A parametrically programmable delay line for quantum information

Abstract

A parametrically programmable delay line device comprises an ensemble of resonators [104, 106, 108, 110, 112, 114, 116, 118]; and a nonlinear superconducting circuit element [120] parametrically coupled via parametric drives with the ensemble of resonators. The nonlinear superconducting circuit element implements a lumped element read-out / buffer mode. The ensemble of resonators may be implemented as superconducting transmission line resonators, lumped element resonators, acoustic resonators, or 3D cavity modes. The nonlinear superconducting circuit element may be implemented as a superconducting nonlinear asymmetric inductive element (SNAIL), a superconducting quantum interference device (SQUID), or an asymmetrically threaded SQUID (ATS) [122], The device may be implemented as a metal on substrate, such as aluminum, niobium, or tantalum on silicon or sapphire.

Description

[0001] A PARAMETRICALLY PROGRAMMABLE DELAY LINE FOR QUANTUM INFORMATION

[0002] FIELD OF THE INVENTION

[0003] The invention relates generally to devices for quantum computation. More specifically, it relates to delay lines that preserve quantum information.

[0004] BACKGROUND OF THE INVENTION

[0005] Delay lines that store quantum information are crucial for advancing quantum repeaters and hardware efficient quantum computers. Traditionally, they are realized as extended systems that support wave propagation but provide limited control over the propagating fields.

[0006] Delay lines that preserve quantum information have been studied as a resource for universal fault-tolerant quantum computing. These are hardware-efficient approaches to quantum computing where the emission from a single well-controlled qubit is captured and stored in a long delay line to be interacted with at a later time.

[0007] Delay lines capable of storing quantum information would significantly reduce hardware overhead for building useful quantum computers. Traditionally, delay lines are realized as extended systems that support propagating waves, such as waveguides, wherein the stored information is encoded in pulses. But existing delay lines provide limited control over the delayed information.

[0008] Several approaches have been developed to delay signals from superconducting quantum circuits without implementing a naive delay line. One approach is to use slow-light structures like metamaterials that effectively slow down the speed of light. Another approach is to use acoustic waveguides, where one can leverage the slower speed of sound. However, none of these approaches allow the experimenter to rapidly control the pulse as it propagates.BRIEF SUMMARY OF THE INVENTION

[0009] The present invention provides a parametrically addressed delay line for microwave photons that enables rapid control of a stored pulse as it propagates. By parametrically driving a three-wave mixing circuit element that is weakly hybridized with an ensemble of resonators, we engineer a spectral response that simulates that of a physical delay line, while providing fast control over the delay line’s properties. We demonstrate this novel degree of control by choosing which photon echo to emit, translating pulses in time, and even swapping two pulses, all with pulse energies on the order of a single photon. For example, we can effectively disconnect our waveguide from the environment by controlling the drive amplitude, or effectively translate our pulse in time by controlling the drive phase, or swap two pulses in time by controlling the drive frequencies. Thus, this programmable delay line provides significantly more control over the dynamics of stored pulses, enabling us to arbitrarily delay or even swap pulses.

[0010] Embodiments of the invention typically include three elements: (1) a readout resonator that is strongly coupled to the environment (and which includes several readout modes), (2) a collection of high-quality-factor resonator modes (which we refer to as storage resonator modes), and (3) a nonlinear element that enables frequency conversion between the readout resonator modes and the storage resonator modes.

[0011] The readout resonator modes can be any resonator modes that can be strongly coupled to the environment or other test equipment, such as lumped-element resonator modes or the modes of a waveguide or cavity. The storage resonator modes can be any low-loss resonator modes, such as the modes of a waveguide or cavity or an acoustic resonator. In the example presented, N storage resonator modes are implemented in N physical resonators, one storage mode in each physical resonator. In some cases, a single physical resonator may be able to provide 2 or more of the storage modes (perhaps even all of them).

[0012] The nonlinear circuit element can be any element that enables parametric coupling, such as a so-called Superconducting Nonlinear Asymmetric Element (SNAIL) or a so-called Asymmetrically Threaded SQUID (ATS). We consider parametrically coupling the storage resonator modes to the readout resonator modes such that the readout resonator simulates a physical waveguide over its bandwidth. The uniqueadvantage of our approach is the ability to alter the characteristics of the simulated delay line simply by modifying the parametric drives, offering a dynamic and physically compact solution for implementing a more versatile delay line. In particular, amplitude and phase of each storage mode can be controlled with a single nonlinear element by using a suitable multi-frequency electrical input to the nonlinear element. In the example presented, this freedom is used to enable the storage resonators to act as a virtual resonator with equally spaced modes (i.e., a delay line). As a result, the input / output behavior of the readout resonator is that of a (programmable) delay line.

[0013] In our first experimental demonstration, we used an ATS that is parametrically coupled to a group of superconducting coplanar waveguide resonators. The ATS both provides the nonlinearity for parametric coupling, as well as the readout modes that simulate the delay line. We also use the ATS to measure the added noise to the delay line from our parametric drives and find that the added noise is much less than one photon, as is required for preserving quantum information.

[0014] Applications include quantum computers with reduced hardware overhead. For example, there are well-known protocols for building extremely hardware efficient quantum computers if only a compact and high-quality-factor delay line existed.

[0015] In one aspect, the invention provides a parametrically programmable delay line comprising: an ensemble of resonators; and a nonlinear superconducting circuit element parametrically coupled via parametric drives with the ensemble of resonators. The parametrically programmable delay line may be implemented as a metal on substrate device, where the metal is aluminum, niobium, or tantalum, and where the substrate is silicon or sapphire. In some implementations, the nonlinear superconducting circuit element implements a lumped element read-out / buffer mode. In some implementations, the ensemble of resonators are implemented as superconducting transmission line resonators, lumped element resonators, acoustic resonators, or 3D cavity modes. In some implementations, the nonlinear superconducting circuit element is implemented as a superconducting nonlinear asymmetric inductive element (SNAIL), a superconducting quantum interference device (SQUID), or an asymmetrically threaded SQUID (ATS). In some implementations, the nonlinear superconducting circuit element is capacitively or inductively coupled with the ensemble of resonators. In some implementations, the nonlinear superconducting circuit elementis configured to allow parametric swapping between nonlinear superconducting circuit element lumped element read-out / buffer mode and modes of the ensemble of resonators.

[0016] In another aspect, the invention provides a parametrically programmable delay line comprising: a readout resonator coupled to an environment and comprising readout modes; a collection of storage resonators comprising storage resonator modes; and a nonlinear element that enables frequency conversion between the readout modes and the storage resonator modes. In some implementations, the readout resonator modes are lumped-element resonator modes or modes of a waveguide or cavity. In some implementations, parametrically programmable delay line of claim In some implementations, the storage resonator modes are modes of a waveguide or cavity or an acoustic resonator. In some implementations, the nonlinear element is a device that enables parametric coupling, such as a so-called Superconducting Nonlinear Asymmetric Element (SNAIL) or a so-called Asymmetrically Threaded SQUID (ATS). In some implementations, the nonlinear element is a Superconducting Nonlinear Asymmetric Element (SNAIL) or an Asymmetrically Threaded SQUID (ATS). In some implementations, the nonlinear element is a device that enables parametric coupling of the storage resonator modes to the readout resonator modes such that the readout resonator simulates a physical waveguide over its bandwidth. In some implementations, the nonlinear element is configured to allow control of amplitude and phase of each storage mode of the storage resonator modes by using a suitable multi-frequency electrical input to the nonlinear element.

[0017] BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0018] Fig. 1A is a frequency mode diagram illustrating parametric coupling of an ensemble of resonators to a readout mode, according to an embodiment of the invention.

