Trading quantum states for temporal profiles: tomography by the overlap

Quantum states and the modes of the optical field they occupy are intrinsically connected. Here, we show that one can trade the knowledge of a quantum state to gain information about the underlying mode structure and, vice versa, the knowledge about the modal shape allows one to perform a complete tomography of the quantum state. Our scheme can be executed experimentally using the interference between the signal and probe states on an unbalanced beam splitter with a single on/off-type detector. By changing the temporal overlap between the signal and the probe, the imperfect interference is turned into a powerful tool to extract the information about the signal mode structure. A single on/off detector is already sufficient to collect the necessary measurement data for the reconstruction of the diagonal part of the density matrix of an arbitrary multi-mode signal. Moreover, we experimentally demonstrate the feasibility of our scheme with just one control parameter -- the time-delay of a coherent probe field.


I. INTRODUCTION
Encoding quantum states in modes of optical signals is the basis of optical quantum information, communication and metrology. A specific example is the use of spectral-temporal modes of light. The art of temporal mode shaping for use in quantum information is already quite advanced [1][2][3], and for example can be used for enhancing the resolution in spectroscopy [4]. One can transform one pulse shape into another preserving its quantum state and single-mode character, for example, using a quantum pulse gate [1,[5][6][7].
Extraction of information on the quantum state encoded into modes of light is the subject of the quantum state reconstruction, in practice performed by mixing the signal with some known reference modes. When the modal profiles and structures of the signal and probes are not the same, this common approach for the state reconstruction becomes increasingly challenging and impractical. Furthermore, it is not the mere increase in required number of copies that renders the scheme impractical for reaching a prescribed accuracy; the problem can be far deeper.
Let us consider, for example, a standard technique of quantum homodyne tomography. The signal state is optically amplified by a bright local oscillator, with information extracted from the noise of the resulting interference pattern. In homodyne detection, the bright local oscillator acts as a perfect mode filter: only information present in the same mode profile as that of the local oscillator is extracted; all other information is lost. Therefore, both precise knowledge of the mode structure of the signal and precise manipulation of the mode of the local oscillator are required. Here the low overlap between the signal and bright probe is equivalent to high loss and might render the reconstruction completely unfeasible [8,9]. Imperfect overlap between the signal and the probe is generally believed to be the main hindrance for any reconstruction scheme relying on the interference. Indeed, an imperfect overlaptypically referred to as "distinguishability" of modes -degrades visibility of interference fringes. Distinguishability of the spatiotemporal modes is also highly detrimental to quantum devices that require for their operation many single photons with identical spatiotemporal profiles, such as Boson Sampling [10], where scalability to higher photon numbers imposes severe restrictions on the overlap of the single photons at the input [11].
In this paper, we argue that the very same distinguishability can be used in a constructive way in the same spirit as noise and losses were exploited in lossy on/off tomography and noise tomography [12,13,[19][20][21]. We propose a new type of state reconstruction scheme exploiting imperfections. This scheme as well -contrary to standard homodyne detection -requires probe and signal of comparable intensities. Recently, non-balanced detection schemes (or homodyning with weak probe) are being extensively studied both theoretically and experimentally (see, for example, Refs. [14][15][16][17][18]). The distinctive feature of our scheme is that it is in fact based on distinguishability, thus turning a hindrance to the state reconstruction into an ally. Such scheme delivers all the information sufficient for the state reconstruction and this is due to some of its intrinsic features explained below. The measured data resulting from the signal-probe interference depends on both the quantum state and the mode overlap between the interfering modes. Importantly, the scheme is sensitive to the whole signal, both parts overlapping with the probe and non-overlapping parts. This is a fundamental trade-off feature of our scheme: one can trade knowledge about a quantum state for its modal profile, and vice versa. Furthermore, it can be done with the same measurement set-up, and from this data, one can determine the fraction of the quantum state that overlaps with the local oscillator, in stark contrast to the standard homodyne detection. Moreover, a non-balanced detection scheme can be used to even extract information on the quantum state of a signal without prior knowledge on its underlying mode structure [21].
