Complete temporal mode characterization of non-Gaussian states
by dual homodyne measurement
Optical quantum states defined in temporal modes, especially non-Gaussian states like photon-number states, play an important role in quantum computing schemes. In general, the temporal-mode structures of these states are characterized by one or more complex functions called temporal-mode functions (TMFs). Although we can calculate TMF theoretically in some cases, experimental estimation of TMF is more advantageous to utilize the states with high purity. In this paper, we propose a method to estimate complex TMFs. This method can be applied not only to arbitrary single-temporal-mode non-Gaussian states but also to two-temporal-mode states containing two photons. This method is implemented by continuous-wave (CW) dual homodyne measurement and doesn’t need prior information of the target states nor state reconstruction procedure. We demonstrate this method by analyzing several experimentally created non-Gaussian states.
Quantum states of light are a promising resource of quantum computation Knill.nature(2001) (); Menicucci.prl(2006) (); Takeda.prl(2017) () and quantum communication Abruzzo.pra(2013) (); Brask.prl(2010) (). They are characterized by optical modes like polarization, spatial, and temporal modes. Among these modes, temporal modes have a lot of flexibility thus are useful for many applications. One prominent example is temporal-mode multiplexing of quantum states to realize large-scale fault-tolerant quantum computation Takeda.prl(2017) (); Menicucci.pra(2011) (); Menicucci.prl(2014) (); Takeda.nature(2013) (); Yoshikawa.qph (); Yokoyama(2013) (); Yoshikawa.apl (). In this scheme, grasping temporal-mode structures of quantum states are essential for basic operations like interference and measurement. Therefore, a methodology to characterize the states’ temporal-mode is in great demand.
A temporal mode is characterized by a complex function , called temporal-mode function (TMF). For example, single-photon states in a temporal mode are given by , where is a multi-mode vacuum state and is a creation operator of temporal-mode . Such non-Gaussian states, which can be used as ancillary states or quantum information carrier in quantum computation, are the main interest of temporal-mode analysis. Although we can calculate TMFs theoretically in some state creation schemes Anne.pra(2007) (), imperfection of experiment varies their actual forms. This mismatch leads to extra photon loss in useful applications. Therefore, experimental estimation of TMFs of optical quantum states is essential to utilize the states with high purity.
So far, Single-temporal-mode states, the states described by one temporal mode, have been actively analyzed experimentally. Especially, continuous wave (CW) homodyne and heterodyne measurements are powerful tools. In Refs. MacRae.prl(2012) (); Morin.prl(2013) (); Yoshikawa.prx(2013) (), the TMFs of single-photon states and Schrödinger’s cat states are estimated by applying principal components analysis (PCA) Adbi(2010) (), which decomposes correlated variables into uncorrelated variables, to CW homodyne signals. This method doesn’t require prior information of the target states and can access the TMFs without a state reconstruction algorithm. We can apply this method to arbitrary single-temporal-mode non-Gaussian states. However, their TMFs are limited to real functions, because PCA gives us only real functions. Ref. Qin.sciap(2015) () estimates complex TMFs of single-photon states by constructing what they call temporal density matrix via CW heterodyne tomography. In the case of general non-Gaussian states, however, we need to consider larger size density matrix having higher-photon number components, and estimation of TMFs is not trivial. On top of that, we need several local oscillator (LO) beams having different frequencies for construction of temporal density matrix.
In this paper, we propose a method what we call complex-number PCA (CPCA); estimation of complex TMFs by applying PCA to complex variables given by CW dual homodyne measurement. Figure 1 is a conceptual diagram of our method. CPCA can deal with arbitrary single-temporal-mode non-Gaussian states characterized by complex TMFs. On top of that, it can be applied to dual-temporal mode states containing two photons to estimate the complex TMFs . These states are the simplest example of multi-temporal-mode states. Although multi-temporal mode states play an important role in applications Chuang.pra(1996) (); Wasilewaki.pra(2007) (); Bergmann.pra(2016) (), this kind of analysis has not been done in previous researches. Our method possibly opens a way to TMF estimation of general multi-temporal mode states. Like previous PCA method, CPCA needs neither prior information, state reconstruction procedure nor LO beams having different frequencies. The simplicity and capability of our method to characterize wide range of quantum states would lead to useful applications not only in state creation experiments but also in quantum communication and quantum computation schemes Knill.nature(2001) (); Menicucci.prl(2006) (); Takeda.prl(2017) (); Abruzzo.pra(2013) (); Brask.prl(2010) (). We experimentally demonstrate this method using several non-Gaussian states characterized by complex TMFs.
