npsm 새물리 New Physics : Sae Mulli

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Research Paper

New Phys.: Sae Mulli 2021; 71: 1082-1089

Published online December 31, 2021 https://doi.org/10.3938/NPSM.71.1082

Copyright © New Physics: Sae Mulli.

Shortcuts to Adiabaticity Using Time-dependent Harmonic Oscillators

Sang Pyo KIM*

Department of Physics, Kunsan National University, Kunsan 54150, Korea

Correspondence to:sangkim@kunsan.ac.kr

Received: October 7, 2021; Revised: October 22, 2021; Accepted: October 25, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Shortcuts to adiabaticity (STA) is a protocol to speed up the quantum adiabatic process through nonadiabatic routes. Time-dependent oscillators, whose specta generate algebra so(2; 1)≈su(1; 1) give quantum invariants and thereby exact quantum states, exhibit a variety of processes from adiabatic to nonadiabatic evolutions. To illustrate how time-dependent oscillators realize the STA, first we use the construction method of an Ermakov-Pinney invariant and find the associated time-dependent oscillator. Then, we introduce a class of time-dependent oscillators whose wave functions are known, and find the condition for the STA for adiabatic and nonadiabatic evolutions.

Keywords: Shortcuts to adiabaticity, Time-dependent oscillators, Quantum invariants, Spectrum generating algebra, Nonadiabatic evolution

Quantum systems can be influenced by environments. The most well-known quantum system is a Paul or ion trap in which charges are controlled by electric and magnetic fields [1,2]. The quantum theory for such a system is governed by the time-dependent Schrödinger equation described by a Hamiltonian that explicitly depends on time or control parameters. Nearly a century ago, Born and Fock proved the adiabatic theorem that the quantum system remains in the same eigenstate during the evolution when the Hamiltonian changes sufficiently slowly [3]. Assuming that a time-dependent Hamiltonian H^ (t) has instantaneous eigenstates corresponding to a discrete energy spectrum as

H^0t|nta=Ent|nta,

the quantum state for the time-dependent Schrödinger equation is given by (in unit of ħ = 1)

ψt=nc nteiα natnta,

where the phase consists of the dynamical phase and the geometric phase:

eiαnat= e i t0 t dt'En t' dynamical phase× e t0 t dt'a nt| n˙ta geometric phase.

Here, the coefficients satisfy a system of linear differential equations:

c˙nt= mn nt| H^ ˙0 t|ntEntEmtei αma tαna t cmt.

Thus, the quantum system remains in the same eigenstate when the change of the Hamiltonian is sufficiently small in comparison to energy separations, and cn(t) are almost constant.

However, when the Hamiltonian rapidly changes, the adiabatic theorem does not hold and the nonadiabatic or exact evolution should be employed. Lewis and Riesenfeld [4] found that a quantum invariant satisfying the Liouville-von Neumann equation

iI^˙t+ I ^ t, H^0t=0,

gives the exact quantum state

ψt=nc neiα ntnt, I^tnt=ϵnnt,

where ϵn is a constant and the phase is given by

αnt=t0t dt' n t|H^ 0 t|n ti nt |n˙t .

Note that the dynamical and the geometric phases are computed with respect to the eigenstates of the invariant, not the eigenstates of the Hamiltonian. One can raise a question about the evolution when the system adiabatically changes around the initial and final times while it evolves nonadiabatically in between. In fact, shortcuts to adiabaticity (STA) is a protocol to speed up quantum adiabatic process through nonadiabatic routes [5]. The invariants are used to study the STA [5,6]. The physics and applications of the STA are discussed in detail in Refs. [7,8] and the references therein.

