Article pubs.acs.org/JPCA
Transient Particle Energies in Shortcuts to Adiabatic Expansions of Harmonic Traps Yang-Yang Cui,† Xi Chen,† and J. G. Muga*,†,‡ †
Department of Physics, Shanghai University, 200444 Shanghai, People’s Republic of China Departamento de Química-Física, UPV-EHU, Apdo 644, 48080 Bilbao, Spain
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‡
ABSTRACT: The expansion of a harmonic potential that holds a quantum particle may be realized without any final particle excitation but much faster than adiabatically via “shortcuts to adiabaticity” (STA). While ideally the process time can be reduced to zero, practical limitations and constraints impose minimal finite times for the externally controlled time-dependent frequency protocols. We examine the role of different timeaveraged energies (total, kinetic, potential, nonadiabatic) and of the instantaneous power in characterizing or selecting different protocols. Specifically, we prove a virial theorem for STA processes, set minimal energies (or times) for given times (or energies), and discuss their realizability by means of Dirac impulses or otherwise.
1. INTRODUCTION The need to shorten the times of adiabatic processes in different quantum or classical systems has motivated a surge of activity to design “shortcuts to adiabaticity” (STA), namely, fast protocols for the external parameters controlling the system that are generically nonadiabatic but provide the same final populations than the adiabatic dynamics; see ref 1 and references therein. The time-dependent harmonic oscillator is a paradigmatic study case, as it represents generically the conditions near equilibrium and has allowed for experimental implementations2,3 and applications in many different classical or quantum systems.4−33 For anharmonic corrections, also see refs 34 and 35. Because a basic aim of the shortcuts is to shorten the process time tf, some “price” in the form of high transient energies may be expected to be paid. Several articles have studied the energies and protocol times involved: their characterization; their mutual relation, which is not necessarily of the simple form of a time-energy uncertainty principle; and also their optimization under different constraints and conditions.12,16,18 We build on these results and continue the analysis of the energies and times in STA processes. We do not aim at being comprehensive but at examining points that either had not been discussed or needed some clarification. Our motivation here is of fundamental nature, but the results will be relevant to quantify the third principle of thermodynamics,6,11,12 as well as the speed, efficiency limits, and costs of quantum engines and refrigerators.5,6,18,28,29,32,36 After a quick review of invariant-based engineering in the Introduction, we shall first put forward in Section 2, a virial theorem for STA processes that relates time-averaged total, kinetic, and potential energies, E, K, and V. We then discuss an idealized protocol that minimizes the time-averaged energy for a given time tf, or vice versa, by means of Dirac-delta impulses © XXXX American Chemical Society
of the spring constant. It is shown that these impulses contribute to the total averaged energy and we specify how much. After a comparison of several protocols in a (E, tf) plane we study the nonadiabatic energy and its time average in Section 3. Its lower bound for given tf is found and its implementation is discussed. We also compare different protocols for this quantity. Finally, the instantaneous power in STA processes is examined in Section 4, and we end up with a discussion. 1.1. Invariant-Based Inverse Engineering. Let us consider a particle of mass m in a harmonic trap, with timedependent Hamiltonian H (t ) =
p2 1 + mω(t )2 q2 2m 2
⏟ K
V
(1)
