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-

/ 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. Since H commutes with the z com- let us look at its general properties. If B=0, then H com- ponent of the total spin operator, Jz =Sz +Iz, it is convenient 2 ͕͉ ͖͘ mutes with both the square of the total spin operator, J , and to arrange the product basis mS ,mI in decreasing order of ϵ ͕͉ 1 1 ͘ ͉ 1 1 ͘ ͉ Jz, where the total spin operator is defined by J I+S. How- the magnetic quantum number m of Jz as 2 , 2 , 2 ,−2 , ͑ ͒ 1 1 ͘ ͉ 1 1 ͖͘ ͑ ͒ ͑ ͒ ever, for B 0, due to the fact that a b, the Hamiltonian 1 − 2 , 2 , − 2 ,−2 . By means of Eqs. 3a and 3b , the Breit- 2 no longer commutes with J , but still commutes with Jz.So Rabi Hamiltonian for a hydrogen atom can be written in the the eigenvalue m of Jz is a good quantum number for the ordered basis

A +2͑a + b͒B 0 00 1 0 − A +2͑a − b͒B 2A 0 H = . ͑4͒ 4΂ 0 2A − A −2͑a − b͒B 0 ΃ 00 0A −2͑a + b͒B

The Hamiltonian ͑4͒ is block diagonal, so it is straightfor- 1 1 ͉ ͘ ͯ Ϯ Ϯ ʹ ͑ ͒ EϮ = , , 5b ward to obtain its eigenvalues and eigenvectors. The sub- 1 2 2 Ϯ ͕͉ 1 1 ͘ ͉ 1 1 ͖͘ space of m= 1 is spanned by 2 , 2 , − 2 ,−2 . The block Hamiltonian on this subspace is already diagonal and has its eigenvalues and eigenvectors Ϯ where the subscripts 1inEϮ1 denote the magnetic quan- A 1 tum number m= Ϯ1. The block Hamiltonian with m=0 is E Ϯ ͑a b͒B ͑ ͒ Ϯ1 = + , 5a ͕͉ 1 1 ͘ ͉ 1 1 ͖͘ 4 2 deﬁned on the subspace of 2 ,−2 , 2 ,−2 and is written as

062106-2 ENTANGLEMENT, BERRY PHASES, AND LEVEL… PHYSICAL REVIEW A 78, 062106 ͑2008͒

2 1 − A +2͑a − b͒B 2A m=±1 ͩ ͪ ͑ ͒ m= 0 H = . 6 1 (a) m=0 ͑ ͒ 4 2A − A −2 a − b B E /A m 0 One can interpret the Hamiltonian Hm=0 as that of a spin in -1 an effective magnetic ﬁeld in the x-z plane, Beff ϵ͑ / ͑ ͒ / ͒ -2 A 2,0, a−b B 2 . The eigenvalues and eigenvectors of 2 Hm=0 can be written easily as 1 (b) A 1 E /A Ϯ Ϯ ͱ͑ ͒2 2 2 ͑ ͒ m 0 E0 =− a − b B + A , 7a 4 2 -1 -2 ␣ 1 1 ␣ 1 1 ͉E+͘ = cos ͯ ,− ʹ + sin ͯ− , ʹ, ͑7b͒ 1 0 2 2 2 2 2 2 (c) S 0.5