[0019] Fig. 1B is a schematic diagram illustrating an analogy between a physical waveguide delay line (illustrated as a ring supporting pulses) and a parametrically addressed delay line under continuous parametric driving.

[0020] Fig. 1C is a schematic diagram illustrating how turning off all parametric drive amplitudes when a photon echo is about to rephase can prevent the signal fromemitting into the environment and thereby selectively emit a later photon echo. Fig. ID is a schematic diagram illustrating how instantaneously translating the delay line mode’s phase can translate the pulse in time.

[0021] Fig. IE is a schematic diagram illustrating how swapping the detunings of the delay line modes can swap the order of two pulses.

[0022] Fig. IF is a circuit diagram illustrating resonators (illustrated by lumped LC resonators) coupled with an asymmetrically threaded SQUID, implementing a parametric delay line, according to an embodiment of the invention.

[0023] Fig. 1G is an optical micrograph of a parametrically addressed delay line, according to an embodiment of the invention, showing the resonators surrounding a lumped element buffer mode, where an asymmetrically threaded SQUID is indicated by a dashed box.

[0024] Fig. 2A shows graphs of magnitude (left) and phase (right) response of a normalized reflection coefficient of a buffer mode in a continuously coupled parametric delay line, according to an embodiment of the invention.

[0025] Fig. 2B shows a graph of analog-to-digital converter (ADC) traces of a pulse that is stored and not stored in a continuously coupled parametric delay line, according to an embodiment of the invention.

[0026] Fig. 3A is a map of ADC traces illustrating how turning off the parametric drive amplitudes for a duration τ, one can selectively emit later photon echos.

[0027] Fig. 3B is a graph of ADC traces for τ = 0.0 μs, 2.5 μs and 4.6 μs, illustrating how turning off the parametric drive amplitudes for a duration τ, one can selectively emit later photon echos.

[0028] Fig. 3C is a map of ADC traces, illustrating how instantaneously translating the phase of the parametric drive, one can translate the pulse in time.

[0029] Fig. 3D is a graph of ADC traces, illustrating how instantaneously translating the phase of the parametric drive, one can translate the pulse in time.

[0030] Fig. 3E is a graph illustrating how instantaneously swapping the detunings in the parametric delay line, one can swap two pulses in time.Fig. 4A is a graph of the number photons vs. normalized two-photon drive strength, illustrating how the steady-state CPW intracavity photon number n changes with the amplitude of the normalized two-photon drive when we operate one of the CPWs as a quantum MPO.

[0031] Fig. 4B is a spectrum of the same CPW resonator as in Fig. 4A when the parametric pump and drives used to generate a parametric delay line are applied on the device.

[0032] Fig. 5 is an optical micrograph of an asymmetrically threaded SQUID used in an embodiment of the invention.

[0033] Fig. 6 is a schematic diagram illustrating an experimental setup used to operate and test a parametrically addressed delay line, according to an embodiment of the invention.

[0034] Fig. 7 is a spectrum illustrating relevant frequencies for an experiment operating and testing a parametrically addressed delay line, according to an embodiment of the invention.

[0035] Fig. 8A is a map of a mode showing a saddle point of a first value of V^.

[0036] Fig. 8B is a map of a mode showing a saddle point of a second value of Vs. Fig. 9 are plots of input photon flux and output photon flux for five different parametric delay line parameters.

[0037] Fig. 10 is a plot of two delayed pulses for a lossless delay line where the parametrically converted detunings are not swapped and when the detunings are swapped, illustrating that swapping the detunings swaps the pulse in time.

[0038] Fig. 11 is a graph of fitted gain at the CPW frequencies from operating them as quantum MPOs near threshold.

[0039] DETAILED DESCRIPTION OF THE INVENTION

[0040] Here we disclose a parametrically addressed delay line (PADL) as a versatile virtual delay line for microwave photons, leveraging pump fields that drive parametric processes to dynamically control the speed, direction, coupling strength, and connec-tion points of the signal within the line. We implement this virtual delay line by parametrically distributing a data pulse that is launched at a lumped element readout mode into excitations in an ensemble of long-lived resonators. Controlling all of the delay line’s properties translates to controlling the parametric drive frequencies, amplitudes, and phases. We show how a parametric delay line gives us more control than a physical waveguide by: (1) controlling the drive amplitudes to selectively choose the number of round trips that a wavepacket makes inside the delay line, (2) controlling the drive phases to translate the pulse in time, and (3) controlling the drive detunings to swap two wavepackets in time.

[0041] A key question is how the delay line will perform for quantum pulses, and whether the PADL’s parametric nature leads to excess noise, such as through parasitic processes, to degrade performance. For this, we measure the added noise from the parametric drives by calibrating the gain of our measurement apparatus using our resonators as quantum microwave parametric oscillators (MPOs) that operate as quantum-calibrated sources of microwave radiation. Specifically, we use the number of photons in our quantum MPO near threshold as an in-situ noise power calibration device, and find that the added noise is much less than one photon per mode.

[0042] OPERATING PRINCIPLES OF A PARAMETRIC DELAY LINE

[0043] The PADL works by parametrically distributing a data pulse that is launched at a readout mode (referred to as the buffer mode) into excitations of a collection of long-lived storage resonators. By controlling the parametric drives, we can engineer the buffer mode’s spectrum to emulate the characteristics of a reflective delay line. As a simple example, consider using PADL to emulate a physical delay line with a free spectral range (FSR) given by fi. This can be achieved with PADL by continuously parametrically coupling the buffer mode and the storage resonators such that the detuning A / ,. between the converted photons from the kthstorage resonator and the buffer form a frequency comb with an FSR (peak spacing) Q and such that the coupling strengths realize the desired loading.

[0044] We implement a parametric delay line by parametrically coupling an ensemble of resonators (with frequencies cu / to a readout mode (with frequency b, which we refer to as the buffer mode).Fig. 1A illustrates the relevant frequencies in this example. The storage resonator frequencies are labeled by ay. and the buffer frequency is labeled by cjb. The parametric couplings / drives are illustrated by dashed lines. We engineer the buffer mode’s spectrum into looking like a delay line with an FSR denoted by Q by choosing the converted storage photons to have detunings A / ,. relative to b and to form a frequency comb with an FSR given by. The parametrically converted storage photons have detunings A&. The storage resonator frequencies do not need to be precisely placed or tuned - we use the parametric drive frequencies to compensate for any disorder to realize the mode spacing.

[0045] Fig. 1B illustrates operation of the device by introducing an analogy to a physical waveguide delay line (illustrated as a ring supporting data pulses). For continuous parametric coupling / driving, the delay line modes evolve by accruing a phase, and they rephase after a round-trip time Trt= 2TF / , such that the output pulse is simply the input pulse delayed by Trt. Specifically, the kthdelay line mode evolves by accruing a phase = A^t.

[0046] However, the PADL provides far more exotic dynamics than a simple physical delay because we can parametrically program the delay line mode couplings, phases, and detunings. Crucially, these couplings are independently tunable; changing the kthparametric drive changes the coupling to the kthstorage resonator.

[0047] In Fig. 1C, we illustrate how we can effectively disconnect the virtual delay line from the input / output by shutting off the parametric drives, i.e. by setting their amplitudes |efc| = 0. This prevents the rephased signal from being emitted into the environment and causes it to propagate around the virtual delay line for longer. In other words, by turning off all parametric drive amplitudes |efc| when a photon echo is about to rephase, one can prevent the signal from emitting into the environment and thereby selectively emit a later photon echo. Turning the drives back on reconnects the waveguide to the environment and causes the pulse to be re-emitted at the next round-trip time WTrtfor some positive integer N. This mode of operation is analogous to fiber loop buffers which may be used for storing quantum information. Full control over the phases of the drives allows us arbitrary access to information stored in the delay line.