where ω(t) is the time-dependent angular frequency of the trap, and q and p are position and momentum operators, with q being just a c number in coordinate representation. We shall deal with processes where the angular frequency is constant for t < 0, ω0, and for t > tf, ωf, and generally time-dependent in between. Lewis and Riesenfeld found the quadratic invariant37 I(t) = (1/2m)π2 + (1/2b2)mω02q2, where π = bp − mḃq is a momentum conjugate to q/b, the dots represent derivatives with respect to time, and b ≡ b(t) is a time-dependent, dimensionless scaling function proportional to the width of the elementary dynamical modes. Because an invariant I(t) must Special Issue: Ronnie Kosloff Festschrift Received: June 25, 2015 Revised: July 31, 2015
A
DOI: 10.1021/acs.jpca.5b06090 J. Phys. Chem. A XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry A satisfy − i[I,H]/ℏ + ∂I/∂ t = 0, so that ⟨Ψ(t)|I(t)|Ψ(t)⟩ remains constant for any |Ψ(t)⟩ that evolves with H(t), b must obey the Ermakov equation b ̈ + ω 2 (t )b =
2. TOTAL, KINETIC, AND POTENTIAL ENERGIES For the nth dynamical mode, the instantaneous energy, En(t) ≡ ⟨ψn|H(t)|ψn⟩, is given by
ω02 3
En(t ) =
(2)
b
Any state evolving with H(t) can be expanded as a superposition of “dynamical modes” |Ψ(t)⟩ = ∑n an|ψn(t)⟩, where n = 0, 1, 2, ..., the an are time-independent amplitudes, 1
t
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⟨q|ψn(t )⟩ =
⎛ mω0 ⎞1/4 e ⎜ ⎟ ⎝ πℏ ⎠
)
t −i(n + 1/2) ∫0 (ω0 / b2)
K n(t ) =
ω2⎞ (2n + 1)ℏ ⎛ 2̇ ⎜b + 20 ⎟ 4ω0 ⎝ b ⎠
(8)
Vn(t ) =
(2n + 1)ℏ 2 2 b ω (t ) 4ω0
(9)
dt ′
(2nn! b)1/2
⎡⎛ mω ⎞ 2 2 ̇ 0⎟ × e(im /2ℏ)(b / b + i(ω0 / b ))q /n⎢⎜ ⎣⎝ ℏ ⎠
1/2
q⎤ ⎥ b⎦
The mean value theorem implies for the interval (0,tf), assuming that the boundary conditions 4 are satisfied38 (10)
The first inequality sets a lower bound for the maximum of the instantaneous kinetic energy, Kn,max ≥ [((2n +1)ℏ)/(4ω0)][(γ − 1)2/tf2]. In general we cannot set bounds for minimal or maximal values of Vn, unless physically motivated bounds apply for both b and ω. 2.1. Virial Theorem for STA Processes. A virial-theorem relation will be proved for STA processes that implies, remarkably, that the time-averaged energy is equipartitioned into kinetic and potential contributions, namely, Kn = Vn= En/2, where the overline denotes a time average over [0, tf], that is, for a generic An(t) function
̇ b(0) = 1, b(0) =0 (4)
are satisfied, where γ = (ω0/ωf)1/2. (This assumes continuity of b and ḃ. ḃ may jump discontinuously at the boundary times if b̈ has a Dirac delta there, as discussed in the next section. In that case the conditions on ḃ are for left (0−) and right (t+f ) limits.) In addition ̈ b(0) = b(̈ tf ) = 0
γ−1 ̈ | ≥ 2γ − 1 , |bmax tf tf2
̇ ≥ bmax
(3)
where /n is a Hermite polynomial. If the commutation relations [H(0−),I(0−)] = 0 and [H(t+f ),I(t+f ) ] = 0 are satisfied (we specify the left and right limits to take into account a possible discontinuity in the frequency and b̈ at 0 or tf), the initial eigenstates of H(0−) are mapped dynamically into final eigenstates of H(t+f ), avoiding final particle excitation.11 These commutation relations indeed hold if the boundary conditions
b(tf ) = γ , b(̇ tf ) = 0
(7)
We may divide the energy into kinetic and potential parts using the separation H = K + V in eq 1. Then, En(t) = Kn (t) + Vn(t), where
|ψn(t)⟩ = eiαn(t)|un(t)⟩, αn(t ) = − n + 2 ω0 ∫ dt ′/b2(t ′) are 0 Lewis-Riesenfeld phase factors, and the |un(t)⟩ are eigenvectors of the invariant, I(t)|un(t)⟩ = λn|un(t)⟩, where λn = (n + 1/ 2)ℏω0 are time-independent eigenvalues. The dynamical modes have the form
(
ω2⎤ (2n + 1)ℏ ⎡ 2̇ ⎢b + ω 2(t )b2 + 20 ⎥ 4ω0 ⎣ b ⎦
An =
1 tf
∫0
tf
A n (t ) d t
(11)
The virial relation for STA processes is not at all an obvious result, as the ordinary quantum-mechanical virial theorem applies to localized (square integrable) stationary waves and a time-independent Hamiltonian.39−41 Generalizations were proposed for time-dependent states and a time-independent Hamiltonian, performing the average over an infinite time or over a period if the motion is periodic.42−44 Also, in ref 45 a virial theorem was found for a charged particle in a timedependent electromagnetic field. Suppose first that the trap expansion is performed adiabatically over a very long time tf so that the state remains in level n from t = 0 to tf. The ordinary virial theorem, that is, the one formulated for stationary states, can then be applied. For the harmonic oscillator in level n, it states that Vn = Kn = En/2. Along the adiabatic process the values of the energies will change slowly in such a way that the relation is preserved. We may thus take a time average and get Vn= Kn= En/2. For an arbitrary nonadiabatic process, however, this relation will not be true in general. According to eqs 7 and 11, the time-averaged energy is
