* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Dotan Davidovich research proposal
Standard Model wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Nuclear structure wikipedia , lookup
Path integral formulation wikipedia , lookup
Photon polarization wikipedia , lookup
Introduction to quantum mechanics wikipedia , lookup
Old quantum theory wikipedia , lookup
Relational approach to quantum physics wikipedia , lookup
Perturbation theory (quantum mechanics) wikipedia , lookup
ATLAS experiment wikipedia , lookup
Double-slit experiment wikipedia , lookup
Renormalization wikipedia , lookup
Electron scattering wikipedia , lookup
Quantum tunnelling wikipedia , lookup
Future Circular Collider wikipedia , lookup
Identical particles wikipedia , lookup
Compact Muon Solenoid wikipedia , lookup
Elementary particle wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Monte Carlo methods for electron transport wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Eigenstate thermalization hypothesis wikipedia , lookup
Probability amplitude wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Adiabatic and Non-adiabatic Transport in Networks Dotan Davidovich Adviser: Doron Cohen Physics Department, Ben Gurion University of the Negev (Dated: October 27, 2011) ====== 1. Background Quantum networks and quantum random networks appear in many different work on various subjects, starting from future quantum computation models[1] ,quantum internet[2] and even in the explanation of photosynthesis[3]. Of particular interest are the currents and counting statistics[4, 5] in driven networks. These were studied in the adiabatic regime[6] for open and closed systems called ”pumping”[7–9] and ”stirring”[10–12] respectably. There are still open question regarding these systems behaviour for non-adiabatic driving and beyond the 2-level landau-zener dynamics[13]. ====== 2. Abstract We would like to study a generalized Landau-Zener crossing problem [13]. In our setup a shuttle level crosses a band of network levels. A particle that is located initially in the shuttle has the probability to be transferred into the network. The questions that we would like to ask are the following: (1) What is the survival probability 𝑃 (𝑡) of the particle in the shuttle? (2) What is the final distribution of the occupation probabilities in real space? (3) What is the final distribution of the probabilities in energy space? (4) What is the statistics of the induced currents in the network. We would like to know how the answers depend on the rate in which the energy of the shuttle is varied, ranging from the adiabatic to the diabatic / sudden regimes. With regard to (4) we are going to analyse the distribution of the circulated charge 𝑄. In the stochastic approximation the distribution is bounded to the range |𝑄| < 1, while in the quantum adiabatic limit |𝑄| might be arbitrarily larger due to a stirring effect. Consequently we would like to develop a phenomenology for the induced stirring in random networks. Modelling .– 𝑁 sites system, the sites are randomly scattered on an 𝐿 × 𝐿 sheet, the on site energies are randomly chosen from a range of [0, Δ]. The system is build using several parameters: 𝐿 𝑟0 = √ = average distance. 