[0048] In Fig. ID, we illustrate how we continuously translate the pulse forwards orbackwards in time by an amount r (modulo Trt) by instantaneously translating each phase by pk—

[0049]

[0050] + A^r. This is analogous to accessing the pulse at positions other than the open port in the virtual delay line. In other words, by instantaneously translating the kthdelay line mode’s phase by pk— ( / >k + A^T, one can translate the pulse by a time r (modulo 2TT / ).

[0051] In Fig. IE, we illustrate how controlling the parametric drive frequencies allows us to swap two pulses in time. Specifically, changing A / ,. — —A / ,. is equivalent to taking t — — t in terms of the phase that the virtual delay line modes accrue. This is analogous to switching the pulse propagation direction in the virtual delay line. In other words, by swapping the detunings of the delay line modes A / ,. — —A / ,. one effectively takes t — — t in the phase accrued by the delay line modes and swaps the order of two pulses.

[0052] In addition to these four ways of manipulating pulse dynamics, one can also engineer more complicated pulse dynamics with the novel degree of control provided by a parametric delay line. Without loss of generality, for the purposes of illustration, our description will focus on these four techniques.

[0053] PHYSICAL IMPLEMENTATION OF A PARAMETRIC DELAY LINE

[0054] The parametric delay line, which includes an ensemble of resonators capacitively or inductively coupled with a nonlinear superconducting circuit element, may have various physical implementations. The ensemble of resonators may be implemented as superconducting transmission line resonators, lumped element resonators, acoustic resonators, or 3D cavity modes. The nonlinear superconducting circuit element may be implemented as a superconducting nonlinear asymmetric inductive element (SNAIL), a superconducting quantum interference device (SQUID), or an asymmetrically threaded SQUID (ATS).

[0055] In one illustrative embodiment, a circuit diagram of this parametric delay line is illustrated in Fig. IF, where we illustrate the resonators 102 by their equivalent lumped element LC models. We weakly hybridized CPW resonators (illustrated by lumped LC resonators) with one asymmetrically threaded SQUID (ATS) 100.

[0056] An optical micrograph of a PADL device according to one embodiment of the invention is shown in Fig. 1G. The device includes an ensemble of CPW resonators104, 106, 108, 110, 112, 114, 116, 118, which are all coupled to a lumped element buffer mode 120, which is provided by the ATS 122. The physical circuit may be implemented on-chip by fabricating quarter-wavelength CPW resonators that are weakly capacitively coupled with an ATS as the nonlinear resonant circuit.

[0057] The ATS circuit 122 is given by two nominally identical Josephson junctions forming a loop with an inductive shunt in the center of the loop. We use an array of Josephson junctions to form the inductive shunt. The ATS provides the lumped element buffer mode that we use to readout our pulses, as well as the necessary nonlinearity to parametrically couple the different modes of our circuit. In this embodiment, we choose to use an ATS for three reasons. Firstly, the ATS provides three-wave mixing as opposed to four- wave mixing - the native nonlinearity of Josephson junctions. Three-wave mixing allows us to parametrically swap between the ATS lumped element mode and the CPW modes while minimizing the adverse effects from four-wave-mixing interactions that are characteristic of a multimode systems connected to junctions. Secondly, the ATS has an inductor, which provides an unconfined parabolic potential and therefore its lumped element mode can be strongly driven before it becomes nonlinear. Finally, we can use the three- wave mixing nonlinearity to implement a quantum MPO in the CPWs, which we can use for noise power calibration.

[0058] When the ATS 122 is precisely biased at its “saddle-point,” the energy associated with the phase drop across the junctions changes from the usual

[0059]

[0060] cos(φ) to sin(φ). This crucial modification shifts the dominant nonlinear term from

[0061]

[0062] to φ3. We use this altered nonlinearity to enable the three-wave mixing processes essential for the operations conducted here. The Hamiltonian at the saddle-point is

[0063] H = huJf'b'b + hoJkO^ak

[0064] - 2EJep(t') sin ypb(b + U) + El-Pk(Slk + tk' ) j ■>

[0065]

[0066] k where φb, is the node flux zero-point fluctuation (ZPF) of the buffer mode across the ATS Josephson junctions with annihilation operator b and frequency

[0067]

[0068] Similarly φkis the ZPF of the kthCPW resonator fundamental mode at the same circuit node with annihilation operator « / ,. and frequency ωk. Ej is the individual junction energy in the SQUID, and ep(t) is a time-dependent parametric flux pump threading the SQUID. Note that we have assumed |e

[0069]

[0070] p(t) | 1.We resonantly select specific interactions by driving the buffer mode while simultaneously flux pumping the SQUID. We focus here on a beamsplitter interaction between the buffer mode and the kthCPW’s fundamental mode. We flux pump the SQUID at a single frequency uy and drive the buffer mode with multiple drives tones, as captured by the following driving Hamiltonian:

[0071] ^drive / n = $2 + h.c.). (2)

[0072]

[0073] k Here, is the frequency of a detuned drive on the buffer with field amplitude |efc|. We choose the drive frequencies to satisfy

[0074] ^

[0075]

[0076] d,k = ~ (^b + Afc) + CUfc. (3) In the frame where both the CPW mode and the buffer are rotated out, our total Hamiltonian approximately becomes:

[0077] H / h = 52 + 52 ^4^ +h-c- (4)

[0078]

[0079] k k where Kgk= Ej6p< >l<^a(3k is the parametric coupling strength between the buffer and the kthCPW’s fundamental mode, and 0kis the small displacement on b generated by the kthdrive. By operating the buffer in the fast-cavity regime, i.e., Kye

[0080]

[0081] gk, we can adiabatically eliminate it (ftb)eis the extrinsic loss rate of the buffer mode). We tune the drive amplitudes and frequencies so that the gkand harmonically placed A / ,. in the resulting effective Hamiltonian closely resemble that of a delay line. Importantly, by controlling the parametric drives’ frequency, amplitude, and phase, we control the corresponding delay line mode’s detuning, coupling, and phase. In principle, the flux pump frequency is arbitrary because the drive frequencies can be chosen to satisfy Eq. 3. In practice, one could be limited by the amount of microwave power that is available to drive the buffer.

[0082] The ATS parameters are chosen such that

[0083]

[0084] = 5.0073 GHz and cufc / 27r ~ 6.91 — 7.46 GHz (see Note 1 for more details). The buffer is capacitively coupled to the environment at a rate Kbje / 27r = 3.95 MHz, and the resonator intrinsic quality factors approximately range from 110 x 103to 300 x 103. We estimate from finite element electromagnetic simulations (see Note 1 for more details) that the buffer impedance and hybridization strengths with the CPWs leads to flux ZPF across the junction for each mode of roughly <?b= 0.336 and tpk0.018 — 0.023.RESULTS: PARAMETRIC CONTROL OF STORED WAVEPACKETS We first start with the simplest PADL experiment - implementing a response that mimics that of a reflectively terminated transmission line probed at the other end. For this, we tune and fix the parametric drives’ amplitudes, phases, and detunings (as illustrated in Fig. 1B) for seven of the CPWs. For all the experiments in this work, we only parametrically couple seven of the eight CPW resonators to the buffer as we observed one of the CPW modes to have larger frequency fluctuations, which we attribute to a nearby two-level system (TLS) defect. We set the FSR to be O / 2TF = 500 kHz so the delay line bandwidth closely matches the buffer mode’s extrinsic coupling rate. The continuous wave (CW) flux pump tone is provided by a signal generator (Keysight E8257D PSG). The parametric drive intermediate frequency (IF) tones are all generated on a single arbitrary waveform generator (AWG) channel (Tektronix 5200) before being up-converted to drive the buffer mode. Before being up-converted, the AWG output is amplified by a room-temperature low-noise amplifier. As discussed later in relation to Fig. 6, pulses sent into the delay line 600 are played and demodulated using an Operator-X (OPX) 602 from Quantum Machines Inc. (QM), and similarly the pulses are up-converted and down-converted using an Octave 604 from QM. The demodulated pulses are also digitized on an analog-to-digital converter in the OPX (see Note 2 for more details). We emphasize that the flux pump tone provides the magnetic flux ep(t) through the ATS loops and is fed to the device via a transmission line that is grounded near the ATS, whereas the parametric drives efc(t) are fed through the readout transmission line that is capacitively coupled to the buffer mode (see Note 2 for more details).

