Wavefunction and spectral properties of single-particle emitters

Transcription

1 Master s thesis Wavefunction and spectral properties of single-particle emitters Elina Locane Supervisor: Peter Samuelsson Assistant supervisor: Vyacheslavs Kashcheyevs University of Latvia Lund June 013

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3 Abstract The past six years have seen the development of fast and accurate single-particle sources, with possible applications in metrology, nanoelectronics, quantum information processing, and single-electron quantum optics. These mesoscopic devices also enable us to gain more insight into the quantum mechanics of interacting fermions. In the present work, we study the energy spectrum of the emitted state from a single-particle source. The model we employ for such a system is a single-level quantum dot with a time-dependent energy level coupled to a non-interacting single-mode lead via a time-dependent tunneling barrier. We use the Floquet scattering approach to obtain an analytic expression for the emitted electron amplitudes as well as the non-equilibrium Green s functions formalism for calculating the average occupation number in the lead. We also show that in the limit when the effect of the Fermi sea can be neglected, these results agree with the single-particle solution to the Schrödinger equation.

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5 Contents 1 Introduction Literature review Statement of the problem Layout of the thesis Single-particle approach 7.1 Hamiltonian Solution to the Schrödinger equation Floquet scattering matrix approach General case with a harmonic potential Oscillating barrier Wavefunctions in the regions I, II, III Matching regions I and II Matching regions II and III Scattering of an incoming particle On-demand coherent single-electron source Time-dependent transparency Non-equilibrium Green s functions approach Hamiltonian Time evolution Interaction picture Evolution of c ki Time- and contour-ordering Matrix-like Hamiltonian Definition of Green s functions Keldysh space Single-level dot Results and discussion Floquet scattering approach Incoming and outgoing amplitudes Note on approximation close to the Fermi level Results for different parameters Comparison with the single-particle approach Non-equilibrium Green s functions approach

6 6 Conclusions and outlook 40 A Derivation of G cc 4 A.1 Expression for G dc A. Wick s theorem A.3 Wick s theorem for finite temperatures A.4 Expressing further G dc A.5 Expression for G cc B Derivation of G kk 48 B.1 Necessary components B. Combining components of G kk Bibliography 56 3

7 Chapter 1 Introduction 1.1 Literature review Single-particle emitters, such as quantum pumps and turnstiles, are devices that allow for time-controlled on-demand flow of charged particles. Over the past two decades, these devices have attracted a lot of interest, both due to fundamental and application aspects. This includes the possibility to gain new insights into the quantum mechanics of interacting electrons, applications in fields like quantum computation [1], nanoelectronics as ingredients in singleelectron circuits [], and metrology as a means towards redefining the ampere [3, 4]. The quantum metrological triangle could allow for a consistency check of the fundamental constants and e [5, 6]. The emerging field of single-electron quantum optics also requires single-particle emitters as sources of a coherent electron and/or hole stream. A recent realization of a Hanbury Brown Twiss interferometer for single electrons was reported in [7], followed by a two-electron collider experiment [8], proposed in [9]. Figure 1.1: Schematic of charge pumping. In step 1, the energy levels of the quantum dot, controlled by the gate voltage V exc, are below the Fermi energy. In step, the upper level is raised above the Fermi energy and an electron escapes to the lead. In step 3, the levels are lowered again, and an electron tunnels into the dot, leaving a hole in the lead. Figure taken from [10]. 4

8 Electron pumps and turnstiles typically consist of a metallic island or a quantum dot connected to one or several leads via tunnel barriers or quantum point contacts. Transparencies of the quantum point contacts can be varied using gate voltages. Usually one or two parameters, like gate voltages, size of the dot, temperature, or magnetic field, are chosen to manipulate and investigate charge transport through the system. A schematic of one possible principle of charge pumping can be seen in Fig The first experimentally realized turnstile was reported in 1990 [11]. It consisted of a linear array of tunnel junctions, and transport through the device was driven by a periodic gate voltage applied to the central electrode, in the direction imposed by an external bias. An early quantum-dot turnstile with oscillating tunnel barriers was realized and the quantized current through it reported in [1]. In contrast to a turnstile, the charge flow through a quantum pump can be reversed and does not require applied static bias. An early quantum pump with phase-shifted modulation of two gate voltages was proposed in [13], and an experimental realization of a quantum pump was reported in [14]. The pump of [14] contained two quantum dots and operated in the Couloumb blockade regime. In the middle of 1990s, a pump with an open quantum dot was proposed [15], and a number of theoretical [16, 17] and some experimental investigations [18] in the area of transport through open quantum systems have been made. Recently, it has become possible to construct fast and accurate charge transport devices. In 007, experimental realizations of single-particle sources operating at frequencies of the order of gigahertz were reported [10, 19]. Since then, many turnstile and quantum pump experiments have been proposed, studied and carried out [0 4]. Increasing the frequency and accuracy of these devices as well as being able to tailor the shape of the emitted wavepacket of an electron are some of the challenges that are encountered in experiments. Among the theoretical challenges related to single-particle emitters is the description of charge capture and release processes [5], the former of which has been analyzed in [6], and the latter is the subject of this thesis. Two powerful theoretical methods used for investigating transport in mesoscopic systems are the Floquet scattering approach and the Green s functions formalism [7, 8]. The Floquet scattering approach for periodically driven systems is well-established [17, 9 34], and applicable for many quantum pump and turnstile experiments, but allows one to work only with non-interacting systems. The Green s functions formalism [35 38] is more general in this sense, because interactions can be included. Both methods can yield analytic results for simple enough models. 5