␣ 1 1 ␣ 1 1 0 ͉E−͘ = − sin ͯ ,− ʹ + cos ͯ− , ʹ, ͑7c͒ 0 2 2 2 2 2 2 -0.2 -0.1 0 0.1 0.2 B (T) ␣ϵ A ͑ where tan ͑a−b͒B . In a weak-magnetic-field limit so called the Zeeman region͒, the Zeeman energy is smaller FIG. 1. ͑Color online͒ Energy levels Em /A of the Breit-Rabi than the hyperfine coupling. At B=0—i.e., ␣=␲/2—the Hamiltonian for a hydrogen atom as functions of the magnetic field ͉ −͘ ͉ −͘ B with ͑a͒ f =1 and ͑b͒ f =−0.5. ͑c͒ The von Neumann entropy S of ground eigenstate E0 becomes the singlet state, E0 1 1 1 1 1 ͑ ͒ Ј = ͱ ͉͑− , ͘−͉ ,− ͒͘. In a strong magnetic field called the the electron or nuclear spin for each eigenstate. Here a 2 2 2 2 2 −1 Ј −1 ͓ ͔ Paschen-Back region, the Zeeman couplings are dominant. =19.767 T and b =−0.03 T are taken from Ref. 22 . →ϱ ␣→ ͉ −͘ That is, in the limit of B , one has 0 and E0 the ground state changes abruptly at the level crossing →͉ 1 1 ͘ − 2 , 2 . points. The eigenvalues and eigenstates of the Breit-Rabi Hamil- tonian for a hydrogen atom, Eqs. ͑5͒ and ͑7͒, depend on two B. Entanglement parameters: the hyperfine constant A and the magnetic field Let us discuss the relation between level crossings and B. The hyperfine constant A of the hydrogen atom in vacuum entanglement. Entanglement refers to the quantum correla- is positive. However, if a hydrogen atom is in an inert gas, the hyperfine constant A could be negative ͓21͔, resembling (a) the spin-spin coupling constant in a Heisenberg model. We ∆/A assume that A as well as B varies and can be negative. To 0.2 this end, A in Eqs. ͑5͒ and ͑7͒ is replaced by fAwith −1 0 ഛ f ഛ1, so A is still kept as a positive constant in vacuum. If 0.2 10 0 5 f is negative, so is the hyperfine constant. f 0 -0.2 -5 Depending on f and B, the ground state of the Hamil- -10 B (T) ͑ ͒ ͉ ͘ ͉ −͘ tonian 4 is given either by EϮ1 or by E0 . It is convenient to plot the energy levels normalized by A. Then, Eqs. (b) Ϯ ͑ ͒ ͑ ͒ / f Ϯ 1 ͑ Ј Ј͒ / S 1 5 and 7 become EϮ1 A= 4 2 a +b B and E0 A f Ϯ 1 ͱ͑ Ј Ј͒2 2 2 Јϵ / Ϸ −1 0.5 =−4 2 a −b B + f , where a a A 19.767 T and Јϵ / Ϸ −1 ͓ ͔ 0 b b A −0.03 T are taken from Ref. 22 , and B is mea- 0.2 / 10 sured in units of tesla. The energy levels Em A are plotted as 0 5 f 0 functions of B for f =1 in Fig. 1͑a͒ and for f =−0.5 in Fig. -0.2 -5 − -10 B (T) 1͑b͒. For f ജ0, the ground level is E . For f Ͻ0, two levels (c) Ϯ 0 E0 and EϮ1 with different magnetic quantum numbers cross Ј Ј at f = 2a b ͉B͉. Figure 2͑a͒ shows the energy gap ⌬/A between aЈ−bЈ β/Ω 1 the ground and first excited states as a function of f and B, 0 −1 −1 where we take aЈ=0.1 T and bЈ=−0.01 T to see clearly -1 the phase diagram of the ground state of the Hamiltonian ͑4͒ 0.2 10 0 5 determined by the magnetic quantum number m. As shown f 0 -0.2 -5 in Fig. 2͑a͒, the energy gap ⌬/A vanishes along the lines -10 B (T) 2aЈbЈ ͉ ͉ defined by f = Ј Ј B and the negative f axis. In the region Ј Ј a −b FIG. 2. ͑Color online͒͑a͒ Energy gap ⌬/A between the ground of f Ͻ 2a b ͉B͉, the ground state becomes either ͉E ͘ or ͉E ͘ aЈ−bЈ +1 −1 and first excited states, ͑b͒ the von Neumann entropy S of the elec- with magnetic quantum number m=1 or m=−1, respectively. tron ͑or nuclear͒ spin, and ͑c͒ the Berry phase ␤/⍀ of the ground On the other hand, the ground state in the region defined by Ј Ј Ј state for a hydrogen atom as a function of f and B. Here a f Ͼ 2a b ͉B͉ is given by ͉E−͘ with magnetic quantum number −1 Ј −1 aЈ−bЈ 0 =0.01 T and b =−0.1 T are taken to see the jumps clearly at m=0. One can see that the magnetic quantum number m of the level crossings.

