Generation of Large Number-Path Entanglement Using Linear Optics and Feed-Forward

TL;DR

This paper presents a linear optics and feed-forward method to efficiently generate large N00N states and Schrödinger cat states, with exponential scalability, for applications in quantum imaging and metrology.

Contribution

It introduces an idealised measurement and feed-forward scheme to produce large number-path entanglement states with improved scalability over previous methods.

Findings

High-probability routing of photons using partial detection

Generation of Schrödinger cat states through dual-port detection

Exponential scalability compared to existing proposals

Abstract

We show how an idealised measurement procedure can condense photons from two modes into one, and how, by feeding forward the results of the measurement, it is possible to generate efficiently superpositions of components for which only one mode is populated, commonly called ``N00N states''. For the basic procedure, sources of number states leak onto a beam splitter, and the output ports are monitored by photodetectors. We find that detecting a fixed fraction of the input at one output port suffices to direct the remainder to the same port with high probability, however large the initial state. When instead photons are detected at both ports, Schr\"{o}dinger cat states are produced. We describe a circuit for making the components of such a state orthogonal, and another for subsequent conversion to a N00N state. Our approach scales exponentially better than existing proposals. Important applications include quantum imaging and metrology.

Figures

Equations

$$P_{\mathrm{cond}}$$

$$|{\psi }_{l,r}⟩$$

$$|{\psi }_{\infty }\left({\Delta }_0\right)⟩\propto {\int }_0^{2\pi }d\theta \exp \left(-iS\theta \right)|\alpha ⟩|\alpha \exp (i{\Delta }_0)⟩,$$

$$U_{\mathrm{bs}}|{\psi }_{\infty }⟩$$

$$\left(\begin{array}[]{c}\hat{o}_{1}\\ \hat{o}_{2}\\ \end{array}\right)\longrightarrow\left(\begin{array}[]{cc}\cos(\gamma)&-\sin(\gamma)\\ \sin(\gamma)&\cos(\gamma)\\ \end{array}\right)\left(\begin{array}[]{c}\hat{o}_{1}\\ \hat{o}_{2}\\ \end{array}\right)$$

$$|{\psi }_{\mathrm{cat}}\left({\Delta }_0,\Lambda \right)⟩\propto |{\psi }_{\infty }\left({\Delta }_0\right)⟩+\exp \left(i\Lambda \right)|{\psi }_{\infty }\left(-{\Delta }_0\right)⟩,$$

$$|l,r⟩{⟨l,r|}_{3,4}{U}_{\mathrm{bs}}{\left(\frac{\pi }{4}\right)}_{3,4}{U}_{\mathrm{bs}}{\left[\mathrm{asin}\left(\sqrt{f}\right)\right]}_{1,3}{U}_{\mathrm{bs}}{\left[\mathrm{asin}\left(\sqrt{f}\right)\right]}_{2,4},$$

$$\mathrm{Circuit}\mathrm{I}:|N{⟩}_1|N{⟩}_2|0{⟩}_3|0{⟩}_4⟶{|{\psi }_{\mathrm{cat}}\left({\Delta }_0,l\pi \right)⟩}_{1,2}.$$

$$U_{\mathrm{ps}}\left(\pi \right)|{\psi }_{\mathrm{cat}}\left({\Delta }_0(l,r),l\pi \right)⟩\propto |{\psi }_{\mathrm{cat}}\left({\Delta }_0(r,l),r\pi \right)⟩.$$

$$U_{\mathrm{bs}}{\left(\gamma \right)}_{2,5}\equiv$$

$$U_{\mathrm{bs}}{\left(\pi /4,\pi /2\right)}_{2,5}{U}_{\mathrm{ps}}{\left(\gamma \right)}_5{U}_{\mathrm{ps}}{\left(-\gamma \right)}_2{U}_{\mathrm{bs}}{\left(\pi /4,-\pi /2\right)}_{2,5}.$$

$$|Q⟩{⟨Q|}_5{U}_{\mathrm{bs}}{\left\{\arccos \left[\tan \left({\Delta }_0/2\right)\right]\right\}}_{2,5}{U}_{\mathrm{bs}}{\left(\pi /4\right)}_{1,2}.$$

$$\mathrm{Circuit}\mathrm{II}:$$

$$|{\psi }_{\mathrm{cat}}\left({\Delta }_0,l\pi \right)⟩⟶|{\psi }_{\mathrm{cat}}\left[\pi /2,l\pi +(2N-l-r-Q)\pi /2\right]⟩.$$

