Wigner negativity in the steady-state output of a Kerr parametric oscillator
WWigner negativity in the steady-state output of a Kerr parametric oscillator
Ingrid Strandberg, G¨oran Johansson, and Fernando Quijandr´ıa Microtechnology and Nanoscience, MC2, Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden (Dated: 2020-09-18)The output field from a continuously driven linear parametric oscillator may exhibit considerablymore squeezing than the intracavity field. Inspired by this fact, we explore the nonclassical featuresof the steady-state output field of a driven nonlinear Kerr parametric oscillator using a temporalwave packet mode description. Utilizing a new numerical method, we have access to the densitymatrix of arbitrary wave packet modes. Remarkably, we find that even though the steady-statecavity field is always characterized by a positive Wigner function, the output may exhibit Wignernegativity, depending on the properties of the selected mode.
I. INTRODUCTION
As opposed to a quantum field confined in a cavity, afield propagating in free space is characterized by a con-tinuum of modes. In order to give a Schrdinger picturedescription of the propagating output state, wave packetmodes are introduced. In quantum optics experiments,it is possible to infer information about a state stabilizedinside a cavity by looking at the output field in a wavepacket mode with a bandwidth matching the inverse cav-ity decay time. The only requirement imposed on thiswave packet function is that it is square-integrable andnormalized to unity, which guarantees that the filteredsignal corresponds to a single bosonic mode. Whereasfor a linear non-driven system, there exists a unique wavepacket mode which maps the cavity state into the outputfield [1], this is not the case in the presence of a contin-uous drive. In this case, nothing restricts us from go-ing beyond the cavity natural bandwidth with arbitraryfunction profiles.Depending on the selected wave packet, properties ofthe cavity field and its corresponding output field canbe quite different. An old and well-known example ofthis difference is given by the degenerate parametric os-cillator (PO). This is commonly employed to generatesqueezed states, i.e., states exhibiting quadrature fluctu-ations below the vacuum level. In the steady-state, themaximum attainable squeezing inside the PO cavity isequal to 50 % of the vacuum noise. Nonetheless, the out-put field squeezing can largely surpass that, and ideallyachieve 100 % squeezing in a narrow frequency bandwidtharound the cavity frequency [2].Squeezed states are a prominent example of nonclassi-cal states of light [3–5]. A signature of the nonclassicalbehavior of a squeezed state is oscillations in the photon-number distribution [6–8]. Such oscillations are not bythemselves a signature of a nonclassical state, since classi-cal states can also display population oscillations. A use-ful nonclassicality criterion which studies the departureof a photon-number distribution from a classical proba-bility distribution was introduced by Klyshko [9]. Morecommonly, the nonclassical nature of a quantum state isdefined in terms of its Wigner function. If the Wigner function is negative, the state is considered nonclassi-cal [10, 11]. On the other hand, Wigner negativity isnot a necessary condition for nonclassicality; for exam-ple, squeezed states are considered nonclassical by theKlyshko criterion but have a positive Wigner function.Still, negativity of the Wigner function plays an essentialrole in quantum computation with continuous variables,as it is a distinguishing feature of states that are resource-ful for a computational advantage [12, 13].The process that generates squeezing in a PO is (de-generate) parametric down-conversion, in which a pumpphoton of frequency 2 ω splits into two photons each withfrequency ω . This process occurs in a medium with asecond-order nonlinear susceptibility [14]. Real-life non-linear materials are however not restricted to second or-der, and higher order nonlinearities may also need to betaken into account. A higher order nonlinearity is also arequirement for the generation of Wigner-negative states.Including a third-order (Kerr) nonlinearity results in theKerr Parametric Oscillator (KPO) which has recentlyfound applications in microwave quantum optics for thedissipative stabilization [15–17] and the adiabatic prepa-ration [18–20] of quantum states of light (cat states),which are useful for quantum computing [21, 22]. Ad-ditionally, by utilizing superconducting circuits it is pos-sible to explore the so-called Kerr single-photon regimein which the strength of the nonlinearity surpasses theinverse cavity decay time by an order of magnitude, en-abling the observation of previously undetected quantumeffects [19, 23–25].In this work we study the steady-state output from theKPO cavity with respect to temporal wave packet modes.We find that the KPO cavity and output fields can havemarkedly different nonclassicality properties, mainly interms of the Wigner function. For comparison, we alsoreview the PO output field in terms of temporal wavepacket modes. While the squeezing spectrum of the out-put is previously known, we analytically solve for thefull output field density matrix, shedding new light onthis well-studied system by investigating how its featuresdepend on the properties of the temporal wave packet.Since an analytical solution is not straightforward forthe KPO output field, we use numerical simulations usingthe ”input-output with quantum pulses” formalism intro- a r X i v : . [ qu a n t - ph ] S e p duced by Kiilerich and Mølmer [26, 27]. We establish thenonclassical character of the KPO output field throughits photon-number distribution using the Klyshko non-classicality criterion. We find that the presence of theKerr