Abstract
CavityQED systems have recently reached a regime where the lightmatter interaction strength amounts to a nonnegligible fraction of the resonance frequencies of the bare subsystems. In this regime, it is known that the usual normalorder correlation functions for the cavityphoton operators fail to describe both the rate and the statistics of emitted photons. Following Glauber’s original approach, we derive a simple and general quantum theory of photodetection, valid for arbitrary lightmatter interaction strengths. Our derivation uses Fermi’s golden rule, together with an expansion of system operators in the eigenbasis of the interacting lightmatter system, to arrive at the correct photodetection probabilities. We consider both narrow and wideband photodetectors. Our description is also valid for pointlike detectors placed inside the optical cavity. As an application, we propose a gedanken experiment confirming the virtual nature of the bare excitations that enrich the ground state of the quantum Rabi model.
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Introduction
The problem of the theoretical description of the photondetection process was addressed by Glauber in ref.^{1}. In this pioneering work, he formulated the quantum theory of photodetection and optical coherence. This theory is central to all of quantum optics and has occupied a key role in understanding lightmatter interactions. In order to discuss measurements of the intensity of light, Glauber described the photon detector as a system that functions by absorbing quanta and registering each such absorption process, e.g., by the detection of an emitted photoelectron. In particular, Glauber defined an ideal photon detector as “a system of negligible size (e.g., of atomic or subatomic dimensions) which has a frequencyindependent photoabsorption probability”. Since the photoabsorption is independent of frequency, such an ideal, small detector, situated at the point r, can be regarded as probing the field at a well defined time t. Glauber showed that the rate at which the detector records photons is proportional to \(\langle i{\hat{E}}^{}({\bf{r}},t){\hat{E}}^{+}({\bf{r}},t)i\rangle \), where i〉 describes the initial state of the electromagnetic field. The operators \({\hat{E}}^{\pm }({\bf{r}},t)\) are the positive and negativefrequency components of the electromagnetic field operator \(\hat{E}({\bf{r}},t)={\hat{E}}^{+}({\bf{r}},t)+{\hat{E}}^{}({\bf{r}},t)\), i.e., the components with terms varying as e^{−iωt} for all ω > 0 (positivefrequency components) or as e^{iωt} for all ω > 0 (negativefrequency components).
In cavity quantum electrodynamics (cavity QED)^{2,3}, where atoms interact with discrete electromagnetic field modes confined in a cavity, it is often the photons leaking out from the cavity that are detected in experiments. To describe the dynamics of the atoms and the photons in the cavity, it is common to adopt a masterequation approach (see, e.g., refs^{4,5}). In this approach, the field modes outside the cavity are treated as a heat bath, whose degrees of freedom are traced out when deriving the master equation. As a consequence, the master equation cannot be directly applied to derive the output field that is to be detected.
This gap between the quantum system and the external detector is typically bridged by inputoutput theory^{6,7}, which can be used to determine the effect of the intracavity dynamics on the quantum statistics of the output field in a very clear and simple way. In particular, if we limit the discussion to a single cavity mode, with annihilation operator \(\hat{a}\), interacting with an external field and apply the rotatingwave approximation (RWA), it is possible to obtain the output field operator \({\hat{a}}_{{\rm{out}}}(t)\) as a function of the intracavity field \(\hat{a}(t)\) and the input field \({\hat{a}}_{{\rm{in}}}(t)\) operators^{6,7,8}:
where κ is an inputoutput coupling coefficient describing the cavity loss rate. Inputoutput relationships can also be obtained for more general finitesize media^{9,10,11,12}.
In recent years, cavity QED has thrived thanks to an increase in the ability to control lightmatter interaction at the quantum level. In particular, owing to the the advances in the detection, generation and emission of photons^{13,14,15,16,17}, quantum systems are increasingly addressed at the singlephoton level. As a consequence, there is a pressing need for a critical analysis of the applicability of the theory of photodetection^{18,19,20}. Moreover, photon correlations are now routinely measured in the laboratory and many experiments, ranging from studying effects of strong and ultrastrong lightmatter coupling to performing quantum state tomography or monitoring singlephoton emitters (see, e.g.^{17,21,22,23,24,25,26,27,28,29,30,31}), have shown their power in characterizing quantum systems. In addition, photodetection is also used for quantumstate engineering^{32} and quantum information protocols^{13,33}.
