A Statistical error on correlation function estimates

Schemes for the observation of photon correlation functions in circuit QED with linear detectors

Abstract

Correlations are important tools in the characterization of quantum fields, as they can be used to describe statistical properties of the fields, such as bunching and anti-bunching, as well as to perform field state tomography. Here we analyse experiments by Bozyigit et al. Bozyigit et al. (2010) where correlation functions can be observed using the measurement records of linear detectors (i.e. quadrature measurements), instead of relying on intensity or number detectors. We also describe how large amplitude noise introduced by these detectors can be quantified and subtracted from the data. This enables, in particular, the observation of first- and second-order coherence functions of microwave photon fields generated using circuit quantum-electrodynamics and propagating in superconducting transmission lines under the condition that noise is sufficiently low.

I Introduction

Field correlations are widely used in the characterization of classical and quantum fields Mandel and Wolf (2008); Walls and Milburn (2008). A particular set of correlations used for such purposes are the coherence functions of a field, as described by Glauber Glauber (1963a, b); Mandel and Wolf (1965). These functions can be used to quantify the ability of a field to interfere with itself, as well as to demonstrate features of quantum fields which cannot be reproduced in a classical system. One of the most famous of these quantum phenomena is known as anti-bunching Paul (1982); Davidovich (1996), and it is frequently used to characterize single-photon sources in the optical regime Kurtsiefer et al. (2000); Yuan et al. (2002); Keller et al. (2004); Birnbaum et al. (2005); Hijlkema et al. (2007). Over the recent years Josephson-junction based superconducting circuits, resonators and transmission lines have emerged as a platform for performing quantum optics experiments in the microwave regime Wallraff et al. (2004); Schuster et al. (2007); Wallraff et al. (2007); Houck et al. (2007); Astafiev et al. (2007); Hofheinz et al. (2008); Fink et al. (2008); Regal et al. (2008); Castellanos-Beltran et al. (2008); Hofheinz et al. (2009); Astafiev et al. (2010). While in the optical regime coherence functions are usually measured using an interferometer where photon number detectors are used, in the microwave regime, linear detectors (i.e. field quadrature measurements) are ubiquitous due to the difficulty of building reliable photon number detectors. This raises the question of how to measure field correlations using linear detectors. This paper answers this question and describes the theory behind the recent experiments performed by Bozyigit et al. Bozyigit et al. (2010), where correlations of a propagating microwave field are measured using only linear detectors, instead of intensity detectors. While the discussion here focuses on the measurement of first- and second-order coherence functions of microwave fields, the analysis can be applied to any correlation of field operators. We note that the measurement of correlation functions of propagating microwave fields using non-linear (i.e. square-law) detectors was theoretically studied in Ref. Mariantoni et al. (2005), under the assumption of negligible correlation in the noise added by the detection chain. In practice, these correlations turn out to be important and are discussed here. Recent work by Menzel et al. Menzel et al. (2010) and Mariantoni et al. Mariantoni et al. (2010) is in a similar direction to the work presented here.

The paper is organized as follows. Section II gives a brief review of coherence functions and how they are measured with non-linear detectors. Section III describes how field correlations, and in particular coherence functions, can be measured using linear detectors. Section IV describes the effects of noise in the experiments, and finally Section V describes how the experimental setup can be simplified in circuit QED experiments.

Ii Coherence functions

The meaning of coherence of a field in a single frequency mode, with corresponding annihilation operator , can be understood by considering interference experiments which use the field leaking out of this mode. Using a double slit, the field can be made to travel two pathways of different lengths which terminate at a single point-like photon detector, as depicted in Fig. 1(a). The combined field that impinges on the detector is made up of fields originally emitted at times and (which depend on the lengths of the paths), so that the observed field intensity at the detector is the sum of the intensities of the two fields plus an interference term which depends on  Mandel and Wolf (2008). Interference effects can only be observed if this correlation is non-zero. It is therefore natural to define

(1)

which is called the first-order coherence function Glauber (1963a), as a measure of the emitted field’s potential to interfere with itself – in other words, a measure of the coherence of the field. One may also consider

(2)

for some time interval in order to obtain an expression that depends only on the time difference between the two paths. If , no interference effects can be observed for a path difference of , where is the speed of light.

