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arXiv:1003.1941 · Europhysics Letters (EPL)

Exact time-reversal focusing of acoustic and quantum excitations in open cavities: The perfect inverse filter

Hernán L. Calvo, Horacio M. Pastawski

Published Mar 1, 2010• 9 citations
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Abstract

The time-reversal mirror (TRM) prescribes the reverse playback of a signal to focalize an acoustic excitation as a Loschmidt echo. In the quantum domain, the perfect inverse filter (PIF) processes this signal to ensure an exact reversion provided that the excitation originated outside the cavity delimited by the transducers. We show that PIF takes a simple form when the initial excitation is created inside this cavity. This also applies to the acoustical case, where it corrects the TRM and improves the design of an acoustic bazooka. We solve an open chaotic cavity modeling a quantum bazooka and a simple model for a Helmholtz resonator, showing that the PIF becomes decisive to compensate the group velocities involved in a highly localized excitation and to achieve subwavelength resolution.

DOI
10.1209/0295-5075/89/60002

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The time-reversal mirror (TRM) prescribes the reverse playback of a signal to focalize an acoustic excitation as a Loschmidt echo. In the quantum domain, the perfect inverse filter (PIF) processes this signal to ensure an exact reversion provided that the excitation originated outside the cavity delimited by the transducers. We show that PIF takes a simple form when the initial excitation is created inside this cavity. This also applies to the acoustical case, where it corrects the TRM and improves the design of an acoustic bazooka. We solve an open chaotic cavity modeling a quantum bazooka and a simple model for a Helmholtz resonator, showing that the PIF becomes decisive to compensate the group velocities involved in a highly localized excitation and to achieve subwavelength resolution.
The reversion of the time evolution of a quantum state can be achieved by changing the sign of the Hamiltonian as in the Loschmidt echo and polarization echo experiment in NMR. In this work we describe an alternative mechanism inspired by the acoustic time reversal mirror. By solving the inverse time problem in a discrete space we develop a new procedure, the perfect inverse filter . It achieves the exact time reversion in a given region by reinjecting a prescribed wave function at the region's boundary.
Echoes are ubiquitous phenomena in several branches of physics, ranging from acoustics, optics, condensed matter and cold atoms to geophysics. They are at the base of a number of very useful experimental techniques, such as nuclear magnetic resonance, photon echo and time-reversal mirrors. Particularly interesting physical effects are obtained when the echo studies are performed on complex systems, either classically chaotic, disordered or many-body. Consequently, the term Loschmidt echo has been coined to designate and quantify the revival occurring when an imperfect time-reversal procedure is applied to a complex quantum system, or equivalently to characterize the stability of quantum evolution in the presence of perturbations. Here, we present the articles which discuss the work that has shaped the field in the past few years.
Recent works [Chong et al., Phys. Rev. Lett. 105, 053901 (2010); Wan et al., Science 331, 889 (2011)] have shown that the time-reversed process of lasing at threshold realizes a coherent perfect absorber (CPA). In a CPA, a lossy medium in an optical cavity with a specific degree of dissipation, equal in modulus to the gain of the lasing medium, can perfectly absorb coherent optical waves at discrete frequencies that are the time-reversed counterpart of the lasing modes. Here the concepts of time reversal of lasing and CPA are extended for optical radiation emitted by a laser operated in an arbitrary regime, i.e., for transient, chaotic, or periodic coherent optical fields. We prove that any electromagnetic signal E(t)E(t)E(t) generated by a laser system SSS operated in an arbitrary regime can be perfectly absorbed by a CPA device ${S}^{\ensuremath{'}}$ which is simply realized by placing inside SSS a broadband linear absorber (attenuator) of appropriate transmittance. As examples, we discuss CPA devices that perfectly absorb a chaotic laser signal and a frequency-modulated optical wave.
In this thesis, we address the long-standing question of time-reversal protocols for the quantum mechanical wave function for continuous degrees of freedom. We propose two new mechanisms - quantum time mirrors - one for two-band systems, the other one for systems described by the nonlinear Schrödinger equation like Bose-Einstein condensates. In both cases, a homogeneous pulse is applied to the system that is supposed to flip the velocity direction and thus invert the motion of the system. In a two-band system, the pulse induces a transition from one band to the other. For bands with opposite group velocity directions, e.g. Dirac cone systems like graphene, the band-transition directly leads to an inversion of the motion. By generalizing the results for arbitrary bands and additionally investigating the effects of perturbations like disorder or electro-magnetic fields, we are able to determine for any (effective) two-band system at hand with a given pulse whether it is a suited candidate for the quantum time mirror. In the Bose-Einstein condensates, the inversion of motion is achieved by quickly varying the nonlinear term. The phases in the wave function acquired during the pulse affect the velocity and for certain parameters it switches sign. Although echoes are induced by this mechanism, the fidelity is rather low. However, time lenses, i.e. refocusing the wave packet by a time-dependent pulse, can be achieved with high fidelity in this setup. Even solitary waves can be created whose refocus fidelity stay above 99% for more than 100,000 pulses. In the last part of the thesis, we investigate zitterbewegung in two-band systems with time-dependent potentials. In the driven setup, we find persistent, i.e. undamped, modes of the zitterbewegung that would decay for a wave packet in a static environment. Moreover, using the quantum time mirror pulses described above, echoes of the zitterbewegung can be achieved that are in some sense similar to the well-known and highly exploited spin echo.
We study the possibility to undo the quantum mechanical evolution in a time reversal experiment. The naive expectation, as reflected in the common terminology ("Loschmidt echo"), is that maximum compensation results if the reversed dynamics extends to the same time as the forward evolution. We challenge this belief and demonstrate that the time t(r) for maximum return probability is in general shorter. We find that t(r) depends on lambda=epsilon(evol)/epsilon(prep), being the ratio of the error in setting the parameters (fields) for the time-reversed evolution to the perturbation which is involved in the preparation process. Our results should be observable in spin-echo experiments where the dynamical irreversibility of quantum phases is measured.