First, we take
and
. Spin operators for spin-
particle are
, where
are the Pauli matrices. These spin operators for spin-
particle form the fundamental irreducible representation of
. Spin operators for the spin 1 particle are the three-dimensional irreducible representation of
. With standard representation in canonical bases [
61], the eigenvalues and the eigenstates of the Hamiltonian given in Equation (
29) are listed in
Table 2. As in the previous cases, for the measurement step of the engine cycle, we have a number of choices for the measurement operators and we explore those options in the previous sections for the case of two spin-
particles. Now, for brevity, we consider one particular measurement setup and observe the effect of higher spin, such that this spin can also be a controlling parameter for the efficiency. We choose the measurement operators as,
where
and
are the eigenstates of the operator
for the spin-half particle
A. In other words, we are doing measurement of the operator
on side
A for the spin-
and measurement of the operator
on the side
B for spin-1. In the plots (see
Figure 6,
Figure 7 and
Figure 8), we compared different scenarios for the same measurement settings on the two sides. By the same measurement settings, we mean that on the spin half side the measurement operators will be the projectors constructed from the eigenstates of the operator
and, on the higher spin side, it will be the projectors of the eigenstates of
. For the above measurement operators, we calculate the quantities as work and heat (during measurement process and thermalization step) and evaluate the efficiency of the heat engine. The expression for the total work extracted for this case is,
From the expression above, one can see that only
will not guarantee the positivity of
for all
J, unlike the previous case of two spin-
particles. Specifically, the work extracted is negative when
. This is not the case for two spin-
particles, where expression of the extracted work is such that
always gives positive extracted work and
gives negative extracted work. However, in the present case, for
, extracted work can be negative and, for
, extracted work can be positive when
. As we show below, this is special for the asymmetric cases only. In
Figure 6, we plot the efficiency and compare it with the spin half scenario for
, with same values of magnetic fields previously considered, i.e.,
and
. We note that (
Figure 6) the efficiency for the spin-1 scenario can be higher than that of spin half scenario for some range of non-zero values of
J. For the uncoupled case, both engines give same efficiencies, which is
. Another point to note is that the efficiency can go to negative for the spin 1 scenario. As discussed above, the negative efficiency comes entirely from the negative extracted work, because
is always positive. The total extracted work
can now be negative starting from a certain value of
J, as shown in
Figure 7, for
, with
and
, whereas
is always positive, as shown in
Figure 8. Now, one can further analyze the work strokes and see which work stroke is contributing more for negative work. From the
Figure 9, one can note that, as before, the first adiabatic work stroke gives the positive work output, whereas, for second adiabatic work stroke, we get negative work. If we take
, the situation is reversed. We have shown the case of
, with
and
, in
Figure 10. The plot is exactly opposite to the previous one with
and
(see
Figure 9). Thus, in the range of
J where the efficiency is negative, the situation appears as follows: the average energy
is entering to the working medium,
work is being done and thereby heat
goes into the heat bath of temperature
T. In some sense, one can associate a refrigerator action for this negative work scenario. In a conventional quantum Otto refrigerator, the system is first prepared in the thermal state with temperature corresponding to the cold bath. Then, in the first and third adiabatic strokes, a total work
W is added to the working medium. In the second and fourth steps,
heat is taken from the cold bath and
heat is added to the hot heat bath and eventually cooling the cold bath more. The co-efficient of performance (COP) for the refrigerator is given as
. In our case,
is always positive and
is always negative. This indicates that we can consider a cold bath of effective temperature
, such that, after the second stroke, we have,
where
is the thermal state at temperature
with the Hamiltonian
, given in Equation (
29), where
. Solving the above equation we can associate an effective temperature
with
. Thus, now one can read the cycle as transferring heat from the cold bath of temperature
to hot bath of temperature
T and, for this,
work has to be done. This means that the COP of this refrigerator action is
. Thus, to get a positive work output, we have to judiciously choose the value of the coupling constant
J, such that efficiency is not negative. Then, we get an advantage for higher efficiency over the uncoupled one. Let us now consider the next asymmetric scenarios and see whether a similar trend, i.e, increase in efficiency and occurrence of negative efficiency is present or not. We start with the case where the spins on two sides are
and
. With standard representation of spin-
in canonical bases [
61], the eigenvalues and eigenstates of the Hamiltonian in Equation (
29) are given in
Table A1 of
Appendix A. Now, we choose the same kind of measurement operators as in the previous case, i.e., on the spin-
side we measure
and on the spin-
side,
:
with
,
, and
,
,
, and
are the eigenstates of
corresponding to the eigenvalues 3, 1, −1, and −3, respectively. We also consider the scenario of
and
. For this case, the eigenvalues and eigenstates of the Hamiltonian in Equation (
29) is given in
Table A3 of
Appendix A. Again, we take the same measurement choices as before, i.e., measurement of
spin operator on the side
A and
spin operator on the side
B. We calculate
W,
and the engine efficiency for each case. As in the previous asymmetric case, the condition
does not guarantee the positivity of the extracted work
in both present cases. Starting from a certain value of
J,
can be negative for
and positive for
. Thus, for asymmetric cases,
and
, both situations give rise to the extracted work to be negative staring from certain ranges of
J. We plot the efficiency of the heat engine for the aforesaid three asymmetric cases together in
Figure 11 for
with
and
.
case can be calculated in a similar way. We observe that, for asymmetric situation, the efficiency becomes negative after a certain value of
J. In addition, in
Figure 11, we can observe that, as the difference of spins increases between the two sides, the efficiency goes to more negative value. Thus, from these observations, it is clear that, if the two spins on both sides are not the same, then efficiency can be negative. Another interesting feature to notice from the plot is that, within the range of
J where efficiency is positive, higher differences of the spin values give larger gain in efficiency over the uncoupled one. We have to take correct coupling strength
J, to have a higher but positive work output from these measurement-based coupled higher spin coupled heat engines. As in the case of
and
, we also plot the work done in two adiabatic strokes for these two asymmetric cases in
Figure 12 and
Figure 13 and note that the negative contribution in the extracted work is due to the second work stroke.