Generation of nonclassical states of light (particularly, spontaneous parametric downconversion [1]) was the impetus of quantum optics development and led to the discovery and demonstration of such quantum effects and techniques as Hong-Ou-Mandel interference [2], ghost imaging [3–6], Schmidt-like correlations [7–11], superresolution with multiphoton interference [12,13], and so on. However, it was shown later that all listed effects can be realized with classically correlated states of light (particularly thermal states), but with low visibility [14–19]. It means that these effects are actually based on the classical intensity correlations and (in contrast with Bell inequality violation and so on) can be explained in terms of classical physics and statistics (as well as Hanbury Brown and Twiss interference, which contrariwise was first demonstrated with thermal light [20] and later become a standard technique for biphoton registration). Continuing this list of thermal light realization of quantum optics effects, we decided to test the recently reported “quantum vampire” effect [21]: is it really quantum or not?

The effect called quantum vampire is a “collapse-free action at a distance by the photon annihilation operator” [see Fig. 1(a)]. Actually, it is related to the multimode problem of photon addition and subtraction, which has become very popular in recent years because of its applicability to the optical quantum computation [22–24]. The quantum state $|\psi ⟩$ comes to the input “1” of the beam splitter (BS), while the vacuum state comes to the other input, “0.” Next, the state $|\psi ⟩$ is distributed between the modes “$A$” and “$B$,” and the photon annihilation process ${\widehat{a}}_A$ takes place in the mode $A$. But the operator ${\widehat{a}}_A$ can be represented as a linear superposition of annihilation operators in modes 0 and 1:

 

Fig. 1. Quantum vampire effect, exploiting (a) Fock states [21] and (b) thermal states. Initial quantum state in the mode 1 is exposed to the annihilation operator in the mode $A$. Due to the direct proportionality between photon annihilation operators in the modes $A$ and 1 (2), the photon subtraction in the mode $A$ leads to the photon subtraction in the mode 1 and therefore in the mode $B$.

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In the original work [21], the experiment was performed with Fock states $|\psi ⟩=|1⟩$ and $|\psi ⟩=|2⟩$. In this case, the modes $A$ and $B$ are entangled in a photon number. Thus, authors explained the quantum vampire effect in terms of “nonlocality.”

Here we should note that actually, there are no experiments demonstrating real nonlocality, i.e., the situation where some action in one point of space immediately causes some measurable changes in the other points. Even subsystem measurement of a complex entanglement system cannot be detected by the measurement of the other subsystem. However, in terms of quantum collapse, one can claim that this subsystem measurement causes the distillation of the mixed state of the other subsystem and this nonmeasurable action on the distance can be called “quantum nonlocality.” Thus the “nonlocality” term usually means “entanglement.”

On the other side, the authors of the work [21] underline that quantum vampire is a “nonlocal effect, occurring without local state collapse by either party.” Moreover, they claim, it does not require entanglement and mention a coherent state as an example: since the coherent state is the eigenstate of the annihilation operator, the photon subtraction cannot disturb it and really casts no shadow. But we should note that there is also no “action at a distance” in this case.

In order to understand whether the quantum vampire effect is really a quantum one, we followed Klyshko’s statement: “All linear optical schemes are described by the classical propagators and, therefore, the quantum specificity, if it is present, is caused only by the light source at the system input” [25]. The experimental setup in Fig. 1 really consists of linear optical elements, including the photon annihilation operator, which can be implemented just probabilistically—typically by using a low-reflective beam splitter combined with the single-photon detector in the reflective channel. Detector clicks corresponds to the photon annihilation [26]. So, all the measured results can be determined by the linear optical transformation of the initial state of light and subsequent postselection.

Since in the original work [21] only two limiting cases of the light sources have been considered (nonclassically correlated Fock states demonstrating “action on a distance” and noncorrelated coherent state demonstrating no action), we decide to test the quantum vampire effect with classically correlated thermal states, in order to understand whether the described nonlocality is based on the quantum specificity or not.

The density matrix of a thermal state has a well-known diagonal form with Bose–Einstein photon number distribution $P_{\mathrm{BE}}$:

This state has the only parameter ${\mu }_0$, a mean photon number. Due to the $k$-photon subtraction, this state transforms as follows [27–30]:

One can find how ${\widehat{\rho }}_{\mathrm{th}-}$ transforms under photon subtraction and under linear loss with a transmission coefficient ${|t|}^2$ (see Supplement 1 for details):

Consider now the quantum vampire effect applied to the thermal state. The sketch of the experiment is shown in Fig. 1(b). Thermal state ${\widehat{\rho }}_{\mathrm{th}}{({\mu }_0)}_1$ comes to the mode 1 After the beam splitter (BS), there are two correlated thermal states in the modes $A$ and $B$: ${\widehat{\rho }}_{\mathrm{th}}({|t|}^2{\mu }_0{)}_A$ and ${\widehat{\rho }}_{\mathrm{th}}({|r|}^2{\mu }_0{)}_B$. It can be shown that the Pearson correlation coefficient [32] of the modes $A$ and $B$ equals

