Full Counting Statistics as the Geometry of Two Planes
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Full Counting Statistics as the Geometry of Two Planes
Y.B. Sherkunov, A. Pratap, B. Muzykantskii, N. d’Ambrumenil
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom (Dated: November 17, 2018)Provided the measuring time is short enough, the full counting statistics (FCS) of the chargepumped across a barrier as a result of a series of voltage pulses are shown to be equivalent tothe geometry of two planes. This formulation leads to the FCS without the need for the usualnon-equilibrium (Keldysh) transport theory or the direct computation of the determinant of aninfinite-dimensional matrix. In the particular case of the application of N Lorentzian pulses, weshow the computation of the FCS reduces to the diagonalization of an N × N matrix. We also usethe formulation to compute the core-hole response in the X-ray edge problem and the FCS for asquare wave pulse-train for the case of low transmission. The experimental and technological importance of afull quantum mechanical treatment of the response ofFermi systems to a time-dependent perturbation, hasgrown as electronic devices have shrunk. In particular,the quantum statistics of charge transfer events inducedby an applied voltage pulse in simple systems such asacross a tunneling barrier or along a wire need to beunderstood if the limits to their capacity to encode ortransmit information are to be found [1, 2]. They arealso excellent examples of non-equilibrium systems forwhich in some cases complete theoretical treatments areknown or are feasible.Theoretically, interest has concentrated on the fullcounting statistics (FCS) or generating function, χ ( λ ),for all moments of the charge distribution [3]. Thereare results for the case of an infinite train of periodicpulses impinging on a tunneling barrier [4, 5, 6] and, forthe case of a low transparency barrier with a constantbias voltage V , the FCS are also known for non-zerotemperatures [2]. Particular attention has been paid tothe case of Lorentzian voltage pulses with ‘integer area’( eh R ∞−∞ V ( t ) dt is an integer). If the voltage pulses allhave the same polarity they generate so-called ‘Minimalexcitation states’ (MES). These have been shown to min-imise the number of excitations in a one-channel Fermigas required to generate a given signal. They have theintriguing property that a train of such pulses retain theminimal noise property [4, 7] independent of the separa-tion or width of the generating voltage pulses providedthe pulses are all of the same polarity. When such signalsimpinge on a tunneling barrier their nature is reflectedin the FCS [4].Here we show that the FCS at zero temperature aredetermined solely by the geometry of two planes. For atrain of N Lorentzian pulses these two planes coincideexcept in an N -dimensional subspace. Computing theFCS in this case reduces to the diagonalization of N × N matrix, which we compute explicitly. The known resultsfor χ ( λ ) follow directly from this geometric formulationcircumventing the need to work with a formulation us-ing Keldysh Green’s functions [6] or solving an auxiliaryRiemann-Hilbert problem to invert singular operators [5]. (b)(a) α xyz FIG. 1: (color online) The geometry of the FCS. In the spaceof single-particle states, orbitals occupied at zero temperaturecorrespond to eigenstates of h with eigenvalue 1 and define amirror plane shown schematically as the x − axis. Unoccupiedstates at zero temperature (with eigenvalue −
