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 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 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 , 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 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. The experimentally determined where Li,α i,α α is the Lindblad operator, α α = , , evolution of the density matrices for the states prepared is the Pauli operator of the ith spin ( x y z)andthe κ as pure double-, single- or zero-quantum coherence constant i,α has the units of inverse time. It has been were obtained via a generalised master equation for- proved that a linear operator on a finite N-dimensional malism. We modelled the inherent noise in the system is the generator of a completely positive by assuming that a correlated phase damping (CPD) dynamical semigroup [47] and hence the Lindbladian acts on both spins and that a GAD is the generator of the semigroup which governs the quantum channel acts independently on each spin. We dissipation of the density operator. obtained good fits of the theoretical model to the exper- In the language of the master equation approach to imental data within reasonable experimental errors. It decoherence, the relaxation of an NMR spin tumbling has been shown that cross-correlated spin relaxation isotropically in a solution can be described by two noise terms arising both from autocorrelation spectral den- channels: a phase damping channel and an amplitude sities for dipolar relaxation as well as from a ‘remote’ damping channel [48–51]. Due to molecular tumbling, CSA–CSA cross-correlation mechanism, contribute dif- the average magnetic field experienced by a spin over ferentially to the relaxation of zero- and double-quantum time is the same but it varies across the sample at a par- coherences in a coupled two-spin system [40]. We ticular time which causes identical spins to slowly lose conjecture that the Redfield description of CSA–CSA phase coherence, which is the process of phase damp- cross-correlated spin relaxation is analogous to the CPD ing. The Kraus superoperator for the phase damping channel in the generalised master equation description (PD) channel acting on a single spin density operator (ρ) can be written as of relaxation. The distinctly different relaxation dynam-   −γ t ics of multiple-quantum coherences in homonuclear PD ρ00 ρ01e E (ρ) = −γ t (4) coupled spin systems as opposed to heteronuclear sys- e ρ10 ρ11 tems is clearly evident from our analysis. The relaxation γ behaviour of homonuclear coupled spin-1/2 nuclei can for the damping rate . The generator of PD channel on a single spin can be written as be explained on the basis of a CPD noise channel acting   on both spins and ties in well with the standard Redfield 0 ρ ZPD(ρ) =−γ 01 . (5) method of studying spin relaxation. ρ10 0 Ordering the matrix elements of ρ in the vector T (ρ00,ρ01,ρ10,ρ11) ,wehave 2. Modelling intrinsic noise in NMR ⎛ ⎞ 00 00 The operator-sum representation is typically used to ⎜ 0 −γ 00⎟ ZPD = ⎝ ⎠ . (6) describe quantum decoherence [41]. A noisy channel 00−γ 0 acting on an input density matrix ρ is given by a com- 00 00 pletely positive trace preserving map The GAD channel for a single spin models the process † where the spin exchanges energy with a reservoir which E(ρ) = i Ei ρE , (1) i is kept at some fixed temperature   where Ei are Kraus operators describing a noisy channel − t/2 † GAD k1ρ00 + k2ρ11 e ρ01 in the sum-operator approach and  E E = 1 ensures E (ρ) = − / , (7) k k k e t 2ρ k ρ + k ρ that the unit trace is preserved. The final noisy state 10 3 00 4 11 − t − t is [42,43] where k2 ≡ (1 −¯n)(1 − e ), k3 ≡¯n(1 − e ), ≡ − ≡ − E(ρ) = ( − μ) ρ † + μ ρ † , k1 1 k3 and k4 1 k2, is the damping rate and 1 i, j Ei, j Ei, j k Ek,k Ek,k (2) n¯ is a temperature parameter where μ is the probability for noise to be correlated and −¯  1 n = E (1 − μ) is the probability for uncorrelated noise. log (8) n¯ kBT 160 Page 4 of 10 Pramana – J. Phys. (2020) 94:160 with E