Finite-time quantum entanglement in propagating squeezed microwaves

Two-mode squeezing is a fascinating example of quantum entanglement manifested in cross-correlations of incompatible observables between two subsystems. At the same time, these subsystems themselves may contain no quantum signatures in their self-correlations. These properties make two-mode squeezed (TMS) states an ideal resource for applications in quantum communication. Here, we generate propagating microwave TMS states by a beam splitter distributing single mode squeezing emitted from distinct Josephson parametric amplifiers along two output paths. We experimentally study the fundamental dephasing process of quantum cross-correlations in continuous-variable propagating TMS microwave states and accurately describe it with a theory model. In this way, we gain the insight into finite-time entanglement limits and predict high fidelities for benchmark quantum communication protocols such as remote state preparation and quantum teleportation.


SAMPLE DETAILS
The basis of our Josephson parametric amplifier (JPA) consists of a quarter-wavelength superconducting microwave resonator in a coplanar waveguide (CPW) geometry (see Fig. 1, Refs. 1, 2). By short-circuiting the microwave resonator on one side to the ground plane via a direct current superconducting quantum interference device (dc-SQUID), the resonant frequency f 0 of the JPA can be tuned by an external magnetic flux applied to the dc-SQUID loop via an external coil or via an on-chip antenna ("pump line"). The JPA samples used in this work were designed and fabricated at NEC Smart Energy Research Laboratories, Japan and RIKEN, Japan. Thermally oxidized silicon with a thickness of 300 µm is used as a substrate. The resonator and the pump line are patterned into a previously sputtered Nb film with a thickness of 50 nm. A coplanar waveguide (CPW) geometry is used for the resonator and the pump line. The dc-SQUID is fabricated using the aluminum shadow evaporation technique. Two JPAs (with the design shown in Fig. 1) have been used for the current experiments of generation of propagating microwave TMS states. Table I summarizes the most relevant parameters of the used JPAs obtained from spectroscopic characterization measurements [3].
The operational principle of squeezing with a flux-driven JPA consists in applying a strong microwave pump tone at port P at the frequency 2f 0 , where f 0 is the flux-tuneable resonant JPA frequency. This results in parametric degenerate (phase-dependent) amplification of an incoming mode f 0 at the signal port. In the case of incoming vacuum (no signal) the JPA effectively produces a squeezed vacuum state which leaks out from port S and can be I: Parameters extracted from fitting of the flux-dependent JPA resonant frequency. The external quality factors Qext and internal quality factors Qint are obtained from independent fits of the JPA spectral linewidths [3]. The critical current Ic is given per Josephson junction, EJ is the corresponding Josephson energy, and βL denotes the screening parameter of the dc-SQUID.

Sample
Ic ( used as a quantum resource for further operations. To separate the incoming (vacuum) and outgoing (squeezed vacuum) signals we employ commercial cryogenic circulators (see Fig. 2).

EXPERIMENTAL SCHEMATICS
The dual-path receiver [1,4] and the cryogenic double-JPA setup are shown in Fig. 2. Squeezed microwave states are produced by the JPAs, which are stabilized at the temperature of T = 50 mK. Then, the resulting states are superimposed by means of a microwave cryogenic hybrid ring beam splitter producing two path-entangled modes containing TMS states. Isolators and circulators provide isolation from spurious reflections. After the first amplification stage with the cryogenic high-electron-mobility transistor (HEMT) amplifiers, the signals are further amplified at room temperature by a chain of rf-amplifiers, which are also temperature-stabilized by a Peltier cooler. Bandpass filters (4.9-6.2 GHz) provide a rough filtering around the desired JPA frequency f 0 .
For the data acquisition, we use a two-stage down-conversion scheme as shown in Fig. 2. First, we downconvert the amplified microwave signals to a fixed intermediate frequency (IF) by mixing input signals at f 0 with a detuned strong local oscillator (LO) such that f IF = f LO − f 0 = 11 MHz. For this mixing process, we use image rejection mixers. The use of these mixers is crucial for filtering out an unwanted signal sideband at frequencies f side = f 0 + 22 MHz, which otherwise contributes to the IF mode and fundamentally limits the reconstructed squeezing levels to 3 dB. We use tunable attenuators for analog pre-balancing of the two channels. Narrow-band filters (9.5-11.5 MHz) further reduce noise and protect the subsequent IF amplifiers from saturation. Finally, an additional chain of IF filters and dc-blocks purges the signals before digitization.
In order to convert the analog signals to the digital domain, we use the Acqiris DC440 analog-to-digital conversion (ADC) card. After transferring the digitized IF data to a computer, we perform a digital downconversion using an efficient, custom-made, multi-threaded acquisition program. The latter also applies fine digital filtering of the incoming signals using a finite-impulse-response (FIR) filter with a tunable bandwidth Ω. The same sofware is used to calculate all correlation moments I n 1 I m 2 Q k 1 Q l 2 with n + m + k + l ≤ 4 for n, m, k, l ∈ N, and to perform averaging. After each averaging cycle, typically consisting of 1.5 × 10 9 samples, we apply the reference state reconstruction code in order to extract the signal operator moments (â † ) nâm and build a corresponding Wigner function. The extracted "on the fly" squeezing angles φ exp1 and φ exp2 are used to calculate a phase correction δφ 1,2 = φ exp1,2 − φ target1,2 in order to stabilize the squeezing angle by correcting the phase of the microwave pump tone by δφ pump1,2 = 2 δφ 1,2 . Finally, the described cycle is repeated approximately 50-100 times in order to obtain sufficient statistics.

