AAWSAP DIRD Quantum Tomography of Negative Energy States in the Vacuum January 11 2011
⚠ Texto extraído por OCR de la fuente oficial — puede contener errores de reconocimiento. El documento original es la autoridad.
UNCLASSIFIED/ /FOA QFFI@IAL l:l!H! 9HL I
Defense
Intelligence
Reference
Document
Defense Futures
11 January 2011
!COD : 10 August 2010
DIA-08-1102-007
Quantum Tomography of
Negative Energy States in
the Vacuum
UNCLASSIFIED// FOR OPPICIJ!ct l:ISE 8P..,¥
UNCLASSIFIED/ /FOA OlifilGIAk WSE 8HLV
Quantum Tomography of Negative Energy States in the
Vacuum
The Defense Intelligence Reference Document provides non-substantive but
authoritative reference information related to intelligence topics or methodologies.
Prepared by:
Technology Warning Division (DW0-4)
Defense Warning Office
Directorate for Analysis
Defense Intelligence Agency
Author:
AAP Person 58
COPYRIGHT WARNING: Further dissemination of the photographs in this publication is not authorized.
This product is one of a series of advanced technology reports produced in FY 2010 under the Defense
Intelligence Agency, Defense Warning Office's Advanced Aerospace Weapons System Applications
AAWSA Pro ram. Comments or questions pertaining to this document should be addressed to
MP Person 1 AAWSA Program Manager, Defense Intelligence Agency, ATTN: JUIAF -
ashington D.C. 20340-5100.
ii
UNCLASSIFIED/ }FQA QFFIGl,t.L: W&ii 9tlb¥
UNCLASSIFIED/ /EOR OFFIEIAk W&liii 8PtLY
Contents
Introduction ........................................................................................................... 1
REVIEW OF NEGATIVE (or SUB-VACUUM) ENERGY ................................................. 3
Overview ........................................................................................................ 3
Examples of Negative (Sub-Vacuum) Energy Found in Nature ....................... 4
Basic Notions of the Quantum Field Theory of Light ................................... 5
Basic Notions on the Origin of the Quantum Vacuum Zero-Point
Fluctuations ............................................................................................... 7
Negative (Sub-Vacuum) Energy in Squeezed Light .................................... 8
Negative (Sub-Vacuum) Energy in the Casimir Effect................................ 13
QUANTUM OPTICAL HOMODYNE TOMOGRAPHY .................................................... 15
Observing Negative Energy in the Lab .......................................................... 15
Basic Notions of Quantum Optical Homodyne Tomography .......................... 16
Wigner Functions ..................................................................................... 17
Beam Splitters.......................................................................................... 24
Photodiodes ............................................................................................. 27
Balanced Homodyne Detection ................................................................. 27
Outline of Experimental Procedure ........................................................... 32
BALANCED HOMODYNE SYSTEMS FOR MEASURING NEGATIVE
(SUB-VACUUM) ENERGY ....................................................................................... 33
Time-Domain Balanced Homodyne System .................................................. 33
Balanced Homodyne System for Casimir Cavities ......................................... 36
CONCLUSION ........................................................................................................ 43
ACKNOWLEDGEMENTS ......................................................................................... 45
REFERENCES ........................................................................................................ 46
iii
UNCLASSIFIED// FOR OFFICIAL U:!I! fJHLY
UNCLASSIFIED/ /FOA QFFI@IAL l:l!H! 9HLY
Figures
Figure 1. Illustration of a Squeezed State of Light ............................................... 13
Figure 2. Schematic of the Casimir Effect ............................................................. 14
Figure 3. Illustration of Quantum Optical Homodyne Tomography ....................... 16
Figure 4. Wigner Function for a Vacuum and for a Coherent State ....................... 19
Figure 5. Wigner Function of a Squeezed Vacuum ................................................ 20
Figure 6. Wigner Function of a Single Photon ....................................................... 21
Figure 7. Quantum Tomography of Schrodinger-Cat States .................................. 22
Figure 8. Schematic of an Ideal Lossless Beam Splitter ........................................ 25
Figure 9. Illustration of a Fictitious Beam Splitter ................................................ 26
Figure 10. Schematic of a Balanced Homodyne Detector...................................... 29
Figure 11. Balanced Homodyne Detector Using Fictitious Beam Splitters ............. 31
Figure 12. Balanced Homodyne Detector Using A Single Effective Fictitious Beam
Splitter ................................................................................................................. 32
Figure 13. Time-Domain Balanced Homodyne Detector........................................ 34
Figure 14. Experimentally Measured Squeezed State ........................................... 35
Figure 15. Balanced Homodyne Detector with a Local Oscillator .......................... 38
Figure 16. Diagram of Casimir Cavity with BHD Photodiodes ............................... 40
Figure 17. Experimental Setup of BHD Photodiodes and LO Field ......................... 40
Figure 18. Detailed Schematic of Experimental BHD Apparatus ........................... 41
Figure 19. Predicted Casimir Spectral Density ...................................................... 41
Figure 20. Predicted Suppression of Vacuum Fluctuations in dB.......................... 42
iv
UNCLASSIFIED/ /EOR OEEICl.1.1. Uiliii ONI.¥
UNCLASSIFIED/ /FOR. OFFICIAL tt.!I!! OHL¥
Quantum Tomography of Negative Energy States in the
Vacuum
Introduction
Future aerospace vehicles could have an advanced propulsion system that uses
negative quantum vacuum energy to modify the spacetime geometry in the immediate
vicin ity surrounding the vehicle in order to induce faster-than -l ight motion via
traversable wormholes or warp drives, or even levitation via antigravity [1, 2]. These
exotic propulsion concepts are well-known in mainstream general relativity and
quantum field theory research. The notion of a physical state with negative energy is
not familiar in the realm of classical physics. However, it is not rare in quantum field
theory to have quantum states with negative energy density or a negative energy flux.
Even for a quantum scalar field in the flat Minkowski spacetime, it can be proved that
the existence of quantum states with negative energy density is inevitable [3].
Although all known forms of classical matter have non-negative energy density, it is not
so in quantum field theory. A general quantum state can be a superposition of particle
number eigenstates and may have a negative expectation value of energy density in
certain spacetime regions due to quantum coherence effects [3]. These considerations
remain true even for quantum fields in a curved spacetime where the effects of
gravitational fields, or equivalently, accelerations, can be observed due to the mass of
astronomical bodies or the motions of astronomical bodies.
There are two key examples of specially prepared quantum vacuum states that are
known to produce small amounts of negative energy density in the laboratory. These
are the well-known Casimir effect and the squeezed vacuum states of the
electromagnetic field. The former is a static quantum vacuum effect wh ile the latter is
a time-domain quantum vacuum effect. There are several other examples of special
quantum vacuum or particle states that produce negative energy density, but they are
beyond the scope of this report because they remain mathematical curiosities or are not
practicable to im plement in the laboratory in the foreseeable future.
We already make small amounts of negative energy in the laboratory via the Casimir
effect and squeezed electromagnetic vacuum states, but we do not yet know if we can
access larger amounts for extended periods of time over extended spatial distributions
for the purpose of modifying spacetime for aerospace propulsion applications. It will be
necessary to first explore the quantum natu re of the Casimir effect and squeezed
electromagnetic vacuum states to determine whether we can measure and spatially
map their negative energy density. This is a necessary first step to take before
beginning any study on producing large quantities of negative energy because we will
first need to know how to measure and spatially map negative energy in order to
properly control it after producing it. This is the motivation for this report.
We need to firm up our understanding of how lab detectors will respond to negative
energy in situ. A first step in this direction was already taken by Hansen et al. [4] in
2001 for the time-domain negative energy pulses in squeezed electromagnetic vacuum
states, and more recently Marecki [5, 6] generalized the analysis of the output of
balanced homodyne detectors (BHDs) for t he case of static negative energy states
1
UNCLASSIFIED// FOR OFFICIAL U.!I!! or~L'
UNCLASSIFIED/ }FOA OFFl&IAl WSE er•tv
inside Casimir cavities. The most important feature of these devices is their abi lity to
quantify the quantum vacuum fluctuations of the electric field because the output of
BHDs provides information on the one- and two-point functions of arbitrary states of
quantum fields. Marecki computed the two -point function and the associated spectral
density for the ground state of the quantum electric field in Casimir geometries, and
predicts a position- and frequency-dependent pattern of BHD responses if a device of
th is type is placed inside a Casimir cavity. The proposed device allows for the direct
detection of quantum vacuum fluctuations and provides a spatial mapping of the
negative energy contained inside the cavity, wh ich will be summarized in this report.
