Magnetometry with spin polarized Hydrogen from molecular photo-dissociation
Konstantinos Tazes, Alexandros K. Spiliotis, Michalis Xygkis, George E. Katsoprinakis, T. Peter Rakitzis, Georgios Vasilakis
MMagnetometry with spin polarized Hydrogen from molecular photo-dissociation
Konstantinos Tazes, Alexandros K. Spiliotis,
2, 1
Michalis Xygkis,
2, 1
George E. Katsoprinakis, T. Peter Rakitzis,
2, 1 and Georgios Vasilakis University of Crete, Department of Physics, Heraklion, Greece Foundation for Research and Technology Hellas, Institute of Electronic Structure and Laser,N. Plastira 100, Heraklion, Crete, Greece, GR-71110
In a recent publication [arXiv:2010.14579], we introduced a new type of atomic magnetometer,which relies on hydrohalide photo-dissociation to create high-density spin-polarized hydrogen. Here,we extend our previous work and present a detailed theoretical analysis of the magnetometer signaland its dependence on time. We also derive the sensitivity for a spin-projection noise limitedmagnetometer, which can be applied to an arbitrary magnetic field waveform.
A broad range of physical objects and processes generate magnetic fields which upon detection can convey importantinformation about the nature and structure of their origin. As a result, magnetic field detection lies at the heart ofmany scientific and technological applications, which can benefit significantly from advances in magnetometry [1].Different magnetic field sensors have been developed which offer distinct advantages and are attractive for particularapplications. In general terms, an ideal magnetometer should present high-sensitivity, wide bandwidth detection, high-performance over a large dynamic range and operating conditions, as well as capability for miniaturization when usedfor magnetic field imaging.Recently, a new type of atomic magnetometer was demonstrated based on high-density spin-polarized atomic H(SPH) [2], which has the potential to address satisfactorily the above requirements for magnetometry. The spin-polarized ensemble is produced by photo-dissociating hydrohalide gas with a circularly-polarized laser pulse [3–5].Magnetic field detection is achieved by monitoring the dynamics of the H hyperfine coherences, which are created inthe optical pumping process without the need for external magnetic fields.This paper is an extension of the work presented in [2], analytically deriving equations for the spin-dynamics, themagnetometer signal and the quantum spin-projection noise.We will consider the magnetometer scheme with mutually orthogonal directions for optical pumping, magneticfield direction and spin-probing, as shown in Fig. 1. Without loss of generality we take the magnetic field to be inthe z direction, the optical pumping along the y axis and the probe axis in the x direction. Monitoring of spins isrealized with an inductive pick-up coil, which detects the magnetic flux generated by the H spins. Since the electronmagnetic moment is more than three orders of magnitude larger than the proton magnetic moment, the coil is toa very good approximation only sensitive to the H electron spins. In the following, we will assume a pickup coilwith a response time much shorter than the hyperfine interaction period and neglect complications arising from anon-spherical polarized region or from geometrical factors in the coupling of the magnetic field from spins to the coil.For simplicity, we will assume that the observable is d ˆ S x dt , where ˆ S i expresses the dimensionless electron spin operatorin the i direction. TIME EVOLUTION