For these complex systems, i.e., realistic atomcavity systems, the theory of photodetection must be applied with great care because the lightmatter interaction may modify the properties of the bare excitations in the system. If the physical excitations in such systems are superpositions of light and matter excitations, it is not immediately clear what really is measured in a photodetection experiment.
More specifically, we observe that the interaction Hamiltonian of a realistic atomcavity system contains socalled counterrotating terms, which allow simultaneous creation or annihilation of excitations in both atom and cavity modes (see, e.g., ref.^{34}). These terms can be safely neglected through the RWA for small atomcavity coupling rates g. However, when g becomes comparable to the resonance frequencies of the atoms or the cavity, the counterrotating terms manifest, giving rise to a host of interesting effects^{17,19,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52}. This ultrastrong coupling (USC) regime is difficult to reach in optical cavity QED, but was recently realized in a variety of solidstate quantum systems^{17,53,54,55,56,57,58,59,60,61,62}. The USC regime is challenging from a theoretical point of view because the total number of excitations in the cavityemitter system is not conserved (only the parity of the number of excitations is)^{38,63}.
In the USC regime, it has been shown that the quantumoptical master equation fails to provide the correct description of the system’s interaction with reservoirs^{39,64}. It was also found^{19} that a naive application of the standard descriptions of photodetection and dissipation fail for thermal emission from a cavityQED in the USC regime. In addition, quantumoptical normalorder correlation functions fail to describe photodetection experiments for such systems^{35}. To understand why an incautious application of Glauber’s idea of photodetection together with standard inputoutput theory will give incorrect results, consider a USC system in its ground state G〉. Due to the influence of the counterrotating terms in the Hamiltonian, \(\langle G{\hat{a}}^{\dagger }\hat{a}G\rangle \ne 0\), and since standard inputoutput theory predicts that \(\langle {\hat{a}}_{{\rm{out}}}^{\dagger }\,{\hat{a}}_{{\rm{out}}}\rangle \propto \langle {\hat{a}}^{\dagger }\hat{a}\rangle \), this would imply that photons could be emitted from the ground state and then detected, which is unphysical. However, with a proper generalization of inputoutput theory^{65}, Glauber’s idea of photodetection can still be applied to the output from a USC system^{19}.
In this article, we present a general and simple quantum theory of the photodetection for quantum systems with arbitrarily strong lightmatter interaction. We show how Glauber’s original results for the quantum theory of photodetection can be applied to systems in the USC regime. In contrast to previous works (e.g., ref.^{19}), our approach does not require the use of inputoutput theory and therefore applies also to more general physical situations, where it is not possible to measure and/or identify the output photons. For example, our approach can describe situations where photodetectors are placed inside electromagnetic resonators. Moreover, we note that in unconventional optical resonators^{66}, like plasmonic nanocavities^{67,68}, the detector could be placed in the nearfield of the system. In this case, the inputoutput theory of ref.^{19} cannot be applied, but the theory of the present work can still be used. Moreover, the inputoutput relations obtained in ref.^{19} relies on many assumptions about the form of the coupling between the system and the bath, and about the form of the bath. Hence, it is far from being general. By contrast, in the present work we do not make any assumptions about which system operator couples to the photon detector; nor do we limit ourselves to a particular form of the bath/detector.
In order to calculate the detection probability of the photoabsorber, we use the more general Fermi’s golden rule. As a consequence, our approach can be applied to measurements of field correlations inside an optical resonator. In such a case, it is not possible to use the inputoutput approach because the interaction strongly modifies the positive and negativefrequency field components. Their explicit expressions, in fact, contain combinations of the bare creation and destruction photon operators, which cannot be treated separately as would be required in inputoutput theory. In addition, using the correct positive and negativefrequency parts of the field dressed by the interaction, we are able to calculate the photodetection probabilities for both narrow and wideband photodetectors.
