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Entanglement, Berry Phases, and Level Crossings for the Atomic Breit-Rabi Hamiltonian

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Entanglement, Berry Phases, and Level Crossings for the Atomic Breit-Rabi Hamiltonian PHYSICAL REVIEW A 78, 062106 ͑2008͒ Entanglement, Berry phases, and level crossings for the atomic Breit-Rabi Hamiltonian Sangchul Oh,1,* Zhen Huang,2 Uri Peskin,3 and Sabre Kais1,2,† 1Max Planck Institute for the Physics of Complex Systems, Noethnitzer Strasse 38, D-01187 Dresden, Germany 2Purdue University, Department of Chemistry, West Lafayette, Indiana 47906 USA 3Technion Israel Institute of Technology, Schulich Faculty of Chemistry, IL-32000 Haifa, Israel ͑Received 16 October 2008; published 9 December 2008͒ The relation between level crossings, entanglement, and Berry phases is investigated for the Breit-Rabi Hamiltonian of hydrogen and sodium atoms, describing a hyperfine interaction of electron and nuclear spins in a magnetic field. It is shown that the entanglement between nuclear and electron spins is maximum at avoided crossings. An entangled state encircling avoided crossings acquires a marginal Berry phase of a subsystem like an instantaneous eigenstate moving around real crossings accumulates a Berry phase. Especially, the nodal points of a marginal Berry phase correspond to the avoided crossing points. DOI: 10.1103/PhysRevA.78.062106 PACS number͑s͒: 03.65.Ud, 03.65.Vf, 32.60.ϩi I. INTRODUCTION crossings or glancing intersections are not the source of Berry phases ͓5͔. Is there any way that Berry phases can Energy levels, eigenvalues of a Hamiltonian, play the detect avoided level crossings? Here we show that the mar- most primary role in determining the properties of a quantum ginal Berry phase of an entangled state could be an indicator system. When energy levels cross or almost cross as param- of avoided level crossings. eters of a Hamiltonian vary, various interesting phenomena The entropy is an another indicator of level crossings. happen. For example, if two instantaneous energy levels of a Since level crossings or avoided crossings are accompanied time-dependent Hamiltonian undergo an avoided crossing, by a drastic change in eigenstates, any contents of informa- nonadiabatic tunneling, called Landau-Zener tunneling, be- tion on relevant eigenstates may also vary. The Shannon en- tween them takes place ͓1͔. Closely related to this, the run tropy of the electron density measures the delocalization or time of adiabatic quantum computation is inversely propor- the lack of structure in the respective distribution. Thus the tional to the square of the energy gap between the ground Shannon entropy is maximal for a uniform distribution—that and first excited levels ͓2͔. An eigenstate encircling adiabati- is, for an unbound system—and is minimal when the uncer- ͓ ͔ cally degeneracy points accumulates a Berry phase in addi- tainty about the structure of the distribution is minimal 12 . tion to a dynamical phase ͓3,4͔. In quantum chemistry, a González-Férez and Dehesa showed that the Shannon en- ͓ ͔ conical intersection of electronic energy surfaces of mol- tropy could be used as an indicator of avoided crossings 13 . ecules plays a key role in understanding ultrafast radiation- The level crossings or avoided crossings of a single particle are well known. Here we focus on the level crossing of a less reactions ͓5,6͔. A quantum phase transition, a dramatic system with two or more particles and its relation to en- change in a ground state as parameters of a system vary, is tanglement. The von Neumann entropy, the quantum version related to crossings or avoided crossings of two lowest- ͓ ͔ of the Shannon entropy, is a good entanglement measure for energy levels 7 . Kais and co-workers have shown that the a bipartite pure state. In quantum information, much atten- finite-size scaling method can be used for studying the criti- tion has been paid to the relation between entanglement and cal behavior—i.e., the level degeneracy or absorption—of a ͓ ͔ ͑␭ ␭ ͒ quantum phase transitions 14,15 . Recently one of the au- few-body quantum Hamiltonian H 1 ,..., k as a function ͕␭ ͖͓ ͔ thors investigated the relation between entanglement, Berry of a set of parameters i 8,9 . These parameters could be phases, and level crossings for two qubits with an XY-type the external fields, interatomic distances, nuclear charges for interaction and found that the level crossing is not always stability of negative ions, cluster size, and optical lattice pa- ͓ ͔ ͓ ͔ accompanied by an abrupt change in entanglement 16 . rameters such as the potential depth 10 . Thus, it is impor- In this paper, in