Double detected spin-dependent quantum dot
We study the dynamics of a spin-dependent quantum dot system, where an unsharp and a sharp detection scenario is introduced. The back-action of the unsharp detection related to the magnetization, proposed in terms of the continuous quantum measurement theory, is observed via the von Neumann measurement (sharp detection) of the electric charge current. The behavior of the average electron charge current is studied as a function of the unsharp detection strength , and features of measurement back-action are discussed. The achieved equations reproduce the quantum Zeno effect. Considering magnetic leads, we demonstrate that the measurement process may freeze the system in its initial state. We show that the continuous observation may enhance the transition between spin states, in contradiction with rapidly repeated projective observations, when it slows down. Experimental issue, such as the accuracy of the electric current measurement, is analyzed.
keywords:Quantum measurement theory, Quantum dot, Quantum Zeno effect, Heisenberg uncertainty principle
The act of a quantum measurement is always performed by an external apparatus and involves complicated interactions with it.
We are going to discuss the the back-action induced by the observer in the framework of unsharp measurements. The unsharp measurement
extracts only partial information from an observable, so we introduce another detector with sharp detection. We are interested in the
output of the sharp detection, which means that the unsharp measurement will be treated in a nonselective picture.
The nonselective description represents a measurement, where our record of data was lost and replaced by an average over the
data ensemble. The sharp detector also
has its back-action, projecting the system randomly into its eigenstate, but now the readout will be kept and the further evolution
of the system will be neglected. The double detected setup gives a possibility to analyze the so-called quantum Zeno effect
(QZE) misra (). Rapidly repeated measurements give rise to the QZE, the suppression of transition
between quantum states. In reality there are more complicated physical processes that take place during a quantum measurement, which can
also cause QZE. The effect can be best understood in terms of the dynamic time evolution of the measured quantum systems.
The double detection scenario is similar to the “indirect measurement” process Mandelstam (), where the back action of a detector on the quantum system is observed by a third party, namely in our case by a sharp detector. In this context, the idea of detecting the measurement back-action related to one type of degree of freedom, with a detection scenario of an another type of degree of freedom, would be suitable for a future experiment. The charge detector is a convenient sharp detector type and its applicability in the semiconductor physics is very high. We choose the other detector to be a spin related, magnetization, detector. The application we have in mind is the spin-dependent single quantum dot, available in high quality due to massive progress in experimental technology. Spin manipulation and magnetization detection in quantum dot was studied in experiment by Ref. Experiment (). An external field, used for spin manipulation, can be viewed as an environment of the subsystem, the quantum dot. The whole of quantum system dynamics is reversible. Tracing out the environment’s degrees of freedom, we arrive at a non-unitary time evolution BreuerPetruccione (). In general, all these non-unitary processes are connected through the Kraus-form (A), related to the completely positive mappings of the density matrices Kraus (). The subsystem’s non-unitary dynamic, imposed by the external field, can be interpreted as an unsharp measurement nuclearmagnetic (). These manipulations are time-continuous, so we will study the magnetization detection in the frame of the continuous quantum measurement theory cont.model (), avoiding the modeling of the detector system as a quantum system.
While the system described above is similar to the spin-to-charge conversion in quantum dots Elzerman (); Barrett (), the model of the unsharp detector is different. Here, the back-action of continuous quantum measurement on magnetization is investigated by a detected electric current, focusing on measurement back-action and on QZE in the spin states. The unsharp detector of the spin-to-charge conversion is modeled as a quantum point contact with a fixed coupling Hamiltonian. The theory presented here is broader, because the coupling between the unsharp detector and the system is choice of will, however has to be subject to a real experimental setup.