9 1. Statement of the problem The aim of the present work is to characterize the emitted state of a singleelectron source. We model the single-electron emitter as a single-level quantum dot coupled to a non-interacting single-mode lead via time-dependent tunneling barriers this model has been considered, for example, in [37, 39]. The specific quantity we are interested in is the occupation in the lead after an electron is released from the dot. To achieve this, we use both of the abovementioned methods the Floquet scattering matrix approach and the non-equilibrium Green s functions formalism. In a certain parameter regime we compare the results obtained with these methods to the analytic single-particle solution of the Schrödinger equation, which is also presented in this thesis, as a way of verifying our results in the limit when the many-body effects due to the Fermi sea can be neglected. A first investigation for the turnstile of Ref. [40] employing the Floquet scattering matrix formalism and the single-particle approach was presented in [41]. We extend these results to the pump of Ref. [10], generalize them to a timedependent tunneling barrier and study a number of different aspects. The nonequilibrium Green s functions formalism was applied to the model we are considering in [37], and an analytic solution was given to the Green s function of the dot. In the present work we extend this result to the Green s function in the contact. 1.3 Layout of the thesis The work is organized as follows. In Chapter we present the single-particle approach to the problem, solving the Schrödinger equation for an ansatz that assumes single-particle states only. In Chapter 3 we introduce the Floquet scattering approach and solve our model with this method. In Chapter 4 we introduce some conventions needed to develop the non-equilibrium Green s functions formalism and give the expression derived in Appendix A for the Green s function in the contact for a more general problem multi-level dot, and then apply it to our specific case. The main results are compared and discussed in Chapter 5, and conclusions are drawn in Chapter 6. Besides Appendix A which contains the mentioned derivation of the Green s function in the contact for a multi-level-dot model, we also include Appendix B with an explicit calculation of the Green s function in the contact for the model with a single-level dot. 6

10 Chapter Single-particle approach.1 Hamiltonian The system we are investigating is a single-level quantum dot, connected to a single-mode lead via a time-dependent tunneling barrier see Figure.1. The lead is assumed to be non-interacting, with time-independent energy levels. Initially, the dot is filled, and the lead is empty. In this chapter we solve the Schrödinger equation of this system in terms of single-particle states. The relevant Hamiltonian is H S = ɛt d d + k ɛ k k k + k V t k d + V t d k,.1 where the subscript S stands for the Schrödinger picture. The ket d denotes the state in the dot, and k denotes a state in the lead, with the corresponding energies ɛt and ɛ k, respectively. V t is the tunneling energy for tunneling from the dot to the lead. It is set to be independent of energy, that is, we are working in the wide-band limit when the energy scales under consideration are assumed to be much smaller than the scales over which the tunneling energy varies significantly.. Solution to the Schrödinger equation We will make the following ansatz for the state ket: ψ = c d t d + k c k t k.. 7

11 Figure.1: Sketch of the model used in the work. ɛt is the energy of the quantum dot level, which is coupled to a Fermi sea of electrons with energies ɛ k via a timedependent tunnel barrier. The tunneling energy for going from the dot to the lead is V t. The time-dependent Schrödinger equation gives i d dt ψ =H S ψ.3 i dc dt dt d + i k dc k t dt k =c d tɛt d + V t k c d t k + + k ɛ k c k t k + V t k c k t d..4 Collecting like terms, we get i dc dt dt d = c d tɛt d + V t k c k t d.5 and i k dc k t dt k = V t k c d t k + k ɛ k c k t k..6 In Eq..6, each of the different c k s must satisfy the relation separately, since the kets { k } are orthogonal. Thus for obtaining the amplitudes c d t and c k t we have to solve the following system of equations: i dc dt = εtc d t + V t k dt c kt i dc.7 kt = V tc d t + ɛ k c k t. dt Laplace-transforming.7 gives us i sc d s 1 i sc k s = [ɛtc d t] s + [V t k c kt] s = [V tc d t] s + ɛ k c k s,.8 8

12 where [...] s denotes the Laplace transform, and for a single variable we write xs [x] s. We have already inserted the initial conditions c k t=0 = 0 and c d t=0 = 1. Rewriting the second equation of.8, we obtain c k s = [V tc dt] s i s ɛ k..9 We now take the inverse Laplace transform of this, which will be denoted by [ ] [V tcd t] c k t = s..10 i s ɛ k t Consider the first equation of.7. In order to obtain a relation containing c d t only, we perform the sum over the levels in the lead V t k c kt. Assuming the spectrum outside the dot is quasi-continuous, the discrete sum can be substituted with an integral V t + c ktρdɛ k, where ρ is the density of states in the lead and is assumed to be independent of ɛ k. V t c k tρdɛ k.10 = V t + [ ] [V tcd t] s i s ɛ k + = V t [[V tc d t] s ρdɛ k t 1 i s ɛ k ρdɛ k ] t = V t [[V tc d t] s iπρ] t Here we have defined = iπρ V t c d t i Γtc dt..11 Γt = πρ V t..1 Inserting Eq..11 into the first equation of.7, we arrive at a first order linear differential equation for c d t: i dc dt = ɛtc d t i dt Γtc dt = ɛt i Γt c d t..13 This is satisfied by c d t = c exp i 0 [ɛτ i ] Γτ dτ..14 Initial condition c d 0 = 1 corresponding to an occupied dot at t = 0 gives c = 1. Let us now substitute c d t in the second equation of.7 with the solution.14: i dc kt = V t exp i [ετ i ] dt Γτ dτ + ɛ k c k t

13 A formal solution can be obtained after some manipulations. exp i ɛ k t dck t = i dt V t exp i i ɛ kc k t exp i ɛ k t 0 [ετ i ] Γτ dτ exp i ɛ k t Rearranging terms and using the chain rule for differentiation yields d c k t exp i ɛ k dt t = i V t exp i 0.16 [ɛτ i ] Γτ dτ exp i ɛ k t. We now perform integration on both sides, and divide by exp i ɛ k t : c k t = i exp i ɛ k t 0 V t exp i 0 [ɛτ i Γτ ] dτ.17 exp i ɛ k t dt..18 Together with.14 and. this gives a complete solution to the Schrödinger equation.3. 10