being the energy level difference between the Table1. NMR parameters of homonuclear two-spin systems ground and excited states of the system. used in this study. An ensemble of NMR spins in thermal equilibrium Molecule (ν1,ν2) (Hz) ν (Hz) J12 (Hz) at room temperature has a Boltzmann distribution of spin populations and in this high-temperature limit, the BTC acid (4602.4, 4287.0) 315.4 4.2 generator of the GAD channel is given by Cytosine (4407.7, 3490.8) 916.9 7.1 Coumarin (4734.0, 3807.9) 926.1 9.5 ⎛ ⎞ 1 − 1 2 00 2 ∞ ⎜ 1 ⎟ ZGAD =− ⎜ 0 2 00⎟ . ⎝ 1 ⎠ (9) 3. Results and discussion 002 0 − 1 00 1 2 2 3.1 System details The simplest extension of these single-spin decoher- ence processes to a two-spin system is to consider a In high-field NMR, the Zeeman interaction causes a phase damping and a GAD channel acting indepen- splitting of the energy levels according to the field dently on each spin. However, as the spin systems we direction and the difference between magnetic quan-  = − have studied are homonuclear, with two proton nuclear tum numbers mrs mr ms defines the order of  = species having slightly different Larmor resonance fre- coherence [2]. For two-spin systems, if mrs 0 quencies, we hypothesise that each spin decoheres under the coherence is a zero-quantum (ZQ) coherence, if  =± the concerted action of a phase-damping channel which mrs 1 the coherence is a single-quantum (SQ)  =± is correlated with that of the other spin. Hence, the nat- coherence and if mrs 2 the coherence is a double- urally occurring decoherence for this system can be quantum (DQ) coherence. modelled as a correlated dephasing channel acting on The Hamiltonian of a weakly-coupled two-spin sys- ω both spins and a GAD channel acting independently on tem in a frame rotating at rf in a static magnetic field each spin [39]. The generator of the CPD channel acting B0 is given by on both spins is given by H =−(ω1 − ωrf )I1z − (ω2 − ωrf )I2z + 2π J12 I1z I2z, CPD Z = diag[0, −γ2, −γ1, −(γ1 + γ2 + γ3), (13) −γ , , −(γ + γ − γ ), −γ , 2 0 1 2 3 1 where Iiz is the zth component of the spin angular −γ1, −(γ1 + γ2 − γ3), 0, −γ2, momentum operator, the first two terms in the Hamil- tonian denote the Zeeman interaction between each −(γ1 + γ2 + γ3), −γ1, −γ2, 0] , (10) spin and the static magnetic field B0 and the last term represents the spin–spin interaction with J being the where γ1 and γ2 are decay rates for independent phase ij 1 damping on spins 1 and 2 and γ3 can be interpreted as scalar coupling constant. We used the H spins of the rate of CPD. The full generator to describe two-spin 5-bromo-2-thiophene-carboxylic (BTC) acid, cytosine decoherence has the form and coumarin as model homonuclear two-spin systems (table 1). The molecular structure of these two-spin Z = ZCPD + ZGAD∞ + ZGAD∞ . systems and the NMR spectra of the spins at thermal 1 2 (11) equilibrium are shown in figures 1a–1c, respectively. Under the action of the full decoherence generator Z, The experiments were performed at an ambient temper- the state ρ decoheres to ature of 298 K on a Bruker Avance III 600 MHz NMR spectrometer equipped with a QXI probe. ⎛ ⎞ α1 β1 β2 β3 ⎜ β α β β ⎟ 3.2 State initialisation schemes E2spin(ρ) = ⎝ 1 2 4 5 ⎠ . (12) β2 β4 α3 β6 β3 β5 β6 α4 We initialise our system into a ‘pseudopure’ state, wherein all the energy levels except one, are uniformly The parameters αi ,βi can be written in terms of the populated. Such special quantum states have interesting decay rates γi , i = 1, 2, 3and i , i = 1, 2ofthe properties and have recently found several applications correlated PD channel and the independent GAD chan- in the area of quantum information processing [41]. nels, respectively. In the next section, we shall proceed While standard schemes for pseudopure state prepara- towards the explicit calculation of the superoperator tion involve a large number of experiments and lead E2spin(ρ) for different input states. to reduced signal, recently a few schemes have been Pramana – J. Phys. (2020) 94:160 Page 5 of 10 160