In the presence of a magnetic field B (which can be time varying), the SPH evolves according to the Hamiltonian(we neglect for the moment the relaxation):ˆ H = ω ˆ S · ˆ I + g s µ B (cid:126) B · ˆ S = ˆ H + g s µ B (cid:126) B · ˆ S z (1)where ˆ I is the dimensionless nuclear spin operator, ω is the hyperfine frequency of H, µ B is the Bohr magneton, g s ≈ g -factor, ˆ H is the Hamiltonian for hyperfine interaction expressed in (angular) frequencyunits. In the above equation, we neglected the coupling of the magnetic field to nuclear spin, as this is three ordersof magnitude smaller than the coupling to the electron spin.For clarity we write the Hamiltonian in the coupled (subscript c) and uncoupled (subscript u) basis taking z (the a r X i v : . [ phy s i c s . a t o m - ph ] F e b x yz Circularly polarizedpump beam lensvacuumchamber pick-up coil SPHmagnetic field
FIG. 1: Schematic of the magnetometer setup. A circularly-polarized laser pulse (red arrow), photo-dissociate molecularhydrohalide gas producing spin-polarized atomic H. A pickup coil detects the evolution of the ensemble magnetization in thepresence of a magnetic field (green arrow). magnetic field direction) as the quantization axis:ˆ H u = ω + γ H B − ω − γ H B ω ω − ω + γ H B
00 0 0 ω − γ H B , ˆ H c = ω + γ H B ω γ H B ω − γ H B γ H B − ω , (2)where γ H = g s µ B / (cid:126) is the gyromagnetic ratio of atomic H. The above matrices are expressed in a basis with thefollowing ordering:uncoupled basis: {| m s = 1 / , m I = 1 / (cid:105) , | m s = − / , m I = 1 / (cid:105) , | m s = 1 / , m I = − / (cid:105) , | m s = − / , m I = − / (cid:105)} , coupled basis: {| F = 1 , m F = 1 (cid:105) , | F = 1 , m F = 0 (cid:105) , | F = 1 , m F = − (cid:105) , | F = 0 , m F = 0 (cid:105)} , (3)where F is the total spin (sum of electronic and nuclear spin) quantum number, m s , m I and m F are respectively theelectronic, nuclear and total spin projection along the quantization axis.Transformation of an arbitrary operator ˆ O or state vector | ψ (cid:105) from one basis to the other can be performed accordingto the following rules:ˆ O c = T cu · ˆ O u · T − , ˆ O u = T − · ˆ O c · T cu , | ψ (cid:105) c = T cu | ψ (cid:105) u , | ψ (cid:105) u = T − | ψ (cid:105) c , (4) T cu = √ √
00 0 0 10 − √ √ . (5)The hydrohalide photo-dissociation occurs with a sub-nanosecond laser pulse and optical pumping effectively trans-fers angular momentum from light to the electronic spin, leaving the nuclear spin degrees of freedom in their thermal(completely unpolarized) state. Following the above basis ordering, after optical pumping half of the polarized Hatoms are in the quantum state (expressed as a column vector): | ψ (0) (cid:105) = | ψ (cid:105) = (1 , , , (cid:124) u,y ˆ R x · (1 , , , (cid:124) −−−−−−−−→ (cid:0) , − ı , − ı , − (cid:1) (cid:124) u T cu −−→ (cid:0) , − ı , − , (cid:1) (cid:124) c , σ = 1(0 , , , (cid:124) u,y ˆ R x · (0 , , , (cid:124) −−−−−−−−→ (cid:0) − , − ı , − ı , (cid:1) (cid:124) u T cu −−→ (cid:0) − , − ı , , (cid:1) (cid:124) c , σ = − , (6)where the subscript y denotes that the quantization axis for spin projections was taken in the y direction (if no axissubscript appears it is implicitly assumed that the quantization axis is in the z direction), (cid:124) is the transpose operation, σ is the helicity of the pumping light pulse and ˆ R x is the rotation matrix around the x -axis applied to the uncoupledbasis (two spins, each of spin 1/2) : ˆ R x = e − ı ˆ S x π ⊗ e − ı ˆ S x π . (7)The other half of the polarized H atoms are in the state: | ψ (0) (cid:105) = | ψ (cid:105) = (0 , , , (cid:124) u,y ˆ R x · (0 , , , (cid:124) −−−−−−−−→ (cid:0) − ı , − , , − ı (cid:1) (cid:124) u T cu −−→ (cid:16) − ı , , − ı , √ (cid:17) (cid:124) c , σ = 1(0 , , , (cid:124) u,y ˆ R x · (0 , , , (cid:124) −−−−−−−−→ (cid:0) − ı , , − , − ı (cid:1) (cid:124) u T cu −−→ (cid:16) − ı , , − ı , − √ (cid:17) (cid:124) c , σ = − . (8)For the observable d ˆ S x dt the contribution to signal of atoms initially at state | ψ (0) (cid:105) is a factor of γ H B/ω smallerthan the contribution of atoms initially at | ψ (0) (cid:105) . In the case of γ H B/ω (cid:28)
