Spin coherent states in NMR quadrupolar system: experimental and theoretical applications R. Auccaise,∗ E. R. deAzevedo, T. J. Bonagamba, and M. H. Y. Moussa Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo, P.O. Box 369, S˜ao Carlos 13560-970, S˜ao Paulo, Brazil. 3 1 0 E. I. Duzzioni 2 Instituto de F´ısica, Universidade Federal de Uberlˆandia, n a J P.O. Box 593, Uberlˆandia 38400-902, MG, Brazil. 4 1 Abstract ] h Working with nuclear magnetic resonance (NMR) in quadrupolar spin systems, in this paper we p - transfer the concept of atomic coherent state to the nuclear spin context, where it is referred to as t n a pseudo-nuclear spin coherent state (pseudo-NSCS). Experimentally, we discuss the initialization of u q the pseudo-NSCSs and also their quantum control, implemented by polar and azimuthal rotations. [ Theoretically, wecomputethegeometric phases acquired by aninitial pseudo-NSCSon undergoing 1 v 2 three distinct cyclic evolutions: i) the free evolution of the NMR quadrupolar system and, by 6 8 analogy with the evolution of the NMR quadrupolar system, that of ii) single-mode and iii) two- 2 . mode Bose-Einstein Condensate like system. By means of these analogies, we derive, through spin 1 0 3 angular momentum operators, results equivalent to those presented in the literature for orbital 1 : angular momentum operators. The pseudo-NSCS description is a starting point to introduce the v i X spin squeezed state and quantum metrology into nuclear spin systems of liquid crystal or solid r a matter. ∗ Electronic address: [email protected] 1 I. INTRODUCTION In a seminal work, R. J. Glauber [1] proposed the concept of a coherent state as an eigenstateoftheannihilationoperatoroftheharmonicoscillator. Thisconcept wasextended to atoms as the well-known atomic coherent states (ACS) [2, 3]. These coherent states, used in many physical processes in the fields of quantum electrodynamics [4], trapped ions [5] and Bose-Einstein condensates (BECs) [6], are of fundamental importance because they minimize the Heisenberg uncertainty [2]. The minimization of uncertainty tends to amplify the quantum effects on the macroscopic scale. One physical system that manifests quantum effects at a macroscopic level is the BEC. In this connection, an interesting application of the ACS was developed by Chen et al. [7], who computed geometric phases of a coupled two-mode BEC system for an adiabatic and cyclic time evolution. Other applications are the generation of macroscopic quantum superpositions [8] and dynamic control [9, 10] in BECs. The topic of geometric phase has recently received considerable attention and it has been employed increasingly in quantum information processing [11–14]. The quantum version of geometric phases was first reported and studied by M.V. Berry in the 1980’s [15] and implemented, thereafter, inmanyphysical systems[16–18]. Ageneraldiscussionofgeometric phases acquired by pure states was published by N. Mukunda andR. Simon [19]. In this new approach, theauthorstookadvantageofquantumkinematicconcepts, sothatthisformalism couldbeappliedinageneralcontext, suchasinanysmoothopenpathdefinedbyunitvectors in a Hilbert space. Many applications using this formalism have appeared, for instance in noncyclic geometric quantum computation [20] and, more recently, nonadiabatic geometric quantum gates using composite pulses [13], both of these applications being discussed in the NMR context. Other applications of the Mukunda-Simon approach have been the study of geometric phases in a nonstationary superposition of atomic states induced by an engineered reservoir[21],andalsothestudyoftheeffectofgeometricphasesonthepseudospindynamics of two coupled BECs [22] and of a spin-orbit-coupled BEC [23]. Likewise, NMR experiments in systems with two coupled spins and quadrupolar spin systems have been published, in which the geometric phase of an adiabatic cyclic evolution of an initial quantum state is analyzed [24]. Other potential NMR applications of adiabatic [11, 25] and nonadiabatic [26–28] geometric quantum computation have also been reported. 