nonlinearity leads to larger population oscillationsthan for the PO. We also find that under circumstancesthat render the steady-state KPO cavity field Wigner-positive, the leaked output field may nonetheless exhibitWigner negativity, and the magnitude of a Klyshko co-efficient directly correlates with the amount of Wignernegativity. Moreover, by numerical optimization we findthe temporal wave packet that maximizes the Wignernegativity.The paper is structured as follows: In Section II wedescribe our system model. In order to make a compar-ison with the KPO, we review the PO output field inSection III before proceeding with the KPO output inSection IV. There, we compare the population statisticsbetween the cavity steady-state and the output state inorder to understand how Wigner negativity can appear inthe output despite the cavity being Wigner-positive. InSection V we take a closer look at the nonclassical prop-erties of the KPO output field for different nonlinearitystrengths. Finally, Section VI summarizes our results. II. THE MODEL
Our system is a parametrically driven nonlinear cavity,and we are interested in its steady-state emission. In aframe rotating at the cavity resonance frequency ω , theHamiltonian of the driven cavity is ( (cid:126) = 1)ˆ H = 12 (cid:16) β ˆ c † + β ∗ ˆ c (cid:17) + K ˆ c † ˆ c , (1)where β is the complex two-photon drive (parametricpump) amplitude, K is the Kerr nonlinearity and ˆ c (ˆ c † )is the annihilation (creation) operator for the photonsinside the cavity. This is an ubiquitous quantum opticsmodel which describes squeezing and parametric ampli-fication [14, 28]. The two-photon drive results from thenonlinear interaction of two modes, typically denoted asthe signal and the pump. Using the the so-called para-metric approximation in which the pump field is assumedto be in a coherent state of large amplitude, i.e. classi-cal, the pump mode operator can be substituted by a c -number. As an alternative to having a second-order non-linear crystal in an optical cavity, this Hamiltonian canalso be obtained via a time-dependent boundary condi-tion such as a movable mirror [29] or a tunable Josephsoninductance in a microwave circuit [30–32].The non-unitary dynamics of the cavity is well de-scribed by the Lindblad quantum master equation whichconsiders a Markovian (memoryless) environment at zerotemperature ∂ t (cid:37) = − i [ ˆ H , (cid:37) ] + γ (cid:18) ˆ c (cid:37) ˆ c † − { ˆ c † ˆ c, (cid:37) } (cid:19) , (2) where {· , ·} denotes the anticommutator and γ is thesingle-photon loss rate of the cavity. In typical quan-tum optics applications, single-photon loss is the maindecay channel of cavity photons into the the continuumof electromagnetic modes which comprise the cavity en-vironment. Here, we consider photon emission into aone-dimensional transmission line. The state of the fieldˆ a out that leaks out of the cavity can be inferred fromthe input-output boundary condition [33], which relatesˆ a out with the drive field and the cavity field operator ˆ c .Considering the two-photon drive as a weakly coupledinput channel, the cavity output field depends solely onthe cavity state as ˆ a out ( t ) = √ γ ˆ c ( t ) . (3) A. Wave packet modes
The cavity output field provides information of the cav-ity state through the input-output relation (3). However,the cavity field corresponds to a single bosonic mode,while the output is comprised by a continuum of fre-quencies. Therefore, in order to do a faithful comparisonof the cavity and output states, it is necessary to definea bosonic mode out of this continuum. This is done interms of wavepacket modes. For convenience, we are go-ing to restrict to a temporal description in the remainderof this work, but the corresponding frequency represen-tation is simply related by a Fourier transform.We are going to characterize the emission from the cav-ity in the Fock space defined by the wavepacket creationoperator ˆ A † f = (cid:90) ∞ d t f ( t ) ˆ a † out ( t ) , (4)with f ( t ) satisfying the normalization condition (cid:82) ∞ d t | f ( t ) | = 1 in order for ˆ A f to fulfill the bosoniccommutation relation [ ˆ A f , ˆ A † f ] = 1. For simplicity, werestrict f ( t ) to be a real-valued function. This creationoperator defines a symmetric wave packet in frequencyspace around the cavity resonance frequency.Experimentally, detection in a given temporal profileis implemented by means of a pulsed local oscillator inhomodyne detection or by processing the measurementsignal by means of a digital filter corresponding to thefunction f . For this reason, we sometimes refer to thewavepacket profile as a filter function . III. PARAMETRIC OSCILLATOR (PO)
Before moving to the Kerr parametric oscillator, westart by studying the linear, or Gaussian, degenerateparametric oscillator (PO) with K = 0 in Eq. (1). It isknown that the steady-state of the field inside of the res-onator exhibits quadrature squeezing . This means thatthe minimum value of the variance of the generalizedquadrature operator ˆ X θ = (ˆ b e − i θ + ˆ b † e i θ ) / √ b is a generic bosonic field operator [ˆ b, ˆ b † ] = 1, i.e.,it can equally refer to the cavity operator ˆ c or the filteredoutput ˆ A f . For our chosen normalization of the general-ized quadratures, the vacuum noise variance is 1 /
2. Welabel the minimum value of the variance as s = min θ (cid:104) (∆ ˆ X θ ) (cid:105) , (5)and consequently, squeezing is indicated by s < /