(a)

(b)

(c) Figure 2. Pulse sequence for the preparation of √1 (|00 +|11 ) and √1 (|01 +|10 ) states from ther- 2 2 mal equilibrium. The sequence of pulses before the vertical dashed red line achieves state initialisation into the |00 pseudopure state. Filled and unfilled rectangles represent π π 2 and pulses respectively, while all other rf pulses are labelled with their respective flip angles. The phase of the rf pulse is written below each pulse, with the phase φ kept along x(− x), depending on the desired coherence order; τ12 denotes a delay fixed at 1/2J . Figure 1. NMR spectra obtained after a π/2 read- 12 out pulse on the thermal equilibrium state of (a)5- bromo-2-thiophene-carboxylic (BTC) acid, (b) cytosine and The two-spin systems were initialised into the |00 pseu- (c) coumarin. dopure state using the spatial averaging technique [56], with the density operator given by proposed that use only one ancilla spin and fewer num- 1 −  ber of experiments [52,53]. The relaxation behaviour of ρ00 = I + |00 00|. (15) 4 two-spin pseudopure states have been investigated and it was noted that cross-correlated spin relaxation plays an The pulse sequence for the preparation of |00 from important role in accelerating or retarding the lifetimes the thermal state is shown in the first part of figure 2. of such states [54]. The relaxation of pseudopure states The pulse propagators for selective excitation were con- in an oriented spin-3/2 system has been described using structed using the GRAPE algorithm [9] to design the Redfield theory and reduced spectral densities [55]. amplitude- and phase-modulated rf profiles. Numeri- The two-spin equilibrium density matrix (in the high- cally generated GRAPE pulse profiles were optimised to temperature and high-field approximations) is in a be robust against rf inhomogeneity and had an average highly mixed state given by fidelity of ≥0.995. Selective excitation was typically achieved with pulses of 10 ms duration for BTC acid ρ = 1 ( + ρ ) eq 4 I eq and 1 ms for both coumarin and cytosine molecules. 2 ρeq ∝ Iiz (14) 3.3 Final density matrix reconstruction i=1 We interrogate our final density matrix via a useful tech- with the thermal polarisation  ∼ 10−5, I being a 4 × 4 nique called tomography, which uses a identity operator and ρeq being the deviation part of set of measurements of the expectation values of spin the density matrix. angular momentum operators, to independently quan- We use notation |0 to denote the eigenstate of a spin- tify all the real and imaginary elements of the density 1/2 particle in the ground state (spin-‘up’) and |1 to matrix. One can hence specifically follow the relaxation denote the eigenstate of the excited state (spin-‘down’). rates of different elements of the density matrix [57]. All 160 Page 6 of 10 Pramana – J. Phys. (2020) 94:160 experimental density matrices were reconstructed using a reduced tomographic protocol [58,59], with the set of operations given by {II, IX, IY, XX} being sufficient to determine all 15 variables for the two-spin system. (a) Here I is the identity (do-nothing operation) and X(Y ) denotes a single spin operator that can be implemented by applying a spin-selective π/2 pulse on the corre- sponding spin.