𝑁 𝑐0 = typical coupling between neighbouring sites. 𝜉 = interaction range 𝜖𝑖 = on-site energy = random ∈ [0, Δ] 𝑟𝑖 = site’s location = random ∈ [0, 𝐿] × [0, 𝐿] The hopping amplitude is determined by the distance between the sites in the following manner: 2 𝑐𝑖𝑗 = hopping amplitude = ±𝑐0 𝑒 𝑟0 −|𝑟𝑖 −𝑟𝑗 | 𝜉 (1) The ± indicates that the sign is random but we also simulate only positive hopping amplitude. Another optional definition of the model is 𝑁 site lattice with random on-site energies taken from [0, Δ], and only nearest neighbours interaction. The interaction is defined as followed: 𝑐𝑖𝑗 = ±𝑐0 e−𝑟𝑎𝑛𝑑𝑜𝑚 , 𝑟𝑎𝑛𝑑𝑜𝑚 ∈ [−𝜎, 𝜎] (2) Δ in We have two dimensionless parameters that describe the system ,𝜉/𝑟0 and Δ/𝑐0 in the first system, and 𝜎 and 𝑐0 the second. These are approximately the same systems because in the second model 𝜎 plays the roll of 𝜉/𝑟0 so we get a random network. Participation number.– The main parameter we use to characterize the eigenstates of the system is the participation number 𝑃 𝑁𝜓 which tells how many sites are involved at a certain state 𝜓 , this is defined by: 𝑃 𝑁𝜓 = ∑︀ 1 * 2 , 𝑖 = site number (𝜓 𝑖 𝜓𝑖 ) 𝑖 (3) This gives 1 if the particle is in 1 site and 𝑁 if the particle is uniformly distributed along the system. The average participation number for the eigenstates of the system is: ∑︀ 𝑃 𝑁𝑛 𝑃 𝑁𝑛 = 𝑛 (4) 𝑁 In Fig.1 we show the dependence of 𝑃¯𝑁 on the dimensionless parameters of the model. Time dependent scenario .– The particle starts at one of the sites with energy 𝜖0 = 𝑢(𝑡). The energy is raised linearly at rate 𝑢˙ until the site’s energy is far above the band. We then have the final state |𝜓(𝑡𝑓 )⟩ which we characterize with 𝑃 𝑁𝑖 and 𝑃 𝑁𝑛 , see Fig.3. Survival probability - 𝒫(𝑡).– The survival probability is the probability to remain in the starting site. If the initial condition was 𝑝0 (0) = 1, 𝑝𝑖 (0) = 0 for 𝑖 ̸= 0, then 𝒫(𝑡) = 𝑝0 (𝑡). In our simulation at moderate 𝑢˙ the particle will ”flow” to the band as seen in Fig.2. ====== 3. Work plan Survival probability at the starting site.– It is of interest to characterize the probability of the particle to stay in the shuttle at the end of the process and how it depends on 𝑢˙ , the adiabatic and sudden regime are well studied ,here we want to know what is the survival probability 𝑃 (𝑡) of the particle in the shuttle , after the shuttle has passed the band. Final site-occupation distribution.– As the shuttle passes through the band the particle ”flows” to the other sites. We would like to find out what is the probabilities of the particle to be at the different sites at the end of the process and the distribution of these probabilities and how they depend on the shuttle rate. We can see a the dependence in Fig.3. Final energy distribution.– As the shuttle passes through the band the particle ”flows” to the other eigenstates. We would like to find out what is the probabilities of the particle to be at the different eigenstates/energies at the end of the process and the distribution of these probabilities and how they depend on the shuttle rate. We can see a the dependence in Fig.3. Induced currents.