Abstract
We propose a simple diatomic system trapped inside an optical cavity to control the energy flow between two thermal baths. Through the action of the baths the system is driven to a nonequilibrium steady state. Using the Large Deviation theory we show that the number of photons flowing between the two baths is dramatically different depending on the symmetry of the atomic states. Here we present a deterministic scheme to prepare symmetric and antisymmetric atomic states with the use of external driving fields, thus implementing an atomic control switch for the energy flow.
Introduction
Understanding how energy flows and how to control this flow has recently attracted considerable attention^{1}. It further prompted the study of transport phenomena in quantum systems including the analysis of fundamental laws like Fourier’s Law^{2,3,4,5} and the second law of thermodynamics^{6}. Furthermore, it was shown that quantum effects can be used to enhance important thermodynamical processes. Examples include quantum thermal machines^{7} and solar cells^{8,9,10}. Another exciting field of research is thermal phenomena and design of electronic and thermal devices at the nanoscale, where several experiments demonstrated electronic switches at the atomic scale^{11,12,13}. Notably, a twoterminal switch to control the charge transport in a junction based on the reversible rearrangement of single atoms was shown in ref. 13.
One of the most promising physical platforms to study quantum transport phenomena is coupled optical cavities with one or more atoms trapped per site. In order to design different topologies, optical cavities can be linked via fibers^{14,15,16} with an effective interaction given by BoseHubbard Hamiltonians^{17}. These cavities are suitable to model transport in systems with nextneighbor interactions with lineal or network topologies^{16}. The control capability of optical cavities make them a very useful tool to study transport phenomena. Applications include the study of quantum transport in different dimensions^{18,19} and the simulation of noiseassisted transport^{20}.
Recently, Buča and Prosen showed that certain symmetries in an open quantum system lead to different nonequilibrium steady states^{21} (see also refs 22,23). These steady states are classified according to the symmetry operator spectrum. The multiplicity of steady states leads to different expected currents. This effect can be used to design a symmetrycontrolled quantum switch to govern the energy current passing through the system by simply selecting its initial state^{24,25}. Subsequently, the role of symmetry in energy transfer was studied in both transient^{26,27} and steady state scenarios28. However, harnessing quantum symmetry to control the energy flux in a steady state scenario was considered only in toy models^{24,25}.
In this paper we propose a feasible design of an atomic symmetrycontrolled thermal switch. Our design comprises a chain of cavities, coupled, at the two ends, to two different temperature baths, T_{1} and T_{2}, while the middle cavity is doped with two laserdriven atoms. The thermal baths drive the system out of equilibrium and it evolves to a steady state scenario, where there is a finite energy current flowing from the hot to the cold bath. We show how a pair of atoms trapped in the middle cavity can be used to control the energy current through the system up to four orders of magnitude. The control is implemented, in a deterministic fashion, by switching between the symmetric and antisymmetric atomic state manifolds with the use of a laser driving field.
Results and Discussion
Our system comprises a chain of three optical cavities coupled to thermal baths at temperatures T_{1} and T_{2}, as it is shown in Fig. 1. The middle cavity is doped with two threestate atoms addressed by external laser fields, while the cavity photons hop between neighboring sites at a rate J. The evolution of the system is given by a Markovian master equation^{29} in the form ( = 1),
where H = H^{ctrl} + H^{hop} is the rotatingwave approximation (RWA) Hamiltonian accounting for the control atomlaser interaction and the hopping of photons, . Here a_{l/2/r} is the anhilitation operator of photons of the left/middle/right cavity, while are Lindblad superoperators, which describe the interaction of the system with the incoherent channels. There are four decay channels associated with the atomic states [see Fig. 1 right], which are accounted for by with i = {1, …, 4} and two decay channels , with c = {l, r}, associated with the thermal bath coupled to the left and right cavity, respectively. The RWA is taken for simplicity, but it does neither affect the multiplicity of steady states nor the control capacity of the system.
To realize a controlled deterministic switching between atomic states with different symmetry we adopt a scheme, where the quantum state preparation is heralded by macroscopic quantum jumps^{30,31}. We assume that two identical threestate λ atoms are coupled to the middle cavity’s mode a_{2} on the 0〉 ↔ e〉 transition with a coupling strength g. Furthermore, the 1〉 ↔ e〉 and 0〉 ↔ 1〉 transitions are driven by two external laser fields with Rabi frequencies Ω_{0} and Ω_{1}, respectively. If the excited states e〉 are far offresonant, i.e. Ω_{1} < g, Γ, , where Δ and Γ are the detuning and decay rate of the excited states, then they can be adiabatically eliminated. Thus, the state space of the diatomic system is reduced to contain only four states, three of which 00〉, 11〉 and , are symmetric, while is antisymmetric, as displayed in Fig. 1 right. The system is then equivalent to a four level system. Then, the control Hamiltonian is given by (see Methods section)
with and .