$$\mathrm{Prob}(Q=0,\cdots ,S-1)$$

$$\mathrm{Prob}(Q=S)$$

$$U_{\mathrm{bs}}{\left(\pi /4,\pi \right)}_{1,2}{U}_{\mathrm{ps}}{\left(\pi /2\right)}_2.$$

$$\mathrm{Circuit}\mathrm{III}:$$

$$|{\psi }_{\mathrm{cat}}\left(\pi /2,l\pi +(2N-l-r-Q)\pi /2\right)⟩⟶|P,0⟩+(-1{)}^l|0,P⟩,$$

$$F$$

Full text

The fundamental limits to optical detection for metrology and imaging are quantum mechanical Giovannetti04 . Of particular interest for reaching such quantum limits are path-entangled states of photons of the form $|N0\rangle+e^{i\phi}|0N\rangle$, in a basis of photon-number states, commonly referred to as “$N00N$” states. A variety of applications have been suggested Kapale05 . For lithography Boto00 and microscopy Teich97 , $N00N$ state light would be used together with multi-photon absorbers to achieve enhanced resolution. This is because the de Broglie wavelength for an $N$-photon state is a factor $1/N$ smaller than the wavelength associated with the single photon, and the absorption rate scales linearly with the incident intensity, rather than as the $N^{\rm{th}}$ power. Regarding applications to precision metrology, whereby an interferometric setup is used to measure small phase shifts, $N00N$ states achieve the Heisenberg limit, for which the phase uncertainty scales as $1/N$ Bollinger96 ; Ou97 ; Boixo06 , and entanglement is a fundamental requirement for achieving this limit. It has been rigorously demonstrated that the cost of improving sensitivity (without using entanglement) is higher intensities or longer coherence times Giovannetti06 . Classically the shot-noise limit applies, attained for example by laser light, for which the uncertainty scales as $1/\sqrt{N}$, already a restraint in applications such as magnetometry Kominis03 and gyroscopy Dowling98 .

However, building a source of $N00N$ states beyond two photons is challenging. Three, four and six photon experiments have been reported Mitchell04 ; Walther04 ; Resch05 , but only in the first two references were $N00N$ states generated. In theory a source could be made using a nonlinear crystal Sanders89 . However, the required optical nonlinearity is not readily available. An alternative is a non-deterministic approach using linear optics, wherein the desired state is generated on condition of a specific outcome at photodetectors. A variety of schemes have been suggested which typically rely on conditional destructive interference Fiurasek02 ; Zou02 ; Kok02 . However, so far none of these scales efficiently, that is they all share the feature that exponentially decreasing success probabilities outweigh the possible gains. Noting that quantum algorithms, exhibiting polynomial and exponential speedups over their classical counterparts, may be implemented scalably in a linear-optics approach KLM01 , we expect that it should be possible to do better. In this Article we address this challenge by adapting a conceptually simple measurement procedure. Our method is as follows. First, we aim to minimise the negative effects of back-action in a sequence of detections, whereby earlier measurements affect the outcomes of later ones. Next, we engineer output states that closely approximate the ideal case. Finally, we exploit feed-forward, for which circuits are actively switched in response to previous photodetections. Feed-forward is an essential ingredient of linear-optics based quantum computing, but is not used in previous proposals for engineering $N00N$ states.

We begin by considering the thought experiment depicted in Fig. 1(a). Here, cavity modes labeled $A$ and $B$ are assumed to start with a well-defined photon number $N$. They are each coupled to an external mode by a weakly transmissive mirror, and these modes are combined at a 50:50 beam splitter, and then subject to partial photodetection. The beam splitter acts to make the origin of the photons indistinguishable. When a photon is registered at the left or right photodetector labeled $D_{L}$ or $D_{R}$, the transformation is given by the Kraus operators $\hat{L}=(\hat{a}-\hat{b})/\sqrt{2}$ or $\hat{R}=(\hat{a}+\hat{b})/\sqrt{2}$ respectively (where $\hat{a}$ and $\hat{b}$ are the annihilation operators for modes $A$ and $B$). To obtain the corresponding probabilities it is necessary to normalise by the total photon number prior to detection. We suppose now that a string of detections occur only at $D_{r}$, say (by adjusting the path length difference of the cavities between detections, with a phase shifter, the same state for the cavity modes can be obtained in every case). After this, the detectors are removed and the system evolves to a final state with all the remaining population at the output ports. Denoting by $|\psi_{AB}\rangle$ the state of modes $A$ and $B$ after $r$ initial detections, we find that $\left|\psi_{AB}\right\rangle=\hat{R}^{r}\left|N\right\rangle\left|N\right\rangle/\sqrt{\left\langle N\right|\left\langle N\right|\left(\hat{R}^{\dagger}\right)^{r}\left(\hat{R}\right)^{r}\left|N\right\rangle\left|N\right\rangle}$, normalising to unity. The probability $P_{\rm cond}$ of finding all the remaining photons “condensed” at the right output port (and none at the left output port) is as follows,