The scheme proposed in this work relies on the interference of the unknown signal field and a reference field of known temporal profile and in a known quantum state. In order to collect data allowing for either complete reconstruction of the signal state or of its temporal profile, there is no need to vary the temporal profile of the reference field. It is sufficient to delay the reference pulse in a controlled way. We show that with the reference field described by a diagonal density matrix it is still possible to infer both the temporal profile and the photon-number distribution of the signal because because the absolute value of the overlap between the signal and the reference fields can be controlled solely by the relative time-delay of the reference.
We present an experimental confirmation of the above with the signal represented by single and two-photon quantum states generated by the spontaneous parametric down-conversion process and with a single on/off detector. Moreover, it is remarkable that our simple detection scheme is able to provide the complete inference of an arbitrary multi-mode quantum state, provided that one can have available a controlled set of multi-mode probes, each in a quantum coherent state. Thus, we show that the simplest on/off detector is a device able to translate a complete information about the arbitrary quantum state into a sequence of binary signals.
The outline of the paper is as follows. In Sec. II we present the theoretical basis for our scheme by describing how the multi-mode probe and signal fields interfere on a beam-splitter. Then, we focus on the particular case of mixing a multimode signal with the probes in coherent states to show the possibility to infer the complete signal state. In Sec. III, the reconstruction scheme is elaborated in detail for the case of the single-mode signal and probe fields, illustrating the implementation of the data pattern approach for the signal state reconstruction. In Sec. IV we discuss how one can infer the overlap and the temporal/spectral profile of the signal. In particular, we show that for the signal/probe with a diagonal density matrix just a single probe state is sufficient to infer the overlap. The experimental results are presented in Sec. V. We have experimentally reconstructed the quantum states of the single and two-photon signals, and then demonstrated the overlap inference from the knowledge of the quantum state.

II. THE SCHEME
First of all, let us show how the state of an arbitrary multi-mode field can be inferred with the simplest measurement set-up with on/off detectors. The signal field is mixed with the probe field on the beam-splitter; the outputs (or just one of them) are impinging on the bucket, on/off type detectors as depicted in Fig. 1.
As illustrated in Fig. 1, the imperfect overlap on the beam splitter (BS) can be modeled by separating the input fields into the modes perfectly overlapping on the BS ("modes mixed by BS") and the modes with zero overlap thus not interacting on the BS ("modes not mixed by BS"). Then let the two inputs and two outputs on a beam splitter be given by the mode creation operators a † k,s and b † k,s , where k = 1, 2 is the operating mode mixed by the BS (solid red lines in Fig. 1), whereas s = 1, . . . , S enumerates the basis of the invariant modes, i.e., not mixed by the BS (various dashed and dash-dotted lines in Fig. 1). The beam splitter itself is represented by a unitary matrix U which relate the output to the input operators: In the matrix form it can be written as b † = (U ⊗ I)a † , where I is the S-dimensional identity matrix and a = (a 1 , a 2 ) with a k = (a k,1 , . . . , a k,S ), etc.