The transitions between the symmetric and antisymmetric atomic states occur as a result of two decay channels,
while there are two more decay channels which link the symmetric states, and , see Fig. 1 right. Here, the decay rates are modified by the external driving fields and are explicitly given by , with Γ_{0} + Γ_{1} = Γ^{30,31}. Note that, there is mixing between the symmetric and antisymmetric atomic subspaces only when Ω_{0} ≠ 0.
The presence of the baths leads to the creation and destruction of cavity photons in the left and right cavities, which we model by the following incoherent channels,
where subscripts 1 and 2 correspond to creation and destruction, respectively, γ_{th} is the interaction rate, n_{1(2)} = 1/[exp(ω/(k_{B}T_{1(2)})) − 1] is the temperaturedependent mean excitation photon number at the resonance frequency in the respective bosonic bath and k_{B} is the Boltzmann’s constant^{29}.
To analise the transport properties of our system we use a Large Deviation approach^{24,25,32,33,34}. The key elements of this technique are the large deviation functions θ(s) and G(q) that account for the statistic of photons interchanged between the system and one of the baths. These functions determine the values for long times of both the momentgenerating function, Z_{s}(t) ~ exp[tθ(s)] and the probability of having a certain value q of the current, P_{q}(t) ~ exp[tθ(s)] (see Methods). The moments of the current distribution can be calculated directly from the large deviation function θ(s) by the expression .
In Fig. 2 the momentgenerator LDF θ(s) is displayed for the cases when the lasers are driving the atoms (Ω_{0,1} ≠ 0) and when they are off (Ω_{0,1} = 0). When the lasers are off there are two different steady states in the system. This is reflected in the kink of θ(s) at s = 0. The nonanalyticity of θ(s) can be interpreted as a dynamical phase transition and it is a consequence of the existence of more than one steady state with different activity. If the atoms are in one of the symmetric states {00〉, 11〉, s〉}, then they facilitate the transfer of photons to the cold reservoir T_{1}. This is described by θ(s) in the region with s < 0. However, if the atoms are in the antisymmetric state a〉, which is a dark state for the system, then they cannot interact with the cavity photons, thereby reducing the energy transfer through the cavities. This scenario is depicted in the LDF θ(s) for the regime s > 0.
The multiplicity of the steady states is a consequence of the symmetries of the system^{24,25}. If Ω_{0} = 0 (laser off) the unitary operator π that interchanges the states of the two atoms (π00〉 = 00〉, πs〉 = s〉, π11〉 = 11〉, πa〉 = −a〉) commutes with both the Hamiltonian and all of the Lindblad operators, , . This leads to a degeneracy of the Liouvillian operator and to multiple steady states, which can be labeled by the eigenvalues of the operator π^{21}. When the laser is on, meaning that Ω_{0} ≠ 0, the degeneracy is broken, because the operator from Eq. (3) does not commute with π because is equal to zero if and only if , that corresponds with Ω_{0} = 0. In this case there is only one steady state that mixes the symmetric and antisymmetric manifolds of the atomic states. Because of this, there is no dynamical phase transition and θ(s) and its derivative are analytical for all values of s, including zero, as it is displayed in Fig. 2.
We recover the large deviation function G(q) from θ(s) by a Legendre transformation. The nonanalyticity of the derivative of θ(s) leads to a nonconvex regime in G(q) that cannot be directly inferred by the Legendre transformation, as it is sketched in the inset of Fig. 2. The probability of having Q events in a long time t can be calculated as P_{Q}(t) = exp[tG(Q/t)].
To compare the efficiency of the symmetric and antisymmetric steady states we calculate the mean number of photons transferred to the reservoir per unit time 〈q〉 directly from θ(s) according to . If there are more than one steady states, the mean flux for the more and less active steady states are given by and , respectively. We calculate the ratio between the maximum and minimum flux α = 〈q〉_{max}/〈q〉_{min}. In Fig. 3 we plot α as a function of the hopping parameter J and we find that α is strongly dependent on the value of J, up to three orders of magnitude. The control capacity of the system increases dramatically when J decreases and vice versa. The reason is that J modifies the coupling of the atoms to the cavity modes – the smaller J is the stronger the atoms are coupled to the photons – which leads to an increase in the control efficiency of the current.
