Coherence and multimode correlations from vacuum fluctuations in a microwave superconducting cavity

The existence of vacuum fluctuations is one of the most important predictions of modern quantum field theory. In the vacuum state, fluctuations occurring at different frequencies are uncorrelated. However, if a parameter in the Lagrangian of the field is modulated by an external pump, vacuum fluctuations stimulate spontaneous downconversion processes, creating squeezing between modes symmetric with respect to half of the frequency of the pump. Here we show that by double parametric pumping of a superconducting microwave cavity, it is possible to generate another type of correlation, namely coherence between photons in separate frequency modes. The coherence correlations are tunable by the phases of the pumps and are established by a quantum fluctuation that stimulates the simultaneous creation of two photon pairs. Our analysis indicates that the origin of this vacuum-induced coherence is the absence of which-way information in the frequency space.


Supplementary Note 1: Derivation of the Heisenberg-Langevin equations of motion
In our analysis we define the Fourier transform as and use the conventions of Ref. [1] for the Fourier transform of the conjugate operator The where ω p with p = 1, 2 are the two pump frequencies. Equivalently, one can replace the two pumps by a single pump operating at the average frequency (ω 1 + ω 2 )/2 but with the amplitude , that is, the amplitude would be modulated at the half-difference frequency (ω 1 − ω 2 )/2. Let us introduce the corresponding detunings with respect to the frequency of the resonator, The dynamics of the system can be solved by writing the Heisenberg-Langevin equations, where κ is the external decay rate of the cavity and a in is the input mode. Next, we write Eq. 5 in a frame rotating at the frequency of the resonator withã = e iωrest a,ã in = e iωrest a in . In the rotating-wave approximation (RWA), by neglecting the terms rotating with high frequencies ±ω p , ±(ω p + 2ω res ) we obtain the equation for the slow component of the fielḋ As in the main paper, we define the Fourier transform with respect to the cavity resonance, with the variable ξ forã and ω for a, such that ξ = ω − ω res , resulting inã . We obtain We note that the procedure above is equivalent to approximating the Heisenberg-Langevin equation Eq. (5) asȧ Eq. (7) can be solved iteratively, by starting at a point ξ, obtaining the first-order reflections 2∆ 1 − ξ and 2∆ 2 − ξ with respect to the pumps, then writing again Eq. (7) at 2∆ 1 − ξ and 2∆ 2 − ξ with second-order reflections 2∆ 2 − 2∆ 1 + ξ (with respect to the second pump) and 2∆ 1 − 2∆ 2 + ξ (with respect to the first pump), and so on. To simplify the results, let us introduce the cavity electrical susceptibility χ(ξ), defined with respect to the cavity resonance ω res , with ξ = ω − ω res . Let us denote byp the relative complement of p in the set {1, 2},p ∈ {1, 2} \ {p}.
A very useful notation is the normalization factor N k (ξ) where k is an index counting the number of reflections, and the superscript [p] is the pump with respect to which the first reflection is performed.
We note that these normalization factors appear naturally whenever a multimode system is studied in the input-output theory, see e.g. Ref. [11]. Then the normalization factor is defined iteratively and The next iterations are obtained as N 2 (2 k ∆p− 2 k ∆ p + ξ). These normalization factors are a generalization of similar factors introduced in the discussion of the dynamical Casimir effect [10] (which can be obtained from those above by truncating the series to k = 2, that is taking N 2 (ξ) = 1). Similar factors also appear in optomechanics [11].
With the above notations we obtaiñ . This can be seen as a phase-insensitive (phase-preserving) amplifier with gain Dissipation can be modeled by introducing a fictive internal port I, distinct from the external port E used for out-coupling the field into the measurement chain. With the internal part, the total decay rate κ consists of an internal component κ I and an external component κ E , with κ = κ E + κ I , and the total input and output fields at any frequency is given bỹ Note that this is a unitary transformation of the beam-splitter type, with probabilities κ E /κ and κ I /κ for a photon to enter/exit the cavity. Knowing a(t) we can obtain the output field in the external mode,ã and similarly for the internal mode. When referenced to the external mode, the gain becomes (18)

Supplementary Note 3: Correlations
To understand the origin and significance of the correlations in this problem, it is better to work with the first-order reflections rather than the full result Eq. (12). Let us truncate the series Eq.