In this paper, we realize the STA by using timedependent harmonic oscillators and investigate the conditions for an adiabatic evolution, an STA and a nonadiabatic evolution. The physical motivation is that as shown in Table 1, the spectrum generating algebra so(2, 1) for harmonic oscillators is isomorphic to those algebras for some noncompact groups and to those algebras for other compact groups through complexification. This means that many quantum systems with isomorphic algebras share common physical characteristics. Another motivation is that any exact quantum state of time-dependent oscillators can be expressed in terms of complex solutions to the classical equation, which was explicitly constructed by Lewis and Riesenfeld [4].

Table 1 . Isomorphic spectrum generating algebras for noncompact and compact groups .

Groupsalgebraalgebraalgebra
Noncompact groupso(2,1)su(1,1)sp(2,R)
Compact groupso(3)su(2)sp(2)


The organization of this paper is as follows. In Section II, we show the spectrum generating algebras for the harmonic oscillators. In Section III, we explain the counterdiabatic driving (CD) part from the spectrum generating algebra and explicitly show the CD part for time-dependent oscillators. In Section IV, we briefly summarize the Lewis-Riesenfeld invariant and the timedependent annihilation and the creation operators, linear quantum invariants. In Section V, we advance two models for the STA: a polynomial amplitude and a hyperbolic amplitude. In Section VI, we introduce an exact model whose wave functions are known, and investigate the condition for the STA. In Section VII, we summarize how time-dependent oscillators can be employed in the STA and discuss possible applications to quantum systems whose spectrum generating algebras are isomorphic to that of oscillators.

Let us consider a time-dependent oscillator of the form (ħ = m = 1)

H^0t=p^22+12ω2tq^2.

We may introduce a spectrum generating algebra

K^0=14p^2+q^2, K^1=14 p ^ q ^ +q ^ p ^ , K^2=14p^2q^2,

which has the commutators for so(2, 1) and su(1, 1) [9]:

K^0,K^1=iK^2, K^1,K^2=iK^0, K^2,K^0=iK^1.

We can also use the oscillator representation

a^=12q^+ip^,  a ^ =12q^+ip^,

to express the algebra

K^0=14a^ a ^ +a ^ a^, K^1=i4a^2a^2, K^2=14a^2+a^2.

The isomorphisms of Lie algebras for noncompact and compact groups are summarized in Table 1.

The inverse engineering is a time-variation of control parameters from a chosen evolution of quantum systems. The complete set of the instantaneous eigenstates (1) of H0(t) leads to an evolution operator

U^t,t0=nψ ntaaψ nt0,

where | n(t)⟩a = eiαnat|n(t)⟩a. A question is raised about which Hamiltonian H^ (t) has the evolution operator (13) as the exact Schrödinger equation

iU^˙t=H^tU^t,

which is the essence of the inverse engineering. Thus, the inverse engineering finds the Hamiltonian

H^t=i U ^ ˙t U ^ t.

In the Berry formulation [10], the given Hamiltonian and the evolution operator have the spectral representation

H^0t=nntE ntnt0 ,
U^t,t0=nnteiα natnt0.

Then, the Hamiltonian necessarily has a counterdiabatic driving (CD) part

H^t= H ^ 0t+ H ^ CDt,

where

H^CD=in n˙ tnt n t | n ˙ t ntnt.

We illustrate the inverse engineering and the CD Hamiltonian by using a time-dependent oscillator (8). The instantaneous eigenstates are constructed by the time-dependent annihilation and the creation operators

a^t=ωt2q^+i12ωtp^, a ^ t=ωt2q^i12ωtp^.

Then, the Hamiltonian has the form

H^0t=K^0+K^2+ω2tK^0K^2=ωta^ta ^ t+12,

and the CD Hamiltonian is given by [11,12]

H^CDt=ddtInωtK^1.

Note that the CD Hamiltonian is a consequence of the spectrum generating algebra for an oscillator; K^1 exhausts the spectrum generating algebra for the Hamiltonian (21).

We briefly summarize the Lewis-Riesenfeld invariant and linear invariants for the time-dependent annihilation and the creation operators. The exact number states and the displaced-squeezed number states were found in Refs. [13,14] and the references therein.

