4.2 Josephson junction coupled to harmonic oscillators
4.2.3 Summary and feasibility
We have proposed and analyzed two thermal machines based on microwave cavities coupled to a Josephson junction. Both are implementations of the smallest thermal machines discussed in the literature [Brunner12].
They have the property to operate with universal efficiencies which can reach the Carnot efficiency in the reversible limit due to the minimal number of state-transitions that are involved in their operation.
The first proposal constitutes a high-power and high-efficiency mesoscopic quantum heat engine. To the best of our knowledge, this proposal constitutes the first thermoelectric heat engine where the heat current is completely separated from the electronic degrees of freedom. This is made possible by the use of a Josephson junction. The sharp energy selectivity of Cooper-pair tunneling along with the peaked spectral
0.0 0.1 0.2 0.3 0.4 0.5 0.75
0.80 0.85 0.90 0.95 1.00
Figure 4.13: Transient cooling. Blue (solid) line: The temperature in thecoscillator oscillates before reaching the steady state temperatureTcS. Green (dashed) line: Switching off the fridge when the temperature reaches its first minimum allows for cooling below the steady state temperature. For low coupling to the cold bath, a temperature belowTcS can be maintained for a substantial amount of time (shaded area). Parameters are as in Tab.4.2except forκh=κc=κr= 0.001Ωc andkBTh= 8Ωc which corresponds to half of the value in Tab. 4.2.
density of the cavities enables the conversion from heat into work with efficiencies above 75 % for realistic system parameters.
The second proposal implements a quantum absorption refrigerator which cools one of the harmonic oscillators below the lowest bath temperature. Moreover, an attractive feature of our model is a built-in on/off-switch, which allows one to take advantage of coherence-enhanced cooling.
Using a set of realistic parameters in Tabs.4.1and4.2, we showed that a substantial thermal effect can be expected for the two proposals. The experimental prospects of these results will now be discussed. In Ref. [Altimiras14], a single oscillator with frequency ∼GHz was coupled to a normal tunnel junction, with coupling λ ≈ 0.5. For Josephson junctions, experiments on two oscillators (with GHz frequencies) with λ≈0.15 have been performed and experiments on four oscillators are in preparation [Parlavecchio15]. We are thus confident that coupling two or three oscillators with λ≈0.3−0.4 is feasible. Note that we kept the couplings (EJ andκα) well below the frequencies in order to remain in the validity regime of our master equation, these parameters could be significantly increased in an experiment for testing different regimes.
Another crucial ingredient for our proposal is the coupling of the harmonic oscillators to thermal baths at different temperatures. Specifically, thehoscillator needs to be coupled to a bath at a temperature that is substantially higher than the temperature of the environment. Using a transmission line to feed the thermal noise from the hot bath to thehoscillator would allow for a spatial separation of the hot bath and the rest of the setup. Finally, an external phase bias is needed for the implementation of the absorption refrigerator.
This could be implemented using a magnetic field in a loop geometry [cf. Fig. 4.9(a)], which is standard, e.g. in rf-SQUIDs.
Our proposals are thus within reach of current experimental capabilities, and will hopefully lead to a better understanding of energy conversion in hybrid mesoscopic structures. One promising avenue to pursue is given by the study of fluctuations of heat, work, and efficiency in these thermal machines. This might help to understand how the first law of thermodynamics should be applied in quantum systems when going beyond average quantities.
Chapter 5
Conclusions & outlook
The goal of this thesis was to investigate mesoscopic systems that show non-classical behavior and thereby contribute to the advancement towards quantum technologies that harvest non-classical resources such as coherence and entanglement. In Chaps.2 and4, we presented proposals for experimental setups that make use of non-classical resources in one way or another. Chapter 2 focuses on devices that rely on single-electron sources. In quantum spin Hall system, such sources were shown to be able to create time-bin entangled particles as well as pure ac-spin currents. Another promising avenue is interferometry with single electron sources. This can reveal striking differences between a coherent, time-dependent bias voltage as compared to a constant dc-bias. Furthermore, interferometry of single electrons can reveal the entanglement of a single electron in a superposition of spatially separated states.
