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Quantum Input-Output Theory for Optical Cavities with Arbitrary Coupling

Strength: Application to Two-Photon Wave-Packet Shaping

Article in Physical review A, Atomic, molecular, and optical physics · September 2013
DOI: 10.1103/PhysRevA.88.043819 · Source: arXiv

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 arXiv:1309.5915 [quant-ph] /Oct 5 2013

Quantum Input-Output Theory for Optical Cavities with Arbitrary Coupling Strength: Application to
Two-Photon Wave-Packet Shaping
M. G. Raymer1 and C. J. McKinstrie2
1
Department of Physics, University of Oregon, Eugene, Oregon 97403, USA
2
Bell Laboratories, Alcatel-Lucent, Holmdel, New Jersey 07733, USA

We develop quantum-optical input-output theory for resonators with arbitrary coupling strength, and for input fields whose
spectrum can be wider than the cavity free-spectral range, while ensuring that the field-operator commutator relations in
space-time variables are correct. The cavity-field commutator exhibits a series of space-time ‘echoes,’ representing causal
connections of certain space-time points by light propagation. We apply the theory to two-photon wave-packet shaping by
cavity reflection, which displays a remarkable illustration of dispersion cancellation. We also show that the theory is
amenable to inclusion of intracavity absorbing and emitting atoms, allowing, for example, dissipative losses within the cavity
to be incorporated in a quantum mechanically correct way.

PACS numbers: 42.50.-p, 42.50.Ar, 42.50.Pq

and assuming that no standing waves are formed, the

optical field is uniformly distributed along the beam axis;
1. Introduction that is, there are no pulse-propagation effects inside the
Input-output theory for optical cavities or resonators cavity. For this limit to hold, the field must have spectral
plays a crucial role in quantum and classical optics because width much smaller than the cavity’s free-spectral range
of the enhancement of coupling between external light fields (FSR). The field necessarily evolves negligibly during one
and the cavity modes, as well as between the cavity modes round trip in the cavity. This high-Q, “good-cavity” limit is
and any medium inside the cavity. By input-output (I-O) commonly used in many experimental situations, and so
theory, one means a differential equation of motion for the the input-output theory developed in the 1980s has seen
field in space-time variables, coupled with appropriate widespread use. On the other hand, there are situations in
cavity boundary conditions, that is amenable to inclusion of which a cavity with lower mirror reflectivity, and thus
intracavity absorbing and emitting atoms. The theory of lower Q, are used. For example, in a cavity with non-
optical cavities was well developed in the decades following negligible dissipative loss, it may be advantageous to
the invention of the laser. Nevertheless, a simple increase the cavity output coupling by decreasing the
formulation treating input-output theory as a scattering mirror reflectivity, in order to increase the efficiency of
problem that is valid in quantum as well as classical extracting field energy from the cavity. A more general
contexts was not developed until the 1980s, when the ideas theory is needed to describe such situations.
of quantum-noise squeezing and cavity-QED were being In this paper we develop input-output theory for
developed. Collett and Gardiner [1], Gardiner and Savage arbitrary coupling-junction strength, and for input fields
[2], and Yurke [3] developed a quantum mechanical linear- whose spectrum can be wider than the cavity FSR, while
systems approach that describes the evolution of the input ensuring that the field-operator commutator relations are
field, cavity field, and output field. That approach crucially correct. A key development is expressing the fields and
ensures correct quantum commutator relations between the field commutators in the space-time variables, and
positive- and negative-frequency components of the extending previous results [7, 4] for these. In particular, we
electromagnetic field, or equivalently the photon find that the cavity-field commutator exhibits a series of
annihilation and creation operators. This ensures that the space-time ‘echoes,’ representing causal connections of
field evolution is quantum mechanically unitary, as it must certain space-time points by light propagation.
be in the absence of dissipative losses. The I-O theory has Figure 1 shows examples of cavity types being
been generalized to include spatially complex cavity considered. Each has two input channels and two output
structures and intracavity losses. [4, 5] channels, with the two channels propagating in opposite
The principal limitation of the most commonly used directions. The input-output coupling is created by a
input-output theories [1, 6] is that they are restricted to the junction, which may be a mirror or a waveguide coupler. In
limit of high cavity quality-factor Q, in which the cavity the common case that only a single input channel is
mirrors or other coupling junctions have very high occupied by light, the system operates in traveling-wave
reflectivity, and no dissipative losses. In the high-Q limit, geometry where standing waves do not form. In that case

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 arXiv:1309.5915 [quant-ph] /Oct 5 2013

and in the absence of backscattering and nonlinear normalized electric field amplitude (or photon annihilation
coupling, the two counter-propagating fields evolve operator) C(z,t) obeys the partial differential traveling-
independently of each other and so we can ignore one of wave Maxwell equation:
them. (However, one should be aware that if there is a
structure in the cavity medium that is capable of acting as a
nonlinear coupling or as a diffraction grating, such as a
  ( ∂ + v ∂ ) C(z,t) = α P(z,t)  ,                                                      (1)  
t z

gain medium periodically modulated on sub-wavelength

scales, then the forward and backward waves are coupled where z is the distance traveled around the cavity path
and the solution must be generalized to account for this.) starting at the junction, and v is the group velocity (speed
of light c for an empty cavity). α is a coupling parameter.
The carrier wave has been factored from the field
amplitude, and all frequencies are specified relative to the
optical carrier frequency. The input coupling and
periodicity of the cavity field are represented by the
boundary condition:

C(0 + ,t) = ρ C(L− ,t) + τ A(t)                                                  (2)  

relating the cavity field to the input field A(t) at the

coupling junction. Here 0 + is the location inside the cavity
immediately following the coupling junction and L− is the
location inside the cavity just before impinging on the
Fig. 1 (Color online) Ring cavity configurations coupling junction. We choose a phase convention in which
with a single input, showing the conventions used for the coupling coefficients τ (for transmission) and ρ (for
amplitude transmission coefficient τ and reflection reflection) are real and thus they must obey τ 2 + ρ 2 = 1 .
coefficient ρ . We choose a phase convention where This relation is the same as used in laser theory, [8, 9, 10,
external reflection creates a phase flip, and internal 11] but generalized to include the input field, which is
reflection does not. normally not considered unless injection locking is being
considered. [12]
There is a close relationship between the present The other boundary condition that must be satisfied
theory and laser theory, and we make use of this. In determines the output field B(t) in terms of the input field
particular, early papers on the theory of multimode optical A(t) and the cavity field:
cavity instabilities and mode locking provide part of the
inspiration for our formulation. [8, 9, 10, 11]
In this paper we present the general formalism for   B(t) = τ C(L− ,t) − ρ A(t)  .                                                  (3)  
arbitrary coupling strength, expressed in both space-
frequency and space-time domains. We apply it to the It is notable that Eqs. (2) and (3) are the same as the
problem of two-photon wave-packet shaping by cavity standard beam-splitter relations [13], familiar in quantum
reflection. We also show how dissipative losses within the optics. They can be written in common matrix form as
cavity can be incorporated in a quantum mechanically
correct way. ⎛ B(t) ⎞ ⎛ τ − ρ ⎞ ⎛ C(L+ ,t)⎞
  ⎜⎝ C(0 ,t)⎟⎠ = ⎜⎜ ρ ⎟
τ ⎟⎠ ⎜⎝ A(t) ⎟⎠
     .                          (4)  
2. Input-Output and Commutation Relations + ⎝

As mentioned above, it is simplest to consider only a These relations enter the theory here as cavity-field
single input beam and a single output beam, in which case boundary conditions rather than as the (sometimes) ad hoc
the field in the cavity is a traveling-wave one with no relations that are postulated in order to maintain
standing-wave effects. It is straightforward to generalize to commutation relations in free space. [14] As in those
the case of two counter-propagating inputs and therefore treatments, the minus sign on − ρ makes the matrix in Eq.
standing waves in the cavity. In addition, only a single (4) unitary. In fact, we will show that, rather than maintain
transverse mode is considered, although it is free-space commutation relations, Eqs. (1) – (3) lead to
straightforward to include more such modes. significant alterations to the commutation relations in a
In the presence of a local electric-dipole polarization way that maintains causality and unitarity in the theory.
P(z,t) (which equals zero for an empty cavity), the

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 arXiv:1309.5915 [quant-ph] /Oct 5 2013

The input field, being a freely propagating field in free τ

space, obeys the familiar commutation relation (for a field   Gca (ω ) =  ,                                                  (14)  
1− ρ exp(iω T )
whose bandwidth is significantly smaller than the optical
so that
carrier frequency):
  c(0 + , ω ) = Gca (ω )a(ω )  .                                              (15)  
  ⎡⎣ A(t), A† (t ') ⎤⎦ = δ (t − t ')  .                                        (5)  
The modulus-square of the Green function Gca (ω ) is
This is a special case of the more general relation that
plotted in Fig. 2, as a reminder of the well-known
applies to a field A(z,t) freely propagating at speed v in enhancement of the density of states that occurs near the
the absence of dispersion: cavity resonances. The free-spectral range (FSR) of the
cavity in radians per second is Ω = 2π / T . (Throughout
  ⎡⎣ A(t, z), A† (t ', z ') ⎤⎦ = δ (t − t '− (z − z ') / v))  ,        (6)   the paper time is measured in units of T , position in units
  of L , and other variables are scaled accordingly.) The
which results simply from A(z,t) = A(0,t − z / v) . This
2
integral of Gca (ω ) over any integer multiple of the free-
shows that the input field (assuming no dispersion) spectral range yields the value 1, showing that the number
commutes with itself at all times, except those that are of states is conserved, while the density of states is
causally connected by the speed of light in free space. redistributed. This is related to an approach picturesquely
Physically, only when a commutator is zero are the two called the ‘modes of the universe.’ [15] In terms of that
operators independently measurable. picture, our cavity is embedded in a nearly infinite-length
The goal is to deduce from Eqs. (1) – (5) the cavity- cavity, and the modes of that large cavity (which have very
field operator and the output-field operator, as well as their small mode frequency spacing) are ‘pulled’ toward the
commutation relations. We first consider the case of an resonances of our cavity by the dispersion it induces (or
empty cavity, so that P(t) = 0 . We introduce the Fourier- equivalently by the boundary conditions). This leads to a
transform fields according to: ‘piling-up’ of mode density near the resonances.
∞

∫ dte
iω t
  f (z, ω ) = F(z,t)  .                                          (7)  
−∞
(Throughout the paper we denote frequency-domain
functions by lower-case letters.) Then Eqs. (1) – (5) imply

  −iω c(z, ω ) + v ∂ z c(z, ω ) = 0                                                (8)  

  c(0 + , ω ) = ρ c(L− , ω ) + τ a(ω )                                            (9)  
  b(ω ) = τ c(L− , ω ) − ρ a(ω ) .                                              (10)  

Equation (8) easily yields for the cavity field:

  c(L− , ω ) = c(0 + , ω )exp(iω T )  ,                                      (11)  

  Fig.2 (Color online) Modulus-square of cavity
where T = L / v is the cavity round-trip time. Equation Green function Eq. (14) versus frequency, for
(11) represents a time shift; by transforming back to the junction reflectivities ρ = 0 (dashed) and
time domain and using the Fourier-shift theorem:
ρ = 0.75 (solid), and T = 1.
  C(L− ,t) = C(0 + ,t − T )  .                                                  (12)  
The output-field spectral amplitude b(ω ) is given in
Inserting (11) into (9), and solving, yields terms of a different Green function Gba (ω ) as:
τ
  c(0 + , ω ) = a(ω )  ,                                    (13)     b(ω ) = Gba (ω )a(ω )  ,                                                  (16)  
1− ρ exp(iω T )
where
a result familiar from classical cavity theory. Equation (13) ⎛ 1 − ρ exp(−iω T ) ⎞
Gba (ω ) = exp(iω T ) ⎜
is written in a linear-response form by defining a Green   ⎝ 1− ρ exp(iω T ) ⎟⎠  ,                (17)  
function Gca (ω )
= exp(i θ (ω ))