1, the magnetometer signal is mainlydetermined by the atoms initially at | ψ (0) (cid:105) .For static magnetic field, Schr¨odinger equation can be solved in a straightforward manner: ı ∂∂t | ψ ( t ) (cid:105) = ˆ H | ψ ( t ) (cid:105) ⇒ | ψ ( t ) (cid:105) = e − ı ˆ Ht | ψ (0) (cid:105) (9)where | ψ ( t ) (cid:105) is the wavefunction at time t after optical pumping. For | ψ (0) (cid:105) = | ψ (0) (cid:105) the wavefunction at time t inthe coupled basis is: | ψ ( t ) (cid:105) = (cid:32) − ie − ı ( ω + γ H B ) t , − ıσ √ γ H Be ı ω t sin (cid:2) ˜ ωt (cid:3) ˜ ω , − ıe − ı ( ω − γ H B ) t , σ e ı ω t (cid:0) ıω sin (cid:2) ˜ ωt (cid:3) + ˜ ω cos (cid:2) ˜ ωt (cid:3)(cid:1) √ ω (cid:33) (cid:124) (10) ≈ − ıe ı ( ω +2˜ ω ) t (cid:16) e − ı ( ω + γ H B + ˜ ω ) t , , e − ı ( ω − γ H B + ˜ ω ) t , ı √ σ (cid:17) (cid:124) , (11)where: ˜ ω = (cid:112) ω + 4 γ B .In the general case of a time varying magnetic field the Schr¨odinger equation cannot be solved analytically, as theHamiltonian does not commute with itself at different times. For an approximate analytical solution it is convenientto work in the interaction picture (denoted by the ˜ symbol): ∂∂t | ˜ ψ ( t ) (cid:105) = − ı ˆ˜ V | ˜ ψ ( t ) (cid:105) , (12)where ˆ˜ V is the Hamiltonian describing the magnetic field coupling to the atoms:ˆ˜ V ( t ) = e ı ˆ H t (cid:104) g s µ B B ˆ S z (cid:105) e − ı ˆ H t = γ H B ( t ) − cos( ω t ) ı sin( ω t ) 00 − ı sin( ω t ) cos( ω t ) 00 0 0 − . (13)In the last equation ˆ˜ V is expressed in the uncoupled basis.The solution of Eq. 12 can be expressed in the form of the Magnus series [6]. For magnetic fields that change slowlywith respect to the hyperfine frequency so that: (cid:90) t B ( t (cid:48) ) e ıω t (cid:48) d t (cid:48) (cid:28) (cid:90) t B ( t (cid:48) ) d t (cid:48) (14)the solution of Eq. 12 can be approximated by keeping only the first term in Magnus expansion. In this case: | ˜ ψ ( t ) (cid:105) ≈ exp (cid:18) − ı (cid:90) t ˆ˜ V ( t (cid:48) ) d t (cid:48) (cid:19) | ψ (0) (cid:105) , (15)and the observable (taking into account the condition 14) can be written as: ddt (cid:104) ˜ ψ ( t ) | e ı ˆ H t ˆ S x e − ı ˆ H t | ˜ ψ ( t ) (cid:105) ≈ σ ddt (cid:26)
12 sin (cid:20) γ H (cid:90) t B ( t (cid:48) ) d t (cid:48) (cid:21) cos( ω t ) (cid:27) (16) ≈ − σ ω (cid:20) γ H (cid:90) t B ( t (cid:48) ) d t (cid:48) (cid:21) sin( ω t ) . (17)The last approximation holds for γ H B (cid:28) ω .The condition stated in 14 implies that the magnetic field does not induce hyperfine transitions and the magnetic fieldcan be treated as a perturbation to the energies of the hyperfine levels. Then, for small magnetic fields ( γ H B (cid:28) ω )the Hamiltonian for H atoms (hyperfine levels with F = 0 and F = 1) can be approximated to be (as before themagnetic field direction is taken to be the quantization axis):ˆ H ≈ ˆ H + γ H B ( t )( ˆ S z + ˆ I z ) . (18)This Hamiltonian commutes with itself at different times and the Schr¨odinger equation can be solved analytically: | ψ ( t ) (cid:105) = e − ı (cid:82) t ˆ H ( t (cid:48) ) d t (cid:48) | ψ (0) (cid:105) (19) Decay
The evolution of spins is also affected by relaxation processes leading to non-Hamiltonian dynamics. We modelthese by introducing a decay term in the density matrix equation: dρdt = ı [ ρ, H ] − T ( ρ − ρ eq ) , (20)where 1 /T is the decay rate, and ρ eq corresponds to the state towards which the decay processes drive the system.We take this quantum state to be the completely unpolarized state, written in the form: ρ eq = 14 I × , (21)where I × is the 4 × t = 0 is ( z quantization axis the direction of magnetic field and optical pumping in the y axis): ρ (0) = 12 | ψ (cid:105)(cid:104) ψ | + 12 | ψ (cid:105)(cid:104) ψ | → σ ı − σ ı σ ı − σ ı u → σ ı √ − σ ı √ − σ ı √ σ ı √ − σ ı √ − σ ı √ σ ı √ σ ı √ c . (22)An analytical solution to the density matrix equation can be found for a static magnetic field or for an arbitraryfield when