3.4 Measuring state fidelity

The fidelity is an estimate of the ‘closeness’ between (b) two pure states or between two density matrices. The fidelity of an experimental density matrix was computed by measuring the projection between the theoretically expected and experimentally measured states using the Jozsa and Uhlmann fidelity measure [60,61]:   √ √ 2 F = Tr ρ ρ ρ , (16) Figure 3. The real (left) and imaginary (right) parts of the theory expt theory experimentally tomographed density matrix of the BTC acid molecule in the (a) √1 (|01 +|10 ) state, with a fidelity of where ρtheory and ρexpt denote the theoretical and exper- 2 0.98 and (b) √1 (|00 +|11 ) state, with a fidelity of 0.99. The imental density matrices, respectively. In our experi- 2 ments, we use fidelity as a measure to evaluate how rows and columns encode the computational basis in binary well our experimental schemes were able to achieve the order from |00 to |11 . theoretically expected final density matrices.

3.5 Experimental creation of multiple-quantum coherences (a) The desired order of multiple-quantum coherence, i.e. zero- or double- was prepared using the latter part (after the red dashed line) of the pulse sequence given in fig- ure 2. A non-selective π/2 pulse was applied along the y-axis, which rotates both spins onto the x-axis, followed by a delay of τ = 1/2J12 (along with refo- cussing pulses being applied at the centre and at the end of the delay). A GRAPE-optimised spin-selective π/2 (b) pulse is applied along −x(x) axis to prepare either the zero-quantum coherence √1 (|01 +|10 ) or the double- 2 quantum coherence √1 (|00 +|11 ).Wewereableto 2 achieve final-state fidelities of ≈0.99 for all the three homonuclear spin systems studied. Parts (a) and (b) of figures 3–5 depict the real Figure 4. The real (left) and imaginary (right) parts of the (left panel) and imaginary (right panel) parts of the experimentally tomographed density matrix of the coumarin experimentally reconstructed density matrices of the molecule in the (a) √1 (|01 +|10 ) state, with a fidelity of 1 2 zero-quantum coherence ( √ (|01 +|10 ) state) and 1 2 0.98 and (b) √ (|00 +|11 ) state, with a fidelity of 0.99. The 2 the double-quantum coherence ( √1 (|00 +|11 ) state), rows and columns encode the computational basis in binary 2 respectively, for the BTC acid, coumarin and cyto- order from |00 to |11 . sine molecules. Computed state fidelities were 0.982 ± 0.011, 0.983 ± 0.017 and 0.983 ± 0.015 for the zero- States with single-quantum coherences, √1 (|00 + quantum coherence and 0.994 ± 0.013, 0.991 ± 0.015 2 . ± . |10 ) or √1 (|00 +|01 ) were prepared by applying a and 0 979 0 016 for the double-quantum coherence. 2 Pramana – J. Phys. (2020) 94:160 Page 7 of 10 160

(a) (b)

(a)

(c) (d) (b)

Figure 5. The real (left) and imaginary (right) parts of the experimentally tomographed density matrix of the cytosine Figure 6. Decay of signal intensity with time of (a)sin- molecule in the (a) √1 (|01 +|10 ) state, with a fidelity of gle-quantum coherence of spin 1, (b) single-quantum coher- 2 ence of spin 2, (c) zero-quantum coherence and (d) dou- 0.98 and (b) √1 (|00 +|11 ) state, with a fidelity of 0.99. The 2 ble-quantum coherence of the BTC acid molecule. rows and columns encode the computational basis in binary order from |00 to |11 . (a) (b)

π/2 selective pulse along the y-axis on the first (sec- ond) spin, respectively, with computed state fidelities of ≈0.99.