This paper is organized as follows. In Sec. II.1, we define temporal modes of light and optical single- and multi-temporal-mode quantum states. In Sec. II.2, we review previous estimation of real TMFs by PCA. In Sec. II.3, we discuss applying PCA to CW dual homodyne measurement signals. In Sec. II.4, we discuss how to analyze two-photon states . In Sec. II.5, we discuss how photon loss affects our analysis. In Sec. III.1, we review a creation method of non-Gaussian states. In Sec. III.2, we explain the experimental setup. In Sec. III.3, we show the experimentally estimated TMFs of several non-Gaussian states.
ii.1 Definition of temporal modes
We introduce optical TMFs and basic operators. A temporal mode is characterized by time spectrum called TMF, in which quantum states can be defined. The TMFs are complex functions in general. We define the instantaneous annihilation and creation operators and which satisfy the commutation relation where is the Dirac delta function. Then, photon annihilation and creation in a temporal mode are described by operators given by
Suppose complex function satisfies so that these operators satisfy . In general,
Therefore, the inner product of TMFs gives the commutation relation of temporal modes.
In experiments, we often treat physical quantities in finite and discrete time. For that, we define time-bin modes which divide time interval into small time-bins. Their TMFs are given by
In many cases, it is useful to use annihilation and creation operators of time-bin modes
instead of instantaneous operators . Their commutation relation is given by .
When has finite values only in and varies slowly during the time change , it is well approximated as
where . Strictly speaking, the both sides of Eq. (5) are not the same, but in the rest of this paper, we use an equal sign in the following for convenience,
Then, by using time-bin modes, annihilation and creation operators of temporal mode are given by
Their commutation relation is given by
In this way, we can describe temporal mode both in infinite-continuous time and finite-discrete time.
When quantum states are described only by creation operators of temporal mode and background vacuum state , we call them single-temporal-mode states in temporal mode . Photon-number states are the examples of such states, and they are orthogonal complete basis of single-temporal-mode states in temporal mode . On the other hand, we need more than one temporal mode to describe multi-temporal-mode states. The basic examples are multi-temporal-mode Fock states . We underline that TMFs are arbitrary complex functions and not orthogonal to each other in general, thus we cannot describe the states as except for the special case when are orthogonal functions. Our goal is to estimate TMFs of those states by experimentally deciding the discrete values in Eq. (6).
ii.2 Estimation of real TMFs
Our goal is to estimate complex TMFs . In this section, however, we concentrate on the case where are real functions. Such functions can be estimated by applying principal components analysis (PCA) to quadratures as introduced in Refs. MacRae.prl(2012) (); Morin.prl(2013) (); Yoshikawa.prx(2013) (). We extend this method to complex number in Sec. II.3.
ii.2.1 Data acquisition
When we try to estimate , one useful information is quadrature values. The quadratures of instantaneous modes are defined by
Ideal homodyne detector can measure the quadrature using CW coherent beam called LO having a phase . The quadratures of a temporal mode is given by
Note that is a real function here. We can obtain by integrating the results of ideal CW homodyne measurement with a weight . In the rest of the paper, we omit the suffix except when it is important.