2 These advances inspired us to study two subjects related to NMR quadrupolar systems. The first is the possibility of transferring the definition of ACS to a nuclear spin state, which will be referred to as a pseudo-nuclear spin coherent state (pseudo-NSCS). The second concerns an application of pseudo-NSCSs to compute geometric phases in three different configurations: i) the free evolution of the NMR quadrupolar system [29] and, by analogy of the evolutions of the NMR quadrupolar system, that of ii) single- and iii) two-mode BEC-like system [30, 31]. Under this cyclic evolution, we apply the Mukunda and Simon formalism and obtained theoretical results analogous to those of Zhang et al. in Ref. [32]. Special attention was given to configurations ii) and iii), in which we took the advantage of the mapping between the quasiparticle description and pseudo-spin momentum operators for any spin value I. In general, the present paper presents experimental results and theoretical applications of the mapping between the atomic and nuclear spin scenarios. This will be a basis for future experimental implementations of quantum control, and also to establish an NMR quadrupolar system as a workbench for a BEC system. This paper is organized as follows: In section II, the main concepts and a review of ACS are presented, while in section III the NMR quadrupolar system is briefly described. Section IV is devoted to the experimental procedure to initialize the pseudo-NSCSs and describes the quantum control of polar and azimuthal rotations. In section V, theoretical applications of the pseudo-NSCS based on the geometric phase concept are elaborated for configurations i), ii) and iii) and in section VI we report our conclusions. II. ATOMIC COHERENT STATES In the theory of ACS, a Bloch vector n = (sinθcosϕ, sinθsinϕ,cosθ) is transformed by the action of a rotationoperator R = e−iθ(J·m). The operator R represents a rotation of θ,ϕ θ,ϕ angle θ around an axis defined by m = (sinϕ, cosϕ,0), where J m = J sinϕ J cosϕ. x y − · − The operator J signifies the total angular momentum, with components J , J , and J , such x y z that j is the total angular momentum quantum number and its projection corresponds to m = j, j +1,...,j 1,j. After a simple algebraic procedure, the application of R to θ,ϕ − − − 3 z Ζ x θ ζ = -e- i ϕ tan θ 2 2 y -y θ n -x x ms y ϕ FIG. 1: (Color online) For a nuclear spin system we define rotation angles θ and ϕ on the Bloch sphere relative to the xyz frame and the corresponding value of ζ in the complex plane denoted by Z, the Greek capital form of ζ. The north pole corresponds to ζ = 0 and the south pole to ζ = . ∞ the fundamental state denoted by j, j produces an excited state represented by [2, 3, 33]: | − i 1 ζ(θ,ϕ) = R j, j = eζJ+ j, j , | i θ,ϕ| − i (1+ζ∗ζ)j | − i j ζj+m (2j)! = j,m , (1) (1+ζ∗ζ)js(j +m)!(j m)! | i m=−j − X where ζ = e−iϕtan θ, with the angles ϕ and θ as shown in Fig. 1, while J = J iJ , − 2 ± x ± y and j,m are eigenstates of the operator J , with eigenvalues m~. The complex phase that z | i characterizes the ACS may be defined in the complex plane Z (Greek capital ζ) or over the Bloch sphere, where the ground state is represented by the south pole and the most excited state is identified with the north pole. Let us consider the density operator of the ACS denoted by ρ = ζ(θ,ϕ) ζ(θ,ϕ) = | ih | j ′ m,m′=−jρm,m′ |j,mihj,m|. From Eq. (1), each element of ρ is given by P ρm,m′ = ( 1)2j+m+m′ei(m′−m)ϕcos2j−m−m′(θ/2)sin2j+m+m′ (θ/2) − (2j)! (2j)! . (2) ×s(j +m′)!