3.6 Decay of populations and single-quantum coherences

= / Spin-lattice relaxation rates 1 T1 was measured (c) (d) ◦ − τ − ◦ using the standard 180y 90x inversion recovery pulse sequence. The spin–spin relaxation rate γ = 1/T2 which is the rate at which a single-quantum coherence decays, was experimentally measured by first rotating the magnetisation of the spin into the transverse plane by a π/2 rf pulse followed by a delay and fitting the resulting magnetisation decay. The decay of single-quantum coherences with time is shown in parts (a) and (b) of figures 6–8,forthetwo- spin homonuclear systems of BTC acid, cytosine and Figure 7. Decay of signal intensity with time of (a)sin- coumarin, respectively. The experimentally measured gle-quantum coherence of spin 1, (b) single-quantum coher- ence of spin 2, (c) zero-quantum coherence and (d) dou- values of spin-lattice relaxation rates 1 and 2 in these = / 1H = . ± . ble-quantum coherence of the cytosine molecule. systems were obtained as 1 1 T1 0 264 0 004 −1 = / 2H = . ± . −1 s and 2 1 T1 0 255 0 003 s for the − = / 1H = . ± . 1 1H −1 BTC molecule, 1 1 T1 0 153 0 002 s turned out to be γ = 1/T = 3.741 ± 0.242 s − 1 2 and = T 2H = 0.152 ± 0.014 s 1 for the cytosine 2H −1 2 1 and γ2 = 1/T = 3.048 ± 0.376 s for spin 1 = / 1H = . ± . −1 2 molecule and 1 1 T1 0 210 0 004 s and and spin 2, respectively in the BTC molecule. In the = / 2H = . ± . −1 2 1 T1 0 135 0 002 s for the coumarin cytosine molecule, the single-quantum coherence decay γ = / 1H = . ± . molecule. The single-quantum coherence decay rates rates were obtained as 1 1 T2 1 618 0 080 160 Page 8 of 10 Pramana – J. Phys. (2020) 94:160

(a) (b) Table 2. CPD factor in homonuclear two-spin systems stud- ied, calculated by fitting the experimental data.

−1 Molecule CPD factor γ3 (s ) BTC acid 5.876 ± 1.825 Cytosine 3.393 ± 1.089 Coumarin 8.6735 ± 1.545

1 − ( + (c) (d) α = (1 − e t 1 2 )) 4 4 β1 = β2 = β3 = 0

1 −t(γ +γ −γ + 1 + 1 ) β = (e 1 2 3 2 2 ) 4 2 β5 = β6 = 0. (17) When the initial state is √1 (|00 +|11 ), i.e a double- 2 quantum coherence state, the parameters αi ,βi in eq. (12)aregivenby Figure 8. Decay of signal intensity with time of (a)sin- 1 gle-quantum coherence of spin 1, (b) single-quantum coher- −t( 1+ 2 α1 = (1 + e )) ence of spin 2, (c) zero-quantum coherence and (d) dou- 4 ble-quantum coherence of the coumarin molecule. 1 − ( + α = (1 − e t 1 2 )) 2 4 1 − ( + −1 γ = / 2H = . ± . −1 α = ( − t 1 2 )) s and 2 1 T2 1 891 0 096 s for spin 1 3 1 e and spin 2, respectively. In the coumarin molecule, the 4 1 −t( 1+ 2 single-quantum coherence decay rates were obtained as α4 = (1 + e )) γ = / 1H = . ± . −1 γ = / 2H = 4 1 1 T2 6 813 0 356 s and 2 1 T2 − β = β = 0 6.761 ± 0.286 s 1 for spin 1 and spin 2, respectively. 1 2

