# Photon statistics of a double quantum dot micromaser: Quantum treatment

###### Abstract

A semiconductor single-atom micromaser consists of a microwave cavity coupled to a gain medium, a double quantum dot driven out of equilibrium by a bias voltage. The masing threshold of this system was recently probed by measuring photon statistics in the cavity [Y-Y. Liu et al, Phys. Rev. Lett. 119, 097702 (2017)]. In this paper, we develop an in-depth, rigorous understanding of this experiment and related works. First, we use a semiclassical theory and study transmission spectroscopy. This approach allows us to derive the masing threshold condition for arbitrary temperature and voltage bias, and expose microscopic principles required for realizing photon gain and thereby a photon amplifier. Next, by employing the quantum master equation approach we extend the Scully-Lamb quantum theory of a laser to the present setup, and investigate the statistics of emitted photons below and above the masing threshold as a function of experimentally tunable parameters. Although our focus is primarily on hybrid quantum dot circuit - quantum electrodynamics systems, our approach is adaptable to other light-matter systems where the gain medium consists of a mesoscopic structure.

## I Introduction

Photon statistics and other characteristics of light provide critical information for understanding fundamental concepts in light-matter interaction singleatom1 (); RevNori (); PNAS1 (); kontoskondo (); kontosnatcom (); kontosprx (); marco14 () and for developing novel devices RevNori (); kulqda (); kulqdb (); KLH2015 (); pettamaserscience (); pettamaserprl (). Particular systems that recently attracted significant experimental and theoretical interests are semiconducting quantum dots integrated with a superconducting qubit architecture Petersson2012 (); Viennot2013 (); petta2014 (); Kulkarni2014 (); Frey2012 (); enslinprl (); Toida2013 (); Deng2013 (); spin-kontos (); gpg (); tk2011 (); Mora1 (); Mora2 (). These so called quantum dot circuit quantum electrodynamics (QD-cQED) setups offer several advantages over standard light-matter (radiation field-atom) systems given their (i) tunability RevNori (), (ii) scalability QCL (); scale1 (); scale2 () and (iii) versatility siv1 (); pd1 (); pd2 (); phaselock1 (). QD-cQED devices combine mesoscopic systems (quantum dots) with quantum optics components. They offer a rich platform for the study of light-matter phenomena, as one can investigate both electronic and photonic properties therein.

Recent remarkable experiments realized parallels to single atom and double-atom masers in QD-cQED systems singleatom1 (); pettamaserscience (); pettamaserprl (). It is to be noted that single atom masers have been previously realized with Rydberg atoms Ryd (), optical cavities coupled to either natural or artificial atoms art1 (); art2 (); art3 (); art4 (), and superconducting junctions super1 (); super2 ().

Considering QD-cQED systems in a nonequilibrium steady state (NESS), several complementary quantities can be experimentally observed and theoretically computed. Recently, in addition to electronic properties, the photonic sector has been thoroughly probed singleatom1 (); pettamaserscience () with measurements reporting on photon transmission, phase response, photon number, and the statistics of emitted photons. The behavior of the electronic degrees of freedom in QD-cQED systems is examined through the NESS charge current, associated current fluctuations, and the quantum dots occupation number petta2014 (); Kulkarni2014 (). Altogether, concurrent studies of the photonic and electronic sectors expose effects related to light-matter interaction.

The realization of masers in QD-cQED systems calls for a rigorous theoretical description. Particularly, the observation of masing, and measurements of photon statistics in Double-Quantum-Dot (DQD) masers are missing a careful, fundamental quantum analysis. In our previous work longNEGF (), we investigated the photonic and electronic properties of a DQD setup employing the non-equilibrium Green’s function (NEGF) approach. However, given the perturbative nature of the analysis in the light-matter interaction energy, the work was limited to the below-masing threshold regime longNEGF (); QCL ().

In this paper, we employ the Lindblad quantum master equation (QME) approach and perform a careful and comprehensive study of light amplification and masing in a cavity coupled DQD setup, see Fig. 1. The QME method allows us to investigate, in a unified manner the statistics of photons in the cavity as we transit from below to above the masing threshold, unlike the perturbative NEGF approach longNEGF (). Traditionally, the Lindblad QME approach has been applied to study systems under an infinite voltage bias, therefore supporting a unidirectional source-drain current Kulkarni2014 (). In contrast, in this paper we extend our investigation to the case of a finite voltage bias, which allows us to observe the passage of the electronic bath from a dissipative element to a driving, gain medium. Given that the source-drain bias can be experimentally tuned, our results, capturing voltage-bias dependent physics, shall be useful for understanding and further developing DQD based masers. The Lindblad QME further allows us to critically examine the detrimental (yet not fatal) impact of substrate phonons on photon amplification, considering it in the limit of weak electron-phonon interaction.