2
UNCLASSIFIED/ /FOR OFFICIAL U.!! 8HLY
UNCLASSIFIED/ /iOR QFFl&IAL l:ISI!!! Brit I
REVIEW OF NEGATIVE (or SUB-VACUUM) ENERGY
Overview
The implementation of faster-than-light (FTL) interstellar travel via traversable
wormholes or warp drives or other antigravity forces for propulsion, generally requires
the engineering of spacetime into very specialized local geometries surrounding the
immediate vicinity of the aerospace vehicle undergoing this type of motion. The
analysis of these via the general relativistic field equation plus the resultant source
matter equations of state demonstrates that such geometries require the use of
"exotic" matter in order to produce the requisite FTL or antigravity spacetime
modification . Exotic matter is generally defined by general relativity physics to be
matter that possesses (renormalized) negative energy density (sometimes negative
stress-tension = outward pressure, a.k.a. gravitational repulsion or antigravity), and
this is a very misunderstood and misapplied term by the non-general relativity
community. We clear up this misconception by defining what negative energy is, where
it can be found in nature, and we also review the two primary experimental concepts
that are known to produce negative energy in the laboratory. Also, it has been claimed
that FTL and antigravity spacetimes are not plausible because exotic matter violates the
general relativistic energy conditions. However, it has been shown that this is a
spurious issue. The identification, magnitude, and production of exotic matter is seen
to be a key technical challenge, however. FTL and antigravity spacetimes also possess
features that challenge the notions of causality and there are alleged constraints placed
upon them by quantum effects. Reference [1] reviews and summarizes these issues
with an assessment on the present state of their resolution.
What exactly is "exotic" matter? In classical physics the energy density of all observed
forms of matter (fields) is non-negative. What is exotic about the type of matter that
must be used to produce traversable wormhole, warp drive, or antigravity spacetimes is
that it must have negative energy density and/or negative flux [7]. The energy density
is "negative" in the sense that the configuration of matter fields we must deploy to
produce a traversable wormhole, warp drive, or antigravity effect must have an energy
density, PE (= pc2, where pis the rest-mass density), that is less t han or equal to its
pressures/tensions, pi [8, 9]. * In many cases, these equations of state are also known
to possess an energy density that is algebraically negative, i.e., the energy density and
flux are less than zero. It is on the basis of these conditions that we call this material
property "exotic." The condition for ordinary, classical (non-exotic) forms of matter
that we are all familiar with in nature is that PE> pi and/or PE;:::,: 0. These conditions
represent two examples of what are variously called the "standard" energy conditions
which are computed from the trace of the matter stress-energy tensort : Weak Energy
Condition (WEC: PE;:::,: 0, PE+ Pi;:::,: 0), Null Energy Condition (NEC: PE+ Pi;:::,: 0),
Dominant Energy Condition (DEC), and Strong Energy Condition (SEC). These energy
conditions forbid negative energy density between material objects to occur in nature,
but they are mere hypotheses. Hawking and Ellis [10] formulated the energy conditions
in order to establish a series of mathematical hypotheses governing the behavior of
• From this point forward, all Latin letters (e.g ., i, j, k = 1... 3) that appear as indices on physical quantities denote
the usual 3-dimensional space coordinates, x•...x3 , indicating the spatial components of vector or tensor quantities .
t The stress-energy-momentum tensor is a matrix quantity that encodes the density and fl ux of energy and
momentum for any type of matter under study .
3
UNCLASSIFIED/ ;<fQA OFFI@IAL l:ISf! 9Htl'
UNCLASSIFIED/ /POI': orr1c11tt U.!I! Oflt I
collapsed-matter singularities in their study of cosmology and black hole physics. More
specifically, classical general relativity allows one to prove lots of general theorems
about the behavior of matter in gravitational fields.
However, real physica l matter is not "reasonable" because the energy conditions are in
general violated by semiclassical quantum effects ( occurring at order Tl) [9]. * More
specifically, quantum effects generically violate the average NEC (ANEC). Furthermore,
it was discovered in 1965 that quantum field theory has the remarkable property of
allowing states of matter containing local regions of negative energy density or negative
fluxes [3]. This violates the WEC, which postulates that the local energy density is non
negative for all observers. And there are also general theorems of differential geometry
that guarantee that there must be a violation of one, some, or all of the energy
conditions (meaning exotic matter is present) for all FTL and antigravity spacetimes.
However, all of the energy condition hypotheses have been experimentally tested in the
laboratory and experimentally shown to be false - 25 years before their formulation
[11].
In quantum field theory, negative energy is a manifestation of what is now called the
"sub-vacuum" levels of the quantum zero-point (or vacuum ground state) fluctuations
that correspond to any particular quantum field of matter under study. Hence, the
energy corresponding to sub-vacuum quantum fluctuations is now called "sub-vacuum
energy": sub-vacuum energy= negative energy. Further investigation into this technical
issue showed that violations of the energy conditions are widespread for all forms of
both "reasonable" classical and quantum matter [12-16]. Furthermore, Visser [9)
showed that all (generic) spacetime geometries violate all the energy conditions. So
the condition that PE> Pi and/or PE~ 0 must be obeyed by all forms of matter in nature
is spurious. Negative energy has been produced in the laboratory and th is wil l be
discussed in the follow ing sections.
Examples of Negative (Sub-Vacuum) Energy Found in Nature
The exotic (energy condition- violating) fields that are known to occur in nature are :
1. Static, radially-dependent electric or magnetic fields. These are borderline exotic,
if their tension were infinitesimally larger, for a given energy density [10, 17].
2. Squeezed quantum vacuum states: electromagnetic and other (non - Maxwellian)
quantum fields [8, 18].
3. Gravitationally squeezed electromagnetic vacuum fluctuations [19).
4. Casimir effect, i.e., the Casimir vacuum in flat, curved, and topological spaces
[20 -28).
5. Other quantum fields/states/effects. In general, the local energy density in
quantum field theory can be negative due to quantum coherence effects [3].
Other examples that have been stud ied are Dirac field states: the superposition
of two single particle electron states and the superposition of two multi-electron
positron states [29, 30]. In the former (latter), the energy densities can be
negative when two single (multi-) particle states have the same number of
* Planck's reduced constant, TJ "' 1.055 x 10-34 J.s.
4
UNCLASSIFIED/ /FOA QFFI&IAL 1:1!11! Oflt I
UNCLASSIFIED/ /EAR AfifilCIAk Wlili 9PtLY
electrons (electrons and positrons) or when one state has one more electron
(electron-positron pa ir) than the other.
Cosmological inflation [9], cosmologica l particle production [9], classical scalar fields
[9], the conformal anomaly [9], and gravitational vacuum polarization [12-15] are
among many other examples that also violate the energy conditions. Since the laws of
quantum field theory place no strong restrictions on negative energies and fluxes, then
it might be possible to produce exotic phenomena such as faster-than- light travel [31-
33], traversable wormholes [8, 9, 34], violations of the second law of thermodynamics
[35, 36], and time machines [9, 34, 37]. There are several other exotic phenomena
made possible by the effects of negative energy, but they lie outside the scope of this
report. In what follows, we consider only items 2 and 4 in the previous list for the
purpose of this report due to their ready applicability and technical maturity. We will
not examine the other items in the list because they are theoretical curiosities that
remain under study by investigators.
Basic Notions of the Quantum Field Theory of Light
Before going further, it will be helpful to briefly outline the basic notions and
terminology of the quantum field theory of light (i.e., quantum optics) because the
content of this report focuses on those aspects.
Classically, light is electromagnetic radiation that can be pictured as waves flowing
through space at the speed of light, c (= 3.0 x 10 8 m/s). The waves are not waves of
anything substantive, but are in fact ripples in the state of a field. These waves carry
energy, and each wave has a specific direction, frequency and polarization state. This
is called a "propagating mode of the electromagnetic field." A simple model for th is is
the electromagnetic oscillator. One complex-valued vector function u(x,t) called a
spatial-temporal mode comprises all classical wave aspects including polarization. The
sim plest example of a spatial-temporal mode is a plane wave
u(x,t) = u0 exp[i(kx-wt)] of polarization vector uo, angular frequency co, and wave
vector k (definition: k 2 =u}/c2 ), where i is the unit complex number, and x is the
space coord inate and tis the time coordinate .
This mode defines a framework in space and time that may be excited by the quantum
field "light." The mode function quantifies the strength of one excitation in space and
time. Also, the mode function obeys the laws of classical waves given by Maxwell's
equations of electrodynamics. The choice of u(x,t) is made by the observer. The
observer singles out one mode, one quantum object from the rest of the world to make
a specific observation or measurement. This object turns out to be a harmonic
oscillator described by the ann ihilation operator a. A useful tool for modeling the
propagating mode of the electromagnetic field in quantum mechanics is the ideal
quantum mechanical harmonic oscillator: a hypothetical charged mass on a perfect
spring oscillating back and forth under the action of the spring's restoring force. The
a
operator stands for the quantized amplitude with which u(x,t) can be excited. In
classical optics it would be just a complex number a of magnitude lal and phase
arg(a) . The quantized amplitude a is neither predetermined nor given by the observer
5
UNCLASSIFIED/ /FOR OFFICIICL tl.!l!! 9HLY
UNCLASSIFIED/ /FOR 8FFl&IAk le:ISE 8Hk¥
but depends on the state of u(x,t). This state exists even if literally nothing is in the
mode chosen by the observer. In this case, the light is just in the vacuum state. §
However, this "nothing" can indeed cause significant physical effects as will be
discussed in later sections.
