Output Field-Quadrature Measurements and Squeezing in Ultrastrong Cavity-QED

We study the squeezing of output quadratures of an electro-magnetic field escaping from a resonator coupled to a general quantum system with arbitrary interaction strengths. The generalized theoretical analysis of output squeezing proposed here is valid for all the interaction regimes of cavity-quantum electrodynamics: from the weak to the strong, ultrastrong, and deep coupling regimes. For coupling rates comparable or larger then the cavity resonance frequency, the standard input-output theory for optical cavities fails to calculate the correct output field-quadratures and predicts a non-negligible amount of output squeezing, even if the system is in its ground state. Here we show that, for arbitrary interactions and cavity-embedded quantum systems, no squeezing can be found in the output-field quadratures if the system is in its ground state. We also apply the proposed theoretical approach to study the output squeezing produced by: (i) an artificial two-level atom embedded in a coherently-excited cavity; and (ii) a cascade-type three-level system interacting with a cavity field mode. In the latter case the output squeezing arises from the virtual photons of the atom-cavity dressed states. This work extends the possibility of predicting and analyzing continuous-variable optical quantum-state tomography when optical resonators interact very strongly with other quantum systems.

It has been shown that, in this USC regime, the correct description of the output photon flux, as well as of higher-order Glauber's normal-order correlation functions, requires a proper generalization of the inputoutput theory for resonators [13,19,20]. Application of the standard input-output picture to the USC regime would predict an unphysical continuous stream of output photons for a system in its ground state ñ |G . This result stems from the finite number of photons which are present in the ground state due to the counter-rotating terms in the interaction Hamiltonian [21]. Specifically, it has been shown [13,22] that the photon rate emitted by a resonator and detectable by a photo-absorber is no longer proportional to á ñ ( )ˆ( ) † a t a t (as predicted by the standard input-output theory), whereâ andˆ † a are the photon destruction and creation operators of the cavity mode, but to á ñ ( ) (ˆ( )) † x t x t , which can be different from the bare photon creation an destruction operators. When the coupling rate is not a negligible fraction of the bare resonance frequencies, the correct separation into positive and negative frequency operators can be performed only by including the influence of the interaction Hamiltonian. This separation can be easily performed after the diagonalization of the total system Hamiltonian.
Direct photon counting experiments provide information about the mean photon number and higherorder normal-order correlations. However a complete quantum tomography of the electro-magnetic field (see, e.g., [23]) requires phase-sensitive measurements which are based on homodyne or heterodyne detection [24,25]. These techniques enable the measurements of the mean field quadratures and their variance, e.g., á ñ x and á ñ -á ñx x 2 2 . More generally, for an electro-magnetic field-mode, it is possible to define two complementary field-quadraturesQ 1 andQ 2 with = [ˆˆ] Q Q , 1 1 2 , as = + (we use  = 1). These zero-point fluctuations represent the standard quantum limit to the reduction of noise in a signal. Other minimum-uncertainty states are possible, and these occur when fluctuations in one quadrature are squeezed at the expense of increased fluctuations in the other one [26]. Light squeezing can be realized in various nonlinear optical processes, such as parametric downconversion, parametric amplification, and degenerate four-wave mixing [27][28][29][30][31] or in presence of timedependent boundary conditions [32][33][34][35]. Squeezed states of light belong to the class of nonclassical states of light. Having a less noisy quadrature, squeezed light has applications in optical communication [36] and measurements [36][37][38][39][40] and is a primary resource in continuous variable quantum information processing [38]. Squeezing of the electromagnetic field has been achieved in a variety of systems operating in the optical and microwave regimes. A noise reduction of −10 dB (−13 dB is the estimation of squeezing after correction for detector inefficiency) has been achieved in the experiment [41]. More recently, a few experiments with superconducting circuits [34,42] have demonstrated the possibility of obtaining much stronger squeezing in microwave fields [43].
Here we present a theory of quadrature measurements of the output field escaping from a resonator coupled to a generic matter system with arbitrary interaction strength, and we apply it to the analysis of squeezing. In cavity-QED systems, the squeezing effect has been usually studied by using the rotating-wave approximation [44][45][46][47][48][49]. While in the USC regime the positive frequency component + x is different fromâ (it may contain contributions from the creation operator of the cavity field), the quadrature operator = + = + + -ˆˆˆ † x a a x x is independent of the light-matter interaction strength. Hence, at a first sight, one may expect that, in contrast to Glauber's correlation functions, quadrature measurements can be analyzed by applying the standard inputoutput theory [50,51]. Here we show that this is not the case: application of the standard input-output picture to the analysis of quadrature measurements in the USC regime leads to incorrect results. We also observe that the calculation of quadrature expectation values by means of the generalized input-output relations (working for arbitrary light-matter coupling) presents some additional complications compared to that of normal-order correlations. In particular, the resulting quadrature expectation values contain products of system and input operators and thus cannot be directly derived within the master equation approach. The present analysis is of particular interest for the description of measurements in circuit-QED systems, where output quadrature measurements are generally employed since efficient microwave photon-counting detectors are not currently available. However, well-developed linear amplifiers allow for the efficient measurement of the field-quadrature amplitudes [25,42,52,53]. Using input-output theory [51], one can show that the full information about the intracavity field-mode is contained in the moments and cross-correlations of the time-dependent output quadrature amplitudes. It has been demonstrated experimentally that correlation-function measurements based on quadrature amplitude detection are a powerful tool to characterize quantum properties of propagating microwave-frequency radiation fields [52]. Hence a general method to calculate these time-dependent moments, when the resonator interacts with one or more artificial atoms in the USC regime, is highly desirable for the analysis of the output microwave field in circuit-QED systems.
We apply the theoretical framework developed here to analyze three different cases: (i) we analyze the output field-quadratures for a system in its ground state. It is know that the ground state of a system in the USC regime is a squeezed vacuum state [19], where the amount of squeezing depends on the coupling strength and on the detuning between the cavity mode and the matter-system resonances. A correlation-function analysis of the quadratures of microwave fields has been exploited for measurements of vacuum fluctuations and weak thermal fields [54]. Hence the question arises if it is possible to detect such vacuum squeezing. Here, under quite general hypotheses, we demonstrate that for arbitrary cavity-embedded quantum systems, independently on the coupling rate, no squeezing can be found in the output field quadratures if the system is in its ground state. (ii) We study a coherently excited cavity interacting with an artificial two-level atom. Recently, it has been shown that superconducting artificial atoms, subject to parity-symmetry-breaking and ultrastrong coupled to superconducting resonators, can display two-photon vacuum Rabi oscillations [22]. However, two-photon correlations cannot directly be detected in these systems. We show that quadrature-noise measurements can provide an alternative direct probe. This process can give rise to a high degree of squeezing in presence of a single two-level system, just exciting the qubit with a classical microwave pulse. (iii) As a further example of the developed framework, we analyze the output squeezing from a resonator interacting with a cascade-type threelevel system.
We start out, in section 2, with the squeezing of the ground state of the Rabi Hamiltonian. In section 3 we present a theoretical framework of the output field quadratures. In section 4 this theoretical approach is applied to single-atom USC cavity-QED systems. Finally, in section 5 we present our conclusions.