(j m′)!s(j +m)!(j m)! − − Hence, if the angular parameters θ and ϕ are known, the amplitude of each element ρm,m′ can be obtained. The inverse procedure, that is, the computation of θ and ϕ from the elements ρm,m′, is also possible. In the case of θ, we use 4 4√j ρj,j = sin θ2, (3) 4√j ρ−j,−j = cos θ2, for any value of j. In the case of ϕ, when j is an integer (ρ +ρ ) (j +1)!(j 1)!(j)! 0,1 1,0 − = cosϕ, (4) 2( 1)2j+1 cops θ 2j−1 sin θ 2j+1(2j)! − 2 2 (ρ ρ ) (j +1)!(j 1)!(j)! 0,1 1(cid:0),0 (cid:1) (cid:0) (cid:1) − − = sinϕ. (5) 2i( 1)2j+1 cops θ 2j−1 sin θ 2j+1(2j)! − 2 2 Similar expressions can be obtain(cid:0)ed in(cid:1)terms(cid:0)of th(cid:1)e density matrix elements ρ and ρ 0,−1 −1,0 (not shown). For j a half integer, these equations become ρ−21,21 +ρ21,−21 j − 12 ! j + 21 ! = cosϕ, (6) (cid:16)2( 1)2j cos θ(cid:17)2(cid:0)j sin(cid:1)θ (cid:0)2j (2j)!(cid:1) − 2 2 ρ−21,21 −(cid:0)ρ12,−21 (cid:1) j(cid:0)− 12 !(cid:1) j + 21 ! = sinϕ. (7) (cid:16)2i( 1)2j cos θ(cid:17)(cid:0)2j sin(cid:1)θ (cid:0)2j(2j)(cid:1)! − 2 2 (cid:0) (cid:1) (cid:0) (cid:1) Therefore a general ACS given by (1) is completely characterized by j, θ, and ϕ. III. NMR QUADRUPOLAR SYSTEMS NMR quadrupolar systems are composed of nuclei with spin I > 1/2 subjected to both a magneticfieldandanelectricfieldgradient, wherethenuclear magneticmoment isquantized as M = I,I 1,..., I. Considering also the interaction with a radio frequency (RF) field − − used for excitation, in a reference frame rotating around z-axis with angular velocity ω , RF the NMR Hamiltonian can be written (for more details see section VII of Ref. [29]): ω = ~(ω ω )I +~ Q 3I2 I2 HNMR − L − RF z 6 z − +~ω (I cosϕ +I sinϕ(cid:0))+ (cid:1), (8) 1 x s y s env H where I , I , I are the x,y,z components of the total spin angular momentum operator. x y z The first term of the Hamiltonian is due to the interaction of the nuclear magnetic moments with the strong magnetic field B (Zeeman term). The second is due to the interaction 0 5 of the quadrupole moments of the nuclei with the internal electric field gradient. (ω ) L and (ω ) stand for the Larmor and Quadrupolar frequencies (coupling strengths), which in Q this case, satisfy the inequality ω ω . The third term represents the external time- Q L | | ≪ | | dependent RF field perturbation of intensity B = ω /γ, applied along the direction of the 1 1 corresponding vector m, with γ being the gyromagnetic ratio. The fourth term represents weak interactions between quadrupolar nuclei and other nuclei, electrons, random fields, referred to here as environment. This term will be neglected as its contribution is weak [34]. A quadrupolar system in the rotating frame will be used to describe two different situa- tions. In the first, the external time-dependent perturbation is present, with ω = ω , L ∼ RF | | | | and ω >> ω specifies a regime where the perturbation equally affects all energy levels 1 Q | | | | of the system, non-selective pulse; the evolution operator is then given by = exp[ iω t(I cosϕ +I sinϕ )]. (9) RF ∼ 1 x s y s U − The physical parameters to be identified are the polar (θ) and azimuthal (ϕ) angles defining the rotation operator R . From the RF operator (t), these angles are θ,ϕ RF U θ ω t, (10) 1 ≡ π ϕ ϕ + . (11) s ≡ 2 Secondly, when the external RF perturbation is absent, the corresponding evolution operator is iω t = exp i(ω ω )tI Q 3I2 I2 , (12) UEv L − RF z − 6 z − (cid:20) (cid:21) (cid:0) (cid:1) which refers to a free evolution regime. Note that the operator cannot be written as Ev U a rotation operator R , since it is not generated by the vectors n and m (see section II). θ,ϕ This approach is similar to that discussed by Kitagawa and Ueda [35] in the study of spin squeezed states. IV. EXPERIMENTAL