To make all this more precise, we postulate that the electric field strength E of the light
field is given by E= u*(x, t)a + u(x, t)a 1 and that the amplitude operator a is a bosonic**
annihilation operator that obeys the quantum mechanical commutation relation
[a, at]=1, where u*(x, t) is the complex conjugate of u(x,t) and at is the adjoint (or
conjugate) of a called the creation operator.t t The hat symbol appearing over
quantities denotes that they are quantum operators (or observables). Another key
element of quantum-oscillator physics is the photon number operator fi, which
accounts for the number of photons (quantized light particles) in the chosen u(x,t) and
is given by the quantum mechanical counterpart of a classical modulus-squared
amplitude: n= a t cz.
Let us now introduce a pair of operators, q and p, called quadratures. They are
defined as q = 2- 112 (at+ a) and jJ = i2- at - a) , which can be inverted to provide the
I12 (
additional useful definitions a= T 112 ( q+ ip) and at =T I12 ( q- ip). In optics q and jJ
correspond to the in -phase and the out-of-phase component of the electric field
amplitude of u(x,t) (with respect to a reference phase). The bosonic commutation
relation demonstrates that q and /J are canonically conjugate observables, [ q,p] = ih.
The quadratures q and p can be regarded as the position and the momentum of the
quantum electromagnetic oscillator. They do not appear in real space but in the phase
space spanned by the complex vibrational amplitude a of the quantum electromagnetic
oscillator, and they have nothing to do with the position and the momentum of a
photon. However, the canon ical commutation relation entitles us to treat q and p as
perfect examples of position- and momentum-like quantities in quantum optics. Finally,
we express the photon number operator n
in terms of the quadratures q and p and
obtain, using the bosonic commutation relation, the standard Hamiltonian (or total
energy) of the quantum harmonic ( electromagnetic) oscillator with unit mass and
frequency:
" - " 1
H ose= n+2
(1)
§ Here we always mean by " vacuum" simply "no light" and not an evacuated system.
•· Boson or bosonic refers to quantum particles that have integer quantum spin .
., In quantum mechanics, the vacuum is defined to be a state of no (or zero) particles and is denoted by the
quantum state eigenvector I0) . By definition a "annihilates" the vacuum state: aI0) = 0 .
6
UNCLASSIFIED/ /FOA OFFIGIAk lel&E 8Ptk\S
UNCLASSIFIED/ /POI\ OPPICIAL YSIE 8Htlf
where the first and second terms in the second line are the kinetic and potential
energies of the oscillator, respectively. The additional 1/2 appearing in the first line of
Eq. (1) is called the vacuum zero-point energy for the reason to be explained in the
next section . The first line of Eq. (1) is more common ly expressed in units of energy
(Joules) in quantum mechanics, which is obtained simply by multiplying the right-hand
side by the photon energy nw so that H ose = hw (ii +½).
It is beyond the scope of this report to elaborate further on the entire subject of the
quantum optics. The reader should consult Reference [38] for more information.
Basic Notions on the Origin of the Quantum Vacuum Zero-Point Fluctuations
Here we discuss the basic notions of the quantum vacuum zero-point fluctuations
(ZPF), which is an important feature in quantum optics. The orig in of the ZPF is
attributed to the Heisenberg Uncertainty Principle. According to this principle, q and
pare any two conjugate observables that we are interested in measuring, and they
obey the commutation relation already shown in the previous section. Their
corresponding uncertainty relation is !1q!1p?:. h/2, where !1q is the variance (a.k.a.
uncertainty) of observable q and !1p is that of the conjugate observable p. This
relation states that if one measures observable q with very high precision (i.e., its
uncertainty !1q is very small), then a simultaneous measurement of observable p will
be less precise (i.e., its uncertainty !1p is very large), and vice versa . In other words,
it is not possible to simultaneously measure two conjugate observable quantities with
infinite precision.
This minimum uncertainty is not due to any correctable flaws in measurement, but
rather reflects the intrinsic fuzziness in the quantum nature of energy and matter.
Substantial theoretical and experimental work has shown that in many quantum
systems the limits to measurement precision is imposed by the quantum vacuum ZPF
embodied within the uncertainty principle. Nowadays we rather see the Heisenberg
Uncertainty Principle as a necessary consequence, and therefore, a derived result of the
wave nature of quantum phenomena. The uncertainties are just a consequence of the
Fourier nature of conjugate pairs of quantities (observables). For example, the two
Fourier-wave-conjugates time and frequency become the pair of quantum-particle
conjugates time and energy and the two Fourier-wave-conjugates displacement and
wave number become the pair of quantum-particle conjugates position and momentum.
The Heisenberg Uncertainty Principle dictates that a quantized electromagnetic
oscillator (a.k.a. a photon state) can never come entirely to rest, since that would be a
state of exactly zero energy, which is forbidden by the commutation relation given in
the previous section. Instead, every mode of the field has liw/2 as its average minimum
energy in the vacuum, and this is called the zero-point energy (ZPE) .** This ZPE term
is added to the classical blackbody spectral radiation energy density p(w)dro [i.e., the
energy per unit volume of radiation in the frequency interval (ro,w + dro)] [25]:
*" hw is the energy of a single mode (or photon).
7
UNCLASSIFIED/ /EOA QFFI&IAL YSE!! 9HL I
UNCLASSIFIED/ /FOR OFFICl"L tJ.!l!!! 8HL¥
p(ro)dro = - oi [ - - - hro - - + -hco ] dco
rc 2 c3 exp(hco/ k 8 T) - 1 2
(2)
3
= - hro-2 3 coth ( -
2n c
J
hro
- dro,
2k8 T
where kp, is Boltzmann's constant ( 1. 3807 x 10- 23 J/K) and Tis the absolute
temperature. The factor outside the square brackets in the first line of Eq. (2) is the
density of mode (or photon) states (i.e., the number of states per unit frequency
interval per unit volume); the first term inside the square brackets is the standard
Planck blackbody radi ation energy per mode; and the second term inside the square
brackets is the quantum zero-point energy per mode. Equation (2) is called the Zero
Point Planck (ZPP) spectral radiation energy density. Planck first added the ZPE term to
the classical blackbody spectral radiation energy density in 1912, although it was
Einstein, Hopf, and Stern who actually recognized the physical significance of th is term
in 1913 [25]. Direct spectroscopic evidence for the reality of ZPE was provided by
Mulliken's boron monoxide spectral band experiments in 1924, several months before
Heisenberg first derived the ZPE for a harmonic oscillator from his new quantum matrix
mechanics theory [39].
Following this line of reasoni ng, quantum physics predicts that all of space must be
filled with quantum electromagnetic ZPF creating a universal sea of zero-point energy.
The other quantum forces of nature also have their own vacuum ZPF which contributes
to the universal sea of zero-point energy. But that is beyond the scope of this report.
Negative (Sub-Vacuum) Energy in Squeezed Light
Substantial theoretical and experimental work has shown that in many quantum
systems the limits to measurement precision imposed by the quantum vacuum ZPF can
be breached by decreasing the noise in one observable (or measurable quantity) at the
expense of increasing the noise in the conjugate observable; at the same time the
variations in the first observable, say the energy, are reduced below the ZPF such that
the energy becomes "negative." "Squeezing" is t hus the control of quantum
fluctuations and corresponding uncertainties, whereby one can squeeze/reduce the
variance of one (physically important) observable quantity provided the variance in the
(physically unimportant) conjugate variab le is stretched/increased. The squeezed
quantity possesses an unusually low variance, meaning less variance than would be
expected on the basis of the equipartition theorem . One can in principle exploit
quantum squeezing to extract energy from one place in the ordinary vacuum at the
expense of accumulating excess energy elsewhere [8].
The squeezed state of the electromagnetic field is a primary example of a quantum fie ld
that has negative energy density and negative energy flux. Such a state became a
physical reality in the laboratory as a result of the nonlinear-optics techn ique of
"squeezing," i.e., of moving some of the quantum-fluctuations of laser light out of the
8
UNCLASSIFIED/ /FOA OFFI&iIAL YSIE 8HLY
UNCLASSIFIED/ /FOR 8FFI@IAL 1::191!! 8HLY
cos[w(t- z!c )] part of the beam and into the sin[w(t- z!c)] part [18, 40-44].§§ The
observable that gets squeezed will have its fluctuations reduced below the vacuum ZPF.
The act of squeezing transforms the phase space circular noise profile characteristic of
the vacuum into an ellipse, whose semimajor and semiminor axes are given by unequal
quadrature uncertainties (of the quantized electromagnetic oscillator operators). This
applies to coherent states in general, and the usual vacuum is also a coherent state
with eigenvalue zero. As th is ellipse rotates about the origin with angular frequency w,
these unequal quadrature uncertainties manifest themselves in the electromagnetic
field oscillator energy by periodic occurrences, which are separated by one quarter
cycle, of both smaller and larger fl uctuations compared to the unsqueezed vacuum .
We digress momentarily by noting that coherent states, also called Glauber states, are
the eigenstates of the annihilation operator a:
ala) =ala), (3)
which have well-defined amplitudes lal and phases arg(a) (recall the discussion in Sect.