Squeezing of the ground state of the Rabi Hamiltonian
The Hamiltonian of the quantum Rabi model ( = 1) [55,56] is given by whereâ andˆ † a are, respectively, the annihilation and creation operators for the cavity field of frequency w c . The Pauli matrices are defined as s = ñá -ñá | | | | e e g g z and s s s , in terms of the atomic ground ( ñ |g ) and excited ( ñ |e ) states. The parameter w q describes the transition energy of the two-level system and W R is the coupling energy between the atomic transition and the cavity field.
Owing to the presence of the so-called counter-rotating terms, s -â and s + † a , in the Rabi Hamiltonian, the operator describing the total number of excitations, = + ñáˆˆ| | † N a a e e , does not commute withĤ R and as a consequence the eigenstates ofĤ R do not have a definite number of excitations [21], however the system described by the Hamiltonian in equation (1) conserves the parity of the number of excitations. For instance the resulting ground state, in terms of the bare cavity and qubit states, is a superposition of an even number of excitations [21], where the second entry in the kets provides the photon number. The coefficients of this expansion can be calculated diagonalizing numerically the Rabi Hamiltonian in equation (1) (see, e.g., [21,22]). When the coupling rate W R is much smaller than the bare resonance frequencies of the two subsystems w c and w q , onlyc g,0 0 is significantly different from zero and the ground state reduces to ñ ñ  |˜|g 0 ,0, which is that of the Jaynes-Cummings model, derived from the Rabi Hamiltonian after dropping the counter-rotating terms. When the coupling rate W R approaches and exceed 10% of the bare frequencies of the subsystems, that is USC regime, contributions with ¹ k 0 in equation (2) become not negligible. One consequence is that the mean photon number in the ground state á ñ|ˆˆ| † a a 0 0 becomes different from zero. Moreover, the ground state displays a certain amount of photon squeezing. Considering the intracavity-field quadrature = - ) turns out to be below the standard quantum limit value 1. as a function of the normalized coupling w W R c and detuning w w For small values of the normalized coupling, the variance approaches the standard quantum limit.
We notice that, increasing w W R c , the variance decreases below the standard quantum limit, reaching a and at a positive detuning Further increasing the coupling, especially at zero and negative detuning, results into an increase of the variance s 2 , caused by quite large contributions in ñ |0 of terms with an odd number of photons. This noise increase can be understood noticing that the noise reduction originates from the terms y y á ñ |ˆ| a 2 , and the operatorâ 2 connects only terms in the quantum states yñ | differing by two photons. Hence squeezing can be larger for states with either an even or an odd number of photons. In the next section we will show that such a ground-state squeezing actually does not give rise to an observable output squeezing.