  3  
 arXiv:1309.5915 [quant-ph] /Oct 5 2013

from which we can see that in the frequency domain the ∞

dω ∞

output field differs from the input field only by a unit- C(0 + ,t) = ∫ 2π e
−iω t
τ a(ω )∑ ρ n einω T
magnitude factor with a frequency-dependent phase θ (ω ) . −∞ n=0
∞ ∞
This fact is consistent with energy conservation.   =τ ∫ dt 'A(t ')∑ ρ δ (t − nT )                              (22)  
n

The inverse relation is easy to write; from Eqs. (16) −∞ n=0

and (17): ∞
= τ ∑ ρ n A ( t − nT ) .
a(ω ) = ( Gba (ω )) b(ω )
−1 n=0

= exp(−i θ (ω ))b(ω )  .                                      (18)   The first term in the sum is the beam-splitter transfer
= Gba (ω ) b(ω ) * function, while the remaining terms are delayed and
attenuated replicas (‘echoes’) of the input. This, too, can be
 
expressed in linear-response form, by introducing the time-
The inverse Green function is simply the complex
domain Green function G  ca (t) , which is the Fourier-
conjugate of the forward one: Gab (ω ) = Gba (ω )* . As can
transform of Gca (ω ) . Then
be seen from (17), the complex conjugate corresponds to
∞
simply replacing T by –T in Gba (ω ) .
The above relations are common lore in cavity theory,
  C(0 + ,t) = ∫ dt 'G
−∞
ca (t − t ')A(t ')  ,                                    (23)  
but to the best of our knowledge they have not previously where
been exploited to derive simple input-output relations for ∞
quantum fields. To further this goal, we derive the   G ∑
 ca (t) = τ ρ nδ ( t − nT )  .                                        (24)  
quantum-mechanical ramifications of the above relations. n=0

The commutator for the input field is, in the frequency  

domain: The commutation relation for the cavity field in the
time domain is derived in Appendix 1 from Eq. (22), and is
  ⎡⎣ a(ω ),a † (ω ') ⎤⎦ = 2πδ (ω − ω ')  .                            (19)   found to be:
 
The commutator for the cavity field in the frequency ⎡⎣C(0 + ,t),C † (0 + ,t ') ⎤⎦ =
domain is, from Eqs. (15) and (19), ∞ ∞
= τ 2 ∑ ∑ ρ m ρ n ⎡⎣ A ( t − nT ) , A† ( t '− mT ) ⎤⎦        (25)  
2 n=0 m=0
⎡⎣ c(z, ω ),c† (z, ω ') ⎤⎦ = Gca (ω ) 2πδ (ω − ω ') ∞

τ
2
 
= ∑ρ |k|
δ (t − t '− kT ) .
k=−∞
= 2πδ (ω − ω ') .
1− ρ exp(iω T )  
  (20)   This shows that the cavity field at position z = 0 +
  commutes with itself at all times except those separated by
This commutator is different from the free-space one, and integer multiples of the cavity round-trip time; that is, at
reflects the increase of the density of states near the cavity those times that are causally connected by the speed of
resonances. light in the cavity. In the limit that there is no cavity, i.e.
The cavity field can be expressed in a different way ρ → 0 , this recovers the free-space relation Eq. (5), as
using a Taylor-series expansion:
expected. For nonzero ρ the commutator decays as ρ |k| ,
indicating loss of memory or correlation between widely
τ a(ω ) ∞
  c(0 + , ω ) = = τ a(ω )∑ ρ n einω T  .              (21)   separated times.
1− ρ exp(iω T ) n=0 The commutator in Eq. (25) can easily be generalized
  to account for different positions in the cavity in similar
Transforming back to the time domain yields manner to Eq. (6) for the input field. Note that Eq. (1) with
P = 0 implies C(z,t) = C(0 + ,t − z / v) ; then Eq. (25)
implies

∞
⎡⎣C(z,t),C † (z ',t ') ⎤⎦ = ∑ρ |k|
δ (t − z / v − (t '− z '/ v) + kT ) .
k=−∞

  (26)  

  4  
 arXiv:1309.5915 [quant-ph] /Oct 5 2013

For equal times this becomes Causality is satisfied. This can also be generalized to
account for different positions in the cavity, as in Eq. (26).
∞ The commutator between the cavity field and the output
⎡⎣C(z,t),C † (z ',t) ⎤⎦ = ∑ρ |k|
δ ((z '− z + k L) / v)
 ,     field can be found by similar means.
k=−∞
The output field B(t) , being a freely propagating field
= δ ((z '− z) / v)
in free space, must obey the same commutation relations as
(27)   does the input field, i.e., Eqs. (5) and (19). This is easy to
where the sum does not contribute because z is contained see in the frequency domain, where the commutator, from
within the interval z ∈[0, L] . This agrees with the Eqs. (16)–(19), is
fundamental field commutator result verified in [7], where
it was noted that Eq. (27) is the same inside and outside the 2
⎡⎣b(ω ),b † (ω ') ⎤⎦ = Gba (ω ) 2πδ (ω − ω ')
cavity, as it must be. What that report left unsaid is that                        (29)  
when different times are considered, as in Eq. (26), the = 2πδ (ω − ω ') .
commutator inside the cavity is not the same as that of the
 
input field. The temporal evolution – here manifested as
On the other hand, the mapping between the input and
echoes – affects the commutator, consistent with causality.
output fields is nontrivial when expressed in the time
Figure 3 illustrates the space-time structure of the
domain. To derive this, express the output-field Green
commutator, from Eq. (26). Slanted white lines indicate
function in the frequency domain, Eq. (17), as
values of t and z where the commutator is non-zero. For
t=0 (horizontal white lines), it can be seen that as z’
increases from 0 to L, the spatial position where the ⎛ τ 2 exp(iω T ) ⎞
Gba (ω ) = − ρ + ⎜
commutator is nonzero (indicated by the line crossings) ⎝ 1− ρ exp(iω T ) ⎟⎠
moves with z’. There is a single crossing for any fixed                                  (30)  
∞
value of z’, in agreement with Eq. (27). In contrast, for = −ρ + τ 2
∑ρ n−1 i nω T
e .
fixed z, z’, and t’ values, there are an infinite number of n=1

times t at which the commutator is non-zero.  