For a possible experimental setup, the idea of the double detection in a spin-dependent quantum dot is reasonable, because quantum dots and also spin manipulations are important for the realization of qubits. The effects of spin decoherence related to quantum computing was studied by Refs. Kane (); Loss (); Jelezko (). On the other hand the field of indirect measurements on quantum dots by means of Coulomb-coupled, quantum point contacts, single-electron transistors, or double quantum dot’s Barrett (); Gur97 ()-Mao04 () and the effects of QZE Gur97 (); QZE () was studied by several works, and the concept of time-continuous measurement has been part of this field Barrett (); Gur97 (); Kor99 (); Goa01 (); BBDG (). Previous research has monitored the sharp or unsharp detection of the observables related to the electron charge. While the work presented here examines the sharp detection of the electric current, it also contains an unsharp detection of a spin observable, the magnetization. The model proposed here is similar to the work of Ref. BBDG (), where the authors studied the effect of the unsharp detection of electric current in a selective measurement scenario. For the difference from that work, we applied the model for the magnetization detection and we studied the evolution of the density matrix as a function of the electron number tunneled through the system.
This article is organized as follows. In Sec. 2 we introduce our model. We derive the many-body Scrödinger equation including the terms of the continuous quantum measurement. We represent the density matrix as a function of the electron number in the right lead. The results of measurement back-action are shown and discussed in Secs. 3, 5. We investigate the accuracy of the electric charge current measurement in Sec. 4. General and continuous quantum measurement theories have a wide literature and to ensure the background of this work we present a short summary of this topic in the A and B using Refs. cont.model (); diosik (); NielsenChuang ().
2 The model
We consider the spin-dependent quantum dot, subject of experimental work KoppensScience309 (); PettaScience309 (), which is coupled to two separate electron reservoirs. The density of states in the reservoirs is very high (continuum), and the dot contains only isolated levels. We consider the highest energy level to be the state of two electrons with different spins () and therefore we include the effects of Coulomb interaction. The split of the one-electron energy level is done by a -directed magnetic field . We include a -directed magnetic field , describing the coherent oscillations between the spin up and spin down levels. The full Hamiltonian of the system reads
is the Hamiltonian of the quantum dot,
is the Hamiltonian of the reservoirs (leads), and
is the coupling Hamiltonian between the reservoirs and dot. The subscripts and
enumerate correspondingly the (very dense) levels in the
left and right leads. () is the annihilation (creation) operator
of spin in the quantum dot. () is the annihilation (creation) operator
of spin in the reservoir or .
is the Coulomb repulsion energy, the energy difference is proportional to and the frequency
and are the one-electron energies with spin in the left and right leads.
and are the respective tunneling
amplitudes of spin between the left or right reservoir and the dot. (Fig. 1)
For simplicity, we restrict ourselves to a low temperature case, . All the levels in the right and left lead are initially filled with electrons up to the Fermi energy and , respectively. This situation will be treated as the “vacuum” state . We consider a large bias and that the energy levels are inside the band, . In the context of these conditions, the electric current flows only from left to right. The evolution of the whole system is described by the many-particle wave function. Taking into account the assumptions, the wave function is represented as
where, for example is the sum over all states with energy and and with the condition . The amplitudes represent the physical situations as: the probability that the system is in the “vacuum” state at time , the probability that one electron with spin up was annihilated in the left reservoir and one electron with spin up created in the quantum dot at time , and so on. The “vacuum” state in this representation has the following properties:
Now, we apply the theory of continuous quantum measurement (B). The time evolution of the system in the presence of a time-continuous measurement of the magnetization, , is given by a modified Schrödinger equation:
where is average detected magnetization and is the Wiener process. In order to derive this equation, we assumed that the detector bandwidth is bigger than the eigenfrequencies of the system, defined by the Hamiltonian . The main parameter of the theory, the detection performance (or detection strength), is defined as
where is the time-resolution (or, equivalently, the inverse bandwidth) of the detector (detecting the magnetization ) and
is the statistical error characterizing unsharp detection of the average value of the magnetization
in the period .