14 Chapter 3 Floquet scattering matrix approach In the scattering formalism for quantum transport through mesoscopic systems usually called the Landauer-Büttiker formalism [4 44], the central region in our case the quantum dot is treated as a scatterer, and the incident particles have some probability to either be reflected from the scatterer or transmitted through it. In particular, if a quantum dot is driven by a time-periodic field with frequency ω, an electron interacting with it can gain energy quanta n ω, n = 0, ±1, ±, ±3,..., 3.1 in accordance with the Floquet theorem for scattering problems [9, 30]. The element S F,αβ E n, E of the Floquet scattering matrix S F is then defined as the amplitude for an electron with energy E coming to the scatterer from channel β to leave through channel α with energy E n, where E n = E + n ω, n = 0, ±1, ±, ±3, We are now going to find the wavefunction of an electron scattering off a harmonically-driven scatterer. 3.1 General case with a harmonic potential Consider an electron subject to a position-independent oscillating potential V t = V cosωt. The time-dependent Schrödinger equation for this electron is m x + V cosωt ψx, t = i ψx, t. 3.3 t 11

15 Since there is no spatial dependence in the applied potential, we can separate variables: ψx, t = k exp ikx ϕ k t. 3.4 Inserting this into the Schrödinger equation and factoring out the spatial dependence, we obtain k m + V cosωt ϕ k t = i t ϕ kt. 3.5 Further, using the ansatz ϕ k t = exp the Schrödinger equation becomes i E k t φ k t, 3.6 k m + V cosωt φ k t exp i E k t = E k φ k t + i exp i E k t t φ kt. We can choose E k to be equal to k, thus arriving at m which has the solution V cosωtφ k t = i t φ kt, 3.7 φ k t = c k exp i V cosωt dt. 3.8 By the Jacobi-Anger identity [45, p. 361], the last expression is equal to φ k t = c k + n= J n V ω where J n ir the n-th order Bessel function of the first kind. exp inωt, 3.9 Combining 3.4, 3.6, and 3.8 gives us the full solution to the Schrödinger equation 3.3: ψx, t = k c k exp ikx i E + k t V J n exp inωt ω n= 3. Oscillating barrier Further, let us look at a case when the oscillating potential V t of Section 3.1 is confined in space, as in Figure 3.1, where it is restricted to the region labeled 1

16 Figure 3.1: Schematic of a one-dimensional oscillating scatterer located in region II, with oscillating potential V t. A X n Bn X denote the amplitude for an electron to be propagating in region X with energy E n in the right left direction. with II. Electron transport in this system is one-dimensional, the boundaries of the region II are defined by x = 0 and x = d, and the amplitude for the electron to be propagating from left right to right left with energy E + n ω in the region I, II, or III, is denoted by A I n, A II n, and A III n Bn, I Bn II, and Bn III, respectively. States with the different n s are called the Floquet sidebands. If we regard the central region as a scatterer, the Floquet scattering matrix S F, defined in the beginning of the Chapter, relates amplitudes for the different sidebands between regions I and III: A III = S F A I, 3.11 where A I and A III are column matrices with elements {A I n} and {A III n } in the order of increasing n, respectively. Similarly, B I = S F 1 B III. 3.1 Due to current conservation the norm of the vectors A I and A III as well as B I and B III have to be the same. This together with the completeness of the scattering states implies the unitarity of the Floquet scattering matrix: S F 1 = S F We are now in a position to write down the wavefunctions in each of the regions of Figure 3.1, using the result 3.10 and applying the Floquet scattering approach. 13

17 3..1 Wavefunctions in the regions I, II, III Since there is no external field in the regions I and III, the wavefunction in these regions will just be the superposition of all possible Floquet sidebands as plane waves: Ψ I x, t = n Ψ III x, t = n A I n expiknx I + Bn I exp iknx I exp A III n expikn III x + Bn III exp ikn III x exp i E + n ω t i E + n ω t The wavefunction in the central region will also be the superposition of all possible Floquet sidebands, with each of the sidebands gaining an extra multiplier as in 3.10 due to the oscillating potential: Ψ II = n m = n,m A II n A II n exp i expikn II x + Bn II exp ikn II x exp ev J m exp imωt ω expikn II x + Bn II exp ikn II x ev J m ω E + n m ω t. i E + n ω t. To obtain a relation between the scattering amplitudes, we will employ the fact that the wavefunction as well as its flux due to probability conservation has to be continuous. 3.. Matching regions I and II Wavefunction continuity At the boundary between the regions I and II, we require that Ψ I 0, t = Ψ II 0, t: A I n + Bn I = A II m + Bm II ev Jn m ω m Here and for the other boundary conditions, we have dropped the sum over n, equating the multipliers in front of exp i E+n ω t on each side of the equation. This means that for each term of the sum on the right hand side of 3.14, the sum of the lower indices at A II m Bm II and J ev n m ω has to be the same as the lower index at A I n Bn I on the left hand side, i.e., m + n m = n. 14

18 Flux continuity Similarly, we require that x Ψ I0, t = x Ψ II0, t, obtaining ik A I n Bn I = ik m A II m Bm II ev Jn m ω Here we have assumed kn I = kn II = kn III = k F = k when the k s enter prefactors as opposed to exponents, because the scattering is assumed to take place in a narrow energy window compared to the Fermi energy E F very close to E F. Combining Eqs and 3.15 gives A I n = m B I n = m ev A II m J n m ω B II m J n m ev ω Matching regions II and III Wavefunction continuity Analogously to Section 3.., for the wavefunction to be continuous we require Ψ II d, t = Ψ III d, t: A III n exp ikn III d + Bn III exp ikn III d = m A II m exp ik II m d + B II m exp ik II m d J n m ev ω Flux continuity The flux continuity requirement x Ψ IId, t = x Ψ IIId, t gives us ik A III n = ik m exp ik III n d B III n exp ik III n d A II m exp ik II m d B II m exp ik II m d J n m ev ω