(12) to the first three terms only. This corresponds also to the real experimental situation, where only these terms are seen clearly at low pumping powers. One can see also from Fig. Supplementary   Figure 1 that, especially if ξ is in-between the pumps ∆ 1 < ξ < ∆ 2 and not too large (that is, close to the cavity resonance), then the second-order reflections at 2∆ 1 − 2∆ 2 + ξ and 2∆ 2 − 2∆ 1 + ξ are rather far away. According to Eq. (12), their input fields will be multiplied by the cavity susceptibility at these frequencies -thus they will be much smaller than the first-order reflections.
We parametrize the pump amplitudes α 1 and α 2 using Here the asymmetry between the pump strengths is accounted for by the angle θ and A is a positive real number characterizing the overall pumping amplitude. For convenience, we took here the variable ξ as referring to frequencies near the origin, |ξ| |∆ p |, which yields χ * (2∆ p − ξ) ≈ χ * (2∆ p ).
Next, we introduce two modes, a 'bright' oneb and a 'dark' oned, defined bỹ These two modes are orthogonal to each other. Note that, the definition of modesb andd resembles a beam-splitter operation in frequency space, where a mode is separated into two branches with a distance 2(∆ 1 − ∆ 2 ) between them.
With these notations we can calculateã out [ξ] using Eq. (12). In order to make the mathematical structure even more transparent, we can further simplify the result by neglecting the terms ξ/κ 1 and (∆ 1 cos 2 θ + ∆ 2 sin 2 θ)/κ that appear in the normalization factor. The justification of this approximation is that, in order to be able to measure any signal at all, the detunings ∆ 1 , ∆ 2 , and ξ have to be in general within the decay rate of the cavity, otherwise they will be filtered out. With Eq. (23) allows us to identify the stability condition A(ξ) < 1. The structure of Eq. (23) suggests the parametrization A = tanh(λ/2), which results iñ Similarly, to find the outputb mode, we write the result Eq. (12) at 2∆ 1 − ξ and 2∆ 2 − ξ, neglect the fields at 2∆ 1 − 2∆ 2 + ξ, 2∆ 2 − 2∆ 1 + ξ and beyond, and use the definition Eq. (21).
Within the same approximation as above, we obtaiñ Let us assume now zero temperature limit, a condition which is satisfied in our experiment.
Then we obtain and in terms of the bright mode, By contrast, applying the same procedure to the dark mode results in The dissapearance of noise power in the dark mode is clearly a coherence effect, which is conceptually different from destructive interference. Indeed, the two modes at 2∆ 1 − ξ and 2∆ 2 − ξ are separated in frequency and they do not really come together to overlap. The coherence is created by the vacuum fluctuations at ξ which triggers correlated spontaneous parametric downconversion two-photon processes in the pumps.
As in the usual interference phenomena, there exist also coherent terms that depend on the phase of the pumps, All these correlations can be extracted from the experimental data by using the field quadratures for the output field, defined at each frequency, ξ, 2∆ 1 − ξ, and 2∆ 2 − ξ, which, when convoluted with the filtering function, give the I-Q quadratures measured in the experiment, see Methods.
Supplementary Note 4: Tripartite structure for the parametric vacuum-induced coherence effect To have a physical understanding of the origin of these correlations, we use the powerful principle of which-way information introduced in quantum optics and used to interpret many fundamental experiments in optics [2]. Our work provides a fully new system to apply these concepts, namely a microwave setting at very low temperatures. As always in microwave-based measurements, we cannot do single-photon experiments due to the lack of single-photon detectors; also we do not have showed that by aligning the idler beams from two nonlinear crystals, where spontaneous parametric downconversion occurs, it is possible to induce coherence between the two remaining signal beams.
More recently, Zeilinger's group in Vienna demonstrated that this principle can be used for optical imaging with undetected photons [6]. In these experiments, it is the lack of which-path information that created the interference in the signal beams. In our case, there is no path in real space. Instead, we can talk about which-color information: for a real photon, at frequency ξ, there is no way to know from which of the two spontaneous parametric downconversion process around ∆ 1 or ∆ 2 it came from. Another peculiarity of our system is that there are not only two downconversions, but an infinite sequence of them: the mode ξ acts as the common idler for both pumps, producing two signals, which in turn act as the idlers for the pumps where they did not originate, etc..