2. Thevariance attains its minimum value for a particular valueof θ . For our setup, it is dependent on the phase of thedrive β in Eq. (1). But for simplicity, from now on weare going to restrict to a real-valued two-photon driveamplitude β ∈ R without loss of generality.The variance is defined as (cid:104) (∆ ˆ X θ ) (cid:105) = (cid:104) ˆ X θ (cid:105) − (cid:104) ˆ X θ (cid:105) .In our particular system we have for the PO steady-state (cid:104) ˆ c (cid:105) ss = 0. Consequently, first-moments of the quadra-ture field also vanish. In general, it is always possi-ble to displace the field in order to meet this condi-tion, so from now on we will ignore first-order moments.Then, the variance can be rewritten as (cid:104) (∆ ˆ X θ ) (cid:105) = (cid:104) ˆ b † ˆ b (cid:105) + (cid:104) ˆ b (cid:105) e − iθ / (cid:104) ˆ b † (cid:105) e θ / /
2. Taking into ac-count that (cid:104) ˆ b † (cid:105) = (cid:104) ˆ b (cid:105) ∗ and noting that ˆ b † ˆ b is a positivesemi-definite operator, we see that there is squeezing ifand only if Re[ (cid:104) ˆ b (cid:105) e − θ ] < − (cid:104) ˆ b † ˆ b (cid:105) . We see that squeez-ing can only appear if the expectation value (cid:104) ˆ b (cid:105) has alarge enough magnitude. This expectation value is as-sociated with off-diagonal terms in the density matrix:coherences between number states | n (cid:105) and | n − (cid:105) .The amount of squeezing of the cavity field increasesas we approach the so-called pump threshold β th = γ/ s = 1 /
4. On the other hand, the field leaking outof the cavity can achieve perfect squeezing s → A. PO output state
With K = 0, the system is Gaussian since the equationof motion (2) is only quadratic in ˆ c and ˆ c † . Hence theoutput field is completely characterized by its first- andsecond-order moments for which it is possible to find aclosed set of equations. This means we can analytically solve for the state of the output field. For the selectionof a wave packet mode, we choose a constant filter withina time interval T , a so-called boxcar filter [34–36]. Thissimplifies the analytical calculations.As already mentioned, we will consider the steady-state emission from the resonator. Using the quantumregression theorem [37–39] we calculate the steady-statetwo-time correlations (cid:104) ˆ c † ( τ )ˆ c (0) (cid:105) ss and (cid:104) ˆ c ( τ )ˆ c (0) (cid:105) ss by as-suming the steady-state condition ∂ t (cid:37) ss = 0 in Eq. (2).Correlations for the filtered output state ˆ A f follow fromintegration: (cid:104) ˆ A † f ˆ A f (cid:105) = ( γ/T ) (cid:82) T d t (cid:48) (cid:82) T d t (cid:104) ˆ c † ( τ )ˆ c (0) (cid:105) ss and (cid:104) ˆ A f (cid:105) = ( γ/T ) (cid:82) T d t (cid:48) (cid:82) T d t (cid:104) ˆ c ( τ )ˆ c (0) (cid:105) ss . From thesecorrelators we can calculate the covariance matrix ele-ments V k(cid:96) = (cid:104) ˆ R k ˆ R (cid:96) + ˆ R (cid:96) ˆ R k (cid:105) /
2, for k, (cid:96) = 1 , R =( ˆ A † f + ˆ A f ) / √ R = ( ˆ A † f − ˆ A f ) / i √
2, which com-pletely determine the state of a Gaussian system. Withthese matrix elements, the Wigner function can be calcu-lated as W ( x, p ) = exp[ − ( x, p ) (cid:62) V − ( x, p )] / (2 π √ det V ),and the Fock space representation of the state is deter-mined following Refs. [40, 41] (more details in AppendixA).The behaviour of the filtered output field of the para-metric oscillator as a function of the filtering time T fortwo different drive strengths β = 0 . β = 0 . T and β are given in units wherethe decay rate is set to γ = 1. In Fig. 1(a) and 1(b)we plot the first six Fock state populations ρ n . For bothdrive strengths, the single-photon state is the first stateto be populated and is the dominant non-vacuum con-stituent of the complete state for T (cid:28)
1. Progressively,two-, three- and higher photon number states becomepopulated. At around T (cid:39)
2, the two-photon popula-tion overcomes the single-photon population. Increasingthe filtering time, we observe the same dynamics for theother even photon number states (the four-photon pop-ulation overcomes the three-photon one, and so on). Aswe go towards T → ∞ we see that the odd Fock statepopulations are suppressed and the state becomes a su-perposition of even Fock states. From the analytical so-lution (not shown), we can see that as we approach β th in the limit T → ∞ the populations of all the even statesbecome identical. This tendency can be seen by com-paring the asymptotic populations in1(a) and1(b): thedifferences between the populations are smaller for thestronger drive.In Figs. 1(c) and 1(d) we plot the squeezing defined byEq. (5) as a function of T . Starting as vacuum noise for T (cid:28)
1, the output field becomes squeezed as soon as westart to generate photon pairs. It reaches the squeezinglevel of the intracavity field for T (cid:39)
1, i.e., around thenatural bandwidth defined by the cavity decay rate, andthen surpasses it for larger T . The maximum squeezing isattained when the state becomes a superposition of evenFock states.A squeezed state which satisfies the minimum uncer-tainty allowed by the Heisenberg uncertainty principle issometimes referred to as an ideal squeezed state [42]. To FIG. 1. Filtered output field populations of the first 6 numberstates | n (cid:105) excluding the vacuum. Even number states in solidlines and odd number states in dashed lines [(a,b)]. Squeezingfor both, the output filtered field in solid lines and the cavityfield in dashed lines [(c,d)]. Determinant of the covariancematrix in solid lines. The dashed lines signal the minimumuncertainty condition [(e,f)] Everything for β = 0 . β =0 .
4, and as a function of the boxcar field width T in unitswhere γ = 1. Note the logarithmic scale on the horizontalaxis. show that the PO output field becomes an ideal squeezedstate in the limit T → ∞ we plot the determinant of thecovariance matrix in Fig. 1 (e) and (f). Because it isa real symmetric matrix, the covariance matrix can al-ways be diagonalized. This diagonal matrix V containsthe variances of the squeezed quadrature and the orthog-onal one. Their product, i.e., the determinant of V , isthe uncertainty product. The minimum uncertainty con-dition in our quadrature normalization corresponds todet V = 0 .