Output field quadratures
According to the input-output theory for general localized quantum systems interacting with a propagating quantum field, the output field operator can be related through a boundary condition to a system operator and the input field operators [57]. In order to be specific, we consider the case of a system coupled to a semi-infinite transmission line [57], although the results obtained can be applied or extended to a large class of systems. While the resonator can be ultrastrongly coupled to a localized quantum system, its interaction with the propagating quantum field (e.g., the transmission line) is weak. To derive the input-output relations we couple the system to a quantum field made of an assembly of harmonic oscillators. The total Hamiltonian of the system can be written as whereĤ S andĤ F are the system and field Hamiltonian and where the interaction between the system and the field can be expressed in the rotating wave approximation as where v is the speed of the traveling field, e.g., the speed of light in the transmission line.
In the above equation, ŵ ( ) b and w ( ) k are the annihilation operator and the spectral density for the harmonic oscillators that describe the output field, + X and -X are the positive and negative frequency components of the generic system operatorX coupled to the field. These components can be obtained expressingX in the eigenvectors basis ofĤ S as Here the eigenstates ofĤ S are labeled according to their eigenvalues such that w w > k j for > k j. We observe that the rotating wave approximation used in equation (A4) is based on the separation into positive and negative frequency operators of the system operatorX after the system diagonalization. The standard RWA is instead based on the separation into bare positive (destruction) and negative (creation) components of the field operator coupled to the external modes, without including its interaction with other components of the system.
The positive frequency component of the input and output fields can be written as is a fixed initial time and ¢ > t t (the output) is assumed to be in the remote future [57]. Formally solving the Heisenberg equations of motion for ŵ ( ) b , the input-output relations for the positive and negative components of the fields can be obtained [22] where for the sake of simplicity the first Markov approximation, w g p = ( ) k 2 , has been adopted. However, the present analysis can be easily extended beyond this approximation. Equation (7) shows that the positive frequency output operator can be expressed in terms of the positive frequency input operator and the positive frequency system operator coupled to the propagating field. If the system consists of an empty single-mode resonator, then µ +X a, beingâ the destruction operator of the cavity mode. If instead the cavity mode is coupled to another quantum system, e.g., an atom, + X will be different fromâ, and may also contain contributions fromˆ † a . In this case, the positive component of the output field may contain contributions from the creation operator of the cavity field, in contrast to ordinary quantum optical input-output relationships [27,51].
We define the output quadrature operatorsˆ( Here Ω and j are, respectively, the frequency and the phase of the local oscillator field employed for the squeezing measurements [29]. It is possible to change from one quadrature to the other by applying a p 2 rotation to one of the two quadratures, e.g., by changing the reference phase j.
Let us now consider the field-quadrature variance . Using equation (9), it can be expressed in terms of the output operators, By using the input-output relation (7), each term in the above equation can be written in terms of input and system operators. For example, the first expectation value in the r.h.s. of equation (10) becomes, We observe that equation (11) contains expectation values involving products of input and system operators.
Even considering the important case of a vacuum input port, the mixed term in is in general different from zero and cannot be directly calculated by the master equation approach, which does not calculate mixed bath-system correlations. This problem can be solved by deriving the commutation relations between system and input operators. By using equation (6) and the expression of the field operator ŵ ( ) b , obtained from solving the Heisenberg equation, we arrive at the following commutation relation between any system variablê ( ) Y t and the input fields 1 2 if t=s, and 0 if < t s. This commutation relation, which holds for arbitrary light-matter couplings, can be considered to be the generalization of an analogous commutation relation obtained within the standard input-output framework [50]. Its derivation is described in appendix A.
Making use of the input-output relations (7) and of the commutation relations (12), we can proceed to calculate the output field quadrature variances in terms of correlation functions involving only input operators or system operators. Here we used á ñ = á ñ -á ñá ñˆˆˆˆÂ . Considering an input in a vacuum or a coherent state, the field-quadrature variances can be expressed as where  is the time-ordering operator that rearranges the creation operators in the forward time, and also the annihilation operators in the backward temporal order. To obtain S 2 we can apply a p 2 rotation to equation (13). For equal-time correlation functions (t = 0), we have The last term in equation (14) á ñ + -Â A , in in describes the quantum noise of the input in the vacuum state. We observe that, according to equation (5), the operator + X can induce only downward transitions from higher energy to lower energy levels. Hence, when it is applied to the ground state, it automatically gives zero: If the system starts in its ground state, in the presence of a vacuum input, it will remain there. Using then equation (15), the term á ñ -+ ( )ˆ( ) X t X t , in equation (14) becomes zero and the output noise in equation (14) coincides with the input one: = á ñ + -