We observe that a key theoretical issue for systems in the USC regime is the distinction between bare (unobservable) excitations and physical particles that can be detected^{34}. Several works^{19,40,65} have shown that the photons in the ground state are not observable, in the sense that they do not give rise to any output photons that can be observed by standard photon detection. However, other works have shown that the photons in the ground state may be indirectly detected (without being absorbed)^{44,69,70,71}. The formalism we develop here allow us to investigate the issue of these groundstate photons more deeply than before, elucidating their virtual nature. In particular, we apply our photodetection theory to investigate the nature of photons dressing the ground state of the quantum Rabi model. As is wellknown from previous work^{31,45,65,72,73,74}, the ground state (but also the excited states^{34}) of the quantum Rabi Hamiltonian contains photons that cannot be detected by photoabsorption in a setup such as the one considered in ref. ^{19}, where the detector is placed outside the cavity. However, the nature of these photons has still not been fully settled in the literature and they remain a subject of great interest (see, e.g., ref. ^{73}). Sometimes they are considered virtual excitations^{34}, sometimes as bound photons^{72}. Our calculation in Sec. IV, beyond confirming, on a general basis, that they cannot be observed in any ordinary photodetection experiment, clarifies that they have all the features of virtual particles: they come into existence only for very short times, compatible with the timeenergy uncertainty principle.
This article is organized as follows. In Sec. II, we derive the photodetection probability for a photoabsorber coupled to a quantum system, which may have arbitrarily strong lightmatter interaction. We then show how to apply this formalism to two representative systems in Sec. III. In Sec. IV, we use the results from Sec. II to analyze the nature of photonic and atomic excitations dressing the ground and excited states in a USC system. We conclude in Sec. V.
Excitation Probability for a Photon Detector
We consider a generic quantum system with lightmatter coupling. This quantum system is weakly coupled to a photoabsorber, which is modelled as a quantum system with a collection of modes at zero temperature, described by the Hamiltonian (we set \(\hslash =1\) throughout this article unless otherwise specified)
The Hamiltonian \(\hat{{\mathscr{V}}}\) describing the coupling between the lightmatter system and the photoabsorber is
where \(\hat{O}\) is some operator of the lightmatter system, \({\hat{c}}_{n}\) (\({\hat{c}}_{n}^{\dagger }\)) is an annihilation (creation) operator for mode n of the photoabsorber, and g_{n} is the strength with which this mode couples to the lightmatter system. Typically, the operator \(\hat{O}\) would be the operator of the intracavity electromagnetic field in a cavityQED setup. However, this is not the only possibility. We could also have a situation where \(\hat{O}\) is an operator belonging to the matter part of the system. As for the operators \({\hat{c}}_{n}\) and \({\hat{c}}_{n}^{\dagger }\), their form will depend on the model of the photoabsorber. If the photoabsorber is a collection of harmonic oscillators, \({\hat{c}}_{n}\) and \({\hat{c}}_{n}^{\dagger }\) are bosonic operators. If the photoabsorber is a generic multilevel quantum system, \({\hat{c}}_{n}=n\rangle \langle 0\), where 0〉 is the ground state and n〉 the nth excited state.
The aim of this section is to calculate the excitation probability of the photoabsorber, which initially is in its ground state 0〉. The matrix element governing this excitation process is \(\langle {F}_{\alpha }\hat{{\mathscr{V}}}I\rangle \), where I〉 = E_{i}, 0〉 and F_{α}〉 = E_{k}, n〉 are the initial and final states, respectively, of the total system (generic lightmatter system plus photoabsorber). Here, we denote the eigenstates of the Hamiltonian \({\hat{H}}_{{\rm{s}}}\) of the lightmatter system by E_{k}〉 (\(k=0,1,2,\ldots \)), and the corresponding eigenvalues by E_{k}, choosing the labelling of the states such that E_{k} > E_{j} for k > j. In order to calculate the excitation probability of the photoabsorber in the longtime limit, following standard photodetection theory, we apply Fermi’s golden rule. Summing over the possible final states, the resulting excitation probability per unit of time for the photoabsorber can be expressed as
If the photoabsorber has a continuous spectrum, the sum over absorber modes can be replaced by an integral over the corresponding frequencies (\({\omega }_{n}\to \omega \)):
where ρ(ω) is the density of states of the absorber. Note that we also limited the summation to k < i, which follows from the deltafunction terms since the continuous spectrum of the absorber only contains positive frequencies ω. Using some further manipulation based on this fact, Eq. (5) can be expressed as
where ω_{i,k} = E_{i} − E_{k}, and we defined χ(ω) = 2πg^{2}(ω)ρ(ω). To further simplify Eq. (6), we define the positivefrequency operator
where
Since in Eq. (6) k < i, we obtain
Inserting Eq. (9) into Eq. (6), we obtain
where we used the identity relation \({\sum }_{k}\,{E}_{k}\rangle \langle {E}_{k}=\hat{1}\).