Figure 1: (Color online) Experimental setups illustrating different degrees of optical coherence with intensity detectors. The red lines represent quantum fields, and the black lines represent measurement records.

The measure of coherence that is most often used to distinguish classical fields from quantum fields is the second-order coherence function given by

(3)

or by the integrated version

(4)

The canonical experiment which gives the physical interpretation of is one with a single light source and two point-like detectors, such that the field takes a time and respectively to reach each detector, as depicted in Fig. 1(b). In that case the correlation between the detected intensities is given by .

For classical fields, where the field operator in the expressions above are replaced by c-numbers, one finds that , while there are quantum states of the field that yield for , a phenomenon known as anti-bunching Paul (1982); Davidovich (1996). The canonical examples of anti-bunched field states are single photon states and squeezed states. In the case of pulsed experiments – where the light field state is prepared with a repetition period of – one writes instead that classical fields obey , and that some quantum states of the field yield for . Only pulsed experiments will be considered in the remainder of this paper, the generalization to continuous experiments being straightforward.

ii.1 Standard experimental setups

Figure 2: (Color online) Standard experiments for the observation of and using intensity detectors: (a) a Mach-Zender interferometer with a variable delay and a variable phase shift , and (b) a Hanbury Brown and Twiss (HBT) interferometer with a variable delay . The light-blue components are balanced beam-splitters. The cavity is taken to be one-sided, with one mirror being perfectly reflective.

As illustrated in Fig. 2, we consider experiments where the source is a single mode of a cavity coupled to a transmission line via a leaky mirror, a situation typical of cavity QED Raimond et al. (2001); Mabuchi and Doherty (2002); Blais et al. (2004). In circuit QED for example, arbitrary superpositions of a single photon and vacuum can be prepared in the dispersive regime via Purcell decay Houck et al. (2007) or by strong coupling to a qubit brought into resonance with the cavity Bozyigit et al. (2010), although details of the state preparation are not important for the remainder of the discussion. The harmonic field in the cavity is associated with an annihilation operators with the usual same-time commutation relation . Using input-output theory Collett and Gardiner (1984); Gardiner and Collett (1985); Walls and Milburn (2008); Gardiner and Zoller (2004), one can show that is related to the modes of the transmission line via

(5)

where is the rate at which photons leak out of , and the input and output fields are given by

(6)
(7)

for transmission line modes at times , and correspond to fields propagating towards or away from the cavity. The commutation relations of the input and output fields are given by

(8)

These definitions lead to an equation of motion for in the interaction frame to be given, for a one-sided cavity, by

(9)

From Eq. (5) it is clear that the correlations of are proportional to the correlations of when is prepared in the vacuum state. The remainder of the discussion will focus on the observation of the coherence functions of the output field only, as they can be taken to be equivalent to the correlation functions of . The “out” subscript will also be dropped when it is clear form the context.

The state of the cavity field is taken to be prepared at times for integer and repetition period , and allowed to decay via the leaky mirror as described above. The repetition period is chosen to obey so that the cavity can be taken to be in equilibrium at the time of the next preparation of the cavity field.

When working with photons in the optical frequencies, is usually observed using a Mach-Zender interferometer with a variable delay of in one of the branches Bachor and Ralph (2004), as depicted in Fig. 2(a). The difference between the intensities in the photo-current detectors can yield the real or the imaginary part of , depending on the phase shift in the lower branch. The standard approach to the observation of the is to use a Hanbury Brown and Twiss (HBT) interferometer Bachor and Ralph (2004), which is illustrated in Fig. 2(b). In order to observe , one simply measures the correlations between the photo-currents of the two detectors.

Both these setups rely on field intensity detectors, which give information about the number of photons, and thus can be modeled by non-linear quantum optical interactions1. Low-noise intensity detectors for optical fields are common, and although non-linear detectors have been demonstrated in the microwave regime Gabelli et al. (2004); Houck et al. (2007) (albeit with higher noise levels than in optics), the main motivation for this paper is to illustrate how the coherence functions of microwave fields in circuit QED may be measured through the use of linear detectors only.

