Pramana – J. Phys. (2020) 94:160 © Indian Academy of Sciences https://doi.org/10.1007/s12043-020-02027-3 Using a Lindbladian approach to model decoherence in two coupled nuclear spins via correlated phase damping and amplitude damping noise channels HARPREET SINGH1,2,ARVIND1 and KAVITA DORAI1 ,∗ 1Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81 Knowledge City, P.O. Manauli, Mohali 140 306, India 2Fakultät Physik, Technische Universität Dortmund, 44221 Dortmund, Germany ∗Corresponding author. E-mail: [email protected] MS received 18 April 2020; revised 25 July 2020; accepted 18 August 2020 Abstract. In this work, we studied the relaxation dynamics of coherences of different orders present in a system of two coupled nuclear spins. We used a previously designed model for intrinsic noise present in such systems which considers the Lindblad master equation for Markovian relaxation. We experimentally created zero-, single- and double-quantum coherences in several two-spin systems and performed a complete state tomography and computed state fidelity. We experimentally measured the decay of zero- and double-quantum coherences in these systems. The experimental data fitted well to a model that considers the main noise channels to be a correlated phase damping (CPD) channel acting simultaneously on both spins in conjunction with a generalised amplitude damping channel acting independently on both spins. The differential relaxation of multiple-quantum coherences can be ascribed to the action of a CPD channel acting simultaneously on both the spins. Keywords. Nuclear magnetic resonance relaxation theory; Multiple quantum coherences; Lindblad master equation; Markovian noisy channels. PACS Nos 03.65.Yz; 76.60.−k; 03.67.a 1. Introduction The interaction with the environment of a quantum system causes loss of coherence and forces the sys- Quantum coherence is associated with a transition tem to relax back towards a time-invariant equilibrium between the eigenstates of a quantum system and most state. This limits the time over which coherences live spectroscopic signals crucially rely on the manipula- and leads to poor signal sensitivity [4]. In solution state tion, transfer and detection of such coherences [1]. NMR the problem is exaggerated when dealing with In nuclear magnetic resonance (NMR), spin coher- larger spin systems such as those encountered in pro- ence resides in the off-diagonal elements of the density teins, where slower rotational tumbling of the molecules operator of the system and a system of coupled spin- leads to faster rates of relaxation and consequently larger 1/2 nuclei can have coherences of different orders losses in signal [5]. Coherence preservation is hence of n (n = 0, 1, 2,...)[2]. NMR can directly access supreme importance in NMR experiments and several only those off-diagonal elements of the density matrix schemes have been designed to suppress spin relax- whose difference in magnetic quantum number is ±1 ation including using long-lived two-spin order states (the single quantum transitions). The direct observation which have lifetimes much longer than T1,termedsin- of multiple quantum transitions (m =±1) is for- glet states [6,7]. Links between NMR coherence orders bidden by quantum-mechanical selection rules (in the and decoherence have been recently investigated [8]. dipole approximation). Multiple quantum coherences Several NMR techniques have benefited from cross- have found several useful applications in NMR includ- fertilisation of ideas from other fields of research such as ing spectral simplification, spin-locking and cross- quantum information processing. For instance, several polarisation experiments [3]. methods that suppress spin relaxation such as optimal 0123456789().: V,-vol 160 Page 2 of 10 Pramana – J. Phys. (2020) 94:160 control theory [9] and dynamical decoupling [10,11], spins are assumed to be quantum mechanical in char- have all drawn on insights from their initial application acter. The Redfield approach is a ‘bottom-up’ approach to general problems of quantum decoherence, algorithm which begins with the allowed degrees of freedom and implementation [12] and quantum entanglement [13]. the relaxation mechanisms which are specific to the sys- Recently, certain special types of correlation functions tem under consideration and then builds a model from called out-of-time-order correlations (OTOC) have been them. The master equation approach on the other hand, used to characterise the delocalisation of quantum infor- is a ‘top-down’ approach which begins by considering mation and linked to multiple-quantum coherences [14– all possible allowed relaxation processes and then con- 17]. Optimal control techniques have also been used to cludes from the data which are the noise channels that control coupled heteronuclear spin dynamics in the pres- are dominant. The insights gained from the Redfield ence of general relaxation mechanisms and to explore and the master equation methods are complementary in how closely a quantum system can be steered to a tar- character and using a combined approach can help