www.nature.com/npjqi ARTICLE OPEN Squeezing and quantum state engineering with Josephson travelling wave ampliﬁers 1 1,2 Arne L. Grimsmo and Alexandre Blais We develop a quantum theory describing the input–output properties of Josephson traveling wave parametric ampliﬁers. This allows us to show how such a device can be used as a source of nonclassical radiation, and how dispersion engineering can be used to tailor gain proﬁles and squeezing spectra with attractive properties, ranging from genuinely broadband spectra to “squeezing combs” consisting of a number of discrete entangled quasimodes. The device’s output ﬁeld can furthermore be used to generate a multi-mode squeezed bath—a powerful resource for dissipative quantum state preparation. In particular, we show how it can be used to generate continuous variable cluster states that are universal for measurement based quantum computing. The favorable scaling properties of the preparation scheme makes this a promising path towards continuous variable quantum computing in the microwave regime. npj Quantum Information (2017) 3:20 ; doi:10.1038/s41534-017-0020-8 INTRODUCTION ampliﬁed signal and an “idler” signal in a two-mode squeezed 13, 18 state. This motivates an alternative viewpoint on the JTWPA: Microwave quantum optics is an emerging ﬁeld where artiﬁcial 1 2 besides using the device to amplify a signal of interest, it can also atoms, such as quantum dots, spin ensembles or superconduct- be viewed as a source of nonclassical radiation. The large ing quantum circuits are placed in engineered electromagnetic bandwidth and simple on-chip integration with coherent quan- environments. Strong light–matter interaction can be achieved by 3, 4 tum systems, such as superconducting qubits and microwave conﬁning the electromagnetic ﬁeld in microwave resonators or 5, 6 resonators, makes the JTWPA an intriguing new resource for one-dimensional waveguides. The ﬂexibility offered by micro- generating quantum radiation with potential applications in wave engineering allows experimentalists to go beyond the limits metrology and quantum information processing, amongst of conventional quantum optics in many ways. Examples include 19–26 others. reaching the so-called ultrastrong coupling regime of light–matter In this paper, we show how the inherent ﬂexibility in the interaction, and using nonlinear microwave resonators to bottom-up JTWPA construction allows us to tailor the quantum simulate relativistic quantum effects. properties of the output ﬁeld leaving the device. In particular, we A recent advancement in microwave quantum optics is the show how to shape the proﬁle of the broadband squeezing bottom-up design of nonlinear, one-dimensional metamaterials spectrum of the output ﬁeld, and how to cut holes in this with strong photon–photon interactions and engineered disper- spectrum such that some frequency ranges are unaffected by the 9–11 sion relations. The nonlinearity in these metamaterials is nonlinear interaction. This type of spectrum engineering is useful provided by Josephson junctions embedded in a transmission for applications of squeezing to quantum informaton processing line, with photon–photon interactions activated by a strong pump 27 tasks, for example to avoid unwanted quantum heating. tone through a parametric four-wave mixing process. These We subsequently demonstrate how the JTWPA can be used as a devices have been dubbed Josephson travelling wave parametric resource for dissipative quantum state preparation. Dissipative 10 (3) ampliﬁers (JTWPAs), and are analogous to one-dimensional χ quantum state preparation has over the last years emerged as an nonlinear crystals. alternative to preparation of entangled states using coherent 28 29 The development of JTWPAs is motivated by their potential use Hamiltonian or gate-based methods. It has been shown that as ampliﬁers for readout of solid-state qubits. The extremely high universal quantum computing can be achieved through dissipa- measurement ﬁdelity necessary for fault-tolerant quantum com- tive processes alone and, in a similar vein, that highly correlated puting requires ampliﬁers with added noise near the fundamental states, such as stabilizer states and projected entangled pair states 13, 14 30, 31 quantum limit. A key advantage to the JTWPA design is the can be created as stable steady states of dissipative processes. operational bandwidth, which is in the GHz range. This is in In this paper, we show that broadband squeezed radiation, such contrast to other near-quantum-limited microwave ampliﬁers as the radiation emitted by a JTWPA, is a particularly potent based on resonant cavity interactions, which typically have resource for dissipative quantum state preparation. The emitted 15–17 bandwidths of a few tens of MHz. An ampliﬁer operating radiation generates a multimode squeezed vacuum, which can be near the quantum limit is, however, very different from a classical used to drive quantum systems placed at the source’s output into ampliﬁer. In particular, quantum-limited phase preserving ampli- entangled states through correlated photon absorption and ﬁcation implies the presence of entanglement between the emission processes. We show that by engineering such a 1 2 Institut quantique and Départment de Physique, Université de Sherbrooke, Sherbrooke J1K 2R1 QC, Canada and Canadian Institute for Advanced Research, Toronto, ON, Canada Correspondence: Arne L. Grimsmo (arne.grimsmo@gmail.com) Received: 4 January 2017 Revised: 6 March 2017 Accepted: 24 March 2017 Published in partnership with The University of New South Wales Squeezing and quantum state engineering AL Grimsmo and A Blais squeezed bath, one can produce pairs of entangled qubits as well as continuous variable (CV) cluster states—the latter being universal resource states for measurement-based quantum computing. The preparation schemes are simple, requiring no Hamiltonian interactions or complicated reservoir engineering. By exploiting the large bandwidth of the JTWPA, the process can furthermore be implemented in a very hardware-efﬁcient manner. The purely dissipative nature of the preparation process distinguishes our proposal from similar approaches for generating 32–37 cluster states in the optical regime. A distinct advantage of a dissipative scheme is that it relaxes constraints on locality, which might allow for a more modular architecture that avoids spurious interactions and increases scalability. Although we focus on JTPWAs as squeezing sources in this work, due to their design ﬂexibility and large bandwidth, we emphasize that the dissipative quantum state preparation schemes we develop are relevant for any type of