– We would like to know what are the statistics of the induced currents and of the circulated charge in the network , and how they depend on the rate of the shuttle from the adiabatic to the diabatic/sudden regimes.The circulated charge is the charge passed through some close path. Optional scenarios.– The starting conditions can be changes so the particle starts at an eigenstate in the middle of the band , and the level we change with time, partially ”scoops” it and partially ”scatters” it over other eigenstates. 3 Geometrical conductance.– The distribution of the geometrical conductance of a current created by the moving of a scatterer through a network is described in Fig.2 [11] but the fitting parameters of this distribution are yet to be explained. We want to find a theory to explain these fitting parameters. ====== 4. Preliminary results In our simulation the particle starts at one of the sites at very low energy 𝜖0 = 𝑢(𝑡) (far from the network band) , then the site’s energy is raised linearly at rate 𝑢˙ until it is far above the band levels. We would like to characterize the system at the end of the process and it’s dependence on 𝑢. ˙ We start by solving the Schrdinger equation with time-dependent Hamiltonian in order to get 𝜓(𝑡). ⎛ ⎞ ⎛ ⎞ 1 𝜖0 𝑐01 𝑐02 · · · ⎟ ⎜ ⎜ ⎟ 𝑐 𝜖 𝑐 · · · 10 1 12 𝜕𝜓 ⎜0⎟ ⎜ ⎟ (5) 𝜖0 = 𝑢(𝑡) , 𝜓(𝑡 = 0) = ⎜0⎟ = −𝑖𝐻𝜓 , 𝐻 = ⎜𝑐20 𝑐21 𝜖2 · · ·⎟ , 𝜕𝑡 ⎝ ⎠ ⎝ ⎠ .. .. .. . . . . ··· . . Where 𝑢(𝑡) ∝ 𝑡. As the shuttle level crosses the band the particle ”flows” to the band , this can be shown by looking at the probability to be at each site as shown in Fig.2. At the end of the process we have an Hamiltonian with eigenstates |𝑛⟩ and the particle state |𝜓(𝑡 = 𝑡𝑓 )⟩ described at the standard basis |𝑖⟩. We would like to find out what are the spatial and energy distribution of the particle at the end, and how these depend on the rate of energy change. First,we have to realize that there are oscillation between the different |𝑛⟩ states the particle has ”flown” to, so we have to average over them. We Start by defining site and energy distributions: 𝑝𝑖 (𝑡) = |⟨𝑖|𝜓(𝑡)⟩|2 , 𝑝𝑛 (𝑡) = |⟨𝑖|𝜓(𝑡)⟩|2 (6) Where 𝑝𝑖 is time dependent and 𝑝𝑛 does not depend on time when the simulation ends 𝑡 > 𝑡𝑓 so we have to average over 𝑝𝑖 . The time average over 𝑝𝑖 is: 2 𝑝¯𝑖 = 𝑝𝑖¯(𝑡) = |⟨𝑖|𝑛⟩ ⟨𝑛|𝜓⟩| (7) This can be proofed as followed: We start from the particle state at a given time: ∑︁ |𝜓(𝑡)⟩ = 𝑒−𝑖𝐸𝑛 𝑡 |𝑛⟩⟨𝑛|𝜓(𝑡)⟩ (8) 𝑛 The probability to be at site 𝑖 in time 𝑡 is: ∑︁ 𝑝𝑖 = |⟨𝑖|𝜓(𝑡)⟩|2 = 𝑒−𝑖(𝐸𝑛 −𝐸𝑚 )𝑡 ⟨𝑖|𝑛⟩⟨𝑚|𝑖⟩⟨𝑛|𝜓(𝑡)⟩⟨𝜓(𝑡)|𝑚⟩ 𝑚,𝑛 = ∑︁ |⟨𝑖|𝑛⟩⟨𝑛|𝜓(𝑡)⟩|2 + 𝑛=𝑚 ∑︁ 𝑒−𝑖(𝐸𝑛 −𝐸𝑚 )𝑡 ⟨𝑖|𝑛⟩⟨𝑚|𝑖⟩⟨𝑛|𝜓(𝑡)⟩⟨𝜓(𝑡)|𝑚⟩ (9) 𝑛̸=𝑚 When we take the time average , the second term averages to zero so we are left with: 𝑝¯𝑖 = 𝑝𝑖¯(𝑡) = |⟨𝑖|𝑛⟩ ⟨𝑛|𝜓⟩| 2 (10) For presentation purposes we define the joint probability distribution: 2 𝑃𝑛,𝑖 = |⟨𝑖|𝑛⟩ ⟨𝑛|𝜓⟩| It follows from the above definitions that: ∑︁ 𝑝𝑛 = 𝑃𝑛,𝑖 (11) (12) 𝑖 𝑝¯𝑖 = ∑︁ 𝑛 𝑃𝑛,𝑖 (13) 4 We can use 𝑃𝑛,𝑖 to calculate the participation number for the sites and eigenstates at the end of the process at different rates: 𝑃 𝑁𝑛 = ∑︀ 1 2 𝑛 𝑝𝑛 , 1 𝑃 𝑁𝑖 = ∑︀ 𝑖 𝑝¯2𝑖 (14) The spatial and energy participation numbers are described at Fig.3, from which we choose an intermediate rate and describe 𝑃𝑛,𝑖 at this rate in Fig.4. Currents and counting statistics