Iii Linear detectors

Field quadrature measurements of microwave signals is a standard technique Pozar (2004) which has been applied very successfully to quantum electrical circuits in the recent years to demonstrate, for example, new regimes of cavity QED Wallraff et al. (2004), high-contrast detection of qubit states Lupaşcu et al. (2006), photon states Schuster et al. (2007), and nanomechanical oscillator states Teufel et al. (2009). Since field quadrature operators are fundamentally different from number operators, different experimental setups are required in order to measure the coherence functions and . Grosse et al. Grosse et al. (2007) have demonstrated how a HBT interferometers can be modified to measure using field quadratures instead of intensity measurements. Here we analyse similar experiments Bozyigit et al. (2010), and consider generalizations and simplifications which exploit features of circuit QED, while at the same time considering the large added noise due to the HEMT amplifiers currently required for measurement in this system.

The details of the implementation of quadrature operator measurements in the microwave regime are different from the standard optical implementation. In particular, homodyne detection in the microwave regime is performed via mixing instead of beam splitting Pozar (2004). For simplicity, we will however consider the optical analogues of the devices we discuss. Common non-idealities in the microwave regime, such as weak thermal states instead of vacuum inputs, can be treated straightforwardly by considering different input states, and thus do not change the analysis significantly.

The measurement of both quadratures of a propagating field, realized in optics through 8-port homodyne Schleich (2001) or heterodyne detection, is performed by an IQ mixer in the microwave regime Pozar (2004). The symbol for the IQ mixer, and its description in terms of its optical analogue are depicted in Fig. 3. The input is any propagating quantum field with annihilation operator , which may stand for any propagating field considered in this paper. The outputs are quadrature measurements of the superpositions of the field with a mode in the vacuum state, where . These outputs are labeled and to emphasize that the measurements are made on different commuting modes, and correspond to the in-phase component and the quadrature component of the measurement respectively.

Finally, it is important to note that, for most circuit QED experiments, only averages of these quadratures over many realizations of the experiment are measured. Here, however, we are interested in experiments where the full time records of these quadratures are recorded, for each realizations of the experiments Bozyigit et al. (2010). Based on these full records, any averages or correlation functions can be reconstructed, as is discussed in the next sections.

Figure 3: (Color online) Optical analog of an IQ-mixer as an 8-port homodyne detector. The field is fed into a balanced beam splitter along with vacuum . The quadrature of one of the beam splitter outputs is measured while the quadrature of the other output is measured. The classical outcomes are the real and imaginary parts of the complex envelope .

iii.1 Complex envelope

Given the two classical outputs and , it is useful to define the complex envelope of as

(10)

which is a random c-number due to the dependence on the measurement records of the quadratures. Noting that

(11)
(12)

one may write that where the complex envelope operator is defined by

(13)

In order to simplify the remainder of the calculations, it is convenient to define in this manner instead of using the quadrature operators explicitly.

Given that the mode is in the vacuum state, the expression for the expectation values take simple forms. The presence of the vacuum mode is indeed important as it leads to the commutation relation

(14)

implying that is normal and therefore diagonalizable. Since is described by the sum of the commuting operators and , its eigenvalues are given by the sum of the eigenvalues of these operators for any fixed eigenvector. This corresponds to the measurement record of being simply the sum of the measurement records and , as claimed earlier. Note that this does not imply that both quadratures of can be measured simultaneously without back-action.

Since and commute at all times, arbitrary correlations of these operators, like the correlations of their measurement records, do not depend on operator ordering. Therefore,

(15)

independently of the ordering of the terms. However, in order to reduce these expressions to a correlation function of alone, one must rewrite the expression such that the modes are in normal ordering in order to immediately evaluate the expectation values, leading to

(16)

or in other words, correlations of the complex envelope of a field correspond to anti-normally ordered correlations of the field operator, under the assumption that the mode is prepared in the vacuum. In this case the measurement of is described by the Husimi-Kano function which is known to give access to anti-normally ordered same-time correlations Husimi (1940); Kano (1965); Schleich (2001). Similar results hold for multi-time correlations.

