indicated by the overlaid △ and □, might be faster. This technique is recursively applicable for further improvement (*6SP → *8SP ) when kmax is sufficiently large.

For trajectories leaving Pi with a kmax-branch, it follows from the fact that the last branch is a kmax-branch as well that the total number of switch points needs to be even. Thus, we need to add two switch points to generate a 6-SP protocol as shown in the left panel of Figure 4. Overall this new control is generated by adding an intermediate lower reversal point PlR between Pi and Pf. The additional kmin-line segment resembles an additional stave of a ladder leading from Pi to the top reversal point. To find the time needed for the trajectory, it is helpful to note that the total path is the concatenation of a 4-SP protocol connecting Pi with PlR and a 4-SP protocol connecting PlR with Pf. The total time is thus the sum of these two 4-SP times. Of course, the position of the lower reversal point as well as the location of the switch points in each 4-SP protocol need to be optimized simultaneously to find the overall fastest trajectory. (2n + 6)-SP Protocols. Increasing kmax will make protocols with even more intermediate switching and reversal points time-optimal. When another lower reversal point is inserted, the 6-SP protocol can be enlarged to become an 8-SP protocol.

Figure 3. An additional reversal point PlR to the right of the final state Pf does not speed up the process.

The trajectory *4 (PuR → PuS → P′S → PlR → PlS → Pf) reaches the switch point PuS on a kmin branch. Without loss of generality we can assume that this path originated at the reversal point PuR. This could in fact represent the initial state Pi or some intermediate reversal point. The time to reach the final point Pf from PuR directly (i.e., with a 3-SP protocol) is calculated from τ(kR,u, kf, kmin, kmax), where kR,u is the equilibrium control at PuR. The time needed to go through the additional reversal point PlR is the sum of two terms: τ(kR,u, kR,l, kmin, kmax) and τ(kf, kR,l, kmin, 3222

DOI: 10.1021/acs.jpca.5b11698 J. Phys. Chem. A 2016, 120, 3218−3224

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This allows one to include another “stave”, as is sketched in Figure 4, right panel. This strategy is recursively applicable, allowing the construction of (2n + 6)-SP protocols with n + 2 staves. The addition of two switch points to a 6-SP protocol leads to a 8-SP protocol, as shown in the right panel of Figure 4. Note that adding branches is always connected with a subsequent optimization of the location of all intermediate switch points and their corresponding reversal points. During the recursive construction of the (2n + 6)-SP protocols, former switch points are shifted toward lower b2 and higher b1 narrowing the staves of the former recursion step. Optimal Cooling Times. As discussed above, the determination of the optimal position of the switching points and the ensuing optimal trajectory times is nontrivial. However, such an optimization has not to be repeated for each combination of ki, kf, kmin, and kmax, as by the choice of units one of the k values can be set to 1. Thus, we are left with three ratios, α = kmax/ki, β = ki/kf, and η = kf/kmin, as independent input variables. To illustrate the previous discussion, we set η = 1 and β to specific fixed values and study the trajectory times as a function of α. In other words, we vary kmax with respect to the other three frequencies and analyze at what values the more complicated protocols with more switch points become timeoptimal. Figure 5 shows a direct comparison of the total time τ for a realistic range of kmax > ki and realistic ratios β = ki/kf of 3, 10,

QUANTIFYING FEATS FOR RB ATOMS We close our exposition of FEATs for the harmonic oscillator by illustrating how our results may be used to measure the quality of STA processes. To this end, we compare a recently reported experiment described in Schaff et. al8 which used a STA protocol to fast cool a cloud of Rb atoms in a quadrupoleIoffe-configuration magnetic trap from 130 nK to 22 nK in 30 ms. The trapping potential was controlled by adjusting radial and axial frequencies according to a fixed scaling law leaving only one control parameter, i.e., once the axial frequency ω∥ is set, the radial frequency needs to follow accordingly. We know from previous work1,31,32 that the larger the ratio of frequencies, the faster the process. Therefore, the smallest ratio in frequency is the speed-limiting factor in the experiment. In particular, this factor is given by the ratio of the axial frequency ω∥ between initial (ω∥/2π = 22.2 Hz) and final state (ω∥/2π = 7.4 Hz), which is about 3. Recall that the ratio of ω∥/ T stays constant for equilibrium states (cf. eq 10). Then, given the initial temperature Ti = 130 nK, the FEAT process would end at the lower temperature of 14.4 nK instead of 22 nK. The difference is due to losses or suboptimal control. The bound on the time required can be found using k min = k f = 2.1618·104mRb 2 k max = k i = 19.457·104mRb 2