IIB-1). They are called coherent states because light fields in these states are perfectly
coherent, and hig h-qual ity lasers generate such fields. This is an important reason why
high-q uality laser light is an excellent tool for experimental quantum optics. Coherent
states come as close as quantum mechanics allows to wave-like states of the
electromagnetic oscillator. Because the wave aspects of light are commonly regarded
as classical, coherent states are often called classical states. Furthermore, fields in
statistical mixtures of coherent states (such as thermal fields) are classical as well,
whereas any state that cannot be understood as an ensemble of coherent states is
called nonclassica/. The experimental generation and application of nonclassical light
fields is the main subject of t his report. Despite much recent progress, producing
nonclassica l states of light is still extremely challenging because they are easily
destroyed (reduced to classical) by any kind of losses. Furthermore, it turns out that
the vacuum is a coherent state as well because it satisfies Eq. (3) for a= 0. In other
words, the vacuum is a zero-am plitude coherent state. With a little algebra we see
directly from Eq. (3) that the mean (i.e., quantum expectation value of the) energy of a
coherent state with unit frequency is
\Ha)= (alata + ½la)
(4)
=lal2 +½-
Equation ( 4) is the sum of the classical wave intensity lal 2 and the vacuum zero-point
energy 1/2. One simply multiplies the rig ht-hand side of Eq. ( 4) by liw to put (Ha)
into units of energy.
§§ z denotes the z-axis direction of beam propagation.
9
UNCLASSIFIED/; PO" OPPICIICL U.!! OHL I'
UNCLASSIFIED/ /EAR AfifilEIAk WSE er•tv
Morris and Thorne [8] and Caves [ 45] point out that if one squeezes the vacuum, i.e., if
one puts vacuum rather than laser light into the input port of a squeezing device, then
one gets at the output an electromagnetic field with weaker fluctuations and thus less
energy density than the vacuum at locations where cos 2[ co(t- z/c)] ~ l and
sin 2 [ co( t- z/c)] < < 1; but with greater fluctuations and thus greater energy density
than the vacuum at locations where cos 2[ co(t-z/c)] << 1 and sin 2 [ co(t - z/c)]~1.
Since the vacuum is defined to have vanishing energy density, any region with less
energy density than the vacuum actually has a negative (renormalized) expectation
value for the energy density. Therefore, a squeezed vacuum state consists of a
traveling electromagnetic wave that oscillates back and forth between negative energy
density and positive energy density, but has positive time-averaged energy density.
In quantum optics the squeezed state is generated by the unitary squeezing operator :
(5)
s
where is a rea l number that parameterizes the deviation of the variances !),,q and !),,p
from their vacuum values and is called the squeezing parameter. From Eq. (5) we
obtain the squeezed vacuum state lcp) = S(~) IO). The squeezing operator S(~) is simply
an evolution operator that describes the result of the nonlinear squeezing interaction
Hamiltonian H im = x( b*a2 - bat2 ). The squeezing parameter~ contains the product of
the amplitude b, the coupling constant X, and the interaction time.
But this is not the entire story . Since we will be dealing with high-quality lasers in what
follows, we also need to know about another important quantum optics operator that
acts on coherent states. We introduce the unitary displacement operator
D(a)=exp(aa: -a· a). D(a) displaces the amplitude a by the complex number a
according to b\a)aD(a)=a + a. To show why D(a) has anything to do with coherent
states, we apply a negative displacement to la). From the basic property of D(a), we
see that
abc-a) la) = D(-a)Dt(-a)GD(-a) la)
= be- a) ( a- a) Ia) (6)
= 0.
Equation (6) equals zero because of the definition Eq. (3) of coherent states. This
result implies that D(-a)la) = I0), which is the vacuum state. Therefore, coherent
states Ja) are displaced vacua la) = D(a) J0). This does not mean that coherent states
10
UNCLASSIFIED/ i PO" OPPICIICL U.!! OHL I'
UNCLASSIFIED/ /FOA QFFIEJIAL YSI!! f>Ht'I"
are physically similar to vacuum states, but instead they have only some quantum
noise properties in common. It is a well known result in the quantum field theory of
light that the vacuum wave function is a simple Gaussian function of the quadratures
(in either q or jJ representation), and thus coherent states are also Gaussian [38].
Furthermore, a proof of Heisenberg's Uncertainty Principle in conjunction with the
application of S(~) and D(a) on the quadrature variances and wave functions showed
that all minimum uncertainty states are displaced Gaussian states such that they have
displaced rescaled vacuum wave functions. Consequently, all minimum uncertainty
states are displaced squeezed vacua [18, 38]:
I\jf) = fJ ca) scs) Io) . (7)
The squeezing interaction Hi"' is realized by the degenerate parametric amplification of
the spatial-temporal mode. A crystal such as potassium titanyl phosphate (KTP) or
lithium niobate (LiNbQ3) is pumped by another laser beam with amplitude band twice
the frequency of the spatial-temporal mode (with amplitude a) of interest. According to
H ;m , the "B" photons (corresponding to b) of the pump beam are converted into pairs
of "A" signal photons (corresponding to a.2 and a,t2 ) with a probability that depends on
the coupling constant X· The KTP or LiNb03 crystal acts like an electromagnetic swing,
and the pump modulates the oscillation of the "A" mode at twice its frequency. The
pump amplifies the signal parametrically much as a swing is amplified by changing the
effective length at twice the frequency of the swing. A classical swing relies on tiny
initial fluctuations (or "wobbles") that are in-phase with respect to the parametric
pump. In this way, the tiny fluctuations are amplified; the swing starts to oscillate. A
quantum swing like the degenerate parametric amplifier experiences at least the
vacuum fluctuations from the very beg inning. Vacuum fluctuations that are in-phase
with respect to the pump are amplified, whereas out-of-phase fluctuations get de
amplified or, in other words, squeezed.
A squeezed vacuum requires a pump for generation, and, hence, when produced it
carries energy. The nonlinear crystal KTP or LiNb03 is a resonator that is shaped like a
cylinder with rounded silvered ends to reflect light. This resonator acts to produce a
secondary lower frequency light beam in which the pattern of photons is rearranged
into pairs. The squeezed light emerging from the resonator will contain pulses of
negative energy interspersed with pulses of positive energy. To quantify the amount of
squeezing energy we 1) apply S(~) to the quadratures and find that it scales their
eigenfunctions; *** 2) we then substitute for a its quadrature decomposition (given in
Sect. IIB-1) and substitute that result into the scaled quadratures; and then 3) do
further algebra to derive how S(~) changes a: stcs)aS(s) =Gcoshs - atsinhs. We
substitute this last result into Eq. (1) and use Eq. (7) to calculate the quantum
expectation value in order to express the mean energy of a squeezed state, and obtain
, .. i.e ., q gets squeezed and p gets stretched .
11
UNCLASSIFIED// FOR OFFICIAL H.!l! 8PU:.lf
UNCLASSIFIED/ /FOA QFFI@IAL l:191! t>flt I
(8)
Equation (8) really describes the mean photon number of a single mode in a squeezed
state, but one simply multiplies the right-hand side by /iro to get the mean energy
(Asqvac ) = hw(la.1 2 + ½+sinh 2 ~ )- We see in Eq. (8) that there are three terms
contributing to the energy: the first term accounts for the coherent energy given by
lal2 , the second term is the vacuum zero-point energy 1/2, and the third term
quantifies the fluctuation energy of squeezed states. The contribution to this squeezing
energy originally comes from the pump used to generate the squeezed light. It is
stored in the enhanced fluctuations of the anti-squeezed component. Because both the
squeezed and the anti-squeezed quadratures contribute to the second line in Eq. (1),
even a squeezed vacuum carries energy.
However, Eq. (8) is not the final result because it only gives the mean energy of a
sin gle mode in a squeezed state, while lasers and nonlinear crystal resonators produce
a very large number of modes. Equation (8) needs to be summed (integrated) over the
infinite number of possible modes; it must then be "renormalized" by sophisticated
mathematical techniques in order to get rid of the divergent (infinite) contribution from
the vacuum zero-point energy (a byproduct of taking an infinite sum of modes); and
then the result must be converted into units of energy density by dividing it by an
appropriate volume element, because Einstein's general theory of relativity requires an
energy density (or pressure, both are in the same units) to induce spacetime bending.
The final result we seek is the energy density, P E-sqvac , given by Pfenning [ 46]:
Pe.sq= =( 21,co) sinh c; [sinh c; + coshi:; cos ( 2ro (t - z/c) +8)] (J / m 3) , (9)
where L 3 is the volume of a large box with sides of length L (i.e., we put the quantum
field in a box with periodic boundary conditions) and 8 is the phase of squeezing.
Equation (9) shows that PE-sqvac falls below zero once every cycle when the condition
cosh~ > sinh~ is met. It turns out that this is always true for every nonzero va lu e of ; ,
so PE-sqvac becomes negative at some point in the cycle for a general squeezed vacuum
state. See Figure 1 for an illustration. Note in the figure that the blue troughs or
valleys are the negative energy pulses. On another note, when a quantum state is
close to a squeezed vacuum state, there wi ll almost always be some negative energy
densities present.