In Chap.4, we presented our proposals for quantum thermal machines. In addition to a heat engine that relies on the interference effect in a MZI, we presented a heat engine and an absorption refrigerator based on microwave resonators coupled to a Josephson junction. Such architectures provide versatile testbeds for investigating quantum systems because of the possibility to design an effective Hamiltonian through a proper choice of a few system parameters. Thermal machines are promising candidates to investigate quantum behavior because of their possible applications for energy harvesting. Furthermore, the field of quantum thermodynamics is fairly young and more theoretical as well as experimental effort is needed to asses the potential of quantum thermal machines.
In addition to these proposals, the present thesis also contributed to the development of the tools that are needed to characterize the systems of interest and their behavior. This is the topic of Chap.3, where we discussed our contributions to the development of WTDs and to the understanding of FCS. By considering the time interval between the excitations that traverse the system, the WTD is an ideal candidate to characterize single-electron sources. However, more research is needed to fully understand how to characterize systems where transport can not be described by the subsequent traversal of excitations.
The FCS is a powerful tool to characterize temporal fluctuations in quantum systems. It shows how interference effects naturally lead to negative quasi-probabilities that render a classical description of the fluctuations, using positive probabilities, impossible. It is at present unclear to what extend such non-classical fluctuations can be harvested for useful tasks (e.g. in quantum computation).
We conclude this thesis by pointing out some promising avenues for future research. The study of fluctuations in quantum thermal machines might reveal ways to exploit quantum fluctuations, e.g. to surpass Carnot efficiency. At the same time, this might help understanding the division of energy into heat and work in quantum thermodynamics better by going beyond mean values. Tying fluctuations of heat and work to quasi-probability distributions might shed light on the utility of negative values in distributions such as the FCS. Furthermore, in order to harness quantum fluctuations, it might be beneficial to develop new quasi-probability distributions that contain information on specific, possibly time-dependent, processes or measurements. Another promising direction is the study of quantum computational tasks using single-electron sources. It is by now fairly clear that entanglement can be created in these systems. However, its manipulation and detection is much less obvious and demands further investigation.
“...how awkward is the human mind in divining the nature of things, when forsaken by the analogy of what we see and touch directly.”
Ludwig Boltzmann
Appendix
A Monte Carlo simulations
For the Monte Carlo simulations, we start with an element of the initial density matrix and perform either a dissipative or a coherent jump with the appropriate probability. At zero temperature, the jumps and their respective probabilities read
|nLihnR| →
|nL−1ihnR−1| with probability pκ= 2κ√nLnRδt i|nL+ 1ihnR| with probability p↑L=f√nL+ 1δt i|nL−1ihnR| with probability p↓L=f√nLδt
−i|nLihnR+ 1| with probability p↑R=f√nR+ 1δt
−i|nLihnR−1| with probability p↓R=f√nRδt
e−i∆(nL−nR)δte−κ(nL+nR)δt
p0 |nLihnR| with probability p0,
(A1)
where
p0= 1−pκ−p↑L−p↓L−p↑R−p↓R. (A2) The coefficients are important to recover the master equation. The coefficient for remaining in the same state can be written as
e−i∆(nL−nR)δte−κ(nL+nR)δt p0
= e−i∆(nL−nR)δt
1−p↑L−p↓L−p↑R−p↓Re−κ(
√nL−√
nR)2δt, (A3) reflecting the fact that the dissipation reduces the off-diagonal elements of the density matrix. In this section, we always assume δt2'0.
To recover the master equation, we write
|nLihnR|(t+δt) = (1 +Lδt)|nLihnR|(t). (A4) We can then verify that the superoperatorLcoincides with the superoperator defined in Eq. (3.121).