broadband squeezing source that can be integrated with coherent quantum systems, such as Fig. 1 Josephson traveling wave parametric ampliﬁer. A chain of 39, 40 other types of traveling wave ampliﬁers, impedance identical coupled Josephson junctions, with Josephson energy E engineered Josephson parametric ampliﬁers, squeezing sources and plasma frequency ω , are coupled in series. Each junction is based on reservoir engineering, or the nonclassical radiation furthermore coupled to ground by a passive, dissipationless 43, 44 emitted by an ac-biased tunnel junction. element described by an impedance Z(ω). By designing this impedance one can engineer the dispersion relation for waves traveling through the device. A strong right-moving pump actives a RESULTS four-wave mixing process through the Josephson potential, which can be used to generate squeezed light To describe the JTWPA’s squeezing properties, we ﬁrst need a quantized theory of its dynamics. Classical treatments of a JTWPA point energy. The label v ∈ {L,R} labels left- and right-moving are presented in refs 9, 11, 45. In the following, we give a modes, respectively. Hamiltonian treatment of the nonlinear dynamics, taking into For the nonlinear Hamiltonian, we systematically perform a account dispersion and the continuum nature of the electro- series of approximations that are ultimately analogous to those magnetic ﬁeld in the waveguide. used in the classical treatment given in refs 9, 11, 45. A quantized The device we consider in this paper is depicted in Fig. 1.It analog of the classical equation of motion found in previous work consists of a series of identical coupled Josephson junctions with is shown to be a limiting case of a more general theory. As Josephson energies E and junction plasma frequencies ω . Each J p detailed in Methods, for a classical right-moving monochromatic junction is coupled to ground by a passive, dissipationless pump at a frequency Ω , and neglecting terms that are smaller element with impedance Z(ω), which is left arbitrary for now. By than second order in the pump, we can write the non-linear engineering Z(ω) one can modify the dispersion relation of waves Hamiltonian in terms of three distinct contributions propagating through the device as shown in ref. 9. We show ^ ^ ^ below how this can be used to tailor the squeezing properties of H ¼ H þ H þ H ; ð2Þ 1 CPM SQ SPM the output ﬁeld leaving the device. Note that other variants of the where H describes cross-phase modulation due to the pump, CPM JTWPA device where the Josephson junctions are replaced by H describes broadband squeezing, and H is a classical 46, 47 SQ SPM SQUIDs have recently been discussed. We do not consider Hamiltonian describing self-phase modulation of the pump. such modiﬁcations here, but the general approach we develop Explicit expressions for these three Hamiltonians in terms of the below can be used to formulate a quantum theory also in these frequency modes a ^ are given in Methods. νω cases. As shown in Methods, we ﬁnd in a scattering limit where the In experimental realizations, JTWPAs have several thousand initial and ﬁnal times of the problem are taken to minus and plus junctions with a unit cell distance much smaller than the relevant inﬁnity, respectively; the following is the expression for the 10, 11 wavelengths. One can therefore approximate the device with asymptotic Heisenberg picture output ﬁeld a continuum description (formally taking the unit cell distance, a, i 2jj β k þΔkðÞ ω =2 z to zero). We furthermore assume that the JTWPA is coupled to out ½ ω † a ^ ¼ e uðÞ ω; z a ^ þ ivðÞ ω; z a ^ ; ð3Þ Rω Rω R 2Ω ω ðÞ p identical, semi-inﬁnite and impedance matched transmission lines to the left and the right, and we neglect any reﬂection of the ﬁeld where the functions u(ω, z) and v(ω, z), deﬁned in Eqs. (28) and 2 2 at the interfaces between the different sections. (29), satisfyjj uðÞ ω; z jj vðÞ ω; z ¼ 1, and As shown in detail in Supplemental Methods 1, a continuum ^ ^ ^ ð4Þ ΔkðÞ ω ¼ 2k k k þ 2jj β k k k ; limit Hamiltonian for the system can be written H ¼ H þ H , p ω 2Ω ω p 2Ω ω ω 0 1 p p where H is a linear contribution containing all terms up to second is the phase mismatch, including a nonlinear correction due to to order in the ﬁelds, and H is a nonlinear contribution due to the the cross- and self-phase modulation of the pump. Here, k is the Josephson junction potential. The linear Hamiltonian can be wave-vector, which due to dispersion inside the JTWPA section diagonalized in terms of a set of frequency modes, following an has a non-linear dependence on ω: approach introduced by Santos and Loudon in ref. 48, leading to sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ the form iωz ðÞ ω l ð5Þ k ¼ ; 2 2 1 ω =ω ^ ^ H ¼ dω hωa a ; where l is the inductance per unit length of the JTWPA section, 0 νω ð1Þ νω −1 −1 ν¼L;R ω is the junction plasma frequency and z (ω)= Z (ω)/a is the admittance to ground per unit cell. The parameter β = I /4I in Eq. P c † 0 where ½a ^ ; a ^ ¼ δ δωðÞ ω , and we have omitted the zero- (4) is the dimensionless amplitude of the classical pump, νω νμ μω npj Quantum Information (2017) 20 Published in partnership with The University of New South Wales Squeezing and quantum state engineering AL Grimsmo and A Blais Fig. 2 Engineered bandgaps. Illustration of the disperion relation when Z(ω) (illustrated in the inset) describes a single resonant mode at a frequency ω linearly coupled to the ﬂux ﬁeld in every unit cell. A bandgap opens up around the resonance frequency, close to 6 GHz in this example. The width of the bandgap is set by the coupling capacitance C shown in the inset expressed in units of the Josephson junction critical current I (see Methods for further details). As Eq. (5) clearly shows, the dispersion relation, and consequently the phase mismatch, can be tuned by engineering the admittance to ground in the JTWPA unit cells. In particular, if the impedance Z(ω) describes a resonant mode, a bandgap will open close to the resonance frequency. The behavior of the dispersion relation close to a bandgap is illustrated in Fig. 2. Note that if dispersion is neglected, Δk(ω) = 0, Eq. (3) reduces to the standard input–output relation for a lossless parametric ampliﬁer (see, e.g., ref. 49). Engineering nonclassical radiation Fig. 3 Gain proﬁle and squeezing spectra. Output ﬁeld properties of The quantum input–output theory developed above allows us a JTWPA with 2000 unit cells and parameters given in the text. a The predict features of the JTPWA’s output ﬁeld, such as the device’s green lines are for a device without RPM. The blue lines are for a gain proﬁle and output ﬁeld squeezing