(5)
may be imposed to make ω(t) continuous at t = 0 and t = tf. To inverse engineer ω(t), b(t) is designed first, interpolating with some convenient function between the boundary conditions at 0 and tf, and ω(t) is deduced from the Ermakov eq 2.11 A simple protocol based on a quintic polynomial for b(t), where the coefficients are found from eqs 4 and 5, is11 b(t ) = 6(γ − 1)s 5 − 15(γ − 1)s 4 + 10(γ − 1)s 3 + 1 (6)
where s = t/tf. For very short times, imaginary frequencies appear for some intermediate times, corresponding to a negative, concave-down potential. Another simple solution is the “bang−bang” (stepwise constant) form of the control function ω(t). It results, in particular, from applying optimal control theory (OCT) to minimize the time for given constraints on the frequency.6,11,15,18 Some relevant expressions are provided in the Appendix. As well, “bang-singular-bang” solutions result from minimizing the time-averaged energy for the same constraints.16
En =
1 tf
∫0
tf
ω2⎤ (2n + 1)ℏ ⎡ 2̇ ⎢b + ω 2(t )b2 + 20 ⎥ dt 4ω0 ⎣ b ⎦
(12)
or, using Ermakov’s eq 2 B
DOI: 10.1021/acs.jpca.5b06090 J. Phys. Chem. A XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry A En =
1 tf
∫0
tf
2 ⎤ (2n + 1)ℏ ⎡ ω0 2 ⎢2 2 + b ̇ − bb̈ ⎥ dt 4ω0 ⎣ b ⎦
⎧ ω02 , t ≤ 0− ⎪ ⎪ D δ(t ), 0− < t < 0+ ⎪ 0 ⎪ω2 b̈ ω 2(t ) = ⎨ 40 − , 0+ ≤ t ≤ tf− b ⎪b ⎪ − + ⎪ Df δ(t ), tf < t < tf ⎪ ωf2 , tf+ ≤ t ⎩
(13)
By using the boundary conditions 4 and partial integration, we t t have ∫ 0f(−b̈b) dt = ∫ 0f b2̇ dt. Then, the time-averaged energy can be rewritten as, see eq 8 En = ,n ≡
1 tf
∫0
tf
2 ⎤ (2n + 1)ℏ ⎡ ω0 2 ⎢ 2 + b ̇ ⎥ = 2 Kn 2ω0 ⎦ ⎣b
(14)
Substituting the δ terms in the Ermakov eq 2, integrating b̈ around (in the immediate neighborhood of) t = 0 and tf, and taking into account the boundary conditions in eq 4, D0 and Df are found to be
(We have introduced a specific notation, ,n , for the integral in eq 14 to emphasize that ,n is only the averaged energy when the boundary conditions are satisfied.) Because En= Kn + Vn, we find, remarkably, the virial-theorem relation
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K n = Vn = En /2
D0 = −
(15)
⎛t ⎞ ⎛ t ⎞2 (B2 − ω02tf2)⎜ ⎟ + 2B⎜ ⎟ + 1 ⎝ tf ⎠ ⎝ tf ⎠
(16)
(19)
En = Δδ + En[0+, tf−]
where B = (ω02tf2 + γ2)1/2 − 1 and the positive root should be taken; this only satisfies the boundary conditions b(0) = 1 and b(tf) = γ at the time limits but not the conditions on the derivatives. Therefore, as such it only provides a bound for the time-averaged energy. Substituting eq 16 into eq 14, this lower bound is EnL
̇ +) b(̇ tf−) b(0 , Df = b(0) b(tf )
(The first impulse always corresponds to a negative delta, while the second may have the two signs.16) The protocol 18 with Dirac impulses provides a formally elegant proof that the bound can indeed be reached, at least in principle, because, as the boundary conditions are satisfied, eq 14 may be used to get eq 17 as the actual time-averaged energy of the process. Note that these protocols minimize the time-averaged energy for a given time tf but also minimize the time tf for a given time-averaged energy. In the idealized processes that realize the bound, the instantaneous potential energy jumps to (plus or minus) infinity due to the Dirac pulses. It might seem that the deltas do not contribute to the averaged energy because ω does not appear explicitly in eq 14; see, for example, ref 16. However, they do. To see why and how much, the time-averaged energy given by the original expression, eq 12, where no partial integration has been carried out, can be separated into two parts, one from the deltas and one from the central time interval