Consider an arbitrary multi-mode state at input k and the non-ideal on-off type detector placed at the outputs of the beam splitter. The quantum state ρ (k) of the input modes k can be conveniently written in Fock basis as Note that in this expression the multi-mode character of the field is encapsulated in quantities in bold, e. g., |n and n ≡ (n 1 , . . . , n S ). Let η k be the sensitivity of the detector at output mode k = 1, 2. We are interested in the probability that the detector k does not click. Such probabilities of no clicks at the outputs (zero-click probabilities) are given by the generating function: where N denotes the normal ordering of the creation and annihilation operators. For example, to calculate the probability of no clicks at detector 1, To compute the probabilies from the family of P (η 1 , η 2 ) in Eq. (3), we use the following identity where A denotes the antinormal ordering of the creation and annihilation operators and H is some positive semidefinite Hermitian matrix with eigenvalues bounded by 1. Eq. (4) can be verified by diagonalizing the matrix H, performing the series expansion of the exponents and checking the equality of both sides using the Fock states. In Eq. (3) we can identify the matrix H and find its inverse as: Note also that det( (4) and (5) into Eq. (3) we then obtain: We can evaluate the trace in Eq. (6) in the coherent state basis |α k ≡ |α k,1 ⊗ . . . ⊗ |α k,S by introducing the generalized (multimode) Husimi function for each input state, The generalized Hiusimi function is normalized, dµ(α)Q (k) (α, α † ) = 1 where dµ(α) ≡ S s=1 d 2 α s with d 2 α s = dReα s dImα s and α = (α 1 , . . . , α S ). From Eqs. (6) and (7) we obtain Here we have introduced the combined vector variable α ≡ (α 1 , α 2 ) with α k = (α k,1 , . . . , α k,S ). The probabilities of no click given by Eq. (8) provide the key relation for the reconstruction of an arbitrary multimode field in quantum state ρ (1) by mixing it with a probe field ρ (2) on a beam splitter with the use of the non-ideal on-off type detector(s). For a set of coherent probes available in different modes the scheme becomes particularly simple. It allows us to infer the multi-mode signal collecting a string of zeros and ones on merely one on/off detector. Let ρ (2) = |β β| for some β = (β 1 , . . . , β S ). We have To derive the expression for the probabilities, we use the identities for the BS transformation matrices: and |U 12 | = |U 21 |, which result from the unitarity of the 2-dimensional matrix U of the beam splitter. For the Gaussian integrals, the following standard relation holds: where α = (α 1 , . . . , α S ), B is a positive semi-definite Hermitian matrix, while g and h are S-dimensional vector rows.
Substituting (9) into (8) and evaluating the Gaussian integral over α 2 using Eq. (10), we obtain Now, notably, for the coherent probes ρ (2) = |β β|, the reconstruction of the multi-mode source field ρ (1) is possible from the no clicks probability even on just one detector, P (η 1 , 0), setting η 2 = 0 in Eq. (11)). This is due to the fact that the expression in Eq. (11) is proportional to a multidimensional Weierstrass transform of the Husimi function of the source field, which allows for the inverse transform by the properties of the Husimi function (see Appendix).

III EXAMPLE: A SINGLE-MODE SOURCE FIELD
We aim at the demonstration of trading the information about the spectral state properties for the knowledge of the quantum state and vice versa. Now let us consider a methodologically and practically important case of the probe and signal fields in single-mode spectrally pure states. We assume that the input temporal modes (TM) of the same polarization in Fig.1 are described by the collective operators where the index x denotes either signal, s, or probe, p, modes; a x (w) is the annihilation operator of the photon in the plane wave with the frequency ω; ∞ 0 dω|f x (ω)| 2 = 1. Again, we assume that both inputs are single mode and in the same spatial mode. Thus, a degree of distinguishabity is described by the overlap We assume that the BS acts similarly on input plane wave modes of arbitrary frequency; the BS action is described by the unitary 2×2 matrix U . Initial states are described by the density matrices ρ x depending only on the corresponding collective mode operators.
To apply the formalism developed in the previous section for our single-mode case, we need to add auxiliary orthogonal vacuum modes to both signal and probe inputs. Thus, the two-mode internal basis (i.e., S = 2) is used in this case. The operator of the original source field mode, c † 1 , is expressed through the internal basis operators as We introduce the auxiliary mode c † 2 orthogonal to c † 1 , and write down the operators of the internal basis, a 1 = (a 1,1 , a 1,2 ) as where c = (c 1 , c 2 ). The single-mode source field reads: (note that the basis c 1,2 is used for the expansion of the state). Eq. (14) leads to an analogous relation between the parameters of the respective coherent states via the identity α 1 a † 1 = α 1 V c † ≡ αc † , i.e., the basis coherent vector α 1 = (α 1,1 , α 1,2 ) in the a-basis is related to the coherent vector α = (α, α ) in the c-basis. In the latter representation, the Husimi function of the above single-mode source (15) becomes where |α is the shortcut notation for the state |α, 0 in the new basis c 1,2 of Eq. (14). The integration measure dµ(...) is invariant under the unitary transformation in Eq. (14). We need to evaluate the Gaussian integral over the variable α in Eq. (11). We need only one detector for reconstruction, select the kth detector. We then obtain the the probability of no clicks at detector k when mixing the single-mode source field ρ (1) (15) with the single-mode coherent probe in the a-basis ρ (2) = |β β| ⊗ |0 0| (i.e., β = (β, 0) in Eq. (11)): Eq. (17) is a generalization for the imperfect overlap of the result obtained in Ref. [9] for the ideal overlap. As we shall see, this equation points directly to the possibility of the signal state reconstruction.
