Probing Quantum Correlation Functions Through Energy Absorption Interferometry

Probing Quantum Correlation Functions Through Energy Absorption Interferometry

Abstract: An interferometric technique is proposed for determining the spatial forms of the individual degrees of freedom through which a many body system can absorb energy from its environment. The method separates out the coherent excitations present at any given frequency; it is not necessary to infer modal content from spectra. The system under test is excited with two external sources, which create generalized forces, and the fringe in the total power dissipated is measured as the relative phase between the sources is varied. If the complex fringe visibility is measured for different pairs of source locations, the anti-Hermitian part of the complex-valued non-local correlation tensor can be determined, which can then be decomposed to give the natural dynamical modes of the system and their relative responsivities. If each source in the interferometer creates a different kind of force, the spatial forms of the individual excitations that are responsible for cross-correlated response can be found. The technique is a generalization of holography because it measures the state of coherence to which the system is maximally sensitive. It can be applied across a wide range of wavelengths, in a variety of ways, to homogeneous media, thin films, patterned structures, and to components such as sensors, detectors and energy harvesting absorbers.

1 Introduction

Quantum correlation functions [1], and their related Green’s functions, play a central role in solid-state physics. They describe dynamical behaviour, and reveal internal order, whilst preserving the exchange symmetries of the constituent particles. The Hermitian and anti-Hermitian parts of retarded correlation functions describe reactive and dissipative processes respectively, and the anti-Hermitian parts also characterise the fluctuations that are present when thermal systems are observed passively [2, 3]. Retarded Green’s functions are central, through Landauer’s formalism [4], to determining the transmissive channels available in multiport quantum networks.

In this paper we propose a technique, called Energy Absorption Interferometry (EAI), for measuring the anti-Hermitian parts of retarded correlation functions. Once this has been done it is possible to determine the spatial forms of the individual coherent excitations through which a many body system can absorb energy from its environment: individual plasma oscillations, current distributions, spin waves, phonon modes, etc. EAI also allows the spatial forms of the individual coherent excitations that connect generalized forces of different kinds to be determined.

The basic idea is to excite the system under test with two external sources, and then to measure the fringe in the total power dissipated as the relative phase between the sources is varied. If the complex fringe visibility is measured for different pairs of source locations, and where appropriate polarisations, the anti-Hermitian part of the complex-valued non-local correlation tensor can be determined, which can then be decomposed to give the natural dynamical modes of the system and their relative responsivities. The method separates out the coherent excitations present at any given single frequency; it is not necessary to infer modal content from spectra. Our proposed technique is essentially a generalization of holography because it measures the state of coherence to which the system under test is maximally sensitive.

There is a wide variety of reasons why it is important to know the allowed, collective excitations of many body systems. In the case of electromagnetic [5, 6, 7, 8, 9, 10], elastic, piezo-electric, and acoustic sensors such as sonar [11, 12, 13, 14], it is essential to know the number, efficiencies and precise forms of the individual modes through which the device can absorb energy. A scanned measurement with a single source can only determine the overall power reception pattern, it cannot determine the amplitude, phase and polarisation patterns of the individual modes that make up the total response.

In the case of microwave and optical photon-counting detectors for quantum communications [15, 16, 17, 18], it is essential to avoid, or at least terminate carefully, electromagnetic modes that can only couple noise and stray light into the detector. In the case of energy harvesting components, antenna arrays and absorbers, including micro-mechanical devices [19], it is essential to maximise the number of degrees of freedom available for collecting power. The same considerations apply to near-field energy and information transfer between separated or overlapping volumes [20, 21, 22]. In the case of qubits for quantum computing [23, 24], which may be based on electromagnetic, spin [25, 26], or mechanical [27, 28] resonators, it is essential to understand the number, nature and origin of the mechanisms that couple the active elements to their passive environments, causing decoherence.

Because we describe EAI in terms of generalized conjugate variables, it has wide applicability. It can be implemented over a wide range of wavelengths, to homogeneous media, thin films, nano-patterned structures, classical and quantum metamaterials [29] , and to individual components and arrays. Unlike passive observations of thermal fluctuations, which through the fluctuation-dissipation theorem [30, 31, 32] also contain information about correlation functions [33, 34, 35, 36], the method achieves high signal-to-noise ratios by driving the system under test with external sources. Low-power sources can be used to probe systems in near-equilibrium, and high-power sources can be used probe the differential behaviour of systems in highly non-equilibrium states. The method can also be used as a convenient tool for exploring and characterising the behaviour of numerical many-body simulations.

