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Particle-particle correlations produced by dynamical scatterer

Particle-particle correlations produced by dynamical scatterer. M. Moskalets Dpt. of Metal and Semiconductor Physics, NTU " Kharkiv Polytechnical Institute ", Ukraine. Keszthely, 2006. Pump is a source of entangled particles.

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Particle-particle correlations produced by dynamical scatterer

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  1. Particle-particle correlations produced by dynamical scatterer M. Moskalets Dpt. of Metal and Semiconductor Physics, NTU "Kharkiv Polytechnical Institute", Ukraine Keszthely, 2006

  2. Pump is a source of entangled particles P.Samuelsson and M.Bűttiker, Phys.Rev.B 71, 245317 (2005) C.W.J.Beenakker, M.Titov, B.Trauzettel Phys.Rev.Lett. 94, 186804 (2005) An arbitrary amplitude pump (a projected state with exactly a single excited electron-hole pair) A weak amplitude pump The current noise (CN) is a measure of non-classical correlations: - the BI’s can be formulated in terms of a CN; - the CN produced by the pump violates BI’s. The entanglement entropy (of spins) for e-h pairs:  relates to CN at weak pumping while it is unrelated to CN at strong pumping Thus a current noise (possibly) gives incomplete information about correlations produced by the pump

  3. Our objectives 1. To explore correlations produced by the dynamical scatterer (a pump) at arbitrary in strength but slow driving 2. To establish a relation between the current noise and the particle-particle correlations at strong driving

  4. pump The set-up  = 3  = 4  = 2 S(t) = S(t+) T = 0  = 2/  =   = 1, 2,…, Nr  = 1  = Nr All reservoirs are uncorrelated  incoming particles are uncorrelated

  5. In a given set up stationary scatterer does not produce correlations while dynamical one (a pump) can produce (to illustrate it we analyze an outgoing state)

  6. Outgoing state We will consider particles in states with definite energy E and describe them via the second quantization operators a(E) (for incoming particles) and b(E) (for out-going particles). Perhaps it is better to speak about incoming and out-going modes (single-particle states). However one can speak about the particles belonging to these states. 1. The stationary case: E(in) = E(out) = E The states with different E  E’ are statistically independent (therefore we consider the states with the same E) =1 E incoming particles out-going states Nr Nr =Nr Since all relevant incoming states are either filled (for E  ) or empty (forE > ) then due to unitarity of scattering all out-going states are either filled or empty. Thus there are no correlations: it means that the probability for the state to be filled/empty is fixed: 1 or 0 (at zero temperature) uncorrelated uncorrelated

  7. Outgoing state 2. The dynamical case: E(in) E(out)= E(in) + nћ, n = 0,±1, ± 2,…, ±nmax The states with E >  + nmaxћ / E <  - nmaxћ are fully empty/filled and thus irrelevant Therefore there are 2nmaxNr relevant state. But only 1/2 incoming states are filled single-particle occupations are shown =1 E(in) ћ  2nmax 2nmaxNr out-going states nmaxNr incoming electrons The (particles belonging to the) partially filled states can be correlated

  8. In general, the dynamical scatterer produces 2-, 3-,…,nmaxNr- particle correlations while what we see (the order of visible correlations) depends strongly on how we look at the system

  9. Registered state 1. To obtain complete information about outgoing particles it is necessary to monitor all the relevant 2nmaxNr outgoing states which contain nmaxNr electrons simultaneous occupations are shown single-particle occupations are shown 1 For instance, one can register the state with exactly a single excited electron-hole pair 2 3ћ 3  4 5 6 Such a (registered) state is a multi-electron (3-electron in our case) state. To characterize it we have to use a multi-electron joint probability which includes multi-particle correlations In a presented case it is P(11;15;16) which includes 2-, and 3- electron correlations

  10. Registered state 2. If we monitor only several (say 2) outgoing states we get incomplete description of a whole (multi-particle) outgoing state. However such a description is useful if only these states are in use. For instance, any 2-particle quantities, e.g. a current noise, “monitor” only the states in pairs. 1  4 P(11;14) P(01;14) P(11;04) P(01;04) Other states can be arbitrary occupied.(and contain 1, 2, 2, and 3 electrons, respectively) Two-particle probabilities P(X1;Y4) include only 2-particle correlations.

  11. Reduction of the order of correlations In a general case there are nmaxNrout-going electrons 1. A weak amplitude pump: nmax = 1  the out-going state is an Nr-electron state. Nr = 2 Nr0 : there are Nr0 orbital channels and 2 spin channels. 2. Spin-independent scattering: the out-going state is a product of two (spin ,) nmaxNr0 -electron states. For weak pumping (nmax = 1), spin-independent scattering, and for Nr0= 2(two single channel leads) the out-going state is effectively a 2-electron state. Therefore, in this case the current noise represents all the correlations produced by the pump. Otherwise, the current noise represent only part of correlations.