25. For values of T much smaller than thecavity characteristic decay time 1 /γ , the minimum uncer-tainty condition is met because the probability of emis-sion is very small, and thus, we are witnessing the mini-mum uncertainty of the vacuum state. As the squeezedstates approach the minimum uncertainty condition forlarge T the populations of the odd Fock states go asymp-totically to zero. In Appendix B we show analyticallythat the odd photon numbers vanish only for a squeezedstate which is also a minimum uncertainty state.Finally, in Fig. 2 we show the first 20 populations ofthe output field for two different widths of the boxcarfilter, T = 4 and T = 500, for β = 0 . β = 0 .
4. We note that vacuum is always the leadingcontribution to the state. For T = 4 we can appreci-ate an enhanced two-photon population for both drivestrengths [Figs. 2(a) and 2(c)], although for the stronger drive in 2(c) many higher number states are already alsopopulated. For both drive strengths, as T becomes verylarge, in Figs. 2(b) and 2(d) we only observe even Fockstates in agreement with Fig. 1. This distinct even-oddoscillation in the populations is a sign of nonclassicality,which will be elaborated further on in section III B. Theinsets in Fig. 2 show the Wigner functions of the corre-sponding states, which become more and more squeezedfor larger β and T . FIG. 2. PO output field Fock state populations for two dif-ferent drive strengths β and boxcar filter times T (in unitswhere γ = 1). For the very large filter time T = 500, onlyeven Fock states are present, and squeezing is enhanced. These results can be intuitively understood by meansof a very simple time-domain argument as follows: thetwo-photon drive places correlated pairs of photons insidethe resonator. However, the photons leave the cavity oneby one. This restricts the number of photon pairs in-side the resonator and therefore, the maximum amountof squeezing that can be sustained. On the other hand,we can access the photon pairs by monitoring the outputfield for a long time compared to the cavity lifetime 1 /γ ,as illustrated in Fig. 3. Thus, large two-photon corre-lations between states | n (cid:105) and | n − (cid:105) can be obtained,and as described in the beginning of this section, there-fore more squeezing is expected in the output. B. Nonclassicality from the PO
There exist several indicators and measures of thenonclassical behaviour of light, including sub-Poissonianstatistics [43], antibunching [44] as well as the previ-ously mentioned photon-number population oscillationsand squeezing. One widely used nonclassicality criterionis defined in terms of the Glauber-Sudarshan P -function:if the P -function is negative or more singular than a Dirac δ -function the state is considered nonclassical [45, 46]. FIG. 3. A drive photon of frequency 2 ω is down-convertedinto two photons each with frequency ω in the cavity withresonance frequency ω and single-photon loss rate γ . For alonger observation/filtering time T , more and more photonpairs are detected. The singular nature of a nonclassical P -function makesit difficult to experimentally verify. A prevalent alter-native is the Wigner function, which is related to the P -function by a Gaussian convolution [47]. The Wignerfunction can be tomographically reconstructed [48], andits negativity indicates nonclassicality.The Wigner functions of the PO cavity and outputfields are always positive. This is a consequence of thestates resulting from a linear, or Gaussian, system [49].Nevertheless, quadrature squeezing can be related to anon-positive P -function [43]. More specifically, the cor-responding P -function is highly singular [50]. Since thesqueezed state exhibits population oscillations as theparametric drive creates excitations in pairs, a perhapsmore useful [51] nonclassicality criterion for squeezed-likestates is given by the Klyshko inequality [9]. B n ≡ ( n + 1) ρ n − ρ n +1 − nρ n < . (6)The coefficient B n is defined in terms of the populationsof three consecutive number states | n − (cid:105) , | n (cid:105) and | n +1 (cid:105) ( n ≥ ρ n in terms of those of its nearest neighbors.If B n < n , the photon-number distribution ofthe corresponding field departs from a classical probabil-ity distribution, i.e., its P -function is negative [52].The nonclassical nature of states produced in paramet-ric down-conversion has been established via the Klyshkoinequality in the optical regime through direct use of pho-ton counters [53]. Similarly, in the microwave regime thenonclassical nature of propagating squeezed states gener-ated through a Josephson parametric amplifier was estab-lished by feeding the field into a 3D cavity and reading itspopulation content via a dispersively coupled transmonqubit [54].The odd Klyshko coefficients are always positive forour system, so in Fig. 4 we show the first three evenKlyshko coefficients B , B and B for the filtered outputof the PO as a function of the width T of the boxcar filterfunction, using two different drive strengths β = 0 . β = 0 .
4. It can be seen that for
T > n =2. As we have already seen in Figs. 1 and 2, for T (cid:38) ( � ) � � � � � � ���� ���� � �� ��� - ���� - ���� - ���� - ���� - ���� - �������� � � � � � ��� � � � �� � � β = ��� ( � ) ���� ���� � �� ��� - ����� - ����� - ����� - ����� - ����� - ��������������� � β = ��� FIG. 4. Klyshko coefficients for n = 2 (solid line), n = 4(dashed line) and n = 6 (dot dashed line) for (a) β = 0 . β = 0 .
4, for a boxcar filter of width T (in units where γ = 1). contribution to the output field, and the inequality (6)for n = 2 implies that ρ > (cid:114) ρ ρ . (7)Therefore, when the population of the two-photon stateovercomes this bound imposed by the populations of thesingle- and three-photon states the output field becomesnonclassical. The dominance of ρ becomes stronger forlarger values of T as the population of the odd numberstates start to decrease. As T is increased, higher evennumber states also become populated, which explains thesuccessively appearing negative values of B and B .Furthermore, we observe a qualitative difference be-tween the behaviour of the Klyshko coefficients for thetwo drive strengths. While the coefficient B exhibits thelargest negativity in both cases, for β = 0 . T → ∞ , when the odd num-ber states are fully suppressed. On the other hand, for β = 0 .