2 Correlation functions

If a generalized external classical force acts on a many body quantum system, the change in the Hamiltonian is

(1)

Superscript denotes a specific generalized force within some set: electric scalar potential, magnetic vector potential, magnetic field, elastic force, etc. Each generalized force is associated, through (1), with a quantum observable , that determines the forces contribution to the total energy. For reasons that will become clear, the domain of integration, , is indicated explicitly. The domains corresponding to different forces can be the same, overlapping or completely disjoint.

According to Kubo [2], the expectation value of the change in at space-time point when generalized force is applied is

(2)

which is a generalized displacement. The spatial vector components of the generalized susceptibility tensor are given by the retarded correlation functions

(3)

We shall use dyadic notation, denoted by a double overline, for spatial vector operators, which does not preclude the possibility that one or both of the generalized forces may be scalars. and are operators in the Heisenberg picture, denotes the expectation over the grand canonical ensemble using the effective Hamiltonian, which includes chemical potential, is the commutator, or anticommutator where appropriate, and the step function ensures causal response. and form conjugate pairs, and so the formalism is general. Kubo’s expression provides a way of calculating macroscopic response functions using quantum-statistical methods, and therefore it contains information about the spatial and temporal excitations allowed.

When a generalized force is applied, the expectation value of the instantaneous rate of work done is given by

(4)

where is the expectation value of the resultant change in the operator on which acts. Using (2) in (4), and allowing for two different kinds of force to be present simultaneously, , the time averaged rate of work done becomes

(5)

The diagonal terms give the powers dissipated by the forces individually, whereas the off-diagonal terms arise because the application of one force can result in a perturbation of the quantum observable associated with the other force. In some cases, the time averaging, over , should be replaced by a convolution integral representing post-measurement filtering.

Often, we are only interested in probing self correlations : for example electric or magnetic susceptibility. Sometimes, we are interested in measuring cross correlations, : for example, ferro-electric susceptibility. One can never measure all correlations associated with all possible physical variables, and so it will be necessary to extract subspaces corresponding to the quantities of interest.

Use the following spectral decomposition for the susceptibility tensor

(6)

and require that the applied forces are time harmonic

(7)

For presentational simplicity, assume that the Hamiltonian of the unperturbed system is constant, so that the correlation function depends only on time differences, , and the spectral representation becomes diagonal . In this case, (5) becomes

(8)

For long integration times, , the first two terms disappear, and

(9)

where we have used .

Each term is a scalar, and so the transpose can be taken without changing the result. Taking the transpose of the second term, and swapping the dummy variables and , and and , gives

(10)

where

(11)

is the adjoint of . is the anti-Hermitian part of the susceptibility tensor, rendered Hermitian by the factor in the denominator. (10) is the average dissipated power when the spatial response is non-local, and the temporal response is stationary. It reduces to well-known expressions in the appropriate limits; for example spatial shift invariance.

Suppose that each applied classical force is itself a statistical quantity defined over an ensemble. (10) is a scalar, and so taking the trace on both sides, rotating to the right, and then calculating the classical average, , gives

(12)

where double-dot notation is used to denote the contraction of the vectorial parts of the tensors. is a tensor field that describes the spatial state of coherence of the applied generalized forces. Strictly, (12) is the average absorbed power when the applied forces are described in terms of slowly varying analytic signals [37]. For broadband forces, (12) is a spectral power, and should be integrated over .

(12) shows that the total dissipated power is given by the full contraction of two tensor fields to a scalar: one of which characterises the ability of the many body system to absorb energy, and the other characterises the spatial state of coherence of the applied forces. (12) is formally an inner product in a mixed tensor space, and so the measured power is given by the projection of a tensor that describes the state of coherence of the applied forces onto a tensor that describes the state of coherence to which the system is maximally receptive. This point will be discussed later.

3 Absorption Interferometry

3.1 Self correlations

Consider the situation where two external, coherent, phase-locked sources of the same kind, , are used to excite a system: Figure 1,

(13)

is the vector force produce by the first source, denoted by the first subscript, when it is placed at sample position and in polarisation state : for example, an electric or magnetic dipole. is the same quantity for the second source, but now its relative phase can be varied by the experimenter. In this case, because there are no sources of the second kind.