spectrum. In this section, device with identical parameters, but where RPM has been used to we show how output spectra can be tailored through dispersion tune Δk Ω ’ 0. The orange lines show a device where in addition engineering. We focus ﬁrst on an ideal, quantum-limited device to RPM, a second resonance has been placed at 9 GHz punching two and discuss the effect of loss below. symmetric holes in the gain and squeezing spectra. b A JTWPA with From Eq. (3), the JTWPA’s amplitude gain is given by u(ω, z), and 19 additional resonances used to generate a “squeezing comb.” The we deﬁne the power gain as GðÞ ω; z ¼jj uðÞ ω; z refs 9, 13, 45. This choice of impedance to ground for each simulated device is illustrated below with color codes corresponding to the plots function grows exponentially with z for small phase mismatch, ΔkðÞ ω ’ 0, but is only of order one if the phase mismatch is large (see ref. 9 and Methods). The squeezing of the JTWPA’s output along the JTWPA transmission line, a technique referred to as ﬁeld is manifested in correlations between frequencies ω and resonant phase matching (RPM). 2Ω − ω, symmetric around the pump frequency. We deﬁne the The effect of RPM on the gain and squeezing spectra is squeezing spectrum of the device as illustrated in Fig. 3 for a simulated device similar to what has been realized experimentally in refs 10, 11: The device length was chosen to be 2000 unit cells with characteristic impedance Z = 0 θ θ ^ ^ SðÞ ω; z ¼ dω ΔY ΔY Rω Rω 50Ω, critical current I =(2π/Φ )E = 2.75 μA, dimensionless pump ð6Þ c 0 J strength β = 0.125 and pump frequency Ω /2π = 5.97 GHz. The ratio of the pump frequency to the junction plasma frequency was ¼ 2NðÞ ω; z þ 1 2jj MðÞ ω; z ; R R −2 Ω /ω = 8.2 × 10 . The green lines in Fig. 3a show the gain proﬁle p p θ iθ=2 out† iθ=2 out ^ ^ and squeezing spectrum of the output ﬁeld for the device without where we have deﬁned quadratures Y ¼ ie a e a , Rω Rω Rω ^ ^ ^ RPM, while the blue lines show results for an identical device with ﬂuctuations ΔY ¼ Y Y and θ the squeezing angle. Rω Rω Rω The parameters N (ω, z) and M (ω, z) introduced on the right hand where RPM has been used to tune Δk(Ω ) = 0. The circuit R R side of Eq. (6), are deﬁned in Eqs. (31) and (32) and can be parameters for the LC resonator are C = 10 fF, C = 7.0 pF, L = c r r 100 pH, giving a resonance frequency of ω /2π = 6.0 GHz. interpreted as the thermal photon number and a squeezing r0 Two-mode squeezing has applications for entanglement gen- parameter for the right-moving ﬁeld, respectively. 19, 20 21 22 eration, quantum teleportation, interferometry, creation The gain and the squeezing at the output depends strongly on the phase mismatch Δk(ω). The phase mismatch can, however, be of quantum mechanics free subsystems, high-ﬁdelity qubit 24, 25 26 compensated for by tuning Z(Ω ), as this allows for tuning the readout and logical operations, amongst others. A broad- pump wavevector k ¼ k according to Eq. (5). As was proposed band squeezing source such as the JTWPA has a great advantage p Ω for scalability, as tasks can be parallelized with many pairs of far- theoretically in ref. 9 and demonstrated experimentally in refs. 10, 11, it is possible to tune the phase mismatch to zero at the pump separated two-mode squeezed frequencies using a single device. frequency, Δk Ω ’ 0, and greatly reduce it across the whole It is, however, not necessarily desirable to have squeezing at all JTWPA bandwidth. This is done by placing LC (or transmission frequencies over the operational bandwidth as this might lead, 25, 27 line) resonators with resonance frequency ω ’ Ω regularly e.g., to unwanted quantum heating. r0 p Published in partnership with The University of New South Wales npj Quantum Information (2017) 20 Squeezing and quantum state engineering AL Grimsmo and A Blais Fig. 5 Squeezing in the presence of loss. Squeezing as a function of gain, GðÞ ω; z ¼ ηjj uðÞ ω; z , in the presence of loss, modeled as a beam splitter with transmittance η placed at the JTWPA output. The solid lines show the maximally squeezed quadrature for three different values of η, while the dashed lines show the corresponding Fig. 4 Engineering ﬂat spectra. A device similar to those in Fig. 3, anti-squeezed quadrature but where RPM has been used to tune Δk(ω) = 0 for ω=ðÞ 2π ’ 1:8 GHz. The larger phase mismatch around ω ’ Ω , shown in the right photon number, NðÞ ω; z !jj ηωðÞ NðÞ ω; z , and squeezing para- panel, gives a ﬂatter proﬁle for both the gain and squeezing spectra R R pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ meter, MðÞ ω; z ! ηωðÞ ηð2Ω ωÞMðÞ ω; z . Taking η = η(ω) R p R frequency independent for simplicity, this gives a reduction in Building on the RPM technique, we consider placing additional squeezing, SðÞ ω; z ! 2jj η NðÞ ω; z þ 1 2jj ηjj MðÞ ω; z . Note that R R R resonances in each unit cell with resonance frequencies ω away rk distributed loss throughout the JTWPA can be taken into account from Ω . This leads to a bandgap and a divergence in k(ω) close to through a simple phenomenological model, but this is beyond each resonance ω , as illustrated in Fig. 2. The huge phase rk the scope of the present discussion. mismatch close to these resonances prohibits any parametric Figure 5 shows the maximum squeezing level as a function of interaction at ω ’ ω and ω ’ 2Ω ω , effectively punching rk p rk gain as the pump strength is ramped up. The parameters are two symmetric holes in the gain and squeezing spectra. This is otherwise identical to those used for the blue lines displayed in illustrated by the orange lines in Fig. 3a, where a single additional Fig. 3a. The solid lines show the maximally squeezed quadrature, resonace has been placed at ω = 9.0 × 2π GHz. The parameters r1 while the dashed lines show the corresponding anti-squeezed are otherwise as before, except that the second LC resonator is quadrature, for three different values η = 0.75 (yellow), 0.99 (dark chosen to have twice the coupling capacitance, 2C . This choice red) and 1.00 (blue). Note that the gain is also reduced by the loss, serves to illustrate how the width of the hole in the spectrum is GðÞ ω; z ¼ ηjj uðÞ ω; z , such that we have attenuation at zero pump determined by the coupling capacitance to the resonator, as is power. clearly seen when comparing the holes at ω and ω . r0 r1 For a non-unity η, the squeezing level saturates with gain, while In Fig. 3b, we demonstrate how this technique can be used to the anti-squeezed quadrature keeps growing proportionally. The engineer a “squeezing comb” where there is considerable gain maximal squeezing depends sensitively on η:while aquantum- and squeezing only