4, as we increase T we involve considerably more(even) number states, each with smaller populations [Cf.Figs. 2(c) and 2(d)] as T → ∞ . The Klyshko B coeffi-cient peaks along with the peak of the two-photon pop-ulation, which is confirmed to occur just before T = 10in Fig. 1(b). IV. KERR PARAMETRIC OSCILLATOR (KPO)
The steady-state of the cavity field for the case of anon-vanishing Kerr nonlinearity has also been exhaus-tively studied in literature. In fact, despite being a non-linear system, the steady-state admits an analytical so-lution. It is a well-established result that the steady-state of our system is characterized by a positive Wignerfunction. Specifically, with successively increased drivestrength, the cavity steady-state field transitions fromvacuum to a weakly squeezed state and finally into anincoherent superposition of coherent states, with quan-tum coherence washed out by interaction with the envi-ronment [25, 55–58].While there has been comprehensive studies of the cav-ity field , to the best of our knowledge, a thorough char-acterization of the properties of the filtered output field is still missing in the literature. In the remainder of thispaper we are going to study the filtered output field ofthe KPO, with special emphasis on its nonclassical prop-erties using the above introduced Klyshko nonclassicalitycriterion as well as negativity of the Wigner function.Characterizing the state of the propagating cavity out-put field is, in general, a difficult problem since it impliesthe calculation of multi-time field correlations for differ-ent time orderings. For a Gaussian or linear system likethe PO studied in Sect. III, the output field is completelycharacterized by its first- and second-order moments forwhich it is possible to find a closed set of equations. Inthe presence of the nonlinearity, this is no longer possible.Instead, we would get an infinite number of equations in-volving every possible order of the output field moments.For this reason we do not attempt to characterize theoutput field by calculating multi-time correlations, butinstead follow a different approach. In order to explorethe features of the output field of the KPO we are go-ing to make use of a technique recently introduced byKiilerich and Mølmer [26, 27]. We implemented it us-ing QuTiP [59], and the code can be found in Ref. [60].The numerical solutions were validated against the ana-lytical solution for the PO. Alternatively, one could relyon stochastic methods to mimic a quantum tomographyexperiment, as done in Refs [35, 36], or numerically solvethe full Schr¨odinger equation for the coupled cavity andoutput fields [20].Before discussing the nonclassicality of the KPO out-put in depth, we will introduce the Poissonian regime ofthe KPO, in which both the cavity and the output areclassical states with Poissonian photon-number statistics.We then then discuss the two-photon population of theKPO output, which will be shown to have a strong con-nection to its nonclassical features.
A. KPO cavity Poissonian regime
The nonlinearity prevents the instability at the drivethreshold β = β th . This means that the mean num-ber of photons in the cavity n cav no longer diverges atthis point, but grows steadily with β . When the photonnumber reaches n cav (cid:29) γ/ K , the cavity steady-stateis an incoherent superposition of coherent states [18].Here the photon-number distribution is Poissonian, ρ n = n n cav exp( − n cav ) /n !. Therefore, we will refer to this typeof field configuration as the Poissonian regime .The maximum of a Poissonian photon-number distri-bution is centered around the average number of pho-tons present in the field. So while the average num-ber of photons in the cavity grows along with the para-metric drive strength β , when the Poissonian regime isreached, individual number state populations reach theirpeak and start decreasing in succession as the peak of thePoissonian distribution shifts to higher and higher num-ber states. As such, we expect that the largest achiev- able population for the n th number state happens whenthe cavity field populations has a Poissonian distribu-tion with n cav = n . Then, it is possible to estimate thelargest two-photon population in the KPO cavity, whichis ρ ∗ (cid:39) . B. Filtered KPO output field—boundary of thePoissonian regime
Intuitively, the output two-photon population is ex-pected to grow with β until the system reaches the Pois-sonian regime, similarly to the cavity state. Here we aregoing to show that, in contrast to the cavity field, for theoutput field it is possible to produce states which exhibitthe same maximum two-photon state population ρ ∗ butwith non-Poissonian photon number statistics. This issignificant, because in Appendix C we show that whenthe cavity is in the Poissonian regime, so is the outputfield. Specifically, the output field is also a classical mix-ture of coherent states, with the average number of pho-tons N f given by the cavity photon number n cav and ascale factor depending on the width of the filter function.For a boxcar filter the mean number of photons inthe output field is given by N f = γn cav T . Thus, inthe Poissonian regime, the largest two-photon popula-tion is achieved when N f = 2. Then, from the photon-number scaling relation, we can calculate the time T ∗ atwhich N f = 2. This corresponds to T ∗ = 2 / ( γ n cav ) [61].Therefore, the largest two-photon population expectedin the output of the KPO in the Poissonian regime is ρ ∗ (cid:39) .
27 for a filtering time T ∗ .We need to be outside of the Poissonian regime to ob-serve nonclassical effects. Below, we show the behavior ofthe photon populations in the filtered output field as wedepart from this regime for different values of the non-linearity strength. As stated in section IV A, not beingin the Poissonian regime implies a limited drive strength β , which in turn requires an increased filtering time T inorder to collect a significant number of photon pairs.In Fig. 5(a) we show the output two-photon populationas a function of β for a weak nonlinearity K = 0 . T (in units of γ = 1). As expected, weobserve that the two-photon population peaks at weakerdrive strengths for larger values of T , and vice versa.Additionally, we note that as T is increased, the photonpopulation statistics change. For the shortest filteringtime T = 0 .
1, the drive strength β at which the maxi-mum two-photon population is attained corresponds tothe cavity field being in the Poissonian regime, and sub-sequently, the output is also Poissonian, as can be seenfrom the photon-number distribution in 5(b). Here thelargest two-photon population observed corresponds to ρ ∗ (cid:39) .