INPUT-OUTPUT THEORY
For the description of the properties of SMS and TMS states, let us first consider the following Hamiltonian of a nonlinear driven resonator (in our case acts as a JPA) with a fundamental frequency f 0 = ω 0 2π: where  is the free Hamiltonian. The first term in Eq. 1 is the squeezing Hamiltonian with nonlinearity χ, the second term is the coupling between the input signal and the transmission line with a coupling rate κ, while the last part is the coupling with a bath with a coupling rate γ. Here and further, for simplicity, we omit operator hats in all equations and set = 1. Defining b(ω) ≡ Be −iω0t , c(ω) ≡ Ce −iω0t , the Heisenberg equations for the modes B(ω) and C(ω) in the interaction picture of H I = ω 0 a † a areḂ The solution of Eq. (3) can be written in terms of two different initial conditions: with respect the input signal B 0 (ω) at time t 0 < t, or with respect the output signal B 1 (ω) at time t 1 > t: Similar equations hold for c(ω). Regarding the resonator mode a, we havė (6) is the dynamical equation for the intra-resonator field, depending on the input signal b in and loss mode c in . If we define the output modes

SQUEEZING
Here, we derive how much squeezing we have in the steady-state of the output signal. Let us consider the equations for the field quadratures q a ≡ (a + a † )/2, p a ≡ − i(a − a † )/2 ([q a , p a ] = i/2, as [a, a † ] = 1).
where q(p) b(c)in have been defined similarly as q(p) a . For the steady state, we obtain thatq a =ṗ a = 0, so Using the boundary conditions, we have that Here, the c in corresponds to the coupled noise mode, and it is usually taken to be the vacuum or a thermal state.

NEGATIVITY IN TERMS OF INTERNAL PARAMETERS AND FILTER FUNCTION
We now calculate the negativity for an entangled system in which one of the paths has suffered a time delay. In order to get an expression for the negativity, it is necessary to compute the terms of the covariance matrix. The system we study is composed of two squeezed states with squeezing phases φ 1 and φ 2 , respectively, which are sent to the two ports of an entangling beam splitter that introduces the following operation: where a ′ 1 and a ′ 2 are the fields emerging from the output ports of the beam splitter. After this, a time delay τ operation is introduced in the second beam. Finally, a measurement is performed in the generalized quadrature x ′ i,λi = (a ′ i e −iλi + a ′ † i e iλi )/2 of path i = 1, 2 with a phase λ i , where we have to take into account the filter function. Here, λ i denotes the angle in which we are measuring and which corresponds to quadrature q when λ i = 0, or p when λ i = π/2. Accordingly, in order to calculate the negativity (or negativity kernel), which is a function of the covariance matrix, which elements should be calculated first. These elements are We now introduce the transformation of a and a † operators for squeezed stateŝ where ξ = re iφ , and C ≡ cosh(r) and S ≡ sinh(r). Hence, Eq. 22 yields Since [a 1 (t), a † 2 (t ′ )] = 0, one gets where 0|a i (0)a † i (τ )|0 = 0|[a i (0), a † i (τ )]|0 = 0|[a(τ ), a † i (0)]|0 = sinc(Ωτ ). So, one finally obtains the expression Following the same formalism, it is possible to obtain covariance matrix entries for the self-correlations q i p i where C i ≡ cosh(r i ) and S i ≡ sinh(r i ). If one, instead of squeezed vacuum states, considers a squeezed thermal state then Eqs. (27)-(29) become, respectively, where n th1 and n th2 are average number of thermal photons referred to the inputs of JPA1 and JPA2 respectively. Notice that for this, we have calculated the terms n|a † (0)a(τ )|m = n sinc(Ωτ )δ n,m , n|a(0)a † (τ )|m = (n + 1)sinc(Ωτ )δ n,m .
In the case of squeezed thermal states we can, based on the derivation of Eq. (36), write the covariance matrix entries of a single-mode squeezed state as For simplicity assuming φ i = 0 one arrives at: where σ 2 s and σ 2 a are the squeezed and antisqueezed quadrature variances in terms of the squeezing factor r and number of noise photons n th .