the Hamiltonian can be approximated by Eq. 18. In the experimentally relevant limit of ω (cid:29) (1 /T , γ H B )the signal is (keeping lowest order terms in the harmonic amplitudes): (cid:104) d ˆ S x dt (cid:105) ( t ) = Tr (cid:20) dρdt ˆ S x (cid:21) ≈ − σ e − t/T sin (cid:20) γ H (cid:90) t B ( t (cid:48) ) d t (cid:48) (cid:21) sin( ω t ) . (23)Eqs. 20-21 implies spin-damping occurs (at equal rate) for both electron and nuclear spin of H. However, the resultsderived here are general for the relevant approximations ( ω (cid:29) (1 /T , γ H B ) and condition 14. For instance, Eq. 23is reproduced also for decay mechanisms that relax only the electronic spin: dρdt = ı [ ρ, H ] − T (ˆ S · ρ − ˆ S · ρ ˆ S · ) . (24) SPIN-PROJECTION NOISE
In the following we will need to know the multi-time correlation (cid:104) d ˆ S x dt ( t ) d ˆ S x dt ( t (cid:48) ) (cid:105) , (25)where we take: ddt ˆ S x = ı (cid:104) ˆ H, ˆ S x (cid:105) − T ˆ S x (26)The last term (not derived from first principles) was introduced to account for the spin decay. We assume thatthe evolution of the density matrix ρ is given by Eq. 20 (though the results are the same -within the relevantapproximations- for the evolution described in Eq. 24).The multi-time correlation can be written operationally in the form: (cid:104) d ˆ S x dt ( t ) d ˆ S x dt ( t (cid:48) ) (cid:105) = Tr (cid:40) d ˆ S x dt (cid:34) ˆ U ( t, t (cid:48) ) (cid:32) d ˆ S x dt ρ ( t (cid:48) ) (cid:33)(cid:35)(cid:41) , (27)where U ( t, t (cid:48) ) is the evolution operator of the density matrix from time t (cid:48) to t ( t > t (cid:48) ), and ρ ( t (cid:48) ) is the density matrixat time t (cid:48) : ρ ( t (cid:48) ) = U ( t (cid:48) , ρ (0). The evolution operator U cannot be written in the form of a matrix and is notassociative. The evolved state ˆ U ( t, t (cid:48) ) (cid:16) d ˆ S x dt ρ ( t (cid:48) ) (cid:17) can be found from the general solution of Eq. 20 taking d ˆ S x dt ρ ( t (cid:48) )as the initial condition for the density matrix. For quantum noise analysis we can take the magnetic field to be zeroin the calculation of the multi-time correlation, since quantum noise affects considerably magnetometry only at lowfields. When ω T (cid:29)
1, it can be found that: (cid:104) d ˆ S x dt ( t ) d ˆ S x dt ( t (cid:48) ) (cid:105) = 18 ω e −| t − t (cid:48) | /T cos [ ω ( t − t (cid:48) )] . (28)In order to find what is the magnetic field uncertainty due to spin-projection noise, we have to specify a method forestimating the magnetic field from the detected signal d ˆ S x dt ( t ). Taking into account Eq. 23, one way to do this (appro-priate for arbitrary magnetic waveforms) is from considering the “quasi-instantaneous” amplitude of the frequencycomponent at the hyperfine frequency: ξ ( nT hf ) = 1 T hf (cid:90) ( n +1) T hf nT hf d ˆ S x dt ( t ) sin( ω t ) d t, (29)where T hf = 2 π/ω is the period of hyperfine oscillation and n is an integer number. We assume that the magnetic fieldcan be written in the form: B ( t ) = B K ( t ), where K is a known (but other than this an arbitrary), time-dependentfunction. Spin-projection noise creates an uncertainty in the estimation of B .We consider the case where the functions K ( t ) and e − t/T evolve in time much slower compared to sin( ω t ) and cantherefore be considered constant during the hyperfine period. Effectively this is the situation for ω T (cid:29) − σ factor in the signal in Eq. 23):1 T hf (cid:90) ( n +1) T hf nT hf d ˆ S x dt ( t ) sin( ω t ) d t ≈ ω e − nT hf /T sin (cid:34) γ H B (cid:90) nT hf K ( t (cid:48) ) d t (cid:48) (cid:35) ≈ ω e − nT hf /T γ H B (cid:90) nT hf K ( t (cid:48) ) d t (cid:48) , (30)where the last approximation holds for small magnetic fields.The magnetic field B can be estimated by minimizing with respect to the parameter B the χ function: χ = M (cid:88) n =0 (cid:34) ω e − nT hf /T γ H B (cid:90) nT hf