19 Now, from Eqs and 3.19 follows A III n B III n exp ikn III d = m exp ikn III d = m A II m exp ik II m d J n m ev ω B II m exp ik II m d J n m ev ω Scattering of an incoming particle Let us now solve the specific problem when there is one incoming electron from the left i.e., in the region I of Figure 3.1 with energy E. Then A I n = δ 0n and Bn III = 0 for all n. From Eq. 3.1 we conclude that Bn II = 0 for all n, since the left hand side is always zero due to Bn III being always zero. Furthermore, Eq implies that all Bn s I are zero as well. We have thus found that there is no back scattering in the system. This result would not have been obtained, had we not chosen to approximate all the diferent k s with k F in the prefactors Eqs and We are now left with a system of equations for all the forward-scattering amplitudes: ev A II m J n m 3. ω A III n A I n = m exp ikn III d = m A II m exp ik II m d J n m ev ω. 3.3 We will from now on drop the superscripts I, II and III at k s for the different regions in line with the assumptions above. We can regard Eqs. 3. and 3.3 as matrix equations. Denoting by J the matrix with the element J n m in the n-th row and m-th column, and defining P = diag..., exp ik n 1 d, exp ik n d, exp ik n+1 d,..., 3.4 these equations become A I = JA II 3.5 P A III = JP A II, 3.6 where A I, A II, and A III are the row vectors introduced previously. Combining 3.5 and 3.6 gives or P A III = JP J 1 A I 3.7 Ã III = JP J 1 A I,

20 Figure 3.: Sketch of our system quantum dot connected to a single-mode lead via a quantum point contact QPC. a Notation for amplitudes. A I and B II contain amplitudes propagating in the lead, while B I and A II are propagating in the dot. Red arrow shows the QPC. b t and t are probability amplitudes for going past the QPC, while r and r for going through the QPC, in the directions shown. where we have introduced à III = P A III. 3.9 We have thus arrived at the desired solution to our scattering problem. 3.3 On-demand coherent single-electron source We will now use the solution of the oscillating barrier problem to describe a system with only one single-mode lead connected to an oscillating scatterer quantum dot. The lead and the dot are connected via a quantum point contact QPC whose transparency is independent of energy, i.e., it can be fully described by two amplitudes t, the amplitude for an incident electron to return to the lead, and r, the amplitude for an incident electron to be transmitted into the dot. This assumption is justified in the case when the oscillation amplitude of the dot is small compared to the energy scale over which the transparency varies. Quantized charge transport with period of the order of nanoseconds between two subsequent emissions through such a system was reported in [10]. The principle of this device can be seen in Figure 1.1, and a sketch of this system is depicted in Figure 3. a and b. In accordance with Figure 3. a, A I now contains the amplitudes incoming from the lead, A II contains the amplitudes incident to the lead from the dot, B I contains the amplitudes transmitted from the lead into the dot, and B II contains the amplitudes outgoing from the quantum dot. See also Figure 3. b for the convention used for the transmission and reflection amplitudes t, r, t and r. Since there is no scattering when going from B I to A II, we know that A II n = exp ik n d B I n,

21 or, in matrix form, A II = P B I, 3.31 which physically corresponds to gaining a phase factor during propagation. The matrix P is P = diag..., exp ik n 1 d, exp ik n d, exp ik n+1 d, as before. The paremeter d can be thought of as the characteristic lengthscale classically circumference of the quantum dot. It will later be gauged out using the approximation 5.6. From the oscillating barrier problem we use the expression 3.17 to express Bn II, where Bm II is now substituted by A II m, and take into account that amplitudes A I are also contributing to B II n : B II n = ta I n + r m J n m A II m 3.33 or, in matrix form, B II = r JA II + ta I Similarly, we use 3.16 to obtain the amplitudes B I, since A II m in 3.16 is now substituted by our Bm, I and we also take into account that the amplitudes A I are now contributing to Bn II as well: B I = rj 1 A I + t A II We have arrived at the following system of equations: A II = P B I B I = rj 1 A I + t A II 3.36 B II = r JA II + ta I Solving 3.36 for B II in terms of A I gives B II = t + r JP 1 t P 1 rj 1 A I, 3.37 thus allowing us to obtain the amplitudes after the scattering has taken place Time-dependent transparency The transparency T of the QPC can be modelled to be time-dependent. We will use the saddle-point contact model [46, 47] for this: 1 T t t sp t = 1 T max e V sinωt+ϕ V 0 V

22 T max is chosen between 0 and 1; ϕ, V 0, and V 1 are arbitrary parameters. The subscript sp serves to distinguish t sp from the transmission amplitude t appearing before, since t relates scattering amplitudes for the different Floquet sidebands in the energy as opposed to time representation. When t sp does not depend on energy, t can be obtained directly from the Fourier transform. Let us note that now t becomes energy-dependent or, to be more accurate, it becomes dependent on the energy difference between the two sidebands it connects: t mn = 1 τ τ The reflection amplitude in the energy domain is 0 e im nωt t sp tdt 3.39 where r mn = 1 τ τ 0 r sp t = e im nωt r sp tdt, t sp t

23 Chapter 4 Non-equilibrium Green s functions approach The present Chapter, in contrast to Chapters and 3, employs the second quantization notation see, for example, [48]. We will start with some definitions and proofs. 4.1 Hamiltonian In the beginning we will consider a model with a multi-level quantum dot. The relevant Hamiltonian in the Schroedinger picture is where H S t = H 0 t + H St, 4.1 H 0S t = k ɛ k tc k c k + n,m h nm td nd m 4. and H St = nk [ V kn tc k d n + V kntd nc k ]. 4.3 d n d n and c k c k are the particle annihilation creation operators in the dot and the lead, respectively. Operators c k c k and c k c k anticommute if k k : {c k, c k } = {c k, c k } = They also obey the anticommutation relation {c k, c k } = δ k,k

24 These relations hold for the operators d n and d n as well. The Hamiltonian 4.1 is equivalent to.1 from Chapter in the case when there is only one energy level in the dot and when the creation and annihilations operators act on a vacuum state 0 with a drained Fermi sea, giving d 0 = d and c k 0 = k. 4. Time evolution The time evolution operator Ut is defined via Ut ψ 0 = ψt, 4.6 where we have denoted the state ket ψ 0 ψ. This is to be accompanied by the requirement Probability conservation implies that Ut is unitary: therefore The Schroedinger equation for a state ket leads to U = ψ 0 ψ 0 = ψt ψt 4.8 ψ 0 ψ 0 4.6,4.8 = ψ 0 U tut ψ 0, 4.9 i d ψt dt U tut = i dut dt the complex conjugate of which is i du t dt = H S t ψt 4.11 = H S tut, 4.1 = U th S t We introduce an operator A H t in the Heisenberg picture through the definition of the expectation value of an operator At with respect to a state: At = ψt A S t ψt 4.14 def = ψ 0 A H t ψ