We perform transmission spectroscopy under the semiclassical approximation, which limits us to the below-threshold masing regime. Furthermore, we study photon statistics using a full quantum approach and observe the entire development of the photon statistics from thermal to Poissonian as we transit from below to above the masing threshold. The two calculations, transmission spectroscopy and photon statistics, agree on the threshold condition, and they provide complementary information on light amplification and masing in our setup.

The paper is organized as follows. In section II, we describe the Hamiltonian of the DQD maser, including the external reservoirs (fermionic and bosonic). In Section III, based on a semiclassical treatment, we present analytical results and simulations for photon transmission and phase response in the system. In order to rigorously examine the rich nature of photon statistics in the cavity, we resort to a fully quantum approach in Section IV, by developing the quantum theory of lasers due to Scully and Lamb slbook (); Agarwal (). We show that the statistics of photons evolves from thermal (with an effective temperature) to Poissonian when experimentally-tunable parameters are varied, such as bias voltage, level detuning, light-matter coupling, cavity decay rate. In Section V, we summarize our work and provide an outlook of future challenges. Technical details are delegated to the Appendix.

## Ii Model Hamiltonian

We consider an open light-matter quantum system with the total Hamiltonian consisting of a matter part which is driven to an NESS by an external voltage bias, a cavity (transmission line resonator) , and a light-matter interaction term ,

(1) |

The matter component consists of a double quantum dot placed between two fermionic leads maintained at different chemical potentials. Electron tunneling between the dots takes place via a direct coherent coupling. The quantum dots further interact and exchange energy with substrate phonons. The matter Hamiltonian therefore consists of the following components,

(2) |

where the bare dots Hamiltonian is

(3) |

The metal leads are included in

and the phononic interaction Hamiltonian is

(5) |

Here, is the detuning parameter and is the direct tunnelling term. and represent the and components of the Pauli matrix, expressed in terms of the localized single electron orbitals, and . We limit ourselves to the Coulomb blockade regime: This implies that at any instant, the DQD is restricted to three possible configurations, namely, the null-electron subspace, denoted by , and the single-electron subspace, with an electron localized either on the left or the right dot, denoted by and , respectively. We set the energy of the unoccupied electronic state at zero.

For the DQD-lead Hamiltonian, is the creation (annihilation) operator for fermions with wave vector in the -th lead (). is the coupling constant between the DQD and the fermionic bath. Similarly, for the DQD-phonon part, is the bosonic creation (annihilation) operator for phonons with wave vector . denotes the interaction energy between the DQD and the phonons.

The photonic component consists of a cavity photon mode of frequency . This so-called primary mode (creation operator ) is coupled to two additional secondary photon baths , which mimic the two ends of a long microwave transmission line (creation operator ). The cavity Hamiltonian is given by

(6) | |||||

Finally, light (cavity)-matter(DQD) interaction is given by the standard dipole coupling term,

(7) |

where is the light-matter coupling constant. This model is extremely compound—and rich—offering different regimes of operation (weak/strong coupling of the DQD to the leads, phonons, cavity mode, weak/strong dissipation of the cavity mode to secondary modes, linear response/far from equilibrium operation).

In what follows, we first diagonalize the DQD Hamiltonian in Eq. (3) and work in the diagonalized basis. This new eigenbasis relates to the localized states via an unitary transformation,

(8) |

Here, represents the ground and excited states for the DQD, and . In the energy basis, the full Hamiltonian transforms to

(9) |

and

(10) | |||||

(11) | |||||

(12) |

Here, is the creation operator for the excited, (ground) state and is the DQD renormalized frequency. We also define the eigenenergies of the DQD as and . To arrive at this matter-cavity Hamiltonian, we perform the rotating wave approximation (RWA) assuming that the cavity frequency is in resonance with the DQD energy gap . However, such RWA is not justified for the DQD-phonon Hamiltonian since the frequency spectrum of the phonon bath can be off-resonant with the DQD energy gap. Note that the cavity Hamiltonian in Eq. (6) is not modified under this transformation.

In what follows, considering this DQD setup, we first discuss photon transmission spectroscopy using a semi-classical approach. This discussion is followed by a quantum treatment for photon statistics.