27 (dotted line) and the filtering time at which itis achieved agrees with T ∗ . For a slightly increased T it ispossible to generate output states with a Poissonian-likephoton-number distribution [cf. 5(c)] but with a smallertwo-photon content than ρ ∗ , as for T = 0 .
4. The photon-number distribution of the output state slowly departsfrom the Poissonian behavior as we further increase T ,as exemplified with T = 1 in Fig. 5(d). But crucially, thetwo-photon population is reduced more and more as T is increased and we move further out of the Poissonianregime. FIG. 5. (a) Output two-photon population for K = 0 . β for T = 0 .
1, 0 . . γ = 1.The dotted line at ρ = 0 .
27 represents the theoretical maxi-mum two-photon population in the Poissonian regime. Notethat the two-photon population is decreased when movingout of the Poissonian regime. Photon number distributionsat the two-photon peaks for (b) T = 0 . n cav = 1 .
9. (c) T = 0 . n cav = 2 .
2. (d) T = 1 . Interestingly, we do not need a strong nonlinearity toget dramatically different behavior. In Fig. 6 we showresults for a moderate nonlinearity K = 0 .
5. Here, as wedecrease the drive strength β and increase the filteringtime T , we depart from the Poissonian regime similarlyas with the weak nonlinearity. But in contrast, even whennot in the Poissonian regime, the largest two-photon pop-ulations for K = 0 . ρ ∗ .In fact, from our numerical simulations the largest two-photon output even can even slightly exceed this value,as can be seen in Fig 6(a).For the longer filtering time in Fig 6(d), populationoscillations start to become evident. But it can be notedthat opposed to the PO, for which we know that the oddnumber states can be completely suppressed for a longfiltering time as shown in Sect. III B, it is not possible FIG. 6. (a) Output two-photon population for K = 0 . β for T = 0 .
5, 2 . . γ = 1. The dotted line at ρ = 0 .
27 represents the theoreticalmaximum two-photon population in the Poissonian regime.The two-photon populations are close to the maximum bothwithin and outside of the Poissonian regime (in stark con-trast to the behavior in Fig 5) with a maximum if ρ = 0 . T = 5 .
0. Photon number distributions at the two-photonpeaks for (b) T = 0 . n cav = 2 .
0. (c) T = 2 . T = 5 . with a nonzero Kerr nonlinearity [c.f. Appendix B].In the next section we are going to study the non-classical features of the filtered output field of the KPO.As described in section III B, nonclassicality is not only aconsequence of the enhanced two-photon population, butby how large it is compared to its nearest neighbors, i.e.,single- and three-photon states. For example, the photondistributions in Figs. 6(b), 6(c) and 6(d) have roughly thesame two-photon populations, yet only 6(b) correspondsto a Poissonian distribution. In 6(c), the state begins todepart from the Poissonian regime and the two-photonpopulation overcomes that of its nearest neighbors. Fur-ther away from the Poissonian regime this asymmetry iseven larger, as in 6(d). We are going to quantify thisusing the Klyshko coefficient B . V. NONCLASSICALITY IN THE KPO OUTPUT
In this section, we will evaluate the nonclassicality ofthe KPO output field in terms of the Klyshko B coeffi-cient and Wigner negativity. Starting with the Klyshkocriterion, we display the minimum B for different non-linearities K in Fig. 7. The Klyshko inequality B < K grows toward 0.5and larger, the enhanced two-photon contribution leadsto a very sharp population oscillation (an example of thisis Fig 6(d)) and consequently, to a very large magnitudeof the Klyshko coefficient. The largest magnitude is ob- K − . − . − . − . M i n B FIG. 7. When the nonlinearity K is increased from zero,the Klyshko B coefficient rapidly drops to its minimum for K = 0 .
7, corresponding to the ”most nonclassical” state asdetermined by the Klyshko criterion. As K is further in-creased, the value of B increases slightly before saturating.The minimum values were obtained by sweeping over a gridof β and T for each K . tained for K = 0 . K (cid:38)
2. Further increasing K doesnot lead to fundamentally different behavior, since thecavity steady-state field is entirely determined by the ra-tio β/K in the strong nonlinearity regime [25, 55, 56, 58].For a weak, or even vanishing, nonlinearity the value of B can remain <
0, but the magnitude is severely dimin-ished compared to what is attainable with a moderate orlarge K .The rather large two-photon populations in the KPOoutput field may also result in more quantum coher-ence, namely, in larger density matrix elements ρ and ρ . This is because the magnitude of these matrix ele-ments is bounded by the populations of the vacuum andtwo-photon states: | ρ | ≤ √ ρ ρ (the equality is onlyachieved for a pure state). Quantum coherence typicallytranslates into negative regions in the Wigner function.Unfortunately there is no simple relation between the ρ coherence and Wigner negativity, as higher-order Fockstates heavily influence the negativity.In Fig. 8 we show the Wigner function and density ma-trix for the K = 0 . β = 0 .
65 steady-state cavity field,and compare this with the output field for T = 2 . T = 5 . T . Negativityin the output increases as a function of T until T = 5 . T is further in-creased. Interestingly, the peak occurs at T = 5 not only FIG. 8. Wigner functions and density matrices for β = 0 . T = 2 . T = 5 boxcar filter, whichgives the largest WLN (0.05). Here, the vacuum population isfurther reduced and the two-photon population is even moreprominent. For clarity in the figure we truncate the densitymatrices at n = 3, but the full states occupy a larger Fockspace. for β = 0 .
65, but for all β (cid:38) .