Another way to generate negative energy via squeezed light would be to manufacture
extremely reliable light pulses containing precisely one, two, three, etc., photons apiece
and combine them together to create squeezed states to order. Superimposing many
such states could theoretically produce bursts of intense negative energy. Photonic
crystal research has already demonstrated the feasibility of using photonic crystal
waveguides (mixing together the classical and quantum properties of optical materials)
to engineer light sources that produce beams containing precisely one, two, three, etc.,
photons. See Reference [1] for more details and for the references cited therein.
12
UNCLASSIFIED/ /fOtt 8Ffl@IAL Y§E 8,.LY
UNCLASSIFIED/ /FOR QFFl&IAL l:ISI!!! 8flt I
ENERGY SQUEEZED STATE
DENSITY
0 POSITION
Figure 1. Illustration of a Squeezed State of Light. (courtesy of Lisa Burnett)
Negative (Sub-Vacuum) Energy in the Casimir Effect
The Casim ir effect originates from the quantum electromagnetic vacuum ZPF. It is by
far the easiest and most well known way to generate (static) negative energy in the
lab. The Casimir effect that is famil iar to most people is the force that is associated
with the quantum vacuum electromagnetic ZPF [47]. This is an attractive force that
must exist between any two neutral (uncharged), parallel, flat, conducting surfaces
(e.g ., metallic plates) in a vacuum. This force has been well measured and it can be
attributed to a minute imbalance in the vacuum electromagnetic ZPE density inside the
cavity between the conducting surfaces versus the vacuum electromagnetic ZPE density
in the free -space region outside of the cavity [ 48- 50]. See Figu re 2 fo r a schematic of
the Casimir effect.
13
UNCLASSIFIED/ /FOR QFFl&IAL l:ISI! 8flt I
UNCLASSIFIED/ /FOR OFFICUtt H.!l! er~tv
Casimir
plates Vacuum
fluctuations
Figure 2. Schematic of the Casimir Effect.
It turns out that there are many different types of Casimir effects found in quantum
field theory [20-22, 26-28, 51]. For example, if one introduces a single infinite plane
conductor into the Minkowski (flat spacetime) vacuum by bringing it adiabatically from
infinity so that whatever quantum fields are present suffer no excitation but remain in
their ground states, then the vacuum (electromagnetic) stresses induced by the
presence of the infinite plane conductor produces a Casimir effect. This result holds
equally well when two parallel plane conductors (with separation distance d) are
present, which gives rise to the familiar Casimir effect inside a cavity. Note that in both
cases, the spacetime manifold is made incomplete by the introduction of the plane
conductor boundary condition(s). The vacuum region put under stress by the presence
of the plane conductor(s) is called the Casimir vacuum. The generic expression for the
energy density of the Casimir effect is PCE = - Ahcd -4 , where A= C,(D)/8rr-2 in
spacetimes of arbitrary dimension O [20-22]. The appearance of the zeta-function (,(D)
is cha racteristic of expressions for vacuum stress-energy tensors, T.J:i~ . t t t I n our
familiar 4-dimensional spacetime (0 = 4) we have that A = rc 2 /720. To calculate T_J:i~
for a given quantum field is to calculate its associated Casimir effect.
We should also point out that the methods used to obtain the quantum vacuum
electromagnetic T;~~ between parallel plane conductors can also be used when the
conductors are not parallel but are joined together along a line of intersection. If the
conductors have curved surfaces instead, then one obtains results that are similar to
the case of intersecting conductors . These geometries have also been evaluated for the
ttt The Greek tensor indices (µ, v = 0... 3) denote spacetime coordinates, >l'... x 3 , such t hat xt...x 3 = space
coord inates and x 0 "' time coordinate . Note in general that T00 = p E (field energy density).
14
UNCLASSIFIED/ ,'FOR OFFI&I.t..k W&li OPtk\S
UNCLASSIFIED/ /FAA QFFl&IAL l:191! 614[1
case of dielectric media. These particular cases will not be considered further since
there are technical subtleties involved that complicate the calculations and application
of the different approaches.
As a final note, negative energy can be created by a single moving reflecting
(conducting) surface (a.k.a. a moving mirror) via the dynamical Casimir effect. A
mirror moving with increasing acceleration generates a flux of negative energy that
emanates from its surface and flows out into the space ahead of the mirror [23, 52].
This is essentially the simple case of an infinite plane conductor undergoing acceleration
perpendicular to its surface. If the acceleration varies with time, the conductor will
generally emit or absorb photons (i.e., exchange energy with the vacuum), even
though it is neutral. This is an example of the well-known quantum phenomenon of
parametric excitation. The parameters of the quantum electromagnetic oscillators
(e.g., their frequency distribution function) change with time owing to the acceleration
of the mirror [53]. However, this effect is known to be exceedingly small, and it is not
the most effective way to produce negative energy for our purposes. We will not
consider this scheme any further.
QUANTUM OPTICAL HOMODVNE TOMOGRAPHY
Observing Negative Energy in the Lab
Negative energy should be observable in lab experiments. A generic, non-optical
scheme for detecting negative energy in experiments was recently reported by Davies
and Ottewill [54] who studied the response of switched particle detectors to static
negative energy densities and negative energy fluxes. Their model is based on a free
(massless) scalar field in flat 4- dimensional Minkowski spacetime and utilized a simple
generalization of the standard monopole detector, which is switched on and off to
concentrate the measurements on periods of isolated negative energy density (or
negative energy flux). The detector model includes an explicit switching factor whereby
five different switching functions (based on data windowing theory) are defined and
evaluated.
In order to isolate the effects of negative energy, a comparison is made for the
response of a detector switched on and off during a period of negative energy density
(or negative energy flux) and that switched on and off in the vacuum. The results shed
light on the response of matter (detectors) to pulses of negative energy of finite
duration, and they showed that negative energy should have the effect of enhancing
de-excitation (i.e., induce cooling) of the detector. This is the opposite of our
experience with detectors that undergo excitation when encountering "normal" matter
or energy, and isolated detectors placed in a vacuum naturally cool due to the usual
thermodynamic reasons. But Davies and Ottewill point out that the enhanced cooling
effect they discovered cannot be used to draw a thermodynamic conclusion because
their modeling was restricted to first order in perturbation theory. It is not possible at
first order to determine whether the enhanced cooling effects are due to the small
violation of energy conservation expected in any process in which a general quantum
state collapses to an energy eigenstate, or whether they predict a systematic reduction
in the energy of the detector which has serious thermodynamic implications. However,
Davies and Ottewill point out that their results are model dependent and they found for
their standard monopole detector model that there is not always a simple relationship
15
UNCLASSIFIED/ /fl61t 6flfllelslct l:l!lfI 8,.LY
UNCLASSIFIED/ /FOR QFFI€ilAL l::l!H! 61\LI
between the strength of the negative energy density/flux and the behavior of the
detector.
It is curious that Davies and Ottewill did not consider using quantum optical homodyne
tomography as a tool to test their hypothesis, because this is already a mature
experimental discipline. In what follows we outline the basics of quantum optical
homodyne tomography and its application to detecting and measuring negative energy
density/flux states in squeezed light and in the Casimir effect.
Basic Notions of Quantum Optical Homodyne Tomography
Tomography, from the Greek word for slice, is a method to infer the shape of a hidden
object from its shadows (or projections) under various angles. Quantum tomography is
the application of this idea to quantum mechanics. In optical homodyne tomography,
the Wigner function or, more generally, the quantum state plays the role of the hidden
object. The observable "quantum shadows" are the quadrature distributions and are
measured using homodyne detection. From these distributions the Wigner function is
reconstructed. See Figure 3 for an illustration of quantum optical homodyne
tomography. The vertical 2-dimensional plane seen in the figure is fictitious and is
shown for illustrative purposes only.
Figure 3. Illustration of Quantum Optical Homodyne Tomography (courtesy of Ulf Leonhardt) .
The Wigner function (3-dimensional hill on the right) is reconstructed in quantum phase space
(gridded plane formed by quadratures q and p) from its experimentally measured projections (curve in
vertical 2-dimensional plane), which represents the scanning process of tomography. The vertical axis
is the magnitude of the Wigner (quasiprobability) function.
Quantum tomography was developed for the simple reason that a fundamental feature
of quantum mechanics prevents us from seeing physical objects in their full quantum
complexity. This is due to the intrinsic fuzziness in the quantum nature of energy and
matter according to the Heisenberg Uncertainty Principle, which prevents us from
simultaneously and precisely measuring the complementary features (e.g., position and
momentum or energy and time) comprising quantum states. For this reason we cannot
16
UNCLASSIFIED/ ;<fQA QFFl&l.t..k YSE 8,.L'&'
UNCLASSIFIED/ j FOR OFFICIAL tJ.!I!! 6HL'f
directly observe quantum states, and so the true nature of an individual quantum
system is hidden. However, no principal obstacle exists to observing all complementary
aspects in a series of distinct experiments on identically prepared quantum objects.
In the sections that follow, we briefly review the several parts that comprise the
tomography machinery, and then put the whole picture together to understand what
the entire process is. No effort will be made for completeness because the subject of
quantum tomography takes up volumes of books. The reader will be referred to the
key literature of importance.