( )ˆ( )ˆ( ) S t A t A t 1 in in
. From equations (14) and (15) we can thus formulate the following general statement: Any open system in its ground state, i.e., ñ = + | X 0 0, does not display any output squeezing, even if its ground state is a squeezed state. Equation (14) holds for general open quantum systems, independently on their composition in subsystems and the degree of interaction among the different subsystems. This absence of output ground-state squeezing has been previously shown in different interacting harmonic systems [19,58,59].
In order to compare this result with previous descriptions for optical resonators, we consider the case wherê X describes the field of a single-mode cavity: . Here X 0 denotes the zero-point fluctuation amplitude of the resonator. Equation (14) can be expressed as , . x a . The noise reduction with respect to the vacuum input can be expressed in terms of the following normallyordered variance

S t S t A t A t X
, .

Squeezing of output field-quadratures in the USC regime
Here we apply the theoretical framework developed in section 3 to study the output field-quadrature variances in single-atom USC cavity-QED systems. We first consider the case of a flux qubit artificial atom coupled to a l 2 superconducting transmission-line resonator, when the frequency of the resonator is near one-half of the atomic transition frequency (see figure 2). Recently it has been shown [60] that this regime can strongly modify the concept of vacuum Rabi oscillations, enabling two-photon exchanges between the qubit and the resonator.
Here we show that such configuration can provide a very large amount of squeezing, although the system has only one artificial atom and displays a moderate coupling rate w W~0.1 R c . Then, we will study the output squeezing of a cascade three-level system where only the upper transition is coupled to the optical resonator.
In order to describe a realistic system, the dissipation channels need to be taken into account. For this reason all the dynamical evolutions displayed below have been numerically calculated solving the master equation [22,61,62], where  i is a Liouvillian superoperator describing the cavity and atomic system losses (see appendix A). All calculations have been carried out by considering zero temperature reservoirs.