It is important to note that, while with this approach arbitrary correlations can be evaluated, the number of statistical samples needed to obtain a desired precision in the estimate grows as the noise power raised to the desired correlation order (see Appendix A for details). In practice, this limits the order of the correlations measured with current amplifier noise levels due to the large number of repetitions of the experiment needed to obtain reasonable error bars. Use of quantum limited amplifiers would greatly improve the situation Regal et al. (2008); Castellanos-Beltran et al. (2008); Clerk et al. (2010).

iii.2 observation

As depicted in Fig. 4, with IQ-mixers, the first order correlation function can be measured from the outputs of a HBT interferometer. Because of the unitary of the beam splitter and the presence of a vacuum port, the complex envelope operators of the outputs labeled and commute.

Figure 4: (Color online) Hanbury Brown and Twiss interferometer with complex envelope measurement used to measure the first-order coherence function . The output of the cavity field is separated by an on-chip beam splitter and the resulting fields measured by an IQ-mixer.

The auto-correlation of one of the complex envelopes, say , is given by

(17)

while the cross-correlation between the complex envelopes is

(18)

where we have used the fact that the expectation values of all the vacuum modes are zero. Thus the first-order coherence function of the field is immediately accessible from cross-correlations of the complex envelopes in a modified HBT interferometer via

(19)

up to non-idealities, such as amplifier noise, which will be treated later. The exact expressions for of the states prepared in Ref. Bozyigit et al. (2010) are given in Appendix B.

Although the divergence of the functions may appear problematic, in reality due to the finite bandwidth of the experiments these delta functions are replaced by smooth bounded functions, while the coherence functions are distorted by a convolution kernel which preserves the relative heights of the peaks in the experiment. This results in the filtered correlation functions

(20)

where and is a function describing the effective action of the filter (see Appendix C for details).

iii.3 observation

The expressions needed to measure the second-order coherence function from the complex envelopes can be constructed by inspection from Eq. (13). Depending on which factors are taken to be complex conjugates or to be displaced in time by , different correlations can be used to extract information about . One such choice is

(21)

so that can be obtained immediately via

(22)

Another choice that leads more directly to is

(23)

so that

(24)

As described earlier, the divergence of the functions is taken care of by filtering in a realistic experiment. The main distinction between these two approaches of measuring the second-order coherence functions is how they are affected by noise in the experiment, as is discussed in the next section.

Iv Rejection and subtraction of noise

The amplitude of microwave signals in a superconducting quantum circuit is small enough that amplifiers are essential for their observation, and so in a realistic experiment, the field is amplified before mixing. Using the Haus-Caves description of a quantum amplifier Caves (1982); Gardiner and Zoller (2004); Clerk et al. (2010), an input operator and an output operator for a phase-preserving amplifier with gain are related by

(25)

where is an added noise mode.

It is clear that if there will be added noise due to amplification, even at zero temperature. However, for thermal white Gaussian noise, one finds that all odd order moments vanish. As a result, the first moments of quadrature fields are not affected by this amplifier noise, just as they are not affected by vacuum noise. The contributions from other moments may be non-zero, however, and must be accounted for. For simplicity, we only consider the case of Gaussian white noise here, but similar results follow straightforwardly for general noise as long as the noise is independent of the inputs. Since the noise moments can be extracted from experimental data, the assumption of Gaussian noise is not essential.

The noise modes from different amplifiers are taken to commute, but in general they may be correlated. While the noise is normally taken to come from the amplification Caves (1982); Gardiner and Zoller (2004); Clerk et al. (2010), formally one may also take to include thermal noise from other sources, such as the vacuum ports of the IQ-mixer and of the beam-splitter, with only minor modifications. Here is taken to have a commutator in order to preserve the bosonic commutation relations of the amplified signals , and auto-correlation . The noise sources are assumed to be independent of the inputs, so that , and . The correlations between and the noise mode from the other amplifier in the experiments described here is taken to be while .

Using this noise model, one can calculate the different correlations using the amplified modes, resulting in

(26)

in the unfiltered case.