An inverse Fourier transform then gives the output Green
function in the time domain:

∞
  G ∑
 ba (t) = − ρδ (t) + τ 2 ρ n−1δ (t − nT )  ,                          (31)  
n=1

 
Therefore, the output field in the time domain is:

∞
Fig. 3 (Color online) Modulus of the cavity-field
commutator, versus z and t, for t ' = 0 , and (a) z ' = 0 ,   B(t) = ∫ dt 'G
−∞
ba (t − t ')A(t ')  ,                                      (32)  
(b) z ' = 0.333 , and (c) z ' = 0.666 . The cavity length
or
L=1; the speed of light v = 1; time t is measured in ∞
units L/v. Horizontal white lines indicate t=0 regions.   B(t) = − ρ A(t) + τ 2 ∑ ρ n−1 A(t − nT )  .                        (33)  
Mirror reflectivity ρ 2 = 0.998 . n=1

 
We can also calculate the commutator between the The first term here is the beam-splitter transfer function,
cavity field at position z = 0 + and the input field: while the terms in the sum are ‘echoes.’ It can be shown
that Eq. (33) is consistent with Eqs. (3), (12), and (22).
∞
From this result, the commutation relation for the output
⎡⎣C(0 + ,t), A† (t ') ⎤⎦ = τ ∑ ρ n ⎡⎣ A ( t − nT ) , A† (t ') ⎤⎦ field in the time domain can be derived as a consistency
n=0 check, and indeed is found to be
∞
 
= τ ∑ ρ δ (t − t '− nT ) .
n
  ⎡⎣ B(t), B† (t ') ⎤⎦ = δ (t − t ')  ,                                              (34)  
n=0

  (28)    
The cavity field C(0 + ,t) commutes with the input field that is, the same free-space commutator that is obeyed by
A (t ') for all times t < t ' because later values of the input
† the input field. The derivation is given in Appendix 2. The
generalized form in Eq. (6) also holds for the output field.
field cannot affect the cavity field at earlier times.
This reflects the fact that the output field, because it is

  5  
 arXiv:1309.5915 [quant-ph] /Oct 5 2013

traveling in free space, can in principle be measured with If the conditions i - iii are met, meaning that κ T << 1 , then
arbitrarily high precision simultaneously at distinct space- the sum in Eq. (39) can be well approximated by an
time points not connected by causal propagation. integral:
The inverse relation can be written in the time domain t
τ
∫ dt ' T e
−κ (t−t ')
as well; from Eq. (18), we saw that it corresponds to   C(0 + ,t) ≈ A(t ')  ,                                    (40)  
simply replacing T by –T. Thus −∞

 
∞ where we used t ' = t − nT and noted that n ≥ 0 implies
  A(t) = ∫ dt 'G ab (t − t ')B(t ')  ,                                          (35)   t ' ≤ t . Comparing this to Eq. (23) shows that in this limit
−∞ the Green function can be effectively replaced by
where  
∞
  G ∑
 ab (t) = − ρδ (t) + τ 2 ρ n−1δ (t + nT )  ,                    (36)  
   ca (t − t ') = τ e−κ (t−t ')Θ(t − t ')  ,                            (41)  
G eff
n=1 T
 
so,
∞
where Θ(x) is the Heaviside step (theta) function.
  A(t) = − ρ B(t) + τ 2
∑ρ n−1
B(t + nT )  .                                  (37)   Transforming this to the frequency domain gives
n=1
∞
τ −κ t τ /T
∫ dte
iω t
This result can be verified explicitly by substituting Eq.   Gca (ω )eff = e Θ(t) =  ,                      (42)  
(33) into Eq. (37) and performing the sums. −∞
T κ − iω
 
3. Reduction to Standard High-Q Input-Output Theory a complex Lorentzian, as expected for a single, narrow
resonance of a cavity. The same result is obtained directly
To verify that the theory above includes the standard I- by considering Eq. (14) in the limit ω T → 0 , and using
O theory [1] as a limiting case, we consider the limit ρ = e−κ T ≈ 1− κ T ≈ 1− τ 2 / 2 , which implies κ ≈ (1− ρ ) / T
regime satisfying three conditions:
i. The junction transmission coefficient τ is very and κ ≈ τ 2 / 2T , which is standard in high-Q cavity theory.
small, so the cavity storage time is long. Near the resonance, the shape of the spectral response
ii. The cavity round-trip time T is very small, in Eq. (42) is similar to the exact result given by Eq. (14),
compared to the duration of the input field pulse. so it might seem tempting to apply the approximate form
iii. The input field is narrow band, so it is well even in the intermediate-loss regime, where ρ is
contained within a single FSR of the cavity. significantly different than 1. The problem with this idea is
For a transform-limited input field, conditions ii and iii are that the approximate form Eq. (42) has a maximum value
equivalent. τ / ln(1 / ρ ) , whereas the exact result has maximum value
τ / (1− ρ ) . These agree only in the limit ρ → 1 , which is
Define a cavity damping rate κ by κ = (1 / T )ln(1 / ρ ) ,
the high-Q regime. Therefore, we restrict application of
so that ρ = e−κ T , which gives two ways to write the Eq. (42) to the high-Q regime.
attenuation factor suffered by the field on each trip around Standard I-O theory in the high-Q regime can be
the cavity. Then we can write, without approximation, recovered easily by noting that the solution in Eq. (40)
(valid in the limit ω T → 0 ) satisfies the following
∞ differential equation:
  G ∑
 ca (t) = τ e−κ nT δ ( t − nT )  .                                    (38)  
n=0
τ
    ∂t C(0 + ,t) = −κ C(0 + ,t) + A(t)  .                        (43)  
T
In order to consider how G  ca (t) behaves in the high-Q
 
limit, note that it is a distribution (not a function), so it has This fundamental equation of motion for the cavity field is
meaning only as a factor inside an integral. The relevant supplemented with the output-field equation, Eq. (3):
integral is Eq. (23), which gives
  B(t) = τ C(L− ,t) − ρ A(t)  .                                      (44)  
∞
  C(0 + ,t) = ∑ τ e −κ nT
A(t − nT )  .                                (39)    
n=0 The goal of standard I-O theory is to be able to treat
  the cavity field as an effective single mode, called a
‘quasimode,’ with annihilation operator C(t) that obeys