1 −t(γ +γ +γ + 1 + 1 ) β = (e 1 2 3 2 2 ) 3 2 3.7 Decay of multiple-quantum coherences β4 = β5 = β6 = 0. (18) The zero (double)-quantum coherences relaxation rates The decay of zero- and double-quantum coherences were experimentally measured by first preparing either with time is shown in parts (c) and (d) of figures 6– the √1 (|01 +|10 ) or the √1 (|00 +|11 ) state from 2 2 8, for the two-spin homonuclear systems of BTC acid, the thermal state, followed by a delay and then rotating cytosine and coumarin, respectively. The experimen- the magnetisation of the first spin by a π/2 rf pulse tally measured values of zero-quantum coherence decay and finally, a measurement of the magnetisation of the rates in these systems were obtained as 0.430 ± 0.062 − − − second spin. The resulting magnetisation decay of the s 1,0.189 ± 0.004 s 1 and 4.247 ± 0.267 s 1 for the second spin was fitted to the noise model to obtain an two-spin systems of BTC acid, cytosine and coumarin, estimate of the multiple-quantum relaxation rates. respectively. The experimentally measured values of When the initial state is √1 (|01 +|10 ), i.e. a double-quantum coherence decay rates in these systems 2 . ± . −1 . ± . zero-quantum coherence state, the parameters α ,β in were obtained as 12 182 1 289 s ,6975 0 465 i i −1 . ± . −1 eq. (12) are given in terms of the decay rates γ of the s and 21 594 0 897 s , for the two-spin systems uncorrelated and correlated PD channels and decay rates of BTC acid, cytosine and coumarin, respectively. The γ of the independent GAD channels by CPD rate 3 obtained by fitting the experimental data to a noise model which incorporates independent and CPD 1 − ( + as well GAD is given in table 2. The plots displayed α = (1 − e t 1 2 )) 1 4 in figures 6–8 show clear evidence of non-exponential 1 − ( + behaviour, with systematic variations above and below α = (1 + e t 1 2 )) 2 4 the best-fit exponential. This implies that the Markovian 1 − ( + model of noise we assumed may not fully capture the α = (1 + e t 1 2 )) 3 4 noise processes active in these systems. Pramana – J. Phys. (2020) 94:160 Page 9 of 10 160