In PCA, the mode conversion of quadratures given by Eq. (10) plays a central role, thus we need precise value of . Unfortunately, homodyne detectors in laboratories are not ideal, preventing us from knowing the exact values of . When the impulse response of the measurement system is given by , CW homodyne measurement signals are given by (under proper normalization)
In the following, we assume is mainly determined by low-pass filter effect due to the finite bandwidth of the homodyne detector. Then, quadratures of temporal-mode we can obtain are given by
When is narrow enough compared to , that is, the homodyne detector is broadband enough compared to the bandwidth of , we can regard as a delta function . It follows that
Therefore, even when our homodyne detectors are not ideal, we can neglect the imperfection if they are broadband enough compared to . In the following, we express and just as and for convenience, thus and means measured and calculated values by using homodyne detectors having finite but broad enough bandwidth.
In experiments, we measure quadrature values at appropriate sampling rate during reasonable time span. Thus, we have to treat finite and discrete time. Let us assume has finite values only in time . Then, we measure quadrature values times at the same interval during . Here, the sampling rate should be large enough that and vary slowly in time change . Note that homodyne measurement is an observation in a system rotating at the carrier frequency of LO beam, thus the time change of and are not so fast. In this situation, time-bin modes introduced in Eq. (3) are useful. From above and Eq. (10), the quadratures of time-bin modes are given by
Similarly to Eq. (6), we use an equal sign in the following,
Thus, we can determine by finite sampling rate homodyne measurement. From Eq. (7), quadratures of temporal-mode is given by
This equation corresponds to Eq. (10). Like above, we can define and measure quadratures both in infinite-continuous time and finite-discrete time.
ii.2.2 Principal component analysis
PCA Adbi(2010) () is an analysis procedure to convert correlated variables into a set of uncorrelated variables called principal components. In PCA, the first principal component has the largest variance in the whole space, and following components are decided to have the largest variance in the subspace which is orthogonal to the components decided before.
In quantum optics, PCA has been applied to the quadratures to estimate real TMFs of non-Gaussian states MacRae.prl(2012) (); Morin.prl(2013) (); Yoshikawa.prx(2013) (). The variables are correlated because autocorrelation functions are not zero in general when . PCA is a procedure to find uncorrelated variables satisfying
The functions have important information about the TMFs we want to estimate.
For example, when we apply PCA to single-photon states , we get , and Morin.prl(2013) (). This is because single-photon states have three times larger quadrature variance than that of vacuum states in uncorrelated modes. Generally, there exists a certain phase where single-temporal-mode states, typically non-Gaussian states, have larger variance of quadratures than other uncorrelated vacuum modes. Therefore, is expected to be the TMF of the single-temporal mode states if we choose a proper phase of LO.
We can carry out PCA by introducing a matrix given by
We can obtain this matrix via CW homodyne measurement. When we measure a target state with a sampling rare during , we get a set of quadratures . By repeating the same measurement, we can calculate the average values . is a real symmetrical matrix thus diagonalized as by a certain orthogonal matrix . In this case, such matrix is given by
Why is diagonalized is shown as follows. is given by
where we use a relation that follows from Eq. (16),
In this way, we can obtain eigenfunctions by getting matrix via CW homodyne measurement and calculating a matrix which diagonalizes . Note that due to the orthogonality of , the eigenfunctions are orthogonal,
Especially, in single-temporal-mode state analysis, TMF is given by
PCA has been applied to experimentally created non-Gaussian states such as single-photon states and Schrödinger’s cat states in a single temporal mode MacRae.prl(2012) (); Morin.prl(2013) (); Yoshikawa.prx(2013) (). In this method, however, the eigenfunctions are limited to real functions because they are given by linear combination of real functions by orthogonal matrix . Therefore, this method is not suitable for the quantum states which have complex TMFs. In the next section, we extend PCA to estimation of complex TMFs.
ii.3 Estimation of complex TMFs
ii.3.1 Dual-homodyne measurement
In the previous section, eigenfunctions are limited to real functions because we apply PCA to quadratures , which are real numbers. In order to estimate complex TMFs, we have to apply PCA to complex variables. CW dual homodyne measurement and CW heterodyne measurement give such variables, measuring the conjugate quadratures and simultaneously. Actually, in Ref. Qin.sciap(2015) (), complex TMFs of single-photon states are estimated by CW heterodyne measurement using several LOs having different frequencies. In Ref. Qin.sciap(2015) (), however, an optimization algorithm is adopted, requiring heavier calculation than PCA. In this section, we discuss applying PCA to CW dual homodyne measurement signals. Dual homodyne can be implemented by LOs having one frequency, unlike heterodyne method used in Ref. Qin.sciap(2015) ().