[0085] In a continuously coupled parametric delay line, we probed the PADL on reflection near the buffer mode frequency and clearly observed the parametrically coupled modes in the normalized reflection coefficient Sn (cu) as shown in Fig. 2 A. These graphs show the reflection coefficient of our buffer mode (S'11(cu)) when seven CPW resonators are parametrically coupled to the mode. We tune the parameters to obtain an approximately linear phase response centered at 5.0321 GHz to emulate the cuTrtphase response of a physical delay line.

[0086] We also measured the time-domain response by sending pulses at the PADL.

[0087] Fig. 2B shows analog-to-digital converter traces of a pulse that is stored and not stored in our parametric delay line. This graph shows the results of reflecting anattenuated Gaussian pulse ( ) 1, with a temporal FWHM of 471 ns; see Note 6 for more details on the calibration of the attenuation). The delayed pulse is delayed by approximately Trt= 2,s ~ 2TF / . The time-domain traces are reported in units of photon flux (see Note 6 for more details). The pulse centered at t = 0 results from reflecting a pulse off of the device in the absence of pump and drives and with the buffer mode far-detuned, so that the device acts as a mirror. The other pulse results from reflecting a pulse off the buffer mode while the device is emulating a delay line. We observe that this pulse is approximately delayed by 2,s ~ Trt. The small reflected pulse at t = 0 is due to mismatched impedance between the environment and the parametric delay line, including any inevitable mismatches from device packaging (see Note 5 for more details). The time-domain traces are reported in units of input photon flux |Z?in|2and output photon flux |

[0088]

[0089] / 3Out |2- These fluxes are related by the input-output boundary condition 3out= / 3in+

[0090]

[0091] where f3 = b).

[0092] By turning off the parametric drive amplitudes for a duration τ, one can selectively emit later photon echos. Turning the parametric drive amplitudes |

[0093]

[0094] e& | off, prevents a stored wavepacket from being emitted, causing the pulse to undertake another round trip in the delay line, as illustrated in Fig. 1C. Turning the amplitudes back on allows the next photon echo to be emitted. In Fig. 3A, we sweep the duration r over which we turn off the parametric drive amplitude. Delay r = 0 corresponds to the drives being on continuously. We see that as r exceeds Trt, the first echo disappears and the second echo at 2Tr, becomes more prominent. Similarly, as r exceeds 2Trt, the second photon echo disappears and the third photon echo at 3Trtbecomes more prominent. In Fig. 3B, we show the ADC data of traces taken with r = 0.s, 2.5.s, and 4.6,s.

[0095] By instantaneously translating the phase of the;thparametric drive by an amount AfcT, one can translate the pulse in time by r (modulo 2TT / ). In other words, we finetune the photon emission time by modifying the parametric drive phases to modifyfc. By changing these phases we continuously translate the pulse in time, as illustrated in Fig. ID. Specifically, by translating (f)k — ( / >k + A^T, we can translate our pulse by a time r modulo Trt. Fig. 3C is a map of ADC traces as we sweep the duration r by which we translate our pulse from —0.7,s to +0.7.s, where r = 0 corresponds continuous parametric driving without phase modification. We see that as we sweep r, the time at which the pulse re-emits translates linearly. In Fig. 3D, we show theADC data of traces taken with r = 0.000 / zs, —0.385 / zs, and +0.385 zs.

[0096] Finally, we show how controlling the detunings of the parametrically converted CPW photons A / ,. allows us to swap two pulses in time. Swapping the detunings A / ,. — — Afc, is analogous to performing time reversal t — — t in the phase accrued by the delay line modes, causing the first stored pulse to be emitted after the second pulse. In Fig. 3E, we show two complex ADC traces: one where the detunings are swapped and one where they are not, illustrating how instantaneously swapping the detunings A& — —A / ,. in the parametric delay line swaps two pulses in time. In this experiment, we send two pulses with FWHM = 283 ns and separation of 1000 ns. We make the first pulse have a TT phase shift relative to the second pulse and plot one quadrature of the measured ADC data to clearly show that the two pulses are swapped in order when the detunings are swapped. The unswapped trace in Fig. 3E plots the case where the detunings are not swapped, corresponding to the case of continuous drives. The swapped trace in Fig. 3E plots the case where the detunings are swapped, corresponding to the case where we take t — — t. We clearly see that the pulses have been swapped. In Note 5, we provide additional numerical simulations considering two pulses with different relative amplitudes and phases.

[0097] RESULTS: FIDELITY AND ADDED NOISE

[0098] An important figure of merit in quantum memories is the fidelity of the state being read out compared to that which was stored. Ideally, the output pulse should be identical in amplitude and phase to the input pulse. In reality, distortions and loss induced by storage in the PADL and thermal noise reduce the fidelity. We will show that the latter is negligible in our system, and first focus on distortion and loss. Given two pulses that are characterized by temporal mode operators A = f dtf (t)d(t) and A2 = J dt / / (t)d(t), we define the fidelity as F = | J dtf*(t) / / (t)|2, where f(t) is the temporal mode profile for the input pulse and / / (t) is the temporal mode profile for the delayed pulse. Here, we focus on the fidelity of the pulse that is delayed in the presence of continuous parametric drives (such as the stored pulse in Fig. 2B) such that g

[0099]

[0100] (t) ~ f(t — Trt). We calculate these mode profiles by normalizing the detected held measured on the ADC. The input mode is always normalized such that f dt|f(t)|2= 1 corresponding to unit detection probability. To isolate the impact of distortions on the pulse while it is stored in the PADL, we first normalize the delayedpulse such that f dt \g(t) |2= 1. We also exclusively focus on the pulse centered at 2,s and neglect contributions from the small reflected pulse at t = 0, which can be caused by impedance mismatches in the device packaging. With this normalization for g(t), which ignores losses and effectively compares the shape of the delayed temporal mode to the input temporal mode, we measure F = 0.95. Time-domain simulations of the semi-classical equations of motion

[0101]

[0102] for tz(i) and &(t) derived from Eq. 4 show that fidelity can be further improved by optimizing the parametric delay line parameters and input pulse bandwidth (see Note 5 more details). To include the effect of losses, we normalize g(t) by the same factor that we normalize f (t) resulting in f dt\g(t) |2< 1. In this case, we find a F = 0.21 that includes both loss and distortion, showing that the infidelity is primarily due to losses in the storage resonators. The infidelity from storage resonator loss is neither limited by the loss of the worst resonator nor the sum of all resonator losses. Given that the input pulse is parametrically distributed into the collection of storage resonators, the input pulse’s frequency components near A / ,, are predominantly attenuated by the / d1' resonator’s loss rate.