Substituting eq. (2) into the equation of motion (7) using the Hamiltonian (1), we find a system of coupled differential equations for the amplitudes
The sharp measurement is represented by a electric current measurement, which is related to the accumulated charge in the right lead. In order to analyze this quantity we introduce the density matrix as a function of , the number of electrons in the right lead. The Fock space of the quantum dot consists of only four possible states, namely: the dot is empty, the dot contains a spin up electron (), the dot contains a spin down electron() and is the fully occupied dot. In our notation, these probabilities are represented as follows:
We are going to investigate a nonselective continuous quantum measurement case, which means that we are only interested in the average over the different realization of the wave function . As a first step, we apply the quantum Ito rules HudsonP () for the product rule of differentiation,
Through this step the stochastic time evolution of the density matrix is defined. Now, we average over the realizations BMZ (), where we use that is the standard Wiener process, a Gaussian random variable with zero mean value () and variance ,
In the context of the nonselective measurement we define the following matrix elements:
where is the difference of the energy levels, which are renormalized by the Lamb-shifts. Due to the large bias condition, the other off-diagonal elements, as , are weakly coupled to the differential equations found above and they are not taken in consideration. However, they have their own dynamics, too.
The left tunneling rates are
and the right tunneling rates are
where is the spin up or spin down density of states in the left (right) lead, .
The energy dependence of the left tunneling amplitudes and is a decreasing function, because the lowest is the energy level, the highest is the probability to be loaded from the left lead. The energy dependence of the right tunneling amplitudes and is a increasing function, because the highest energy levels are more likely emptied to the right lead then the lower ones. Using the relation we have the following properties for the incoherent tunnelings:
We may assume without loss of generality that the probability of filling up the state from the left lead is equal to the probability of emptying the state to the right lead, which leads to the assumptions:
We remind the reader that we also considered a low temperature case and the large bias condition
was used. The following calculations will be based
entirely on the parameters of eqs. (29), (30).
The time evolution of the density matrix is represented in the terms of the number of electrons tunneled through the dot. The convenient measurement would be the number of the accumulated electrons, but the number operator has a spectral decomposition, where the different spin states projectors has the same eigenvalue. If this eigenvalue is detected, the detector does not give an information as to which state belong and therefore we can not determine the state of the dot. Instead of the charge measurement we consider the measurement of electric current, which is given by a commutator of with the total Hamiltonian of the system,
where is the elementary charge. Using eq. (1) we obtain
It follows form eq. (32), that by measuring the electric current, the projections into different state correspond to different
observed value of the current. This implies that the states of the quantum dot can be determined by monitoring directly the electric
The model describes the spin-dependent quantum dot, where the magnetization is continuously detected with an averaged output gained and at the same time there is a sharp detection of the electric current. The sharp detection gives information about the system, and the result will depend on the interaction strength of the continuous detection.
We are going to analyze two cases to show the presence of measurement back-action and QZE. We remind the reader that the unsharply
detected operator, the magnetization, is diagonal in the Hilbert space of the dot and this is the
reason why the damping mechanism has effect only on the internal coherent motion. The theory of the continuous quantum measurement allows
the discussion of more general operators, which may introduce a complex damping mechanism, although they have to be subject of
possible experimental realizations.
QZE is where the repeated observations of the system slow down transitions between quantum states. As a result of a continuous observation the system cannot leave its initial state misra (). The original paper formulates the problem in very general way, but only proves for projective measurements, which prevails negligibly small time evolution during the time when the number of repetitions . The theory of continuous quantum measurement is built up from sequence of unsharp measurements, such that each measurement is increasingly weak by the increase of the repetition . This construction BLP1 (); BLP2 () allows QZE only for measurement strength . In order to investigate the effect, let us consider the case that our system is initially in the spin up state.
In the first step we study our model in the presence of magnetic leads. We define the left lead with spin up, and the right lead with spin down states: . Before any further discussion we need a reminder that the direction of the current was fixed, and is flowing through the dot from left to right. Here, the electrons are coming from the left and fill up the state then they hop to the state, from where they leave the dot to the right lead. When the measurement-induced damping parameter is smaller than the spin flip frequency , the system oscillations are maintained. If is increased, the coherent oscillations die out, see Figs. 2. We find that for small the rate of transition form the spin up to the spin down state slows down with the increase of . If tends to infinite then the off-diagonal density matrix elements, eq. (24) and its complex conjugate, do not contribute to the dynamics and the system remains frozen in its initial state, the state . This implies that continuous measurements would localize the system and our equations reproduce the QZE. If we consider the spin down state as the initial state, then the system will not remain there. As is increased the transition from the spin down state to the the spin up state is slowed down, although the system will be localized in the state at .