062106-3 OH et al. PHYSICAL REVIEW A 78, 062106 ͑2008͒ tion between subsystems and has no classical analog ͓23,24͔. axis by an angle ␾. The SO͑3͒ rotation of the magnetic field When level crossing happens as the parameter of the Hamil- described above corresponds to the SU͑2͒ SU͑2͒ transfor- tonian varies, the ground state changes drastically. Entangle- mation on the Hamiltonian ͑1͒; thus, one obtains the Breit- ment as a physical quantity may also undergo a significant Rabi Hamiltonian in the magnetic field B=Bnˆ : change. However, entanglement is not always a good indica- ͑␪ ␾͒ ͑ ͒ tor of level crossing as shown in Ref. ͓16͔. H , = AI · S + aB · S + bB · I. 9 First, let us examine the relation between entanglement The hyperfine interaction AI·S is spherical symmetric, so the and level crossings for each eigenstate. The von Neumann eigenvalues and eigenvectors of the Hamiltonian ͑9͒ are entropy S of a subsystem is a good entanglement measure for identical to those of the Hamiltonian ͑4͒ except replacing ͉␺ ͘ 1 1 1 a pure bipartite system. If AB is a quantum state of a sys- ͉Ϯ ͘ ͉ Ϯ ͘ ͉ Ϯ ͘ 2 by nˆ ; 2 . Here nˆ ; 2 are eigenstates of nˆ ·S or tem composed of two subsystems A and B, the entanglement nˆ ·I. If the magnetic field B is rotated slowly about the z axis between A and B is measured by the von Neumann by 2␲ to make a cone with a solid angle ⍀=2␲͑1−cos ␪͒, ͑␳ ͒ ͑␳ ␳ ͒ ͑␳ ͒ entropy of the subsystem, S A =−tr A log A =S B then the instantaneous eigenstate ͉nˆ ; Ϯ 1 ͘ follows it and ac- ͑␳ ␳ ͒ ␳ 2 =−tr B log2 B , where the reduced density matrix A of the ␤ ϯ 1 ⍀ cumulates the Berry phase Ϯ = 2 . The total Berry phase subsystem A is obtained by tracing out the degrees of free- ␤ ␳ ͉͑␺ ͗͘␺ ͉͒ of electron and nuclear spins is the sum of two phases dom of B as A =trB AB AB . If the ground state is given ͉ ͘ acquired by each one. It depends on the magnetic quantum by EϮ1 —i.e., a product state—then the von Neumann en- number m: tropy S of the electron ͑or nuclear͒ spin is zero. On the other ͉ Ϯ͘ ϯ⍀ Ϯ hand, for the quantum state E0 of the electron and nuclear for m = 1, ͑ ͒ ␤ = ͭ ͮ ͑10͒ spins, the von Neumann entropy of the electron or nuclear 0 for m =0. spin can be written as As expected, the Berry phase is nonzero only for real 1 + cos ␣ 1 + cos ␣ Ϯ ͑ ͒ S͑␳ ͒ =− log crossings—i.e., m= 1. Figure 2 c plots the total Berry A 2 2 2 phase as a function of B and f and shows that the total Berry ␣ ␣ phase jumps at the level crossings. The zero Berry phase of 1 − cos 1 − cos ͉ Ϯ͑␪ ␾͒͘ − log . ͑8͒ the eigenstates E0 , can be understood in two ways. 2 2 2 Ϯ First, two levels E0 avoid crossing at B=0, so it is zero. ͉ Ϯ͘ Figure 1͑c͒ shows the von Neumann entropy of the electron Another view is as follows. Since E0 is a superposition of ͉ 1 1 ͘ ͉ 1 1 ͘ ͑or nuclear͒ spin for each eigenstate as a function of B. For 2 ,−2 and − 2 , 2 , the Berry phase of the electron spin is ͉ Ϯ͘ opposite to that of the nuclear spin and they cancel each