How to Detect Level Crossings Without Looking at the Spectrum

arXiv:0705.1958v1 [physics.atom-ph] 14 May 2007 semiconductors ra h osqec fvriga nenl(..in- (e.g. internal an varying field electric) of coordinate, (e.g. consequence external ternuclear) the an tuning as of or result a as cur es htte r o evident not are they that sense atoms, as such systems physical many of otx fsml hsclsses n htcudbe could : that are level, the and (under)graduate in systems, the arise at physical that considered simple questions Some of (avoided) context detail. treat any texts in introductory crossing few physics, in uitous auscoscrepn o‘idn ymtiso the of eigen- symmetries of ‘hidden’ number system. to large physical a correspond where cross spectrum values the in points rnpre nacoe ici rudacosn nthe in phase. crossing eigenstate Berry a a an around up circuit example picks closed spectrum For a in transported crossings. with (avoided) associated been such have phenomena physical teresting ec fqatmcasi hsclsystem. physical emer- a in the chaos signals quantum spectrum of example gence the third in a As avoidance eigenvalue Hamiltonian. the with commute that rsig n vie rsig cu ntespectra the in occur crossings avoided and Crossings npt fisiprac sabscpeoeo ubiq- phenomenon basic a as importance its of Inspite .I hr a opeittettlnme of number total the predict to way a there Is 1. .Smtmstecosnsaehr odseni the in discern to hard are crossings be the spectrum Sometimes the 5. in - any if - degeneracies the Can 4. mechanisms physical the identify to way a there Is 3. (avoided) the all locate to way systematic a there Is 2. aodd rsig ntesetu ? spectrum the in crossings (avoided) pcrm-cnte evsaie nacerrway clearer a in visualized be ? they can - spectrum ? crossings as of thought cross- ? (avoided) spectrum of the in occurrence ings the for responsible ? spectrum the in crossings ASnmes 37.e 37.m 44.z 73.21.Cd 84.40.Az, field. 03.75.Lm, magnetic 03.75.Be, uniform numbers: a PACS in atom d hydrogen to state Hamiltonian ground Breit-Rabi the the superi of using sometimes is method which handlin the crossings illustrate of visualize to capable way is a offers crossings, matrices (avoided) by to prope representable this introduction Hamiltonians use quantum then of We spectra eigenvalues. the calculating without ermn h edrta ti osbet eli w rmr e more or two if tell to possible is it that reader the remind We 3 .INTRODUCTION I. o odtc ee rsig ihu okn ttespectru the at looking without crossings level detect to How 8 n irwv cavities. microwave and hs ymtisaehde nthe in hidden are symmetries These 6 eateto hsc,Uiest fAioa usn Ariz Tucson, Arizona, of University Physics, of Department o ntne ubro in- of number A instance. for priori a 7 saohrexample, another As 4 sobservables as hycnoc- can They 1 molecules, 9 Dtd coe 2 2018) 22, October (Dated: .Bhattacharya M. 5 2 nasre farticles, of series discriminant. a the In perhaps tool, but useful mathematical very to a neglected reintroduce crossings (avoided) we of approach It study (avoided) this the to others. describing approach In some systematic as but crossings. simple well a as for questions allows basic these dresses n nyta h opiae oeua oetasare potentials molecular complicated the real. that only assum- the molecules ing of for breakdown approximation the detect Born-Oppenheimer to us allows methods gebraic is determined. Hamiltonian the completely they when not known - crossings well (avoided) physicists find to fact also unknown a can not - and Hamiltonian re- mathematicians the to without not of crossings are (avoided) solution they locating quiring that quantum of way shown of capable have powerful spectra only We a the is in systems. crossings mechanical techniques (avoided) algebraic locate of to use how tail unu ehnc swl slna algebra. linear as of well understanding strik- as mechanics their a quantum enhances in as that included (under)graduates demonstration be for ing easily curriculum could physics article the this believe we in Indeed they exposition research, that tool. the pedagogical advanced below effective of an show form context we also the resonances, Feshbach in namely used were niques Icnan ipemteaia nrdcinuiga using introduction mathematical simple 2 a contains II n opeeifrainaotteprmti symmetries Hamiltonian. parametric invariants the the of of encode about invariants class These information new complete Hamiltonian. a Breit-Rabi derive the to of us allow techniques ie o h edr eto Isple discussion. a supplies VI exer- some Section in reader; suggests atom the V hydrogen for Section state cises field. ground magnetic the uniform on a far so developed × × nti ril eepo nagbacmto htad- that method algebraic an employ we article this In lhuhi h okjs etoe leri tech- algebraic mentioned just work the in Although h eto h ae sarne sflos Section follows. as arranged is paper the of rest The arx eto I eeaie hst h aeo an of case the to this generalizes III Section matrix. 2 16 n elsi aitnasaayial,and analytically, Hamiltonians realistic g rt htpoie ytesetu.We spectrum. the by provided that to or arx.ScinI eosrtstetechnique the demonstrates IV Section . matrix hsapoc rvdsapedagogical a provides approach This . srb h yefieZea structure hyperfine-Zeeman the escribe the in crossings (avoided) detect to rty gnauso arxaeequal, are matrix a of igenvalues 16 14,15,16 n 85721 ona saohreape h s fal- of use the example, another As 15 ehv eosrtdi de- in demonstrated have we soeeape algebraic example, one As m 10,11,12,13 II. A 2 × 2 MATRIX III. AN n × n MATRIX

