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Proceeding Paper

Continuous and Pulsed Quantum Control †

1
Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari, Italy
2
INFN, Sezione di Bari, I-70126 Bari, Italy
3
Department of Physics and Astronomy, Center of Engineered Quantum Systems, Macquarie University, Sydney, NSW 2109, Australia
4
Istituto Nazionale di Ottica (INO-CNR), I-50125 Firenze, Italy
*
Author to whom correspondence should be addressed.
Presented at the 11th Italian Quantum Information Science conference (IQIS2018), Catania, Italy, 17–20 September 2018.
Proceedings 2019, 12(1), 15; https://doi.org/10.3390/proceedings2019012015
Published: 24 June 2019
(This article belongs to the Proceedings of 11th Italian Quantum Information Science conference (IQIS2018))

Abstract

:
We consider two alternative procedures which can be used to control the evolution of a generic finite-dimensional quantum system, one hinging upon a strong continuous coupling with a control potential and the other based on the application of frequently repeated pulses onto the system. Despite the practical and conceptual difference between them, they lead to the same dynamics, characterised by a partitioning of the Hilbert space into sectors among which transitions are inhibited by dynamical superselection rules.

1. Introduction

Coinsider a quantum system with a finite dimensional Hilbert space H , whose evolution U ( t ) = e i t H is generated by the Hamiltonian H. We are interested in some protocols which dynamically induce a partition of H into superselection sectors H μ = P μ H , in the sense that if the system is initially in some state belonging to one of the superselection sectors, i.e: | ψ H μ , it will remain in that sector during its evolution | ψ ( t ) H μ , as shown pictorially in Figure 1. More precisely, given a complete set of orthogonal projections { P μ } satisfying
μ = 1 m P μ = I , P μ P ν = δ μ ν P μ = δ μ ν P μ ,
we want to engineer an effective dynamics generated by the block-diagonal Hamiltonian
H Z = μ = 1 m P μ H P μ .
This evolution is a manifestation of a Quantum Zeno Dynamics (QZD), a generalisation of the quantum Zeno effect [1], consisting in the freezing of the state of a quantum system when it is subject to frequent measurements aimed at ascertaining if it is still in its initial state. In the case of non-selective measurements onto multi-dimensional subspaces H μ = P μ H a non-trivial evolution can take place inside each subspace, generated by the Hamiltonian (2), with P μ being the measurement projections. In this context the superselection sectors H μ are called quantum Zeno subspaces (QZSs) [2].

2. Strong Continuous Coupling

The first protocol consists in adding to the Hamiltonian H a strong coupling to a control potential V, so that the dynamics is generated by a total Hamiltonian H K = H + K V , where K > 0 is the coupling strength. As K grows to infinity, the evolution generated by H K is equivalent to a QZD, with the QZSs determined by the eigenprojections of the control potential V. Such result is expressed formally in Theorem 1, where we also bound the error between the actual evolution of the system and the controlled evolution when K is large but finite.
Theorem 1.
Let H and V be Hermitian operators acting on a finite dimensional space H , with V having the spectral decomposition
V = μ = 1 m λ μ P μ .
Then, defining H Z as in Equation (2), we have
e i t ( H + K V ) = e i t K V e i t H Z + O 1 K ,
as K . (Here and in the following the notation O ( x ) will stand for an operator A ( x ) depending on the real parameter x such that A ( x ) C | x | for x sufficiently small and nonvanishing, and for some positive constant C).
The proof of the theorem makes use of an adiabatic theorem [3,4,5].

3. Pulsed Decoupling

The second protocol consists in the application of periodic pulses to the system, implemented by an instantaneous unitary transformation U kick applied to the evolving state at time intervals t / n , as shown in Figure 2a. The idea at the basis of this procedure—and of the proof of Theorem 2—can be understood by looking at each step as an effective “rotation” of the Hamiltonian (see Figure 2b), so that the global effect over the whole time interval ( 0 , t ) is to average out of the Hamiltonian the off-diagonal part with respect to the eigenprojections of the unitary kick [3,6]. Such result is expressed formally in Theorem 2.
Theorem 2.
Let H be a Hermitian operator on a finite dimensional Hilbert space H , and let U kick be a unitary operator with the spectral decomposition
U kick = μ = 1 m e i λ μ P μ .
Then, by defining H Z as in Equation (2), we have
U kick e i t n H n = U kick n e i t H Z + O 1 n ,
as n .