order to study how entanglement and tant to develop a way of finding level crossings and to un- Berry phases vary at level crossings, we consider the Breit- derstand how eigenstates or relevant physical quantities Rabi Hamiltonian describing a hyperfine interaction of elec- change at crossing or avoided crossings. tron and nuclear spins in a uniform magnetic field ͓17͔.Itis Recently Bhattacharya and Raman presented a powerful shown that the von Neumann entropy of the electron ͑or algebraic method for finding level crossings without solving ͒ ͓ ͔ nuclear spin is maximum at avoided crossings. It is demon- an eigenvalue problem directly 11 . Along with this math- strated that the significant changes in Berry phases and en- ematical way, it is necessary to understand what physical tanglement are closely related to level crossings. We show quantities can be used to detect or characterize crossings or that the marginal Berry phase of the electron ͑or nuclear͒ avoided crossings. First of all, the measurement of a Berry spin could be a good indicator of avoided level crossings. phase could be a good way to detect level crossings because The marginal Berry phase has nodal points at the avoided it is due to level crossings. It is well known that avoided crossing points. The paper is organized as follows. In Sec. II, the Breit- Rabi Hamiltonian is introduced. In Sec. III, as a specific *[email protected] application of the Breit-Rabi Hamiltonian, we consider a hy- †Corresponding author: [email protected] perfine interaction between a nuclear spin 1/2 and an elec- 1050-2947/2008/78͑6͒/062106͑7͒ 062106-1 ©2008 The American Physical Society OH et al. PHYSICAL REVIEW A 78, 062106 ͑2008͒ / Ϯ tron spin 1 2 of a hydrogen atom in a magnetic field. We Breit-Rabi Hamiltonian. With ladder operators SϮ =Sx iSy Ϯ ͑ ͒ analyze the close relation between entanglement, Berry and IϮ =Ix iIy, the Hamiltonian 1 can be rewritten as phases, and level crossings. In Sec. IV, we make an similar A analysis for a sodium atom with a nuclear spin 3/2 and an ͑ ͒ ͑ ͒ ͑ ͒ H = AIzSz + S+I− + S−I+ + B aSz + bIz . 2 electron spin 1/2. In Sec. V, we summarize the main results. 2 ͉ ͘ Let us use a simple notation mS ,mI to represent the product ͉ ͘ ͉ ͘ ͉ ͘ 2 II. BREIT-RABI HAMILTONIAN state S,mS I,mI , where S,mS is an eigenstate of S and S , and ͉I,m ͘ is an eigenstate of I2 and I . The first and third Let us consider an atom with a single valence electron in z I z terms in Eq. ͑2͒ give the diagonal matrix elements the ground state with orbital angular momentum L=0.Inthe ͗ ͉ ͉ ͘ ͑ ͒ presence of a uniform magnetic field B in the z direction, its mSmI H mSmI = f mS,mI = AmSmI + mSaB + mIbB. atomic spectrum is described by the Breit-Rabi Hamiltonian ͑3a͒ ͓17͔, which is given by the sum of the hyperfine interaction between a nuclear spin I and an electron spin S and their The second term in Eq. ͑2͒ corresponds to the off-diagonal Zeeman couplings to the magnetic field matrix elements ͑ ͒ ͑ ͒ ͗ Ј Ј͉ ͉ ͘ ͱ͑ ͒͑ ͒ H = AI · S + aSz + bIz B, 1 mSmI S+I− mSmI = S − mS S + ms +1 ␥ ប ϫͱ͑ ͒͑ ͒␦ ␦ where A is the hyperfine coupling constant, a= e , and b I + mI I − mI +1 mЈ,m +1 mЈ,m −1. ␥ ប ␥ ␥ S S I I = n . Here e and n are the electron and nuclear gyromag- netic ratios, respectively. The electron spin operator S and ͑3b͒ ប the nuclear spin operator I are measured in the unit of . Since mЈ−m =1 and mЈ−m =−1 ͑or vice versa͒, one has the ͑ ͒ S S I I The Breit-Rabi Hamiltonian 1 is well studied to describe selection rule ⌬m=͑mЈ+mЈ͒−͑m +m ͒=0; that is, the mag- ͓ ͔ S I S I double resonance in nuclear magnetic resonance 18 and netic quantum number m=m +m is conserved. This implies ͓ ͔ S I muon spin rotation in semiconductors 19 . Although simple that the Hamiltonian ͑1͒ is block diagonal in the basis set and well understood, it still continues to provide new in- ͕͉m ,m ͖͘ ordered by m. sights. Recently Bhattacharya and Raman found a new class S I of invariants of the Breit-Rabi Hamiltonian ͓11͔. As will be III. HYDROGEN ATOM IN A UNIFORM MAGNETIC shown here, it is a prime example for showing the close FIELD relation between level crossings, entanglement, and geomet- A. Eigenvalues and eigenstates ric phases. Also it is related to a Hamiltonian of electron spin qubits in quantum dots ͓20͔ where the Heisenberg interaction As a simple but real system described by the Hamiltonian between two electron spins can be turned on and off to ͑1͒, let us consider the interaction between the nuclear spin implement the controlled-NOT gate. I=1/2 and the electron spin S=1/2 of a hydrogen atom in a Before applying the Hamiltonian ͑1͒ to specific systems, uniform magnetic field.
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