This implies a number of consequences. For example, minimizing any of the time-averaged energies automatically minimizes the others. Also, nontrivial bounds may be set because bounds for one of the averaged energies work for the others as well. Thus, despite the fact that Vn may be negative during some time interval, the time-average Vn must be positive, as Kn ≥ 0; even more, it will be bounded from below by a positive number, as described in the next subsection. Further applications of the virial relation in this work may be found in Section 3.1 and in the Appendix. 2.2. Energy Contribution from Dirac Impulses. Using the Euler−Lagrange equation12 to minimize eq 14 or optimal control theory (OCT),16 the following quasi-optimal function is found b(t ) =
(18)
(20)
where Δδ =
∫0
0+
−
En(t ) dt +
En[0+, tf−] =
⎪ (2n + 1)ℏ ⎧ ⎨(B2 − ω02tf2) − 2ω0tf = 2 2ω0tf ⎪ ⎩
∫0
t −f
+
∫t
t f+ − f
En(t ) dt
En(t ) dt
(21)
(22)
and En(t) is given in eq 7 with the frequencies given by eq 18. Making use of the coefficients 19, the contribution from the Dirac impulses to the time-averaged energy is
⎡ ⎛ B 2 + B − ω 2t 2 ⎞ ⎛ B ⎞⎤⎫ ⎪ 0 f ⎬ ⎟ −arctanh⎜ × ⎢arctanh⎜ ⎟⎥ ⎪ ω0tf ⎢⎣ ⎝ ω0tf ⎠⎥⎦⎭ ⎝ ⎠
Δδ =
(17)
which becomes EnL ≈ [(2n + 1)ℏ]/[2ωftf2] for tf ≪ 1/ (ω0ωf)1/2 and γ ≫ 1. Chen and Muga12 added polynomial “caps” around a central time segment where eq 16 holds to satisfy all the boundary conditions. A numerical example for tf ≪ 1/(ω0ωf)1/2 and γ ≫ 1 showed that when the cap duration τ went to zero, En→ EnL; however, the caps did contribute to the integral. Actually, half of the total time-averaged energy was due to them, but the significance of this fact was not discussed. An alternative to the polynomial caps was put forward by Stefanatos and Li,16 namely, Dirac-delta impulses of the control function ω2(t) that switch the derivatives ḃ(0+) and ḃ(t−f ) to ḃ(0−) = ḃ(t+f ) = 0
(2n + 1)ℏ ̇ − ̇ +)b(0)] [b(tf )b(tf ) − b(0 4ω0tf
(23)
The other term in eq 20 can be rewritten using partial integration as En[0+, tf−] = ,n + Δboundary
(24)
where Δboundary = −
(2n + 1)ℏ ̇ − ̇ +)b(0)] [b(tf )b(tf ) − b(0 4ω0tf
(25)
Compared with eq 23 Δδ = −Δboundary C
(26) DOI: 10.1021/acs.jpca.5b06090 J. Phys. Chem. A XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry A Ena(t ) =
so, according to eqs 20, 24, and 26 En = ,n = EnL
n
(27)
(2n + 1)ℏ 2 (B − ω02tf2) 2 4ω0tf
Ena(t ) = E0na ∑ Pn(0−)(2n + 1)
(28)
When tf ≪ 1/(ω0ωf)1/2 and γ ≫ 1 Δδ ≈
(2n + 1)ℏ = EnL /2 4ωf tf2
(31)
n
E0na(t ) =
(29)
which agrees with the contribution of the caps found numerically in ref 12 as τ → 0. The reason for this result, however, had not been explained. 2.3. Comparison of Protocols. Figure 1 depicts tf versus the time-averaged energy for different protocols. A log−log
ω2⎞ 1 ℏ ⎛ 2̇ ⎜b + ω 2(t )b2 + 20 ⎟ − ℏω(t ) 4ω0 ⎝ 2 b ⎠
(32)
na
Its time average, E , for initially thermal states, has been proposed to quantify the cost to implement a quantum engine based on the harmonic oscillator.32 To minimize the time average for any initial state it is enough to minimize E0na =
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(30)
Using eq 7, the nonadiabatic energy factorizes as a constant factor that depends on the initial state times the nonadiabatic energy for n = 0
Substituting the quasi-optimal protocol 16 in eq 23 Δδ =
∑ [En(t ) − ϵn(t )]Pn(0−)
ℏω0 4tf
∫0
tf
⎛ b2̇ ω 2 (t )b 2 2ω(t ) ⎞ 1 ⎜⎜ 2 + ⎟⎟ dt + − ω0 ⎠ b2 ω02 ⎝ ω0 (33)
Note that the minimization of this quantity cannot be reduced trivially to the one performed in the previous section because both E0 and Ead 0 depend on ω(t). Using Ermakov’s equation, partial integration and the conditions in eq 4, we get from eq 33 a simpler form E0na = , 0na ℏω0 ≡ 2tf
Figure 1. Protocol time tf versus time-averaged energy En for a quintic polynomial protocol (dotted brown), bang−bang for ω1 = ω2, which decreases for larger times, see eq 52 (solid black), and lower bound (dashed red). The filled black point is for the bang−bang corresponding to −ω20 ≤ ω2(t) ≤ ω20. The bang−singular−bang form (dotted-dashed blue) is for the same bounds, with the bang− bang control of the point as its limiting, minimal-time case. Parameters: ω0 = 2π × 2500 Hz and ωf = 2π × 25 Hz.