[0103] In addition to loss and distortion, an important concern is whether the parametric driving of the PADL leads to excess noise being added to the microwave held. We estimate the added noise by measuring the power spectral density of the held emitted at the CPW mode frequencies when the drives are on. The key challenge is to calibrate the gain and loss in the readout signal path with sufficient precision to be able to infer the microwave huctuations at the device from the held detected outside the fridge. Previously, this has been accomplished by using in-situ calibrated sources, such as shot noise tunnel junctions or qubits. In both these cases, the physics of the source is sufficiently well-understood to provide a signal with a well-dehned photon hux or noise power without requiring a component-by-component accounting of gain and loss. Here we use the device itself, operated as a parametric oscillator, as such a quantum calibrated source. In a classical model of a parametric oscillator there is a discontinuity at the oscillation threshold. This discontinuity is smoothed away by the huctuations of the electromagnetic held in a more accurate quantum model of a parametric oscillator. The dependence of number of intracavity photons n vs. drive power obtains a characteristic shape which we use to infer the number of photons and calibrate the gain of our amplihcation chain (see Note 6 for more details).

[0104] We implement the MPO on the same device by pumping the

[0105]

[0106] ATS at = 2cc / ,. — bto resonantly select a 2-photon swapping term given by a +a^b. After adiabatically eliminating the fast-decaying buffer mode and applying a resonant drive, the effective dynamics of the;thCPW mode are governed by a Lindblad master equation p = —i\H,p\ + T>[Li]p + T^L^p, with the Hamiltonian and loss operators given by H

[0107]

[0108] = ie^a2+ h.c., = S / K^O2, and L± = (5) In these equations, e2is the two-photon drive strength, a2is the two-photon loss rate, and Ki is the single-photon loss rate.

[0109] We now consider calibration of added noise. Fig. 4A is a plot of the number photons n vs. normalized two-photon drive strength e2 / fti as one MPO crosses threshold. Both axes are log scale. The figure shows how the steady-state CPW intracavity photon number n changes with the amplitude of the normalized two-photon drive when we operate one of the CPWs (the resonator at 6.975562 GHz) as a quantum MPO. In the following experiments, we directly readout the individual CPWs through the readout port, which is possible due to weak parasitic capacitances. We measure the integrated power spectral density (PSD), which is proportional to n, on an RF spectrum analyzer and plot the result versus the amplitude of the driving held sent to the buffer mode. A smooth transition is clearly visible and the data agrees closely with the quantum model (solid line) of the MPO (see Eq. 5) with three fitting parameters: (1) the proportionality constant between e2 / / t1and the driving held at the instrument, (2) the proportionality constant between n and the integrated PSD at the instrument, which is related to the gain, and (3) K2 / KI.

[0110] With the gain calibrated, we can determine the number of added noise photons.

[0111] Fig. 4B shows the spectrum of the same resonator when the pump and drives used to generate a parametric delay line are applied on the device. The hgures shows the spectrum of the same CPW resonator as in Fig. 4A but now with the parametric drives and pump for the delay line experiments turned on. By htting this spectrum to a Lorentzian (solid line) to extract the spectral area and by using our calibrated gain, we conclude that 0.11 noise photons are added to this mode when it is operated as the PADL. We find that the added noise in all modes is always less than ~ 0.15 photons for the reported drive intensities. The added noise is probably due to heating from our strong parametric flux pump. This pump power is comparable to previous quantum-limited parametric amplifiers and is likely comparable to previous ATS-based devices with higher quality-factor resonators demonstrating a superposition ofmacroscopically distinct states. Therefore, we do not believe this small added noise will hinder future quantum applications for the PADL.

[0112] DISCUSSION

[0113] Unlike a waveguide delay line, the PADL gives us complete control over the detunings, phases, and coupling rates of the delay line modes. Furthermore, delays that are comparable to a several kilometer long waveguide can be achieved for microwave photons in a small footprint. Unlike catch-and-release methods that require a single resonant mode and precise mode matching via dynamic and precisely timed control of cavity parameters, PADL offers far greater flexibility. By using a three-wave mixing circuit element we sidestep issues due to parasitic processes that arise more frequently in four-wave mixing schemes. In lieu of performing process tomography on encoded qubits, which would enable the evaluation of the delay as a quantum process, we characterize the bosonic channel by measuring the number of photons of noise that are added into our delay line as well as the overlap between the input and output wavepackets. We demonstrate dynamic programmability by selecting emission of later photon echos, translating pulses continuously in time, and swapping two pulses stored in the emulated delay line.

[0114] Finally we stress that more intricate control can be engineered, given that the PADL is fully programmable and that integration with qubits is a possibility. Furthermore, by integrating qubits on the same chip as the PADL, one can address any impedance mismatches that arise from packaging. Nonetheless, practical use as a quantum memory will require much higher fidelities. One simple way to improve fidelity is to use larger bandwidth pulses and shorter delays such that the delay is much less than the CPW resonator lifetime. Ultimately, improving the resonator lifetime is important for improving fidelity. In future work, integration with recently developed high-Q Tantalum CPW resonators or microwave cavities coupled to ATSs may open the route to long programmable delays in quantum processors with much higher fidelity.

[0115] METHODS: DEVICE FABRICATION

[0116] A parametrically programmable delay line of the present invention may be fabricated using a variety of different material systems, but is generally physically realizedas a metal on substrate. For example, the metal may be aluminum, niobium, or tantalum, and the substrate may be silicon or sapphire.

[0117] In one embodiment, a parametric delay line device is patterned in aluminum on 525 m thick high-resistivity silicon ( > 10 k -cm). The sample is first solvent cleaned in acetone and isopropyl alcohol, followed by the following four-mask process:

[0118] 1. Etched alignment marks: Alignment marks are patterned with photolithography (Heidelberg MLA150 direct-writer) and etched into the sample using XeF2- The sample is then cleaned in baths of piranha (3:1 H2SO4: H2O2) and buffered oxide etchant.

[0119] 2. Circuit patterning: Ground planes, CPWs, flux lines, and the ATS island are patterned with photolithography, followed by a gentle oxygen plasma. Aluminum is deposited in an electron beam evaporator (Plassys) and lifted off in N-Methyl-2-pyrrolidone (NMP).

[0120] 3. Junction patterning: The ATS itself (including the SQUID loop and the superinductor) are patterned by electron beam lithography (Raith Voyager), followed by a gentle oxygen plasma. Aluminum is deposited at an angle of 62°, followed by oxidation at 50 Torr for 10 minutes, followed by aluminum deposition at an angle of 0°. Liftoff is performed in NMP. The junction-array inductor is formed from 21 junctions using a Dolan-bridge method, whereas the single junctions in the SQUID are formed using a T-style process.

[0121] 4. Bandaging: We use a bandage mask to ensure a superconducting connection between the previous two masks. The bandages are patterned with electron beam lithography and overlap both masks. Prior to aluminum deposition (at 0°), we ion-mill in-situ to clear away any oxide, thus ensuring a superconducting connection. Liftoff is performed in NMP.

[0122] METHODS: CIRCUIT ANALYSIS

[0123] In one embodiment of the device, the ATS is implemented as a SQUID (with individual junction energies Ej) that is threaded by an inductor (with energy EL. In terms of the node flux operator of the ATS node, the potential from the inductorand SQUID can be written as:

[0124] U = ~ELbip2- 2Ej cos(^s) cos(<^ + ( / ?A) (6)

[0125]

[0126] where:

[0127] AS=(^ext,! +(dext,2) / 2 (7) A A = (<dext,l - <dext,2) / 2 (8) and where <2ext,i and <?ext,2 are the external magnetic fluxes threading the left and right loops formed by the inductor and a Josephson junction.

