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arXiv:1112.0446 · Physical Review A

Frequency-modulated pulses for quantum bits coupled to time-dependent baths

Benedikt Fauseweh, Stefano Pasini, Götz Silvester Uhrig

Published Feb 7, 2012• 13 citations
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Abstract

We consider the coherent control of a quantum bit by the use of short pulses with finite duration ${\ensuremath{\tau}}_{\mathrm{p}}$. By shaping the pulse, we perturbatively decouple the dynamics of the bath from the dynamics of the quantum bit during the pulse. Such shaped pulses provide single quantum bit gates robust against decoherence which are useful for quantum-information processing. We extend previous results in two ways: (i) we treat frequency-modulated pulses and (ii) we pass from time-independent baths to analytically time-dependent baths. First- and second-order solutions for $\ensuremath{\pi}$ and $\ensuremath{\pi}/2$ pulses are presented. They are useful in experiments where amplitude modulation is difficult to realize.

DOI
10.1103/physreva.85.022310

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We consider the coherent control of a quantum bit by the use of short pulses with finite duration \ensuremathτp{\ensuremath{\tau}}_{\mathrm{p}}\ensuremathτp​. By shaping the pulse, we perturbatively decouple the dynamics of the bath from the dynamics of the quantum bit during the pulse. Such shaped pulses provide single quantum bit gates robust against decoherence which are useful for quantum-information processing. We extend previous results in two ways: (i) we treat frequency-modulated pulses and (ii) we pass from time-independent baths to analytically time-dependent baths. First- and second-order solutions for \ensuremathπ\ensuremath{\pi}\ensuremathπ and \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/2 pulses are presented. They are useful in experiments where amplitude modulation is difficult to realize.
Reliable long-time storage of arbitrary quantum states is a key element of quantum information processing. In order to dynamically decouple a spin or quantum bit from a dephasing environment by non-instantaneous pulses, we introduce an optimized sequence of N control π pulses that are realistic in the sense that they have a finite duration and a finite amplitude. We show that optimized dynamical decoupling is still applicable and that higher-order decoupling can be reached if shaped pulses are implemented. The sequence suppresses decoherence up to the order , with T being the total duration of the sequence and τ mx the maximum length of the pulses. The exponent depends on the shape of the pulse. Based on existing experiments, a concrete setup for the verification of the properties of the advocated sequence is proposed.
In the framework of nuclear magnetic resonance, we consider the general problem of the coherent control of a spin coupled to a bath by means of composite or continuous pulses of duration ${\ensuremath{\tau}}_{\text{p}}$. We show explicitly that it is possible to design the pulse in order to achieve a decoupling of the spin from the bath up to the third order in ${\ensuremath{\tau}}_{\text{p}}$. The evolution of the system is separated in the evolution of the spin under the action of the pulse and of the bath times correction terms. We derive the correction terms for a general time dependent axis of rotation and for a general coupling between the spin and the environment. The resulting corrections can be made to vanish by an appropriate design of the pulse. For $\ensuremath{\pi}$ and $\ensuremath{\pi}/2$ pulses, we demonstrate explicitly that pulses exist which annihilate the first- and the second-order corrections even if the bath is fully quantum mechanical, i.e., it displays internal dynamics. Such pulses will also be useful for quantum information processing.
The coherent control of a small quantum system is considered. For a two-level system coupled to an arbitrary bath we consider a pulse of finite duration. We derive the leading and the next-leading order corrections to the evolution operator due to the noncommutation of the pulse and the bath Hamiltonian. The conditions are computed that make the leading corrections vanish. The pulse shapes optimized in this way are given for $\ensuremath{\pi}$ and $\frac{\ensuremath{\pi}}{2}$ pulses.
We consider pulses of finite duration for coherent control in the presence of classical noise. We derive the corrections to ideal, instantaneous pulses for the case of general decoherence (spin-spin relaxation and spin-lattice relaxation) up to and including the third order in the duration τ_p of the pulses. For pure dephasing (spin-spin relaxation only), we design πand π/2 pulses with amplitude and/or frequency modulation which resemble the ideal ones up to and including the second order in τ_p. For completely general decoherence including spin-lattice relaxation the corrections are computed up to and including the second order in τ_p as well. Frequency modulated pulses are determined which resemble the ideal ones. They are used to design a low-amplitude replacement for XY8 cycles. In comparison with pulses designed to compensate quantum noise less conditions have to be fulfilled. Consequently, we find that the classical pulses can be weaker and simpler than the corresponding pulses in the quantum case.