## Iii Semiclassical theory of Photon transmission

### iii.1 Threshold condition for masing

We first investigate the cavity response using transmission spectroscopy, calculating the transmission amplitude and the phase response for emitted microwave photons. In transmission spectroscopy measurements, the cavity is driven with a coherent microwave field; the output field is measured via heterodyne detection petta2014 (). In this section we assume that light-matter interaction is weak, and that the driving field is weak. The resulting QME then correspond to the semiclassical limit, as we explain below.

We employ the quantum master equation approach to derive an expression for the transmission. We write down an equation of motion for the reduced density matrix for the DQD + cavity mode, by tracing out all reservoirs’ degrees of freedom, that is the electronic, photonic and phononic reservoirs. We make use of the standard set of approximations, namely, the system-bath decoupled initial condition for the density matrix, the Markov approximation, and a weak-coupling treatment between the system and the different reservoirs. Similar approximations have been used in other studies of QD-cQED systems when exploring full-counting statistics, transport and photonic properties Vavilov-fcs (); Schon_noise (); ss1 (); ss2 (); Nori-bistable (); siv2 (); Schon (); Child (); Jin (); Muller (). We then arrive at the following equation,

(13) |

The first term represents the quantum coherent time evolution governed by the Hamiltonian . The last term in is a coherent driving term, which represents the transmission measurement petta2014 (); Kulkarni2014 (). The last three terms in the QME correspond to different Liouvillians capturing the effects of the electronic, photonic, and phononic reservoirs, respectively. The electronic Liouvillian collects the effect of the two metals on the system,

(14) |

where

(15) |

Here,

(16) |

is the spectral function (or hybridization) of the electronic lead with the state . For simplicity, it is chosen to be flat (wide-band limit). is the Fermi distribution for the lead with chemical potential and inverse temperature . The dissipator for an operator is defined as

(17) |

We can similarly write down expressions for and . Assuming the cavity is in contact with a very low temperature transmission line, the photonic Liouvillian is

(18) |

where is the cavity decay rate per port (), and is the total decay rate. Here is the density of states of the -th photonic bath and is the average coupling between the cavity and photon bath modes. For simplicity, we assume a symmetric decay rate i.e., . Lastly, the Liouvillian due to the phonon environment is given as

(19) | |||||

Here, , and are the phonon pumping, relaxation and pure dephasing rate constants, , and . Recall that the DQD frequency is defined as . The phononic rate constants are given in terms of , the Bose-distribution function with inverse temperature , and , which is the phonon spectral function. It is chosen to be of the following form petta2014 (); Kulkarni2014 (); petta-phonon (); weber-phonon (),

(20) |

where and are the scaling parameters, is the coupling strength. The typical parameter values are given in our Table. 1.

We now write down equations of motion for the population and coherences of the DQD states and the photon mode under the semiclassical approximation, given by, . We receive,

(21) | |||||

(22) | |||||

(23) | |||||

(24) |

Here we use the following compact notation, , ; the superscripts follow from the sin (s) and cos (c) functions in Eq. (16). The bar symbol replaces the Fermi function by . For example, . is the effective damping constant and is a function of the different tuning parameters of the electronic medium (DQD+leads) such as the chemical potentials, temperature of the electronic leads, coupling energy between the DQD and the electronic and phononic environments.

We next solve the above set of equations in steady state by moving to a rotating frame with respect to the driving frequency and assuming and . We get

(25) |

Substituting this expression into the equation of motion in Eq. (24) and solving it in steady state we obtain the transmission function,

(26) |

where we identify as the charge susceptibility, given as

(27) |

The population of the DQD states, , are evaluated in the nonequilibrium steady state and in absence of light-matter interaction. The susceptibility, which contain information about population imbalance, has a non-trivial dependence on experimentally tunable parameters. Note that arriving at Eq. (26) following Eq. (24) is valid only in the limit

(28) | |||||

We identify the right hand side of this inequality as the masing threshold. When using the NEGF formalism, a masing threshold limit for a similar model had been derived in our previous work longNEGF (), based on the causality condition for the Green’s function.

We rewrite the transmission as ; gain in the cavity photon is and the phase response is included in . It is immediately clear from Eq. (26) that to achieve photon gain one has to counteract two different sources of dissipation: the cavity decay to the ports (rate constant ) and the imaginary component of the electronic medium induced charge susceptibility. At equilibrium, , , which immediately implies that photon gain is impossible to achieve in this limit: At equilibrium, in addition to the photon bath, the electronic degrees of freedom further act as a dissipative channel for cavity photons.