To infer a set of discrete parameters, such as density matrix coefficients in the Fock states basis, it is not necessary to revert to the inverse Weierstrass transform. In general, one can reconstruct an unknown state of the signal mode by taking a discrete set of the amplitudes, β, of the coherent probe for a fixed overlap, φ 1 , between the modes (which can be equal to 1 as a special case). However, with any given accuracy, one can also reconstruct an unknown state just by varying the overlap for a fixed amplitude β of the coherent mode. To that end, let us consider representation of ρ s in some discrete basis. For example, in Fock-state basis the Q-function can be represented as where the matrix elements ρ n,m ≡ n|ρ s |m . From Eqs. (17)(18) it straightforwardly follows that the probability of zero clicks as a function of parameter z is given by: where P(z, z * ) is an expansion in non-negative powers of z and z * , which reads Eq. (19) points to some important conclusions. First of all, for the realistic finite reconstruction subspace (corresponding to the truncation of the Fock state expansion in (15)), one can indeed infer the signal state for a finite number of z values. It is always possible to choose such a number of z values as to interpolate a continuous function P(z, z * ) by the Lagrange polynomials with sufficient accuracy. This allows then to infer the Q-function. Of course, it is more practical to infer a discrete set of expansion coefficients, ρ m,n . This can be done in a number of already well-established ways (for example, by inverting Eq. (14) using constrained least-square methods [22,23], or maximum-likelihood methods [21]). From Eq. (14) it also follows that one can get the set of z-points sufficient for reconstruction of any finite-subspace expansion of ρ s for an arbitrarily small probe state amplitude, β.
To realize the scheme in practice, it is necessary to devise a way to change the overlap to produce the set of values, {z j }, sufficient for the inference. Nowadays there are quite sophisticated methods for shaping the spectral profile of the mode (see, for example, Refs. [24,25]). However, one really does not need to manipulate precisely a shape of the probe pulse to obtain the necessary set; a simple time-delay arrangement is sufficient.
For the inference it is illustrative to represent an arbitrary signal as the finite sum of the coherent projectors, where the coefficients c j are real, but not necessarily positive. Such a representation is usually implemented in datapattern tomography schemes [22,23,[26][27][28]. Thus, for different values of the overlap, φ 1,l , the no-click probability is given by the following expression: where ρ s = j c j |α j α j |, α j = √ η|U 1,1 |α j , and β l = − √ η exp{i arg(U 1,1 )}U * 1,2 φ * 1,l β. That is, the density matrix ρ s in (22) is represented by coherent projectors sitting in the smaller region on the phase plane than in case of the original matrix (21). Note that the matrix ρ s can be arbitrary as well. Eq.(22) links the probability of no clicks with the signal density matrix in terms of coherent projectors (21). That means, if it is possible to represent accurately the signal in terms of the coherent state projectors with the amplitudes β l , then our scheme allows for the reconstruction of the coefficients c j in Eq.(21).

FIG. 2:
The zero-click probability distribution as a function of the overlap φ. The panel (a) shows zero-click probability p0 for different overlaps for fixed coherent signal and varied probe. The dotted red, dash-dotted green, dashed purple and solid black lines show p0 for the overlap |φ| 2 = 1, 0.9, 0.7, 0.5, respectively, and arg(φ) = 0. The panels (b,c) show zero-click probability for the fixed coherent signal and varied probe, β, for the two different values of the overlap. The panel (b) corresponds to the intensity overlap equal to 1 and the overlap phase of 3π/2 with respect to the phase between the signal and the probe fields; the amplitude of the signal is 2. The panel (c) is for the overlap equal to 0.95 and the zero overlap phase. For all the graphs the efficiency, η = 0.59; |U1,2| 2 = 0.544; the amplitude of the signal is 2/ √ η|U1,1|; U1,2 and U1,1 are taken to be real.