[0128] When the device is flux-biased to

[0129]

[0130] = A = TT / 2 and a small RF modulation ep(t) is applied to

[0131]

[0132] the potential (to first order in ep) becomes:

[0133] U = ^ELb<p2- 2Ejep(t) sin(<£) (9)

[0134]

[0135] Since the ATS is capacitively coupled to other modes, the node flux operator p representing the flux across the junction can be written in terms of the normal modes of the linear circuit as:

[0136] ip = + + ^2<pk(ak+ 4) (10)

[0137]

[0138] k where 925 is the node flux ZPF of the “buffer-like” normal mode, and

[0139]

[0140] is the node flux ZPF of the “CPW-like” normal mode of the kthCPW mode. The full Hamiltonian is then:

[0141] H = fiw$b + hbJkalak U.( 1 1 )- 2Ejep(t) sin p,(6 + &') + 2. + iff,) J

[0142]

[0143] k as discussed above.

[0144] Microwave Parametric Amplification: When ep(t) is pumped at a frequency ay = 2cufc — we need to look for terms in the Hamiltonian that can resonate with this frequency. They are of the form:

[0145]

[0146] + aJ,2S).

[0147] These terms represent processes where two photons from the kthCPW mode are either absorbed or emitted in combination with the emission or absorption of a photon from the buffer-like mode.Beamsplitter operation: Alternatively, when ep(t) is pumped at a frequency uy = ~k, and a second drive at frequency

[0148]

[0149] is applied to the buffer, we will implement a beam splitter interaction between the buffer-like mode and the kthCPW mode. In this setup, the interaction Hamiltonian can be simplified to terms that resonate with the combined effect of both drives. The resonant terms in this case are:

[0150] ^2b^k(b ak+ h.c.)

[0151] NOTE 1: DEVICE PARAMETERS

[0152] Device parameters for one embodiment of the parametric delay line are summarized in Table 1. To determine tpkfor the CPW resonators and φb, for the buffer mode, we simulate the capacitive coupling between the CPWs and the buffer and fit the inductance of the ATS junction array that reproduces the measured buffer mode frequency. We leave the ATS junction array and SQUID junctions open in our capacitance simulation. We find the ATS junction array inductance to be Lb= 7.50 nH, corresponding to Ekb / h = 21.8 GHz. This values agrees within 8% of values computed from room-temperature resistance measurements (using the Ambegaokar-Baratoff formula) of nominally identical junctions that are fabricated near the device. We use an array of 21 junctions to form the ATS junction array. The SQUID junction energy is also inferred from room-temperature measurements. An ATS that was fabricated in parallel with the measured device is pictured in the optical micrograph of Fig. 5. This is an image of an identical device that was fabricated in parallel with the measured device, with the exception that the Dolan-style junctions in the ATS superinductor are nominally 0.975 m narrower. The ATS components shown in the figure include a silicon substrate 500 and Aluminum bandages 502, 504, 506 used to connect the Josephson junctions of the ATS to the readout mode.

[0153] Given the capacitance matrix, the ATS inductance, and by treating the quarterwavelength CPW resonators as effective lumped element LG resonators (based on their designed characteristic impedances and lengths), we can diagonalize the circuit to find the eigenfrequencies and ZPFs at the ATS node. We find that our calculated eigenfrequencies agree well with our measured frequencies (within 1%). The resulting eigenvectors are used to compute <band <k.

[0154] The buffer mode and the CPW modes are characterized using spectroscopy with aUb / ^TV 5.0073 GHz ± 600 kHz

[0155] Lb7.50 nH

[0156] ‘■Pb 0.336

[0157] ^b,e / ^ 3.95 MHz ± 60 kHz

[0158] Kb^TX 130 kHz ± 40 kHz

[0159] Ej / h 5.28 GHz

[0160]

[0161] k wfc / 27F [GHz ± kHz] P>k Kfcie / 27F [kHz] Hk,i / ^ [kHz] Qi,k x 1031 6.904939 ± 2 0.0186 37 ± 1 32 ± 3 220 ± 20 2 6.975562 ± 3 0.0228 40 ± 2 29 ± 2 240 ± 20 3 7.156324 ± 9 0.0210 70 ± 5 41 ± 7 170 ± 30 4 7.247145 ± 3 0.0175 57 ± 1 34 ± 4 210 ± 30 5 7.318975 ± 4 0.0179 74 ± 1 39 ± 4 190 ± 20 6 7.389379 ± 2 0.0186 102 ± 2 25 ± 3 300 ± 40 7 7.460333 ± 4 0.0211 132 ± 4 67 ± 6 110 ± 10

[0162]

[0163] Table 1: Table of device parameters.

[0164] vector network analyzer when the buffer is flux biased at the saddle point. We are able to directly measure the CPW modes due to parasitic capacitances between the CPW resonators and the readout transmission line. The measured values are reported in Table 1. The buffer frequency, extrinsic loss, and intrinsic loss are respectively labeled by b, b,e-> Kb,i- The reported uncertainties are standard deviations from repeatedly measuring the buffer over 95 samples. The CPW frequencies, extrinsic losses (due to parasitic capacitance to the readout line), intrinsic losses, and intrinsic quality factors are respectively labeled by Wfc, Kk,e, k,i, and Qi,k- The reported uncertainties are standard deviations from repeatedly measuring the CPWs over 81 samples, all at single-photon powers.

[0165] NOTE 2: EXPERIMENTAL DETAILS AND SETUP

[0166] An experimental setup used for operating the PADL device is shown in Fig. 6.

[0167] The components include a parametric delay line device 600, an Operator-X (OPX)602 from Quantum Machines Inc. (QM), which is used to play and demodulate pulses sent into the PADL device 600, and an Octave 604 from QM, which is used to up-convert and down-convert pulses from the PADL device 600.

[0168] In Fig. 7 we illustrate all the relevant frequencies for our experiment. We show the CPW frequencies 706 (from Table 1) and the buffer frequency 702 (from Table 2). We also show the parametric drive frequencies 704, which are (in GHz): 4.719403, 4.789553, 4.971818, 5.062160, 5.132469, 5.202372, 5.276428. Finally, we show the pump frequency 700, which is 2.84654 GHz.

[0169] Here, we describe the main components of our experiment: (1) parametric driving of the buffer mode, (2) playing and digitizing pulses, (3) DC flux biasing, and (4) flux pumping.

[0170] Parametric driving of the buffer mode: We use a 5 GS / s AWG to simultaneously play the seven drive tones that parametrically couple the CPW resonators to the buffer mode. Each individual drive tone had a Vppranging from 3.3 mV to 27 mV depending on the detuning. These tones are amplified after up-conversion to provide strong enough driving on the buffer mode. The output is combined with the output of the QM OPX and Octave. The on-chip power of each individual drive tone approximately ranged from —99.6 dBm to —120.0 dBm depending on the detuning.

[0171] Playing and digitizing pulses: We use a QM OPX and Octave to generate Gaussian pulses that we store in the PADL. The pulses are synthesized and played from the Analog Outputs on the OPX, which are then up-converted using mixers and local oscillators (LOs) in the Octave. The Octave conveniently calibrates its internal mixers to remove spurious tones from the LO. The up-converted pulse is played from the RF Outputs on the Octave. Similarly, the Octave down-converts pulses after they interact with our device. Specifically, pulses incident on the RF Inputs are down-converted and played from the IF Outputs, which are then sent to the Analog Inputs on the OPX. For Gaussian pulses

[0172]

[0173] with (n) ~ 1 and temporal FWHM of 471 ns, the peak power on-chip is approximately —140.3 dBm.

[0174] DC flux biasing: As shown in Fig. 1G, we have two flux lines 126 and 128 that are symmetrically placed on either side of the ATS 122. We use two voltage sources (SRS SIM928) to bias these two flux lines. The flux lines are low-pass filtered at the 4 K stage of a dilution refrigerator (Aivon Therma-24G).Flux pumping: As shown in Fig. 1G, we have one additional flux line 124 that is placed directly underneath the ATS (where the ATS is grounded). This flux line provides a magnetic flux that symmetrically threads both ATS loops, thereby providing the parametric flux pump ep(t). This flux pump is sourced from a power signal generator (Keysight E8257D PSG). The flux pump power on-chip is approximately -57.6 dBm.