To discuss the connection with these fundamental aspects of few-photons physics, let us consider only the first-order reflections and simplify the notations by using the bandwidth-averaged modes a 0 ,ã 1 , andã 2 . Suppose we work at such low intensities that at most two photons of either of the frequencies ξ, 2∆ 1 − ξ, 2∆ 2 − ξ are present in the system at a time. One of these photons must always be in the modeã 0 : otherwise, the two photons should be one inã 1 , the other inã 2 , and there is no term in the Hamiltonian that corresponds to this process. Now, for the photon atã 0 there is no way, even in principle, of determining where it came from -pump 1 or pump 2. The probabilities of originating from the downconversions in either one of these pumps might be different (for example if the pumps have different intensities) but for a particular occurrence of a photon inã 0 there is nothing distinctive to allow us to say where it came from. This lack of which-way information implies that, in the two-photon subspace, the state of the resonator at each frequency ξ must be a superposition, where |β ã 1 ;ã 2 = (1/ √ 2) |1 ã 1 |0 ã 2 + |0 ã 1 |1 ã 2 is a Bell state. For simplicity, we assume here that the two processes have equal probability amplitude, which can be achieved by considering two pumps with properly adjusted amplitude and phase. Thus, even in the low-intensity limit, one does expect the appearance of correlations between the photons in the modesã 1 andã 2 .
The appearance of the state Eq. (39) can be obtained more formally by analyzing the structure of the input-output equations. Let us consider a resonator with a large enough external coupling κ, ensuring that the field is nonzero at some finite detuning, in other words we take |ξ|, ∆ 1 , ∆ 2 κ.
Then, we can write Eqs. (24,25) asã where the squeezing transformation S is defined as This imposes a quite specific structure for the output vacuum which follows from the requirementsã j,in |0 in = 0ã j,out |0 out = 0, where j = 0, 1, 2. When the pumps have identical amplitudes and phases, from Eq. (43) it follows immediately that the output state in the subspace with two photons has the structure Eq. (39), which was derived from general quantum-information concepts. The state Eq. (43) is a two-mode squeezed state when considering the modes (ã 1,in +ã 2,in ) / √ 2 andã 0,in , while in the subspace of the modesã 1,in andã 2,in the double pumping yields a beam-splitter transformation, producing a superposition. This is called a bisymmetric state [7,8]. When truncated to the subspace containing no more than two photons (a regime that can be achieved experimentally by lowering the pump powers), the state obtained from Eq. (43) is a superposition of the vacuum and the two-photon state Eq. (39). This is a tripartite state in the W class [9], as can be checked by applying an X gate to the photon in modeã 0 .

Supplementary Note 5: Single-photon effects
The coherence effect seen in the experiment has a quantum-mechanical origin. It stems from the wavelike-character of quantum particles, which cannot be seen here as discrete lumps of energy that are randomly created by parametric downconversion processes occuring in each pump separately.
To see this more clearly, we will rule out one by one all reasonable semiclassical type of explanation based on the particle character of the photons. First, let us introduce the cavity density of states As expected, in a lossless cavity D(ξ) = δ(ξ), corresponding to a single mode at the resonance frequency. With this notation, we now investigate the power spectrum produced by a single parametric downconversion process, which can be obtained immediately from our general result Eq.
(12) by switching off one of the pumps, with p = 1, 2. This of course is the result for the single-pump dynamical Casimir effect (DCE), see Eq. (1) in Ref. 10. The quantity DCE p (ξ) is the adimensional noise spectrum that specifies how much power <ã † outã out > is in the output mode relative to the background vacuum level, at frequency ξ. Here |α p | is the rate of pump downconversion from the Hamiltonian Eq. (3). The rate of power conversion from the pump (the rate of pump depletion) is then |α p | 2 , which has to be multiplied with the densities of states at the frequency of the idler and of the signal.
The coherence persists down to average cavity photon numbers of the order of unit and even below, see Eq. (39). This equation shows that the superposition in the bright mode exists even for a single photon present in the modesã 1,in andã 2,in . This is important to understand, because one might try the following semi-classical description: suppose that one spontaneous parametric downconversion process occurs in pump 1: as a result, two photons are created, one at ξ and the other at 2∆ 1 − ξ. Next, the photon at ξ acts as signal for the second pump, which triggers another downconversion process, with the creation of another pair, with one photon at ξ and another one at Thus, the minimum number of photons required would be four, and also this process would be second-order: its probability would depend on the product of the squared pump amplitudes.
This would imply that the total power spectrum is DCE 1 (ξ) × DCE 2 (ξ) ≈ |α 1 | 2 |α 2 | 2 . This is clearly not the case, and the formalism developed in Sect. III shows that the process depends linearly on the power of the pumps.