25 At time operators in the different Heisenberg and Schroedinger pictures coincide: Using 4.6, the equality 4.15 can be rewritten as from which follows A H = A S At = ψ 0 U ta S tut ψ 0 = ψ 0 A H t ψ 0, 4.17 A H t = U ta S tut The rate of change of the expectation value of an operator At is given by d 4.14 At = d dt dt ψt A St ψt 4.15 = ψ 0 d dt A Ht ψ The expectation value in the Schroedinger picture can be expanded using the chain rule as d dt At = d dt ψt A St ψt = ψt A S t 1 i H St ψt + ψt 1 i H StA S t ψt + ψt d dt A St ψt. This can be rewritten, substituting A St for da St, since in the Schroedinger t dt picture operators are constant unless explicitly made time-dependent: d dt At = ψt A St 1 i H St ψt + ψt 1 i H StA S t ψt + ψt t A St ψt. 4.1 Since this last expression, according to 4.19, must be equal to ψ 0 d dt A Ht ψ 0, we conclude or d dt A Ht 4.19,4.1 = i d dt A H 1 i U t [A S t, H S t] Ut + U t A St Ut 4. t 4.10 = U A S tuu H S U U H S UU A S U + i U A S t U 4.18 = [A H, H H ] + i U A S t U def = [A H, H H ] + i A H t. 4.3 The last equality is in accordance with the Heisenberg equation of motion and defines the partial derivative of A H.

26 4.3 Interaction picture The decoupled Hamiltonian in the Schrödinger picture H 0S t at differen times commutes with itself, i.e., [H 0S t, H 0S t ] = If we define and U 0 = e i/ H 0S τdτ 4.5 A I t = U 0 ta StU 0 t, 4.6 it follows that and H 0I = U 0 H 0SU 0 = H 0S U 0 U 0 = H 0S 4.7 i d dt A I = i d U 0 dt A SU 0 d = i dt U 0 A S U 0 + i U d 0 A S dt U 0 = U 0 H 0SA S U 0 + U 0 A SH 0S U 0 + i U 0 + i U 0 t A S t A S U 0 U 0 = [A I, H 0S ] + U A S 0 t U def = [A I, H 0I ] + A I t, where in the last line we have defined the partial derivative of A I t in an analog way to 4.3. The subscript I stands for interaction picture Evolution of c ki We can now calculate the commutator [ ] [ ] c k, c k c k = c k, c k c k + c k [c k, c k ] = c k 4.9 and use this result to determine the time evolution of the operator c k in the interaction picture ɛ k is now assumed time-independent: i d ] [ ] dt c kit 4.,4.8,4.9 = ɛ k [c ki t, c ki tc kit = ɛ k c ki t, c k c k ] [ ] = ɛ k [U 0 tc ku 0 t, c k c k + ɛ k U 0 t [c k, c k c k = ɛ k U 0 tc ku 0 t = ɛ k c ki, = ɛ k U 0 tc k ] [ U 0 t + ɛ k U 0 t, c k c k U 0 t, c k c k ] c k U 0 t 3

27 which gives c ki t = e i/ ɛ kt c k Similarly we can derive d ni t = e i/ ɛ td t d n Time- and contour-ordering Now, using 4.18 and 4.6, we express and from here define the operator It obeys the equation i du I = i d U 0 dt dt U du 0 = i dt where we have defined A H = U tu 0 ta I U 0 tut 4.3 U I t = U 0 tut ,4.5 = U 0 H 0SU + U 0 H SU U + i U = U 0 H 0SU + U 0 H 0SU + U 0 H SU d dt U = U 0 H SU = U 0 H SU 0 U 0 U = H IU I, 4.34 H It = U 0 H SU The time dependence of H I t is well understood, since both H S t Eq. 4.3 and U 0 t Eq. 4.5 are given explicitly. Eq is the Schwinger- Tomonaga equation. Solution to 4.34 is given by cf. Eq..3.. from [49] { U I t = T exp i } H Iτdτ The time-ordering operator T, by definition, orders all operators inside it according to their time argument latest comes first leftmost in the expression. Time-ordering is important here because H I at different times does not necessarily commute with itself. Each time two fermionic operators are swapped inside T, the overall sign in front of the time-ordered product changes. Consequently, { U I t = T exp + i } { H Iτdτ = T exp i 0 } H Iτdτ. t

28 Figure 4.1: Complex contour C used in Eq is the moment in time when Schrödinger, Heisenberg and interaction pictures coincide. This contour will be employed when evaluating non-equilibrium Green s functions. The anti-time-ordering operator T acts in the opposite way to T it orders all operators so that the ones with the latest time arguments come last rightmost in the expression. It arises from the fact that complex conjugation of the operator U I reverses the order of the operators H I inside the integral. The sign change before the integral also comes from complex conjugation, and it is switched once more when changing the limits of integration. Now, from 4.3, 4.33, 4.36, and 4.37 we see that A H t = T or, equivalently, { exp i 0 } { H Iτdτ A I tt exp i } H t Iτdτ, 4.38 A H t = T C {exp i } H Iτdτ A I t, 4.39 C where T C is the contour-ordering operator, meaning that it orders all operators inside it according to where their time argument appears on the contour that goes from to t above the real axis, and comes back from t to below the real axis see Fig. 4.1 latest comes first leftmost in the expression. For each time we interchange two fermionic operators inside the contour-ordering operator, the overall sign changes. This result will be used later, to express Green s functions in interaction picture operators. 4.4 Matrix-like Hamiltonian The tunneling part of the Hamiltonian, in accordance with 4.3, is H = D V C + C V D,

