Generalised Counterdiabatic Driving in Open Systems

This is a Perspective on "Shortcuts to Adiabaticity in Driven Open Quantum Systems: Balanced Gain and Loss and Non-Markovian Evolution" by Sahar Alipour, Aurelia Chenu, Ali T. Rezakhani, and Adolfo del Campo, published in Quantum 4, 336 (2020).

By Anthony Kiely (School of Physics, University College Dublin, Belfield 4, Ireland).

All real quantum systems are embedded in an external bath which may lead to a loss of coherence over time. However, the coherence of the state is critical to exploit many quantum effects needed for future quantum technologies. Shortcuts to adiabaticity (STA) [1] describes a variety of control methods, typically analytical, which are all intended to achieve the same goal. This goal is to achieve the same outcome as quantum adiabatic processes in significantly shorter times, i.e., much faster than the typical decoherence time.

Previous work on STA has focussed on combating detrimental noise effects using the multiplicity of control protocols offered by STA [2,3], dissipationless solutions [4] and minimal dephasing trajectories [5]. However, in S. Alipour et al. [6], the authors generalise a specific STA-technique known as counterdiabatic driving or transitionless tracking [7], where a combined control of the Hamiltonian and dissipator are relied on to control the state of the system. The original counterdiabatic driving method uses the addition of an extra driving Hamiltonian which guarantees exact adiabatic behaviour for short time scales. The exact form of this Hamiltonian is known analytically but requires knowledge of the spectrum of the original Hamiltonian and may contain non-local terms which are difficult to implement in practice. Using a different approach (specifically a formulation of adiabaticity in open systems), counterdiabatic driving was previously generalised in [8].

In [6], an inverse engineering strategy is applied by first starting with a general ansatz for the density matrix and then constructing the corresponding master equation from it. There are two parts to the resulting master equation which can be naturally associated with the coherent and decoherent evolution. The coherent part has a unique interpretation as the implementation of a particular system Hamiltonian. The authors of [6] then show how the part pertaining to the decoherent evolution can be interpreted either as an additional non-Hermitian Hamiltonian or as a Lindblad master equation. This incoherent control allows for a change of the von Neumann entropy. These results are illustrated for fast thermalisation of a two-level atom and for a harmonic oscillator but are model-independent and will hence be of interest for researchers in many fields.

In subsequent work, it has been discussed how the use of stochastic parametric driving or continuous quantum measurements [9] can be used to modulate the dephasing rate in order to implement this approach. Similar ideas have been applied to engineer dissipators for fast preparation of squeezed thermal states [10]. This theoretical framework, later termed “trajectory-based shortcut to adiabaticity”, has also been exploited to separate the internal energy change of the system into a contribution of pure entropy change (heat) and another part with no entropy change (work) [11]. This new method complements previous efforts on fast thermalisation using Floquet-engineering [12], sped up isothermal processes using smoothly changing system-bath interaction [13] and reverse engineering of a non adiabatic master equation [14,15].

The original counterdiabatic driving Hamiltonian first appeared as the term neglected in the adiabatic approximation. After being proposed as a method to implement nonadiabatic unitary dynamics, it was applied to many different quantum systems such as finite-level systems [16,17,18], the harmonic oscillator [19], and the Ising spin chain [20]. The formalism has been extended to different interaction pictures [21], reformulated as a minimisation problem [22], and used in combination with Floquet-engineering [23]. The energetic cost required has also been connected to the minimal allowed evolution time (i.e., the quantum speed limit) [24]. Considering how fruitful counterdiabatic driving has proven to be in closed systems, this new extension will surely have a similarly significant impact.

► BibTeX data

► References

[1] D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J.G. Muga, Shortcuts to adiabaticity: Concepts, methods, and applications, Rev. Mod. Phys 91, 045001 (2019). https:/​/​​10.1103/​RevModPhys.91.045001.

[2] A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, Optimally robust shortcuts to population inversion in two-level quantum systems, New J. Phys. 14, 093040 (2012). https:/​/​​10.1088/​1367-2630/​14/​9/​093040.