IMPLEMENTATION OF THE PSEUDO-NSCS The NMR experiments were carried out at room temperature in a VARIAN INOVA 400 MHz spectrometer. The pseudo-NSCS was implemented with the 23Na nuclei (I = 3/2) present in sodium dodecil sulfate (SDS) in a lyotropic liquid crystal, prepared with 21.3 wt 6 % SDS, 3.7 wt % decanol, and 75 wt % D O. The strength of the Larmor frequency and 2 quadrupolar couplings were 105.85 MHz and 15 kHz, respectively. Typical π-pulse lengths of 8 µs and recycle delays of 500 ms were used. The T and T relaxation times of the 23Na 2 1 nuclear spins are 2.6 0.3 ms and 12.2 0.2 ms, respectively. ± ± Room temperature thermal equilibrium NMR systems are almost maximum mixture states, represented by a density matrix which deviates only slightly from the normalized identity matrix (1 β~ω I ) L z ρ − , (13) ≈ Z where β = 1/k T and = Tr e(−βHNMR) is the partition function, T the room temper- B Z ature and kB the Boltzmann co(cid:2)nstant. Usi(cid:3)ng suitable spin rotations – RF pulses and free evolutions – and a temporal average technique (so-called strong modulated pulses - SMP [36]), this thermal state can be transformed into a state of the form 1 ρ ǫ 1+ǫ ψ ψ , (14) ≈ − | ih | (cid:18)Z (cid:19) where ψ ψ is called a pseudo-pure state, having the form of a pure state density matrix | ih | with unitary trace, and ǫ = β~ω / . By the technique of quantum state tomography L Z (QST) [37], we obtain experimentally the quantum state ψ that will be used to represent | i the pseudo-NSCS I,I ζ(0,0) . This fundamental pseudo-pure state represents, from | i ≡ | i the NMR point of view, the precession of the magnetic moment around an axis defined by the direction of the strong static magnetic field B . To obtain other excited pseudo-NSCSs, 0 we just need to apply the operator (Eq. (9)) to I,I . In Fig. 2 we show how to RF U | i implement the fundamental and excited pseudo-NSCSs. A. Initializing pseudo-NSCSs To produce excited pseudo-NSCSs from I,I , we apply non-selective pulses of strength | i ω and phase ϕ during time interval t = θ/ω (see Eq. (9)). As a first example, the excited 1 s 1 pseudo-NSCS ζ(π/2,0) = 1 canbegeneratedbyapplyinganon-selectiveπ/2pulsethat | i |− i produces a rotation about the negative y–axis. On the other hand, a rotation of π/2 about thepositive y–axisofthevector state I,I generates thestate ζ(π/2,π) = 1 . Weobserve | i | i | i that the unitary transformation(ofEq. (9)) associated with non-selective pulses, acting on a 7 QST Azimuthal SMP R(θ ,ϕ ) Pulse Rotation I I I FID x y -x π ϕ π 2 τ 2 τ t FIG. 2: (Color online) The pulse sequence has four steps. In the first step the pseudo-NSCS I,I | i is implemented by the SMP technique. The second step consists of the non-selective pulses with parameters (θ,ϕ) that produce excited pseudo-NSCSs ζ(θ,ϕ) . The third step is the azimuthal | i rotation through time τ, which can be used to produce cyclic evolutions. The last step involves the read-out of the pseudo-NSCS (QST procedure). given pseudo-NSCS, produces excited pseudo-NSCSs analogously to those discussed in Ref. [22]. In Fig. 3, we present experimental results for the deviation density matrix, in the form of bar charts of the real (left) and imaginary (middle) components, for the quantum states ζ(0,ϕ) = 0 , ζ(π/2,π) = 1 , ζ(π/2,0) = 1 , ζ(π/2,3π/2) = i , ζ(π/2,π/2) = | i | i | i | i | i |− i | i | i | i i , and ζ(π,ϕ) = (see Tab. IVA). The intensity of the elements of each deviation |− i | i |∞i density matrix was obtained by QST. On the right of Fig. 3, the Husimi -distribution Q was used to characterize these implemented quantum states, which is sketched on the Bloch sphere [8]. The highest positive normalized intensity corresponds to the red color and the most negative intensity corresponds to the orange