  6  
 arXiv:1309.5915 [quant-ph] /Oct 5 2013

the commutator [C(t),C † (t)] = 1 . To this end, we note that Eq. (47), using κ ≈ (1− ρ ) / T . (a) ρ = 0.97 , (b)
in the limit ω T → 0 , the effect of one round trip is ρ = 0.70 . (Delta functions are represented by narrow
negligible, so Eq. (11) and (12) imply that Gaussians for visualization.)
c(L− , ω ) ≈ c(0 + , ω ) and C(L− ,t) ≈ C(0 + ,t) , so we define
4. Cavity Shaping of Time-Frequency Entangled Two-
C(t) = T C(0 + ,t) , where we also introduced a scaling
Photon Wave-Packets
factor T . This makes the cavity field dimensionless.
Then, also using κ ≈ τ 2 / 2T , we find A standard example of non-classical light is the time-
frequency-entangled photon pair. [16, 17, 18, 19] It can
exhibit violations of Bell inequalities [16], violation of
  ∂t C(t) = −κ C(t) + 2κ A(t)  .                                    (45)   classical Maxwell electromagnetic theory [20, 21], and is
  useful in quantum cryptographic key distribution [22],
The output-field equation becomes, in the limit ρ → 1 among other applications. It is easily created using
spontaneous parametric down-conversion in crystals [23,
24] or spontaneous four-wave mixing in fibers [25, 26, 27,
  B(t) = 2κ C(t) − A(t)  .                                          (46)   28], and since it contains only two photons is fully
  characterized by its fourth-order electric-field correlation
The commutator of the (rescaled) quasimode operator is function,
easily found from Eq. (40) to be:
  f (t1 ,t 2 ) = Ψ A† (t1 )A† (t 2 )A(t 2 )A(t1 ) Ψ  ,                  (48)  
  [C(t),C (t ')] = exp(−κ | t − t ' |)  .                                    (47)  
†

  where the field operator is (if the light is not too broad
This reduces to [C(t),C (t)] = 1 for equal times, justifying
† band)
the scaling factor that we used. Equations (45) - (47) are
dω
the standard I-O theory for high-Q cavities, originally   A(t) = ∫ a(ω )e−iω t  .                                                (49)  
derived using a master-equation method. [1] Note that the 2π
commutator Eq. (34) is exactly upheld even with the  
approximations made in arriving at Eqs. (45) and (47). The state can be expressed equivalently in the frequency or
It is helpful to compare graphically the forms of the time domains as
commutator in the exact and approximate theories, as in
Fig. 4. For high junction reflectivity, ρ = 0.97 , the Ψ = (2π )−2 ∫ dω ∫ dω 'ψ (ω , ω ')a † (ω )a † (ω ') vac
approximate commutator (using κ ≈ (1− ρ ) / T ) acts like  
= ∫ dt ∫ dt 'ψ (t,t ') A† (t)A† (t ') vac ,
an accurate envelope for the exact commutator, which is a
sum of delta functions. However, for ρ = 0.70 , the   (50)  
approximate result deviates significantly from the true 
where ψ (t,t ') is the double Fourier transform of ψ (ω , ω ')
envelope of the delta functions. By using the exact and the field operators obey the commutators Eq. (19) or
expression for the damping rate, κ = (1 / T )ln(1 / ρ ) , the Eq. (5). (Note that we are working in the Heisenberg
approximate commutator can be made to decay at the exact picture, where the state Ψ is time independent.) The
same rate as the exact commutator. But then, as stated modulus-squared of the two-photon probability amplitude
above, the magnitudes of the Green functions in the 2
frequency domain do not agree quantitatively unless ψ (t1 ,t 2 ) gives the joint probability to detect photons at
(1− ρ ) << 1 , which is the high-Q limit. both times t1 and t 2 , and is determined by the properties
of the down-conversion crystal and the laser field used to
pump it. [29, 30, 31] Likewise, ψ  (ω , ω ') 2 gives the joint
spectral density – the probability to detect photons at both
frequencies ω and ω ' .
The correlation function in Eq. (48) is the inner
product of A(t 2 )A(t1 ) Ψ with its hermitian conjugate. So
Fig. 4 (Color online) Cavity-field commutator we evaluate:
†
[C(t),C (t ')] versus time difference; solid - exact
from Eq. (28); dashed - standard approximation from

  7  
 arXiv:1309.5915 [quant-ph] /Oct 5 2013

  The output state in the time domain is found by

A(t 2 )A(t1 ) Ψ = ∫ dt ∫ dt 'ψ (t,t ')A(t 2 )A(t1 )A (t)A (t ') vac
† † writing Eq. (35) for the creation operator (note the Green
  function is real):
= (ψ (t1 ,t 2 ) + ψ (t 2 ,t1 )) vac ,
∞
    (51)  
 
  A† (t) = ∫ dt 'G
−∞
ab (t − t ') B† (t ')  ,                                      (54)  
where we used the commutator Eq. (5) repeatedly inside
the integral to put the operators into normal order
 
 ab (t − t ') is given by Eq.(36). Then Eq. (50) implies
and G
(annihilation operators to the right). Then the correlation
2
function becomes f (t1 ,t 2 ) = Φ(t1 ,t 2 ) , where we defined ∞
the ‘two-photon wave function’ as  ab (t − τ ) B† (τ )
Ψ = ∫ dt ∫ dt 'ψ (t,t ') ∫ dτ G
  −∞
∞
Φ(t1 ,t 2 ) = vac A(t 2 )A(t1 ) Ψ  ab (t '− τ ') B† (τ ') vac
× ∫ dτ 'G     (55)  
                                       (52)  
= ψ (t1 ,t 2 ) + ψ (t 2 ,t1 ) . −∞
∞ ∞
 