It is only when the DQD is driven far from equilibrium, population inversion happens, changes sign, and photon gain is achieved. Therefore, by driving the DQD out of equilibrium one can realize a photon amplifier. More interestingly, one can further derive a sum rule for the transmission function, following the definition in Eq. (26), given as

(29) |

valid in the regime , as mentioned before. The general analytical expression for the transmission function, the threshold condition and the sum rule are the first set of central results of this paper.

Table 1: Typical parameter values from experiments (Refs. petta2014, ; Kulkarni2014, ) Cavity loss rate eV MHz Light-matter coupling eV MHz Cavity frequency eV GHz Elastic tunneling eV GHz detuning eV GHz Drain tunneling rate eV GHz Source tunneling rate eV GHz (unknown) eV GHz Scaling frequency eV GHz Phonon bath cutoff 35 eV GHz Temperature mK 0.16 GHz

### iii.2 Numerical results for photon transmission

We present numerical results for photon transmission as a function of experimentally tunable variables: voltage bias, DQD parameters, driving frequency; values for relevant parameters are given in Table 1. Unless otherwise stated, we set the Fermi energy of the electronic leads at zero, and symmetrically adjust the voltage around it, . We set the temperature of all the baths to be the same. For the spectral function of the electronic leads we use the wideband approximation and choose symmetric couplings, .

In Fig. 2(a) we plot the absolute value of the photon transmission as a function of the incoming coherent microwave frequency . The corresponding phase is displayed in Fig. 2(b). We observed the following:

First, in the absence of the matter-cavity interaction, the transmission is exactly unity at the cavity frequency and it displays a broadening proportional to (dashed line). Correspondingly, the phase response is zero at the resonant frequency and asymptotically it reaches in the off-resonant regime ().

Once the cavity-matter interaction is switched on—yet keeping all the baths at equilibrium with the same temperature and chemical potentials—the maximum value of the transmission drops below unity with corresponding frequency value shifting from the bare cavity frequency (dotted line). This shift is due to charge fluctuations in the dots, and it is directly proportional to the real part of the charge susceptibility . The broadening of the transmission function is related to the difference between and ; recall that the latter is negative when it acts as a dissipative medium. Since at equilibrium , the broadening is large. In other words, the increased broadening at equilibrium compared to the isolated cavity-matter case implies that the electronic component acts as a dissipative channel for the cavity photons. It is easy to note that the phase response is zero when the transmission is maximal.

Next, the DQD is voltage biased. Once it is driven sufficiently far from equilibrium (here ), the absolute value of the transmission exceeds unity (dashed line). This enhancement is accompanied by a reduction in broadening as due to population inversion in the DQD states. This can also be understood from the sum rule formula in Eq. (29), which indicates that a reduction in broadening leads to an enhancement in the peak value of the transmission.

To better understand the photon signal, we display in Fig. 3 the transmission amplitude, phase, and the real and imaginary components of the charge susceptibility, all as a function of bias voltage at the (fixed) cavity frequency . In the close to equilibrium regime, , the absolute value of the transmission is less than unity, as expected, while it shows an enhancement for higher bias . This shows as a sudden dip in the phase. The sudden jump takes place when both the real and imaginary components of the change sign. In particular, when becomes positive, photon gain is observed.

In Fig. 4 we plot the photon signal at the cavity frequency as a function of the detuning of the DQD for a large bias voltage. Both absolute value of transmission and the phase display the peak and dip structure. In the transmission amplitude, for positive detuning (negative detuning ), electron transport through the DQD is assisted by light, reaching resonance condition via the emission (absorption) of photons and thereby reflected as a peak (dip). At large detuning, the cavity and matter units effectively decouple and the transmission value settles to unity with zero phase response.

The unfavorable role of the phonon environment is illustrated in Fig. 5. Here, we display a contour plot for the absolute value of transmission as a function of incoming frequency and detuning in the absence (a) or presence (b) of phonons. Ignoring the substrate phonons reduces the broadening and results in a further gain in the photon signal. Nevertheless, we find that the transmission can exceed unity even under the dissipative action of phonons. It should be reminded that in this work electron-phonon interaction is assumed weak; Lindblad dissipators agree with treatments based on second order system-bath perturbation theory. Specifically, the additive nature of the Lindblad dissipators in the different baths reflects the absence of bath-cooperative effects. As such, phonons are detrimental to photon gain since they assist in the dissipation of electronic excitations within the DQD. In contrast, when cooperative photon-electron-phonon processes are realized (through strong coupling interactions), phonon-assisted gain, beyond the strict resonance condition, , is achieved pettamaserprl ().