This is a direct consequence of the conditions imposed, which result that the state is the preferred steady-state
due to the combination of right lead magnetization and the direction of tunneling.
While for small the transition is slowed down, after some critical time an enhanced decay can be seen (Figs. 2) for values comparable to the energy level displacement . This enhanced decay results faster transition between the spin up and the spin down states than in the unmeasured system. Nevertheless the effect is just the opposite of the QZE. These behaviours are also reflected in the steady-state. If the enhanced decay occur then the probability of the state is smaller than in the case of the unmeasured system.
In spite of great progress made in microfabrication techniques, the construction of magnetic leads to a tiny quantum dot is still a challenge and we also study the considered system with normal leads. The localization into the state as in the magnetic lead model is impossible, because both the spin up and spin down states can be filled from left and emptied to right. Examining the results in Figs. 3 the effect of suppressed transition can be found for small values of . Here, the system cannot be frozen in its initial state as a consequence of the normal leads. When is comparable with then the effect of the enhanced decay characterize the time evolution of the state . The steady-state behaviour shows a different aspect than in the case of magnetic leads. If the strength of the measurement increases then the probability of the state is increasing. In the case of the unmeasured system, the probability of the steady state may decrease as a function of , but for large values of is an increasing function of . This means, when the suppression of the coherent oscillations is weak and the energy difference is large enough, then the probability of filling up the state is less likely. As we expect, the probability of the steady state , the lower energy level, has the opposite character as a function of . These behaviours are shown in Fig. 4. Here, exists a mixed state, namely , which provides the QZE. If this state can be prepared as an initial condition, then for the case of no magnetization measurement the system evolves into another mixed state, but for infinite accuracy measurement (eq. (8)), , the system remains frozen in its initial condition.
Next we analyze the electric current flowing through the system. This quantity is the only measurement result retained and we study it, to show the presence of measurement back-action. Using eqs. (2), (32) we obtain the average electric current averageinner ():
In Fig. 5 we find the suppression effect induced by the continuous measurement. In the case of no measurement the
current is increasing and oscillating and then is followed by a decrease and relaxation to the steady-state. If the enhanced
decay occur in the spin up state then the current shows a slow increase and a quick relaxation to its stationary value, which is
bigger than the value found for the unmeasured case. If the measurement is extremely strong then the current is increasing slowly
compared to the other cases, the relaxation is longer and the asymptotic value is the highest. This means that the suppression of
transitions can be determined
directly after a short time evolution of the system or indirectly by examining the relaxation mechanism. The electric current in the
magnetic leads case is zero,
when QZE occurs.
In a future experimental setup where this double detection setup will be applied, changing the conditions of the unsharp detection and analyzing the sharp detection output, the scope of the continuous quantum measurement theory may be demonstrated. The substrate induced decoherence was treated, so a possible charge current change will only be the result of a measurement back-action.
4 The accuracy of the measurement
We analyzed the detection of measurement back-action by the average electric current, which can be determined from ensemble
measurements. The latter involves the problem of accuracy. If the current can be detected directly, via its magnetic field Levitov (),
then the states of the dot can be monitored with any accuracy.
If such a measurement cannot be performed, one can make an indirect measurement. Obtaining the charge counting statistics in the right lead is the most plausible in this systems and from the statistics can be deduced the average current. Recently, there was another suggestion, where the variation of the right lead charge was discussed Gur2008 (). In this cases the standard deviation of the electric current from its mean can be best understood from the uncertainty principle. The uncertainty principle gives a lower bound for the standard deviation. The upper bound of the standard deviation is the square root of the second moment. The standard deviation could be any number between these bounds. However, we focus on the lower bound, because this quantity tells us when the measurement of the current could be less uncertain.
For the observables and the pure state the following inequality holds:
where (see averageinner ())
In order to evaluate the standard deviation of the charge number, we have to rewrite the related averages in the following way