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[6] S. Alipour, A Chenu, A. T. Rezakhani, and A. del Campo, Shortcuts to Adiabaticity in Driven Open Quantum Systems: Balanced Gain and Loss and Non-Markovian Evolution, Quantum 4, 336 (2020) https:/​/​​10.22331/​q-2020-09-28-336.

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[10] L. Dupays and A. Chenu, Dynamical engineering of squeezed thermal states, arXiv:2008.03307 https:/​/​​abs/​2008.03307.

[11] S. Alipour, A. T. Rezakhani, A. Chenu, A. del Campo, and T. Ala-Nissila, Unambiguous formulation for heat and work in arbitrary quantum evolution, arXiv:1912.01939 https:/​/​​abs/​1912.01939.

[12] T. Villazon, A. Polkovnikov, and A. Chandran, Swift heat transfer by fast-forward driving in open quantum systems, Phys. Rev. A 100, 012126 (2019). https:/​/​​10.1103/​PhysRevA.100.012126.

[13] N. Pancotti, M. Scandi, M. T. Mitchison, and M. Perarnau-Llobet, Speed-ups to isothermality: Enhanced quantum thermal machines through control of the system-bath coupling, Phys. Rev. X 10, 031015 (2020). https:/​/​​10.1103/​PhysRevX.10.031015.

[14] R. Dann, A. Tobalina, and R. Kosloff, Shortcut to Equilibration of an Open Quantum System, Phys. Rev. Lett. 122, 250402 (2019). https:/​/​​10.1103/​PhysRevLett.122.250402.

[15] R. Dann, A. Tobalina, and R. Kosloff, Fast route to equilibration, Phys. Rev. A 101, 052102 (2020). https:/​/​​10.1103/​PhysRevA.101.052102.

[16] X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, Shortcut to Adiabatic Passage in Two- and Three-Level Atoms, Phys. Rev. Lett. 105, 123003 (2010). https:/​/​​10.1103/​PhysRevLett.105.123003.

[17] A. Benseny, A. Kiely, Y. Zhang, T. Busch, and A. Ruschhaupt, Spatial non-adiabatic passage using geometric phases, EPJ Quantum Technol. 4, 3 (2017). https:/​/​​10.1140/​epjqt/​s40507-017-0056-x.

[18] N. V. Vitanov, High-fidelity multistate stimulated raman adiabatic passage assisted by shortcut fields, Phys. Rev. A 102, 023515 (2020). https:/​/​​10.1103/​PhysRevA.102.023515.

[19] J. G. Muga, X. Chen, S. Ibáñez, I. Lizuain, and A. Ruschhaupt, Transitionless quantum drivings for the harmonic oscillator, J. Phys. B 43, 085509 (2010). https:/​/​​10.1088/​0953-4075/​43/​8/​085509.

[20] A. del Campo, M.M. Rams, and W.H. Zurek, Assisted Finite-Rate Adiabatic Passage Across a Quantum Critical Point: Exact Solution for the Quantum Ising Model, Phys. Rev. Lett. 109, 115703 (2012). https:/​/​​10.1103/​PhysRevLett.109.115703.

[21] S. Ibáñez, X. Chen, E. Torrontegui, J.G. Muga, and A. Ruschhaupt, Multiple Schrödinger Pictures and Dynamics in Shortcuts to Adiabaticity, Phys. Rev. Lett. 109, 100403 (2012). https:/​/​​10.1103/​PhysRevLett.109.100403.

[22] D. Sels and A. Polkovnikov, Minimizing irreversible losses in quantum systems by local counterdiabatic driving, Proc. Natl. Acad. Sci. U.S.A. 114, E3909 (2017). https:/​/​​10.1073/​pnas.1619826114.

[23] P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Floquet-Engineering Counterdiabatic Protocols in Quantum Many-Body Systems, Phys. Rev. Lett. 123, 090602 (2019). https:/​/​​10.1103/​PhysRevLett.123.090602.

[24] S. Campbell and S. Deffner, Trade-Off Between Speed and Cost in Shortcuts to Adiabaticity, Phys. Rev. Lett. 118, 100601 (2017). https:/​/​​10.1103/​PhysRevLett.118.100601.

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