Landau-Zener evolution under weak measurement: Manifestation of the Zeno effect under diabatic and adiabatic measurement protocols

The time evolution and the asymptotic outcome of a Landau-Zener-Stueckelberg-Majorana (LZ) process under continuous weak non-selective measurement is analyzed. We compare two measurement protocols in which the populations of either the adiabatic or the non-adiabatic levels are (continuously and weakly) monitored. The weak measurement formalism, described using a Gaussian Kraus operator, leads to a time evolution characterized by a Markovian dephasing process, which, in the non-adiabatic measurement protocol is similar to earlier studies of LZ dynamics in a dephasing environment. Casting the problem in the language of measurement theory makes it possible for us to compare diabatic and adiabatic measurement scenarios, to consider engineered dephasing as a control device and to examine the manifestation of the Zeno effect under the different measurement protocols. In particular, under measurement of the non- adiabatic populations, the Zeno effect is manifested not as a freezing of the measured system in its initial state, but rather as an approach to equal asymptotic populations of the two diabatic states. This behavior can be traced to the way by which the weak measurement formalism behaves in the strong measurement limit, with a built-in relationship between measurement time and strength.


Introduction
The quantum Zeno effect -the suppression of time evolution between discrete quantum states under frequent repeated measurement -is well understood as a consequence of the general theory of the time evolution of a quantum system that interacts with its environment. In the simplest manifestation of this effect, interstate transitions in an interacting two-level system are shown to slow down under repeated interrogation of the level populations. When discussed in the framework of measurement theory, this behavior reflects the wavefunction collapse upon determination of the quantum state. In the more general weak measurement theory the effect on the system of a continuous weak measurement can be cast as a decoherence process whose rate reflects the measurement weakness. Indeed, the time evolution of a quantum system interacting with its environment is usually discussed without making connection to an underlying measurement process. Still, it is sometimes useful to make this connection for its conceptual value as well as its experimental implication. To elaborate, consider a two level system that represents an electron tunneling between the two minima of a double-well potential and assume that temperature is low enough so that only the two lowest electronic states in this potential can be occupied. We may choose to measure the charge state of one of the wells using a nearby point contact device or we may devise a spectroscopic tool that monitors the population of the true system ground state (a linear combination of the two states localized in each wells). These different measurement protocols have different effects on the system dynamics and their consideration may provide insight on the interrelationship between measurement, decoherence and quantum time evolution.
In this paper we consider the effect of continuous weak measurement on the time evolution of a Landau-Zener-Stueckelberg-Majorana [1][2][3][4] process. In the so-called diabatic representation, the Hamiltonian of this well-known model describes two coupled levels with time dependent energy spacing The superscript dia indicates that this Hamiltonian is represented in the so called diabatic basis.
Denoting the general time dependent solution of the Schrödinger equation in this representation There is a large body of work that address the effect of coupling to an external thermal environment on this evolution, [5][6][7][8][9][10][11][12][13][14] [15,16] [ [17][18][19][20][21] including the possibility of externally affected control. [22] As pointed out above, and further demonstrated below, the effect of continuous weak quantum measurement on a system can be cast as a dephasing process. As such, its description is strongly related to the above studies. Indeed, some of these works address detailed properties of the external bath, including its temperature, that are not usually included in standard descriptions of measurement. On the other hand, discussing this time evolution as a consequence of a measurement process can highlight issues that are not naturally considered otherwise. For example, most of the works cited above focus on a particular model of system bath coupling, where the diagonal matrix elements of the Hamiltonian in the diabatic representation are randomly modulated or otherwise linearly coupled to a harmonic thermal bath. In the framework of measurement theory it is natural to define first the nature of the measurement. In particular, we may consider monitoring the populations of the adiabatic levels or of the diabatic levels, with possibly different consequences on the ensuing time evolution. This distinction may come up in specific experimental situations. For example, in many applications, the two diabatic states represent electron localization on different sites in the system (in which case the coupling V in Eq. (1) is the interaction responsible for electron transfer between the two sites). Measurement of the corresponding population may be done by monitoring the charge on one of these sites using a nearby quantum point contact whose transmission (hence the corresponding monitored current) is sensitive to this charge, see e.g., Ref. [23]. On the other hand, it is possible to monitor the instantaneous population of the (adiabatic) electronic eigenstates of a system, as was done in the possibly first experimental demonstration of the quantum Zeno effect [24] [25] (see also Ref. [28] for a recent demonstration of such measurement).
In this paper we discuss the realization of the Zeno effect under these two types of measurement. In the next Section we briefly review the theory of continuous weak measurement in the Kraus operator formalism [29,30] and discuss the time evolution showing the different manifestations of the Zeno effect in the strong measurement limit of these two schemes. Section 4 concludes. in the case of a continuous spectrum of measurement outcomes. We are interested in a concrete form of a measurement operator, which is able to describe a fuzzy measurement process. We expect that this operator [32,33] depends on a parameter  , which defines the strength of measurement and is hence related to its resolution. This parameter should provide the ability to interpolate continuously between the hard projective measurement and a fuzzy measurement with very few impact on the system. Intuitively we expect that it is more probable that an actual eigenvalue of Â lies close to the measured value a and that the probability to be the actual value then decreases smoothly by growth of | | A a  . Hence, the measurement operator is approximated by a Gaussian form with a single parameter 

