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Landau–Zener formula

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Landau–Zener formula

The Landau–Zener formula is an analytic solution to the equations of motion governing the transition dynamics of a 2-level quantum mechanical system, with a time-dependent Hamiltonian varying such that the energy separation of the two states is a linear function of time. The formula, giving the probability of a diabatic (not adiabatic) transition between the two energy states, was published separately by Lev Landau,[1] Clarence Zener,[2] Ernst Stueckelberg,[3] and Ettore Majorana,[4] in 1932.

If the system starts, in the infinite past, in the lower energy eigenstate, we wish to calculate the probability of finding the system in the upper energy eigenstate in the infinite future (a so-called Landau–Zener transition). For infinitely slow variation of the energy difference (that is, a Landau–Zener velocity of zero), the adiabatic theorem tells us that no such transition will take place, as the system will always be in an instantaneous eigenstate of the Hamiltonian at that moment in time. At non-zero velocities, transitions occur with probability as described by the Landau–Zener formula.

Landau–Zener approximation

Such transitions occur between states of the entire system, hence any description of the system must include all external influences, including collisions and external electric and magnetic fields. In order that the equations of motion for the system might be solved analytically, a set of simplifications are made, known collectively as the Landau–Zener approximation. The simplifications are as follows:

  1. The perturbation parameter in the Hamiltonian is a known, linear function of time
  2. The energy separation of the diabatic states varies linearly with time
  3. The coupling in the diabatic Hamiltonian matrix is independent of time

The first simplification makes this a semi-classical treatment. In the case of an atom in a magnetic field, the field strength becomes a classical variable which can be precisely measured during the transition. This requirement is quite restrictive as a linear change will not, in general, be the optimal profile to achieve the desired transition probability.

The second simplification allows us to make the substitution

\Delta E = E_2(t) - E_1(t) \equiv \alpha t, \,

where \scriptstyle{E_1(t)} and \scriptstyle{E_2(t)} are the energies of the two states at time \scriptstyle{t}, given by the diagonal elements of the Hamiltonian matrix, and \scriptstyle{\alpha} is a constant. For the case of an atom in a magnetic field this corresponds to a linear change in magnetic field. For a linear Zeeman shift this follows directly from point 1.

The final simplification requires that the time–dependent perturbation does not couple the diabatic states; rather, the coupling must be due to a static deviation from a \scriptstyle{1/r} coulomb potential, commonly described by a quantum defect.

The Landau–Zener formula

The details of Zener's solution are somewhat opaque, relying on a set of substitutions to put the equation of motion into the form of the Weber equation[5] and using the known solution. A more transparent solution is provided by Wittig[6] using contour integration.

The key figure of merit in this approach is the Landau–Zener velocity:

v_{LZ} = {\frac{\partial}{\partial t}|E_2 - E_1| \over \frac{\partial}{\partial q}|E_2 - E_1|} \approx \frac{dq}{dt},

where \scriptstyle{q} is the perturbation variable (electric or magnetic field, molecular bond-length, or any other perturbation to the system), and \scriptstyle{E_1} and \scriptstyle{E_2} are the energies of the two diabatic (crossing) states. A large \scriptstyle{v_{LZ}} results in a large diabatic transition probability and vice versa.

Using the Landau–Zener formula the probability, \scriptstyle{P_D}, of a diabatic transition is given by

\begin{align} P_D &= e^{-2\pi\Gamma}\\ \Gamma &= {a^2/\hbar \over \left|\frac{\partial}{\partial t}(E_2 - E_1)\right|} = {a^2/\hbar \over \left|\frac{dq}{dt}\frac{\partial}{\partial q}(E_2 - E_1)\right|}\\ &= {a^2 \over \hbar|\alpha|} \end{align}

The quantity a is the off-diagonal element of the two-level system's Hamiltonian coupling the bases, and as such it is half the distance between the two unperturbed eigenenergies at the avoided crossing, when E_1 = E_2.

Multistate Landau–Zener problem

The simplest generalization of the two-state Landau–Zener model is a multistate system with the Hamiltonian of the form H(t)=A+Bt, where A and B are Hermitian NxN matrices with constant elements. There are exact formulas that provide analytical expressions for special elements of the scattering matrix in any multi-state Landau-Zener model. These include the Brundobler–Elser (BE) formula (noticed by Brundobler and Elser in numerical simulations[7] and rigorously proved by Dobrescu and Sinitsyn,[8] following the contribution of Volkov and Ostrovsky[9]), the no-go theorem (formulated by Sinitsyn[10] and rigorously proved by Volkov and Ostrovsky[11]).