∫0
tf
⎛ b2̇ ω(t ) ⎞ 1 ⎜⎜ 2 + 2 − ⎟⎟ dt ω0 ⎠ b ⎝ ω0
(34)
where, again, we have introduced a special notation for the last integral, , 0na , to emphasize that it becomes Ena 0 provided the boundary conditions in eq 4 are satisfied. 3.1. Lower Bound for Time-Averaged Nonadiabatic Energy. A useful analogy exists between the Ermakov equation
representation is chosen to show the global behavior and the domains where the protocols are applicable. As announced, the fastest protocols for a given En are the ones that implement the bound 17. Other constraints lead to different winners. For example, if |ω2| is limited by some predetermined value the fastest solutions are of bang−bang form.11,18 If in addition to this bound the time-averaged energy is fixed, they are of bang− singular−bang form,16 where the protocol in eq 16 applies in a central segment flanked by constant-frequency intervals. The bang−bang protocol considered in Figure 1 assumes the two intermediate frequency steps iω1 and ω1 for durations t1 and t2, respectively; ω1 decreases for larger times, and a maximal time (minimal En) exists, as discussed in the Appendix and shown in the Figure.
Figure 2. Classical particle (blue dot) moving along the bottom of an expanding potential U(t) = C[ω(t)2b2 + ω20/b2]/2 (red line); see the text. The two parts of U(t) are shown, Cω(t)2b2/2 (blue dashed line) and Cω20/(2b2) (black dashed line).
3. NONADIABATIC ENERGY The “nonadiabatic energy” is defined as the difference between the total energy and the energy of a corresponding adiabatic process Ena(t) = ⟨H(t)⟩ − Ead(t), where Ead ≡ ∑n 7n(0−)ϵn(t ), ϵn(t) = (n + 1/2) ℏω(t) are the instantaneous eigenenergies, and Pn(0−) are the initial probabilities of the eigenstates. We must now assume ω(t) ≥ 0 to have a meaningful real quantity. Further assuming that the initial density operator is diagonal (and therefore stationary) in the eigenbasis of H(t < 0)
and the dynamical equation of a fictitious (classical!) particle with position x. In terms of the dimensionless “position” b = x/ (ℏ/(2ω0m))1/2 it moves in a potential of the form15,47 U (t ) = D
C 2 [ω (t )b2 + ω02 /b2] 2
(35) DOI: 10.1021/acs.jpca.5b06090 J. Phys. Chem. A XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry A with kinetic energy Cḃ2/2, where C = ℏ/(2ω0). Newton’s equation takes indeed the form of the Ermakov equation b ̈ = ω02 /b3 − ω 2(t )b
Thanks to the factorization in eq 31, the positivity of Ena 0 (t) means that the nonadiabatic energy is non-negative for all initial states considered (diagonal in the energy representation, that is, stationary states of H(0−)), even if they are not passive. This does not contradict the “minimal work principle”,46 which establishes passivity (7n(0−) ≥ 7n + 1(0−)) as a sufficient condition for ⟨H(t)⟩ ≥ Ead(t), provided no level-crossings occur. 3.2. Comparison of Protocols. The linear trajectory of b(t) in eq 41 does not satisfy all boundary conditions in eq 4, and thus the protocol ω(t) = ω0/b2 based on it leads to excitation. Dirac impulses are not an option now, as the one at t = 0 would have to be negative, but ω2 < 0 is not allowed. We may instead complement the protocol with caps of durations τL and τS (for “launching” and “stopping” respectively) that connect the linear function with the proper boundary conditions, similarly to ref 12. A hybrid protocol defined in this manner is
