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Cost of transitionless driving and work output

Yuanjian Zheng, Steve Campbell, Gabriele De Chiara, Dario Poletti

Published Sep 7, 2015• 2 citations
OpenAlex

Abstract

Unitary processes allow for the transfer of work to and from Hamiltonian systems. However, to achieve non-zero power for the practical extraction of work, these processes must be performed within a finite-time, which inevitably induces excitations in the system. We show that depending on the time-scale of the process and the physical realization of the external driving employed, the use of counterdiabatic quantum driving to extract more work is not always effective. We also show that by virtue of the two-time energy measurement definition of quantum work, the cost of counterdiabatic driving can be significantly reduced by selecting a restricted form of the driving Hamiltonian that depends on the outcome of the first energy measurement. Lastly, we introduce a measure, the exigency, that quantifies the need for an external driving to preserve quantum adiabaticity which does not require knowledge of the explicit form of the counterdiabatic drivings, and can thus always be computed. We apply our analysis to systems ranging from a two-level Landau-Zener problem to many-body problems, namely the quantum Ising and Lipkin-Meshkov-Glick models.

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Unitary processes allow for the transfer of work to and from Hamiltonian systems. However, to achieve nonzero power for the practical extraction of work, these processes must be performed within a finite time, which inevitably induces excitations in the system. We show that depending on the time scale of the process and the physical realization of the external driving employed, the use of counterdiabatic quantum driving to extract more work is not always effective. We also show that by virtue of the two-time energy measurement definition of quantum work, the cost of counterdiabatic driving can be significantly reduced by selecting a restricted form of the driving Hamiltonian that depends on the outcome of the first energy measurement. Lastly, we introduce a measure, the exigency, that quantifies the need for an external driving to preserve quantum adiabaticity which does not require knowledge of the explicit form of the counterdiabatic drivings, and can thus always be computed. We apply our analysis to systems ranging from a two-level Landau-Zener problem to many-body problems, namely, the quantum Ising and Lipkin-Meshkov-Glick models.