  12. To investigate particle-particle correlations we calculate a joint probability to find several out-going channels occupied and compare it to the product of occupation probabilities of individual channels The single-channel occupation probability is a one-particle distribution function. The joint multi-channel occupation probabilities are multi-particle distribution functions.

  13. Single-particle distribution function b(En)  En = E + nћ ,  < E <  + ћ adiabatic driving: SF(En,Em) = Sn-m() S(t)  a(Em) incoming particles: out-going particles: (it is a sum of squared single-particle scattering amplitudes)

  14. Two-particle distribution function B,(En,Em) a two-particle operator: (the order is irrelevant)  a two-particle distribution function (a joint probability):  an electron-electron correlation function (a covariance): while incoming electrons are not correlated:

  15. Such a process is prohibited by the Pauli exclusion principle (the level  -Ep contains only single indivisible electron and thus can not be a source for two electrons)  En  Ep  Em Two-particle distribution function i) Contribution of non-correlated particles: Thus we need subtract something to fulfill the Pauli principle

  16. - =             Two-particle distribution function ii) Whole contribution: If there are several incoming particles then the quantum-statistical interaction (due to the Pauli principle) results in existing of two (direct and exchange) amplitudes for any two-particle scattering process In our case it becomes evident if we rewrite a two-particle distribution function in terms of 22 Slater determinants (it is a sum of squared two-particle scattering amplitudes)

  17. Electrical noise and two-particle correlations The zero-frequency current noise power produced by the pump at arbitrary driving amplitude can be expressed in terms of electron-electron (2-particle) correlations: The factor /2 counts all the statistically independent sets of states within the interval 0 < E -  < ћ. M. Moskalets, M. Büttiker,PRB 73, 125315 (2006)

  18. Multi-particle correlations i) a multi-particle (N-particle) operator: ii) a multi-particle distribution function in terms of NN Slater determinants: (it is a sum of squared multi-particle scattering amplitudes) iii) a multi-particle correlation function: PN = (n1,n2,…,nN) is a permutation of integers from 1 to N. The cyclic permutations are excluded.

  19. Multi-particle correlations A generating function: Here: A pair correlator: The unit matrix: A diagonal matrix:

  20. stationary driven 2-particle 3-particle > < Multi-particle correlations A three-particle distribution function: The sign of correlations: - , < 0 2, , 0

  21. Higher order current cumulants and multi-particle correlations Nth-order current correlation function (symmetrized in lead indices): (the sum runs over the set of all the permutations PN=(r1,…rN) of integers from 1 to N; 1=0) The zero frequency current cross-correlator (different leads) can be expressed in terms of the N-particle correlation functions for outgoing particles:

  22. Higher order current cumulants and multi-particle correlations The multi-particle correlation functions ( i.e., irreducible parts of multi-particle probabilities ) are the quantities which are directly related to the higher order current cumulants

  23. Example: a resonant transmission pump V1(t) V2(t) IL IR

  24. Single-particle distribution function weak pumping, a single-particle distribution function n = -1 fL ћ n = 0 E-, ћ

  25. weak pumping, a single-particle distribution function - the dependence on  fL(n=0) f(h)L(n=-1) , 2

  26. strong pumping, a single-particle distribution function fL no dc current  = 0 IL 1e/cycle  = /2 E-, ћ

  27. strong pumping, a single-particle distribution function - the dependence on  fL(n=0) f(h)L(n=-1) , 2

  28. Two-particle correlationsweak pumping, the dependence on  at transmission resonance f(1L,n=0 ;0R,m=-1) fL,n=0 f(h)R,m=-1 , 2

  29. strong pumping, the dependence on  at transmission resonance f(1L,n=0;0R,m=-1) fL,n=0 f(h)R,m=-1 , 2

  30. Three-particle correlationsstrong pumping, the dependence on , at transmission resonance f (L,0; L,+1; R,-1) , 2

  31. strong pumping, the dependence on , at transmission resonance fL, 0 fL, +1 f(h)R, -1 + 2-particle correlations f(1L,0;1L,+1;0R, -1) , 2

  32. Conclusion • The current noise generated by the pump can be expressed in terms of two-particle correlations at arbitrary strength of a drive • The N-particle distribution functions depends on multi- particle correlations up to Nth order • The multi-particle correlations can be experimentally probed via the Nth-order cross-correlator of currents flowing into the leads attached to the pump

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