To control the switch we need to be able to distinguish between the symmetric and antisymmetric (dark) states. For our parameter selection the dark state current is close to zero and the probability of photon absorption for this state is negligible. This means that if one photon is absorbed by the cold reservoir we are in a bright state with a very high probability. To be able to detect it we need to ensure that the time between absorption of photons in the bright state is smaller than the mean length of the dark period. The probability of transition from a dark into a light period is and the mean length of a dark period equals^{31},
The mean time between photon absorption can be calculated as the inverse of the mean flux T_{C} = 1/〈q〉. As the mean flux is an average of the bright and dark fluxes we can consider it as a lower bound of the flux when the atoms are in the bright manifold. In Fig. 3 (left) we can see that the ratio T_{C}/T_{D} increases with the hopping between cavities J. There is a competition between the capacity to distinguish the bright and dark periods that require a high current and the ratio between the bright and dark flux that is higher for smaller fluxes. Nonetheless, for a broad range of the parameters the time between photon absorption in the bright state is long enough, compared to the mean length of the dark period and the photon flux of the bright state is more than two orders of magnitude larger than the photon flux in the dark state.
The maximum flux, 〈q〉_{max}, is displayed in Fig. 3 (right) as a function of the cavities hopping parameter J. Due to the small value of the coupling parameter J, the calculated values for 〈q〉_{max} are also small but, notably, they are measurable. To assess this, we performed a simple calculation and obtained a 〈q〉_{max} ~ 10^{−5} (in atomic units) that corresponds to a flux of the order of 10^{13} photons per seconds. Given such light intensity for the bright period, it will be easily distinguished from the dark period when the flux is several orders of magnitude smaller. The proposed design has several potential applications. It can be used as an energy current switch, an atomic memory and also an entangling scheme due to the indirect measurement of the atomic state through the measurement of the current. This is particularly useful for memory and entanglement generation, as the antisymmetric state of the atoms is a maximally entangled state. Thus, by monitoring the energy current we can control the separability properties of the atoms. The readout of the atomic state can be performed without directly interacting with the atoms themselves, as their state affects the macroscopic light flux. The proposed system can also be extended to realize better control by placing more pairs of atoms in the cavities. By connecting these cavities with optical fibers we can create networks of cavities^{16}, each of them acting as a switch, thereby increasing the flux control.
In this paper we have presented a realistic model of an atomic thermal switch based on coupled optical cavities. We showed that by placing two laserdriven atoms in an optical cavity it is possible to control the energy current trough the system when it is driven out of equilibrium by the action of two thermal baths at different temperatures.
The control capacity of the system depends on the system parameters, increasing when the hopping between cavities decreases. For small values of the hopping parameter between the cavities the current of the bright state can be three orders of magnitude bigger than the current of the dark state. This can be used also as a quantum memory as the state of the atoms that can be measured without distorting the atoms, just by measuring the current between the full system and the thermal bath. The importance of this design relies on the simplicity of manipulating an internal degree of freedom of a system in comparison with modifying the system parameters. Similar symmetrycontrolled switches can be designed for many different quantum setups such as quantum dots, optical lattices and trapped ions.
Methods
Calculation of the Master Equation
The symmetry control of our system is given by a pair of Λ atoms at the second cavity driven by a laser. This scheme was first proposed to create entangled states by macroscopic quantum jumps^{30,31}. The Hamiltonian of the system can be decomposed in the way H = H^{ctrl} + H^{hop}, were H^{ctrl} is the part of the Hamiltonian that models the interaction between the atoms and the laser field and H^{hop} is the photon hopping between cavities. The atoms are trapped in the cavity in a way such that the effect of the laser is the same in both of them. The coupling strength of the 0 − 2 transition to the cavity field is g and the laser Rabi frequencies for the 0〉 − 1〉 and 1〉 − 2〉 transitions are Ω_{0} and Ω_{1},
where is the bosonic ladder operator that creates and destroys a photon at cavity 2. It is assumed that Ω_{0} < g, Γ, . As the detunning Δ is bigger than the other parameters of the system the excited atomic states can be adiabatically eliminated. The system is then equivalent to a four level system. As both atoms are equally affected by the laser, it is convenient to use the Bell basis for describing the state of the two atoms, with and . The control Hamiltonian becomes
with and . The Lindblad operators that correspond to the atoms incoherent channels are
where .