the eigenstates E0 , it is maximum at B=0—i.e., at the avoided crossing point. This is analogous to the sharp change other. ͉ Ϯ͘ in Shannon entropy at the avoided crossing in Ref. ͓13͔. Although the entangled states E0 of electron and nuclear Now, let us look at how entanglement changes at level spins accumulate no Berry phase, each subsystem ͑electron crossings as the parameters of the Hamiltonian vary. Figure spin or nuclear spin͒ can get nonzero marginal Berry phases 2͑b͒ plots the von Neumann entropy S of the electron ͑or of mixed states. Following studies on the geometric phase of nuclear͒ spin for the ground state as a function of f and B. mixed states ͓25,26͔ and the relation between entanglement Ј Ј ͓ ͔ Across the level crossing line f = 2a b ͉B͉, the von Neumann and marginal Berry phases 26–28 , we investigate the rela- aЈ−bЈ tion between avoided level crossings, marginal Berry phases, entropy changes abruptly. For f Ͼ0, S becomes 1 as B goes and entanglement. For an adiabatic cyclic evolution param- to 0. Along the line of f =0, the von Neumann entropy S etrized by x, an instantaneous eigenstate of a bipartite system vanishes even though there is no level crossing. It will be AB can be expressed in a Schmidt decomposition ͉␺͑x͒͘ interesting to investigate the more general case that entangle- =͚M ͱp ͉e ͑x͒͘ ͉f ͑x͒͘, where ͕͉e ͑x͖͒͘NA is an orthonormal ment does not work as an indicator to level crossings. i=1 i i i i i=1 ͕͉ ͑ ͖͒͘NB basis for a subsystem A, fi x i=1 for a subsystem B, M ഛmin͕N ,N ͖, and ͚M p =1. Here our attention is restricted C. Berry phase A B i=1 i ͱ to the case that the Schmidt coefficients pi are independent An instantaneous eigenstate encircling the energy level of x. After an adiabatic cyclic evolution implemented by crossing points acquires the Berry phase in addition to the x͑0͒=x͑T͒, the total Berry phase of the bipartite system AB dynamical phase. The information on the level crossings is is given by encoded in the Berry phase. At B=0 and f =1, the two levels Ϯ M EϮ1 cross and the other two levels E0 avoid crossing. Also ␤ ͑␤A ␤B͒ ͑ ͒ − 2aЈbЈ = ͚ pi i + i , 11 EϮ and E cross at f = ͉B͉. Here we focus on the Berry 1 0 aЈ−bЈ i=1 phase due to the level crossing or avoided crossing at B=0. A ͛ ͗ ͑ ٌ͉͒ ͉ ͑ ͒͘␤ Due to the fact that aӷ͉b͉, an electron spin rotates much where i =i Cdx· ei x x ei x . Then the marginal mixed- ⌫ faster than a nuclear spin. We assume that the magnetic field state Berry phase A of a subsystem A is defined by B is rotated slowly enough for both the electron and nuclear ⌫ ͚ ͑ ␤A͒ ͑ ͒ spins to evolve adiabatically. The magnetic field B=Bnˆ in A = arg pi exp i i . 12 i the direction of nˆ =͑sin ␪ cos ␾,sin ␪ sin ␾,cos ␪͒ is con- structed starting from B=Bzˆ. First, it is rotated about the y With Eqs. ͑11͒ and ͑12͒, let us analyze how the total Berry ␪ ͉ −͘ axis by an angle . And it is subsequently rotated about the z phase and the marginal Berry phase of E0 depend on B. The