So far, we have considered the semiclassical limit. We acquired the threshold condition and analyzed photon amplification below threshold. However, a full quantum approach is required to understand the properties in the above-threshold regime, which is related to the masing phenomenon. We address this issue in the next Sec. IV.

## Iv Quantum Theory of Photon Statistics: Scully-Lamb approach

In this section, we focus on the statistics of the cavity mode and follow its behavior as we transit from below to above the masing threshold. We define the reduced density operator for the cavity photon (ph) as and investigate its population dynamics, . Following the standard QME procedure, as done before, we obtain

(30) | |||||

The first term (explicit dependence) describes coherent evolution and it consists of joint cavity and DQD density matrix elements. The latter part (explicit dependence) is due to the interaction of the cavity mode with the transmission line (photon bath) and it is responsible for the decay of cavity photons with rate . Here is the Bose-Einstein distribution function of the photon mode at the photon bath temperature and frequency .

In order to close equation (30), we need to express the joint cavity - DQD density matrix elements and in terms of the cavity mode populations . To achieve that, we write down equations of motion for these elements, then make a crucial approximation that the DQD relaxes to the nonequilibrium steady state much faster than the cavity mode, which is indeed the case within our parameters, , . The equations for these combined density matrix elements are

(31) | |||||

(32) | |||||

(33) | |||||

(34) | |||||

and

(35) |

To close the equations, we also make use of the following two equations,

(36) | |||||

We now employ the adiabatic approximation and solve for the steady state of the DQD. We first set in Eq. (35) and express or in terms of using Eqs. (LABEL:ph-norm). We then find,

(38) |

Here depend on the various parameters of the DQD, electronic and phononic baths; their expressions are given in Appendix A. We next substitute the solution for into Eqs. (31)-(34) to get closed set of equations, which can be written in the following matrix form,

(39) |

where once again , , , , are combination of parameters related to previously defined constants, , , etc. Explicit expressions are given in the Appendix. In the steady state limit we set the left side of Eq. (39) to zero and invert the matrix to obtain and , as required by Eq. (30), which are solely expressed in terms of the cavity mode population,

(40) |

The combined constants , and are given in the Appendix. These set of equations are substituted into Eq. (30) and we reach our central equation,

(41) |

The constants and , expressed in terms of , , , , and are included in the Appendix. The small-letter constants combine the electronic decay rates and the phonon-induced energy relaxation constants. Note that, and have the dimension of inverse time (rate) whereas is dimensionless. They quadratically depend on the light-matter coupling .

Equation (41), our central result, is quite compound, and we now discuss the behavior of the photon statistics in steady state in two cases. First we focus on the below-threshold regime and compare our result with the previously-obtained semiclassical prediction. Then, we go back to the general result but simplify it at low temperature, thereby deriving closed expressions for photon statistics. We further explain below through figure 6 the physical information contained in and , which are composite bias-driven light-matter rates determining the onset of masing in the system.

Below threshold behavior. For simplicity, we consider the resonance situation . When , which takes place when is smaller than bias-dependent electronic excitation and relaxation processes (see Appendix), one can ignore the denominator in the first two terms of Eq. (41) and get

(42) | |||||

This equation can be solved in steady state by invoking the detailed balance principle slbook (), namely,

We then receive the solution for the photon mode population as

(44) |

The normalization condition (or in other words, the validity of the solution) requires that

(45) | |||||

We conclude that Eq. (44) is valid as long as , which precisely matches the threshold condition obtained using the semiclassical analysis in Eq. (28). This is the so-called below-threshold regime. The steady state photon mode distribution is given by an exponentially decaying incoherent thermal distribution, , with an effective temperature defined as

(46) |

As expected, reduces to the photon-bath temperature when the light-matter interaction is switched off, with .

We can further generalize the threshold condition (45) by relaxing the resonance requirement . In the general case, the photon mode population is obtained by replacing and in Eq. (44). This generalized threshold condition again matches with the corresponding semiclassical prediction.

General Solution. We go back to Eq. (41) and solve it,

(47) |

This solution describes both below- and above-threshold regimes. Assuming a resonance setup, , and that the temperature is low such that photon bath-induced excitation of the cavity are missing, , one simply obtains

(48) |

The maximum probability appears at the number

(49) |

The condition corresponds to the above threshold regime; Again we find that is the threshold condition.

The population distribution in this lasing regime is sharply peaked about , which is equivalent to the average photon number