This definition is standard notation in the quantum-field
= ∫ dτ
−∞
∫ dτ 'ψ
−∞
out (τ , τ ')B† (τ )B† (τ ') vac ,
theory of massive particles. In quantum optics, it can be
thought of as simply a function that is proportional to the
 
where the two-photon amplitude at the output is
two-photon detection amplitude [32, 33] or as a true
photon wave function. [34, 35] Note that this function
automatically has the correct boson symmetry under  ab (t − τ )G
ψ out (τ , τ ') = ∫ dt ∫ dt ' G  ab (t '− τ ')ψ (t,t ').      
photon label exchange (t1 ↔ t 2 ) . [35] This function can be (56)  
engineered to be very narrow in the time difference t1 − t 2 ,  
which implies that in the frequency domain there is strong The correlation function at the output is
anticorrelation between observed frequencies, with their
sum equaling that of the pump laser. [36]   fout (t1 ,t 2 ) = Ψ B† (t1 )B† (t 2 )B(t 2 )B(t1 ) Ψ  ,          (57)  
As an example of the input-output theory, consider
what happens when such a two-photon state is incident on
 
a cavity of the type in Fig.1. One might expect one of a which equals the inner product of B(t 2 )B(t1 ) Ψ with its
few possibilities: the tight temporal correlation will be hermitian conjugate. In Appendix 3 we show that
disrupted because a given photon in the pair may take any
number of round trips around the cavity before emerging; B(t 2 )B(t1 ) Ψ =
the tight temporal correlation will be maintained because  (58)  
the effect of the cavity is only to introduce dispersion, and ∫ dt ∫ dt 'G ab
 ab (t '− t )Φ(t,t ') vac .
(t − t1 )G 2

it is known that dispersion in bulk-media propagation is  

cancelled in certain situations for time-frequency entangled The correlation function for the output is thus
photon pairs [37]; or some combination of these two 2
effects might occur. fout (t1 ,t 2 ) = Φ out (t1 ,t 2 ) , where the two-photon wave
The quantum state, Eq. (50), when expressed in the function is, using Eq. (36),  
output-mode variables in the frequency domain, can be  
found using the inverse relation Eq. (18) to write  ab (t − t )G
Φ out (t1 ,t 2 ) = ∫ dt ∫ dt 'G  ab (t '− t )Φ(t,t ')
1 2
a † (ω ) = Gba (ω )b † (ω ) . Then ∞
= ρ 2 Φ(t1 ,t 2 ) − τ 2 ∑ ρ m Φ(t1 ,t 2 − mT )
{
Ψ = (2π )−2 ∫ dω ∫ dω ' ψ (ω , ω ')Gba (ω )Gba (ω ') }   ∞
m=1
     
× b (ω )b (ω ') vac .
† † −τ 2 ∑ ρ n Φ(t1 − nT ,t 2 )
n=1
    (53)   ∞ ∞
The quantity in brackets inside the integral is the output +τ 4 ∑ ∑ρ n+m−2
Φ(t1 − nT ,t 2 − mT ) .
wave function. Because the green function Gba (ω ) is n=1 m=1

(59)  
unimodular, this confirms that there is no change of the
joint spectral density as a consequence of passing through
the cavity.

  8  
 arXiv:1309.5915 [quant-ph] /Oct 5 2013

This clearly holds the possibility for photon-counting remarkable example of the cancellation of dispersion that
coincidences to occur at any combinations of delays is known for two-photon light with perfect frequency
suffered separately by the two photons. But two-photon anticorrelation. [37]
quantum interference can eliminate some of these The case of a non-stationary two-photon source is also
possibilities under certain conditions. of interest. This occurs if the pump field is pulsed. In a
As an example, first consider the common case of a special case we can model the wave function as a two-
stationary down-conversion source, pumped by a constant dimensional Gaussian, with parameter σ giving the
(cw) laser field. In this case the two-time wave function correlation time, and a second parameter β giving the
depends only on the time difference, Φ(t1 ,t 2 ) = D(t1 − t 2 ) , pulse duration,
with the width of the function D being the coherence time.
This implies a two-time wave function in the frequency Φ(t1 ,t 2 ) = exp[−(t1 + t 2 )2 / 2 β 2 ]exp[−(t1 − t 2 )2 / 2σ 2 ]  .
domain proportional to
  (63)  
Then, from Eq. (59) and Eq. (63),
ϕ (ω , ω ') ∝ δ (ω + ω ') ∫ dτ ei(ω −ω ')τ D(τ )  ,                (60)  
 
(
Φb (t1 ,t 2 ) = τ 2 F0 + ρ 2 exp[−(t1 + t 2 )2 / 2 β 2 ] )
× exp[−(t1 − t 2 ) / 2σ ]
2 2
showing the perfect frequency anticorrelation that is
∞
characteristic of this form of time-frequency entanglement.
To evaluate the two-time wave function of the output field (
+τ 2 ∑ Fm − ρ m exp[−(t1 + t 2 − mT )2 / 2 β 2 ] )
in this case, we use the math relation
m=1
    (64)  
×exp[−(t1 − t 2 + mT )2 / 2σ 2 ]
∞

( )
∞ ∞

∑ ∑ ρ n+m D(t1 − t2 + (n − m)T ) +τ 2 ∑ Fm − ρ m exp[−(t1 + t 2 − mT )2 / 2 β 2 ]

n=1 m=1 m=1
∞ ∞ × exp[−(t1 − t 2 − mT )2 / 2σ 2 ] ,
  = ∑ ∑ ρ 2 s D(t1 − t 2 + kT )                                    (61)  
 
k=−∞ s=|k|/2+1

ρ 2 ∞ where
= ∑ ρ |k| D(t1 − t2 + kT ) . ∞
1− ρ 2 k=−∞   Fm = τ 2 ∑ ρ 2 s−2 exp[−(t1 + t 2 − 2sT )2 / 2 β 2 ]  .(65)  
s=|m|/2+1

 
Then we find The function Fk goes to ρ |k| in the limit β → ∞ , thus
∞
recovering the result Eq.(62).
Φ out (t1 ,t 2 ) = ρ 2 D(t1 − t 2 ) − τ 2 ∑ ρ m D(t1 − t 2 + mT ) Consider the case of equal σ and β , so the wave
m=1 function of the input field is separable (expressible as a
∞
product of two function, one in t1 and one in t 2 ). The
−τ 2 ∑ ρ n D(t1 − t 2 − nT )
n=1   magnitude of the two-photon wave function, Eq. (64) is
τ 4 ∞ ∞ plotted in Fig. 5.
+ ∑ ∑ ρ n+m D(t1 − t2 − (n − m)T )
ρ 2 n=1 m=1
= D(t1 − t 2 ) .
    (62)  