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Unitary processes allow for the transfer of work to and from Hamiltonian systems. However, to achieve non-zero power for the practical extraction of work, these processes must be performed within a finite-time, which inevitably induces excitations in the system. We show that depending on the time-scale of the process and the physical realization of the external driving employed, the use of counterdiabatic quantum driving to extract more work is not always effective. We also show that by virtue of the two-time energy measurement definition of quantum work, the cost of counterdiabatic driving can be significantly reduced by selecting a restricted form of the driving Hamiltonian that depends on the outcome of the first energy measurement. Lastly, we introduce a measure, the exigency, that quantifies the need for an external driving to preserve quantum adiabaticity which does not require knowledge of the explicit form of the counterdiabatic drivings, and can thus always be computed. We apply our analysis to systems ranging from a two-level Landau-Zener problem to many-body problems, namely the quantum Ising and Lipkin-Meshkov-Glick models.
Unitary processes allow for the transfer of work to and from Hamiltonian systems. However, to achieve nonzero power for the practical extraction of work, these processes must be performed within a finite time, which inevitably induces excitations in the system. We show that depending on the time scale of the process and the physical realization of the external driving employed, the use of counterdiabatic quantum driving to extract more work is not always effective. We also show that by virtue of the two-time energy measurement definition of quantum work, the cost of counterdiabatic driving can be significantly reduced by selecting a restricted form of the driving Hamiltonian that depends on the outcome of the first energy measurement. Lastly, we introduce a measure, the exigency, that quantifies the need for an external driving to preserve quantum adiabaticity which does not require knowledge of the explicit form of the counterdiabatic drivings, and can thus always be computed. We apply our analysis to systems ranging from a two-level Landau-Zener problem to many-body problems, namely, the quantum Ising and Lipkin-Meshkov-Glick models.
Many years have passed since the conception of the quintessential method of shortcut to adiabaticity known as counterdiabatic driving (or transitionless quantum driving). Yet, this method appears to be energetically cost-free and thus continually challenges the task of quantifying the amount of energy it demands to be accomplished. This paper proposes that the energy cost of controlling a closed quantum system using the counterdiabatic method can also be assessed using the instantaneous excess work during the process and related quantities, as the time-averaged excess work. Starting from the Mandelstam-Tamm bound for driven dynamics, we have shown that the speed-up of counterdiabatic driving is linked with the spreading of energy between the eigenstates of the total Hamiltonian, which is necessarily accompanied by transitions between these eigenstates. Nonetheless, although excess work can be used to quantify energetically these transitions, it is well known that the excess work is zero throughout the entire process under counterdiabatic driving. To recover the excess work as an energetic cost quantifier for counterdiabatic driving, we will propose a different interpretation of the parameters of the counterdiabatic Hamiltonian, leading to an excess work different from zero. We have illustrated our findings with the Landau-Zener model.
We apply the method of transitionless quantum driving for time-dependent quantum systems to spin systems. For a given Hamiltonian, the driving Hamiltonian is constructed so that the adiabatic states of the original system obey the Schrödinger equation. For several typical systems such as the XY spin chain and the Lipkin-Meshkov-Glick model, the driving Hamiltonian is constructed explicitly. We discuss possible interesting situations when the driving Hamiltonian becomes time independent and when the driving Hamiltonian is equivalent to the original one. For many-body systems, a crucial problem occurs at the quantum phase transition point where the energy gap between the ground and first excited states becomes zero. We discuss how the defect can be circumvented in the present method.
Transitionless quantum driving, also known as counterdiabatic driving, is a unique shortcut technique to adiabaticity, enabling a fast-forward evolution to the same target quantum states as those in the adiabatic case. However, as nothing is free, the fast evolution is obtained at the cost of stronger driving fields. Here, given the system initially gets prepared in equilibrium states, we construct relations between the dynamical evolution speed and the cost rate of transitionless quantum driving in two scenarios: one that preserves the transitionless evolution for a single energy eigenstate (individual driving), and the other that maintains all energy eigenstates evolving transitionlessly (collective driving). Remarkably, we find that individual driving may cost as much as collective driving, in contrast to the common belief that individual driving is more economical than collective driving in multilevel systems. We then present a potentially practical proposal to demonstrate the above phenomena in a three-level Landau-Zener model using the electronic spin system of a single nitrogen-vacancy center in diamond.
Abstract We quantitatively assess the energetic cost of several well-known control protocols that achieve a finite time adiabatic dynamics, namely counterdiabatic and local counterdiabatic driving, optimal control, and inverse engineering. By employing a cost measure based on the norm of the total driving Hamiltonian, we show that a hierarchy of costs emerges that is dependent on the protocol duration. As case studies we explore the Landau–Zener model, the quantum harmonic oscillator, and the Jaynes–Cummings model and establish that qualitatively similar results hold in all cases. For the analytically tractable Landau–Zener case, we further relate the effectiveness of a control protocol with the spectral features of the new driving Hamiltonians and show that in the case of counterdiabatic driving, it is possible to further minimize the cost by optimizing the ramp.
A system undergoes adiabatic evolution when its population in the instantaneous eigenbasis of its time-dependent Hamiltonian changes only negligibly. The realization of such dynamics requires slow-enough changes of the parameters of the Hamiltonian, a task that can be hard to achieve near quantum critical points. A powerful alternative is provided by the counterdiabatic modification of the Hamiltonian allowing for an arbitrarily quick implementation of the adiabatic dynamics. Such a counterdiabatic driving protocol has been recently proposed for the quantum Ising model (del Campo et al 2012 Phys. Rev. Lett. 109 115703 ). We derive an exact closed-form expression for all the coefficients of the counterdiabatic Ising Hamiltonian. We discuss two approximations to the exact counterdiabatic Ising Hamiltonian quantifying their efficiency of the dynamical preparation of the desired ground state. In particular, these studies show how quantum criticality enhances finite-size effects in counterdiabatic dynamics.