In order to motivate the general case we first consider For a real symmetric n × n matrix H(P ) all of whose a real-symmetric 2×2 matrix entries are polynomials in P the characteristic polynomial is E1 V H(P )= , (1) n V E2 m |H(P ) − λ| = Cmλ , (8) mX=0 with (unknown) eigenvalues λ1,2. The notation implies that all the matrix elements depend on some tunable pa- where the coefficients C are all real. The discriminant rameter P . Also E could be the bare energies of a m 1,2 is defined as17 two-level quantum system, which are mixed by the perturbation V . Usually to find λ1,2 we solve the equation n 2 D[H(P )] ≡ (λi − λj ) , (9) |H(P ) − λ| =0, (2) Yi

It is important to note that we did not actually calculate 3. A crossing is defined as the intersection of two λ1,2 at any point in the discussion so far. Clearly, ∆ = 0 eigenvalues. Hence the simultaneous intersection whenever a level crossing occurs in the spectrum of H(P ). m of m eigenvalues gives rise to = m(m−1)/2 For example, if E1 = E2 = P , and V = 2P, then ∆ = 2 4P 2, and there is a level crossing at P = 0. This may crossings. be verified by calculating the eigenvalues λ1,2 = P, 3P . Note that a single crossing in the spectrum corresponds 4. Every (avoided) crossing contributes, as in Section to a double root of the discriminant. II, a factor quadratic in P to D[H(P )]. For a full We see from this example that use of the discriminant proof, see Ref.15. transforms the problem of finding crossings in the spectrum to a polynomial root-finding problem. Further, it 5. Since H(P ) is real symmetric, the eigenvalues λi enables us to avoid calculating the eigenvalues. Lastly, are all real. It follows from Eq.(9) that D[H(P )] ≥ it provides the locations of all the crossings in the spec- 0. Hence Log[D[H(P )] + 1] ≥ 0 and goes to zero at trum. In the next section we generalize these statements every crossing. We will plot this function in order to the case of an n × n matrix. to visualize crossings.