If single photon subtraction takes place in the mode $A$, ${\widehat{\rho }}_{\mathrm{th}}({|t|}^2{\mu }_0{)}_A$ converts to ${\widehat{\rho }}_{\mathrm{th}-}({|t|}^2{\mu }_0,1{)}_A$. Since the photon annihilation operator ${\widehat{a}}_A$ is proportional to the operator ${\widehat{a}}_1$ (2), the photon subtraction in the mode $A$ corresponds to the photon subtraction in the mode 1, which leads to the transformation ${\widehat{\rho }}_{\mathrm{th}}{({\mu }_0)}_1\to {\widehat{\rho }}_{\mathrm{th}-}{({\mu }_0,1)}_1$. The last means that there should be the attenuated photon-subtracted state ${\widehat{\rho }}_{\mathrm{th}-}({|r|}^2{\mu }_0,1{)}_B$ in the mode $B$. It seems that the above description meets some trouble with causality, but we should note again that the photon annihilation operator can be implemented just probabilistically: the detector click realizes the postselection of photon-subtracted states in all the modes (see Supplement 1 for details). Moreover, double-click means double-photon subtraction and leads to the registration of the double-photon-subtracted states ${\widehat{\rho }}_{\mathrm{th}-}({|t|}^2{\mu }_0,2{)}_A$ and ${\widehat{\rho }}_{\mathrm{th}-}({|r|}^2{\mu }_0,2{)}_B$ in the modes $A$ and $B$.

Experimental tests based on the conditional photon subtraction from the thermal light have been realized both in the pulsed [33,34] and in the cw [29,30] regime. The last case enables multiple photon subtraction with the use of just one single-photon detector by means of the multiple temporal mode registration. The principal scheme of the experimental setup shown in Fig. 2 is similar to the setup described in [30], so all the technical details of the experiment as well as the data processing and quantum state tomography algorithm can be found there.

 

Fig. 2. Experimental setup. The He–Ne cw laser radiation is asymmetrically distributed by the FBS between the local oscillator and the state preparation modes. The thermal state is prepared with use of GGD [35,36]. BS1 splits the input mode to the modes $A$ and $B$. Conditional photon subtraction is realized by a low-reflective BS2 combined with a single-photon detector (D). The FM selects the mode ($A$ or $B$) to be measured by the homodyne detection technique [37].

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An initial single-mode thermal state of light was prepared by the passing cw He–Ne laser radiation at the wavelength of 633 nm through the rotating ground glass disk (GGD) [35,36] and subsequent spatial mode filtering with the use of the single-mode optical fiber. The corresponding coherence time ${\tau }_{\mathrm{coh}}=120\mathrm{\mu s}$ approximately equals the time it takes for the disk grain to cross the laser beam. The other part of the laser radiation was separated by the fiber beam splitter (FBS) and served as a local oscillator for the following homodyne detection.

Prepared thermal states were sent to the input of the beam splitter (BS1) and distributed between channels $A$ and $B$.

Conditional photon subtraction in channel $A$ was realized by beam splitter BS2 with reflectivity ${|r|}^2=1\%$, combined with a homemade single-photon detector D, based on the avalanche photodiode (APD) with 500 Hz dark count rate and a dead time ${\tau }_d=2\mathrm{\mu s}$ placed in the reflection channel. Since ${\tau }_d\ll {\tau }_{\mathrm{coh}}$, it was possible to register several photocounts during the coherence time, which corresponds to multiple photon subtraction [30].

Next, both channels were directed to the homodyne detection scheme [37]. Selection of the channel to be measured was implemented by the flipped mirror (FM).

Density matrix reconstruction under the measured quadrature distributions was done by the maximum likelihood estimation technique [38] with the simple two-parameter model of the quantum state (4), i.e., just two parameters, ${\mu }_0$ and $k$, were estimated. This model enables both good fit of the experimental data and a high-precision parameter estimation. To account for the homodyne efficiency $\eta =78\%$, the model quadrature distribution $P(q)$ was additionally smoothed with a Gaussian function $\exp [-q^2\eta /(1-\eta )]$.

The quantum vampire effect has been tested with the thermal state under single- and double-photon subtraction in the mode $A$. The mean photon number of the initial thermal states in the modes $A$ and $B$ equals ${\mu }_{0A}=4.60\pm 0.02$ and ${\mu }_{0B}=4.55\pm 0.03$.

The results of the experiment are presented in Fig. 3. The measured quadrature distributions are shown as histograms in the insets; the acquired sample size was varied from 5,800 to 110,000, depending on the number of subtracted photons. The reconstructed photon number distributions, which completely define the density matrix ${\widehat{\rho }}_{\mathrm{th}-}$ (4), are also shown as histograms on the main plots. Red solid lines correspond to the initial thermal states, blue dashed dotted lines correspond to one-photon-subtracted thermal states, and green dashed lines to double-photon-subtracted thermal states.

 

Fig. 3. Photon number distributions and quadrature distributions for the initial thermal state and states with one- and two-photon subtraction. Experimental data are plotted as histograms; theoretical distributions are figured as lines.

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The left column corresponds to the state in $A$ mode and the right column in $B$ mode. The top row indicates initial thermal states while no photocounts have been detected, the middle row to one-photon subtraction in the mode $A$, and the bottom row to the double-photon subtraction in the mode $A$. It is easy to see that photon subtraction in mode $A$ modifies not only the state ${\widehat{\rho }}_{\mathrm{th}-A}$, but ${\widehat{\rho }}_{\mathrm{th}-B}$ as well. Thus, we conclude that the quantum vampire effect can be demonstrated with thermal states of light as well.

We also estimated the agreement between theoretical and experimental density matrices by calculating the fidelity:

In conclusion, we have shown both theoretically and experimentally, that the quantum vampire effect can be included in the list of quantum optics effects, based on the photon number correlations, both quantum and classical. It does not need any “quantum nonlocality” or entanglement; it demonstrates just conditional postselective nonlocality, which can be completely explained in terms of classical physics and statistics. However, we hope our implementation of a quantum vampire with thermal light will raise interest in this effect because of its accessibility for quantum optics experimentalists.