A model for measurement of the states in a coupled-dot qubit

Transcription

1 A model for measurement of the states in a oupled-dot qubit H B Sun and H M Wiseman Centre for Quantum Computer Tehnology Centre for Quantum Dynamis Griffith University Brisbane 4 QLD Australia h.sun@griffith.edu.au Abstrat. We propose a quantum trajetory analysis of a sheme to measure the states of a oupled dot devie (qubit) where there is a flutuating energy gap Δ between the two states. The system onsists of the qubit and a readout dot oupled to soure and drain leads. The tunnel rate through the detetor is onditioned by the oupation number of the nearer quantum dot (target) of the qubit and therefore probes the states of the qubit. We derive a Lindblad-form master equation to alulate the unonditional evolution of the qubit and a onditional stohasti master equation alulating the onditional evolution for different tunneling rates. The results show the effets of various devie parameters and provide the optimum seletion and ombination of the system struture.. Introdution There have been wide interests and numerous proposals in the area of quantum transport and measurement in mesosopi eletroni system [- ]. Coupled quantum dots have been suggested as qubits: the basi element of a quantum omputer. In addition to manipulations of quantum states a readout devie is required to perform quantum measurements of the resulting state of the qubit. The aurate readout of data enoded in the qubit states is an important part of the performane of a quantum omputer. In this paper we analyse a method to measure the states of a oupled dot qubit based on the theory of open quantum system []. The measurement postulate of quantum mehanis states requires that the unitary quantum evolution to be applied to the total system whih inludes the measurement apparatus and the measured system. However the measurement proess automatially introdues a statistial desription of the system dynamis. A density matrix desribing a pure state has the property: ρ = ρ whih is not the ase for a density matrix desribing a statistial ensemble. The Lindblad equation [3] is the quantum master equation of the redued dynamis that still preserve the three basi properties of the density matrix: positivity Hermitivity and the norm required to desribe a pure state as well as markoviaity desribing a subsystem that undergoes an irreversible dynamis to equilibrium. The whole system in this study inludes a measurement devie of a quantum tunneling readout-detetor (RD) suh as the quantum ontat point single eletron transistor or quantum dot oupled to soure and drain leads and a oupled-quantum dot of the harge qubit as the system being measured. We derive a onditional stohasti master equation to desribe the onditioned evolution of the qubit. The ensemble-averaged evolution of

2 the qubit state is alulated for various parameter ombinations to estimate the optimum seletion. In setion we introdue the system and modeling followed by setion 3 alulating trajetories in the ases of without and with variable energy gap between two dot states. We disuss the results in setion 4 and summarize in setion 5.. The System and modeling The system studied is depited shematially in figure. There is a single eletroni bound state that an be oupied in eah dot of the qubit. The energy differene between these two bound states is Δ and the eletron an tunnel between two dots at rate t. In the RD the eletron-tunneling rate is onditioned on the oupation of the nearer dot (target) at D 0 and D 0 +D for non-oupied and oupied ases respetively.. Figure Illustration of the system The total Hamiltonian of the qubit system for oherent oupling ase (Δ = 0) is t H = h#! ii i + ih ( " ) () i= where i represent the Fermi annihilation and reation operators for the i single eletron state of the ith dot and t is the tunneling rate between two dot states. For the readout dot the bakground tunnelling urrent when the target is not oupied is D 0 and the rate of the deteted signal of the oupation of the target is D 0 + D with D > 0. We assume that the tunneling through the RD is one way only ( diretion as shown in figure ) and the esaping tunneling rate is large ompared to other rates and based on these we an derive a Lindblad-form master equation [4]: d" = #i H" dt [ ] + de " # ( [ " + " ]) * where γ de = D 0 +D is the deoherene rate. The stohasti reord of measurement omprises a sequene of times at whih eletrons tunnel through the RD. In the zero response-time limit the urrent onsists of a