The output field has the same narrow temporal correlation

function as does the input field! The photons of a pair
emerge together after scattering from the cavity.
To see the origin of this result, note that the second
term canceled with the positive-k parts of the fourth term
(written in the form of Eq. (61)), whereas the third term
canceled with the negative-k parts of the fourth term. The
k=0 part of the fourth term combined with the first term to
yield the result. That is, quantum amplitudes with zero
cavity transits for one photon and m transits for the other
(second and third terms) are cancelled by amplitudes with
k transits for one photon and k+m for the other. This is a

  9  
 arXiv:1309.5915 [quant-ph] /Oct 5 2013

 
Fig. 6 (Color online) Magnitude of two-photon wave
Fig. 5 (Color online) Magnitude of two-photon wave function Φb (t1 ,t 2 ) vs. t1 and t 2 , from Eq.(64), for
function Φb (t1 ,t 2 ) vs. t1 and t 2 , from Eq.(64), for
parameter values σ = 0.2, β = 0.7 , T = 1 , and the
parameter values σ = 0.3, β = 0.3 , T = 1 , and the junction transmission varied as: (a) τ = 0.999 , (b)
junction transmission varied as: (a) τ = 0.999 , (b) τ = 0.95 , (c) τ = 0.85 , (d) τ = 0.60 .
τ = 0.95 , (c) τ = 0.85 , (d) τ = 0.60 .
5. Inclusion of Dissipative Loss
For τ = 0.999 the dominant correlation peak occurs at To include dissipative loss in the arbitrary-coupling-
(t1 ,t 2 ) = (1,1) because the light takes one full round trip in strength model, we introduce an absorbing molecular
the nearly nonreflecting cavity before emerging. For medium throughout the cavity. Then Eq. (1) reads:
τ = 0.60 the dominant correlation peak occurs at
(t1 ,t 2 ) = (0,0) because the light reflects from the junction,
 
( ∂ + v ∂ ) C(z,t) = α P(z,t)
t z
                   (66)  
without delay, into the output beam. The output wave ∂t P(z,t) = −γ P(z,t) − β C(z,t) + F(z,t) ,
function is separable, as it was at the input. See Appendix
 
4. (This can be understood by noting that the output wave
where α , β are coupling parameters, γ is the damping
function in the frequency domain, in Eq. (53) retains its
separability if the input state is separable. And by noting rate for the molecular electric dipole polarization P(z,t) ,
that separability in frequency implies separability in time.) and F(z,t) is a quantum Langevin fluctuation operator
Figure 6 shows a case in which the input field’s wave obeying the commutator [38]
function is non-separable. The wave function retains this
non-separability at the output, as it develops ‘echoes.’
  ⎡⎣ F(z,t), F † (z ',t ') ⎤⎦ = 2γ δ (t − t ')δ (z − z ')  .                    (67)  
 
 
Integrate the equation for the dipole polarization, assuming
the molecular damping is fast, making the absorber broad
band:
t

P(z,t) = ∫ dt 'e
− γ (t−t ')
( − β C(z,t ') + F(z,t '))
  −∞
             (68)  
≈ −(β / γ )C(z,t) + FP (z,t) ,
 
where the effective Langevin fluctuation operator for the
dipole polarization is

  10  
 arXiv:1309.5915 [quant-ph] /Oct 5 2013

inversion and/or coherent control fields. This might be

t useful, for example, as a model for a quantum memory,
∫ dt 'e
− γ (t−t ')
  FP (z,t) ≡ F(z,t ')  .                              (69)   and will be considered in a following paper.
−∞

 
Calculate the commutator for the dipole fluctuation We thank Ian Walmsley, Josh Nunn, Steven van Enk,
operator: Kartik Srinivasan, and an anonymous reviewer for helpful
comments. MGR was supported by the National Science
⎡ FP (z,t), FP † (z ',t ') ⎤⎦ = exp(−γ | t − t ' |)δ (z − z ') Foundation, EPMD (ENG) and AMOP (Physics).
  ⎣          (70)  
→ 2γ δ (t − t ')δ (z − z ') ,
  Appendix 1: Commutator for cavity field
where the final step results from assuming the molecular
damping is fast. That is, in the limit we consider, one can To verify Eq. (25), write
idealize the P-fluctuations as delta-correlated. Absorber
models were introduced previously in I-O theory. [4, 5] ⎡⎣C(0 + ,t),C † (0 + ,t ') ⎤⎦ =
Here it leads to the modified cavity-field propagation ∞ ∞
equation, = τ 2 ∑ ∑ ρ m ρ n ⎡⎣ A ( t − nT ) , A† ( t '− mT ) ⎤⎦ (A1)
n=0 m=0

  ( ∂t + v ∂ z ) C(z,t) = −(αβ / γ )C(z,t) + α FP (z,t)  ,          (71)   ∞ ∞

= τ 2 ∑ ∑ ρ n+mδ (t − t '− (n − m)T ) .
  n=0 m=0

where αβ / γ plays the role of the attenuation rate of the

cavity field. The key point is that dissipative loss always For t ≥ t ' it is required that n ≥ m ; so
brings with it additional fluctuations, and these are
accounted for by the Langevin operator. ⎡⎣C(0 + ,t),C † (0 + ,t ') ⎤⎦t≥t ' =
It is straightforward to solve Eq. (71), along with the ∞ n
boundary conditions Eqs. (2) and (3), in steady state to = τ 2 ∑ ∑ ρ n+mδ (t − t '− (n − m)T )
study the effects of attenuation and fluctuation on the n=0 m=0

cavity and output fields. We leave this as an exercise. ∞ ∞ (A2)