IV. INFERRING THE OVERLAP AND TEMPORAL PROFILE OF THE SIGNAL
For our set-up, the knowledge trade-off between the quantum state of the signal and the spectral shape of the signal TM is realized through the estimation of the overlap. In Ref. [21] is was shown it is possible to infer the modulus of the overlap even for an unknown signal through the implementation of a sufficiently large set of coherent probes, so that it is large enough for the reconstruction of the Wigner function with acceptable accuracy. However, it is possible to reconstruct the complete temporal/spectral profile of the signal TM using much smaller set of the coherent states (probes) if a simple time-delay arrangement is used.
The estimation of the overlap value is based on the analysis of the zero-click probability dependence on the value of the overlap. Eq. (22) shows that altering the amplitude or phase of the overlap leads to changes in the zero-click probability distribution. For the signal and probe in coherent states this is illustrated in Fig. 2. Lowering the absolute value of the overlap and changing its phase leads to shifting and damping of the zero-click probability, p 0 (φ, β).
We suggest the following general strategy to estimate the overlap: 1. measure p 0 (φ, β l ) for a set of the amplitudes β l and fixed overlap; 2. calculate p 0 (φ, β l ) for different assumed values of the overlap aiming at fitting the measured values.
For the set of known overlap values, it is straightforward to infer the product of spectral envelopes of the signal and probe modes (for the experimental results see next section). If we vary the overlap using the time-delay for one of the pulses, the product of the frequency profiles of the modes of the signal state, f s (ω), and the probe state, f p (ω), is connected to the overlap by the Fourier transform: where δt is the value of time-delay. The problem of the profile inference remains feasible even in the case of when only the modulus of the overlap is estimated. It is a particular case of much discussed phase-retrieval problem commonly encountered in the image reconstruction (see, for example, the recent brief review [29]). One of the simplest ways to do the profile inference is to represent f s (ω)f * p (ω) as a superposition of some localized basis functions, typically the Hermite-Gaussian functions are used. Such a representation is quite useful, for example, for the reconstruction of the states generated in a spontaneous down conversion schemes [30].

V. EXPERIMENTAL RESULTS AND INFERENCE
The setup used for this experiment is shown in Figure 3. The signal is generated in an engineered periodically poled potassium titanyl phosphate (KTP) waveguide with a type-II parametric down-conversion process [31]. The source can generate a nearly spectrally decorrelated state at 1550 nm. The down conversion is pumped at 775 nm with frequency doubled light from an OPO system. The residual pump light from the OPO at 1550 nm is used as a weak probe field. To adapt the spectrum of the probe field to the signal, spectral shaping involving bandwidth and spectral phase is employed. An automated free space variable delay line is used to change the relative timing between signal and probe. Both fields are then combined on a fibre integrated beam splitter. The idler mode from the parametric down conversion is split on a second 50/50 beam splitter allowing for heralding on one or two clicks. The light is detected with superconducting nano-wire detectors with a detection efficiency of 90%. The repetition rate of the experiment is limited to 1 MHz. An acousto-optic modulator is used to lower the repetition rate for the 775 nm light whereas a fibre integrated electro-optic modulator is used for the probe field.
A total of 81 different delays between signal and probe where scanned with an acquisition time of 150 seconds each. Between these measurements automated shutters where used to either measure the PDC signal or the probe field to account for long term drifts. While only using the PDC signal (shutter 2 is closed) the Klyshko efficiency can be measured giving the product of all efficiencies after the PDC [32]. In addition a calibrated coherent state is used to The results of the quantum state reconstruction for the single-photon and two-photon states are displayed in Fig. 4. Estimation of the photon-number distributions was performed using Eq. (22) for least squares fit with linear constraints [33]. The crucial part of the procedure is the experimental inference of the actual value of the overlap as was described in Sec. IV. The overlap profile inferred for the single-photon states is shown in the inset of Fig. 4a.