in eqs 23 and 24. We find the FEAT process time of 14.8 ms, which is about a factor of 2 smaller than the experimentally reported process time of 30 ms. Note that for kmax = ki and kmin = kf, the 3-SP protocol is optimal so no further calculations are needed. If in fact the allowed control range is larger, the time required can only decrease further. In that case, the numerical formalism described here (or in Stefanatos et al.26) must be employed to get a better bound. This gives the analysis of potential savings from using a FEAT instead of a STA process. Such use of FEATs to quantify the potential improvements of a process is important in devices where the jumps in frequency are deemed impractical.

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SUMMARY We presented a treatment of fastest effectively adiabatic transitions (FEATs)20 for an ensemble of harmonic oscillators with bounded real frequencies limited to a finite frequency interval. The derivation can be extended to repulsive potentials, for which similar results have been obtained.26 These timeoptimal trajectories give the upper speed limit for frictionless processes in finite time. The FEAT time has a very thermodynamic feel; it provides bounds and allows us to define a time eff iciency that measures the quality of performance of STA processes. Our presentation is based on the 2-dimensional graphical analysis made possible through the use of the Casimir companion X.29 For the case kmax = ki and kmin = kf, the 3-SP protocol is optimal.25 Our motivation in undertaking the present work was a hope that the less restrictive control would be similarly amenable to pure geometrical analysis. This was true, but only to an extent. One advantage of the graphical representation is its connection with the potential energy (∼b1) and kinetic energy of the system (∼b2). In light of the expression for the time as an integral over b1 and the fact that b1,f > b1,i, we would not expect to move left, because this will necessitate moving back again

Figure 5. FEAT time τ (left scale) for processes with initial equilibrium values of k(t = 0) = ki ∈ {3, 10, 30} and final equilibrium values k(t = τ) = kf = 1 shown as a function of the ratio of kmax/ki. At distinct values for kmax/ki = α, optimized paths with four switch points (r > 0) become faster than paths with three switch points (r = 0). The optimal location of the additional switch point (cf. Figure 2) is given through the value of r (right scale).

and 30. For instance, Schaff et al.8 reported an axial frequency ratio ωi/ωf = 3; hence, ki/kf = 9. We observe that for a specific value αcrit a 4-SP protocol with r ≈ 1 becomes time-optimal. For instance, if β = ki/kf = 10, the upper bound of the control interval kmax has to be about 4.59 times larger than ki. At this value, the optimal strategy changes from r = 0 (3-SP protocol) to r = 0.96 (4-SP protocol). As kmax increases further, time-optimal 6-SP and (2n + 6)-SP protocols become possible. The exact location of the transition to higher-order FEATs can be calculated numerically. 3223

DOI: 10.1021/acs.jpca.5b11698 J. Phys. Chem. A 2016, 120, 3218−3224

Article

The Journal of Physical Chemistry A

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over that same b1 range. The fact is, however, that such motion counts for less time (is faster) when the kinetic energy b2 is large enough. Thus, investing some time up front by going backward in potential energy while increasing the kinetic energy can pay off, but only when the maximum frequency (kmax) is sufficiently large to make the up−down trajectories fast. Indeed, we know from Schmiedl and Seifert28 that in the limit kmax → ∞ we can effectively jump to infinite kinetic energy by going backward just a little. At the large kinetic energy we can then move toward the right and to the desired b1 value in very short time before we descend back down in kinetic energy to the required b2,f. In the limit, this all happens in zero time. Thus, our geometrical picture, closely linked to physically interpretable coordinates, enables a full understanding of the optimal protocols. Determining the number and location of the optimal switch points for specific cases is a much harder matter that requires numerical optimization. One further observation is that the optimized trajectories can all be characterized by the feature that local maxima of the kinetic energy are always left using branches with the minimum value kmin of the control. Correspondingly, local minima of the kinetic energy are always left using the maximum control value kmax with the exception of the initial point, which is left on a kmin-branch for the 3-SP trajectory. In summary, we find that our geometrical representation gives physical understanding of the more complicated optimal controls required as the control interval is enlarged. The fact that finding such controls still requires numerical optimization is disappointing but appears to be required even in our simpler representation.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: hoff[email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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DOI: 10.1021/acs.jpca.5b11698 J. Phys. Chem. A 2016, 120, 3218−3224