29 where D and C are column vectors containing annihilation operators d 1, d,..., d n and c 1, c,..., c m, respectively: d 1 c 1 d D =. ; C = c d n c m Their complex conjugates are the corresponding row matrices: D = d 1 d... d n ; C = c 1 c... c m 4.4 The tunneling matrix element V ij connects the j-th level of the dot to the i-th level in the lead, whereas V ij connects the j-th level in the lead to the i-th level of the dot. 4.5 Definition of Green s functions We define Green s functions as the following correlation functions all ordering and integration is done elementwise after matrix multiplication, and not with the full D, D, and C, C matrices: G dd t, t = 1 TC DtD t i 4.43 TC CtC t 4.44 G cc t, t = 1 i G cd t, t = 1 i G dc t, t = 1 i TC CtD t 4.45 TC DtC t 4.46 Angle brackets denote the quantum statistical average, which is taken with respect to the initial state operators are in the Heisenberg picture. We also introduce the zeroth-order in tunneling Green s functions: G 0 ddt, t = 1 i G 0 cct, t = 1 i G 0 cdt, t = 1 i G 0 dct, t = 1 i T C D I td I t T C C I tc I t T C C I td I t T C D I tc I t According to 4.39, these can be viewed as the full Green s functions for the specific case of V = 0 and, therefore, H = 0. 6

30 4.6 Keldysh space According to 4.39, 4.40, and 4.44, G cc can be expressed as G cc = 1 T C i n=0 n { } n 1 1 D I n! i τv I τc Iτ + C I τv IτD I τdτ C I tc I t. C 4.51 Here integration on the complex contour Figure 4.1 has to be performed. There are four distinct combinations for the order in which t and t can be placed on the contour. If both t and t are on the upper lower part of the contour, we denote Gt, t with G ++ G. If t is on the upper lower part of the contour while t is on the lower upper part of the contour, we denote Gt, t with G + G +. In line with this reasoning, we can formally introduce a matrix-like notation for Gt, t : [ ] Gt, t G ++ t, t G + t, t G + t, t G t, t. 4.5 Gt, t is now a four-component object in the Keldysh space. According to the definition of the contour-ordering operator T C Section 4.3., these four components of the Green s function G cc, for example, are given by i G cc t, t = T C CtC t [ T Ct + C t + C t Ct + ] = Ct C t + T Ct C t 4.53 The components T Ct + C t + and T Ct C t can be expressed further: T Ct + C t + = θt t CtC t θt t C t Ct 4.54 and T Ct C t = θt t C t Ct θt t CtC t Two objects At, t and Bt, t in the Keldysh space can be multiplied to obtain Ct, t, as defined below: Ct, t At, t 1 Bt 1, t dt C Explicitly in the matrix-like notation this looks as follows: [ ] C ++ C + [ ] A C + C = ++ B ++ A + B + A ++ B + A + B A + B ++ A B + A + B + A B dt

31 The last relation becomes clear when we consider separately integration on the upper denoted by C + and lower denoted by C parts of the contour. To obtain C xy, where x, y = ±, we integrate A x+ B +y over C + and subtract because the integration variable decreases along the direction of integration the integral of A x A y. Now we define the objects G R G ++ G G A G ++ G G < G and arrange them in a matrix G R G < 0 G A It can be verified that and holds. i G R cct, t = θt t CtC t + C t Ct 4.6 i G A cct, t = θt t C t Ct + CtC t 4.63 G R G + G 4.64 G A G + G 4.65 We can now make the observation that C R C < A 0 C A = R A < B R B < A 0 A A 0 B A = R B R A R B < + A < B A 0 A A B A, 4.66 which are the Langreth multiplication rules. As an example, let s calculate C R : C R 4.58 = C ++ C A ++ B ++ A + B + A ++ B + + A + B dt = A ++ = B ++ B + A + B + B dt ,4.64 = A R B R dt C Similarly we arrive at other components of the matrix R C < 0 C A and observe that ordinary matrix multiplication rules indeed apply for these objects. Now we define the inverse A 1 of the object A: AA 1 =

32 We claim that the identity operator in this space is the following: δt t 1 = 0 0 δt t. 4.7 This can be verified by checking that A 1 is indeed equal to A: A A 1 = R A < δt t 0 0 A A 0 δt t A R t, t 1 δt 1 t dt 1 A < t, t 1 δt 1 t dt 1 = 0 A A t, t 1 δt 1 t dt A = R A < 0 A A Single-level dot In Appendix A, we derive the following expression for the Green s function in the contact: G cc t, t = G 0 cct, t + G 0 cct, τ 1 Vτ 1 G dd τ 1, τ V τ G 0 ccτ, t dτ 1 dτ. C C 4.76 We will now use this expression for our specific case a single-level dot coupled to a single-mode lead, with time-dependent tunneling in the wide-band limit WBL approximation i.e., when tunneling does not depend on energy. The Hamiltonian of such a system is a specific case of 4.1: with H S t = H 0 t + H St 4.77 H 0S t = k ɛ k c k c k + ɛtd d 4.78 and H St = k [V tc k d + V td c k ] Thus, in rewriting 4.76 we will take into account that V and V are now one-by-one matrices and will write V instead of V. We will also replace the indices dd by d for simplicity, and write the equation 4.76 for a specific pair of indices k and k, rather than for the whole matrix CC. This leads to G kk t, t = G 0 kkt, t δ kk + G 0 kkt, τ 1 V τ 1 G d τ 1, τ V τ G 0 k k τ, t dτ 1 dτ. C C