3 sequene of δ funtion spikes: i(t) = edn/dt where dn(t) is a lassial point proess defined by the following onditions [ dn( t) ] = dn( t) E[ dn( t) ] /dt = D 0 + D Tr " t [ ( ) ] () where E[x] indiates a lassial average of a lassial stohasti proess x. The first ondition states that dn(t) equals zero or one. The seond means that the rate of events is equal to the quiesent rate D 0 plus an additional rate D if and only if the eletron is in the target dot. Applying the theory of open quantum systems [] we obtain the stohasti master equation onditioned on the observed event in time dt as [4] d" = dn D 0 + D T[ ] + D 0 D[ ] # D 0 + D Tr[" )" + dt #D " ] ( { } + D Tr " { } = AB + BA. where T[ A] B = ABA AB and D[ A] B = T[ A] B! ( A AB + BA A) / 3. Calulations 3. Coherent tunneling ase [ ]" # i[ H" ] ( ) (3) To simplify the alulations we introdue the Bloh representation of the state matrix: The Pauli matries are defined as " = ( I + x# + y# + z# x y z) (4) " x = + " y = i( # ) " z = # (5) The moments of the Pauli matries are given by <σ α > = α (α = x y z) whih provide physial meanings. For example when the system is in a definite state (dot or dot ) the average population differene z is equal to ±. The set of oupled stohasti differential equations for the Bloh sphere variables an be expressed as: / # dx = "t.z " D z x ( dt " x dn( t ) dy = " D 0 z y dt " y dn( t) ) dz = tx + D " z ) ( ) * + -. dt " D " z + * + D 0 + D " z ( ) / ( ) /. -. (6)

4 Detailed derivation and approximation are referred in ref. [4]. The subsript indiates that these variables refer to the onditional state. Calulated the trajetories at various oupling rates when D 0 = 0 are plotted in figure. When the oupling between the dots is small (t <γ de /) the eletron is loated in a fixed dot (z = - at dot and z = + at dot ) for a long time till a sudden transition as shown in figure (a). For strong oupling ase as shown in figure () when t > > γ de / the trajetory shows nearly sinusoidal osillations with jumps ourring at an average rate of γ de / this means that the eletron is not loalized but shared by two dots through the strong tunnelling. In figure (b) with the moderate oupling strength the trajetory shows that the eletron is neither well loalized nor regular harmonially osillating between two dots. Figure Trajetories for various oupling rates: t = (a) 0.; (b) 0.5; and () 5 D. 3. Energy gap Δ 0 ase We extend the appliation to the ase that there is an energy differene of Δ 0 between two dot states whih is a model of for example the qubit system proposed by Kane [4]. The relevant Hamiltonian an be written as #" t H = where t is the tunnelling between two states. The stohasti ( t 0 differential equations for the Bloh sphere variables desribing onditional dynamis now beome: * dx = "tz " y # " D x z ) dt " x ( dn( t) dy = x # " D y z + ) dt " y ( dn( t) dz = tx + D " z D ( ) " z ) dt " - ( D 0 + D " z ( ) / ( ) / dn t ( ) (7) The numerial alulation results are presented in figure 3. By omparison with figure one an see the effet of the energy gap. It takes muh longer time to tunnel through the gap from one dot to the other for the low tunneling rate ase (note: the time sales on the horizontal axes are

5 different in these two figures) while quasi-harmoni osillation features are kept in high tunneling rate region (tunneling rate >> gap Δ). For the moderate oupling rate the plot shows none-loalization and nonsinusoidal osillations between two dots with lower frequeny ompared to those in figure. In order to investigate the influenes of various parameters of devies on the system dynamis (performane) we investigate the unonditional ensemble average properties of the system in detail. Figure 3 Trajetories for various t with Δ = the parameters are shown on top of eah plot and the rates are all normalized by D. 3.3 Ensemble average properties The relevant Hamiltonian an be diagonalized by rotating an angle of " = t ( tg# ) * to H = " 0 ) 0 #( with " = ( # + # + 4t ) ; = ( # # + 4t ) (8) the transformation from the original representation to the new representation is given by: " x " os(() 0 sin(() " x y = 0 0 y # z #)sin(() 0 os(() # z In the new representation the evolution of the ensemble-averaged Bloh sphere variables is desribed by

6 " " x y () = ( + + 4t de # z # () de os (*) + + 4t () de () de 4 sin(4*) 0 () de 4 sin(4*) " x 0 y # z sin (4*) (9) We an monitor the state of the qubit from the evolution of the moment z(t). The alulated results of ensemble-averaged evolution of the system state are plotted in figures 4-8 illustrating the influene of various parameters. When a partiular parameter is hosen to vary in a plot all other parameters in the figure are fixed. 4. Results and disussion For the ensemble evaluation of the qubit state we use the approximate values of real Δ and t given by Kane [4] ie. Δ/h = 0 GHz and t/h = GHz. The values of the ratio of t/δ in all plots are therefore hosen as 5x0-3. Figure 4 Loality of the eletron at D 0 = 0.5 Δ = 0. and t = 0-4 D. Figure 5 Influene of Δ: from top Δ = D. Figure 4 shows a typial evolution of the loality of the state. It is obvious that the system is not osillating but deviates from the initial state to a mixed state as time approahes infinity whih is different from the oherent tunneling ase. The insert is enlarged details of the early stage whih shows a sharp deviation followed a flatter slope. Figures 5-7 show