Here one should emphasize the different role played by the overlap in different measurement schemes, in particular, in a common homodyning with a strong probe and non-balanced scheme with the weak probe as the one described and implemented in our work. In balanced homodyning with a strong probe the role of the finite overlap is equivalent to additional detection loss. From Eq. (17) it is seen that the role of the overlap for the non-balanced scheme with a weak probe is quite different.
First of all, the phase of the overlap is rather important (this one can see, for example, in Fig. 2). As we have demonstrated here, it is possible to build a set of measurements sufficient for the complete reconstruction of the signal by varying the phase and the modulus of the overlap for the fixed coherent probe. Then, the value of the overlap comes into the equation for the zero-click probability in a different way as the detector efficiency. In fact, the overlap additionally damps and rotates the part of the probe coherent state interfering with the signal, whereas the signal is only damped by the imperfect efficiency (see Eqs. (17,22)). Zero overlap is not the same as zero efficiency for our measurement scheme. Zero efficiency gives unit zero-click probability irrespectively of other parameters. Zero overlap "factorizes" a zero-click probability, which for a zero overlap becomes a product of probabilities of probe and signal independently interfering with the vacuum. Notice that for the case of no phase correlation between the signal and the probe, when only the diagonal elements can be inferred, the overlap phase is washed out just like the phase of the probe.
Finally, with the weak probe one can infer the modulus of the overlap even without knowing the signal [21]. So, the non-balanced scheme with the weak probe indeed gives the possibility to detect appearance of distinguishability between the signal and probe modes.

CONCLUSIONS
Here we have shown that the state of an arbitrary multi-mode field can be inferred using the imperfect overlap between the signal and the reference fields using just one on/off detector. Interference of the signal with the controlled probe gives possibility to translate the state of the signal into merely a chain of binary numbers (via the zero-click probability, i.e. probability that the detector does not click). Moreover, information about the temporal structure of the field can be traded for the information about the quantum state of this field, and vice versa and reconstruction of both can be realized with the same measurement set-up. The key for such a measurement scheme is to vary the imperfect mode overlap in a controlled way.
For the single mode signal and reference fields, we show experimentally that this can be achieved by the simple time delay of the probe. An important step in the procedure is the experimental estimation of the actual value of the mode overlap φ 1 (13). Remarkably, for the single-mode case and for probe or signal in a diagonal state, for the known signal the overlap can be inferred with just one fixed probe state. The experimental results for the quantum state inference from the knowledge of the spectral profile by changing the overlap, and the reverse problem of estimating the overlap from the knowledge of the signal state, are depicted in Fig. 4. The suggested scheme can be used for diagnostics of devices (such as quantum pulse gate) which can possibly alter both temporal profile and quantum states of the field.
Thus we have shown that the distinguishability of modes can be a valuable resource. It can be implemented for quantum state diagnostics and tomography. Actually, distinguishability and imperfect overlap of the probe and signal is a bridge which allows to connect spacial, temporal and spectral features of wave-package carrying the quantum state and parameters of this state.

APPENDIX
The multidimensional Weierstrass transform of a function Q(α) of a complex variable α = (α 1 , . . . , α S ) ∈ C S is defined as a convolution with the following multidimensional Gaussian (for σ > 0) W(α, α † ) = dµ(β)e −σ|α−β| 2 Q(β, β † ), where β = (β 1 , . . . , β S ) ∈ C S . Assuming that there is the (multidimensional) Fourier transform of Q(α) (as in the case of the Husimi function Eq. (7)), one can show that there is the inverse transform to Eq. (24). Let us write the Fourier transform using the complex variables: Substituting the Fourier transform of Eq. (25) into Eq. (24) and evaluating the Gaussian integral with the help of Eq. (10), we can put the Weierstrass transform in the differential operator form (i.e., an infinite series expansion): Eq. (26) has the inverse transformation (in the form of an infinite series) The infinite series representation of Eq. (27) (derived by using Eq. (10)) into Eq. (27) and setting z s = x s + iy s , we obtain: where W R (Reα, Imα) is the Weierstrass transform W(α, α † ) expressed as a function of the real and imaginary parts of the complex vector α.