The detector excitation rate is thus proportional to the initialstate expectation value of the Hermitian operator \({\hat{x}}^{}{\hat{x}}^{+}\). We can extend this result to a more general situation where the initial state is mixed, described by the density matrix \(\hat{\rho }={\sum }_{j}\,{P}_{j}j\rangle \langle j\), where P_{j} is the probability that the initial state is j〉. In this case, the excitation probability rate becomes
where
If the frequency dependence of χ(ω) can be neglected, we can set \(\chi (\omega )\equiv \chi \) and write the excitation probability rate as
where \({\hat{O}}^{+}={\sum }_{i}\,{\sum }_{j < i}\,\,{O}_{ji}{E}_{j}\rangle \langle {E}_{i}\).
We now consider the case of a narrowband photodetector, which only absorbs excitations in a narrow band around a frequency ω_{d}. Setting ω_{n} = ω_{d} and \({g}_{n}=g({\omega }_{d})\equiv g\) in Eq. (4), we obtain
where
Observing that \({O}_{ki}^{}={O}_{ik}^{+\dagger }\) and \({O}_{ki}^{+\dagger }(t+\tau )={O}_{ki}^{+\dagger }(t){e}^{i({E}_{i}{E}_{k})\tau }\), Eq. (14) becomes
Performing the summation over the possible final states k, we finally obtain
where \({\langle {\hat{O}}^{}(t){\hat{O}}^{+}(t+\tau )\rangle }_{i}\) is the twotime expectation value with respect to the initial state.
Applications
We now show how this formalism for photon detection can be applied to two typical quantum systems with lightmatter interaction: cavity QED with natural atoms and superconducting circuits with artifical atoms and microwave photons. Once the interaction Hamiltonian with the correct system and photoabsorber operators has been identified, applying the results from Sec. II is straightforward.
An atom as a detector for the electromagnetic field in a cavity
We first consider the electromagnetic field in a cavity, interacting with arbitrary strength with some quantum system, e.g., one or more natural atoms situated in the cavity. As our photoabsorber, we take an atom that is weakly coupled to the field (and not coupled at all to the quantum system that the cavity interacts with; such setups are recently being considered for probing systems in the USC regime^{30,44}). The interaction Hamiltonian describing the field and the absorber can then be written as
where \(\hat{{\bf{p}}}\) is the atomic momentum operator, \(\hat{{\bf{A}}}\) is the vector potential of the electromagnetic field, e is the charge of the electron orbiting the atom, and m is the mass of the electron. For the sake of simplicity, we are considering a oneelectron (hydrogenlike) atom. We are adopting the Coulomb gauge and we neglected the \({\hat{A}}^{2}\) term, which is a good approximation in the weakinteraction regime (strength of the detectorfield interaction much lower than the cavity frequency). Using Eq. (4), labelling the atomic eigenstates by n〉 and the energies of these states by ω_{n} (we set the energy of the ground state 0〉 to zero), we obtain an expression for the atomic excitation rate:
By using the relationship \([\hat{{\bf{r}}},{\hat{H}}_{{\rm{d}}}]=i\hat{{\bf{p}}}/m\), where \({\hat{H}}_{{\rm{d}}}\) is the Hamiltonian of the photodetector in the absence of the interaction with light, we obtain
where \({{\bf{d}}}_{n}=e\langle n\hat{{\bf{r}}}\mathrm{0}\rangle \). Introducing the matrix element from Eq. (37) into Eq. (19) leads to
which, after introducing the positivefrequency electricfield operator
using the Dirac delta function, and assuming a constant dipole moment d_{n} = d (wideband detector), can be expressed as
where the Greek letters indicate the cartesian components of the dipole moment and of the electricfield operator, and repeated indices are summed over. Note that Eq. (22) follows Glauber’s prescription^{1} for the positivefrequency operator. Specifically, switching to the Heisenberg picture, from the spectral decomposition in Eq. (22), it is clear that the operator \({\hat{{\bf{E}}}}^{+}\) includes only positivefrequency terms oscillating as exp[−i(E_{m} − E_{j})t], with E_{m} > E_{j}.
We note again that if the strength of the coupling between the cavity field and the quantum system it interacts with (not the photoabsorber) is arbitrarily large, the positive and negativefrequency electricfield operators appearing in the final expression for the photodetection probability in Eq. (23) may not correspond to the bare creation and annihilation operators a and a^{†} of that field. Instead, the photodetection probability is set by transitions between the eigenstates of the full system (cavity field plus the quantum system it interacts with).