Creating and controlling well-defined single-photon states is important for many quantum enhanced technologies. Information can be encoded in any degree of freedom associated with a single-photon field excitation, for example in the polarization or transverse spatial mode structure. The ability to encode multiple qubits in these states is desirable for high data transmission rates and increased information processing capacity. The spectral-temporal domain offers a large Hilbert space for encoding information, well suited to integrated optical architectures owing to the low cross-talk between channels. Indeed, time-frequency encoding is an integral component of existing information technologies infrastructure. Thus, complete, coherent control of the spectral-temporal mode structure of light is essential to advancing optical quantum technologies. Arbitrary control of single-photon states has been demonstrated in the polarization and spatial-momentum degrees of freedom however, is yet to be established in the spectral-temporal domain. Recently it was shown that spectral-temporal pulse shaping is experimentally feasible using nonlinear optical methods. However, such techniques can introduce noise photons, are challenging to implement deterministically and require specially prepared auxiliary pump pulses. Furthermore, they are typically only possible at relatively low repetition rates to avoid damage of nonlinear optical materials. Here, we demonstrate deterministic pulse shaping of single-photon wave packets by introducing a time-varying phase. Applying linear or quadratic phase to the wave packet introduces a spectral shift or spectral broadening, respectively. Achieving significant spectral manipulation requires temporal-phase modulation on the order of π. The phase must also vary on the same time scale as the optical pulses, approximately 1 ps in duration here. This relatively rapid phase variation is achieved using a fast electro-optic phase modulator. The modulator is driven by a time-varying voltage that is synchronized with a pulse train of single-photon wave packets for accurate temporal phase control. In this thesis, we experimentally demonstrate control of the spectral-temporal state of single-photon wave packets using this method. Heralded single-photon wave packets are generated by spontaneous parametric down conversion pumped by a frequency-doubled Titanium-Sapphire laser oscillator with 80 MHz repetition rate. To achieve significant modulation of single-photon pulses a 10 V peak-to-peak radio-frequency signal of 40 GHz drives the phase modulator, which requires approximately 3.6 V to achieve a π phase shift. The signal is synchronized with the oscillator pulse train using custom designed electronics. Details of these electronics are presented, resulting in a robust phase and amplitude controllable 40 GHz voltage source, phase locked to the single-photon pulse train. To demonstrate the utility of this pulse-shaping method we experimentally show spectral shearing of almost 1 nm for single-photon wave packets with 830 nm central wavelength and 1 nm bandwidth. Measurement of the second-order intensity correlation before and after the wave-packet manipulation remains constant within uncertainty of the measurement, showing that the quantum nature of the single-photon source is not deteriorated by the modulator. Preservation of the wave-packet coherence is verified by two-photon interference between spectrally sheared and non-modulated single-photons.
Abstract Quantum optimal control theory (QOCT) can be used to design the shape of electromagnetic pulses that implement operations on quantum devices. By using non-trivially shaped waveforms, gates can be made significantly faster than those built by concatenating monochromatic pulses. Recently, we applied this idea to the control of molecular spin qudits modeled with Schrödinger’s equation and showed it can speed up operations, helping mitigate the effects of decoherence (Castro et al 2022 Phys. Rev. Appl. 17 064028). However, short gate times require large optimal pulse amplitudes, which may not be experimentally accessible. Introducing bounds to the amplitudes then unavoidably leads to longer operation times, for which decoherence can no longer be neglected. Here, we study how to improve this procedure by applying QOCT on top of Lindblad’s equation, to design control pulses accounting for decoherence already in the optimization process. We define the control signal in terms of generic parameters, which permits the introduction of bounds and constraints. This is convenient, as amplitude and frequency limitations are inherent to waveform generators. The pulses that we obtain consistently enhance operation fidelities compared to those achieved with the optimization based on Schrödinger’s equation, demonstrating the flexibility and robustness of our method. The improvement is larger the shorter the spin coherence time T 2.