for a discrete set of narrow quasimodes. With limited device with η = 1 would produce more than 25 dB of a larger number of closely spaced resonance frequencies—either squeezing at 20 dB of gain, a device with η = 0.75 only gives about using individual lumped LC circuits or the resonances of a multi- 6.5 dB of squeezing for the same gain. For a realistic device, further mode transmission line resonator—it is possible to have phase reductions in squeezing might arise due to disorder, the distributed matching only over narrow frequency bandwiths. In Fig. 3b, we show the gain proﬁle and squeezing spectrum where 19 nature of loss throughout the device, and the frequency dependence of the attenuation leading to asymmetry between additional resonances at ω = ω + k × ω /20, k =1, 2,…,19 has rk r0 r0 the signal and idler [Kamal, A. Private communication (2016)]. been used to create a squeezing comb with 38 quasimodes. Slightly different parameters were chosen for this device, to get similar gain and squeezing proﬁles as before: Z =14Ω, I = 2.75 μA, 0 0 Probing the output β = 0.069, while the additional coupling capacitances were chosen The examples discussed above demonstrate how the ﬂexible to be 3.0C . Note that it is not necessary to place the LC resonators JTWPA design allows for generating nonclassical light with in every unit cell in an experiment. In practice, RPM has been interesting and useful squeezing spectra. realized by repeatedly placing identical LCs every few unit cells. The squeezing spectrum can be found experimentally by For certain applications, it might also be of interest to have a measuring the variance of ﬁltered two-mode quadratures (see squeezing spectrum with a ﬂatter proﬁle than what is shown in Supplemental Methods 1 and, e.g., refs. 43, 52–54). However, this Fig. 3. This can be achieved by suitably engineering the phase necessarily includes insertion loss and noise from subsequent mismatch. In Fig. 4, we show a device where RPM has been used 10 parts of the ampliﬁcation chain, which can make the detection to tune Δk(ω) = 0 for ω=2π ’ 1:8 GHz, with the pump frequency of two-mode squeezing challenging. For a more direct probing of close to the resonance frequency at ω /2π = 6 GHz. The simulated r0 the JTWPA’s performance, we propose placing two superconduct- device otherwise has parameters Z =60Ω, I = 1.75 μA, β = 0.113. 0 0 ing qubits capacitively coupled directly to the transmission line at This choice of dispersion engineering leads to larger phase the output port. mismatch in the center region of the spectrum, close to the pump, For two off-resonant qubits with respective frequencies ω ≠ ω , 1 2 giving the ﬂatter proﬁle shown in the ﬁgure. and ω + ω ≄ 2Ω , the qubits will be in uncorrelated thermally 1 2 p populated states. If, however, ω + ω =2Ω , the qubits become 1 2 p Reduction in squeezing due to loss entangled and information about the JTWPA’s squeezing spec- Internal loss in the JTWPA, as well as insertion loss, is likely to be a trum is encoded in the joint two-qubit density matrix. This source of reduction in squeezing from the ideal results shown in information can then be extracted by measuring qubit–qubit Fig. 3. A simpliﬁed loss model is a beam splitter with transmittance correlation functions. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ηωðÞ placed after the JTPWA, with vacuum noise incident on the Assuming for simplicity a single right-moving pump and a left- beam splitter’s second input port. This leads to a reduction in moving ﬁeld in the vacuum state, we have that for ω + ω ≄ 2Ω , 1 2 p npj Quantum Information (2017) 20 Published in partnership with The University of New South Wales Squeezing and quantum state engineering AL Grimsmo and A Blais Fig. 6 Modes of operation for a JTPWA. a Ampliﬁcation mode: quantum systems (here depicted as two-level systems to illustrate) are placed at the device input. b Probing mode: quantum systems placed at the output absorbs correlated photons from the JTWPA’s output ﬁeld and become entangled. c Reﬂection mode: higher degrees of entanglement can be reached by avoiding the left-moving vacuum noise. A circulator can be added to avoid back scattering into the JTWPA the steady state of the two qubits is the product state ρ = ρ ⊗ρ , 1 2 where ρ is a thermal state with thermal population N (ω )/2 and m R m ðÞ m inversion hσ i¼1=ðN ðω Þþ 1Þ. On the other hand, for ω + z R m 1 ω =2Ω the qubits become entangled. Under the simplifying 2 p symmetric assumptions N (ω ) ≡ N and γ ≡ γ we ﬁnd that R m m DE DE Re½ M ðÞ 1 ðÞ 2 ðÞ 1 ðÞ 2 ^ ^ ^ ^ hi σ σ ¼ σ σ ¼ ; x x y y ð7Þ 2 2 ðÞ N þ 1ðÞ N þ 1 jj M and DE Im½ M ðÞ 1 ðÞ 2 hi σ ^ σ ^ ¼ ; x y ð8Þ 2 2 Fig. 7 Generating entanglement with squeezing. Concurrence of ðÞ N þ 1ðÞ N þ 1 jj M two qubits in a two-mode squeezed bath as a function of the gain of the squeezing source, GðÞ ω; z ¼ ηjj uðÞ ω; z , for three different source in steady state, where M(ω ) ≡ M. More general expressions are loss levels η = 0.75, 0.99, 1.00. No thermal noise at the squeezing given in Supplementary Method 2. Hence, by measuring source input is assumed. The inset shows the behavior of the qubit–qubit correlation functions and single-qubit inversion using spectral gap of the Lindbladian 55–57 standard qubit readout protocols, one can map out the squeezing spectrum of the source. θ entanglement entropy E Ψ ¼ tr½ ρ log ðÞ ρ ’ 1 1=4N , 1 2 1 We can also turn this around and, rather than view the two θ θ where ρ ¼ tr Ψ Ψ . qubits as a probe of the JTWPA’s performance, view the JTWPA as Of practical importance is the steady state entanglement’s a source of entanglement for the qubits. To achieve maximal dependence on the degree of loss, and the behavior of the degree of entanglement between the qubits, it is desirable to spectral gap of the Lindbladian in Eq. 34. The latter is important avoid the vacuum noise of the left-moving ﬁeld. This can be because it sets the time-scale for approaching the steady state. It achieved by squeezing the left-moving ﬁeld with a separate is deﬁned as ΔðÞ L ¼jj Reλ , where λ is the non-zero right- 1 1 JTWPA section, or more simply by operating the device in eigenvalue of L with real part closest to zero. In Fig. 7, we plot reﬂection mode, as illustrated in Fig. 6c. the steady state entanglement, quantiﬁed by the concurrence, Assuming ideal conditions where the qubits couple symme- and the spectral gap as a function of gain for different values of η trically to equally squeezed left-moving ﬁelds and right-moving (as deﬁned above). These results show that the achievable ﬁelds, N (ω)= N (ω ) ≡ N/2, M (ω)= M (ω ) ≡ M/2, and an ideal L i R i L i R i entanglement is very sensitive to loss, but an upshot is that lossless squeezing source, the steady state of the two qubits is the relatively modest gains are needed to achieve high degree of pure state (see Supplementary Method 2 for more information) entanglement. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ θ iθ Ψ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N þ 1ji gg þ e Nji ee ; ð9Þ 2N þ 1 CV cluster states where θ is the squeezing angle. For large N, this pure The two-qubit dynamics considered above demonstrates the state approaches a maximally entangled state with JTWPA’s potential for entanglement generation. To go beyond bi- Published in partnership with The University of New South Wales npj Quantum Information (2017) 20 Squeezing and quantum state engineering AL Grimsmo and A Blais partite entanglement, one can add multiple pump tones, such modes from distinct OPOs. We adopt this scheme in the following, that a single frequency can become entangled with multiple other using JTWPAs (other types of broadband squeezing sources can “idler” frequencies in a multi-mode squeezed state, resulting in also be used) in place of OPOs. The main difference between our 32, 33 potentially complex patterns of entanglement. Together with its proposal and that of ref 35 and previous proposals is that our broadband nature and the potential for dispersion engineering, scheme is purely dissipative: the CV modes of the cluster state this turns the JTWPA into a powerful resource for dissipative never interact directly, but rather become entangled through quantum state preparation, as we demonstrate in the following. absorption and stimulated emission of correlated photons from To exemplify the potential of broadband squeezing as a their environment. We focus primarily on a situation where the resource for quantum computing and state preparation, we show modes are embodied in multimode resonators, which is a below how CV cluster states can be generated through a particularly hardware efﬁcient implementation. We emphasize, dissipative and deterministic process, using the output ﬁeld of however, that due to the dissipative nature of the scheme, this is multiple broadband squeezing sources. The cluster states are a not a necessary constraint. The modes could in principle all be powerful class of entangled many-body quantum states that are embodied in physically distinct and remote resonators, removing resource states for measurement-based quantum computing. any constraints on locality. This is an attractive advantage of such Given a cluster state, an algorithm is executed using only single- a dissipative scheme. 59–63 site measurements and classical feed forward on the state. Following ref. 35, the modes of the cluster states are resonator A CV cluster state is deﬁned with respect to a (weighted) simple modes with equally spaced frequencies ω ¼ ω þ mΔ, where m m 0 graph G =(V,E), with V the set of vertices and E the set of edges. A is an integer, ω is some frequency offset and Δ the frequency pﬃﬃﬃ CV quantum systems with quadratures ^ x ¼ð^c þ ^c Þ= 2 and separation. We require a number of degenerate modes for each v v pﬃﬃﬃ v † † ^ y ¼ið^c ^c Þ= 2, where ^c ð^c Þ is a bosonic annihilation frequency ω : to create a D-dimensional cluster state requires a v v v m v v (creation) operator, is associated to each vertex v. The ideal CV 2× D-fold degeneracy per frequency. This can be achieved, e.g., by cluster state (with respect to G)isdeﬁned as the unique stateji ϕ using 2 × D identical multi-mode resonators, as illustrated for D =1 61, 63, 64 satisfying in Fig. 8. Each resonator mode will be a vertex in the cluster state 0 1 graph, and as will become clear below, a set of degenerate modes can be thought of as a graph “macronode”. It is convenient to @ A ^ ^ y a x ji ϕ ¼ 0 8v 2 V; ð10Þ m vw w v G relabel the frequencies with a “macronode index” M¼ðÞ 1 m. w2NðÞ v We show in Supplementary Method 3 that a master equation with Lindbladian of the form Eq. (11) is realized for a single where N (v) is the neighborhood of v, i.e., all the vertices resonator interacting with a bath generated by the output ﬁeld of connected to v by an edge in E and a = a ∈[−1, 1] is the vω ωv a JTWPA with a single pump frequency Ω = ω + pΔ/2, where p = weight of the edge {v, w}. Note thatji ϕ is an inﬁnitely squeezed p 0 m + n for some choice of frequencies ω ≠ ω . The graph is in this state, and thus not physical. In practice, one has to work with m n case a trivial graph consisting of a set of disjoint pairs of vertices Gaussian states that approaches ji ϕ in a limit of inﬁnite connected by an edge, i.e., a set of two-mode cluster states, which squeezing. We can deﬁne an adjacency matrix [a ] for the graph, vw where a = 0 if there is no edge {v, w} ∈ E. Since the adjacency vw can be represented as G = matrix uniquely deﬁnes the graph, and vice versa, we use the … The edges have weight +1, under the assumption of a quantum symbol G to interchangeably refer to both the graph and its pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ adjacency matrix in the following. limited, ﬂat squeezing spectrum MðÞ ω ¼ iM ¼ i NNðÞ þ 1 with N We focus here on a class of graphs, ﬁrst studied in refs 32, 33, (ω)= N over the relevant bandwidth. satisfying two simplifying criteria: (1) The graph is bicolorable. This More complex and useful graphs can be constructed using means that every vertex can be given one out of two colors, in these two-mode cluster states as basic building blocks. Taking such a way that every edge connects vertices of different colors a number of JTWPAs, each labeled by i and acting as a (the square lattice is an example). (2) The graph’s adjacency matrix −1 squeezing source independently generating a disjoint graph is self-inverse, G = G . The latter constraint has a simple geometric G = … as above, universal interpretation described in ref. 33. We show in Supplementary i cluster states can be created by combining the output ﬁelds of the Method 3 that for a graph G satisfying these critera, the Lindblad different sources on an interferometer. The action of the equation ρ_ ¼L ρ, with Lindbladian interferometer can be written as a graph transformation, where ðMÞ G ¼ G transforms to G ! RGR , where R ¼ R represents i i M P D ^ ^ L ¼ κðÞ N þ 1D½ c þ κND c G v an interferometer acting independently on each macronode M, v2V ð11Þ i.e., each set of 2 × D-fold degenerate modes. R has to be −1 orthogonal for the transformed graph to be self-inverse (G = G ), † † ^ ^ κa S c ; c ; vw iM v w which we recall is one of the criteria for Eq. (11) to generate the fg v;w2E corresponding cluster state. As shown in ref. 35 this is the case if pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where M ¼ NNðÞ þ 1 and S [A, B]isdeﬁned in Eq. (35), has a iM the 2D ×2D matrix R is a Hadamard transformation R ¼ H D D unique steady stateji ϕ ðÞ M that approachesji ϕ as M !1. The built from 2 × 2 Hadamard matrices G G 0 1 existence of graphs satisfying all the listed criteria, with associated cluster states,ji ϕ , that are universal for quantum computing is ð12Þ G B C H ¼ pﬃﬃﬃ : @ A shown in refs 32, 33. Equation (11) is a remarkable result, because it implies that CV cluster states can be prepared simply by placing 1 1 oscillators in a multi-mode squeezed bath, i.e., broadband squeezing is the only necessary resource for the preparation. In Physically such a transformation can be realized by pairwise the following, we detail how multi-mode squeezed baths with an interfering the output ﬁelds of the JTWPAs on 50-50 beam splitters entanglement structure giving rise to universal cluster states can with beam splitter matrix as in Eq. (12). The network of beam be engineered adapting a simple scheme from. splitters needed for the case D = 1 is illustrated in Fig. 8, for D =2 In ref. 35, Wang and coworkers showed how cluster states with in Fig. 9, and for higher dimensions in ref. 35. graphs of the type considered here could be generated through In ref. 35, it was shown that graphs G constructed in this way Hamiltonian interactions between the modes of optical parametric can give rise to D-dimensional cluster states that are universal for oscillators (OPOs), followed by an interferometer combining measurement-based quantum computing. Let us consider an npj Quantum Information (2017) 20 Published in partnership with The University of New South Wales Squeezing and quantum state engineering AL Grimsmo and A Blais Fig. 8 Dissipative generation of a linear cluster state. a Two JTWPAs are used as squeezing sources. The output ﬁelds of the two devices are combined on a 50-50 beam splitter enacting a Hadamard transformation, before impinging on two identical multi-mode resonators. b Each JTWPA is pumped by a single pump tone, generating entanglement (curved arrows) between pairs of frequencies satisfying ω þ ω ¼ 2Ω .We n m i focus here on center frequencies corresponding to the frequencies of the resonator modes, illustrated by the pink and blue arrows. The numbers show the macronode index of each frequency. c Linear graph deﬁning the steady state cluster state of the resonator modes. The horizontal edges are generated by the two pumps, while the diagonal edges are generated by the Hadamard transformation (see Supplementary Method 3 for details). The numbers show the macronode index, and the circle shows macronode M¼ 2 in the graph example with D = 1 in some more detail to illustrate the basic principles. First, take two JTWPAs pumped individually with respective pump frequencies Ω and Ω , with i ¼j ¼ ΔM.On i j the macronode level, this gives exactly one edge between macronodes separated byjj ΔM , as illustrated by the horizontal edges in Fig. 8 for ΔM¼ 1. By interfering the output ﬁelds of the two JTWPAs on a beam splitter deﬁned by Eq. (12), every node in each macronode becomes entangled with every node in the neighboring macronode, as illustrated by the diagonal edges in Fig. 9 Schematic setup for a universal microwave quantum the ﬁgure. This gives a graph G with a linear structure, computer. Four JTWPAs are used as squeezing sources to corresponding to a one-dimensional cluster state that is universal dissipatively prepare the modes of four identical multi-mode 33–35 for single-mode quantum computation. resonators in a two-dimensional cluster state. The quantum The scheme can straight-forwardly be scaled up to arbitrary D- computation is subsequently performed through Gaussian and dimensional cluster states using 2 × D squeezing sources and the non-Gaussian (e.g., photon-number resolving ) single-mode mea- same number of beam splitter transformations as shown in ref. 35. surements on the resonators. D = 2 is sufﬁcient for universal quantum computation; a possible setup of JTWPAs and resonators is illustrated in Fig. 9.As emphasized in ref. 35, the relative ease of creating even higher Moreover, a JTWPA can be thought of as an artiﬁcial non-linear dimensional cluster states is a very attractive property of the crystal, introducing a new non-linear element to the ﬁeld of scheme. microwave quantum optics. The JTWPA is qualitatively different from previous, essentially point-like, non-linear elements used in microwave circuits and represents, in this respect, a major DISCUSSION departure from the conventional paradigm of “circuit quantum We have shown how the recently developed JTWPAs are powerful electrodynamics” based on localized electromagnetic modes. sources of nonclassical radiation. The design ﬂexibility and broad The possibility of dispersion engineering together with control- bandwidth allows us to tailor the properties of the quantum lable non-linear parametric interactions opens new possibilities for radiation emitted by the device through dispersion engineering. quantum optics in the microwave regime, for example studying In this way, the output ﬁeld can be optimized for speciﬁc the interplay between light and matter in structured non-linear applications where broadband squeezing is useful. Furthermore, 65, 66 media. we have illustrated how the output ﬁeld of one or multiple JTWPAs can be used for reservoir engineering: By placing quantum systems at the output of a broadband squeezing source, METHODS the systems can be driven into non-trivial entangled states through correlated photon absorption and emission processes. Asymptotic input–output theory We have shown both how to prepare pairs of entangled qubits, as As shown in Supplementary Methods 1, the position-dependent ﬂux, ϕðÞ x well as CV cluster states that are universal for quantum computing (in the Schrödinger picture), along a transmission line with a JTWPA in this manner. section extending from x =0 to x = z can in the continuum limit be Published in partnership with The University of New South Wales npj Quantum Information (2017) 20 Squeezing and quantum state engineering AL Grimsmo and A Blais expanded in terms of a set of left-moving modes and right-moving modes, dropped fast rotating terms and the highly phase mismatched left moving sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬁeld. h For notational convenience, we have deﬁned the phase matching ϕðÞ x ¼ dω g ðÞ x a ^ þ H:c:; 72, 73 νω νω ð13Þ function 2cxðÞω ν¼L;R hi z † 0 where a ^ ; a ^ ¼ δ δωðÞ ω and the mode functions are given by ik½ ω ðÞ x kω ðÞ x þkω ðÞ x kω ðÞ x x νω 0 νμ μω 1 2 3 4 ΦðÞ ω ; ω ; ω ; ω¼ dxe ; ð19Þ 1 2 3 4 