Large deviation approach
To analyse the statistics of the current flowing between the system and the thermal reservoirs we use a fullcounting statistics method^{24,25,32,33,34}. For simplicity, we focus only on the current between the system and the T_{2} reservoir. We first introduce the reduced density matrix ρ_{Q}(t), which is the projection of the full density matrix onto the subspace with Q photons interchanged between the system and the T_{2} bath. It is not normalised and the probability of having a certain value of the current Q in a period of time t is given by P_{Q}(t) = Tr[ρ_{Q}(t)]. This probability scales for long times following a large deviation principle, P_{Q}(t) ~ exp[tG(q)] (q = Q/t), where G(q) is the LDF of the current.
The direct calculation of G(q) is complicated, but it becomes simpler after a change of ensemble. We make a Laplace transform on ρ_{Q}(t) and obtain a new reduced density matrix , where we have introduced a counting field s, that is the conjugate field of the number of photons Q. The fields Q and s are dynamical variables with the same relation as the thermodynamical variables pressure and volume. This procedure is formally equivalent to the thermodynamic transformation between the canonical and microcanonical potentials. The utility of this transformation comes from the fact that the evolution of the reduced density matrix ρ_{s} unravels into a set of equations in the form
where and and they account for the introduction of the counting field s in the right current.
The Laplace transform fulfills a large deviation principle in the form Z_{s}(t) ~ exp[tθ(s)], with θ(s) being a large deviation function related to G(q) by a Legendretype transform θ(s) = max_{q}[G(q) − sq]. Here we have introduced Z_{s}(t), which is the generating function of the current momenta. The LDF θ(s) corresponds to the eigenvalue of the superoperator with the largest real part. The existence of more than one steady states with different currents leads to a nonanalytic behavior of the LDF θ(s) at s = 0^{24,25,35}.
To calculate the statistical properties of the current we need to diagonalize the superoperator . However, this is a numerically intractable problem as the dimension of our system is infinite due to the infinite number of photon modes. To overcome this challenge, we adopt a low photon number approximation, where the number of photons in the system is small, which can be realized if the temperatures of both baths are very small and the frequency of the laser Ω_{0} is also small. In this regime we assume that there can be at most one photon in the total system. Under this assumption, the Hilbert space required to describe the system is equivalent to the Hilbert space of a four qubitsystem. Note that, we make the low photon number approximation only for numerical tractability. Our scheme will operate as a current switch independent of the number of photons in the system.
Additional Information
How to cite this article: Manzano, D. and Kyoseva, E. An atomic symmetrycontrolled thermal switch. Sci. Rep. 6, 31161; doi: 10.1038/srep31161 (2016).
References
Dubi, Y. & di Ventra, M. Colloquium: Heat flow and thermoelectricity in atomic and molecular junctions. Rev. Modern Phys. 83(1), 131 (2011).
Žnidarič, M. Exact solution for a diffusive nonequilibrium steady state of an open quantum chain J. Stat. Mech. L05002 (2010).
Sun, K. W., Wang, C. & Chen, Q. H. Heat transport in an open transversefield Ising chain. EPL (Europhysics Letters) 92, 24002 (2010).
Manzano, D., Tiersch, M., Asadian, A. & Briegel, H. J. Quantum transport efficiency and Fourier’s law. Phys. Rev. E 86, 061118 (2012).
Asadian, A., Manzano, D., Tiersch, M. & Briegel, H. J. Heat transport through lattices of quantum harmonic oscillators in arbitrary dimensions. Phys. Rev. E 87, 012109 (2013).
Brandao, F., Horodecki, M., Ng, N., Oppenheim, J. & Wehner, S. The second laws of quantum thermodynamics. Proc. Natl. Acad. Sci. 112, 3275 (2015).
Linden, N., Popescu, S. & Skrzypczyk, P. How small can thermal machines be? The smallest possible refrigerator. Phys. Rev. Lett. 105, 130401 (2010).
Scully, O. Quantum photocell: Using quantum coherence to reduce radiative recombination and increase efficiency. Phys. Rev. Lett. 104, 207701 (2010).
Creatore, C., Parker, M. A., Emmott, S. & Chin, A. W. Efficient biologically inspired photocell enhanced by delocalized quantum states. Phys. Rev. Lett. 111, 253601 (2013).
Wang, C., Ren, J. & Cao, J. Optimal tunneling enhances the quantum photovoltaic effect in double quantum dots. New Journal of Physics 16, 045019 (2014).
Fuechsle, M. et al. A singleatom transistor. Nature Nanotechnology 7, 242 (2012).
Park, J. et al. Coulomb blockade and the Kondo effect in singleatom transistors. Nature 417, 722 (2002).
Schirm, C. et al. A currentdriven singleatom memory. Nature Nanotechnology 8, 645 (2013).