[TABLE]

where $S\equiv 2N-r$ denotes the total remaining photon number, $C$ denotes a binomial coefficient, and we assume that $r<N$. Evaluating the value of $P_{\rm cond}$ numerically for initial states of increasing size, we find that its value is determined asymptotically by the proportion of the input that is measured. For example, setting $r$ either as one quarter or one third of $2N$ suffices for ${P_{\rm cond}}>0.6$ or ${P_{\rm cond}>0.7}$, respectively.

We have found for our thought experiment, that later detections tend to strongly reinforce earlier ones. Hence, the effect of measurement back-action here is useful for state engineering, and in what follows we adapt the measurement process for $N00N$ state generation. We divide our analysis into three stages. First, we translate our thought experiment into a mathematically equivalent procedure based purely on linear optics, and consider the general case for which photons are detected at both photodetectors. The localisation phenomena resulting from this measurement process, have been studied extensively in the context of the debate over the existence of absolute optical coherence in common quantum-optical experiments Molmer97 . It has been demonstrated that well-defined correlations in the relative optical phase evolve, for the remaining population, and play a central role in the ongoing dynamics. Hence, in the second stage of our analysis, we investigate simple procedures for manipulating phase correlations, and relate states with well-defined correlations in the relative phase and $N00N$ states. Finally, we identify a method based on feed-forward to enable $N00N$ states to be generated efficiently.

First, we translate our thought experiment into a mathematically-equivalent procedure, based on linear optics, as depicted in Fig. 1(b). We label this optical circuit, Circuit I. Here all modes are propagating, and a source is assumed to supply dual Fock states $|N\rangle|N\rangle$ to the principal modes one and two. Beam splitters of reflectance $f$ couple modes one and two to ancillae modes three and four, which are combined at a 50:50 beam splitter. They are then measured by number-resolving photodetectors labeled $D_{l}$ and $D_{r}$, where on average a fraction $f$ of the input photons are registered. We now consider the state $|\psi_{l,r}\rangle$ generated in modes one and two after $l$ photons are registered at $D_{l}$ and $r$ at $D_{r}$. Following Ref. Sanders03 , it is convenient to adopt a representation in terms of coherent states, which are of the form, $|\alpha\rangle\!\equiv\!||\alpha|\exp(i\theta)\rangle\propto\sum_{k=0}^{\infty}\sqrt{|\alpha|^{2k}/k!}\exp\left(ik\theta\right)|k\rangle$ in a basis of Fock states. It has been shown that,

[TABLE]

where $S\equiv 2N-l-r$, $\alpha_{j}=|\alpha_{j}|\exp(i\theta_{j})$, $\theta_{\rm av}=(\theta_{1}+\theta_{2})/2$ and $\Delta\equiv\theta_{2}-\theta_{1}$. The superposition phase $\sigma$ takes the value $l\pi$, and hence the measurement record must be known exactly. The scalar function $G(X)$ is given to good approximation by the Gaussian expression $\exp\left[-(l+r)X^{2}/4\right]$. The total photon number is equal to $S$. There are well-defined correlations in the relative-optical phase parameter $\Delta$ at values plus and minus $\Delta_{0}$, determined only by the ratio of $l$ to $r$. These correlations are multi-valued whenever photons are registered at both photodetectors, and a Schrödinger cat state is generated. We can see that cats are generated as a result of the symmetry of the setup. Specifically, $\hat{L}$ and $\hat{R}$ are invariant under an exchange of the labeling of the modes, a transformation which reverses the sign of the relative phase. The generation of cat states therefore also requires precise phase stability between the modes. Turning to the source, we see that the state of the input can be a mixture of the form $\sum_{N}P_{N}|N\rangle|N\rangle\langle N|\langle N|$, since $\Delta_{0}$ is independent of $N$, and standard linear optical elements obey a superselection rule for the photon number. Several two-mode squeezing processes strongly suppress relative number fluctuations, and hence might serve as practical sources of light described by these mixed states.