062106-4 ENTANGLEMENT, BERRY PHASES, AND LEVEL… PHYSICAL REVIEW A 78, 062106 ͑2008͒

phase of the electron spin jumps at ␪=␲/2 and B=0. The node at B=0 corresponds to the avoided crossing.

Γ e π -0.2 0 IV. SODIUM ATOM IN A UNIFORM MAGNETIC FIELD −π 0 0 B (T) A. Energy spectrum π/4 π/2 3π/4 23 θ π 0.2 Now we consider an Na atom in its 3S1/2 ground state in the presence of a uniform magnetic ﬁeld B along the z axis. 23 / FIG. 3. ͑Color online͒ Marginal Berry phase ⌫e of the electron The nuclear and electron spins of an Na atom are I=2 3 ͉ −͘ / spin for the entangled eigenstate E0 of a hydrogen atom as a and S=1 2, respectively. Although the Breit-Rabi Hamil- function of B and the azimuthal angle ␪. Here we take f =1, aЈ tonian of an 23Na atom is similar to that of a hydrogen atom =19.767 T−1, and bЈ=−0.03 T−1. in Sec. III, the former shows different features from the lat-

␣ ter. The ground state of the Breit-Rabi Hamiltonian of an 2 23 two Schmidt coefﬁcients are given by p1 =sin 2 and p2 Na atom has both real crossing at B=0 and avoided cross- 2 ␣ =cos 2 . It is easy to obtain the marginal Berry phase of the ing at B 0 as will be shown. As in Sec. III, it is convenient ⌫ ͑ ␣ ⍀ ͒ ͕͉ ͖͘ electron spin, e =arctan cos tan 2 , and the average Berry to arrange the product basis mS ,mI in decreasing order of ␤ ϵ ␤e ␤e ⍀ ␣ ͕͉ 1 3 ͖͘ phase of the electron spin, e p1 1 +p2 2 = 2 cos . In the the magnetic quantum number m of Jz as follows: 2 , 2 , ӷ ␣→ ͉ −͘→͉ 1 1 ͘ ␤ ͕͉ 1 1 ͘ ͉ 1 3 ͖͘ ͕͉ 1 1 ͘ ͉ 1 1 ͖͘ ͕͉ 1 3 ͘ ͉ 1 1 ͖͘ limit of B 1—i.e., 0—one has E0 − 2 , 2 and e 2 , 2 , − 2 , 2 , 2 ,−2 , − 2 , 2 , 2 ,−2 , − 2 ,−2 , and ⍀/ ͕͉ 1 3 ͖͘ ͕͉ 1 1 ͘ ͉ 1 3 ͖͘ = 2. Also the marginal Berry phase of the electron spin is − 2 ,−2 . For example, 2 , 2 , − 2 , 2 spans the subspace ⌫ ⍀/ ഛ␪Ͻ ␲ ⌫ ͑ ͒ given by e = 2 for 0 2 . Figure 3 plots e as a func- of m=1. In this ordered basis set, the Hamiltonian 1 for the tion of B and the azimuthal angle ␪. The marginal Berry sodium atom can be represented by a block-diagonal matrix

1 3 fͩ , ͪ 000000 0 2 2 ⎛ ͱ ⎞ 1 1 3 0 fͩ , ͪ A 0 00 0 0 2 2 2 ͱ3 −1 3 ⎜ 0 A fͩ , ͪ 000 0 0⎟ 2 2 2 1 −1 A 00 0fͩ , ͪ 00 0 2 2 2 H = ⎜ ⎟, ͑13͒ A −1 1 00 0 fͩ , ͪ 00 0 2 2 2 1 −3 ͱ3 00000fͩ , ͪ A 0 ⎜ 2 2 2 ⎟ ͱ3 −1 −1 00000 A fͩ , ͪ 0 2 2 2 ⎝ −1 −3 ⎠ 0000000fͩ , ͪ 2 2

where f͑m ,m ͒ϵAm m +m aB+m bB. Each block is at 1 3 S I S I S I ͉ ͘ ͯ Ϯ Ϯ ʹ ͑ ͒ ϫ EϮ = , . 14b most a 2 2 matrix and can be easily diagonalized. First, 2 2 2 consider the subspace of m= Ϯ2. The corresponding eigenvalues and eigenvectors can be written as Notice that Eqs. ͑14͒ are comparable to Eqs. ͑5͒. Second, in ͕͉ 1 1 ͘ ͉ 1 1 ͖͘ the subspace with m=1 spanned by 2 ,−2 , − 2 , 2 , one obtains the eigenvalues and eigenvectors

3 1 Ϯ A 1ͱ 2 2 EϮ = A Ϯ ͑a +3b͒B, ͑14a͒ E =− + bB Ϯ ͓A + ͑a − b͒B͔ +3A , ͑15a͒ 2 4 2 +1 4 2