Circuit QED
As our second example, we consider a circuitQED setup. In circuit QED, artificial atoms formed by superconducting electrical circuits incorporating Josephson junctions can be strongly coupled to LC and transmissionline resonators^{17,28,75}. These circuits can be designed to explore new regimes of quantum optics. In particular, recent circuitQED experiments^{59,61} hold the current record for strongest lightmatter interaction, having reached not only the USC regime but also the regime of deep strong coupling, where the coupling strength exceeds the resonance frequencies of both the (artificial) atom and the electromagnetic mode(s). CircuitQED systems are also being used to investigate virtual and real photons in other settings than ultrastrong lightmatter interaction^{76}, e.g., in the dynamical Casimir effect^{77,78,79,80}.
As sketched in Fig. 1, we treat our (possibly quite complex) quantum circuit as a “black box”^{81}. The quantum circuit will contain both electromagnetic modes and artificial atom(s), but for our purposes it is sufficient to know how this system as a whole couples to an absorbing photon detector. We assume that the coupling is through an inductor L_{c} that connects a node flux Φ of the circuit to a node flux Φ_{in} of the photoabsorber. Analogous results can be obtained for a quantum circuit capacitively coupled to a transmission line; see, e.g., ref.^{82}.
From standard circuit quantization methods^{83}, it follows that the interaction Hamiltonian for our setup is^{84}
where the node fluxes have been promoted to quantum operators and thus acquired hats. The operator \({\hat{{\rm{\Phi }}}}_{{\rm{in}}}\) represents the measurement system that we hook up to our quantum circuit; it can be rewritten as a weighted contribution of absorber modes:
Similarly, the flux operator \(\hat{{\rm{\Phi }}}\) can be expressed as^{84}
where \({{\rm{\Phi }}}_{{\rm{ZPF}}}^{(m)}\) is the quantum zeropoint fluctuations in flux for mode m of the quantum circuit. Using Eqs (24–26), the interaction Hamiltonian can thus be expressed as
Neglecting the quadratic terms in the last two lines of Eq. (27) if they can be considered small, or including them in the bare Hamiltonians of the quantum circuits and the photoabsorber, we obtain
where g_{n} = k_{n}/L_{c} and \(\hat{O}={\sum }_{m}\,{{\rm{\Phi }}}_{{\rm{ZPF}}}^{(m)}({\hat{a}}_{m}+{\hat{a}}_{m}^{\dagger })\).
Observing that the operatorial form of the interaction Hamiltonian in Eq. (28) is the same as that given in Eq. (3), the results from Sec. II imply that the probability to absorb a photon from the quantum circuit in the initial state E_{i}〉 is proportional to the mean value of the operator \({\hat{x}}^{}{\hat{x}}^{+}\), where, in this case,
Of course, to find the eigenstates E_{j}〉 of the quantum circuit, a more detailed description of that system is needed. In general, these eigenstates will include contributions from both artificial atoms and resonator modes in the circuit. Thus, the operators \({\hat{x}}^{+}\) and \({\hat{x}}^{}\) may not correspond to the bare creation and annihilation operators \({\hat{a}}_{m}\) and \({\hat{a}}_{m}^{\dagger }\).
At optical frequencies, the radiation produced by a source is frequently characterized by analysing the temporal correlations of emitted photons using singlephoton counters. At microwave frequencies, however, it is difficult to develop efficient singlephoton counters^{16,17}. In this spectral range, signals are generally measured by using homodyne or heterodyne linear detectors. The analysis of measurements on circuitQED systems in the USC regime requires a description of output fieldquadrature measurements beyond the standard approach^{31}. However, in the last years, methods able to measure normalorder correlation functions, like those considered in standard photodetection, have been developed^{26,85}. The results presented here also apply when these detection methods are employed. Specifically, the method developed in ref.^{26} is based on heterodyne detection able to extract the complex envelope of the positivefrequency amplified electric field (∝\({\hat{x}}^{+}\)). In ref.^{85}, each signal photon deterministically excites a qubit coupled to a resonator. The subsequent dispersive readout of the qubit produces a discrete “click”. This method can be regarded as a standard photoabsorption process, and the Fermi golden rule derived in Sec. II can be directly applied.