B FCS for the dissipative emptying of a cavity
The FCS for a cavity that is being emptied by coupling to a bath can be calculated using only occupation probabilities (i.e. diagonal elements of the density matrix). To this end, we first consider a Fock state and introduce the following quantities
Π|n0i(τ) =e−n02κτ, F|n0i→|n0−1i(τ) =−∂τΠ|n0i(τ) =n02κe−n02κτ. (B1) Here Π|n0i(τ) gives the probability that no photon is emitted during the timeτ if the system is in the state
|n0i. Its negative derivativeF|n0i→|n0−1i(τ) gives the probability density of emitting a photon from the state
|n0iafter waiting a timeτ. Having introduced these quantities, we can write for the FCS of an initial Fock
state
Here the term for a fixedpis the contribution where eventuallypphotons leak out of the cavity. Thej-th photon leaves the cavity at tj and the delta function makes sure that the integrated photon number adds up to m. We definedt0= 0 andtp+1=t. Inserting Eqs. (B1) and expressing the Dirac delta distribution as
δ(t) =
To solve the integrals over time, we note that
t
With the help of the above relation and the binomial expansion, we find P|n0i(m) =
Analytical expressions for the first three Fock states read
P|0i(m) =δ(m), (B7)
P|1i(m) =δ(m−t)e−2κt+ 2κe−2mκΘ(m)Θ(t−m), (B8) P|2i(m) =δ(m−2t)e−4κt+ 4mκ2e−2mκΘ(m)Θ(t−m)
+ 4κ[1 +κ(2t−m)]e−2mκΘ(m−t)Θ(2t−m). (B9) In the long-time limit, we find
P|n0i(m;t→ ∞) =
From Eq. (3.87), we deduce that each Fock state contributes independently to the FCS P(m) =
∞
X
n0=0
hn0|ρˆ|n0iP|n0i(m). (B11)
For a coherent state |αi, the probability distribution for the incoherent emptying of the cavity then reads
which coincides with a calculation along the lines of Refs. [Clerk07;Clerk11] and is plotted in Fig.3.15.
Using Eq. (B10), we find in the long-time limit P(m;t→ ∞) =e−|α|
where I1(x) is the modified Bessel function of the first kind.
C Connection between FCS and weak values
To connect weak values to FCS, we closely follow Ref. [Dressel15] who showed that the stark shift and the dephasing rate of a qubit coupled to a cavity can be related to the real and imaginary parts of the weak value of the cavity photon number. As in the main text, we couple the system of interest dispersively to a qubit
Hˆtot = ˆH+λ
2nˆˆσz. (C1)
We first consider pure states. Since the coupling is switched on att= 0 we have
|Ψ(0)i= (c↑| ↑i+c↓| ↓i)⊗ |ψ(0)i, (C2) where |ψ(0)idenotes the initial state of the system. After evolution for a timet, we have
|Ψ(t)i=c↑| ↑i|ψ↑λ(t)i+c↓| ↓i|ψλ↓(t)i. (C3) The off-diagonal element of the qubit reduced density matrix then fulfills the equation
∂tρ↑↓(t) =−iλnw(t)ρ↑↓(t), (C4)
where nwis the weak value of the observable ˆn
nw(λ;t) =hψλ↓(t)|nˆ|ψ↑λ(t)i
hψ↓λ(t)|ψλ↑(t)i , (C5) with pre- and post-selection on the spin states | ↑i and | ↓i. Note that pre- and post-selection happen at timet right before and after the measurement of ˆn.
For the moment generating function, we find Λ(λ;t) = Trρ↑↓(t) The FCS is thus fully characterized by the time evolution of the weak value. For the cumulants, we find
hhm(t)jii=j notably, the mean is simply given by
hm(t)i=
t
Z
0
dt0hˆn(t0)i. (C8)
Fluctuations around the mean value are thus encoded in the coupling strength dependence of the weak value.
These results can straightforwardly be generalized to mixed states. From the master equation in Eq. (3.112), we reproduce Eq. (C4) with the generalized weak value
nw(t) = Tr{σˆ−ˆnˆρ(t)}
Tr{σˆ−ρ(t)ˆ } . (C9)
D Expressions beyond linear response for the thermally biased MZI
Here we give the charge and heat currents for a three-terminal MZI and a four-terminal double MZI obtained from Eqs. (4.12) and (4.13). The expressions for the two-terminal setup are obtained by choosing the appropriate boundary conditions discussed in the main text.