The current operator describing the current in a single bond is obtained by introducing a flux and then taking it to zero: (︀ )︀ (︀ )︀ * 𝐼𝑖→𝑗 = 𝑖 |𝑖⟩𝐻𝑖𝑗 ⟨𝑗| − |𝑗⟩𝐻𝑖𝑗 ⟨𝑖| = 𝑖𝑐𝑖𝑗 |𝑖⟩⟨𝑗| − |𝑗⟩⟨𝑖| (15) ⟨𝐼𝑖→𝑗 (𝑡)⟩ = ⟨𝜓(𝑡)|𝐼𝑖𝑗 |𝜓(𝑡)⟩ = 𝑖(𝜓𝑖 𝑐𝑖𝑗 𝜓𝑗* − 𝜓𝑗 𝑐𝑖𝑗 𝜓𝑖* ) = −2𝑐𝑖𝑗 Im{𝜓𝑖 𝜓𝑗* } From the currents we can define the counting operator: ∫︁ 𝑡 𝐼𝑖→𝑗 (𝑡′ )𝑑𝑡′ 𝑄𝑖→𝑗 (𝑡) = (16) (17) 0 We would like to characterize the counting statistic in our system. If a stochastic picture were applicable, then 𝑄 would satisfy |𝑄| ≤ 1 But in the quantum adiabatic limit (for slow rate) |𝑄| might be arbitrarily large due to a stirring effect.This is shown at Fig. 5. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] Phys. Rev. Lett. 78, 32213224 (1997) arXiv:0806.4195v1 [quant-ph] 25 Jun 2008 S. Engel et. al, Nature 446, 782-786 (12 April 2007) M. Chuchem and D. Cohen, J. Phys. A 41, 075302 M. Chuchem and D. Cohen, Physica E 42, 555 (2010) J.E. Avron, A. Raveh and B. Zur, Rev. Mod. Phys. 60, 873 (1988). D. J. Thouless, , Phys. Rev. B 27 6083 (1983). Niu, Q. & Thouless, D. J. J. Phys. A 17 2453 (1984). M. Buttiker, H. Thomas and A Pretre, Z. Phys. B-Condens. Mat., 94, 133-137 (1994). D. Cohen, Phys. Rev. B 68, 155303 (2003). D. Cohen, T. Kottos and H. Schanz, Phys. Rev. E 71, 035202(R) (2005). I. Sela and D. Cohen, J. Phys. A 39, 3575 (2006). C. Zener, Proc. R. Soc. Lond. A 317, 61 (1932). 5 18 0.5 16 ∆/c0 1 1.5 14 2 12 2.5 10 3 8 3.5 6 4 4 4.5 2 1 2 3 σ 4 FIG. 1: Average Participation number for different 𝜎 and Δ/𝑐0 . We would like our system to be at the ”green” zone, so we don’t miss the geometrical aspects of the model. Later we use 𝜎 = 1 and Δ/𝑐0 = 1. 1 0.8 Pi 0.6 0.4 0.2 0 −30 −20 −10 0 10 20 30 u FIG. 2: The Probability to be at any site as a function of time(parametrized by the starting site’s energy 𝑢) at an intermediate rate, 𝑢=20. ˙ 6 40 PN i 30 20 10 0 0 10 20 30 40 50 60 70 80 90 100 60 70 80 90 100 rate PN n 15 10 5 0 0 10 20 30 40 50 rate FIG. 3: Average Participation numbers at the end of the process for different rates for 𝜎 = 1 and Δ/𝑐0 = 1. one can identify several regimes: slow, intermediate and fast. In the slow regime we have 𝑃 𝑁𝑛 ≈ 1. In the fast regime both 𝑃 𝑁𝑛 ≈ 1 and 𝑃 𝑁𝑖 ≈ 1, as expected from the sudden approximation. |i> , Distance −−−> <−−− energy ,|n> 5 0.025 10 15 0.02 20 25 0.015 30 0.01 35 40 0.005 45 10 20 30 40 FIG. 4: Probability distribution for sites and eigenstates sorted by distance from the starting site and state’s energy for 𝜎 = 1 and Δ/𝑐0 = 1. Intermediate rate (𝑢=20), ˙ averaged over 1000 realizations. The spatial dispersion is narrow, energy dispersion is wide. For slow rates (𝑢˙ ≤ 1), the energies involved are only the low ones and the spatial dispersion is wide. For fast rate (𝑢˙ ≥ 90), the particle mainly stays at the starting site. 7 1.8 1.6 Maximal Q 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 70 80 90 100 rate FIG. 5: Maximal |𝑄| for different rates(𝜎 = 1 and Δ/𝑐0 = 1), one can recognize a low rate region there is no zero probability to have |𝑄| ≥ 1. Red dots are maximal |𝑄| for different realizations with the same parameters. 200 # 150 100 50 0 −0.5 0 0.5 1 Q max FIG. 6: Histogram for maximal |𝑄| for 𝑢˙ = 20 (10,000 values). 1.5