Analysis of the Nature of Photons Dressing the Ground State of the Quantum Rabi Hamiltonian
As another application of our results from Sec. II, we now study in more detail the ground state of an ultrastrongly coupled lightmatter system. We wish to clarify the question of the virtual nature of excitations in parts of the system that contribute to the ground state of the system as a whole. Using the formalism from Sec. II, we will perform a gedanken experiment which, in principle, lets us estimate the lifetime of such excitations.
The quantum Rabi model
We consider the quantum version of the Rabi model^{86}, which describes a twolevel atom interacting with a single electromagnetic mode. The full system Hamiltonian is
where a (a^{†}) is the annihilation (creation) operator for the electromagnetic mode, ω_{0} is the resonance frequency of said mode, ω_{a} is the transition frequency of the twolevel atom, \({\hat{\sigma }}_{z}\) and \({\hat{\sigma }}_{x}\) are Pauli matrices, and g is the strength of the lightmatter coupling.
If the coupling strength g is much smaller than the resonance frequencies ω_{0} and ω_{a}, the RWA can be applied to reduce \({\hat{H}}_{{\rm{R}}}\) to the Jaynes–Cummings (JC) Hamiltonian^{87}
where \({\hat{\sigma }}_{}\) (\({\hat{\sigma }}_{+}\)) is the lowering (raising) operator of the atom. The JC Hamiltonian is easy to diagonalize and has the ground state g, 0〉, where g〉 is the ground state of the atom and the second number in the ket indicates the number of photons in the electromagnetic mode.
However, if the coupling strength increases, the full quantum Rabi Hamiltonian must be used. This Hamiltonian can also be solved^{63}; the eigenstates can be written in the form
where e〉 denotes the excited state of the atom. In particular, the ground state of \({\hat{H}}_{{\rm{R}}}\) is
with nonzero coefficients \({c}_{g,k}^{0}\) and \({d}_{e,k}^{0}\) for states that contain an even number of bare atomic and photonic excitations. Thus, if we calculate the expectation value of the bare photon number, the result is
As mentioned in the introduction, several theoretical studies^{19,30,31,34,40,65} have shown that these photons that are present in the ground state cannot be observed outside the system, since they do not correspond to output photons that can be detected. The diagrammatic approach to the quantum Rabi model in ref. ^{34} also suggests that these photons should be thought of as virtual. However, there are theoretical proposals^{44,69,70} for indirect, nondemolition detection of the photons in the ground state. The question may thus arise whether these photons should be termed virtual or real.
We now compare the results obtained by using the standard photodetection theory with those obtained using our approach. We consider a system described by the quantum Rabi Hamiltonian in Eq. (34) and study the vacuum Rabi oscillations. Specifically, we begin with the system in its ground state. Then, we assume that the system is excited by a resonant optical pulse driving the resonator, described by the timedependent Hamiltonian
where \({\mathscr{G}}\) describes a normalized Gaussian pulse arriving at time t = t_{0} with variance σ^{2}, and \({\mathscr{A}}\) is the effective amplitude. We consider a central frequency ω_{d} = (E_{1} + E_{2})/2 − E_{0}, and analyse the resonant case: ω_{a} = ω_{0}. Figure 2 displays the time evolution of the dressed \(\langle {\hat{X}}^{}{\hat{X}}^{+}\rangle \), and bare \(\langle {\hat{a}}^{\dagger }\hat{a}\rangle \) photonic populations, using three different values of the normalized coupling strength \(\eta \equiv g/{\omega }_{0}\). For reference, the upper panel also shows the Gaussian pulse \({\mathscr{G}}(t)\) (note that its height has been arbitrary scaled).
When the normalized coupling strength is much lower than 1 (η = 0.05), the difference between the two quantities is very small. We can notice in the time evolution of \(\langle {\hat{a}}^{\dagger }\hat{a}\rangle \) some small fast oscillations superimposed on the main signal, which is absent in \(\langle {\hat{X}}^{}{\hat{X}}^{+}\rangle \). When increasing the coupling (η = 0.1), the differences become more pronounced. It is also possible to observe that \(\langle {\hat{a}}^{\dagger }\hat{a}\rangle \) is different from zero even before the pulse arrival. Still increasing the coupling (η = 0.3), the differences become drastic.