Since the thermal noise in the amplifiers is independent of the inputs, a steady-state experiment with the input mode in the vacuum state can be used to estimate the noise strengths and subtract the corresponding terms from to obtain an estimate of . When the noise cross-correlation is expected to be zero or negligible compared to the noise auto-correlations and , the approach to the estimation of based on provides noise rejection without additional post-processing.

The second-order coherence function for the amplified fields has similar properties. One finds

(27)

while

(28)

where all odd moments of the noise modes where taken to be zero (if such an assumption cannot be made, similar expressions involving the odd moments are easily derived but are omitted here for brevity). The recovery of the second-order coherence function from noisy signals is clearly more involved, but requires only the estimation of first-order coherence functions, as well as two and four-point noise correlations in an experiment where the input mode is prepared in the vacuum. Since the filters are taken to be linear and time-invariant, is a scaled and distorted version of , preserving the relative heights of the peaks, so that the non-classical properties of the field can still be verified. Once again we see that provides a more direct estimation of by rejecting contributions from uncorrelated noise up to four-point noise correlations.

V Two-sided cavities

Figure 5: (Color online) The setup for the observation of coherence functions using a two-sided cavity.

Strictly speaking, the beam splitter is not necessary for the observation of the coherence functions described above. If one considers a two-sided cavity, illustrated in Fig. 5, the correlations between the cavity outputs behave in a manner similar to the outputs of the beam splitter in the HBT interferometers. In particular, using causality as well as the boundary conditions of the input and output fields of the two-sided cavity, Appendix D shows that

(29)

where is defined in a manner analogous to , with a mirror leakage rate , and an amplifier with gain being applied before mixing and measurement. It is thus possible to measure the complex envelopes of the two cavity outputs and calculate the correlations in the same manner as in the modified HBT setup without the need for an additional beam splitter. This can lead to simpler and smaller experimental setups, as beam splitters in the microwave regime can occupy a significant area in coplanar devices.

Calculating the correlations using the two cavity outputs and one finds

(30)
(31)
(32)
(33)

where and are now the coherence functions of the cavity field instead of the cavity output fields, leading to the introduction of additional factors which depend on the cavity leakeage rates . These expressions are directly analogous to Eqs. (26), (27), and (28).

Vi Summary

We have analysed experiments for the measurement of field correlations using only field quadrature detectors and in the situation where the full record of many repetitions of the experiment are available. The combination of the quadrature measurements into complex envelopes gives direct access to anti-normally ordered field correlations. While re-ordering of the operators in the correlations and the use of phase-preserving amplifiers introduces additional noise into these measurements, we demonstrated that the noise can be accounted for and subtracted in order to reveal only the field correlations of interest. Although there are indications that the number of statistical samples scales exponentially with the order of the correlation function, the measurement of low order correlations is possible for current amplifier noise levels.

Vii Acknowledgments

M.P.S. was supported by a NSERC postdoctoral fellowship. D.B. and A.W. were supported by ERC and ETHZ. A.B. was supported by NSERC, the Alfred P. Sloan Foundation and is a CIFAR Scholar.

Appendix A Statistical error on correlation function estimates

In order to estimate the minimal number of repetitions of the experiment which must be performed to extract a given correlation function, consider the product of uncorrelated Gaussian random variables with zero mean and identical variances . These random variables correspond to the measurements of different outputs at steady-state after the cavity state has decayed, and the variances are given by the noise power of the measurement record (including vacuum noise). In order to illustrate the argument, we consider real valued random variables first, and generalize to complex valued random variables . Since these random variables are uncorrelated, it follows that . However, given a finite number of statistical samples, the sample average will deviate from zero due to statistical fluctuations. Signal features which are comparable with the typical size of these fluctuations cannot be reliably observed. As the typical size of these fluctuations decreases with the increasing number of repetitions, this is in principle not a fundamental problem.