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ± ik ðxÞx g ðÞ x ¼ e : ð14Þ νω and a dimensionless pump amplitude, βðÞ Ω , which as shown in 2πη ðÞ x vxðÞ Supplementary Methods 1, can be written in terms of the ratio of the Here, + (−) corresponds to ν ¼ R(ν ¼ L), k ðÞ x ¼ η ðÞ x ω=vxðÞ is the ω pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pump current to the Josephson junction critical current, wavevector, with η ðÞ x the refractive index, and vxðÞ¼ 1= cxðÞlxðÞ. The I ðÞ Ω x-dependent parameters are deﬁned such that they take one (constant) ð20Þ βðÞ Ω ¼ ; 4I value inside the JTWPA section, and another value outside this section. We c have deﬁned c(x), the capacitance to ground per unit cell, and l(x), the where I ¼ðÞ 2π=Φ E . c 0 J linear inductance of the transmission line. We emphasize that this simple It should be noted that H can also in principle include frequency form of the mode functions assumes that we can neglect reﬂection at the conversion processes where a photon with frequency close to 2Ω þ ω is interfaces between the JTWPA and the linear transmission line sections. created by absorbing a single photon at frequency ω and two pump The only difference from the usual prescription for the quantized ﬂux in photons at Ω . We have left out such contributions in Eq. (2) under the 67, 68 a linear, homogeneous and dispersion free transmission line is the x- assumption that appropriate steps have been taken to ensure that these dependent wavevector, which now takes a different form inside and frequency conversion processes are heavily phase mismatched. In outside the nonlinear section. Explicitly, the dispersion relation is found to particular, current experiments use low-pass ﬁlters before and after entry be (see Supplementary Methods 1 and ref. 9) to the JTWPA section of the transmission line, which ensures that no plane- wave solutions exist above 2Ω [Gustavsson, S. & Krantz, P. Private >qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 communication (2016)]. > iωz ðÞ ω lxðÞ for 0<x<z 2 ^ 1ω =ω The dynamics according to the non-linear Hamiltonian H is in general P 1 k ðÞ x ¼ ð15Þ 72, 73 difﬁcult to treat analytically to all orders, and we therefore take a : perturbative approach treating the non-linearity to ﬁrst order. An otherwise; vxðÞ input–output relation linking the ﬁeld entering the JTWPA to the emitted 71–75 1 1 where z ðÞ ω ¼ Z ðÞ ω =a is the admittance to ground per unit cell in the output ﬁeld can then be derived in the usual asymptotic scattering limit. JTWPA section. Equations of motion similar to those we derive here have been used previously by Caves and Crouch in a study of wideband traveling wave Implicit in the continuum description is that we are considering ampliﬁer, where they were taken as operator versions of macroscopic sufﬁciently low frequencies, such that the wavelengths are large compared Maxwell’s equations for a nonlinear, homogeneous and dispersionless to the unit cell distance, a. Furthermore, plane wave solutions only exists medium. We here justify similar input–output relations by deriving them when the right hand side of Eq. (15) is real. In practice, we are interested in 2 2 from a microscopic theory, taking into account the ﬁnite extent of the frequencies ω ω such that the dispersion due to the plasma nonlinearity as well as dispersion. For the JTWPA the latter stems from oscillations of the junctions is relatively small. If, however, the admittance both junction plasma oscillations, non-linear phase-modulation and z ðÞ ω describes a linear element with a resonant mode, a bandgap opens engineered bandgaps in the medium. around the resonance frequency for which no plane wave solutions exists. Treating H as a perturbation, it is natural to go to an interaction picture Physically, such a resonant mode behaves as a “matter ﬁeld” in the with respect to H . The time-evolution operator for the problem in this continuum limit, and the excitations of the systems resemble light-matter 69–71 picture is polaritons. As long as we are away from any bandgap, however, these “matter ﬁelds” slave the photonic ﬁeld, ϕðÞ x; t , and only modiﬁes the R1 dtH ðÞ t dielectric properties of the medium, manifest in the dispersion relation Eq. 1 ð21Þ ^ t Ut ð ; t Þ¼T e ; (15). 0 1 We assume the presence of a strong right-moving classical pump ^ ^ ^ ^ where H ðÞ t ¼ expðÞ i=h H t H exp ðÞ i= h H t and T is the time-ordering 1 0 1 0 centered at a frequency Ω with corresponding wavevector k , and replace P p operator. a ^ ! a ^ þ bðÞ ω , with b(ω) a complex valued function centered at Ω . Rω Rω P Solving the time-dynamics according to Eq. (21) is difﬁcult in general. The ﬁelds, a , are assumed to be sufﬁciently weak so that we can drop in νω However, if we neglect any backaction from the ﬁelds onto the pump, i.e., H terms that are smaller than second order in the pump. As shown in take the undepleted pump approximation and disregard any quantum Supplementary Methods 1, a Hamiltonian for the system can then be ﬂuctuations around the pump frequency, we can solve for the pump ^ ^ ^ ^ ^ found of the form H ¼ H þ H where H and H are given in Eqs. (1) and 0 1 0 1 separately according to the classical Hamiltonian H and substitute this SPM (2), respectively, with back into the remaining parts of H . The remainder is then effectively a quadratic Hamiltonian for the quantum ﬁelds. We will in the following also pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ make several further simpliﬁcations that greatly reduces the complexity of h 0 0 0 H ¼ dωdω dΩdΩ k k β ðÞ Ω βðÞ Ω CPM ω ω ^ 2π the problem. First of all, we treat H as a perturbation to ﬁrst order only, in ð16Þ which case the time-ordering in Eq. (21) can be dropped. See, however, 0 † refs 72, 73 for a discussion on the breakdown of this approximation. ^ ^ 0 ´ ΦðÞ ω; ω ; Ω; Ω a a þ H:c:; Rω Rω Subsequently we take a “scattering limit” and let the initial and ﬁnal times describing cross-phase modulation, to t ¼1 and t ¼1, respectively. The time-integral then gives rise to 0 1 delta-functions in frequency space, and we are left with approximate 1 asymptotic evolution operator, or scattering matrix, R pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h 0 0 0 H ¼ dωdω dΩdΩ k k 0βðÞ Ω βðÞ Ω SQ ω ω 4π i ^ ð17Þ K1 ^ ^ h ð22Þ 0 U UðÞ 1; 1¼e ; 0 0 † † where ´ ΦðÞ ω; Ω; ω ; Ω a ^ a ^ þ H:c:; Rω Rω ^ ^ ^ ð23Þ K ¼K þ K þ K : describing broadband squeezing, and 1 CPM SQ SPM Explicit expressions for K for a general classical pump can be found in R pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Supplementary Methods 1. We from now on make one last simplifying h 0 0 0 H ¼ dωdω dΩdΩ k k 0β ðÞ Ω βðÞ Ω SPM ω ω 4π approximation