(36)
This analogy provides useful intuition and will help us to discuss the lower bound and properties of E na . The corresponding energy is
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/=
ω2⎞ C ⎛ 2̇ ⎜b + ω 2(t )b2 + 20 ⎟ 2⎝ b ⎠
(37)
It has the same form as the total energy 7, E0 = / . Beware that the kinetic and potential energies of the classical particle do not correspond, in general, to the quantum counterparts in eqs 8 and 9. Specifically, the term Cω20/(2b2) is “kinetic” in the quantum scenario and “potential” in the classical analogy. Let us now calculate the excitation energy of the particle, Eex, measured from the potential minimum. From dU(t)/db = 0, at the minimum ω0 b 2 (t ) = ω(t ) (38)
3 ⎧ ⎪ ∑ f sn , ⎪ n=0 n ⎪ b(t ) = ⎨(γ − 1)s + 1, ⎪ 3 ⎪ n ⎪ ∑ gns , ⎩ n=0
so that Umin = Cω0ω, and E ex ≡ / − Umin ⎞ ω2 C 2 C⎛ = b ̇ + ⎜ω 2b2 + 20 − 2ω0ω⎟ 2 2⎝ b ⎠
(40)
with solution γ−1 b=1+ t tf
(41)
that satisfies the boundary conditions b(0) = 1 and b(tf) = γ. This linear b also minimizes the time-averaged kinetic energy for these boundary conditions, as it can be seen from the Euler−Lagrange equation; however, the particle trajectory described by eq 41 is not at rest at t = 0 and tf, so for the more restricted family of trajectories satisfying eq 4 it only provides a lower bound for Eex. Of course the same lower bound is valid for the analogous quantum system, namely E0,naL =
2 ℏ ⎛ γ − 1⎞ ⎟ ⎜ 4ω0 ⎝ tf ⎠
τL ≤ t ≤ tf − τS tf − τS ≤ t ≤ tf (43)
where s = t/tf and the coefficients f n are found from the equations that match b and ḃ at 0 and τL and gn from the matching at tf − τS and tf. It can be proved that in the cubic interpolation b > 0. The cap durations τL,S can be adjusted by a subroutine that minimizes Ena with the constraint ω ≥ 0. Because of the constraint the cap times cannot be made zero, so, unlike Section 2.2, the bound is not reached. The caps can only be constructed in this way until a minimal tf for which imaginary frequencies appear. We also consider bang−bang protocols that provide a meaningful nonadiabatic energy with ω1 = 0 and ω2 = ω0β, β > 0 (see the Appendix), with
(39)
This has exactly the same form as the nonadiabatic energy 32, ex Ena 0 = E . To complete the analogy, we consider an ω(t) that changes from ω0 to ωf and trajectories from the initial to the final points of the potential minimum, b(0) = 1 to b(tf) = γ. Note that both the first term (kinetic energy) and the second one (potential energy measured from the minimum) are positive. The potential part of eq 39 can in fact be made zero if the particle moves all the time at the bottom of the potential without ever being affected by a force, then eq 38 should hold and, substituted in eq 36, this gives the equation b̈ = 0
0 ≤ t ≤ τL
t1 =
1 ω0
(γ 2 − 1)(γ 2β 2 − 1) γ 2β 2
(44)
t2 =
⎤ ⎡ γ2 − 1 ⎥ 1 arcsin⎢ ⎢⎣ β 2γ 4 − 1 ⎥⎦ ω0β
(45)
Figure 3 shows the scaling of tf and the time-averaged nonadiabatic energy. Note that there is some complementarity among the different protocols, which are applicable in different domains. Whereas the bang−bang protocol only satisfies the boundary conditions in a small-time, high-energy regime, the quintic polynomial 6 or hybrid protocols apply rather in a largetime, low-energy scenario.