The time-dependent oscillator in Eq. (8) has the Lewis-Riesenfeld invariant [4],

I^t=ξp^ξ˙q^2+ q ^ 2ξ2,

where ξ is a solution to the Ermakov equation [15]

ξ¨+ω2tξ=14ξ3.

The time-dependent annihilation and the creation operators are linear invariants [16,17]

A^t=iu*tp^u˙*tq^, A ^ t=iu*tp^u˙tq^,

where u is a complex solution to the classical equation

u¨t+ω2tut=0, Wrut,u*t=i.

The magnitude ξ of the complex solution u(t)

ut=ξtei 1 2ξ2

is the solution to the Ermakov equation (24). The exact number states are [13]

Ψnq,ξ=12π2nn!ξein+1/2 1/ 2ξ 2 ×Hnq2ξe1/ξ2iξ˙/ξq2/2,

where Hn is the Hermite polynomial.

In this section, we introduce a stratagem to construct the amplitude of the Ermakov equation (24), and then to find the associated time-dependent frequency for oscillator. Pinney obtained the most general solution to the Ermakov equation [18]. In Ref. [19], provided that the amplitude ξ(t) is given, the frequency for the timedependent oscillator is constructed as

ω2t=14ξ4tξ¨tξt.

The first model is a polynomial amplitude and the second model is a hyperbolic amplitude below.

1. Polynomial Amplitude

The first model has a polynomial amplitude [20]

ξt=12ω0+12ω112ω0ft,

where

ft=1015tT+6tT2tT3.

Note that the amplitude satisfies the boundary condition at the initial time

ξ0=12ω0, ξ˙0=ξ¨0=0,

and another boundary condition at the final time

ξT=12ω1, ξ˙T=ξ¨T=0.

Note that I^(0) = H^0(0)/ω0 and I^(T) = H^0(T)/ω1. Thus, the exact states approach the instantaneous eigenstates of the Hamiltonian both at t = 0 and t = T.

Setting ω0 = 1, Fig. 1 shows that the amplitude ξ(t) exhibits constancy for a ω1 range [1, 3] around the initial time t = 0 and the final time t = T. As shown in Fig. 2, the frequency can change from ω0 = 1 to above ω1 = 3 keeping the STA. The elongation of T does not change the dynamics of the STA because it corresponds to a rescaling of time.

Figure 1. (Color online) The amplitude ξ(t) against ω1 for the polynomial model with T = 1.

Figure 2. (Color online) The frequency ω2(t) against ω1 for the polynomial model with T = 1.

2. Hyperbolic Amplitude

The second model has a hyperbolic amplitude

ξt=12ω0+1212ω112ω0tanhtT +1V4tanh2tT 1.

This model has the asymptotic amplitude

ξ=12ω0, ξ=12ω1

And

ξ˙=ξ¨=0.

In fact, the amplitude ξ(t) approaches 1/ 2ω0 exponentially as e2t/T in past infinity and 1/2ω1 exponentially as e−2t/T in future infinity. The derivatives vanish exponentially as 1/ cosh2(t/T). This means that the variation of amplitudes at t = ∓5T from the asymptotic value is exponentially suppressed.

The constancy of the amplitude ξ(t) is shown around the initial time t = −5T and final time t = 5T in Fig. 3 for T = 1, and the corresponding frequency is shown in Fig. 4. In the case of large adiabatic parameter T = 10, Fig. 5 shows the constancy of the amplitude, and the corresponding frequency is shown in Fig. 6. Note that the large adiabatic parameter is nothing but a rescaling of time in the construction method for the frequency and the oscillator.

Figure 3. (Color online) The amplitude ξ(t) of the hyperbolic model against ω1 with T = 1 and V = −1.

Figure 4. (Color online) The frequency ω2(t) of the hyperbolic model against ω1 with T = 1 and V = −1.