Figure 1: Energy Absorption Interferometer. Two phase-locked sources produce generalized forces and . The time-averaged total power displays a fringe as the differential phase is varied. The complex visibilities for different source locations enable the complex-valued system response tensor to be determined.

Because the external sources are fully coherent and phased locked, the dissipated power is given by (10),

(14)

which follows because only one term is present in the sum over . The domains of integration are now the same. is Hermitian, and so the first and last terms are real, and independent of the phase difference between the sources. The first term is the total power absorbed from the source at position and in polarisation , whereas the last term is the total power absorbed from the source at position in polarisation . The second and third terms are complex scalars, and the complex conjugates of each other.

Figure 2: Fringe in the dissipated power as relative phase between sources is varied. The average is given by the sum of the powers dissipated by the sources individually , the height of the fringe is given by the magnitude of the cross correlation , and the phase of the fringe is given by the phase of the cross correlation .

The dissipated power can be written

(15)

where and are the amplitudes and phases of

(16)

which are the matrix elements of the dissipative part of the susceptibility tensor in the vector space, strictly the dual space, of the sources. As the phase is varied, the dissipated power displays a fringe, Figure 2, which gives the complex matrix elements. In practice, it is not necessary to sweep out each fringe explicitly, but it is sufficient to switch between two states and to record the real and imaginary parts of the matrix element directly. In experimental work, we have found it convenient to run the sources at slightly different frequencies, and then to use a lock-in amplifier to measure the real and imaginary parts of the modulation directly [38].

If the fringe is recorded for enough source locations and polarisation states, to be quantified later, the complex-valued response tensor can be found in the vector space of the sources. Suppose that the sources are moved throughout some scanning region, volume or surface, leading to a total of sample positions and polarisations. The impressed forces then form a basis , where different correspond to different combinations of and . The resulting basis is general: it is not necessary to use the same polarisation states, or indeed orthogonal polarisation states, at the sample positions, which helps devise simple scanning strategies.

In cases where the sources produce point-like unidirectional forces, say mechanical probes,

(17)

can be substituted into (16) to yield

(18)

and the experiment measures directly the corresponding vector component of the spatial response tensor at the positions of the sources. The spatial coherence function can be traced out by moving the probes.

At the other extreme, where the sources provide spatially uniform forces in orthogonal directions, as in the case of the magnetic fields produced by orthogonal Helmholtz pairs,

(19)

(16) then gives

(20)

which shows that fringes are formed in the total dissipated power as the phase between the fields, currents in the orthogonal Helmholtz pairs, is varied. As will be seen later, this allows the directional forms of the individual degrees of freedom that make up the total, spatially integrated, directional response to be determined.

In general, the basis functions are neither orthogonal nor uniform over : for example the sampling fields produced by a scanned electric or magnetic dipole. The spatial susceptibility tensor must then be reconstructed through

(21)

where is the dual of . The dual set can be found numerically, see later, once the functional forms of the impressed force are known. This scheme applies even if the two sources in the interferometer do not produce the same force distributions, say because they are not identical.

The source fields and their duals span the same vector space. (21) may, however, be an approximation because it is not generally known whether is complete, over complete or under complete with respect to the degrees of freedom in . Reconstruction using the dual functions covers all possibilities, giving the best orthogonal metric projection when the basis is under complete. The process of reconstructing using the dual set amounts to ‘deconvolving’ the probe field patterns from the measurements.

3.2 Cross correlations

In some cases, the primary need is to determine the response tensor corresponding to two different kinds of generalized force. Interferometry is then carried out using two different kinds of source:

(22)

and (10) becomes

(23)

The first and last terms are the powers dissipated by the two sources individually, into their respect loss mechanisms. The second and third terms lead to a fringe, which only exists when there is a cross coupling in the system. Notice the mixed domains on the integrals. The dissipated power can be written

(24)

We have used the fact that the overall tensor is Hermitian, from which it follows that , which is Onsager’s reciprocity [39]. It follows that . The complex visibility of the observed fringe gives the real and imaginary parts of , which are the matrix elements of the cross response tensor in the basis of the source fields:

(25)

The matrix elements in this case are evaluated with respect to two different vector spaces, not least because and can be different.

The cross response tensor is then reconstructed through

(26)

where is the dual set of , over the appropriate domain.