1) are in thecomplement of plane, one of these is the y-axis and the re-mainder are shown as the z-axis. In the insets the dashed linedivides states into those above and below the unperturbedFermi level but no other ordering by energy is implied. Ap-plication of a voltage pulse, biasing one electrode with respectto the other, transforms h → e h = UhU † . Case (a): ( N u = 1, N b = 0 see text) The corresponding plane has an added di-mension and is shown as the shaded plane. Case (b): ( N u = 0, N b = 1) The plane rotates by α/ h . The axis of rotation and the new occupied state arefound from [ h, e − iφ ( t ) ] (see text). All other states are rotatedas a function of time but remain eigenstates of both h and e h . The mirror planes are defined by the operators h =2 n − h = U hU † , where n is the ground state den-sity matrix and U is the unitary operator which describesthe effect of applying a voltage pulse between differentcomponents of the sytem. Directions corresponding tooccupied single-particle states are in the correspondingmirror plane (eigenvalue +1) while directions correspond-ing to unoccupied states are in its complement and re-flected (eigenvalue − both the initial and transformedstates and which therefore contribute nothing to the FCS.The remaining states are rotated with respect to one an-other by angles which we denote by α k . The directionswhich are rotated can be identified from the commutator[ h, e − iφ ( t ) ], which defines a spanning set of states for therotated space.We consider as a model system a quantum point con-tact between two ideal single channel conductors coupledby a tunneling barrier localised around x = 0 and treatthe effect of a voltage pulse, V ( t ), applied between thetwo conductors (see Figure 1). We assume that we canneglect the contributions to the noise and other momentsof the distribution of transferred electrons which scale aslog t ξ where t is the observation time and ξ is an ultra-violet cutoff set by the Fermi energy. These are presenteven in equilibrium and are associated with the fact thatthe charge number in the left or right electrode is not aconserved quantity [3]. The FCS of transmitted chargeacross the point contact (which we take to have reflectionand transmission probabilities R and T = 1 − R ) can beobtained from the generating function χ ( λ ) = ∞ X n = −∞ P n e iλn (1)where P n is the probability of n particles being transmit-ted (from the left electrode to the right one).The principle result of [6] is that χ ( λ ) = ( R + T e iκ l λ ) N u N b Y (cid:16) RT sin α e iλ + e − iλ − (cid:17) , (2)where N u is the number of unidirectional events with κ l = ± N b the number of bidirectional events. The geome-try for the two types of event is illustrated in Figure 1.Consider first the case of a simplified system consisting ofa single state on either side of the barrier. Suppose nowthat, after taking account of the voltage pulse, the stateon the left side is occupied and empty on the right. Thisis an example of a single MES [7] or unidirectional [6]event and is the case illustrated in Fig 1(a): the h -plane(shown as the x − axis) is actually zero-dimensional forthis simplified one-state system and the e h plane is one-dimensional. The FCS for this situation is clearly givenby χ ( λ ) = R + T e iλ : P = R and P = T in (1).The bidirectional event is illustrated in Fig 1(b).Again consider first the case of a simplified system ofa single occupied state on either side of the barrier. Ifthe occupied states are equivalent ( i.e. they have thesame energy or the same decomposition by energy), thentransfer across the barrier is blocked and χ ( λ ) = 1. Ifthe two states are orthogonal to one another (have dif-ferent energies or orthogonal decompositions by energy)then each state behaves independently. The FCS for thiscase is a product over that for two independent eventseach of which is of the type illustrated in Fig 1(a). This would give χ ( λ ) = ( R + T e iλ )( R + T e − iλ ). In general,the projection of the two occupied states onto one an-other is equal to cos α , where α is the angle betweenthe two planes in Fig 1. Now, the FCS for this sim-plified system is just the weighted average of the twoprevious cases: cos α + sin α ( R + T e iλ )( R + T e − iλ ) =1 + RT sin α ( e iλ + e − iλ − h e h is uni-tary and therefore has an orthonormal basis which canbe contructed from its eigenvectors [6]. For positive meantransfer from left to right electrode N u > e h is N u greater than that of h . Thereare therefore N u directions with eigenvalue of h e h equalto -1. The remaining eigenvalues come in pairs ( e ± iα j with j = 1 . . . N b ) and the 2D plane defined by the cor-responding basis vectors, e j − × e j , intersects the twomirror-planes defined by h and e h in two lines inclinedat an angle α j /