4. POWER We finally consider the instantaneous power during the STA process dEn ̇ 2 Pn(t ) ≡ = knKb (46) dt where we have used eq 7 and defined kn ≡ [(ℏ(2n + 1))/ (4ω0)] and 2(t ) ≡ ω 2 . This is a remarkably simple expression
(42)
2 When γ ≫ 1, Ena L ≈ ℏ/(4ωftf ), which is half the lower bound of the time-averaged energy for the ground dynamical mode if, in addition, tf ≪ 1/(ω0ωf)1/2. Under these conditions the quantum kinetic energy is dominated by the first term in eq 8, which corresponds to the classical kinetic energy, and the virial theorem implies K0= E0/2.
E
DOI: 10.1021/acs.jpca.5b06090 J. Phys. Chem. A XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry A
Figure 3. Dependence of tf on the time-averaged nonadiabatic energy for the hybrid protocol in eq 43 (dotted brown line), quintic polynomial protocol (solid blue line), bound in eq 42 (dashed black line), and bang−bang protocol with different bounds (dotted-dashed red line). The parameters are ω0 = 2π × 2500 Hz and ωf = 2π × 25 Hz.
Figure 4. Relative power versus s = t/tf for the polynomial 6 (dashed red line) and for the polynomial 51 (solid blue) with c3 = 78.5088, c4 = −459.7638. Parameters: ω0 = 2π × 2500 Hz, ωf = 2π × 25 Hz, and tf = 8 ms.
for Pn. (Defining the time-independent operator v by V = 2(t )v , and using eq 9, it may also be written as K̇ ⟨v⟩ which is the form given in ref 48.) For any STA process its integral is
where s = t/tf.
∫0
tf
5. DISCUSSION Different transient energies in shortcuts to adiabatic harmonic expansions for a single particle have been studied as well as the relations with the process time. This is important to establish operational limits, which are different from the naive application of a time-energy uncertainty relation. General bounds are provided, and protocols that realize or approach them have been discussed. The specific experimental conditions, that is, the actual realization of the system and the potential by different interactions (optical, magnetic, electrostatic), will determine their realizability, because optimal protocols often imply Dirac deltas (e.g., in the potential or in the power). The virial theorem for time-averaged energies emerges as an important relation for STA processes. We have given examples where it is instrumental in providing bounds and interpreting the results. Possible extensions will be discussed elsewhere. The power in STA has also been brought to the forefront. It provides a further criterion to choose among different shortcuts. Even though its time integral is invariant for all STA, the maximal values could be minimized.
dEn ⎛ 1⎞ dt = En(tf ) − En(0) = ⎜n + ⎟ℏ(ωf − ω0) ⎝ dt 2⎠ (47)
As a possible use of power for shortcut design, suppose that we are interested in minimizing the power peak (maximum) of |Pn|, distributing the power homogeneously from 0 to tf to facilitate the power extraction. We then set ̇ 2 = Cn Pn = knKb
(48)
where Cn is a constant adjusted to satisfy eq 47, namely Cn = (n + 1/2)ℏ(ωf − ω0)/tf
(49)
note that Cn/kn does not depend on n, so that the relative power Prel ≡ Pn/Cn does not depend on n. According to the mean value theorem, 1 is a lower bound for the maximum of Prel in an arbitrary STA expansion process. 2 and b must satisfy the Ermakov eq 2, so eq 48 becomes ω2 ω bb ⃛ − bb̈ ̇ + 4 30 b ̇ = 2(ω0 − ωf ) 0 tf b
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APPENDIX: BANG−BANG PROTOCOLS A “bang−bang” protocol is composed by time segments with constant frequency so that the energy rate transfer (power) from or to the system consists of Dirac deltas at the switching times. For applications and discussion of their relation to optimal control theory, see refs 6, 11, 15, 16, and 18. Typically the frequency is chosen at the extreme values allowed to achieve optimal results. We consider here processes with two intermediate steps of the form (more switching times may be required to find the true minimal time in general15)
(50)
This is a third-order equation. To avoid a sudden jump in ω(t) at time t = 0, and thus a Dirac delta in the power, we must impose b̈(0) = 0, so, in fact, with b(0) = 1 and ḃ(0) = 0, we already fix the three possible constants at time t = 0. The numerical solution may satisfy b(tf) = γ for a specific tf, but the conditions ḃ(tf) = b̈(tf) = 0 fail in general. We may thus resort to more modest objectives. As an example of many possible strategies, in Figure 4 we plot Prel for the fifth-order polynomial 6 and for a polynomial that satisfies the boundary conditions 4 and 5 with two free parameters that are chosen to minimize numerically the maximum of Prel 3
4
b(t ) = 1 + c3s + c4s − (21 + 6c3 + 3c4 − 21γ )s
⎧ ω0 , t = 0 ⎪ ⎪ iω1 , 0 < t < t1 ω(t ) = ⎨ ⎪ ω2 , t1 < t < t1 + t 2 ⎪ ⎩ ωf , t = tf = t1 + t 2
5
+ (35 + 8c3 + 3c4 − 35γ )s 6 − (15 + 3c3 + c4 − 15γ )s 7
(51) F
(52) DOI: 10.1021/acs.jpca.5b06090 J. Phys. Chem. A XXXX, XXX, XXX−XXX
The Journal of Physical Chemistry A
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where ω1 ≥ 0 and ω2 > 0 are real constants, and t1 and t2 are the durations of each part. For 0 < t < t1, the solution of the Ermakov equation satisfying b(0) = 1 and ḃ(0) = 0 is b(t ) =
ω02
1+
+
ω12
ω12
*E-mail:
[email protected]. Notes
sinh2(ω1t )
b(t ) =
γ2 +
−γ
ω22
The authors declare no competing financial interest.