In summary, interferometry can be used to find the matrix elements of the anti-Hermitian part of the generalized susceptibility tensor in the vector space of the field patterns of the applied forces. Dual functions can then be used to reconstruct the response tensor in the space domain. One may only be interested in the spatial correlations corresponding to one kind of force, in which case it is sufficient to carry out an experiment with two sources of the same kind; or one may be interested in finding the spatial correlations corresponding to two kinds of force, in which case it is possible to use two different kinds of source. Two different sources create fringes that isolate and extract information relating to cross-correlated response.

4 Response tensor decomposition

What information is contained in the susceptibility tensor, and how many degrees of freedom need to be found? The susceptibility tensor and force correlation tensor are, by definition, Hermitian when considered over all variables: position, polarisation, and type. The response tensor only appears as the kernel of an integral equation (10), and so it is appropriate to look for a discrete decomposition. It can be shown that a tensor field, , is Hilbert Schmidt [40] if

(27)

Every physical system must satisfy this condition. According to (11), comprises a forward and time-reversed process, both of which map a set of generalized forces onto a set of responsive perturbations. Because there is only a finite number of physical degrees of freedom available for effecting this mapping, (27) follows. Equivalently, the response tensor has a finite coherence volume, wherever it is measured, and the system occupies a finite region, and therefore there is a finite number of degrees of freedom available. A truly local response having the form is not physically possible because there would be an infinite number of degrees of freedom in every finite volume. A similar condition holds for the force correlation tensor:

(28)

Because any physical system must satisfy (27) and any realisable force must satisfy (28), the following Hilbert-Schmidt decompositions exist:

(29)
(30)

The basis set spans fields of type over the domains and respectively. The same is true of . However, orthogonality is only guaranteed over the whole of the vector space, including the sum over :

(31)

which is undesirable in some circumstances, as will be discussed. The integrals in (27) can be evaluated by substituting (29) and using the orthogonality condition (31). This process gives , and therefore (27) essentially states that the number of channels for absorbing power is limited. Likewise, (28) states that the number of channels available in the source that can do work is limited. For all systems, the eigenvalue spectrum, , tends rapidly to zero as some threshold value of is exceeded, and only a finite number of degrees of freedom need to be found when carrying out interferometry.

(29) and (30) can be substituted into (12) to give

(32)

where

(33)

(32) describes power absorption in terms of a scattering process, , that projects the natural modes of the forces, having weightings , onto the natural modes of the system, having responsivities . When the system is driven by an incoherent superposition of its natural modes , and the system is maximally responsive with respect to spatial variations in the force.

(29) and (30), where a single set of basis functions spans both domains, are the most suitable decompositions in many cases. For example, if corresponds to an electric field and to a magnetic field, then these would be correlated if an electromagnetic wave is incident on the system. In this case, the diagonal block terms should be retained in . Alternatively, the impressed field may, for example, comprise a physical force and magnetic vector potential, in which case the two generalized forces can be regarded as independent, and the block off-diagonals are not needed. Later we shall discuss the situation where one force is a scalar and the other a vector, as in the case of the electric scalar potential and magnetic vector potential.

Rather than using (29) and (30), there is a different approach, which seems better suited to decomposing data when only part of the susceptibility tensor is measured. Because (27) and (28) hold, the individual terms under the sum must also be Hilbert Schmidt, and because the block diagonal terms are each Hermitian, they can be diagonalised separately:

(34)
(35)

In this case, forms a complete orthonormal basis over . Different orthogonal basis sets are therefore generated for the two domains. The same is true of the force basis .

Consider what happens when only one kind of force is present, say . Substituting (34) and (35) into (12) gives

(36)

(35) describes the partially coherent generalized force in terms of an incoherent superposition of fully coherent fields, with weighting factors . These are the natural modes of the illumination, as introduced in the context of optics by Wolf [37]. (34) describes the absorptive response in terms of a set of orthogonal modes, each having responsivity . According to (36), the natural modes of the force scatter, with efficiencies , into the modes to which the system is responsive. This representation constitutes the coupled-mode model [5, 6] of power absorption. Again, maximum coupling is achieved when the modes of the field match those of the system, over the appropriate domain, which defines the state of coherence to which the system is maximally receptive as the spatial form of the impressed field is varied.