2. The FCS, χ ( λ ), is then given by theproduct over the appropriate factors for the N u and N b events to give (2).The case where N b and N u are both finite correspondsto a train of Lorentzian pulses [8] and has attracted alot of attention [4, 7]. Here we show that it is possibleto identify explicitly the sub-space for which the eigen-states of h e h have eigenvalues different from unity. Thecomputation of the FCS reduces to the diagonalizationof an N × N matrix where N = N u + 2 N b to find theangles α i from its eigenvalues ( e iα i ). The flux throughthe circuit, φ ( t ) = − ( e/h ) R t V ( t ′ ) dt ′ , can be written inthis case as e iφ ( t ) = N Y m =1 t − p ∗ m t − p m = 1 + N X m =1 A m t − p m . (3)The poles at p m = t m + iτ m have residues A m , which aredetermined by the system of equations − N X m =1 A m p ∗ n − p m , n = 1 , . . . , N. (4)To within overall phase factors, the effect of the voltagepulse is equivalent to a multiplication of left electrodestates by the factor e iφ ( t ) .The states which define the space in which the eigen-values of h e h are different from 1 are | ψ m ( t ) i = 1 / ( t − p m )for m = 1 , . . . , N . This follows from considering explic-itly the effect of U . All single particle states, ψ , satis-fying ( h e h − | ψ i = he iφ [ h, e − iφ ] | ψ i = 0, do not changeoccupancy under the effect of U (and hence make no con-tribution to χ ( λ )). For an arbitrary single-particle state ψ ( t ), we find using (3) and (4)( h e h − | ψ i = X m,m ′ sign( τ m ′ ) 2 A m ′ A ∗ m p ∗ m − p m ′ f m t − p m ′ . (5)Here f m is the projection of ψ ( t ) onto | ψ m ( t ) i : f m = i π Z dt t − p m ) ∗ ψ ( t ) . (6)The right-hand side of (5) vanishes for all ψ ( t ) orthogonalto the set of states { ψ n ( t ) , n = 1 , . . . N } .The FCS for the case of a train of N Lorentzian pulsescan then be found by considering the effect of h e h onthe subspace of the states { ψ n ( t ) } . We find h e h | ψ n i = P m M mn | ψ m i , where: M mn = X m ′ sign( τ m τ m ′ ) A m A ∗ m ′ ( p m − p ∗ m ′ )( p n − p ∗ m ′ ) . (7)The angles α i which determine the FCS in (2) are foundfrom the eigenvalues ( e iα i ) of the N × N matrix M mn .Analytic results for the FCS are known in some simplecases [4]: i) The { τ n } all have the same sign and ii) N =2 with sign( τ τ ) = −
1. The case of N = 4 has alsobeen solved analytically [9]. Case i) corresponds to thecase of MES. All poles of e iφ are in the same half-planeand all N states, | ψ n ( t ) i = 1 / ( t − p m ) are eigenstates of h e h with eigenvalue -1. This can be seen from (7) usingsign( τ m τ ′ m ) = 1, (4) and by considering de − iφ ( t ) dt . Thesestates define the N non-coplanar directions which are inone mirror plane and in the complement of the other.The FCS for this case of MES corresponds to N u = N unidirectional events in (2).Case ii) corresponds to a single bidirectional event.Here the FCS are just found from the eigenvalues of the2 × M mn . Explicit calculation gives the resultsin α/ | p − p ∗ p − p | [4]. In this case, because the states | ψ i and | ψ i are orthogonal to each other, it is also par-ticularly simple to compute the effect of U on the single-particle states explicitly. We work with normalized states | ¯ ψ i i = C i | ψ i i where C i = p | p i − p ∗ i | . We find that theeffect of U on the state | ¯ ψ ∗ i = C /t − p ∗ is to transformit into a linear combination of ¯ ψ and ¯ ψ : e iφ ( t ) | ¯ ψ ∗ i = C ( p ∗ − p ) C ( p − p ) | ¯ ψ i + p − p ∗ p − p | ¯ ψ i . (8)If | ψ ∗ i is in the mirror plane corresponding to h then thecomponent of the transformed state proportional to | ψ i is in the complement of the mirror plane and its ampli-tude is given by sin α e iφ = p − p ∗ p − p a result previouslyobtained using an operator formalism [7].In the general case when N b is not finite [8], the ge-ometric approach simplifies calculations when the trans-parency, T , (or the scattering phase shifts in other prob-lems) are low, it is possible to deduce the results withoutthe need for any direct diagonalization. We illustratethis by rederiving the standard result for the orthogonal-ity catastrophe or Fermi edge singularity (FES) problemand then use the method to generalise results for the FCS for a system subjected to a sudden pulse of finiteduration.The FES problem arises in the context of the X-rayabsorption spectrum of a metal. Here, the form of thespectrum is determined by the response of the conduc-tion electrons to the sudden appearance of a core hole [10]and has been shown to be the consequence of the orthog-onality of the ground states of the Fermi gas with andwithout the local potential due to the core hole [11]. Thecore hole spectral function is the Fourier transform withrespect to t f of the overlap between the state, U ( t f ) | i ,where | i is initial ground state at t = 0 and U ( t f ) theunitary time-development operator of the system. U ( t f )desribes the effect of the new potential which switches onat t = 0. (All states are written in the interaction ba-sis with H the Hamiltonian of the unperturbed system.)This is a one-channel version of the problem we have con-sidered. The overlap h | U | i = Q k cos α k /
2, where the e ± iα k are the eigenvalues of h e h . When the rotation an-gles α k are small, we can expand the cosine to obtain h | U | i ≈ exp (cid:16) Tr( h e h − (cid:17) . In the usual one-channelFES problem, U = e iφ ( t ) where φ ( t ) = 2 φ θ ( t ) θ ( t f − t )and where φ is the phase shift of the core hole poten-tial computed on the scattering (unperturbed) states ofthe system. If we work in the time-domain, h ( t, t ′ ) = iπ P( t − t ′ ) where P denotes Cauchy principal part, we findTr (cid:16) h e h − (cid:17) = Z dtdt ′ h ( t, t ′ ) (cid:16) e iφ ( t ) e − iφ ( t ′ ) − (cid:17) h ( t ′ , t )= − π sin φ log ξt f , (9)( ξ is the usual short-time cutoff of order the Fermi en-ergy) and we recover the standard result for this problem: h | U | i = ( ξt f ) − φ /π .The FCS for a train of step pulses in the low trans-parency limit is known to be Poissonian [12]. Herewe will consider a periodic signal with pulses of length t p and period S applied to the left electrode with re-spect to the right electrode: V ( t ) = 2 φ ( δ ( t − nS ) − δ ( t − nS − t p )). The geometry of this situation isvery similar to that of the core hole Green’s functionin the FES problem. The general result (2) then giveslog χ ( λ ) = P k log[1 + T R sin α k / e iλ + e − iλ − T R ≪
1, we can expand the logarithm anduse P k sin α k = − Tr( h e h − (cid:16) h e h − (cid:17) = − π X m,n (cid:20) log (cid:18) t p + S ( m − n − S ( m − n −
1) + τ (cid:19) + log (cid:18) t p + S ( m − n ) S ( m − n ) + τ (cid:19)(cid:21) sin φ , (10)where τ is short time cut-off used to characterise thedelta-function δ ( t ) = τπ ( t + τ ) and N u = 0 for this peri-odic train of pulses. In (10), the terms with m = n and m = n + 1 are divergent for τ →
0, while all other termsare convergent. With a measurement time
N S we ob-tain the usual result log χ ( λ ) = 2 N RTπ sin φ log t p τ ( e iλ + e − iλ − T = 0. The scattering matrix is taken to be differentfrom unity only during the observation time t when thetransmission becomes non-zero ( T >
0) and the numberof particles on a particular side of the barrier is no longera good quantum number. The geometry of this situa-tion is similar to the case of the step pulse in voltage ex-cept that the unitary transformation acting on the initialstates mixes states on both sides of the barrier [13]. Inpractice, these logarithmic corrections imply that the ob-servation time t for the MES and other effects discussedhere to be observable above the equilibrium noise mustbe short. For a system with ǫ F = 10meV and settingthe equilibrium noise at low temperatures ( π log ǫ F t /h )equal to the minimal noise obtained using from a MEScorresponds to working at frequencies ν >