3. The ρ coherencesare similar for the two output fields, and they are onlyslightly larger than for the cavity field. But the maincontribution to the cavity state coherence is from thevacuum population, which inhibits Wigner negativity.As the source of quantum coherence is the two-photondrive and the output state is favoured towards even num-ber states, it is no surprise that the KPO output Wignerfunction resembles a two-component kitten or cat state.Nevertheless, the resulting nonclassical states have a verylow purity, roughly 0.617 for the most Wigner-negativestate observed.A possible measure of the negativity of the Wignerfunction is the integrated Wigner negativity [11], or al-ternatively the more recently introduced Wigner Loga-rithmic Negativity (WLN) [63] W = log (cid:18)(cid:90) | W ( x, p ) | d x d p (cid:19) . (8)It has the property W > W ( x, p ) has a negative part, and has the benefit of beingan additive resource monotone [64]. In Fig. 9(a) we showa map of the WLN as a function of both the two-photondrive strength β and the boxcar width T for K = 0 . T = 5. This holds for K ≥ .
3. Inthe strong nonlinearity regime the maximum negativityoccurs for β/K (cid:39) B and the WLN. This is shown inFig. 9(b). As it can be seen, the nonclassical populationoscillations around the two-photon state which result innegative values of B directly translate into nonclassical-ity of the Wigner function. We show this correspondencefor T = 5, but it holds for every value of T . FIG. 9. (a) Contour map of the WLN as a function of β and T for K = 0 .
5. The dashed line indicates T = 5. (b) showsthe WLN and Klyshko B coefficient as a function of β for afixed T = 5. The WLN and B are clearly related. These results establish that, whereas the cavity de-cay rate imposes a detection bandwidth for which thecavity state can be output with high fidelity, detectionwith a wave packet beyond this natural limit may reveala complete different nature of the output field. In ad-dition, while single-photon losses may destroy quantumcoherence inside the cavity, it does not represent a lossmechanism for the output field. In the next section, wewill briefly study the effects of the temporal profile of thewave packet.
A. Impact of the filter function on the Wignernegativity
So far, we have for convenience selected a temporalmode of the cavity output field with a boxcar filter, butany square-integrable function can define a bosonic modein accordance with Eq. (4). Since the filter response cansignificantly change the nature of the detected state [65],it is reasonable to suspect that the choice of filter func-tion can have an impact on the observed Wigner negativ-ity. In this section, we are going to target the temporal mode profile which gives the maximum WLN by meansof numerical optimization.Since there is literally an infinite number of possibletemporal modes, the best filter could easily be left outif a selection of filter functions were tested manually. Toensure that the best filter function was found, even if itwas not a well-behaved, smooth function, we performedan optimization of the numerical array that representsthe filter using the scipy.optimize package [66]. Thepart of the filter that is zero until steady-state has beenreached was fixed and not included in the optimization.Besides the first and final points being zero, the initialfilter was random (but properly normalized, i.e. (cid:80) i f i =1). These constraints on the first and final points as wellas normalization was enforced during the optimization.Due to the randomness of the initial filter, the resultsof different optimization runs were not identical. But ingeneral, the optimized filter obtained a Gaussian shape.A representative example is displayed in Fig 10. In fact, t i . . . . . . | f ( t i ) | OptimizedInitialGaussian fit
FIG. 10. Numerical optimization of the filter function at dis-crete time points t i . The initial filter function array was con-structed by generating random numbers in the interval [0 , σ = 2 . µ = 17 .
0. Inthis example with system parameters β = 0 .
65 and K = 0 . the maximum WLN obtained with a Gaussian filter istwice the maximum of the boxcar. A comparison betweenthe two is shown in Fig. 11 for T = 5 which gives themaximum WLN for the boxcar filter, and σ = 2 . VI. SUMMARY & CONCLUSIONS
In this paper we have studied the nonclassicality of theoutput field of the steady-state Kerr parametric oscilla-tor, in terms of the Wigner function and Klyshko coeffi-cients. Our main result is that whereas the KPO cavityis Wigner-positive in the steady state, the output fieldcan be Wigner-negative, depending on the properties of0 . . . . . . β . . . . . W L N GaussianBoxcar
FIG. 11. A comparison of the maximum WLN for the boxcarand Gaussian filters, as a function of β with the filter widthsfixed to the values that give the largest possible WLN: T = 5for the boxcar and σ = 2 . the selected field mode. In order to obtain the state ofthe output field we have defined bosonic modes in termsof wave packet functions, utilizing the new ”input-outputwith quantum pulses” formalism introduced by Kiilerichand Mølmer [26], which allows us to obtain the densitymatrix of the output field.We also revisited the well-studied linear parametric os-cillator. The linear PO is instrumental for the genera-tion of quadrature-squeezed states of light. It is also aparadigmatic example of the different properties exhib-ited by the cavity and output fields of a continuouslydriven setup. While the results for the KPO were ob-tained by numerical simulations, for this linear system wecould study the properties of the filtered output field byreconstructing its density matrix analytically from two-time output field correlations. Here we also explored theso-called even-odd population oscillations as a function ofthe temporal width of the wave packet function. To thebest of our knowledge, these oscillations have previouslyonly been studied by direct photon detection, which isinsensitive to the mode structure of the field.We could then contrast the output of the PO to thatof the KPO. The nonlinear Kerr parametric oscillatoris also ubiquitous in quantum optics literature. Buteven though its cavity steady-state has been analyti-cally solved, to the best of our knowledge the outputfield has not been studied beyond its squeezing proper-ties. We found that the presence of the nonlinearity leadsto stronger population oscillations, which is expressed bythe larger magnitude of the Klyshko coefficient. This iswhat gives rise to the Wigner negativity in the KPO out-put, as the magnitude of the Klyshko coefficient directlycorrelates to the integrated Wigner logarithmic negativ-ity (WLN). Furthermore, by numerical optimization wehave verified that the nonclassical properties of the out-put field are dependent on the chosen wave packet func-tion, and that a Gaussian wave packet maximizes the WLN.Typically, the important parameter responsible fordriven nonlinear oscillators to reach quantum regimes isthe ratio between the Kerr parameter and cavity decayrate, i.e., efficiency of quantum nonlinear effects requiresa high nonlinearity with respect to dissipation [67]. Incontrast, there is no need for a strong nonlinearity to ob-serve Wigner negativity in the KPO steady-state output,as K/γ = 0 . VII. ACKNOWLEDGMENTS
The authors would like to thank Klaus Mølmer andChris Wilson for valuable discussions. IS acknowledgessupport from Chalmers Excellence Initiative Nano. FQand GJ acknowledge the financial support from the Knutand Alice Wallenberg Foundation through the Wallen-berg Center for Quantum Technology (WACQT).