Cirac, J. I., Zoller, P., Kimble, H. J. & Mabuchi, H. Quantum state transfer and entanglement distribution among distant nodes in a quantum network. Phys. Rev. Lett. 78, 3221 (1997).
van Enk, S. J., Kimble, H. J., Cirac, J. I. & Zoller, P. Quantum communication with dark photons. Phys. Rev. A 59, 2659 (1999).
Kyoseva, E., Beige, A. & Kwek, L. C. Coherent cavity networks with complete connectivity. New J. Physics 14, 2 (2012).
Jaksch, D., Bruder, C., Cirac, J. I., Gardiner, C. W. & Zoller, P. Cold bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 3108 (1998).
Luo, X. W. et al. Quantum simulation of 2D topological physics in a 1D array of optical cavities. Nature Communications 6, 7704 (2015).
Manzano, D., Chuang, C. & Cao, J. Quantum transport in ddimensional lattices, Arxiv:1507.05705 (2015).
Caruso, F., Spagnolo, N., Vitelli, C., Sciarrino, F. & Plenio, M. B. Simulation of noiseassisted transport via optical cavity networks. Phys. Rev. A 83, 01381 (2011).
Buča, B. & Prosen, T. A note on symmetry reductions of the Lindblad equation: Transport in constrained open spin chains. New J. Physics 14, 073007 (2012).
Popkov, V. & Livi, R. Manipulating energy and spin currents in nonequilibrium systems of interacting qubits. New J. Physics 15, 023030 (2013).
Baumgartner, B. & Narnhofer, H. Analysis of quantum semigroups with GKS–Lindblad generators: II. General J. Phys. A: Math. Theor. 41, 395303 (2008).
Manzano, D. & Hurtado, P. I. Symmetry and the thermodynamics of currents in open quantum systems. Phys. Rev. B 90, 125138 (2014).
Thingna, J., Manzano, D. & Cao, J. Dynamical signatures of molecular symmetries in nonequilibrium quantum transport. Sci. Rep. 6, 28027 (2016).
Chin, A. W., Datta, A., Caruso, F., Huelga, S. F. & Plenio, M. B. Noiseassisted energy transfer in quantum networks and lightharvesting complexes. New J. Phys. 12, 065002 (2010).
Wang, C., Ren, R. & Cao, J. Optimal tunneling enhances the quantum photovoltaic effect in double quantum dots. New J. Phys. 16, 045019 (2014).
Manzano, D. Quantum Transport in Networks and Photosynthetic Complexes at the Steady State. Plos One 8, e57041 (2013).
Breuer, H. P. & Petruccione, F. The theory of open quantum systems. Oxford University Press (2002).
Metz, J., Trupke, M. & Beige, A. Robust Entanglement through Macroscopic Quantum Jumps. Phys. Rev. Lett. 97, 040503 (2006).
Metz, J. & Beige, A. Macroscopic quantum jumps and entangledstate preparation. Phys. Rev. A 76, 022331 (2007).
Esposito, M. & Mukamel, S. Fluctuation theorems for quantum master equation. Phys. Rev. E 73, 046129 (2006).
Esposito, M., Harbola, U. & Mukamel, S. Nonequilibrium fluctuations, fluctuation theorems and counting statistics in quantum systems. Rev. Mod. Phys. 81, 1665 (2009).
Garrahan, J. P. & Lesanovsky, I. Thermodynamics of quantum jump trajectories. Phys. Rev. Lett. 104, 160601 (2010).
Bertini, L., De Sole, A., Gabrielli, D., JonaLasinio, G. & Landim, C. Current Fluctuations in Stochastic Lattice Gases. Phys. Rev. Lett. 94, 030601 (2005).
Acknowledgements
This work has been supported by an SUTDMIT International Design Centre (IDC) Grant, Project No. IDG 31300102. D.M. acknowledges financial support from MITSUTD program and the Junta de Andalucía and EU Project TAHUB/II148 (Program ANDALUCÍA TALENT HUB 291780). E.K. acknowledges financial support by an SUTD StartUp Research Grant, Project No. SRG EPD 2012029 and by the project COPQE by H2020 Marie ScklodowskaCurie Actions Individual Fellowships with Project No. 705256.
Author information
Authors and Affiliations
Contributions
E.K. designed the system and performed the analytical analysis. D.M. carried out the numerical simulations and data analysis. E.K. and D.M. contributed equally to the manuscript preparation.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Manzano, D., Kyoseva, E. An atomic symmetrycontrolled thermal switch. Sci Rep 6, 31161 (2016). https://doi.org/10.1038/srep31161
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/srep31161
This article is cited by

Dynamical signatures of molecular symmetries in nonequilibrium quantum transport
Scientific Reports (2016)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.