color. The highest intensity (red color) of the quasi-probability distribution function indicates the orientation of the spin nuclei magnetization. The large dispersion of the Husimi -distribution function is due to the Q small spin value I = 3/2, or analogously, the small number of particles N = 2I = 3 of an atomic system. In order to compare the theoretical predictions with the experimental values of the de- viation density matrices of Fig. 3, we present in Tab. IVA the complex phase ζ(θ,ϕ) and the Bloch vector n. The experimental values for θ and ϕ were computed with Eqs. (3), (6), and (7). 8 FIG. 3: (Color online) Experimental results for the 23Na nuclei in the sample of SDS. The QST results are represented by bar charts (left and middle column) for the implemented pseudo-NSCSs andontheBlochSphere(rightcolumn)bytheHusimi -distribution. Thetopbarchartrepresents Q the pseudo-NSCS 0 , the left (right) bar chart showing the real (imaginary) part of the deviation | i density operator, and similarly for the other bar charts. The pseudo-NSCS 0 is implemented as | i explained in section IV by the SMP technique. The bottom bar chart shows the pseudo-NSCS , a state that is implemented by applying a rotation operator R with θ = π and ϕ = 3π/2 θ,ϕ |∞i to the pseudo-NSCS 0 . The pseudo-NSCS 1 is implemented by applying a π/2 non-selective | i |− i pulse in the negative y direction to transform 0 . Analogous remarks apply to the bar charts − | i labeled as 1 , i , and i . In the right column, we present the calculated Husimi -distribution | i | i |− i Q of each deviation density matrix and, at the bottom, the intensity of the –distribution function Q is encoded in the colour bar. 9 B. Controlling pseudo-NSCSs Besides knowing how to initialize a quantum system, i.e., how to prepare the desired quantum state, an important task in quantum physics is the control of the quantum system with high accuracy. The implementation and control of pseudo-NSCSs are important for applications in quantum computing and geometric phases [22], squeezed states [35], and quantum metrology [38]. In the present case, such control is performed through polar and azimuthal rotations. Polar rotations – In the context of polar rotations, we have demonstrated the experimen- tal control of angle θ by taking the initial quantum state ζ(0,0) = 0 and applying the | i | i rotation operator R , with θ = 0,π/18,2π/18, ...,π (total of nineteen steps) and fixed θ,ϕ ϕ = π. After each step, the state ζ(θ,ϕ) was reconstructed by QST (see pulse sequence in | i Fig. 2). First, the average values of the magnetic angular momenta I were calculated x,y,z h i for each deviation density matrix ζ(θ,ϕ) ζ(θ,ϕ) . The experimental results (symbols) | ih | are shown in Fig. 4 together with the values obtained by numerical simulation (solid lines) and theoretical prediction (dashed lines). We use the term numerical simulation when we apply numerical recipes to the dynamics of evolution given by Hamiltonian (8), while the theoretical prediction is given by the evolution operator of Eq. (9). The differences between simulated data and theoretical prediction are caused by the quadratic term of Hamiltonian Theoretical Experimental θ ϕ ζ(θ,ϕ) n ζ(θ,ϕ) n | i | i 0 0 0 (0,0,1) 0.02 0.01i ( 0.05, 0.02,0.99) | i | − i − − π 0 1 (1,0,0) 0.88 0.12i (0.98, 0.14,0.12) 2 |− i |− − i − π 3π i (0, 1,0) 0.88i (0, 0.99,0.13) 2 2 |− i − |− i − π π 1 ( 1,0,0) 0.79+0.25i ( 0.94,0.30,0.18) 2 | i − | i − π π i (0,1,0) 0.09+0.84i (0.1,0.98,0.16) 2 2 | i |− i π 0 (0,0, 1) 7.75+1.16i ( 0.25,0.04, 0.97) |∞i − | i − − TABLE I: Comparison between theoretical and experimental values obtained from the deviation density matrix of Fig. 3 by means of equations (3), (6) and (7). Experimental values are rounded to two decimal places. 10