2 IV. THE HYDROGEN ATOM We now systematically locate all the (avoided) crossings in the spectrum. There is a 6-fold real root at B = 0, A simple but real example of a physical system whose which points to the only crossings in the spectrum (Sec- spectrum exhibits both crossings and avoided crossings tion III 1.). Conventionally these ‘zero-field’ crossings are is a ground state hydrogen atom in a uniform magnetic called degeneracies and are not considered to be cross- field.20 Such an atom is accurately described by the Breit- ings. Hence we will maintain that there are no crossings b=0 Rabi Hamiltonian21 in the spectrum of HBR . Eq.(14) has two complex roots: I S A HBR = A · + B(aSz + bIz), (11) B = ± i. (15) a where I and S indicate the proton and electron spin respectively, and B is the magnetic field along the z- This conjugate pair corresponds to a single avoided cross- axis. A equals the hyperfine splitting and a = geµB and ing at B = 0 (Section III 2. and 4.). From Eq.(15) we can b = gpµN where ge(p) are the electron(proton) gyromag- see that A, the hyperfine coupling of the two spins, is the netic ratios and µB(N) are the Bohr (nuclear) magnetons physical mechanism responsible for the avoided crossing. respectively. The numerical values for these constants Setting A = 0 turns the avoided crossing into a zero-field were obtained from Ref.22 and are to be found in the fig- degeneracy. ure captions in this article. In the basis |MI ,MSi which We can confirm the predictions of the discriminant denotes the projections of I and S along B, the states (Eq.14) by looking at Fig.1(a). At B = 0, three states are 3 in the upper manifold coincide, giving rise to =3 2 1 1 −1 1 1 −1 −1 −1 | , i, | , i, | , i, | , i. (12) crossings (Section III 3.). Further, each crossing con- 2 2 2 2 2 2 2 2 tributes a factor quadratic in B (Section III 4.) to the In this basis, the representation of Eq.(11) is discriminant (Eq.14), which therefore should - and does - contain a factor of B6. This shows that the discriminant A − 2(a + b)B 1 0 −A + 2(a − b)B 2A 1 accounts for the (hyperfine) degeneracies of the spectrum HBR = . B 2A −A − 2(a − b)B C 4 B C @ A + 2(a + b)BA by counting them as crossings. (13) An avoided crossing occurs at B = 0 between the states In order to calculate discriminants the basis (12) is all drawn with dotted lines. In Fig.1(b) a logarithmic rep- we need but we would also like to correlate our results b=0 resentation of D[HBR ] shows a dip corresponding to the to the spectrum, where we will use the basis |F,MF i degeneracies at B = 0. to label the eigenstates of (13). Here F = I + S is the total angular momentum and MF its projection along the magnetic field. This labelling is loose as strictly speaking B. The b 6= 0 case in the presence of a magnetic field MF is a good quantum number but F is not. When b 6= 0, we find from Eq.(13)

2 A. The b = 0 case D[H ]= 1 (a + b)2B6 A2 + (a − b)2B2 BR 16 (16) × [(a + b)A +2 abB]2 [(a + b)A− 2abB]2 . To begin we recollect that |b| ≪ |a|, since the proton is more massive than the electron.22 We set b = 0 in The 6-fold real root at B = 0 persists from the b = 0 Eq.(13), calculate its characteristic polynomial, and find case (Eq.14), but now there are (‘authentic’) crossings the discriminant to be for B 6=0 at 1 D[Hb=0]= A4a6B6 A2 + a2B2 . (14) (a + b)A BR 16 B = ± . (17) 2ab b=0 Considering D[HBR ] as a polynomial in B, the total ‘Switching’ b off and on therefore reveals the physical number of its roots equals its degree, namely 8. Using mechanism behind the appearance of crossings in the the results 1., 2. and 4. from Section III we can then Breit-Rabi spectrum : it is the interaction of the pro- say that the total number of the avoided crossings plus ton spin with the external magnetic field B. b=0 the number of crossings in the spectrum of HBR is ex- The complex roots of Eq.(16) are actly 8/2=4, a fact verified below. Thus the discriminant allows us to predict the number of (avoided) crossings be- A B = ± i (18) fore they are actually found. a − b

3 BHGL BHTL -1000 -500 0 500 1000 20 -15 10 -5 0 5 10 15 20 60 7.5 L 40 È1, 1\ L È \ È \

1 1, 1 , 1, 0 1 - 5. 20 È1, 0\ - cm cm

3 HaL 2.5 3 -

0 -

10 0. 10 H - - È1, 1\ H 20 -2.5 HaL -40 -5. Energy È0, 0\ Energy -60 -7.5 È0, 0\, È1, -1\

0.5

D 50 D 1 1 + + D 0.4 D 0 40 = b BR HbL BR H H @ 30 @ 0.3 D b ND

@ 20 0.2 N

@ HbL

Log 10 Log 0.1 0 - - 0. 1000 500 0 500 1000 - - BHGL 20 15 10 5 0 5 10 15 20 BHTL

0.4 b=0 FIG. 1: (a) The spectrum of HBR for atomic hydrogen (S = 1/2,I = 1/2) with states labelled using the |F, MF i basis. L 0.2 − È1, 1\ The parameters used were A = 0.0473cm 1 and a = 9.35 × −5 −1 −1 10 cm G . The spectrum was calculated by analytically 0. È1, 0\ arb. units