Two-photon Rabi oscillations
We now consider a flux qubit ultrastrongly coupled to a coplanar resonator [2]. This quantum circuit is analogous to a cavity-QED system, where the flux qubit with its discrete anharmonic energy levels represents the artificial atom and the coplanar resonator the optical cavity (see figure 2). Recently it has been shown that this system paves the way to anomalous vacuum Rabi oscillations, where two or more photons are jointly and reversibly emitted and reabsorbed by the qubit [60,63].
This quantum circuit can be described by the following extended x z R c q R In this system both the number of excitations and parity symmetry are no longer conserved and transitions which are forbidden in natural atoms become available [64]. The angle θ as well as the qubit resonance frequency depend on the flux offset dF º F -F q e x t 0 , where F ext is the external magnetic flux threading the qubit and F 0 is the flux quantum. A flux offset dF = 0 q implies q = 0. In this case ¢ H R reduces to the standard Rabi Hamiltonian (1). We choose the labeling of the eigenstates ñ |ĩ and eigenvalues wj of ¢ H R such that w w >k j for >k j.
The lowest eigenenergy offsets with respect to the ground energy w w -j 0 as a function of the qubit transition frequency w q are shown in figure 2. Looking at the numerically calculated eigenvectors, the first excited state, ñ |1 , contains a dominant contribution from the bare state ñ |g, 1 , ( ñ ñ  |˜|g 1 ,1). The figure also shows an avoided crossing when w w » 2 q c . The splitting can be attributed to the resonant coupling of the states ñ |e, 0 and ñ |g, 2 , although the USC regime implies that the resulting dressed states ñ |2 and ñ |3 contain also small contributions from other bare states, as ñ |g, 1 and ñ |e, 1 . This splitting cannot be found in the rotating wave approximation, where the coherent coupling between states with a different number of excitations is not allowed, nor does it occur with the standard Rabi Hamiltonian (q = 0). We consider a system initially in the ground state. Excitation occurs by direct optical driving of the qubit via a microwave antenna. The corresponding driving Hamiltonian is describes a Gaussian pulse. Here A and τ are the amplitude and the standard deviation of the Gaussian pulse, respectively. We consider the zero-detuning case, corresponding to the minimum energy splitting W 2 eff between the two split levels ( ñ |2 and ñ |3 ) in figure 2(b). The central frequency of the pulse has been chosen to be in the middle of the two split transition energies: 0 . If τ is much smaller than the effective Rabi period, t p = W  T 2 R e f f , the driving pulse is able to generate an initial superposition with equal weights of the states ñ |2 and ñ |3 , which will evolve displaying two-photon quantum vacuum oscillations [60]. Figure 3(a) displays the resulting qubit population (red dashed curve) and mean photon number (blue continuous) after a pulsed excitation with an effective pulse area  p = 3. Figure 3(b) shows the normally-ordered variance of the two orthogonal output field quadratures ( ) S n 1 (blue continuous curve) and ( ) S n 2 (dotted red). Both the two quadratures display a significant amount of squeezing when the mean photon number is maximum. It is interesting to see that the periodicity of the two variances is twice the Rabi period T R . This can be understood noticing that after the excitation, the quantum state is a superposition of the ground state ñ |0 and the excited states ñ |2 and ñ |3 . After one Rabi oscillation, the excited states acquire a π phase shift. A second Rabi oscillation is needed to recover the initial phase. The dynamics of the corresponding variances (not shown here) calculated by usingâ andˆ † a , instead of + x andx , are affected by fast oscillations. Considering the qubit initially prepared in the superposition state y ), the resulting time evolution of the system state is, to a good approximation, where W 2 eff is the minimum energy splitting in figure 2 which is a squeezed photon state, reaching a maximum squeezing for a  1 3.

Cascade three-level system
We consider a three-level ñ ñ (| | s g , and ñ | ) e atom-like system with the upper transition ñ « ñ (| | ) g e ultrastrongly coupled with a mode of the resonator and a lower transition which does not interact with the resonator, as schematically shown in figure 4. The peculiar optical properties of this system have been analyzed calculating the dynamics of the populations and of normal-order correlation functions [15,65,66]  . In order to obtain large squeezing, we choose the driving amplitude such that f » tan 2 2, corresponding to the angle where squeezing for this superposition state is maximal. Figure 5 shows the time evolution of the normally-ordered variances ( ) S n 1 (blue upper curve) and ( ) S n 2 (red lower curve) calculated using the correct positive and negative operators + X and -X , with reference frequency w W = c . The squeezing displayed in figure 5  , and it does not present any fictitious fast oscillation. On the contrary, with the use of the standard operators, see appendix C, the squeezing starts with a fictitious value less than zero and shows large fictitious oscillations.

Conclusions
We have derived a generalized theory of the output field-quadrature measurements and squeezing in cavity-QED systems, valid for arbitrary cavity-atom coupling rates. In the USC regime, where the counter-rotating terms cannot be ignored, the standard theory predicts a large amount of squeezing in the output field, even when the system is in its ground state. Here we have shown that, in this case, no squeezing can be detected in the output field-quadratures, independently of the system details. We have applied our theoretical approach to study the output squeezing produced by an artificial two-level atom embedded in a coherently excited cavity. We showed that, a large degree of squeezing can be obtained with this elementary quantum system. We also studied the output field-quadratures from a cavity interacting in the USC regime with the upper transition of a cascade-type three-level system. The numerical results have been compared with the standard calculations of output squeezing (see figure 5). The approach proposed here can be directly applied also to resonators displaying ultrastrong optical nonlinearities [67]. This work extends the possibilty of predicting and analyzing output-field correlations when optical resonators interact very strongly with other quantum systems.    figure C1(a) has been calculated by using the reference frequency W = 0. Figure C1(b) has been obtained by using w W = c . Figures C1(a) and (b) show that it is not possible to eliminate fast and large-amplitude fictitious oscillations, as well as the fictious initial squeezing within the standard approach.