Wigner Functions
In classical optics the state of an electromagnetic oscillator is perfectly described by the
statistics of the classical amplitude a. The amplitude may be completely fixed (then
the field is coherent), or a may fluctuate (then the field is partially coherent or
incoherent). In classical optics as well as in classical mechanics, we can characterize
the statistics of the complex amplitude a or, equivalently, the statistics of the
component position q and momentum p by introducing a phase space distribution called
the Wigner function, W(q,p). "*" W(q,p) quantifies the probability of finding a particular
pair of q and p values in their simultaneous measurement. Knowing W(q,p) for a
particular quantum state that is under study, all statistical quantities of the
electromagnetic oscillator can be predicted by calculation. In this sense W(q,p)
describes the state in classical physics. The motivation for introducing the Wigner
function was the desire to find a quantum mechanical description similar to that in
classical statistical physics. However, in quantum mechanics Heisenberg's Uncertainty
Principle prevents one from observing position and momentum simultaneously and
precisely. In addition to this, we also cannot directly observe quantum states either.
Nevertheless, we are perfectly entitled to use the concept of quantum states as if they
were existing entities. We use their properties to predict the statistics of observations.
It is well known that the quantum mechanical wave function depends exclusively on
either the position or the momentum and contains nevertheless a// the information
about the quantum system under study. However, E. Wigner showed that it is possible
to define a formal quantum mechanical analog to the classical distribution function. He
showed that we could use W(q,p) as a quantum phase space distribution exclusively to
calculate observables in a classical-like fashion. Wigner discovered that W(q,p) is a
real-valued function, but it is usually not just positive; it can also become negative.
This is a very nonclassical behavior for a probability distribution. It is for this reason
that W(q,p) came to be called a quasiprobability distribution.
W(q,p) has several properties and mathematical postulates, but it turns out that just
one postulate is sufficient for t he purposes of quantum tomography [38). Using this
postulate, it is assumed that W(q,p) behaves like a joint probability distribution for q
and p without ever mentioning any simultaneous observation of position and
momentum. The reduced, or marginal, distributions J: W(q, p)dp or J: W(q, p)dq
*" Recall in Sect. IIB- 1 that the real and the imaginary parts of the complex amplitude o. can be regarded as the
position and the momentum of the electromagnetic oscillator.
17
UNCLASSIFIED/ ifOR Offlf:!IAL Y.!I!! er•t I
UNCLASSIFIED/ /POI\ OPPICIJIIL YSIE 8Hllf
must give the position or the momentum distribution, respectively. Furthermore, if one
performs a phase shift 8 all complex amplitudes a are shifted in phase, §§§ meaning that
the components q and p rotate in the 2-dimensional phase space (q,p). A classical
probability distribution for position and momentum values would rotate accordingly.
This fact leads to the postulate that the position probability distribution pr(q,8) after an
arbitrary phase sh ift 8 should be [38]
pr(q,0) = (qJOce)p 0\ e)J q)
(10)
= f: w( qcos0-psin0,qsin0+pcos0)dp,
where p is the quantum density operator (or density matrix) which describes the
statistical (or most general) state of a quantum system. The first line in Eq. (10) is the
quantum expectation value of the phase-shifted p, which simply gives the probability
distribution for the q-eigenstates to occu r with probabilities p q (the elements of p).
This single formula joins W(q,p) with quantum mechanics. It ties W(q,p ) to observable
quantities, and it links quantum states to observations .
It is beyond the scope of this report to repeat the entire mathematical development of
the explicit functional representations, identities, transformations and modifications of
W(q,p). The reader should consult Reference [38] for more information. However,
Figures 4 through 7 provide an example of what the experimentally reconstructed
Wigner function visually looks like from the quantum optical homodyne tomography of
the following cases of interest: a vacuum state, a coherent state, a squeezed vacuum
state, a single photon, and Schrtidinger cat states. The Schrtidinger cat states are a
very interesting case study of unusual nonclassical states of light that have been
experimentally measured via quantum optical homodyne tomography.
§§§ U(0) = exp(- i0ii), where ii is the photon number operator and e is the
The unitary phase shifting operator is
phase shift angle . Its action on the amplitude a is: u \ 8)11U(8) = 11e.>.p(- i0) .
18
UNCLASSIFIED/ /FOR 0661CIAk W&IE 8HLY
UNCLASSIFIED/ /FOR 8FFl@lsllt U.!I! 014Li
Figure 4. Wigner Function for a Vacuum (top) and for a Coherent State (bottom). This
clearly shows that coherent states are just "displaced vacua " (Sect. IIB-3). Optical homodyne
tomography was used to reconstruct the Wigner functions from experimental data (courtesy of Ulf
Leonhardt).
19
UNCLASSIFIED/ /FOR OFFICIAL U.!I! 9HLY
UNCLASSIFIED/ /FOR OFFl@IAL Y!H! 8HL'I
0 50 100 150 200
Time rmsl
Figure S. Wigner Function of a Squeezed Vacuum. Wigner function (top) and quadrature
fluctuations (bottom) . This shows the experimentally reconstructed Wigner function of a
significantly squeezed vacuum generated by parametric amplification (Sect. IIB-3). The noise
trace (bottom) shows a part of the experimental data used to reconstruct the depicted Wigner
function via optical homodyne tomography (courtesy of Ulf Leonhardt).
20
UNCLASSIFIED/ i POlt OPPlelJcL Y.!l!! 8HLY
UNCLASSIFIED/ /FOR 8ffl@IJ!tt U.!I! 014Li
0.1
0
-0.l
-0.2 .
-0.3 .
-.... ......"--...
-2 ·,, p
-......__
-/ ,,"-....
o~ .....
q ----
1~
-....
2 .....___
0.1
0
_-:.·: l
-0.3 -~------
·-"- ... ____.__,.__
-2 1 ~..,____
=-=- ---·~- - "~ ....
........ ~
- - - -·-
....
• 0 -, -"? - - - - - - -
p
Figure 6. Wigner Function of a Single Photon. The figure shows the experimentally
reconstructed Wigner function as seen from above (top) and from below (bottom).
Negative "probabilities" are clearly visible near the origin of the phase space, which demonstrates
the nonclassical aspect of photons (courtesy of Ulf Leonhardt).
21
UNCLASSIFIED/ /POlt OPPl@IJ!tt li!lf! 8HLY
UNCLASSIFIED/ /PO" orr1e1J11t l:l!H! e .. tv
----------
----------
---------
0.2
0
-0.2
-4 '----- q
- ~----
0 ,.______
/J
----- 4
-------------------
-----
-0.2\
'--..
q
-5 -------~
-2.5
0
p 2. s"'-...._____
.....
5
Figure 7. Quantum Tomography of Schrodinger-Cat States. Top : q 0 = 3. Two separated
coherent amplitudes (peaks) are clearly visible. Bottom: qo = 4. The larger the separation of the
amplitudes, the more rapid is the oscillation in t he quantum interference structure between the
t wo peaks. Negat ive probabil ities appear within the quantum inte rference st ructure. The
experimental data used to reconstruct the depicted Wigner functions was provided by A. Furusawa
and H. Yonezawa , University of Tokyo.
We digress for the moment to explain what Schrodinger cat states are . Schrodinger's
cat is a famous illustration of the principle of superposition in quantum theory that was
proposed as a thought experiment by Erwin Schrod inger in 1935 . Schrodinger's cat
22
UNCLASSIFIED/ /fOtt 8ffl@IAL Y§E 8,.LY
UNCLASSIFIED/ /FOR OFFICiltt tJ.!l! f>HLY
serves to demonstrate the apparent conflict of what quantum theory tells us is true
about the nature and behavior of matter on the quantum (atomic or subatomic) level
compared with what we actually observe to be true about the nature and behavior of
matter on the macroscopic level.
Schrodinger's thought experiment is as follows: One places a living cat into a steel
chamber along with a device containing a vial of hydrocyanic acid. There is also a very
small amount of a radioactive substance inside the chamber. If even a single atom of
the substance decays during the test period, then a relay mechanism will trip a
hammer, which will in turn break the vial and kill the cat.
The observer cannot know whether or not an atom of the radioactive substance has
decayed, and consequently, cannot know whether the vial has been broken, the
hydrocyanic acid released, and the cat killed. Since one cannot know, the cat is both
dead and alive in a superposition of quantum states according to the quantum
superposition principle. It is only when one breaks open the box and learns the
condition of the cat that the superposition is lost, and the cat becomes either dead or
alive. This situation is sometimes called quantum indeterminacy or the observer's
paradox: the act of observation or measurement itself affects the outcome, so that the
outcome as such does not exist unless, and until, the measurement is made. (That is,
there is no single outcome unless it is observed.)
According to the fundamental superposition principle of quantum mechanics, we are
entitled to think of quantum superpositions of coherent states. These are states that
contain simultaneously two coherent components (or states), one pointing in one
direction in phase space and the other pointing in another direction. We label the
former component the "alive-cat" state and the latter component the "dead-cat state."