To begin with, we explain CW dual homodyne measurement. Figure 2 is a conceptual diagram of CW dual homodyne measurement. In this measurement, the target state (quadratures ) is divided into two outputs by a 50:50 beam splitter. Another input of the beam splitter is a vacuum state (quadratures ). The two outputs are measured by CW homodyne detectors by using orthogonal phases of LOs. Measurement operators of these homodynes are given by
where we put . We treat these values as a complex number given by
This value corresponds to complex amplitude of the state . The vacuum term means the uncertainty of simultaneous measurement of and . We define similar value about temporal-mode as below,
ii.3.2 Complex-number principal components analysis
Next, let us discuss applying PCA to the complex variables . This process converts correlated variables into uncorrelated variables which satisfy
Because CW dual-homodyne measurement has two-mode input , the average value is given by
Therefore, shows the average photon number the state has in a temporal-mode . For example, when applying PCA to single-photon states , we get and . Generally, when we analyze single-temporal-mode states, only is larger than zero, and . To distinguish from previous PCA, we call this method as complex-number PCA (CPCA) in this paper.
We can carry out CPCA by introducing a matrix given by
We can obtain this matrix via CW dual homodyne measurement. We measure a target state with a sampling rate during to get a set of values . By repeating the same measurement, we can calculate . is an Hermite matrix thus diagonalized by a unitary matrix as follows,
Like PCA, we can obtain eigenfunctions and average photon numbers through diagonalizing process. The important thing is that are complex functions because the matrix is unitary. It also follows
In single-temporal-mode state analysis, TMF is given by
Like above, we can estimate complex TMF of single-temporal-mode states via CPCA.
ii.4 Dual-temporal-mode state analysis
In the previous section, we discussed single-temporal-mode state analysis by CPCA. However, useful states are often defined in multi-temporal modes. For example, some quantum error correction codes use multi-mode states to protect fragile quantum information Chuang.pra(1996) (); Wasilewaki.pra(2007) (); Bergmann.pra(2016) (). Therefore, we should develop mode characterization tools for multi-mode states. As a simple case, we treat two photons distributed in two temporal modes. As shown in Yoshikawa.qph (), such states always can be given by
where are complex functions not orthogonal in general. We will characterize the temporal-mode structures of by estimating and . In the following, we assume .
When we apply CPCA to , in no longer zero, but has positive value. Thus, and have the information of temporal-mode structures of . From Eq. (38), the modes and are orthogonal. Note that in most cases are not the same modes as , because are not orthogonal in general. The two photons in are distributed in , thus can be described by
where and satisfy . The quadratic polynomial of the creation operators in Eq. (41) can be decomposed into a product of linear polynomials,
where . From Eq. (1), we can use the next relation for arbitrary normalized orthogonal functions and ,
Thus, we can get
This is the same form as Eq. (40), thus are given by
Our goal is to calculate and experimentally. We can get by CPCA, thus we express the matrix by experimentally obtainable values.
Thus, when ,
There still exists uncertainty among and . Therefore, we cannot decide only from the CPCA results.
Interestingly, we can overcome this problem by introducing 4-th order moments. Firstly, we utilize
The latter case corresponds to the situation when and are orthogonal, because the columns of are orthogonal. In this case, the phase of is still not unique. We can determine the phase by using another -th order moment , which reveals .