062106-5 OH et al. PHYSICAL REVIEW A 78, 062106 ͑2008͒

␣ 1 1 ␣ 1 3 (a) m=±2 ͉ + ͘ 1 ͯ ʹ 1 ͯ ʹ ͑ ͒ 4 m=+1 E+1 = cos , + sin − , , 15b m= -1 2 2 2 2 2 2 m= 0 2 Em/A ␣ 1 1 ␣ 1 3 0 ͉ − ͘ 1 ͯ ʹ 1 ͯ ʹ ͑ ͒ E+1 = − sin , + cos − , , 15c 2 2 2 2 2 2 -2 ͱ ␣ ϵ 3A where tan 1 A+͑a−b͒B . Third, the Hamiltonian of m=−1 is -4 ͕͉ 1 3 ͘ ͉ 1 1 ͖͘ 1 defined on the subspace spanned by 2 ,−2 , − 2 ,−2 . Its (b) eigenvalues and eigenstates are given by S 0.5 Ϯ A 1 E =− − bB Ϯ ͱ͓A − ͑a − b͒B͔2 +3A2, ͑16a͒ −1 4 2 0 -0.2 -0.1 0 0.1 0.2 ␣ 1 1 ␣ 1 3 B (T) ͉E+ ͘ = cos 2 ͯ− ,− ʹ + sin 2 ͯ ,− ʹ, ͑16b͒ −1 2 2 2 2 2 2 FIG. 4. ͑Color online͒͑a͒ Energy levels Em /A and ͑b͒ the von Neumann entropy S of the electron ͑or nuclear͒ spin for the eigen- ␣ 1 1 ␣ 1 3 ͉ − ͘ 2 ͯ ʹ 2 ͯ ʹ ͑ ͒ states of the sodium atom as a function of B. The two parameters of E = − sin − ,− + cos ,− , 16c −1 −1 −1 2 2 2 2 2 2 a sodium atom, aЈ=32.091 T and bЈ=−0.012709 T , are taken ͓ ͔ ͱ from Ref. 22 . ␣ ϵ 3A ͑ ͒ where tan 2 A−͑a−b͒B . Note that Eqs. 16 can be obtained ͑ ͒ from Eqs. 15 by replacing B with −B. Finally, the subspace rotating the magnetic field Bnˆ slowly. For an adiabatic rota- ͕͉ 1 1 ͘ ͉ 1 1 ͖͘ of m=0 is spanned by 2 ,−2 , 2 ,−2 . The corresponding tion keeping the azimuthal angle ␪ constant and varying the eigenvalues and eigenvectors are given by polar angle ␾ from0to2␲, the instantaneous eigenstates A 1 accumulate the total Berry phases proportional to the mag- Ϯ Ϯ ͱ͑ ͒2 2 2 ͑ ͒ ␤ ϯ ⍀ E0 =− a − b B +4A , 17a netic quantum number m, = m . In contrast to a hydro- 4 2 ͉ − ͘ ͉ − ͘ gen atom, the ground state is given either by E+1 or by E−1 with m= Ϯ1, so it acquires the total Berry phase ␤= ϯ⍀. ␣ 1 1 ␣ 1 1 ͉E+͘ = cos 0 ͯ ,− ʹ + sin 0 ͯ− , ʹ, ͑17b͒ Let us analyze how the marginal Berry phase of the en- 0 2 2 2 2 2 2 tangled state is related to the avoided crossings. We focus on ͉ − ͘ the eigenstate E . It has two Schmidt coefficients p1 ␣ +1 ␣ ␣ 1 1 ␣ 1 1 =sin2 1 and p =cos2 1 . With Eq. ͑11͒, one obtains the total ͉E−͘ = − sin 0 ͯ ,− ʹ + cos 0 ͯ− , ʹ, ͑17c͒ 2 2 2 0 2 2 2 2 2 2 phase as a sum of the Berry phases acquired by nuclear and electron spins with weights of the Schmidt coefficients: ␣ ϵ A ͑ ͒ where tan 0 ͑a−b͒B . As expected, Eqs. 17 is very similar to Eqs. ͑7͒ in the case of a hydrogen atom. ␣ ⍀ ⍀ ␣ ⍀ 3⍀ ␤ = sin2 1 ͩ− − ͪ + cos2 1 ͩ+ − ͪ =−⍀. B. Entanglement 2 2 2 2 2 2 Let us examine the relation between entanglement and ͑18͒ level crossings or avoided crossings for a sodium atom. With 23 the values of the parameters A, a, and b of the Na atom in From Eq. ͑12͒, one obtains the marginal Berry phases of an ͓ ͔ EϮ /A ͑ ͒ ⌫ ⌫ Ref. 22 , the energy levels m are plotted in Fig. 4 a . electron spin e and of nuclear spin n: The von Neumann entropies of the electron ͑or nuclear͒ spin for each eigenstates are shown in Fig. 4͑b͒. The ground state ␣ ␣ ͉ − ͘ Ͼ ͉ − ͘ Ͻ ⌫ ͩ 2 1 −i⍀/2 2 1 −i3⍀/2ͪ ͑ ͒ is given by E+1 for B 0 and E−1 for B 0. Two levels n = arg sin e + cos e , 19a + − 2 2 E+1 and E+1 are avoided crossing and maximally entangled at ͑ ͒ ͱ + − A− a−b B= 3A. Another two levels E−1 and E−1 are avoided crossing and maximally entangled at A+͑a−b͒B Ϯ ⍀ =ͱ3A. Two levels with m=0, E are avoided crossing and ⌫ = arctanͩcos ␣ tan ͪ. ͑19b͒ 0 e 1 2 maximally entangled at B=0. Two levels EϮ2 show real crossing at B=0 and have zero von Neumann entropies. ⌫ Again one can see that the eigenstate is maximally entangled Figure 5 plots the marginal Berry phase of a nuclear spin n at the avoided crossing point. This is analogous to the results as a function of B and the azimuthal angle ␪. In the limit of ͓ ͔ ӷ ␣ → ͉ − ͘→͉ 1 3 ͘ ⌫ ⍀/ in Ref. 13 , where Shannon entropy is used as an indicator B 1—i.e., 1 0—one has E+1 − 2 , 2 , e = 2, and ⌫ ⍀/ of avoided crossings. n =−3 2. It is clearly seen that the node of the marginal Berry phase of a nuclear ͑or electron͒ spin corresponds to the C. Berry phase avoid crossing at A+͑a−b͒B=ͱ3A. Thus it could be ex- As in Sec. III C, let us consider an adiabatic cyclic evo- pected that the marginal Berry phase of a subsystem for an lution of nuclear and electron spins of a sodium atom by entangled state has a node at avoided crossings.