For the second stage in our analysis, we identify the outputs of Circuit I as examples of quantum reference frames — reference frames for a classically defined parameter composed of finite quantum resources. Quantum reference frames are subject to depletion and degradation as they are used, and are currently of interest for protocols in the field of quantum information, in which they are regarded as a resource Bartlett06 . By making an analogy to classical phase references we can now identify simple ways in which states of the form Eq. (1) can be manipulated. For the current purposes we can assume that a large number of detections have been performed and define,

[TABLE]

where $\alpha=|\alpha|e^{i\theta}$, for a state with a total photon number $S$ and a relative phase of $\Delta_{0}$ (assumed to be normalized). Relative phase correlations between more than two modes are transitive and are transformed additively by phase shifters. A phase reference can be extended to additional modes by combining it with the vacuum at a beam splitter. As an example, a 50:50 beam splitter, which we denote here by $U_{\rm bs}$, beating light in a Fock state with $S$ photons against the vacuum yields, $U_{\rm bs}|S\rangle|0\rangle\propto|\psi_{\infty}(0)\rangle$, and $U_{\rm bs}|0\rangle|S\rangle\propto|\psi_{\infty}(\pi)\rangle$. Therefore, we see that a simple circuit, consisting only of a beam splitter and a phase shifter, can convert a cat state generated by Circuit I to a $N00N$ state, whenever the relative phase correlations differ by $\pi$. This happens when $l=r$ and $\Delta_{0}=\pi/2$. We label this circuit, Circuit III (anticipating an intermediate process modifying the cat states for the general case).

Before proceeding to the final stage of our analysis, we consider a simple $N00N$-state generator, that attempts to convert every cat state generated by Circuit I using Circuit III. This method might be expected to yield close approximations to $N00N$ states, whenever the the relative phases of the cat state are close to plus and minus $\pi/2$. The situation is summarized in Fig. 2(a). To measure the quality of the output state we adopt the fidelity, denoting it by $F$. For the measurement-induced condensation, considered at the start, $F$ takes the same same value as $P_{\rm cond}$. For schemes generating $N00N$ states, it is necessary to account for the phase of the superposition, and we define F=\rm{max}_{\phi}\Big{|}\Big{(}\langle 0S|+\exp\left(-i\phi\right)\langle S0|\Big{)}|\psi_{\rm output}\rangle\Big{|}^{2}/2, where $S$ is the total photon number of the state $|\psi_{\rm output}\rangle$. Evaluating $F$ for our $N00N$-state generator, when Circuit I generates a cat state with relative phase components at $\pm\Delta_{0}$ and total photon number $S$, we find to first approximation that $F\sim\cos^{2S}\left[\left(\Delta_{0}-\pi/2\right)/2\right]$. As with other proposals, this scheme in fact scales exponentially poorly whenever the relative phase correlations are less than $\pi$ apart, as is typically the case. Inspecting the overlap for different relative phase components, as in Eq. (2) with total photon number $S$, we find that \Big{|}\langle\psi_{\infty}(\Delta_{1})|\psi_{\infty}(\Delta_{2})\rangle\Big{|}=\Big{|}\cos\left[\left(\Delta_{2}-\Delta_{1}\right)/2\right]\Big{|}^{S}. The poor scaling can be attributed to the non-orthogonality of the cat state components.

We now proceed to the final stage of our analysis. Our previous $N00N$-state generator is effective when Circuit I generates cat states with components which are orthogonal. However, this occurs with low probability. It is not possible to improve the situation with any combination of (idealised) beam splitters and phase shifters, since these implement unitary transformations. Hence, we now devise a circuit, labeled Circuit II, to make input cat components orthogonal, using additional processes of measurement and feed-forward. This more sophisticated scenario is depicted in Fig. 2(b). To identify a suitable circuit, we investigate how a beam splitter transforms phase references, starting with two classical fields. Here the field in each mode is represented by a complex number, with the square amplitude corresponding to the intensity, and the phase to the optical one. A 50:50 beam splitter, configured so as not to impart additional phase shifts to the modes, outputs two classical fields described by the sum and difference of the values for the inputs (altering both the square amplitudes and the phases). If the input has a relative phase of [math] or $\pi$, and equal intensities for each mode, the population is transferred entirely into one mode. On the other hand, if the input has a relative phase of plus or minus $\pi/2$, and equal intensities in each mode, the relative phase and intensities are preserved. Moving to the quantum case, we consider the action of the beam splitter for a state, defined as in Eq. (2), with a relative phase of $\Delta_{0}$, a total photon number $S$, and an intensity $S/2$ in each mode. Computing the final state explicitly, we find a scenario similar to the classical case,