(53)
sin 2[ω2(tf − t )]
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The matching conditions for b(t1) and ḃ(t1) determine t1 =
ω 2(γ 2 − 1)(γ 2ω22 − ω02) 1 arcsinh 21 2 ω1 γ (ω2 + ω12)(ω02 + ω12)
t2 =
ω22(γ 2 − 1)(γ 2ω12 + ω02) 1 arcsin ω2 (ω22 + ω12)(γ 4ω22 − ω02)
π , 2 ω0ωf
Enmin = (2n + 1)ℏ
ω0 + ωf 4
(56)
To calculate En, we use the fact that the total energy remains constant during the constant-frequency intervals. It can be calculated with eqs 8 and 9, at t = 0+ for the first segment and at t = t−f for the second segment 2 ⎧ 2 (n + 1/2)ℏ ⎪(ω0 − ω1 )/ω0 , 0 < t < t1 ⎨ En = ⎪ 2 2 2 ⎩ (ωf + ω2 )/ωf , t1 < t < tf
(57)
If ω1 = ω2 ≫ω0 with γ ≫ 1, tf approaches zero, and the corresponding time-averaged energy is En ≈
(2n + 1)π ℏ ln(2γ ) 16ωf tf2
(58)
This is of course larger than the asymptotic value of lower bound for the time-averaged energy, EnL ≈ [((2n + 1)ℏ)/ (2ωftf2)]. If we set ω1 = 0, ω2 ≫ ω0, and γ ≫ 1, then tf ≈ t1 ≈
1 ω0ωf
En ≈ (n + 1/2)ℏω0
ACKNOWLEDGMENTS
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REFERENCES
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There is an upper bound for tf. According to eq 55, ω2 ≥ (ω0ωf)1/2 must be satisfied to make t1 ≥ 0. With this condition, t2 will be positive. At ω2 = (ω0ωf)1/2, t1 = 0, which gives the upper bound for tf and the corresponding minimal timeaveraged energy tfmax =
■
We thank D. Stefanatos for commenting on the manuscript. This work was partially supported by the NSFC (61176118 and 11474193), the Shanghai Shuguang and Pujiang Programs (14SG35 and 13PJ1403000), the program for Eastern Scholar, the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 2013310811003), the Basque Government (Grant IT472-10), MINECO (Grant FIS201236673-C03-01), and the program UFI 11/55 of UPV/EHU.
4
γ 2ω22
AUTHOR INFORMATION
Corresponding Author
and for t1 ≤ t ≤ tf, the one that satisfies b(tf) = γ, ḃ(tf) = 0 is ω02
Article
(59)
or, making use of the relation between tf and ω0 in eq 59, we find again the bound value, namely, En ≈ [((2n + 1)ℏ)/ (2ωftf2)]. We may interpret this result in view of the virial theorem. Under these conditions, most of the protocol time is just a free expansion ended up by a short segment with a strong potential. The frequency switch at time zero suddenly turns off the initial harmonic potential so that half the initial energy vanishes, and for most of the time the energy (purely kinetic) is En(0)/2. Thus, Kn ≈ En(0)/2, and according to the virial theorem the last segment must be such that Vn= En(0)/2 as well. That explains why En ≈ En(0). G
DOI: 10.1021/acs.jpca.5b06090 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
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The Journal of Physical Chemistry A
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