and focus on the monochromatic pump limit, taking ð18Þ bðÞ ω ! bðÞ ω δω Ω . In this limit we have 0 0 0 ´ ΦðÞ ω; ω ; Ω; Ω b ðÞ ω bðÞ ω þ H:c:; K ¼2 hz dωβjj k a ^ a ^ ; ð24Þ CPM ω Rω Rω describing self-phase modulation of the pump. Here we have furthermore 0 npj Quantum Information (2017) 20 Published in partnership with The University of New South Wales Squeezing and quantum state engineering AL Grimsmo and A Blais iθ and angle, which is given through MðÞ ω; z ¼jj MðÞ ω; z e . We emphasize that R R Eqs. (31)–(33) are deﬁned exclusively in terms of the right-moving ﬁeld. The left-moving ﬁeld also contributes vacuum noise and might add to the † † ^ ^ K ¼ dωλðÞ ω Φ½ Δk ðÞ ω ´ a a þ H:c:; ð25Þ SQ L Rω R 2Ω ðÞ pω total photon number, but will have zero squeezing parameter in the absence of left-moving pump ﬁelds. The squeezing spectrum is typically probed in experiments by heterodyne measurement of ﬁltered ﬁeld and K ¼ hzjj β k b b , where we have deﬁned b ¼ b Ω , β ¼ SPM p p p p 43, 52–54 quadratures. We discuss how Eq. (33) is probed in some more detail β Ω and qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ in Supplementary Methods 1. 2 † 0 0 ^ ^ λωðÞ¼ β k k ; ð26Þ For a vacuum input ﬁeld, where aðÞ ω; 0 aðÞ ω ; 0 ¼ δωðÞ ω and all ω 2Ω ω R p R other second order moments vanish, it follows that NðÞ ω; z ¼ GðÞ ω; z 1 ¼ iΔkðÞ ω z jj vðÞ ω; z and MðÞ ω; z ¼ iuðÞ ω; z vðÞ ω; z e . These expressions satisfy Δk ðÞ ω ¼ 2k k k : ð27Þ L p ω 2Ωpω 2 jj MðÞ ω; z ¼ NðÞ ω; z½ NðÞ ω; z 1 , the maximum value allowed by the R R R Here, Δk ðÞ ω quantiﬁes a phase-mismatch due to the linear dispersion in Heisenberg uncertainty relation and also imply quantum-limited ampliﬁca- the JTWPA section. As we show below there is also an additional nonlinear tion. This, of course, assumes that there is no internal loss in the JTWPA contribution to the phase-mismatch that must be taken into account. device. out † ^ ^ Deﬁning asymptotic output ﬁelds, a ^ ¼U a ^ U,we ﬁnd the Rω Rω input–output relation in Eq. (3), where Two qubits in a squeezed bath iΔkðÞ ω Assuming for simplicity that the qubits are both located at the JTWPA uðÞ ω; z ¼ cosh½ gðÞ ω z sinh½ gðÞ ω z ; ð28Þ 2gðÞ ω output, x > z, their reduced dynamics after tracing out the bath is governed by a Markovian master equation, ρ_ ¼Lρ. The form of L for the λωðÞ general case is given in Supplementary Method 2, while we here focus on vðÞ ω; z ¼ sinh½ gðÞ ω z ; ð29Þ the most interesting situation when the two qubits are tuned in with the gðÞ ω squeezing interaction, such that ω + ω =2Ω . We can then write the 1 2 p sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 Lindbladian in the interaction picture ΔkðÞ ω no hi ð30Þ gðÞ ω ¼ jj λωðÞ : P γ γ ðmÞ ðmÞ m m L¼ N þ 1 D σ ^ þ N D σ ^ m;ν m;ν 2 2 ν¼L;R 2 2 A straightforward calculation shows thatjj uðÞ ω; z jj vðÞ ω; z ¼ 1, and m¼1;2 ð34Þ out out† 0 hi that the modes satisfy the commutation relation a ^ ; a ^ ¼ δωðÞ ω . pﬃﬃﬃﬃﬃﬃﬃ Rω Rω γ γ ð1Þ ð2Þ 1 2 ^ ^ S σ ; σ ; To summarize, this limiting equation is valid for weak nonlinearity and M ν þ þ weak ﬁelds, only treating the nonlinear Hamiltonian H to ﬁrst order, a where strong monochromatic classical pump at a frequency Ω , and in an asymptotic large time limit where t ¼1 and t ¼1. 0 1 S ½ A; B ρ ¼ MAðÞ ρB þ BρA fg AB; ρþ H:c:; ð35Þ How can we interpret the asymptotic limit where the initial and ﬁnal describes a dissipative squeezing interaction, and D½ A ρ ¼ AρA times are taken to minus and plus inﬁnity, respectively? If we consider a A A; ρ =2 is the usual dissipator. γ is the decay rate of qubit m and situation where an initial wave packet is localized far away at x 0atan σ ^ ¼ji g hj e (σ ^ ¼ ji e hj g ) is the qubit lowering (raising) operator. The early time t 0, this can be interpreted as a scattering limit, where we let Lindbladian has two contributions coming from the left-moving ﬁeld and the wave packet propagate through the nonlinearity and consider the 71, 75 the right-moving ﬁeld, respectively. In general, both ﬁelds can have non- asymptotic ﬁeld at x z for a late time t 0. Since the initial zero thermal photon number N ¼ NðÞ ω and squeezing parameter m;ν ν m evolution before the wave packet enters the nonlinear section is governed M ¼½ MðÞ ω þ MðÞ ω =2. If, on the other hand, the qubits are tuned out ^ ν ν 1 ν 2 by H , it is trivial to propagate the wave packet forward towards the of resonance with the squeezing interaction, ω + ω ≄ 2Ω , the last line in 1 2 p nonlinearity. The late evolution after the wave packet has left the nonlinear Eq. (34) will be fast rotating and can be dropped in a rotating wave section is similarly trivial. We can therefore think of a as a frequency Rω out approximation (see Supplementary Method 3 for more details). domain input ﬁeld entering the JTWPA and a as the corresponding Rω output ﬁeld leaving the device. This is similar to the deﬁnition of input and output ﬁelds used in the description of damped quantum optical 75, 77 ACKNOWLEDGEMENTS systems. One should keep in mind, however, that the validity of this A. L. G thanks N. Quesada and J. Sipe for helpful discussions on quantization in interpretation depends on the problem one is trying to solve: it is clearly dispersive and inhomogeneous media, and N. Menicucci and O. Pﬁster for helpful not appropriate if, for example, the initial state of the ﬁeld is delocalized comments regarding CV cluster states. We also thank A. Clerk, L. Govia and A. Kamal over the nonlinear section. for useful discussions. This work was supported by the Army Research Ofﬁce under Grant No. W911NF-14-1-0078 and NSERC. This research was undertaken thanks in The squeezing spectrum part to funding from the Canada First Research Excellence Fund. The squeezing of the JTWPA’s output ﬁeld is manifest in correlations between frequencies ω and 2Ω ω, symmetric around the pump AUTHOR CONTRIBUTIONS frequency. It is convenient to deﬁne for the right moving ﬁeld the thermal photon number All authors conceptualized the project. A.L.G. proposed to use JTWPAs to generate 1 cluster states and worked through the detailed calculations. A.L.G. wrote the 0 out† out out† out manuscript with contributions from A.B. 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Resolving photon number states in a superconducting circuit. © The Author(s) 2017 Nature 445, 515–518 (2007). Supplementary Information accompanies the paper on the npj Quantum Information website (doi:10.1038/s41534-017-0020-8). Published in partnership with The University of New South Wales npj Quantum Information (2017) 20
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Published: Jun 6, 2017
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