33 We have also taken into account the fact that to the zeroth order there are no correlations between different states in the contacts. Applying the Langreth multiplication rules 4.66 to 4.80, we get the Keldysh equation for the lesser Green s function in the contact G < kk t, t = G 0< kk t, t δ kk + dt 1 dt G 0< kk t, t 1V t 1 G A d t 1, t V t G 0A k k t, t + dt 1 dt G 0R kk t, t 1 V t 1 G < d t 1, t V t G 0A k k t, t + dt 1 dt G 0R kk t, t 1 V t 1 G R d t 1, t V t G 0< k k t, t, where the integrals are now on the real time axis. In Appendix B we rewrite 4.81 more explicitly, and arrive at: 4.81 G < kk t, t = i e i ɛ kt t fɛ k δ kk [ +e ɛ i k t ɛ k t gt, ɛ k, ɛ k fɛ k g t, ɛ k, ɛ k fɛ k ] +iρ dωfωgt, ɛ k, ωg t, ɛ k, ω 3 e ɛ i k t ɛ k t i t nd dt 1 dt e i ɛ k t ɛ kt 1 V t 1 V t e i 1 t d tɛ t e Xt 1 +Xt, 4.8 where fɛ k is the initial energy distribution in the lead, ρ is the density of states in the lead, n d is the initial occupation of the dot, and the functions gt, ɛ k, ɛ k and Xt are the following: 3 i t gt, ɛ k, ɛ k = dt dt 1 e i ɛ k t ɛ kt 1 V t 1 V t e i and Γt, just like in Eq..1 of Chapter, is defined as t 1 ɛ td t e Xt 1 Xt 4.83 Γ t Xt= d t Γt = πρ V t We can now calculate the average occupation number in the contact defined as nk, t = c k tc kt = i G < kk t, t 4.8 = fɛ k 1 i [gt, ɛ k, ɛ k g t, ɛ k, ɛ k ] + ρ 3 dωfω gt, ɛ k, ω n d i dt 1 dt e i ɛ kt t 1 V t 1 V t e i 1 t d tɛ t e Xt 1 +Xt

34 The function gt, ɛ k, ɛ k is: gt, ɛ k, ɛ k = 3 i t dt dt 1 e i ɛ kt t 1 V t 1 V t e i t ɛ td t 1 e Xt 1 Xt

35 Chapter 5 Results and discussion 5.1 Floquet scattering approach Here we present some of the numerical results obtained with the Floquet scattering approach and compare them with the result of Chapter. We focus on the type of problem discussed in Section 3.3 Fermi sea of electrons in a lead at zero temperature and Fermi energy E F is scattered from a quantum dot with a harmonically oscillating energy level V cosωt. The probability for an electron to go through the quantum point contact connecting the dot and the lead is time-dependent Eq Incoming and outgoing amplitudes The incoming amplitudes contained in A I were modelled with the Fermi-Dirac distribution at zero absolute temperature and energies were counted from the Fermi energy E F see also 5.4: A I =.,

36 where the first zero-valued element is A I 1. In theory, the number of Floquet sidebands elements in the column vectors A I, A II, B I, and B II is infinite, but for problems suitable to be described within the Floquet formalism, only a limited number of the sidebands is important, allowing one to have finite vectors when performing numerical calculations. The outgoing amplitudes B II were obtained from Eq derived in Section 3.3: B II = t + r JP 1 t P 1 rj 1 A I Note on approximation close to the Fermi level To calculate the matrix P = diag..., exp ik n 1 d, exp ik n d, exp ik n+1 d,..., 5.3 we use the fact that we operate close to the Fermi level E F and can thus linearize E ε E F : The wave number is consequently m m k n = E n = E + n ω m = E F + ε + n ω = mef 1 + ε + n ω E F E = E F + ε. 5.4 me F 1 + ε + n ω E F k F 1 + ε + n ω E F The phase that an electron acquires due to a spatial displacement d is k n d = k F d ε + n ω E F /k F d = k F d + ε + n ω, 5.6 where we have introduced = E F k F d, the spacing between subsequent energy levels in the quantum dot. It is a convenient unit of energy, describing all other energies in comparison with the characteristic energy scale in the dot Results for different parameters Depending on the values chosen for the different parameters V and ω amplitude and frequency of the dot level oscillations, respectively, and T max, ϕ, V 0, and V 1 parameters determining the quantum point contact transparency, Eq. 3.38, we obtain different outgoing amplitudes B II. Below are the main qualitative characteristics of the parameter space of our model. 33

37 Figure 5.1: Outgoing energy distribution for the parameter value ϕ = π the probability for a state in the lead with a given energy to be occupied after one cycle of dot level oscillations. Inset shows the QPC transparency dashed black line along with the energy of the dot level solid red line as functions of time. Note that the peak at positive energies is centered around the energy of the dot level when it is being opened this is emphasized with the red circles. When the energy of the dot level reaches the Fermi level and starts decreasing below it, the dot is capturing states from the Fermi sea, which we can see from the deficit in the distribution at negative energies this is emphasized with the blue ellipses. Other parameter values are: = 10, V = 4, ω = 0.1, V 0 = 1, V 1 = 0.1, T max = 0.. For numerical calculations, the step in energy for the states in the lead was chosen to be Transparency of the contact Parameters V 0 and V 1 allow to manipulate the shape of the quantum point contact transparency T t between the dot and the lead. They were chosen so that the shape of T t is close to rectangular, see, for example, the inset of Figure 5.1. The phase ϕ was crucial in finding the optimal pumping regime. As can be seen in Figure 5.1, a localized electron wavepacket was emitted if the contact between the dot and the lead was opened when the dot level was near the peak. If the dot level was at its minimum when the dot was opened, a hole was emitted. The latter situtation is shown in Figure 5.. For cases in between, emission of a particle did not appear in the the outgoing energy distribution see Figure 5.3. The maximum transparency T max of the quantum point contact also has a significant influence on the outgoing energy profile. Decreasing T max makes the width of the resonance decrease as well. When T max is much smaller than 1, 34

38 Figure 5.: Outgoing energy distribution for ϕ = 0. Other parameters and explanation is the same as in Figure 5.1, except that here we observe emission of a hole at energies near the value at which the dot is being opened red circles, and then filling of states near the Fermi level when the dot energy starts to increase above it blue ellipses. 1 Occupancy T Time t, arb. units 4 0 Vcosω t, / Energy, /10 Figure 5.3: Outgoing energy distribution for ϕ = π/. Other parameters are the same as in Figure 5.1. Particle emission is not achieved with this pumping regime. 35