Our doubly pumped DCE system illustrates some of the standard paradoxes of quantummechanical interference in a new setting. Let us postulate that we would be in possession of a single-photon detector, with enough frequency resolution so that it could discriminate between the frequencies ξ, 2∆ 1 − ξ, and 2∆ 2 − ξ. Such a detector, when placed at the output of the cavity, would detect one by one the photons, as they were produced in the cavity. This would lead immediately to an apparent paradox: as the detector would detect a photon at the frequency 2∆ 1 − ξ, we could infer that this was the result of a parametric downconversion in pump 1, accompanied by the creation of a photon at ξ. This would ensure energy conservation: a downconversion in pump 2 would have produced one photon at energy ξ and another one at 2∆ 2 − ξ, and the sum of these frequencies would equal 2∆ 2 instead of the required 2∆ 1 . This means that we were able to extract which-color information, and as a result the coherence, when counting large enough number of photons, should disappear. The situation is analogous to the vanishing of the interference fringes in a two-slit interferometer where the transversal momentum kick onto the panel with the slits is measured in order to extract which-way information. According to this reasoning, the interference should not exist even in the case of our experiment, simply because it would be in principle possible to use a single-photon detector.
The apparent paradox discussed above is solved immediately when we apply the time-energy uncertainty principle to the resonator. The decay time of the cavity sets the time uncertainty to δt ∼ κ −1 , which yields ∆E κ. This means that it is not possible to infer from the existence of a detection event at 2∆ 1 − ξ that the downconversion occurred in pump 1. Attempting to increase the quality factor of the cavity (while keeping the detunings ∆ 1 and ∆ 2 fixed) will result in the overall loss of signal at ξ, 2∆ 1 − ξ, and 2∆ 2 − ξ. Thus the coherence does eventually dissapear while attempting to extract which-color information, though the mechanism is somewhat different from the case of standard optical interferometer experiments.

Supplementary Note 6: Pulsed parametric pumping
We analyze here the case when the pumps act only during finite time intervals T p , p = 1, 2 centered at delay times τ p , respectively. In other words, the pumping amplitudes α p can be approximated as rectangular pulses The Fourier transform is where the sinc function is defined as sinc(x) = sin(x)/x. We now wish to solve approximately the Heisenberg-Langevin equation with time-dependent pumps, Taking the Fourier transform we obtain where the effect of parametric pumping on the modes is obtained as a convolution. Next, we write this equation at the first reflections 2∆ p − ξ with respect to the pumps, .
This corresponds to the regime of small gain, similar to a linear approximation in λ in the continuous case discussed before cosh(λ) ≈ 1, sinh(λ) ≈ λ. Using the cavity susceptibility we can estimateã One can notice that, indeed, the first term after the approximately sign yields a gain of 1 and a phase shift which corresponds to the well-known reflection −(κ/2 + iξ)/(κ/2 − iξ) from a cavity in the absence of pumping. Using this expression, we can calculate the vacuum-induced correlation We assume that the response of the cavity is rather flat χ ≈ 2/κ in the region between 2∆ 1 and 2∆ 2 , in other words κ >> ∆ p . In this case φ p ≈ ϕ p according to the definitions used before. We can use the fact that the convolution of two sinc functions is also a sinc function, and we find where T 0 is the duration of overlap of the two pulses and τ 0 is the center of this overlap. Assuming τ 2 > τ 1 , In the experiment, this correlation is calculated at ξ 1 ≈ ξ 2 where it reaches the maximal value Note that in the case of large T 0 we recover the results of continuous pumping, by using the sinc representation of the Dirac function, with → 0. When applied to Eq. (52) we get the previous result,  The experiment relied on three phase-coherent sources, the two pumps from two low phase noise microwave generators and the LO of the signal analyzer. The coherence provided by the common 10 MHz frequency reference from a rubidium standard provided sufficient stability for obtaining the results presented in this paper.

Supplementary Note 8: Additional information about data analysis
The signal analyzer digitizes I(t) and Q(t) at a sampling rate of 50 MHz which also yields a data bandwidth of 50 MHz (Nyquist limit for quadrature information). Additionally, the signal analyzer has a built-in sharp-edged hardware filter limiting the analog bandwidth to 31.25 MHz thus eliminating potential aliasing artifacts. Due to computing limitations (as well as hardware phase drift related issues) we transform and process at most N = 2 23 ≈ 8M samples at a time.
Altogether we collect approximately 20 GB of data.
This data vector is Fourier transformed in the complex form yielding a complex vector F N 1 = FFT(I + iQ) containing the amplitude and phase representation of the signal in frequency domain.
Here subscript and superscript are used to indicate the index of the first and last element. This vector is constructed such that the first index (1)  Two mode squeezing can be extracted by simply repeating the previously described procedure for two separate frequencies and calculating the correlation separately for all pairs and combinations of real and imaginary parts. This was done to obtain the histogram presented as the inset of Fig. 3.
in the main paper.