=τ 2
∑∑ ρ δ (t − t '− kT )
k+2 s

k=0 s=0
6. Discussion ∞
= ∑ ρ kδ (t − t '− kT ) ,
The main results of this study are: (1) Showing that k=0
the standard classical field propagation equations, Eqs. (1)-
(3), provide a proper quantum mechanical description of where we used τ 2 + ρ 2 = 1 and the general relation
input, cavity and output fields when the input coupling
strength takes on arbitrary values; (2) Deriving explicit ∞ n ∞ ∞
formulas for Green functions and commutators for the ∑ ∑ f (n + m)g(n − m) =∑ ∑ f (k + 2s)g(k) . (A3)
three fields in both space-frequency and space-time n=0 m=0 k=0 s=0

domains; (3) Confirming that the commutator Eq. (26)

agrees with the fundamental equal-time field commutator For t ≤ t ' , it is required that n ≤ m ; so
Eq. (27), which must always be respected, and (4) Deriving
the effects of a reflecting cavity on a two-photon wave- 0

packet state.
⎡⎣C(0 + ,t),C † (0 + ,t ') ⎤⎦t≤t ' = ∑ρ −k
δ (t − t '+ kT ) . (A4)
k=−∞
The I-O theory formulated here is amenable to
inclusion of intracavity absorbing and emitting atoms. Combining the two cases gives:
Given that the equal-time field commutator agrees with the
fundamentally required one Eq. (27) [7], the effects of
atoms in the cavity may be accounted for by using the ⎡⎣C(0 + ,t),C † (0 + ,t ') ⎤⎦ =
standard minimal-coupling atom-field interaction ∞ ∞ . (A5)
Hamiltonian. The theory can also account for dynamical = ∑ρ δ (t − t '+ kT ) =
|k|
∑ρ δ (t − t '− kT )
|k|

absorbing media in the cavity by generalizing the equation k=−∞ k=−∞

of motion for the dipole polarization P(z,t) in Eq. (66) to

include multilevel media with or without population

  11  
 arXiv:1309.5915 [quant-ph] /Oct 5 2013

Appendix 2: Commutator for output field

And the output one is:
To verify Eq. (34), use Eqs. (2) and (3) to write Φ out (t1 ,t 2 ) = vac B(t 2 )B(t1 ) Ψ =

B(t) = −
1 τ
A(t) + C(0 + ,t) . (A6)
= ∫ dt ∫ dt ' G(
 ab (t − t )G
1
 ab (t '− t ) + G
2
 ab (t − t )G
2
 ab (t '− t ) ψ (t,t ')
1 )
ρ ρ
(A12)
Use this and Eq. (28) to write This can be written as

⎡⎣ B(t), B† (t ') ⎤⎦ = Φ out (t1 ,t 2 ) = ∫ dt ∫ dt 'G  ab (t '− t ) (ψ (t,t ') + ψ (t ',t))

 ab (t − t )G
1 2

1 τ   (A13)
= 2 ⎡⎣ A(t), A† (t ') ⎤⎦ − 2 ⎡⎣C(0 + ,t), A† (t ') ⎤⎦
ρ ρ
Appendix 4: Separability of two-photon state
τ τ2
− 2 ⎡⎣ A(t),C † (0 + ,t ') ⎤⎦ + 2 ⎡⎣C(0 + ,t),C † (0 + ,t ') ⎤⎦
ρ ρ Proof that if the state is separable at the input, then it is
1 τ 2 ∞ separable at the output: If Φ(t,t ') = φ1 (t)φ2 (t ') , then from
= δ (t − t ') − 2 ∑ ρ nδ (t − t '− nT ) Eq. (59),
ρ2 ρ n=0
τ2 ∞ n τ2 ⎛ ∞ ⎞
2 ∑
− ρ δ (t − t '+ nT ) + 2 ⎜ ∑ ρ |k|δ (t − t '− kT ) ⎟   Φ out (t1 , t 2 ) = ψ 1out (t1 )ψ 2out (t 2 )                            (A14)  
ρ n=0 ρ ⎝ k=−∞ ⎠
 
1 τ2 ⎛ ∞
⎞ where
= 2 δ (t − t ') − 2 ⎜ δ (t − t ') + ∑ ρ nδ (t − t '− nT )⎟
ρ ρ ⎝ n≠0 ⎠  ab (t '− t)φ (t)
ψ 1out (t) = ∫ dt 'G 1

τ ⎛
2 ∞
⎞ τ2 ∞ n
− δ (t − t ') + ∑ ρ nδ (t − t '+ nT )⎟
2 ⎜
ρ ⎝ n≠0 ⎠ = − ρφ1 (t) + ∑ ρ φ1 (t − nT ) ,
ρ n=1
               (A15)  
τ2 ⎛ ∞ ∞
⎞  ab (t '− t)φ (t ')
ψ 2out (t) = ∫ dt 'G
+ 2 ⎜
ρ ⎝
δ (t − t ') + ∑ ρ n
δ (t − t '− nT ) + ∑ ρ |k|δ (t − t '+ kT ) ⎟
⎠
2
n≠0 k≠0
τ2 ∞ n
⎛ 1 τ2 ⎞
= ⎜ 2 − 2 ⎟ δ (t − t ') = δ (t − t ')
= − ρφ2 (t) + ∑ ρ φ2 (t − nT )
ρ n=1
⎝ρ ρ ⎠
Plotting this form Eq. (A14) for the example shown in Fig.
(A7) 5 gives results identical to those shown there.

Appendix part 3: Two-photon correlation function

To verify Eq. (58), write References

B(t 2 )B(t1 ) Ψ = [1] M. J. Collett and C. Gardiner, Phys. Rev. A 30, 1386
∞ ∞
,      (A8)   (1984).
= ∫ dτ ∫ dτ ' F(τ ,τ ')B(t
−∞ −∞
2 )B(t1 )B† (τ )B† (τ ') vac [2] C. W. Gardiner and C. M. Savage, A multimode
quantum theory of a degenerate parametric amplifier in a
cavity, Opt. Commun., 50, 173 (1984)
where
 ab (t − τ )G
F(τ , τ ') = ∫ dt ∫ dt 'G  ab (t '− τ ')ψ (t,t ') .        (A9)   [3] B. Yurke, Phys. Rev. A 29, 408 (1984).
[4] Carlos Viviescas and Gregor Hackenbroich, Field
Repeated use of commutators gives quantization for open optical cavities, Phys. Rev. A 67,
013805 (2003)
B(t 2 )B(t1 ) Ψ = ( F(t1 ,t 2 ) + F(t 2 ,t1 )) vac (A10)   [5] M. Khanbekyan, L. Knöll, D.-G. Welsch, A. A.
Semenov, and W. Vogel, QED of lossy cavities: Operator
The input two-photon function is: and quantum-state input-output relations, Phys. Rev. A.,
72, 053813 (2005).
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