4. Example: Four-Level System

As a particular example, consider a 4-level system, where H = C 4 , and a Hamiltonian H inducing Rabi transitions between adjacent levels (this scheme is very similar to that implemented in [7]):
H = k = 1 4 Ω k ( | k k + 1 | + | k + 1 k | ) .
Such Hamiltonian will determine a time oscillation of the populations (see Figure 4a) P k ( t ) = | k | e i t H | 1 | 2 . Using Theorem 1, we can show now that it is possible to decouple levels | 1 and | 2 from | 3 and | 4 with a strong coupling between | 3 and | 4 :
K V = K ( | 3 4 | + | 4 3 | ) .
The eigenprojections of this potential are P 0 = | 1 1 | + | 2 2 | and P ± = 1 2 | 3 ± | 4 3 | ± 4 | , so that the Zeno Hamiltonian, H Z = P 0 H P 0 + P + H P + + P H P , is block-diagonal with respect to the QZSs
H 1 = span { | 1 , | 2 } , H + = span { | 3 + | 4 } , H = span { | 3 | 4 } .
The situation is pictorially represented in Figure 3. Figure 4b shows the behaviour of occupation probabilities P k ( t ) in the strong coupling regime: we can see oscillations between states | 1 and | 2 which belong to the same QZS, but the probability of a transition towards the states | 3 and | 4 vanishes since they do not belong to the initial QZS. The same result can be obtained by using instead the protocol considered in Theorem 2, with e.g. the unitary kick
U kick = e i λ ( | 3 4 | + | 4 3 | ) .
In this example we have considered a particular Hamiltonian H generating the evolution of the system to be controlled. Note however that there are no assumptions on the structure of the Hamiltonian in our theorems, which are therefore valid in completely general situations, as long as we consider finite dimensional quantum systems.

Author Contributions

All authors have contributed equally to this paper.

Acknowledgments

This work was partially supported by Istituto Nazionale di Fisica Nucleare (INFN) through the project “QUANTUM”, and by the Italian National Group of Mathematical Physics (GNFM-INdAM).

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Misra, B.; Sudarshan, E.G. The Zeno’s paradox in quantum theory. J. Math. Phys. 1977, 18, 756–763. [Google Scholar] [CrossRef]
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  3. Burgarth, D.; Facchi, P.; Gramegna, G.; Pascazio, S. Generalized Product Formulas and Quantum Control. arXiv 2019, arXiv:1906.04498. [Google Scholar] [CrossRef]
  4. Facchi, P. Quantum Zeno effect, adiabaticity and dynamical superselection rules. In Fundamental Aspects of Quantum Physics, Proceedings of the Quantum Probability and White Noise Analysis XVII, Tokyo, Japan, 27–29 September 2001; World Scientific: Hackensack, NJ, USA, 2001; Volume 197. [Google Scholar] [CrossRef]
  5. Kato, T. On the Adiabatic Theorem of Quantum Mechanics. J. Phys. Soc. Japan 1950, 5, 435–439. [Google Scholar] [CrossRef]
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  7. Schäfer, F.; Herrera, I.; Cherukattil, S.; Lovecchio, C.; Cataliotti, F.S.; Caruso, F.; Smerzi, A. Experimental realization of quantum Zeno dynamics. Nat. Comm. 2014, 5, 3194. [Google Scholar] [CrossRef] [PubMed]
Figure 1. A pictorial representation of the partitioning of the Hilbert space H into QZSs H μ = P μ H . If the system is in a given QZS at the initial time t 0 , it will evolve coherently in this subspace and will never make a transition to the other QZSs.
Figure 1. A pictorial representation of the partitioning of the Hilbert space H into QZSs H μ = P μ H . If the system is in a given QZS at the initial time t 0 , it will evolve coherently in this subspace and will never make a transition to the other QZSs.
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Figure 2. (a) Alternating free evolutions of duration t / n with instantaneous unitary kicks U kick (b) is equivalent to a sequence of infinitesimal evolutions of duration t / n generated by Hamiltonians H = U kick H U kick rotated at each step by the unitary kick.
Figure 2. (a) Alternating free evolutions of duration t / n with instantaneous unitary kicks U kick (b) is equivalent to a sequence of infinitesimal evolutions of duration t / n generated by Hamiltonians H = U kick H U kick rotated at each step by the unitary kick.
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Figure 3. Effect of the strong coupling between states | 3 and | 4 . The other two QZSs have not been highlighted in the figure since they are made of linear combinations of states | 3 and | 4 .
Figure 3. Effect of the strong coupling between states | 3 and | 4 . The other two QZSs have not been highlighted in the figure since they are made of linear combinations of states | 3 and | 4 .
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Figure 4. Populations P k with Ω 1 = Ω 2 = Ω 3 Ω without control potential (a) and with the control potential turned on with K = 100 Ω (b).
Figure 4. Populations P k with Ω 1 = Ω 2 = Ω 3 Ω without control potential (a) and with the control potential turned on with K = 100 Ω (b).
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MDPI and ACS Style

Gramegna, G.; Burgarth, D.; Facchi, P.; Pascazio, S. Continuous and Pulsed Quantum Control. Proceedings 2019, 12, 15. https://doi.org/10.3390/proceedings2019012015

AMA Style

Gramegna G, Burgarth D, Facchi P, Pascazio S. Continuous and Pulsed Quantum Control. Proceedings. 2019; 12(1):15. https://doi.org/10.3390/proceedings2019012015

Chicago/Turabian Style

Gramegna, Giovanni, Daniel Burgarth, Paolo Facchi, and Saverio Pascazio. 2019. "Continuous and Pulsed Quantum Control" Proceedings 12, no. 1: 15. https://doi.org/10.3390/proceedings2019012015

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