In this way, we can characterize the temporal-mode structures of experimentally. What we have to do is executing CPCA to obtain and , and calculating the -th order moments from the dual homodyne signals and . Then, following Eqs. (45) and (52) or (54), we can calculate and in Eq. (40). In the next section, we will discuss the case when the analysis objects have errors due to a lossy optical channel.
ii.5 Mixed states analysis
We discussed pure states so far. In experiment, however, what we can prepare is mixed states due to the coupling between quantum states and the environment. Usually, the most dominant error is photon loss. In this section, we show that the analysis method discussed in Sec. II.3 and II.4 can work even when photon loss exists.
When we treat pure states , the component of is given by
The photon loss process is usually described by a beam splitter model. When one photon is lost with a probability , the mode is mixed with a vacuum mode by a beam splitter whose transmittance is ,
Similarly, in Eq. (28) is changed into given by
where and are vacuum terms due to the photon loss and dual-homodyne measurement. Photon loss degrades pure states into mixed states , then the component of in Eq. (34) is given by
Thus, we get
where is an -dimensional identity matrix. When is diagonalized by like Eq. (36), is also diagonalized by as follows,
Therefore, photon loss only changes into , and we can assume CPCA gives the same eigenfunctions in pure state case and mixed state case.
Let us discuss how photon loss affects our analysis method. In single-temporal-mode state analysis, what we want is the eigenfunction as explained in Sec. II.3, thus photon loss doesn’t affect the analysis procedure.
In Sec. II.4, we calculated from and matrix given by Eqs. (52) and (54). We can still obtain , but we need slight modification about . We introduced a -th order moment in Eq. (49). When photon loss exists, what we actually obtain is given by
Thus, the phase of the moment is not affected. Because , we can modify the norm of the moment,
We can decide the sign of the Eq. (64) by the phase of another moment , which is also not affected by photon loss. Therefore, we can calculate the functions experimentally even when photon loss exists.
In this section, we showed photon loss doesn’t affect our TMF estimation essentially. In the next section, we demonstrate these methods.
iii.1 Heralded creation of optical non-Gaussian states
High purity non-Gaussian states have been created by heralded scheme Lvovsky.prl(2001) (); Lvovsky.prl(2002) (); Yukawa.optexp(2013) (); Dakna.pra(1997) (); Ourjoumtsev.science(2006) (). In this method, entangled two modes (idler and signal modes) are prepared and photon detection in the idler mode heralds non-Gaussian states in the signal mode. So far, such states as single-photon states, superposition of photon number states, and Schrödinger’s cat states have been created Lvovsky.prl(2001) (); Lvovsky.prl(2002) (); Yukawa.optexp(2013) (); Dakna.pra(1997) (); Ourjoumtsev.science(2006) (). These created states are defined in wave packet temporal modes, called time-bin modes. The envelopes of the wave packets give the TMF of the modes.
In this scheme, we can engineer the TMFs by the configuration of idler path. Especially, TMFs can be complex when we put an asymmetric Mach-Zehnder interferometer in idler path Takeda.nature(2013) (). Therefore, heralded creation of non-Gaussian states using interferometers is a good way to demonstrate CPCA. To verify our temporal mode estimation method, we conduct 3 types of heralding experiments with an interferometer. First one is analysis of time-bin qubit as the simplest example. Second one is analysis of what we call dual-rail cat qubit, qubit consisting of Schrödinger’s cat states. This experiment shows our method’s ability to deal with phase-sensitive and multi-photon states in a single temporal mode. Last one is analysis of time-bin qutrit containing two photons to verify our dual-temporal-mode analysis explained in Sec. II.4. In the following, we explain how these qubits and qutrits are related to our complex TMF estimation method.
Generation of time-bin qubits and qutrits has already been realized in Refs. Takeda.nature(2013) (); Yoshikawa.qph (). In those studies, the idler and signal modes are realized by two-mode squeezed vacuum emitted from non-degenerate optical parametric oscillator (OPO). One- (two-)photon detection after interferometer(s) in the idler mode heralds time-bin qubits (qutrits) in the signal mode. Time-bin qubits Takeda.nature(2013) () are generally recognized as two-temporal-mode states, where one photon is distributed in two orthogonal time-bin modes and . They are described as