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We calculate the work done by a Landau-Zener-like dynamical field on two- and three-level quantum system by constructing a quantum power operator. We elaborate a general theory applicable to a wide range of closed-quantum system. We consider the dynamics of the system in the time domain ]-
*t*
_{LZ},
*t*
_{LZ}[ (where is the LZ transition time in the sudden limit) where the external pulse changes its sign and its action becomes relevant. The statistical work is evaluated in a period [0,
T] where
*T* ≤t
_{LZ}. Our results are observed to be in good qualitative agreement with known results.

The pioneering work of Jarzynski establishes a non-trivial relation between the non-equilibrium work performed on a thermally insulated classical system and the change in its equilibrium free energy [

Here,

where

Though Equation (1) is extended to quantum systems, a key and natural question arised: does it still hold in a more realistic situation where the system remains in thermal contact with its environment while the forcing protocol is in action? An affirmative answer to this question was given by Crooks based on classical arguments [

The experimental measurements of the proper free energy of a system lead to the average exponentiated work using Equation (1). This measurement is not always easily performed experimentally. The determination of the proper work has turned out to be a non-trivial task [

The present paper is devoted to the calculation of the work done by an external field of constant amplitude on two- and three-level isolated systems. The systems are assumed to be thermally isolated from their environ- ments. We consider as in Ref. [_{i} and t_{f}. Consider the difference Hamiltonian operator

Here, the overdot denotes the time derivative. The average statistical work done during a period T on any quantum system is statistically defined as:

This formula is employed throughout this paper.

The paper is organized as follows: In Section 2, we present a general theory for calculating the work done on a coherently driven system. In Section 3, the theory is applied and tested on two-level system subjected to inter- band LZ transitions [

The procedure for calculating the work done during transitions between Zeeman multiplet is illustrated. We consider systems on which act simultaneously a strong time-dependent diagonal field and a slowly varying perpendicular field. The prototype Hamiltonian describing these effects are written in the diabatic basis (basis of the eigen-states of the Hamiltonian in the absence of couplings) as follows:

The dynamical symmetry associated with (5) is referred to as

dependent control protocol

states

The protocols

During the work, the system passes through a sequence of several configurations (non necessarily equili- brated). If the states of the system are described by the reduced density matrix operator:

then, the statistical average of an arbitrary time-independent operator

(disordered average). For our case, the eigen-spectrum is discrete and characterized by the

the total wave-function

where

where H indicates the Heisenberg picture. The thermal and statistical averages are taken as:

Our goal is thus achieved once the evolution operator

The first and the second moments of the work whatever the process involved are respectively given in the Heisenberg picture by:

and

Here, the power operator is basically a function of the fermionic occupation number

such that

As an important remark, evaluation of

and

where we have defined the transition amplitudes

and

The transition amplitude

The measurement of the

Once the eigenvalues are obtained, the transfer matrices for the intermediates trajectories

where

Between measurements, the system propagator describing a set of transitions through the j-crossing points is expressed as follows:

Consider the

Here,

transition is

Here,

A point of concern for introducing

with

The average occupation number

We illustrate the theory presented above by considering the simplest case of the spin-1/2 two-level system.

The model Hamiltonian considered is deduced from Equation (5) as,

The two instantaneous eigenvalues and eigenfunctions relevant to (22) should be evaluated. The results read:

where

is the level-separation energy and

where the normalization factors

For spin-1/2 considered, adiabatic

with

The projections

Here,

sentation,

Our analyzes of the work done on a two-level system are mainly performed in the limits

which is achieved in the sudden limit while

is the one obtained for the counterpart. It is instructive to note that the matrix

These data are helpful to evaluate the work done on a two-level system by an external field of constant amplitude.

The LZ process describes the dynamics of two states which come close by linear variation of a control protocol:

The energies

The time-evolution of the transition probability function during the rapid and slow drives show that nothing happens to the system before the crossing. It mainly remains in its initial state exhibiting an insensitivity to the external sweeping protocols

Considering the Hamiltonian (22), the power operator for a two-level system is explicitly evaluated as:

where

The average work done during a period T to transfer a population from the state

derived using the formula:

where

Here,

The average of the square fluctuations of the work,

The full propagator for the two-level system driven by the traditional LZ process (single crossing time

Here,

with

being the phase accumulated by the components of the wave-function from

where the angle

is the Stockes phase. The function

In the sudden limit of transition,

Substituting the instantaneous eigenstates (25) and (26) into the above expressions yields the transition amplitudes. Another way to find the transition amplitudes is to consider the projections of the states (43) and (44) onto the diabatic basis (

and

In these relations, projections of instantaneous eigenstates read:

In the regime

Thus, the statistical average works done on the two-state system are given by:

and

In principle, for the Landau-Zener drive,

An algebraic character can be associated with the quantum work. The work is antisymmetric by path reversal. By changing the protocol

In addition, it should also be noted that

Because of the link between work and heat, the properties in Equations (52) and (53) can be attributed to the heat.