Attempting to detect groundstate photons through absorption
Our approach to photon detection allows us to elucidate the nature of the groundstate photons in the quantum Rabi model in two ways. First, we consider whether the photons can be detected with a photoabsorber. From the treatment in Sec. II, we know that a photoabsorber coupled to the electromagnetic mode will have an excitation probability proportional to \(\langle {E}_{0}{\hat{x}}^{}{\hat{x}}^{+}{E}_{0}\rangle \) when the system described by the quantum Rabi model is in its ground state. Since
because there are no terms with k < 0, we conclude that a photoabsorber is not able to detect any photons in E_{0}〉. Note that this is a more general result than what has been obtained with inputoutput theory. It does not only hold for photon detectors placed outside the resonator hosting the electromagnetic mode; it also holds for photon detectors placed inside the interacting lightmatter system. This apparently trivial result confirms that \({\hat{x}}^{+}\) is a good annihilationlike operator for photons, in contrast to the bare photon annihilation operator \(\hat{a}\), which yelds \(\hat{a}{E}_{0}\rangle \ne 0\) (when counterrotating terms in the system Hamiltonain are included), as shown in Eq. (34).
However, it is important to point out that the presence of virtual photons in the ground state can still be indirectly probed by observing their effects on the renormalization of the energy levels^{34}, or detecting the energy shifts induced on a probe qubit^{44}. The latter is an example of quantum nondemolition measurement. It has also been shown that, if the paritysymmetry of an artificial atom coupled to a resonator (in the USC regime) is broken, the virtual photons in the ground state can induce parity symmetry breaking on a probe qubit weakly coupled to the resonator^{69}. In each of these cases, however, no virtual photon is directly detected through photonabsorption. Finally, we observe that, in the presence of timedependent Hamiltonians (e.g., an abrupt switchoff of the interaction)^{34,35,88}, or spontaneous decay effects^{40}, virtual photons can be converted (not directly detected) into real ones. These considerations apply to artificial atoms coupled to resonators, and, in general, to any (natural or artificial) twolevel system coupled to any bosonic mode.
Probability of photoabsorption for short times
The above would seem to further strengthen the case for calling the groundstate photons virtual, but another objection to that would be that virtual particles only exist for very short times, while the excitations considered here are always present in E_{0}〉. As a further application of our approach to photon detection, we therefore calculate the lifetime of the excitations present in the ground state. We begin by noting that the photondetection theory used here is based on Fermi’s golden rule, and as such it gives the probability of photoabsorption for long times. We now extend this theory to short times.
Applying standard firstorder perturbation theory and using Eq. (3), we can calculate the probability that a photon disappears from the state E_{0}〉 and one of the absorber modes is excited. Assuming \(\hat{O}=\hat{a}+{\hat{a}}^{\dagger }\), the matrix element describing this process is
where the second label in the kets indicates the absorber state. Due to the presence in E_{0}〉 of states with a nonzero number of photons, this transition matrix element is nonzero. It is interesting to observe that this matrix element would be zero for the JC model, i.e., replacing E_{0}〉 with g, 0〉. The resulting transition probability in the case of the quantum Rabi Hamiltonian is^{89}
where \({\omega }_{k,0}^{(n)}={\omega }_{n}+({E}_{k}{E}_{0})\), and
If t is sufficiently large, the function F(t, ω) can be approximated to within a constant factor by the Dirac delta function δ(ω). In that case, we obtain that the transition rate for times \(t\gg {({E}_{1}{E}_{0})}^{1}\)
Since \({\omega }_{k,0}^{(n)} > ({E}_{1}{E}_{0})\) is strictly larger than zero and is of the order of ω_{0}, no transitions will be observed for large times. However, for \(t\ll 1/{\omega }_{k,0}^{(n)}\),
and thus the photons in E_{0}〉 can induce transitions with a very small probability (due to the t^{2} term) during a small time interval. This means that the groundstate photons are coming into existence for very short times, on the order of a period of the electromagnetic mode, in agreement with the timeenergy uncertainty principle. This is consistent with the interpretation of the groundstate photons as virtual rather than real. This result represents a direct manifestation of the general energytime uncertainty principle.