The position wave function '¥ of such a state would be the superposition of two
coherent state (Gaussian) wave functions [38]:
(11)
The normalization factor has been omitted in Eq. (11) because it is not important here.
Equation (11) shows that'¥ has two peaks, one at +qo (alive-cat state) and the other at
- qo (dead-cat state) according to the superimposed coherent amplitudes. Also, Eq.
(11) has nothing to do with optical interference. When two fields interfere, their
amplitude may be enhanced or canceled, producing, for example, coherent states of
enhanced or zero amplitude (vacuum). The quantum superposition shown in Eq. (11)
still contains both coherent amplitudes ±qo. It is also much different from an
incoherent superposition of ±qo, where the field has either the amplitude +qo or the
amplitude -qo with certain probabilities. The quadrature amplitude of'¥ is +qo as well
as -qo (simultaneously!), with a resolution given by the vacuum fluctuations.
This strange behavior of'¥ being simultaneously at +qo and - qo turns out to be the best
representation of Schrodinger's famous thought experiment in the quantum field theory
of light. Schrodinger cat states are difficult to observe in the optical domain because
23
UNCLASSIFIED/ /PO" OFFIClltt tJ.!l! Oflt I
UNCLASSIFIED/ j FOR OFFICiltt tJ.!l! 8HLY
they are extremely vulnerable to quantum decoherence. Quantum decoherence is
caused by linear losses, and it is the main reason why the extremely strange quantum
phenomena allowed in quantum t heory are very difficult to observe in practice.
However, the good news is that investigators have successfully controlled or
suppressed quantum decoherence to such a high degree that Schrodinger cat states
were experimentally observed and measured using optical homodyne tomography [55,
56]. Figure 7 shows the experimentally reconstructed Wigner functions for two
Schrodinger cat states that have different amplitude values ±qo. The observed peaks at
±qo seen in the figure are of small magnitude, so investigators euphemistically call
these "Schrodinger kitten states." As seen in the figure, the interference structure
halfway between the peaks displays the quantum superposition of both amplitudes,
showing rapid oscillations with a frequency given by the distance 2lqol of the
superimposed amplitudes. Also seen in the figure is that the two reconstructed Wigner
functions become negative (i.e., negative "probabilities"), indicating the nonclassical
behavior of Schrodinger cat/kitten states.
Beam Splitters
A very important device that is used to demonstrate the quantum nature of light is the
simple optical beam splitter. A large number of strange quantum effects have been
experimentally observed by splitting or recombining photons using a small cube of
glass. The beam splitter also serves as a theoretical model for other linear optical
devices such as interferometers, semitransparent mirrors, dielectric interfaces, wave
guide couplers, and polarizers. The beam splitter model can also be used to account for
the effect of absorption, mode mismatch, and other linear losses.
An ideal beam splitter is a reversible, lossless device in which two incident beams of
light may interfere to produce two emerging beams [38]. For example, a dielectric
interface inside a cube or plate of glass splits a light beam into two. This situation may
be reversed by sending the two beams back to the cube (or plate) where they interfere
constructively to restore the original beam. However, if the phases of the two beams
are changed, then their mutual interference generates two emerging beams in general.
So four beams might be involved, two incident lig ht modes and two outgoing light
modes, and the splitting of just one beam is a special case. Therefore, the most
general theoretical beam splitter model is a four-port device, which is simply a "black
box " with two input and two output ports having certain mathematical and physical
properties [38). See Figure 8 for a schematic of an ideal lossless four-port beam
splitter.
The beam splitter is quantum mechanically described by a simple unitary
transformation operator (or matrix), based on an analog transformation matrix in
classical optics,**** which mathematically transforms the two input light modes into the
two output light modes. This operator is unitary, which reflects the fact that a lossless
beam splitter conserves energy and that the total light mode intensity at cii + ai a2 is an
invariant quantity. Since the incoming and the outgoing light modes are both
independent boson ic modes, their annihilation operators must satisfy the follow ing
·•·• In classical optics, the components of the transformation matrix of a real beam splitter are simply the
transmissivity and reflectivity, which account for the transmission and reflection probabilities of photons passing
through the glass cube or plate.
24
UNCLASSIFIED/ {FOR OFFICIO ls Ulilii 8PU:¥
UNCLASSIFIED/ /POI': orr1c11tt U.!I! Oflt I
boson ic commutation relations: [a;' a:,;] = [ a, 'a,;,] 81 and [a;' a:,,]
= Ill = [ al 'a,,.] = 0 /
where 6c111 (= l if e= m and 0 if t ":/: m) is the Kronecker delta and the indices (t , m) are
integers [38].
A beam splitter is a four-port device not only in the case of two incoming light modes
interfering to produce two emerging light modes; a beam splitter is always a four-port
device. Even if only one beam is split into two beams, if literally nothing behind the
semitransparent mirror is interfering with the incident beam, quantum mechanically this
nothing means a vacuum state. The very possibility that the second light mode behind
the mirror might be excited makes a difference. The vacuum fluctuations carried by the
empty mode (and entering the apparatus via the so-called unused input port of the
beam splitter) do cause physical effects. Therefore, the vacuum fluctuations entering
the second (unused) input port of the beam splitter must always be assigned a formal
mode operator, a2, in order for the system to conserve energy, 2) obey the beam
1 )
splitter's aforementioned bosonic commutation relations and 3) guarantee that the two
outgoing beams are independent bosonic light modes.
first input
a1
second output
A/
a2
second input
a2
first output
A/
al
Figure 8. Schematic of an Ideal Lossless Beam Splitter. Two
incident spatial-temporal light modes (with the annihilation operators a 1
and a2) interfere optically to produce two emerging light modes (with
the annihi lation operatorsa; and a; ) (courtesy of Ulf Leonhardt).
In Figure 9 we illustrate the effect of vacuum fluctuations for the case of a fictitious
beam splitter, which is a model for describing linear absorption or, equivalently,
25
UNCLASSIFIED)) FOR orrtClltt U.!I! OHL¥
UNCLASSIFIED/ /POI': OPPIClltt U.!I! Oflt I
a
detection losses. The input signal is attenuated and, simultaneously, contaminated
by the vacuum fluctuations entering the second (unused) input port of the fictitious
beam splitter. The absorber acts like a fictitious beam splitter, and when light is
attenuated it can be imagined as being split into a transmitted part and an absorbed
part. On the other hand, we know from the fluctuation-dissipation theorem that losses
are always accompanied by fluctuations [57]. At least the vacuum fluctuations of the
absorbing medium must be taken into account. In the simple absorber model, these
fluctuations come into play via the second (unused) input port of the fictitious beam
splitter as shown in Figure 9. The annihilation operator a of the partially absorbed
(input signal) mode is transformed by the fictitious beam spl itter according to
a,,
a,' =ri'12 a+(l - riY' 2 where the factor 17 (0 < 17 s; 1) reduces the intensity of any initial
coherent state [a) to lri l/2a) after undergoing partial absorption, a' is the output signal
mode that goes to the detector (which counts the number of photons it absorbs,
fi' = £,,'t cl), and d2 is the mode operator of the vacuum fluctuations entering the second
(unused) input port of the fictitious beam splitter. The second term (l - ri/ 2 a2 in a' is
essential to guarantee that the attenuated light field remains a proper bosonic mode,
otherwise energy conservation and the aforementioned bosonic commutation relations
would be violated.
Finally, we note without further elaboration that the mode operators, quadrature wave
functions, and Wigner functions are all rotated through some angle under the action of
a beam splitter. And the Wigner function of a signal is smoothed during absorption
under the action of a fictitious beam splitter. This provides additional models to
develop the properties of other types of optical instruments and understand their
behavior on incoming light modes (or input signals).
signal
a
---)---- absorption
vacuum
a2
detector
Figure 9. Illustration of a Fictitious Beam Splitter.
( courtesy of Ulf Leonhardt)
26
UNCLASSIFIED/ /POK OPPICIAL OSI! 014[1
UNCLASSIFIED/ /FOR OFFICIAL 03E 014L I
Photodiodes
Most photodetectors apply a version of the photoelectric effect to operate in which
incident light radiation ionizes a piece of photosensitive material in the detector and
produces freely moving electrons, i.e., an electric current is created that can be
amplified and hand led by electronic means. A commonly used type of detector is the
linear-response photodiode. In most cases, the photosensitive part of the detector is a
P-1-N structure, a sandwich of Positively doped, Intrinsic, and Negatively doped
semiconductor material. Commonly, silicon (Si) or indium gallium arsenide (lnGaAs)
are used where Si detects light out to a 1 µm wavelength and InGaAs operates in the
range 0.19 µm to 2.6 µm. A bias voltage of about 10 Volts is applied to drain the
majority carriers (electrons in N and holes in P) out of the intrinsic zone. In this
depletion region an unstable situation is created for the minority carriers. As soon as
electron-hole pairs are present in the intrinsic zone, the bias voltage produces a current
that is proportional to the number of carriers. Electrons in the valence band are lifted
into the conduction band by the absorption of light radiation, i.e., the absorption of a
single photon lifts one electron into the conduction band, which creates electron-hole
pairs in the depletion zone. This process can be made highly efficient because the
applied voltage is very low so that no avalanche of charge carriers into the conduction
band (via collisions) is formed. The current response of the detector is linear in the
intensity of the detected light. However, thermal fluctuations cause Nyquist noise in
the photocurrent. Thermal effects also create electron-hole pairs in the depletion zone
thus producing dark current, which is electronic noise. Because of this electronic noise,
linear-response photodiodes do not reach single-photon resolution. They are suitable
for relatively high intensities, greater than about 100 photons per microsecond.