LZ dynamics under continuous measurement
It is easy to check that the completeness relation is satisfied. Furthermore it is clear that for    we obtain an operator which describes a strong, exact measurement as the Gaussian becomes very narrow and peaked for the eigenvalues of Â , while 0   corresponds to a very weak measurement with fuzzy observations and the Krausoperator become almost 1 .
The probability density ( ) a  to obtain a result a of a measurement is in general given by †( The normalized density matrix after such a measurement is † after,ˆ= The density matrix formalism provides the ability to treat nonselective measurements. We perform a measurement on a system and the output is registered but not used. Accordingly, we obtain for the nonselective post measurement density matrix: We have to sum over the unnormalized selective density matrix to conserve the normalization of nonsel after  . This is the same as the sum over all normalized selective matrices weighted with the probability ( ) a  .
Continuous weak measurement. The previous definition of a measurement can be easily generalized to a continuous measurement. Naivly, continuous projective measurement would cause a total suppression of the dynamics analogously to the quantum Zeno effect [31]) due to the continuous collapse of the wave function into an eigenstate. Alternatively we can consider continuous weak measurements which provide less information but do not disturb the system to such an extent. The question is whether it is possible to obtain sufficient information with continuous weak measurement while leaving the system as undisturbed as needed.
A general description of the time evolution The time evolution of the system after time measurements separated by intervals of free time evolution is then given by: Furthermore, it is assumed that the measurement strength is inversely proportional to its frequency [32] = t   with constant  . Then we have the following form of the Kraus operator: And in the continuum limit 0 t   ( N   ), we obtain up to a normalization factor The discrete results i a become a function (  In Ref. [32] it was shown (see also [31] In applying this general result to the LZ evolution under continuous measurement we can choose to monitor populations in the diabatic states or in the adiabatic states. The former measurement mode can be accomplished by choosing In Eq. (12). This leads to 11 For the other possibility, continuous measurement of the adiabatic populations, the measurement observable is the transformed operator where   U t is the unitary trasformation that diagonalizes the instantaneous Hamiltonian (1) Alternatively (and equivalently) we can represent the dynamics in the adiabatic basis, where The evolution of the density operator is similarly modified. Eq. (12) becomes The explicit form of the transformation matrix U is given by

Results and Discussion
It is convenient to display the results in terms of dimensionless parameters. Define In terms of these variables Eqs. (14)   [35] More interesting is the way in which the Zeno effect is manifested in the weak and strong measurement regimes as best seen in the z=0.5 results. In the absence of measurement this evolution is fairly adiabatic, and the population of the initially populated diabatic level goes from 1 to ~ 0.2 as the system evolves across the avoided crossing. As   increases from zero the measurement affects an increased non-adiabatic character of the time evolution -an increased probability to remain in the initial non-adiabatic level. However, as   increases further (stronger measurement) this probability assumes the asymptotic value of 0.5. Further increase in   does not change this asymptotic limit, however the typical Zeno behavior is seen in the slowing down of the approach to this limit.

Discussion and conclusions
We have found that the time evolution associated with the Landau-Zener process under continuous weak population measurement depends on the character of the measurement process: When population of the adiabatic states is monitored, the time evolution exhibits a quantum Zeno effect behavior, becoming more adiabatic for stronger measurement. Interestingly, close to the adiabatic limit (   These results should not be surprising in view of past work on the dynamics of the LZ process in a system interacting with a dissipative environment, [5][6][7][8][9][10][11][12][13][14] [15,16] [ [17][18][19][20][21] including the possibility of externally affected control. [22] however viewed in the framework of measurement theory can yield some new insight. First is the strong dependence of the dynamics on the character of the measurement. Most of the papers cited above consider the effect on the LZ process of decoherence in the diabatic basis. In the present context, monitoring the population of the adiabatic states has a markedly different effect on the system dynamics than following the corresponding non-adiabatic states. Obviously, this difference just reflects the fact that environmental effects on system dynamics depend on the way the environment is coupled to the system, however viewed from the perspective of a measuring process this points to a way to controlling the system dynamics by engineering processes that affect its decoherence. [37,38] Secondly, the manifestation of the Zeno effect when the measurement is done in the nonadiabatic basis calls into question the standard measurement theory argument for this effect. This  Which becomes 1 as N   . This argument, however, disregards the question whether projective measurements can be made at arbitrarily short time intervals, and arguments against this possibility were made. [39] Without getting into this discussion we note that the theory of continuous weak measurement implicitly assumes that the strength of individual measurements is inversely proportional to the measurements frequency, see Eq. (9). Indeed, it is easy to show that the argument that leads to Eq. (29) and consequently to 1  , is obtained as the T   limit of Eq. (30) under the assumption that inverse frequency t T N   of projective measurements must be finite.
It is of interest to consider scenarios for experimental realization of such different measurements. Diabatic poulations can in principle be monitored in systems where the two diabatic states correspond to two molecular (or dot) charging states. It is harder to envisualize a measurement of adiabatic populations: The standard tool for such measurement is optical spectroscopy which is inherently a destructive measurement. Identifying a property of the eigenstates of a system's Hamiltonian that can be detected without destroying the state is an interesting challenge.