It is also interesting to point out some similarity with processes in quantum field theory (QFT). One analogy consists in the nonconservation of the number of excitations. For example, in QFT, it is possible to create additional particles from the collision of two sufficiently energetic particles. Analogously, in the lightmatter USC regime, if energy is conserved, one excitation can be converted into two or more excitations^{43,46,51,54}. Another interesting analogy concerns virtual particles. Differently from the USC regime, in QFT the vacuum state does not explicitly contain virtual excitations. However, the interaction in the Lagrangian still allows for the creation of virtual particles from the vacuum. Specifically, the QED interaction (see, e.g.^{90})
(ψ is the electron field operator, γ_{μ} are the Dirac matrices, \(\bar{\psi }={\psi }^{\dagger }{\gamma }_{0}\), and A_{μ} is the vector potential) contains energy nonconserving terms describing the simultaneous excitation of one photon and one electronpositron pair (note that ψ contains both destruction operators for electrons and creation operators for positrons). Hence the matrix element of V between the vacuum state and a state with one photon and one electronpositron pair is nonzero just like the matrix element in Eq. (37). However, since the initial state (the vacuum) and the final state (one photon and one electronpositron pair) have very different energies (ω_{F} and ω_{I} respectively), owing to the energytime uncertainty principle, these threeparticle states can only exist for a very short time Δt~1/(ω_{F} − ω_{I}). This can be verified directly by using firstorder perturbation theory as in Eq. (38). More generally, in QFT virtual particles can appear in any process described by a Feynman diagram where at some intermediate point energy conservation is violated.
Conclusions
We have explored photon detection for quantum systems with arbitrarily strong lightmatter interaction. In these systems, the very strong interaction makes light and matter hybridize such that a naive application of standard photodetection theory can lead to unphysical results, e.g., photons being emitted from the ground state of a system. While some previous works have shown how to amend inputoutput theory to arrive at correct expressions for the photon output flux, we have presented a more complete theory for photon detection in these systems without relying on inputoutput theory. We followed Glauber’s original approach for describing photon detection and found, using Fermi’s golden rule, the correct excitation probability rate for a photoabsorber interacting with the lightmatter quantum system. Calculating this rate requires knowledge of the system eigenstates, such that the system operator coupling to the photoabsorber can be divided into negative and positivefrequency components. The difference with standard photon detection arises because the strong lightmatter interaction dresses the system states such that the aforementioned components no longer correspond to bare annihilation and creation operators.
We presented results for both wideband and narrowband photon detectors. We then showed in detail how the formalism can be applied to two representative quantum systems that can display strong lightmatter interaction: cavity QED with an atom acting as the photoabsorber, and a circuitQED setup with inductive coupling to a photon detector. Although the results we derived here were limited to secondorder correlation functions, they can be directly generalized to higherorder normalorder correlation functions.
We also applied our photondetection formalism to the quantum Rabi Hamiltonian, which describes a twolevel atom interacting with a single electromagnetic mode. For large lightmatter interaction, the ground state of this model contains photons, and whether these photons are virtual or real has been subject to debate. Using our formalism, we were able to clarify the nature of the groundstate photons in two ways. First, we showed that the groundstate photons, in the limit of long times, will not be detected by a photoabsorber, no matter where this photoabsorber is placed. Unlike previous results obtained with modified inputoutput theory, our result also holds for a detector placed inside the system (e.g., inside an optical cavity). Second, we considered a gedanken experiment, where the excitation probability of the photoabsorber is calculated for short times. We found that there is a small excitation probability for such short times, which is consistent with an interpretation where virtual photons are flitting in and out of existence on a timescale set by the timeenergy uncertainty relation. Our results thus provide further evidence for the virtual nature of the photons present in the ground state of the quantum Rabi Hamiltonian.
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Acknowledgements
F.N. is supported in part by the: MURI Center for Dynamic MagnetoOptics via the Air Force Office of Scientific Research (AFOSR) (FA95501410040), Army Research Office (ARO) (Grant No. W911NF1810358), Asian Office of Aerospace Research and Development (AOARD) (Grant No. FA23861814045), Japan Science and Technology Agency (JST) (QLEAP program, ImPACT program, and CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (JSPSRFBR Grant No. 175250023, and JSPSFWO Grant No. VS.059.18N), RIKENAIST Challenge Research Fund, and the John Templeton Foundation.
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O.D.S. and S.S. developed the idea, O.D.S. performed the analytical calculations with the help of A.F.K. and A.R., S.S. and F.N. supervised the project. O.D.S. and A.F.K. wrote the paper with input from all authors.
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Di Stefano, O., Kockum, A.F., Ridolfo, A. et al. Photodetection probability in quantum systems with arbitrarily strong lightmatter interaction. Sci Rep 8, 17825 (2018). https://doi.org/10.1038/s41598018360561
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DOI: https://doi.org/10.1038/s41598018360561
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