062106-6 ENTANGLEMENT, BERRY PHASES, AND LEVEL… PHYSICAL REVIEW A 78, 062106 ͑2008͒

between level crossings, entanglement, and Berry phases. It is shown that entanglement between nuclear and electron spins is maximum at avoided crossing points. The Berry Γn π -0.2 phase and the von Neumann entropy change abruptly at level 0 crossings as the parameters of the Breit-Rabi Hamiltonian −π 0 for a hydrogen atom vary. An entangled state encircling the 0 B (T) π/4 π/2 avoided crossing acquires the marginal Berry phase of an 3π/4 θ π 0.2 electron ͑or nuclear͒ spin like an eigenstate moving around the real crossing accumulates a Berry phase. We have shown FIG. 5. ͑Color online͒ Marginal Berry phase of the nuclear spin that the nodal points of the marginal Berry phase of an en- ⌫ ͉ − ͘ ␪ n for the eigenstate E+1 as a function of B and . tangled state corresponds to the avoided crossing points.

V. CONCLUSIONS ACKNOWLEDGMENTS We have considered the Breit-Rabi Hamiltonians for hy- This work was in part supported by the visitor program of drogen and sodium atoms, describing the hyperfine interac- Max Planck Institute for the Physics of Complex Systems. tion between a nuclear spin and an electron spin in the pres- We would like also to thank the Binational Israel-U.S. Foun- ence of a magnetic field. We have examined the relation dation ͑BSF͒ for financial support.