39 1 Occupancy T max = 0.05 T max = 0. T max = Energy, /10 Figure 5.4: Outgoing energy distribution for different T max values. Inset: close-up of the plot for T max = 0.05 between energy values 0 and 5. Note that the interval between peaks in the inset is ω = 0.1, which corresponds to one Floquet quantum = 1. Other parameter values are as in Figure 5.1. the width of the resonance becomes smaller than one Floquet quantum, and Floquet fringes separated by n ω appear in the outgoing energy distribution, as can be seen for the blue line T max = 0.05 in Figure 5.4. This is a similar effect as described in Section IV of [41], where the authors discuss different frequency regimes for an electron-hole turnstile. There the Floquet fringes arise for frequencies around ω = 0.1, because the oscillation period starts to become comparable or smaller than the electron tunneling time, and the pumped charge starts to decrease. If, on the other hand, T max is close to one, the electron can enter and leave the dot every oscillation cycle with large probability, and the Floquet fringes are smeared out. The larger T max, the larger is the energy value around which the outgoing distribution is centered. This is to be expected, because for larger T max s emission will take place earlier in the oscillation cycle. A comparison of the outgoing energy distribution for different values of T max is given in Figure 5.4. Oscillation amplitude of the dot level To obtain the desired operating regime of the device, oscillation amplitude V of the quantum dot energy level was chosen to be smaller than half of the level spacing of the dot. 36

40 1 Occupancy V = 0.1 V = 0.4 V = T Time t, arb. units Vcosω t, arb. units Energy, /10 Figure 5.5: Outgoing energy distribution for different values of V. Inset: timedependence of QPC transparency versus quantum dot energy in arbitrary units, same for all V values. Other parameter values are the same as in Figure 5.1. Larger oscillation amplitudes led to greater energies at which the electron was emitted, as displayed in Figure 5.5. For small values of V the emission took place close to the Fermi level and created disturbances in the Fermi sea the blue line V = 0.1 in Figure 5.5. See also Figure 5.1 where it can be seen that emission occurs when the quantum point contact is opened by comparing the energy of the dot level when the transparency is being increased which can be read from the inset with the energy corresponding to the peak of the outgoing distribution. The ratio V/ ω determines the number of Floquet sidebands that will be excited in the scattering process, and increasing this ratio has a considerable effect on the computational costs. We therefore kept the ratio below Comparison with the single-particle approach In Chapter we solved the Schrödinger equation.3 with the single-particle ansatz., which corresponds to a situation with no Fermi sea. With the Floquet approach, the Fermi sea is taken into account, but its effect is minimal if the emission of an electron occurs high above the Fermi energy E F. This is achieved if the dot is opened when the energy of the dot level is at the peak, and if the oscillation amplitude is sufficiently large, as in Figure 5.1. The correspondence between the analytical single-particle solution.18 and results from the Floquet scattering approach is very good, if we choose parameters appropriately a detailed discussion on the connection between the Floquet scattering and the 37

41 1 Occupancy Floquet Single particle Energy, /10 Figure 5.6: Comparison between a Floquet-approach simulation all parameters as in Figure 5.1 and an analytic solution Eq..18 with the same parameters. Inset shows a close-up of the profiles in energy range between 0 and 5. single-particle wavefunction parameters is given in [41]. A comparison is shown in Figure Non-equilibrium Green s functions approach In Chapter 4 we derived an expression for the average occupation number in the lead: nk, t = fɛ k 1 i [gt, ɛ k, ɛ k g t, ɛ k, ɛ k ] + ρ 3 dωfω gt, ɛ k, ω n d i dt 1 dt e i ɛ kt t 1 V t 1 V t e i 1 t d tɛ t e Xt 1 +Xt. 5.7 As a consistency check, let us compare this result with our earlier results for the single-particle no Fermi sea case Eq..18 in the appropriate limit. Disregarding the effect of the Fermi sea by setting fɛ k = 0 is physically justified in a situation when the release of the electron happens at energies much higher than the Fermi energy. Inserting fɛ k = 0 and n d = 1 the dot is initially filled into 5.7 gives 38

42 nk, t = i t dt 1 dt e i ɛ kt t 1 V t 1 V t e i 1 t d tɛ t e Xt 1 +Xt 5.8 = i t dt 1 dt e i ɛ kt t 1 V t 1 V t e i 1 t d tɛ t e 1 Γ t d t Γ t d t, 5.9 which is equivalent to the single-particle result c E t = i for = 0. 0 V t exp i 0 [ ετ iγτ ] dτ exp i E t dt

43 Chapter 6 Conclusions and outlook The aim of this thesis was to characterize the emitted state of a single-electron source. This has been done with three different approaches the single-particle approach, Floquet scattering matrix approach, and the non-equilibrium Green s functions approach. In all cases analytic expressions were obtained, and numerical results were presented for the single-particle and the Floquet scattering matrix approaches. The model used for the single-electron source was a singlelevel quantum dot coupled to a non-interacting single-mode lead with a timedependent tunneling barrier. The single-particle approach led to the expression.18 for the amplitudes in the lead. It was verified Figure 5.6 by comparing to the Floquet scattering result that the single-particle result works well for emissions high above the Fermi level. The general analytic result of the Floquet scattering matrix approach 3.37 is exact if we take into account an infinite number of Floquet sidebands. For numerical calculations, only a finite number of contributing sidebands needs to be taken into account, and this is determined by the ratio between the oscillation amplitude and frequency of the energy level of the dot. Several parameter regimes of the Floquet scattering were compared in Chapter 5, and it was shown that for certain set of parameters a compact wavepacket is emitted as in Figure 5.1. The expression obtained for the Green s function in the contact 4.8 allows us to calculate the average occupation number in the lead 5.7. This result is an extenstion to the previous analysis of this non-interacting resonant-level model presented in [37]. We showed in Chapter 5 that in the limit of a drained Fermi sea it is identical to the single-particle result of Chapter. An advantage of the Green s functions formalism is that in principle the Green s function 4.80 could be evaluated for different approximations of the Green s function for the dot, and interactions could be included. In that case the problem would have to be solved numerically. 40