7 the effets of the energy gap Δ the oupling rate between the two dotes t and the quiesent rate of urrent tunneling through the RD D 0 respetively. Figure 6 Influene of t from top: t = 0-4 5x0-4 D Figure 7 Influene of D 0: from top D 0 = D. Figure 8 Comparison of measurement quality for various D. The dashed lines are orresponding harateristi times.

8 Figure 9 Comparison of measurement quality for different values of D 0 at a fixed ratio of D 0 / D = 0.5. In the plots all parameters are normalized by the rate D. Both top lines (a) in figure 6 and figure 8 are very lose to the top frame edge. As expeted we see that the larger Δ (figure 5 (a)) and the smaller t (figure 6 (a)) the slower deviation and the smaller bakground rate of the detetor D 0 (() in figure 7) the better measurement quality. The interesting feature in figure 7 is that with a small D 0 z(t) shows a sharp first slope followed a flatter seond slope whih is most desirable ondition as it may be interpreted as that the state is distinguished quikly with less deviation from the initial state. Now we reah a question naturally: how would one judge the quality of a measurement? One parameter determining the quality of a measurement is the loalisation rate whih is related to the signal-to-noise ratio. The harateristi time is defined as the minimum time when the two possibilities of the eletron loality are distinguishable. In our system it is given by T = (D 0 +D )/D whih is twie inverse of the loalisation rate [4]. Within the harateristi time the loser to the initial state the better measurement. Figures 8 and 9 illustrate the omparisons of the measurement qualities with various parameter ombinations. Figure 8 shows that the larger D (strong oupling between the qubit and the detetor) the more sensitive detetion the RD reads out the state of the qubit in a shorter time with less disturbane. In figure 9 D 0 and D vary in their absolute values at the fixed ratio of D 0 /D = 0.5. It is lear from the graph that the larger rates of RD (urve ) make better measurement and strong oupling is therefore preferred. The above outomes may provide referene for the devie designers when they takle optimum seletion of the parameters. For example if the tehnology limits the redution of quiesent urrent of a non-ideal detetor one ould inrease the measurement tunneling rate D by devie designing or bias setting in experiments to ompensate and ahieve better measurement quality. 5. Summary It has been suggested to use mesosopi eletroni systems suh as oupled quantum dots superonduting juntions and single spinpolarised eletrons as qubits. We model the quantum measurement of states of suh systems using the theory of open quantum system. The requirements to perform quantum alulations and a quantum measurement (readout) appear to ontradit eah other. During the manipulations the dephasing should be minimised while a quantum

9 measurement should dephase the state of the qubit as far as possible. We propose a measurement sheme to study the dynamis of the system. To guarantee the alulated evolution representing the state of a real physial system we derive the Lindblad-form master equation. We alulate the onditional evolution of the states and the ensemble-averaged evolution of the states of the oupled quantum dots as the qubit. The results show the effets of various devie parameters on the quality of the measurements. These may ontribute to the devie parameter seletion and experimental designing of the readout proesses of a solid-state quantum omputer for the better performane. Referenes [] Gurvitz S A 997 Phys. Rev. B [] Shninman A and Shhon G 998 Phys. Rev. B (998) [3] Sun H B and Milburn G J 999 Phys. Rev. B [4] Wiseman H M et al 00 Phys. Rev. B [5] Korotkov A N 00 Phys. Rev. B [6] Gurvitz et al 003 Phys. Rev. Lett [7] Jordan A N and Buttiker M 005 Phys. Rev. Lett [8] Oxtoby N P et al 006 Phs. Rev. B [9] Luo J Y at al 007 Phys. Rev. B [0] Kieblih G et al Phys. Rev. Lett [] Dong B et al 008 Phys. Rev. B [] Carmihael H An Open Systems Approah to Quantum Optis (Springer Verlag Berlin 993); Wiseman H M and Milburn G J 993 Phys. Rev. A (appendix) [3] Lindblad G Commun. Math. Phys ) [4] Kane B E et al 000 Phys. Rev. B 6 96