Recall that

A particular characteristic for a quantum work similar to that of classical work should be pointed out. Basi- cally, the work done on a classical system does not depend on the followed path but only on the initial and final positions. Relations (50) and (51) show a contrasted situation in the regime of sudden transitions. Namely, the work done on a quantum two-level system does not depend on the followed path. It does not depend yet on the initial and the final states. The initial state can be chosen arbitrary, the efficiency remaining the same.

In the regime

The occupation probability,

plete transfer. Both diabatic states remain constantly coupled and the total population is preserved,

An alternative way to find the work done on a system is defined through the two-measurement process (TMP) [

ning and at the end of the evolution. The work done during the process is predetermined by the corresponding energy difference,

procedure for a rapid LZ drive process(non-adiabatic evolution). The work is then defined as:

Considering the Hamiltonian

As already shown, the transition amplitudes do not depend on time in the sudden limit. The work is obtained as follows:

This result exactely coincides with the one derived from Equation (35) under the same assumptions.

From a quantum mechanical view point it is more convenient to find the Hamiltonian difference

which is nothing but the power operator in Equation (33).

Here, an additional level position is present. It might evolve with time or not. States are coupled by this intermediate position via a constant coupling. The model is of the form:

In this representation,

The associated eigenvalues are expressed as follows

Here,

with

where

and

Similarly, we have defined

The instantaneous eigenfunctions are calculated. The results are written as follows:

where

Here,

and

The normalization factor

where

For spin-1, we do

We obtained this matrix by direct calculations. Indeed, the angle

the diabatic basis are of the form (28), namely,

relations serve for derivation of transition amplitudes as we did for the case of two-level.

A projection matrix can be constructed. The extreme limits

and

for adiabatic limit. As for the case of two-level, these two matrices obey

The definition of the work given in the first section is used. The power operator is expressed here as:

where,

The average of the work can be evaluated with aid of the formula:

The average

and

The components

These representations help to approximate the work done on a three-level system for the sudden and adiabatic limits of transition. For instance, in the sudden limit, it can be shwon that, populations transfered between the three levels correspond to those for spin-1 LZ problem:

The works in (82) are decomposed as follows:

and correspond each to a diabatic state. As already explained,

been transferred from the diabatic states

Considering the works done on two-level systems, that for three-level in Equations (86)-(88) are the sum of works between intermediate diabatic positions. These works could be constructed intuitively considering intermediate works separately.

Equations (86)-(88) can be transformed with the aid of the components of the matrix in Equation (85). Thus, one obtains:

We have exploited the fact that

We have presented a theory for evaluating the work done on a multi-level system. Two particular cases (two- and three-level) are considered and permit to illustrate the theory. The obtained results for two-level spin-1/2 system were shown to be simple functions of the Landau-Zener probability function. Thus, the work depends on control protocol which can be experimentally manipulated. We have demonstrated that forward work and backward were absolutely identical and differ algebraically by a sign in the sudden limit. The efficiency of the work done has been observed as being independent on the initial state chosen. It has been pointed out that an adiabatic variation of the protocol cannot lead to a complete population transfer when the system is isolated from its environment. The half of the initial population corresponds to the maximum of the population trans- ferable. Both states remain constantly coupled. If one allows the internal energy of such a system to flow out of it or an external energy source to flow towards the system, it will be entangled and its states will no longer be expressible as linear superposition of the states of the subsystem. An equilibrium would not be achieved. The system will mostly evolve out of equilibrium. The work done will be accompanied by an additional work due to the perturbation:

Here,

For three-level system on the other hand, the work to be done in order to achieve a transfer of population from one of the upper (lower) to another lower (upper) diabatic states appeared as being the sum of intermediate works performed independently.

The authors thank M. Tchoffo, A. J. Fotue, Kenfack Sadem and F. Ngoran for careful reading of the manuscript and valuable suggestions.

IssofaNsangou,Lukong CorneliusFai, (2015) Work Done on a Coherently Driven Quantum System. Journal of Quantum Information Science,05,89-102. doi: 10.4236/jqis.2015.53011