In order to estimate the number of samples needed for the reliable estimation of two-point correlations, consider the product of two Gaussian random variables. The characteristic function of this product is given by

(34)
(35)

The characteristic function of the average of samples is given by

(36)

Given some error and a number of repetitions , the probability that the sample average obeys is given by the integral of the inverse Fourier transform of over this range and simplifies to

(37)

which can be evaluated by numerically. Thus it is straightforward to calculate the number of repetitions required to observe a feature larger than with confidence .

Another approach that provides a looser bound, but is more readily generalized to higher order correlations, is based on Chebyshev’s inequality Rosenthal (2009). The variance of the product of independent random variables with zero mean is the product of the variances of each of the random variables. In the case of samples of the product of independent random variables one finds that

(38)

Note that in order to obtain this bound no assumption was made about the form of distribution of the random variables, other than the fact that the random variables are independent. Solving for one obtains the worst-case upper bound

(39)

which makes clear the exponential relationship between the order of the correlation and the number of samples needed to have a statistical error of less than with some fixed probability.

In order to generalizing this to complex-valued random variables – where the real and imaginary parts of are independent with variance , and the are mutually independent – simply consider the real and imaginary parts of the correlations separately. In that case, because a larger number of terms contribute to the real and imaginary parts of the correlation, the variance has a larger bound, and one finds

(40)

where is the probability that the absolute value of the real or imaginary parts of are greater than .

There is no indication that taking into account the Gaussian statistics of the random variables leads to better scalings. Thus the ratio of the number of statistical samples needed to estimate vs. for some fixed noise variance and desired accuracy is at worse proportional to the noise power in the experiments. As a result the noise added by the amplifier can be the crucial element in determining the feasibility of a correlation function experiment. It becomes even more important for higher order correlations, where the number of samples depends on the noise power raised to some larger exponent.

Appendix B Coherence functions for states with at most one photon

In the experiments described here Bozyigit et al. (2010), the cavity is periodically prepared in the state , with a period such that . This ensures that, to a very good approximation, the cavity returns to the vacuum state before the superposition is prepared again.

The coherence functions can be calculated straightforwardly via their definitions in terms of the field correlations, while the correlations can be calculated by solving the Heisenberg equations of motion for the cavity field, and using the quantum regression theorem Lax (1963); Carmichael (1993); Gardiner and Zoller (2004). This procedure can be greatly simplified by noting that, if and are in different preparation periods, then

(41)

and

(42)

due to the assumption .

In the case were and are between and for some integer , one finds that

(43)
(44)

while if and are in different preparation periods starting at and , one finds that

(45)
(46)

After integration over , the first-order coherence function can be shown to be well approximated by

(47)

This can be interpreted as a series of time-shifted copies of , where the peak centered at has a height equal to , while the peaks centered at non-zero multiples of have a height equal to .

Under similar assumption, the second order correlation function can be shown to be well approximated by

(48)

such that the center peak has a height proportional to while the other peaks have heights proportional to .

For the superpositions of vacuum and a single photon considered in Bozyigit et al. (2010), we find that

(49)
(50)
(51)
(52)

indicating that the center peak of is abscent, while the other peaks are non-zero, which is a signature of the purely quantum effect known as anti-bunching Paul (1982); Davidovich (1996).

Appendix C Filtering

The finite bandwidth of the detection chain can be modeled by considering the insertion of a bandpass filter in an ideal (infinite bandwidth) detection chain. In order to calculate the effect of filtering on correlation functions one can consider a general framework which describes what happens to multi-time, multi-channel correlations when measurement signals are filtered. Assume a system with channels where each channel is filtered individually. One can write the filtered outcome of each channel in terms of the input signal and the filter function by using the relations for linear time-invariant systems Proakis and Manolakis (2006)

(53)

Each channel has a separate time variable to capture the case of multi-time correlations. This also clarifies with respect to which variable the convolution is done. The goal is now to express the filtered coherence function

(54)

in terms of the unfiltered coherence function

(55)

This can be done straightforwardly by substituting Eq. (53) into Eq. (54).

(56)

Realizing that all convolutions are related to different time variables one can rearrange this expression as

(57)

The integral form clarifies this expression

(58)

This expression can be seen as a generalized convolution with respect to more than one time variable. Introducing the global filter function