͓1͔ L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 2nd ed. ͓15͔ O. Osenda, Z. Huang, and S. Kais, Phys. Rev. A 67, 062321 ͑Pergamon Press, Oxford, 1965͒; C. Zener, Proc. R. Soc. Lon- ͑2003͒. don, Ser. A 137, 696 ͑1932͒. ͓16͔ S. Oh, e-print arXiv:0807.4662v1. ͓ ͔ 2 E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, ͓17͔ G. Breit and I. I. Rabi, Phys. Rev. 38, 2082 ͑1931͒. ͑ ͒ and D. Preda, Science 292, 472 2001 . ͓18͔ C. P. Slichter, Principles of Magnetic Resonance ͑Springer- ͓3͔ M. V. Berry, Proc. R. Soc. London, Ser. A 392,45͑1984͒. Verlag, Berlin, 1990͒. ͓4͔ A. Shapre and F. Wilczek, Geometric Phases in Physics ͓19͔ B. Patterson, Rev. Mod. Phys. 60,69͑1988͒. ͑World Scientiﬁc, Singapore, 1998͒. ͓ ͔ ͑ ͒ ͓5͔ D. Yarkony, Rev. Mod. Phys. 68, 985 ͑1996͒. 20 D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 1998 . ͓ ͔ ͓6͔ A. Kuppermann, in Dynamics of Molecules and Chemical Re- 21 S. N. Foner, E. L. Cochran, V. A. Bowers, and C. K. Jen, J. ͑ ͒ actions, edited by R. E. Wyatt and J. Z. H. Zhang ͑Marcel Chem. Phys. 32, 963 1960 . Dekker, New York, 1996͒, pp. 411–472. ͓22͔ E. Arimondo, M. Inguscio, and P. Violino, Rev. Mod. Phys. ͓7͔ M. Vojta, Rep. Prog. Phys. 66, 2069 ͑2003͒. 49,31͑1977͒. ͓8͔ J. P. Neirotti, P. Serra, and S. Kais, Phys. Rev. Lett. 79, 3142 ͓23͔ J. Gruska, Quantum Computing ͑McGraw-Hill, London, ͑1997͒. 1999͒. ͓9͔ P. Serra, J. P. Neirotti, and S. Kais, Phys. Rev. Lett. 80, 5293 ͓24͔ S. Kais, in Reduced-Density-matrix Mechanics with Applica- ͑1998͒; Phys. Rev. A 57, R1481 ͑1998͒. tions to Many-electron Atoms and Molecules, Advances in ͓10͔ S. Kais and P. Serra, Adv. Chem. Phys. 125,1͑2003͒; Int. Chemical Physics, Vol. 134, edited by D. A. Mazziotti ͑Wiley, Rev. Phys. Chem. 19,97͑2000͒. New York, 2007͒. ͓11͔ M. Bhattacharya and C. Raman, Phys. Rev. Lett. 97, 140405 ͓25͔ E. Sjöqvist, A. K. Pati, A. Ekert, J. S. Anandan, M. Ericsson, ͑2006͒; Phys. Rev. A 75, 033406 ͑2007͒. D. K. L. Oi, and V. Vedral, Phys. Rev. Lett. 85, 2845 ͑2000͒. ͓12͔ Q. C. Shi and S. Kais, J. Chem. Phys. 121, 5611 ͑2004͒. ͓26͔ E. Sjöqvist, X. X. Yi, and J. Åberg, Phys. Rev. A 72, 054101 ͓13͔ R. González-Férez and J. S. Dehesa, Phys. Rev. Lett. 91, ͑2005͒. 113001 ͑2003͒. ͓27͔ X. X. Yi, L. C. Wang, and T. Y. Zheng, Phys. Rev. Lett. 92, ͓14͔ L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. 150406 ͑2004͒. Phys. 80, 517 ͑2008͒. ͓28͔ X. X. Yi and E. Sjöqvist, Phys. Rev. A 70, 042104 ͑2004͒.

062106-7