build get state [18–22]. Synthesiser noise can lead to severe a complete picture of coupled spin relaxation. dephasing effects akin to a decohering environment and In the master equation formalism, the NMR longi- new methods have been recently proposed to eliminate tudinal T1 and transverse T2 relaxation processes are such noise using two single-spin systems in opposite described by two different noise channels, namely the static magnetic fields [23,24]. Transverse relaxation amplitude damping and the phase damping channel, times in systems of coupled spins have been accurately respectively [34]. The effect of the phase damping chan- measured and the noise profiles of multispin coherences nel on a single spin is to nullify the coherences stored in and their scaling with respect to coherence order has the off-diagonal elements of the spin density matrix. The been studied [25,26]. In order to devise techniques to generalised amplitude damping (GAD) channel leads obviate the deleterious effects of spin relaxation, one to energy loss through dissipative interactions between first needs to gain a deeper understanding of the mecha- the spin and the lattice at finite temperatures, where nisms underlying this complex phenomenon. Molecules the spin in the excited state decays to its ground state. in a liquid freely tumble and undergo stochastic Brow- The Lindblad operators were delineated for a system of nian motion which is the main source of NMR spin two coupled spin-1/2 nuclei by measuring the density relaxation, where the spin lattice degrees of freedom operator at multiple time points [35]. The phase damp- include all the molecular rotational and translational ing, amplitude damping and depolarising noise channels motions. The semiclassical Redfield approach is typi- have been implemented in NMR using two and three cally used to describe NMR spin relaxation which uses heteronuclear coupled spins [36]. the density matrix formalism and second-order pertur- It has long been known in NMR that the relaxation of bation theory; the noisy spin environment is treated multiple quantum transitions contains useful informa- classically by a spin lattice model while the spins are tion about correlated fluctuations occurring at different treated as quantum mechanical objects and a weak nuclear sites as well as about molecular motions [2]. system-environment coupling is assumed [27]. The bath In contrast to single-quantum experiments on coupled correlations decay much more rapidly than the evolu- spins, the relaxation dispersion profiles of multiple- tion of spins and the Markovian approximation remains quantum relaxation rates are sensitive to the chemical valid. For two coupled spins 1/2, the major relaxation environment of the involved nuclei and can hence mechanisms in NMR are the dipole–dipole (DD) relax- be used to gain insights about millisecond time-scale ation and the relaxation arising from the chemical shift dynamics in large biomolecules [37]. Multiple quan- anisotropy (CSA) of each spin. In general, interference tum relaxation has been used to probe protein–ligand terms between the DD and CSA relaxation mechanisms interactions, conformational exchange processes and can give rise to another mechanism for relaxation called side-chain motions in proteins [38]. cross-correlated spin relaxation [28,29]. Extensions of A spin system consisting of coupled spins of the same Bloch–Redfield relaxation theory have developed a uni- nuclear species is called a homonuclear system while fied picture by including contributions from dipolar cou- a coupled spin system consisting of different nuclear pling between remote spins [30,31] and by considering species is called a heteronuclear system. In this work, a two-state Markov noise process which includes lat- we focus on studying the relaxation dynamics of quan- tice fluctuations and chemical exchange dynamics [32]. tum coherences in homonuclear systems of coupled The most general form for the non-unitary evolution of spin-1/2 nuclei. On the other hand, in a heteronuclear the density operator of an open quantum system can coupled two-spin system, the noise was fitted using be described by a master equation [33]. In the mas- several noise models and it was shown that such sys- ter equation approach, both the environment and the tems can be treated as being acted upon by independent Pramana – J. Phys. (2020) 94:160 Page 3 of 10 160 noise channels [39]. We use the Lindblad master equa- For a special class of noisy channels where the tion for Markovian relaxation to set up and analyse Markovian approximation is valid, one can write the the relaxation of coherences of different order, namely master equation governing decoherence in a Lindblad zero-, single- and double-quantum coherences. We first form [33,44–46] experimentally prepared states with different orders of ∂ρ † 1 † quantum coherences, tomographed the state and com- = Li,αρL − {L Li,α,ρ} , (3) ∂t i,α 2 i,α puted state fidelity. We then allowed the state to decay i,α and experimentally measured the decay rates of different √ ≡ κ σ (i) σ (i) quantum coherences.