[0175] NOTE 3: DC FLUX BIASING

[0176] We use the two DC flux lines to find the so-called saddle point where

[0177]

[0178] = = TT / 2. In Fig. 8A and Fig. 8B, we show the frequency of the buffer mode in the vicinity of the saddle point as we sweep our two voltage supplies. We tune the buffer mode frequency by sweeping our two voltage sources. The point where As = A A = 7T / 2 is easily identified as a saddle point in a color map of the mode. We work in the basis o

[0179]

[0180] f = (B + V2) / 2 oc and IA = (FL — Fs) / 2 oc where V± and V2 are the voltages of the individual voltage supplies. In reality, slight junction asymmetry between the two SQUID junctions can lead to slight differences in the buffer frequency at different saddle points given by As, A = ±7T / 2. In Fig. 8 A and Fig. 8B, we show two different saddle points taken at two different values of

[0181]

[0182] Fortunately, we find these two saddle point frequencies agree within a linewidth of the buffer mode. We also confirmed this at saddle points taken at two different values of Therefore, we neglect contributions from junction asymmetry in this work.

[0183] NOTE 4: PARAMETRIC DELAY LINE PARAMETERS

[0184] The scattering parameters in Fig. 2A are fit to the following model An model:

[0185]

[0186] where G J, is the buffer frequency, nb eis the buffer extrinsic loss rate, Kb=

[0187]

[0188] + nbiis the total loss rate of the buffer (including extrinsic and intrinsic loss), gkis the parametric coupling between the buffer and the kthCPW,k' is the frequency of the photons from the kthCPW that are being parametrically coupled to the buffer, and Kkis the total loss rate of the kthCPW.

[0189] The fit parameters are listed in Table 2. In the first column, we list the CPW frequencies from Table 1 to index each row. The reported uncertainties are stan-wb / 27F [GHz ± kHz] 5.03123 ± 70

[0190] Kbie / 27r [MHz ± kHz] 3.37 ± 70

[0191] Kb)i / 27r [kHz ± kHz] 440 ± 60

[0192]

[0193] CUfc / 27F [GHz] 4 / 27F [GHz ± kHz] Kfc / 271" [kHz ± kHz] gfc / 27r [kHz ± kHz] 6.904939 5.032140 ± 1 67 ± 6 530 ± 20 6.975562 5.032625 ± 4 78 ± 8 530 ± 20 7.156324 5.031112 ± 9 109 ± 7 540 ± 10 7.247145 5.031607 ± 4 90 ± 8 560 ± 10 7.318975 5.033132 ± 5 130 ± 10 520 ± 20 7.389379 5.033617 ± 4 140 ± 20 520 ± 10 7.460333 5.03058 ± 10 210 ± 30 520 ± 20

[0194]

[0195] Table 2: Table of fit parameters in S'11(ω).

[0196] dard deviations from repeatedly measuring S11(ω) 500 times overnight and fitting the parameters.

[0197] NOTE 5: DOMINANT SOURCES OF INFIDELITY AND ADDITIONAL SIMULATIONS

[0198] Given two temporal modes that are defined by annihilation operators Ai = f dtf (t)dwg(t) and A2= f dtg(t)dwg(t), we define the fidelity as F = | J dtf*(t)g(t)\2. This is the fidelity associated with two single-photon wavepackets A

[0199]

[0200] j|vac) and A^vac). Even though we are not encoding quantum information in single-photon wavepackets, we can still estimate this fidelity from the detected field, i.e., the complex digitized data collected by the analog-to-digital converter. We first turn off the pumps and tune the buffer mode off resonance. The device acts as a mirror which reflects the input pulse. Our ADC records the reflected signal and averages the measured field to find VinM =in[t]

[0201]

[0202] corresponding to the mean field in the reflected pulse. We then operate the PADL in the delay line mode, and record VoutM = βoutM

[0203]

[0204] + corresponding to the mean field of delayed pulse. We normalize both of these averaged traces by the same constant such that ∫Vin[t]Vin[t]* = 1. Fidelity for a given( \2

[0205] delay τ is then calculated by F[τ] = |∫Vin[t − τ]Vout[t]* dt|². We find that fidelity

[0206]

[0207] is maximized for τ ~ Trt, so that F = maxτ(F[τ]) which is very nearly F[Trt]. Note that in the above formulation, since we are using the same normalization constant for both the input and output fields, the fidelity captures the effects of both loss and distortion.

[0208] We can simulate this whole process by numerically integrating the equations of motion for d(t) and b(t) derived from the Hamiltonian in Eq. 4. This allows us to estimate the dominant sources of fidelity loss in our delay line. The equations of motion are given by:

[0209] dak / dt = -(κk / 2 + iΔk)ak - igkb + √κk,e ain

[0210]

[0211] k Since these differential equations are linear, we can take the expected values of the fields, and obtain exact mean field equation for αk =

[0212]

[0213] (dk) and f3 = b). The input and output photon fluxes incident on the buffer mode can be obtained from the input-output boundary condition βout = βin +

[0214]

[0215] Fig. 9 shows plots of input photon flux and output photon flux for five different parametric delay line parameters. Plot 900 shows simulated input and output pulses for the Hamiltonian parameters from our experiment as reported in Table 2 and with an input Gaussian pulse (temporal FWHM of 471 ns). Plots 902 and 904 are from running identical simulations but with Kb,i = 0 and Kb,i = Kk = 0, respectively. In plot 906 we not only consider no intrinsic loss, but we also fix Δk to be evenly spaced from −κb,e / 2 to +κb,e / 2 over 7 modes and fix g / 2π = 562.4 kHz. In plot 908 we consider the same delay line as in plot 906 but consider nonzero Kk given by state-of-the-art CPW resonators quality factors.

[0216] In plot 900, we simulate the Hamiltonian parameters from our experiment, which are reported in Table 2. We choose / 3jn(t) to be a Gaussian pulse with temporal FWHM of 471 ns to match the experiment. The output trace agrees well with the results measured by our ADC in Figs. 2A,2B, and we calculate a state fidelity of F = 0.24. For our fidelity calculation, we normalize the input mode profile ( / (t)) such that f dt|f(t) |2= 1, and we normalize the delayed mode profile (g(t)) by the same factor. In plot 902, we set = 0 and find that the fidelity improves toF = 0.31. In plot 904, we set all intrinsic loss channels to zero, i.e.,

[0217]

[0218] = 0 and find that the fidelity improves to F = 0.86. In 906, we consider a lossless delay line where the Hamiltonian parameters are chosen such that A^ is evenly spaced from — / p>,e / 2 to +K(,,e / 2 over 7 modes and fix g / F = 562.4 kHz (roughly corresponding to a frequency comb with finesse = 1.5). We also increase the pulse temporal FWHM to 942 ns, which improves the fidelity to F = 0.996. Finally, in plot 908, we consider the same delay line parameters as in plot 906 but we add CPW loss again. We consider state-of-the-art CPW quality factors with Qi of 15e6 and find that F = 0.991.

[0219] In plot 900, we also observe a small reflected pulse at t = 0 in the parametrically delayed (output) trace. This small reflected pulse was present throughout our experiments. This is due to the slight impedance mismatch that arises from our nonideal phase response. We observe significant reduction in this reflected pulse when we optimize our Hamiltonian parameters and increase our pulse bandwidth, as can be seen in plot 906. One challenge we faced in preparing a perfect parametric delay line (i.e. a perfectly linear phase response) is the challenge of rapidly and reliably fitting Sn(cj) while tuning the parametric drive amplitudes and detunings, especially given the large number of parameters to fit. Another source of reflections at t = 0 could be inevitable impedance mismatches in our device packaging, most notably from the wirebond connecting our readout line to the PCB.