diagonalizing the matrix in Eq.(13) and numerically plotting H b=0 the eigenvalues. (b) Log[N0D[HBR ] + 1] plotted using the È1, -1\ 35 -0.2 HcL same parameters as in (a) and the scaling factor N0 = 10 Energy used to optimize visibility. The overall logarithmic trend is È0, 0\ clearly visible. -0.4 -7.5 -5. -2.5 0. 2.5 5. 7.5 BHarb. unitsL and imply an avoided crossing at B = 0 as in Section IV A (Eq.15). FIG. 2: (a) The spectrum of HBR for atomic hydrogen (S = − 1/2,I = 1/2). The parameters used were A = 0.0473cm 1 , − − − − − − We can conﬁrm the predictions of the discriminant (16) a = 9.35 × 10 5cm 1G 1 and b = −1.4202 × 10 7cm 1G 1. by looking at Fig.2. The crossings which occur at ±16.6 The spectrum was calculated by analytically diagonalizing the T (from Eq.17) for actual values of the hydrogenic param- matrix in Eq.(13) and numerically plotting the eigenvalues. eters a, b, and A cannot be seen in the spectrum [Fig.2(a)] The crossings at B = ±16.6 T cannot be resolved on this scale. as the energy level pairs are separated by less than the (b) Log[NbD[HBR] + 1] plotted using the same parameters as 10 width of the lines used to plot them, a point made ear- in (a) and the scaling factor Nb = 10 . All three crossings lier in this journal (see Fig.2 in Ref.20). However the are distinctly indicated by the minima of the discriminant; logarithmic representation [Fig.2(b)] of the discriminant those corresponding to crossings at B = ±16T actually do on the same scale clearly shows dips at all the crossings. reach zero, but appear shorter here due to the resolution of the graphics. The overall logarithmic trend of the plot is A scaled spectrum has been shown in Fig.2(c) using a evident. (c) The same spectrum as in (a) but redrawn using much larger relative value of b in order to display the the parameters A = 0.1,a = 0.01, and b = −0.1. The larger crossings clearly. This example illustrates how the dis- relative value of b ensures that the crossings at B 6= 0 can be criminant can sometimes prove superior to the spectrum seen in the plot. in displaying crossings. For another example see Ref.16.

4 V. SUGGESTED EXERCISES the discriminant offers an alternative to locating crossings in the spectrum. It must be noted however that the 1. Prove that the product in Eq.(9) contains n(n − discriminant yields no information about which eigenval- 1)/2 factors. ues avoid or intersect, or about the eigenvectors. Also shallow avoided crossings do not show up in the logarith- 2. Justify the presence of A4 in Eq.(14). mic representation, especially if they are near to crossings, which give rise to strong features in the discrimi- 3. For the parametric symmetry a = b in Eq.(16) show nant. For all such information the spectrum has to be that there are crossings but no avoided crossings in consulted. the spectrum of HBR. The technique we have presented can be used algorith- 4. Plot D[HBR] as a function of B and try to iden- mically on Hamiltonians which are polynomial in some tify the zeros that correspond to crossings. This parameter P , and which can be represented by finite di- is an exercise designed to show that the highly mensional matrices. Examples are a spin 1/2 particle nonlinear nature of the discriminant implies that in a magnetic field23 (the archetypal two-level system) each of its terms dominates in a different regime and the hydrogen atom in an electric field23 (usually pre- of B. This makes it difficult to include all the fea- sented as an example of degenerate perturbation theory). tures of the discriminant on a single scale unless However the method can also yield insight into physi- a smoother representation - such as a logarithmic cal systems whose Hamiltonians are usually truncated one - is adopted. to a finite dimension for practical calculations such as the nucleus modeled as a triaxial rotor,24 and a polar 5. The Wigner von-Neumann non-crossing rule6 says molecule in an electric field.25 Another interesting appli- “States of the same symmetry (i.e. quantum num- cation is the calculation of critical parameters of quan- ber) do not cross, except accidentally.” Verify the tum systems,26 since the critical point occurs at a cross- rule for the |M i states in Figs.1(a) and 2(c). That F ing. An example using the Yukawa Hamiltonian has been is, show that states which (avoid) cross possess treated in Ref.27. A list of physical systems in atomic (same) different M ’s. F and molecular physics to which algebraic methods can be applied is provided in Ref.16. VI. DISCUSSION To conclude we have presented an algebraic technique for systematically analysing (avoided) crossings in the The examples presented above illustrate that the dis- spectra of physical systems and pointed out its useful- criminant is an elegant but simple device for locating and ness as a pedagogical device. It is a pleasure to thank counting (avoided) crossings. Further, it is an effective P. Meystre for support. This work is supported in part tool for investigating the physical mechanisms behind the by the US Army Research Office, NASA, the National occurrence of (avoided) crossings. Lastly, visualization of Science Foundation and the US Office of Naval Research.