[TABLE]

This has an intensity $SI_{1}/\left(I_{1}+I_{2}\right)=\left[1-\cos(\Delta_{0})\right]/2$ in mode one and $SI_{2}/\left(I_{1}+I_{2}\right)=S\left[1+\cos(\Delta_{0})\right]/2$ in mode two, and a relative phase of plus $\pi/2$ when $0<\Delta_{0}\leq\pi/2$ and of minus $\pi/2$ when $-\pi/2\leq\Delta_{0}<0$ (we consider cases for which the intensity is increased in favour of mode two).

The symmetry of the beam splitter transformation makes it useful for altering the cats generated by Circuit I, so that the relative phases are different by $\pi$. However, it creates a difference in the intensities between the modes. To correct this, we propose beating mode two against the vacuum, so as to to move the difference of the intensities to an ancillary mode, which can be removed by a photodetection. This method depends critically on feeding forward the result of the detections performed by Circuit I, so that a variable beam splitter can be set according to the value of $\Delta_{0}$. A variable beam splitter can be implemented with 50:50 beam splitters and variable phase shifters. The cost of correction is a decrease in the total photon number, which varies non-deterministically. As can be seen from Eq. (3), a fraction of $\cos\left(\Delta_{0}\right)$ of the photons are lost on average. Overall, Circuits I through III constitute a complete $N00N$-state generator. Additional mathematical analysis is given in the supplementary online text. Runs for which Circuit I fails to generate a Schrödinger cat state, or too many photons are lost in the detection process are discarded. The fidelities at the output are, on average, $0.87$, $0.94$ or $0.98$, when a fraction of one third, one half, or two third respectively of the input photons are detected by Circuit I. Higher fidelities are possible when the photon number at the input is small. If allowance is made for sufficient input photons to be detected by Circuit I, and a further half to be detected in Circuit II, the probability of failure is not too large.

Finally, we suggest some possibilities for experimental implementation. For the source, we propose an optical parametric oscillator setup for which the two mode squeezed output of an optical parametric amplifier is enhanced by a cavity Zhang02 . Note, however, that the current purposes require twin beams of a much lower intensity than is typical in many experiments, and that the beams must be rendered frequency degenerate. Techniques of feed-forward and photodetection are being developed with a view to quantum information technologies Kok07 ; Prevedel07 . For the source, an important problem is imperfect correlation between the modes. If, for example, two independent lasers of equal intensity provide the input, the scheme generates the intended relative phase correlations, but no entanglement Cable05 . Photodetectors are subject to loss and dark counts. Losses will act to degrade the source, reducing the relative number correlation and increasing the uncertainty in the total photon number. Dark counts are more problematic, mixing over the phase for the superposition in Eq. (1). An alternative suggestion is using trapped bosonic atoms. One possibility might be to work in a regime for which the atomic wave-packets are much longer than the typical scattering length, as proposed in Ref. Popescu06 . Another is to use Bose-Einstein condensates, for which a variety of coherent operations have been demonstrated. Number-resolved condensates might be obtained from the Mott Insulator phase, while relative-number squeezing can be achieved by different techniques.

In conclusion, we have proposed for the first time a linear-optics based scheme that generates large $N00N$ states efficiently, the photon number at the output scaling with that of the source — all the while maintaining high fidelities, high success probabilities and a fixed number of circuit components. As well as being of immediate interest for a range of applications, our results have connections with other topics. For example, the scaling we derive for our measurement-induced condensation procedure is of relevance to the study of the interference of light from independent sources and localizing relative optical phase, phenomena with analogs in different physical systems Rau03 . We have left as an open question the extent to which the scaling can be attributed to Bose statistics. Regarding our $N00N$-state generators, the creation of macroscopic entangled states is of interest for exploring the quantum-classical transition. Finally, our study of Schrödinger cat states may have application to quantum computing, where Schrödinger cat states, defined for one mode only, have been proposed to encode qubits, which may be manipulated using standard experimental techniques Jeong05 .

Acknowledgements

The authors would like to acknowledge support from the Hearne Institute, the Army Research Office, and the Disruptive Technologies Office. H. C. would like to thank Terry Rudolph, Ryan Glasser, Sonja Daffer and Yuan Liang Lim for helpful discussions.