For systems of heteronuclear coupled spin-1/2 nuclei, Acknowledgements it was previously shown that the intrinsic NMR noise acting on the spins can be modelled completely by All experiments were performed on a Bruker Avance-III considering uncorrelated PD channels acting indepen- 600 MHz FT-NMR spectrometer at the NMR Research dently on both spins [39]. Our results indicate that this Facility at IISER Mohali. does not hold true for homonuclear systems of coupled spin-1/2 nuclei, where the spins are physically prox- imate and have identical gyromagnetic ratios. In such cases, the true picture of noise that emerges is a ‘corre- References lated’ one, wherein a new PD channel acts on both spins together, in addition to the independent channels acting [1] A Streltsov, G Adesso and M B Plenio, Rev. Mod. Phys. on each spin separately. Furthermore, this CPD channel 89, 041003 (2017) contributes differentially to the relaxation rates of the [2] R R Ernst, G Bodenhausen and A Wokaun, Principles multiple-quantum coherences inherent in the system. of NMR in one and two dimensions (Clarendon Press, This noise model hence provides a plausible explanation 1990) [3] J Tang and A Pines, J. Chem. Phys. 72, 3290 (1980) as to why the double-quantum coherences in homonu- [4] M Schlosshauer, Rev. Mod. Phys. 76, 1267 (2005) clear spin systems decay much faster than the zero- [5] J Cavanagh, W Fairbrother, A Palmer and N Skelton, quantum coherences. On the other hand, heteronuclear Protein NMR spectroscopy: Principles and practice spin systems do not exhibit such effects, indicating that (Elsevier Science, 1995) such systems do not have appreciable correlated phase [6] G Pileio, Prog. Nucl. Magn. Reson. Spect. 98–99,1 noise. (2017) [7] D Khurana and T Mahesh, J. Mag. Res. 284, 8 (2017) [8] D P Pires, I A Silva, E R deAzevedo, D O Soares-Pinto and J G Filgueiras, Phys. Rev. A 98, 032101 (2018) 4. Conclusion [9] Z Tosner, T Vosegaard, C Kehlet, N Khaneja, S J Glaser and N C Nielsen, J. Magn. Reson. 197, 120 (2009) [10] A M Souza, G A Álvarez and D Suter, Philos. Trans. Weused a previously designed model by Childs et al [39] Roy. Soc. A 370, 4748 (2012) for intrinsic NMR noise in homonuclear two-spin sys- [11] H Singh, Arvind and K Dorai, Phys. Rev. A 90, 052329 tems arising from a CPD channel acting on both the (2014) spins and a GAD channel acting independently on each [12] S Pal, S Moitra, V S Anjusha, A Kumar and T S Mahesh, spin. Our results suggest that the major contribution to Pramana – J. Phys. 92: 26 (2019) spin relaxation in coupled homonuclear two-spin sys- [13] D Das, R Sengupta and Arvind, Pramana – J. Phys. 88: tems comes from CPD noise. The theoretical model used 82 (2017) to describe multiple quantum relaxation in homonuclear [14] K Hashimoto, K Murata and R Yoshii, J. High Energy two-spin systems is in good agreement with our exper- Phys. 138, 1 (2017) imental data. We conjecture that the CPD behaviour [15] M Niknam, L F Santos and D G Cory, Phys. Rev. Res. 2, 013200 (2020) exhibited by the multiple quantum coherences has its [16] J Li, R Fan, H Wang, B Ye, B Zeng, H Zhai, X Peng and origins in contributions from dipolar autocorrelated JDu,Phys. Rev. X 7, 031011 (2017) relaxation as well as from ‘remote’ cross-correlated [17] D Khurana, V R Krithika and T S Mahesh, arXiv interference terms between two different CSA relax- e-prints, arXiv:1906.02692 (2019), arXiv:1906.02692 ation mechanism that are present in such systems. Our [quant-ph] results have potential applications in NMR relaxation [18] N I Gershenzon, K Kobzar, B Luy, S J Glaser and T E dispersion experiments in large proteins and recent Skinner, J. Magn. Reson. 188, 330 (2007) quantum information processing studies which utilise [19] N Khaneja, T Reiss, B Luy and S J Glaser, J. Magn. out-of-time-order (OTOC) correlators, where multiple- Reson. 162, 311 (2003) quantum coherences play a key role. The more general [20] N Khaneja and S Glaser, in 42nd IEEE Interna- theories of master equations which are used to describe tional Conference on Decision and Control (IEEE Cat. decoherence processes in open quantum systems can No.03CH37475), Vol. 1 (2003) pp. 422–427 [21] D Stefanatos, N Khaneja and S J Glaser, Phys. Rev. A provide deeper insights into the mechanisms which gov- 69, 022319 (2004) ern the relaxation of NMR multispin relaxation. The [22] N Khaneja, B Luy and S Glaser, Proc. Natl. Acad. Sci. validity of the CPD noise model is by no means limited USA 100, 13162 (2003) to the two-spin case, but can be extended to higher-order [23] G Long, G Feng and P Sprenger, Quantum Engineering multiple quanta as well as larger networks of magneti- 1, e27 (2019) cally equivalent homonuclear spins. [24] K Li, Quantum Engineering 2, e28 (2019) 160 Page 10 of 10 Pramana – J. Phys. (2020) 94:160