Several classes of completely solvable multistate Landau–Zener models have been identified and studied, including:

  • Demkov–Osherov model[12]
  • Generalized bow-tie model[13]
  • Reducible multistate Landau–Zener models[14]
  • Landau–Zener transitions in a linear chain.,.[15][16]

Noise in the Landau–Zener problem

Applications of the Landau–Zener solution to the problems of quantum state preparation and manipulation with discrete degrees of freedom stimulated the study of noise and decoherence effects on the transition probability in a driven two-state system. Several compact analytical results have been derived to describe these effects, including the Kayanuma formula[17] for a strong diagonal noise, and Pokrovsky–Sinitsyn formula[18] for the coupling to a fast colored noise with off-diagonal components. The effects of nuclear spin bath and heat bath coupling on the Landau–Zener process were explored by Sinitsyn and Prokof'ev[19] and Pokrovsky and Sun,[20][21][22] respectively.

Exact results in multistate Landau–Zener theory (no-go theorem and BE-formula) can be applied to Landau-Zener systems which are coupled to baths composed of infinite many oscillators and/or spin baths (dissipative Landau-Zenertransitions). They provide exact expressions for transition probabilities averaged over final bath states if the evolution begins from the ground state at zero temperature, see in Ref. for oscillator baths[23] and for universal results including spin baths in Ref.[24]

See also

References

  1. ^ L. Landau (1932). "Zur Theorie der Energieubertragung. II". Physikalische Zeitschrift der Sowjetunion 2: 46–51. 
  2. ^ C. Zener (1932). "Non-Adiabatic Crossing of Energy Levels".  
  3. ^ E. C. G. Stueckelberg (1932). "Theorie der unelastischen Stösse zwischen Atomen".  
  4. ^ E. Majorana (1932). "Atomi orientati in campo magnetico variabile".  
  5. ^ Abramowitz, M.; I. A. Stegun (1976). Handbook of Mathematical Functions (9 ed.). Dover Publications. p. 498.  
  6. ^ C. Wittig (2005). "The Landau–Zener Formula".  
  7. ^ S. Brundobler and V. Elser (1993). "S-matrix for generalized Landau–Zener problem".  
  8. ^ B. Dobrescu and N. A. Sinitsyn (2006). "Comment on 'Exact results for survival probability in the multistate Landau–Zener model'".  
  9. ^ M. V. Volkov and V. N. Ostrovsky (2004). "Exact results for survival probability in the multistate Landau–Zener model".  
  10. ^ N. A. Sinitsyn (2004). "Counterintuitive transitions in the multistate Landau–Zener problem with linear level crossings".  
  11. ^ M. V. Volkov and V. N. Ostrovsky (2005). "No-go theorem for bands of potential curves in multistate Landau–Zener model".  
  12. ^ Yu. N. Demkov and V. I. Osherov (1968).  
  13. ^ Yu. N. Demkov and V. N. Ostrovsky (2001). "The exact solution of the multistate Landau–Zener type model: the generalized bow-tie model".  
  14. ^ N. A. Sinitsyn (2002). "Multiparticle Landau–Zener problem: Application to quantum dots".  
  15. ^ N. A. Sinitsyn (2013). "Landau-Zener Transitions in Chains".  
  16. ^ V. L. Pokrovsky and N. A. Sinitsyn (2002). "Landau–Zener transitions in a linear chain".  
  17. ^ Y. Kayanuma (1984),  
  18. ^ Eq. 42 in V. L. Pokrovsky and N. A. Sinitsyn (2004). "Fast noise in the Landau–Zener theory".  
  19. ^ N. A. Sinitsyn and N. Prokof'ev (2003). "Nuclear spin bath effects on Landau–Zener transitions in nanomagnets".  
  20. ^ V. L. Pokrovsky and D. Sun (2007). "Fast quantum noise in the Landau–Zener transition".  
  21. ^ D. Sun and A. Abanov and V. L. Pokrovsky (2008). "Molecular production at a broad Feshbach resonance in a Fermi gas of cooled atoms".  
  22. ^ D. Sun and A. Abanov and V. L. Pokrovsky (2009). "Static and Dynamic properties of a Fermi-gas of cooled atoms near a wide Feshbach resonance". arXiv:0902.2178. Bibcode 2009arXiv0902.2178S.
  23. ^ M. Wubs, K. Saito, S. Kohler, P. Hanggi, and Y. Kayanuma (2006). "Gauging a quantum heat bath with dissipative Landau-Zener transitions".  
  24. ^ K. Saito, M. Wubs, S. Kohler, Y. Kayanuma, and P. Hanggi (2007). "Dissipative Landau-Zener transitions of a qubit: Bath-specific and universal behavior".  
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