There are inefficiencies and noise associated with realistic photodetection. A convenient
model to understand the effect these have on experiments is provided by imagining a
fictitious beam splitter placed in front of an ideal detector. See Figure 9. Only the
transmitted photons are counted, so t hat the transmissivity of the fictitious beam
splitter corresponds to the detection efficiency. Dissipation is always accompanied by
fluctuations. These degrade the quantum noise properties of the detected light. The
fluctuations are modeled by a vacuum entering the unused port of the fictitious beam
splitter. This analysis shows how the nonclassical features of light are lost when the
detectors are inefficient.
Balanced Homodyne Detection
Under idealized conditions the photon number is measured in direct photodetection.
However, another method of detection exists, in which the light field amplitudes (the
quadrature components) are measured instead of the quantized light intensity.
Intensity (photon number) and field amplitude (quadrature) are distinct quantities.
There is no simple relationship between the photon statistics and the quadrature
distributions in the quantum regime, but the two are shown to be related via the
mathematics and procedures of quantum state sampling [38]. Furthermore, the field
amplitudes contain phase information, and so they are dependent on phase.
27
UNCLASSIFIED/ /FOA &FFI@IAL U.!E ONE I
UNCLASSIFIED/ /FOR 8FPl€1J!tt U.!I! l>NL I
Quadrature components q0 are definedtttt with respect to a certain reference phase 8
that ca n be varied experimentally.
The principle scheme of a balanced homodyne detect or is depicted in Figure 10. The
signal interferes with a coherent laser beam at a well - balanced 50 : 50 beam splitter.
The laser light field is called the local oscillator {LO), and it provides t he phase
reference 8 for the quadrature measurement. It is assumed that the signal and the LO
have a fixed phase relation , as is the case in most experiments applying homodyne
detection , because both fi elds are ultimately generated by a common master laser. The
LO should be intense with respect to the signal for providing a precise phase reference.
It is also assumed that the LO is powe rfu l enough to be treated classically, i.e., we
totally neglect the quantum fluctuations of the LO. After the optical mixing of the signal
with the LO, each emerging beam is directed to a linear-response photod iode. The
photocurrents Ii and h are measured, electron ically processed, and finally subtracted
from each other. The difference current h 1 = h - / 1 is the quantity of interest because
it contains the interference term of the LO and the signal. It is assumed for simpl icity
that the measured photocu rrents Ti and h are propo rtional to the photon numbers ii 1
and n2 of the beams striking each detector, which are given by ,'i 1 = a;ta; and
Yli =ll~ta~ in terms of the mode operators a; =T (a- aw) and a~ = T (a +aw) of
112 112
the fields emerg ing from the beam sp litter [38] . Here a denotes the annihilation
operator of the signa l and aw is the comp lex amp litude of the LO .
The difference current hi is proportional t o the difference photon number (assum ing
perfect quantum efficiency) ii 21 = ii 2- ii 1 = a ~0 a+ a w at , where a~ is the complex
conjugate of aw. The phase of the LO is 0, and so we note from the definit ion of q0
that the measured quantity hi is indeed proportional to q0 because n21 =2' 12 la wl q9 ,
which is a resu lt that has been verified by more sophisticated theories of homodyne
detection [38 ). A balanced homodyne detector measures q0 . The reference phase 0 is
provided by the LO and can be varied by adjusting the LO using a piezo-electrically
movable mirror, for example. An experimental method for find ing the scaling of q0 in
the difference current hi is to keep a record of the sum current because the sum of ft
and h is proportional to la w l2 to leading order [38]. Th is can be experimentally
important because the intensit y of the LO is usually an unknown quantity.
tm We note th at phase shifting rotates th e quad rat ures, q0 = (;t (0) q U (0) = qcos0 + psin0 and
Pe= rJt(0) p U(0) = -qsin0 + pcos0 , via the quad rature decomposition defi ned in Sect. IIB- 1 and t he phase
shifting property of th e annihi lation operator defi ned in Sect. IIIB-1.
28
UNCLASSIFIED// FOR OFPICIAL 091! er•t t
UNCLASSIFIED/ /P91t 9Ffl@IAL l:ISIE 8HLY
signal
detector
local
oscillator
(aLO)
Figure 10. Schematic of a Balanced Homodyne
Detector. (courtesy of Ulf Leonhardt)
Furthermore, the balanced homodyne detector is also an amplifier. The LO amplifies
the signal by the mutual optical mixing of the two. In other words, the homodyne
detector is an interferometer that can be measurably imbalanced by a single photon in
the signal mode because the reference field is very intense. A very important technical
advantage of this is that the amplified signal is well above the electronic noise floor of
the photodiodes. The signal amplitude is enhanced so that even the noisy linear
response photodiodes can detect the quantum features of the signal with single photon
resolution. Because the LO serves as a coherent amplifier, it also chooses the signal
mode. The LO singles out one spatial-temporal (bosonic) mode from the rest of the
contin uous quantum field "light" (that matches the LO field). In this way the observer
separates the quantum object (a single optical mode) from the rest of the world. The
mode function is given by the spatial-temporal shape of the LO beam at the detector
surface and during the measurement time interval [0, T]. The overall phase and
intensity of the LO is comprised in the complex amplitude ULO. Shifting the phase 0 =
arg(aLO ) rotates the measured q0 . The observer defines via the LO the frame in space
and time that is subject to the field-quadrature measurement. By tailoring the shape of
the LO beam high spatial-temporal resolution can be achieved.
Photodetection is usually not completely efficient in practice so it is important to
describe the influence of inefficiencies on homodyne detection. This is easily done by
using the simple model for losses in direct photodetection that was given in Section
IIIB-2. We imagine fictitious beam splitters to be placed in front of the two (assumed
ideal) detectors in the measurement setup (see Figure 11). We use
29
UNCLASSIFIED// FOR OFFICIAL tJ.!! 9HLY
UNCLASSIFIED/ /FOR 055ICIAl 1481!! er•t I
a'=rt\1 + (l - ri) a from Section IIIB-2 to define the annihilation operators of the
1' 2
2
. , ,, 1/ 2 • 1 1/ 2 - - · 1/ 2 • 1 1/ 2 - -
detected light modes a 1 = ri a 1 +(1- ri) b 1 and a 2 =ri a 2 +(1 - ri) b2 , where b, and
b2 are the annihilation operators of the vacua entering the second unused ports of the
fictitious beam sp litters. The ann ihilation operators a; and a~ describe the light modes
(or fields) emerg ing from the 50:50 beam splitter where the signal is optically mixed
with the LO. Again, the LO is an intense field compared with the signal so it can be
treated classically. Therefore, we do some algebra to compute the difference photon
number ii 2 1 = n; - n7 = a;ta; - a7t
a7 , but retain on ly the lead ing terms with respect to
a w , and obtain the final result [38]:
(12)
The symbol H C in Eq . (12) denotes the Hermitian conjugate of the other part of an
b b -b
expression and = i- 112 ( 2 1) . The fluctuat ion mode operator b
corresponds to the
optical mixing of the fictit ious vacuum -noise modes b1 and b2, and it obeys the bosonic
commutation relation [ b, b'] = 1 (e.g., see Sect. 111B-2). Because the interference of
vacuum with vacuum yields vacuum, the fluctuation mode b can be regarded as a
bosonic mode, being in the vacuum st ate as well.
30
UNCLASSIFIED/ /FOR OFFI@IAl YSfI or•t'&'
UNCLASSIFIED/ /FOR OFFICl"L U.!I! er•tY
detector
vacuum
I vacuum
"V ~
I b1
signal
a
detector
local
oscillator
(aw)
Figure 11. Balanced Homodyne Detector Using Fictitious
Beam Splitters to Account for Detection Losses. (courtesy
of Ulf Leonhardt)
Equation (12) provides an additional model for detection losses. Similar to direct
photon counting, a fictitious vacuum field has to be added to the attenuated signal in
homodyne detection. This means that we can replace the arrangement of two fictitious
beam splitters in front of the photodetectors with just one effective beam splitter in
front of an ideal homodyne detector (see Figure 12). This effective beam splitter
accounts for other kinds of losses including mode mismatch, whereby the quantum
effects of both detection losses and mode mismatch are comprised in an effective TJ.
31
UNCLASSIFIED/ /FOA OFFI@IAL li!II! er•t I
UNCLASSIFIED/ /PO" OPPICIJ!tt l:191! OHL'f
detector
I vacuum
'Y.
Ib
signal
a
Fuente: archivo UAP oficial del gobierno de EE.UU. (dominio público) · war.gov/ufo ↗ · ver en el archivo de Nodriza