[25] I Chakraborty, A Chakrabarti and R Bhattacharyya, [45] S Daffer, K Wóodkiewicz, J D Cresser and J K McIver, Phys. Chem. Chem. Phys. 17, 32384 (2015) Phys. Rev. A 70, 010304 (2004) [26] D Khurana, G Unnikrishnan and T S Mahesh, Phys. Rev. [46] ARivas,ADKPlato,SFHuelgaandMBPlenio,New A 94, 062334 (2016) J. Phys. 12, 113032 (2010) [27] A Redfield, Advances in magnetic and optical resonance [47] V Gorini, A Kossakowski and E C G Sudarshan, J. Math. edited by J S Waugh (Academic Press, 1965) Vol. 1, pp. Phys. 17, 821 (1976) 1–32 [48] F M Paula, I A Silva, J D Montealegre, A M Souza, E [28] A Kumar, R C R Grace and P Madhu, Prog. Nucl. Magn. R de Azevedo, R S Sarthour, A Saguia, I S Oliveira, D Reson. Spectr. 37, 191 (2000) O Soares-Pinto, G Adesso and M S Sarandy, Phys. Rev. [29] D Canet (ed.), Cross-relaxation and cross-correlation Lett. 111, 250401 (2013) parameters in NMR, New Developments in NMR (The [49] R Auccaise, L C Celeri,´ D O Soares-Pinto, E R Royal Society of Chemistry, 2018) pp. P001–324 deAzevedo, J Maziero, A M Souza, T J Bonagamba, [30] J Jeener, A Vlassenbroek and P Broekaert, J. Chem. R S Sarthour, I S Oliveira and R M Serra, Phys. Rev. Phys. 103, 1309 (1995) Lett. 107, 140403 (2011) [31] M Goldman, J. Magn. Reson. 149, 160 (2001) [50] H Singh, Arvind and K Dorai, Europhys. Lett. 118, [32] D Abergel and A G Palmer, J. Phys. Chem. B 109, 4837 50001 (2017) (2005) [51] I A Silva, A M Souza, T R Bromley, M Cianciaruso, R [33] G Lindblad, Commun. Math. Phys. 48, 119 (1976) Marx, R S Sarthour, I S Oliveira, R L Franco, S J Glaser, [34] D O Soares-Pinto, M H Y Moussa, J Maziero, E R E R de Azevedo, D O Soares-Pinto and G Adesso, Phys. deAzevedo, T J Bonagamba, R M Serra and L C Celeri,´ Rev. Lett. 117, 160402 (2016) Phys. Rev. A 83, 062336 (2011) [52] T Xin, L Hao, S-Y Hou, G-R Feng and G-L [35] N Boulant, T F Havel, M A Pravia and D G Cory, Phys. Long, Sci. China-Phys. Mech. Astron. 62, 960312 Rev. A 67, 042322 (2003) (2019) [36] T Xin, S-J Wei, J S Pedernales, E Solano and G-L Long, [53] J W Wen, X C Qiu, X Y Kong, X Y Chen, F Yang and Phys. Rev. A 96, 062303 (2017) G L Long, Sci. China-Phys. Mech. Astron. 63, 230321 [37] T Yuwen, A Sekhar, A J Baldwin, P Vallurupalli and L (2020) EKay,Angew. Chem. Int. Ed. 57, 16777 (2018) [54] A Ghosh and A Kumar, J. Magn. Reson. 173, 125 [38] Y Toyama, M Osawa, M Yokogawa and I Shimada, J. (2005) Am. Chem. Soc. 138, 2302 (2016), pMID: 26855064 [55] R Auccaise, J Teles, R Sarthour, T Bonagamba, I [39] A M Childs, I L Chuang and D W Leung, Phys. Rev. A Oliveira and E deAzevedo, J. Magn. Reson. 192,17 64, 012314 (2001) (2008) [40] P Kumar and A Kumar, J. Magn. Reson. A 119, 29 (1996) [56] D Cory, M Price and T Havel, Physica D 120,82 [41] M A Nielsen and I L Chuang, Quantum computation (1998) and quantum information (Cambridge University Press, [57] G L Long, H Y Yan and Y Sun, J. Opt. B: Quantum Cambridge, UK, 2000) Semiclass. Opt. 3, 376 (2001) [42] C Macchiavello and M F Sacchi, Phys. Rev. A 94, [58] G M Leskowitz and L J Mueller, Phys. Rev. A 69, 052302 052333 (2016) (2004) [43] C Addis, G Karpat, C Macchiavello and S Maniscalco, [59] H Singh, Arvind and K Dorai, Phys. Lett. A 380, 3051 Phys. Rev. A 94, 032121 (2016) (2016) [44] E Jung, M-R Hwang, Y H Ju, M-S Kim, S-K Yoo, H [60] R Jozsa, J. Mod. Opt. 41, 2315 (1994) Kim, D Park, J-W Son, S Tamaryan and S-K Cha, Phys. [61] A Uhlmann, Rep. Math. Phys. 9, 273 Rev. A 78, 012312 (2008) (1976)