Figure 5. (Color online) The amplitude ξ(t) of the hyperbolic model against ω1 with T = 10 and V = −1.

Figure 6. (Color online) The frequency ω2(t) of the hyperbolic model against ω1 with T = 10 and V = −1.

As an exact model, we consider the time-dependent Frequency

ω2t=ω0212ω02ω12tanhtT +1V4tanh2tT 1.

The complex solution is known as [21]

ut=N1+tanhtTiω0T/21tanhtTiω1T/2×Fα,β; 1+iω1T; 1tanht/T2,

Where

α=12+iT2ω0+ω1+121+VT2,β=12+iT2ω0+ω1121+VT2.

Here, N is a normalization constant such that the asymptotic solution is given by u(−∞)/N = e−iω0t0t/2ω0.

The Hamiltonian is expressed in terms of A^(t) and A^(t) as

H^0t= u˙ *u˙ +ω2tu*u12A^tA ^ t+A ^ tA^t+Ut12A^2t+U*t12A^2t.

Here, U(t)

Ut:=u˙2+ω2tu2u˙*u˙+ω2tu*u,

measures the condition for the STA:

H^0tI^tI^t,ddt u˙ *u˙ +ω2tu*u1

both at the initial time t = 0 and final time t = T.

For numeral results, we set ω0 = 1. The frequency is shown for ranges [1, 5] of ω1 and [−5, 5] of V in Figs. 7 and 8 for a short period T = 1. The real part of an exact solution in Fig. 9 and the imaginary part in Fig. 10 exhibit oscillatory behaviors as expected. The measure of the STA is numerically shown in Figs. 11 and 12. For a range [1, 5] of ω1, the real and the imaginary parts of U(t) approach zero and ||H^0(t) − I^(t)|| = 0 at the final time. The frequency itself can be rescaled by a time t˜ = t/T. However, the time-dependent Schrödinger equation has an adiabaticity

Figure 7. (Color online) The frequency ω2(t) of the exact model against ω1 with T = 1 and V = 2.

Figure 8. (Color online) The frequency ω2(t) of the exact model against V with ω1 = 2 and T = 1.

Figure 9. (Color online) The real part of an exact solution u(t) for T = 1 and V = 2.

Figure 10. (Color online) The imaginary part of an exact solution u(t) for T = 1 and V = 2.

Figure 11. (Color online) The real part of U(t) with T = 1 and V = 1.

Figure 12. (Color online) The imaginary part of U(t) with T = 1 and V = 1.

it˜ψ t ˜ =TH^0t˜ψ t ˜ .

Indeed Fig. 13 shows that the evolution is adiabatic as expected.

Figure 13. (Color online) The real part (yellow curve) and imaginary part (blue curve) of U(t) with V = 1 for the adiabatic parameter T = 10.

Shortcuts to adiabaticity (STA) is a protocol to speed up adiabatic evolutions in the same eigenstates. We studied the STA by using time-dependent oscillators. The fact that oscillators have spectrum generating algebras so(2, 1) ≈ su(1, 1) for noncompact groups, which through complexification, can be made isomorphic to spectrum generating algebras so(3) ≈ su(2) ≈ sp(2), generators for compact group, implies that time-dependent oscillators may provide useful information about other quantum systems, such as spin systems.

Time-dependent oscillators have the Lewis-Riesenfeld invariants whose Ermakov-Pinney solutions are expressed in terms of the classical solution for the corresponding oscillators. Furthermore, the time-dependent annihilation and the creation operators, which are invariants, constructed all quantum states. By finding the frequencies from the Ermakov equation, we constructed time-dependent oscillators that realize the STA. We also investigated the STA for a time-dependent oscillator that has an adiabaticity parameter as well as a parameter for adiabaticity or nonadiabaticity and showed that the STA could be realized by time-dependent oscillators with a wide range of parameters.

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