Appendix A: Density matrix from the covariancematrix for a Gaussian state
A Gaussian state is defined by its covariance matrix V with matrix elements: V = (cid:104) ˆ x (cid:105) (A1) V = (cid:104) ˆ p (cid:105) (A2) V = V = 12 (cid:104) ˆ x ˆ p + ˆ p ˆ x (cid:105) , (A3)with ˆ x and ˆ p the position and momentum quadraturesrespectively ˆ x = 1 √ b † + ˆ b ) (A4)ˆ p = i √ b † − ˆ b ) , (A5)with ˆ b (ˆ b † ) the bosonic annihilation (creation) opera-tion which might refer to a cavity or filtered propagatingmode. Here we are assuming that (cid:104) ˆ x (cid:105) = (cid:104) ˆ p (cid:105) = 0 which1is true for the models studied in the main text. Recallthat in the steady-state of (2) with Hamiltonian (1) wehave (cid:104) ˆ c (cid:105) ss = 0 and consequently, (cid:104) ˆ A f (cid:105) = 0 for the filteredoutput fieldThe density matrix elements in the Fock or numberbasis can be recovered from the covariance matrix V bymeans of the relation (cid:104) m | ρ | n (cid:105) = (cid:18) d + t (cid:19) − / √ m ! n ! H { R } mn (0 , , (A6)with d = det V and t = tr V the determinant andthe trace of the covariance matrix, respectively [40].The so-called multidimensional Hermite polynomials H R mn ( x , x ) are defined in terms of a 2 × R with elements [41] R = (cid:18) d + t + 12 (cid:19) − ( V − V − V ) (A7) R = (cid:18) d + t + 12 (cid:19) − ( V − V + 2i V ) (A8) R = R = (cid:18) d + t + 12 (cid:19) − (cid:18) − d (cid:19) . (A9)The arguments x and x are related to the first momentsof the field, which in our case are always zero. We get H { R } mn (0 ,
0) = m ! n ! (cid:114) R m R n m + n min( m,n ) (cid:88) k =0 (cid:18) − R √ R R (cid:19) k ×× k !( m − k )!( n − k )! H m − k (0) H n − k (0) , (A10)with H n ( x ) being the n th order Hermite polynomial. Appendix B: Suppression of odd Fock statespopulations
The Heisenberg uncertainty principle puts a lowerbound on the minimum value of d = det V , where V is the covariance matrix of a quantum state, defined byEqs. (A7)-(A9). Under the quadrature normalizationused here we have d ≥ /
4. A minimum uncertaintystate is by definition a state for which d = 1 /
4. Exam-ples of these are coherent states and the so-called ideal squeezed states [42]. In a squeezed state, noise (the vari-ance) in one quadrature is reduced below the vacuumlevel at the expense of increased noise in the orthogonal quadrature. For an ideal squeezed state, the product ofthe quadrature variances equals the lower bound.A quintessential example of an ideal squeezed state isthe squeezed vacuum state, that is, the state that resultsfrom the action of the unitary squeezing operator ˆ S ( ξ ) =exp[( ξ ∗ ˆ b − ξ ˆ b † ) / ξ = r exp( iθ ) ( r, θ ∈ R ) on thephoton vacuum state | (cid:105) . For θ = 0, the variances satisfy V = (cid:104) ˆ x (cid:105) = exp( − r ) and V = (cid:104) ˆ p (cid:105) = exp(+2 r )and thus, d = 1 / d = 1 / R = 0 in Eq. (A9). If this is the case,only the term k = 0 will contribute to the summation inEq. (A10). For m = n the latter reduces to ( H m (0) /m !) .All of the odd-order Hermite polynomials are identical tozero at the origin ( x = 0). Consequently, (cid:104) n | ρ | n (cid:105) = 0 inEq. (A6) for odd n . Appendix C: Time-filtering of a coherent state
Let us assume that the steady-state of the cavity field isa coherent state. A coherent state is completely definedby the first moment of the bosonic field operator, i.e., (cid:104) ˆ c (cid:105) ss with higher-order moments factorizing in terms ofit. Following the input-output relation, we have for thefiltered output field moments: (cid:104) ˆ A f (cid:105) = √ γ (cid:90) ∞ d t f ( t ) (cid:104) ˆ c ( t ) (cid:105) ss , (C1) (cid:104) ˆ A † f ˆ A f (cid:105) = γ (cid:90) ∞ d t d t (cid:48) f ( t ) f ( t (cid:48) ) (cid:104) ˆ c ( t ) † ˆ c ( t (cid:48) ) (cid:105) ss , (C2)where for simplicity we are assuming the filter function f to be real. 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