[0220] Fig. 10 show results of a pulse-swapping simulation of two delayed pulses for a lossless delay line where the parametrically converted CPW photon detunings Δk are not swapped (i.e.,

[0221]

[0222] Δk → Δk and when the detunings are swapped (i.e., Δk → −Δk). In this simulation, we control the detunings of the parametrically converted CPW photons A / ,. to swap two pulses in time. We consider a lossless delay line (with parameters identical to those in Fig. 9, plot 906) and we consider two pulses with temporal FWHM = 377 ns and separation 1000 ns. One trace in Fig. 10 plots the pulse when the detunings have not been swapped, and the other trace plots the pulse when the detunings have been swapped. We clearly see that swapping the detunings swaps the pulse in time. The two pulses have a relative 7r phase shift and the earlier pulse has an amplitude that is twice as large as the later pulse.

[0223] We can use these simulations to approximate the minimum number of resonators needed to perform the simplest possible experiment: delaying a pulse by more than its temporal FWHM. The figure of merit with delay lines is the delay-bandwidthproduct (DBP). Assuming Δk is evenly spaced from −κb,e / 2 to +κb,e / 2, we have that the DBP = N − 1 for the PADL, where N is the number of resonators that are being parametrically coupled to the buffer. The narrowest temporal pulse that one could use will have a temporal FWHM TFWHM — 2TF / 1. Therefore, we have that Tdelay / TFWHM = N − 1. To visually differentiate the stored pulse from the delayed pulse, we require 2TFWHM ~ Tdelay and thus one would require a bare minimum of 3 resonators.

[0224] NOTE 6: GAIN, ATTENUATION, AND ADDED NOISE CALIBRATION To bound the number of added noise photons caused by strongly driving our device, we first need to carefully calibrate the gain of our measurement apparatus. To do this, we need an in-situ noise source, i.e. some way to relate the number of on-chip quanta to a measurable value at room-temperature.

[0225] In our system, we leverage the three-wave mixing interaction of the ATS to operate our CPW resonators as MPOs. We use the number of resonator photons n near threshold as our in-situ noise source. The smooth increase in n vs. driving field observed in our quantum MPO is similar to the behavior of a laser near threshold.

[0226] To operate the kthCPW as an MPO, we pump our device at a frequency ωp = 2ωk − ωb and drive our buffer at a frequency ωd = ωk. To find where the pump and drive frequencies precisely satisfy these energy conservation relations, we sweep them and measure the spectrum of the CPW resonator. Far below threshold, n is only nonzero if the pump and drive frequencies are resonant.

[0227] Once we have determined our resonant pump and drive frequencies, we measure the spectrum of the CPW resonator at different drive strengths. We fit the integrated PSD vs. the drive strength to a master equation model with Hamiltonian H and jump operators L

[0228]

[0229] ± and given by

[0230] H = iε₂â² + h.c.,

[0231] L"2 = (14) L

[0232]

[0233] i = where ε₂ is the two-photon drive strength, κ₂ is the two-photon loss rate, and κ₁ is the single-photon loss rate. An example for the mode at 6.975562 GHz is shown in Fig. 4A.CUfc / 27F [GHz] Gain [dB] K2A1 (n)

[0234] 6.904939 95.70 8.406e-4 0.143

[0235] 6.975562 96.53 22.09e-4 0.092

[0236] 7.156324 94.72 6.323e-4 0.109

[0237] 7.247145 94.59 5.523e-4 0.143

[0238] 7.318975 96.07 7.165e-4 0.083

[0239] 7.389379 96.78 12.73e-4 0.065

[0240] 7.460333 99.11 70.65e-4 0.021

[0241]

[0242] Table 3: Table of the fitted gain, HQ Hi, and added noise.

[0243] Our only fit parameters are: (1) the proportionality constant between

[0244]

[0245] ε₂ / κ₁ and the driving field, (2) the proportionality constant between n and the integrated PSD, which is related to the gain, and (3) K2 / KI. By normalizing everything with respect to κ₁, our fits are robust to pump-induced effects that may worsen κ₁.

[0246] In Fig. 11, we plot our measured gain (in dB) at the different CPW frequencies (error bars are the standard deviation from repeated measurements and fits) from operating them as quantum MPOs near threshold. The gain is the proportionality constant between the integrated PSD at room temperature and the power leaking out of the MPO (PMPO,out = κk,e⟨n⟩) We also report our fitted gains and K2 / KI in Table 3.

[0247] With the gain calibrated near the CPW frequencies, we can bound the noise added from strongly driving our device. Specifically, we turn on the parametric pump and drives that we use for our delay line experiments while measuring the CPW spectra. An example of such a spectrum is shown in Fig. 4B. In Table 3 we report the added noise in each CPW mode and observe it to be much less than 1 photon per mode. In the first column, we list the CPW frequencies from Table 1 to index each row.

Claims

1. CLAIMS1. A parametrically programmable delay line comprising:(a) an ensemble of resonators; and(b) a nonlinear superconducting circuit element parametrically coupled via parametric drives with the ensemble of resonators.

2. The parametrically programmable delay line of claim 1, wherein the parametrically programmable delay line is implemented as a metal on substrate device, wherein the metal is aluminum, niobium, or tantalum, and wherein the substrate is silicon or sapphire.

3. The parametrically programmable delay line of claim 1, wherein the nonlinear superconducting circuit element implements a lumped element read-out / buffer mode.

4. The parametrically programmable delay line of claim 1, wherein the ensemble of resonators are implemented as superconducting transmission line resonators, lumped element resonators, acoustic resonators, or 3D cavity modes.

5. The parametrically programmable delay line of claim 1, wherein the nonlinear superconducting circuit element is implemented as a superconducting nonlinear asymmetric inductive element (SNAIL), a superconducting quantum interference device (SQUID), or an asymmetrically threaded SQUID (ATS).

6. The parametrically programmable delay line of claim 1, wherein the nonlinear superconducting circuit element is capacitively or inductively coupled with the ensemble of resonators.

7. The parametrically programmable delay line of claim 1, wherein the nonlinear superconducting circuit element is configured to allow parametric swapping between nonlinear superconducting circuit element lumped element readout / buffer mode and modes of the ensemble of resonators.

8. A parametrically programmable delay line comprising:(a) a readout resonator coupled to an environment and comprising readout modes;(b) a collection of storage resonators comprising storage resonator modes; and (c) a nonlinear element that enables frequency conversion between the readout modes and the storage resonator modes.

9. The parametrically programmable delay line of claim 8, wherein the readout resonator modes are lumped-element resonator modes or modes of a waveguide or cavity.

10. The parametrically programmable delay line of claim 8, wherein the storage resonator modes are modes of a waveguide or cavity or an acoustic resonator.

11. The parametrically programmable delay line of claim 8, wherein the nonlinear element is a device that enables parametric coupling, such as a so-called Superconducting Nonlinear Asymmetric Element (SNAIL) or a so-called Asymmetrically Threaded SQUID (ATS).

12. The parametrically programmable delay line of claim 8, wherein the nonlinear element is a Superconducting Nonlinear Asymmetric Element (SNAIL) or an Asymmetrically Threaded SQUID (ATS).

13. The parametrically programmable delay line of claim 8, wherein the nonlinear element is a device that enables parametric coupling of the storage resonator modes to the readout resonator modes such that the readout resonator simulates a physical waveguide over its bandwidth.

14. The parametrically programmable delay line of claim 8, wherein the nonlinear element is configured to allow control of amplitude and phase of each storage mode of the storage resonator modes by using a suitable multi-frequency electrical input to the nonlinear element.

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