1 J. R. Walkup, M. Dunn, D. K. Watson and T. C. Germann, spectra of triangles”, Proc. R. Soc., London Ser. A 392, “Avoided crossings of diamagnetic hydrogen as functions 15 (1984). of magnetic ﬁeld strength and angular momentum”, Phys. 8 E. A. Yuzbashyan, W. Happer, B. A. Altshuler and S. Rev. A 58, 4668-4682 (1998). B. Sastry, “Extracting hidden symmetry from the energy 2 D. R. Yarkony, “Diabolical conical intersections”, Rev. spectrum”, J. Phys. A 36, 2577-2588 (2003). Mod. Phys 68, 985-1013 (1996). 9 F. Haake, Quantum Signatures of Chaos (Springer, Berlin, 3 A. Sibille, J. F. Palmier and F. Laruelle, “Zener intermini- 2006). band resonant breakdown in superlattices”, Phys. Rev. 10 W. D. Heiss and W. -H. Steeb, “Avoided level crossings Lett. 80, 4506-4509 (1998). and Riemann sheet structure”, J. Math. Phys., 32, 3003- 4 J. Wiersig, “Formation of long-lived, scarlike modes near 3007 (1991). avoided resonance crossings in optical microcavities”, 11 V. V. Stepanov and G. Muller, “Integrability and level Phys. Rev. Lett. 97, 253901-1–4 (2006). crossing manifolds in a quantum Hamiltonian system”, 5 G. Moruzzi, The Hanle Eﬀect and Level-Crossing Spec- Phys. Rev. E, 58, 5720-5726 (1998). troscopy (Kluwer Academic Publishers, Norwell, 1991). 12 T. J. Davis, “2D magnetic traps for ultra-cold atoms: a 6 J. von Neumann and E. P. Wigner, “Behaviour of eigenval- simple theory using complex numbers”, Eur. Phys. J. D ues in adiabatic processes”, Z. Phys. 30, 467-470 (1929). 18, 27-36 (2002). 7 M. V. Berry and M. Wilkinson, “Diabolical points in the 13 I. Freund, “Coherency matrix description of optical polar-

5 ization singularities”, J. Opt. A 6, S229-S234 (2004). 22 E. Arimondo, M. Inguscio and P. Violino, “Experimen- 14 M. Bhattacharya and C. Raman, “Detecting level crossings tal determinations of the hyperfine structure in the alkali without looking at the spectrum”, Phys. Rev. Lett. 97, atoms”, Rev. Mod. Phys. 49, 31-75 (1977). 253901-1–4 (2006). 23 E. Merzbacher, Quantum Mechanics (John Wiley and 15 M. Bhattacharya and C. Raman, “Detecting level crossings Sons, New York, 1970). without solving the Hamiltonian I. Mathematical back- 24 J. L. Wood, A-M. Oros-Peusquens, R. Zaballa, J. M. All- ground”, Phys. Rev. A 75, 033405-1–15 (2007). mond and W. D. Kulp, “ Triaxial rotor model for nuclei 16 M. Bhattacharya and C. Raman, “Detecting level crossings with independent inertia and electric quadrupole tensors”, without solving the Hamiltonian II. Applications to atoms Phys. Rev. C 70, 024308-1–5 (2004). and molecules”, Phys. Rev. A 75, 033406-1–14 (2007). 25 J. Brown and A. Carrington, Rotational Spectroscopy of 17 P. D. Lax, Linear Algebra (John Wiley and Sons, New Diatomic Molecules (Cambridge University Press, Cam- York, 1997). bridge, 2003). 18 S. Basu, R. Pollack, and M. F. Roy, Algorithms in Real 26 S. Kais, C. Wenger and Qi Wei, “Quantum criticality at Algebraic Geometry (Springer-Verlag, Berlin, 2003). the infinite complete basis set limit : A thermodynamic 19 In Mathematica the Discriminant is defined in terms of analog of the Yang and Lee theorem”, Chem. Phys. Lett. the Resultant function. 423, 45-49 (2006). 20 R. S. Dickson and J. A. Weil, “Breit-Rabi Zeeman states 27 P. Serra, J. P. Neirotti and S. Kais, “Finite-size scaling of atomic hydrogen”, Am. J. Phys. 59, 125-129 (1991). approach for the Schrodinger equation”, Phys. Rev. A 57, 21 G. Breit and I. I. Rabi, “Measurement of nuclear spin”, R1481 (1998). Phys. Rev. 38, 2082-2083 (1931).