K ( t (cid:48) ) d t (cid:48) − ξ ( nT hf ) (cid:35) , (31)where M T hf is the total measurement time. The above equation can be viewed as a curve fitting problem withunknown parameter B for a noisy signal. The solution to the curve fitting problem is: B = (cid:80) Mn =0 ξ ( nT hf ) e − nT hf /T (cid:82) nT hf K ( t (cid:48) ) d t (cid:48) γ H (cid:80) Kn =0 18 ω e − nT hf /T (cid:16)(cid:82) nT hf K ( t (cid:48) ) d t (cid:48) (cid:17) , (32)and the estimation uncertainty in B due to the spin-projection noise is: δB = (cid:80) Mn (cid:48) =0 (cid:80) Mn =0 (cid:104) ξ ( nT hf ) ξ ( n (cid:48) T hf ) (cid:105) e − ( n + n (cid:48) ) T hf /T (cid:82) nT hf K ( t (cid:48) ) d t (cid:48) (cid:82) n (cid:48) T hf K ( t (cid:48)(cid:48) ) d t (cid:48)(cid:48) γ H (cid:20)(cid:80) Kn =0 18 ω e − nT hf /T (cid:16)(cid:82) nT hf K ( t (cid:48) ) d t (cid:48) (cid:17) (cid:21) . (33)From Eqs. 28 and 29 we find (in the limit of ω (cid:29) γ ): (cid:104) ξ ( nT hf ) ξ ( n (cid:48) T hf ) (cid:105) = 132 ω e − | ( n − n (cid:48) ) | T hf /T , (34)so that: δB = 2 (cid:82) T m (cid:82) T m d t d t (cid:48) e − ( t + t (cid:48) ) /T e − | t − t (cid:48) | /T (cid:82) t K ( x (cid:48) ) d x (cid:48) (cid:82) t (cid:48) K ( x ) d x (cid:20) γ H (cid:82) T m d te − t/T (cid:16)(cid:82) t K ( x ) d x (cid:17) (cid:21) , (35)where T m is the measurement time for a single run of the experiment.The above derivation applies to a measurement a single H atom. For N SPH independent (absence of spin-squeezing)H atoms the uncertainty in magnetic field estimation is: δB = 2 N SPH (cid:82) T m (cid:82) T m d t d t (cid:48) e − ( t + t (cid:48) ) /T e − | t − t (cid:48) | /T (cid:82) t K ( x (cid:48) ) d x (cid:48) (cid:82) t (cid:48) K ( x ) d x (cid:20) γ H (cid:82) T m d te − t/T (cid:16)(cid:82) t K ( x ) d x (cid:17) (cid:21) . (36) SUMMARY
We developed analytical equations describing the operation of the SPH magnetometer shown in Fig. 1. The timeevolution of the quantum state for a time-varying magnetic field was derived and the magnetometer signal was shownto exhibit first order dependence to the sensed field. Finally, the effect of spin-projection noise on the estimation ofmagnetic field was considered, deriving a formulation for an arbitrary waveform.
ACKNOWLEDGEMENTS
This work was supported by the Hellenic Foundation for Research and Innovation (HFRI) and the General Sec-retariat for Research and Technology (GSRT), grant agreement No HFRI-FM17-3709 (project NUPOL) and by theproject “HELLAS-CH” (MIS 5002735), which is implemented under the “Action for Strengthening Research and Inno-vation Infrastructures”, funded by the Operational Programme “Competitiveness, Entrepreneurship and Innovation”(NSRF 2014-2020) and cofinanced by Greece and the European Union (European Regional Development Fund). [1] Pavel Ripka.
Magnetic Sensors and Magnetometers . Artech House Publishers, 2001.[2] Alexandros K. Spiliotis, Michail Xygkis, Konstantinos Tazes, George E. Katsoprinakis, Georgios Vasilakis, and T. PeterRakitzis. A nanosecond-resolved atomic hydrogen magnetometer, 2020.[3] T. P. Rakitzis, P. C. Samartzis, R. L. Toomes, T. N. Kitsopoulos, Alex Brown, G. G. Balint-Kurti, O. S. Vasyutinskii, andJ. A. Beswick. Spin-polarized hydrogen atoms from molecular photodissociation.
Science , 300(5627):1936–1938, 2003.[4] Dimitris Sofikitis, Luis Rubio-Lago, Lykourgos Bougas, Andrew J. Alexander, and T. Peter Rakitzis. Laser detection ofspin-polarized hydrogen from HCl and HBr photodissociation: Comparison of H- and halogen-atom polarizations.
TheJournal of Chemical Physics , 129(14):144302, 2008.[5] Rakitzis T. P. Pulsed-laser production and detection of spin-polarized hydrogen atoms.
Chemphyschem : a European journalof chemical physics and physical chemistry , 5:1489–1494, 2004.[6] S. Blanes, F. Casas, J.A. Oteo, and J. Ros. The magnus expansion and some of its applications.