If the system responds in an entirely local way, , a Hilbert Schmidt decomposition does not exist, but (12), still results in finite power, because the number of channels available for absorbing power is limited by the smoothness of the impressed force. Because, in this case, the natural modes of the system span any force distribution over , it behaves as a near-perfect absorber; the generalized equivalent of a ‘light bucket’.

Now consider the case where two different kinds of force are present simultaneously. In order to calculate the absorbed power, it is necessary to calculate the Hilbert Schmidt decomposition of the cross terms :

(37)
(38)

Primes have been used to indicate that the natural basis functions that describe the cross response may be different to those that describe the self response. (37) and (38) have the forms needed to ensure that the overall response tensor is Hermitian. (38) describes the cross correlations in terms of an incoherent superposition of fully coherent field pairs. In other words for every basis function in domain , , there is a unique, associated basis function in , . These basis functions are in one-to-one correspondence, revealing generalized force distributions in the two domains that are mutually fully coherent and uniquely related. For example, one might correspond to an electric field and the other to a magnetic field. The Hilbert-Schmidt decomposition of the off-diagonal block , therefore describes the cross correlations between two different vector spaces as a weighted linear combination of field pairs. This approach generalizes Wolf’s formalism to include cross correlations between different vector spaces.

The same decomposition can be carried out on the susceptibility tensor. The reason for decomposing the on-diagonal and off-diagonal blocks individually, is that it is only necessary to carry out partial interferometric measurements—say using two sources of the first kind, or two of the second kind, or one of each—in order to reveal collective behaviour. It should also be appreciated that the force can be described in terms of one scheme, say (30), and the system in terms of the other, say (34) and (37), and (12) still returns the correct result for the absorbed power.

The process of decomposing the self- and cross-subspaces can be summarised as follows:

(39)

which is shown schematically in Figure 3.

Figure 3: Two domains corresponding to two different generalized forces. They may be fully separated, partially overlapping, or the same. The dissipative response in domain is spanned by the orthonormal collective excitations having responsivties . The dissipative response in domain is spanned by the orthonormal collective excitations having responsivities . The cross-correlated response between the two domains is described by the function pairs and with relative weightings .

The top left block in (39) corresponds to the decomposition obtained when interferometric measurements are made using sources of type 1 only. The Hilbert-Schmidt decomposition, which is a diagonalisation in this case, gives the individual natural modes through which the structure can absorb power from a partially coherent force of type 1; the eigenvalues are the associated responsivities. In addition, the bottom right block can be measured and decomposed in the same way, giving a full description of the system’s ability to absorb power from a partially coherent source of type 2. If only the off-diagonal blocks in (39) are measured, the Hilbert-Schmidt decomposition describes cross-correlated response.

An interesting question is how do the functions and relate to each other? Ordinarily it might be expected that the modes responsible for absorbing power from the sources individually are the same as the modes responsible for modulating the behaviour when two different kinds of force are applied simultaneously: in other words, and are the same. The real elements on the leading diagonal of the whole tensor account for all energy dissipation mechanisms present. The off-diagonal blocks account for work done by one kind of force on the conjugate variable associated with the other source, and therefore account for the modulation in the dissipated power when sources of two kinds are present. They do not represent power dissipation mechanisms in their own right, and certainly, the absorbed power, given by (20), cannot become negative as the phase between the interferometric sources is varied.

and have null spaces corresponding to those force distributions that cannot dissipate power in the system. In other words the tend rapidly to zero above some critical index . Likewise the tend rapidly to zero above some critical index . The null spaces of the off-diagonal blocks span, at least, the null spaces of the diagonal blocks, and , which can lessen the amount of experimental work needed if the whole tensor is measured. In fact, the sampling strategy can be chosen to ensure that any cross-correlations present will be found. The natural basis functions, and , do not have to be the same, but the must span the . The cross-correlated response can be described in terms of the modes of the self correlations. Ultimately, the precise relationship between the decompositions depends on the nature of the physical system being studied. To keep the analysis general, we prefer to calculate the natural modes in the two domains on the basis of the diagonal blocks, giving , and then to project the natural modes of the cross correlations, onto those basis sets to look for spatial relationships between the self and cross correlations.

To this point, we have assumed that both generalized forces are vector fields, but consider what happens when one force is a scalar and the other a vector. The overall generalized force is then described by a four-vector. In the case of an electric scalar potential and a magnetic vector potential, the use is clear. For any general four vector, the block decomposition becomes