Ex) Article Title, Author, Keywords
Ex) Article Title, Author, Keywords
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.
Sang Pyo KIM*
Department of Physics, Kunsan National University, Kunsan 54150, Korea
Correspondence to:sangkim@kunsan.ac.kr
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
the quantum state for the time-dependent Schrödinger equation is given by (in unit of
where the phase consists of the dynamical phase and the geometric phase:
Here, the coefficients satisfy a system of linear differential equations:
Thus, the quantum system remains in the same eigenstate when the change of the Hamiltonian is sufficiently small in comparison to energy separations, and
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
gives the exact quantum state
where
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
Table 1 . Isomorphic spectrum generating algebras for noncompact and compact groups .
Groups | algebra | algebra | algebra |
---|---|---|---|
Noncompact group | so(2,1) | su(1,1) | sp(2,R) |
Compact group | so(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 (
We may introduce a spectrum generating algebra
which has the commutators for
We can also use the oscillator representation
to express the algebra
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
where |
which is the essence of the inverse engineering. Thus, the inverse engineering finds the Hamiltonian
In the Berry formulation [10], the given Hamiltonian and the evolution operator have the spectral representation
Then, the Hamiltonian necessarily has a counterdiabatic driving (CD) part
where
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
Then, the Hamiltonian has the form
and the CD Hamiltonian is given by [11,12]
Note that the CD Hamiltonian is a consequence of the spectrum generating algebra for an oscillator;
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],
where
The time-dependent annihilation and the creation operators are linear invariants [16,17]
where
The magnitude
is the solution to the Ermakov equation (24). The exact number states are [13]
where
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
The first model is a polynomial amplitude and the second model is a hyperbolic amplitude below.
The first model has a polynomial amplitude [20]
where
Note that the amplitude satisfies the boundary condition at the initial time
and another boundary condition at the final time
Note that
Setting ω_{0} = 1, Fig. 1 shows that the amplitude
The second model has a hyperbolic amplitude
This model has the asymptotic amplitude
And
In fact, the amplitude
The constancy of the amplitude
As an exact model, we consider the time-dependent Frequency
The complex solution is known as [21]
Where
Here,
The Hamiltonian is expressed in terms of
Here,
measures the condition for the STA:
both at the initial time
For numeral results, we set ω_{0} = 1. The frequency is shown for ranges [1, 5] of ω_{1} and [−5, 5] of
Indeed Fig. 13 shows that the evolution is adiabatic as expected.
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
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.
This work was supported by National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1I1A3A01063183).