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The QASER revisited: insights gleaned from analytical solutions to simple models
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2014 Laser Phys. 24 094014
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Laser Physics
Astro Ltd
Laser Phys. 24 (2014) 094014 (6pp)
doi:10.1088/1054-660X/24/9/094014
The QASER revisited: insights gleaned from
analytical solutions to simple models*
Marlan O Scully
Princeton University, Princeton, NJ 08544, USA Texas A&M University, College Station, TX 77843, USA
Baylor University, Waco, TX 76798, USA
E-mail: [email protected]
Received 8 March 2014
Accepted for publication 6 April 2014
Published 15 August 2014
Abstract
Lasers and masers typically require population inversion. But with phase coherent atoms
(phasers), we get lasing without inversion (e.g. 10% of the atoms excited). However, in recent
work we found that it is possible to get coherent light emitted with no atoms excited, via
Quantum Amplification of Superradiant Emission of Radiation (QASER). In particular, we
found that by utilizing collective superradiant emission, we can generate coherent light at high
frequency in the UV or x-ray bands by driving the atomic system with lower frequency source.
Here, we present a simple analysis based on near-resonant QASER operation and on a multiphoton Hamiltonian obtained by, e.g. a canonical transformation.
Keywords: multiphoton Hamiltonian, quantum amplification of superradiant emission of
radiation
1. Introduction
motivation for our QASER studies was the interesting connection between the single atom maser and single photon
superradiance. The equations for the probability of finding an
atom excited in these two problems are intriguingly similar, as
explained in table 1.
The Nphoton which appears in the single atom maser
expression is the essence of stimulated emission and is the
harbinger of maser-laser Light Amplification by Stimulated
Emission of Radiation. On the other hand, the Natom which
appears in the single photon entry in table 1 is the essence of
superradiant emission. To make a laser, we must add a drive
field and go to the many-atom case. One naturally asks: by
adding a drive field and going to the many photon limit can we
likewise get superradiant amplification? The answer is 'yes'
and we proceed by extending the usual Maxwell–Schrödinger
equation to treat the problem of many-atom cooperative emission, i.e. driven superradiance.
Lasing without inversion (LWI) is a quantum phase sensitive
process requiring some population in the excited state and careful attention to the relative atomic decay rates [1–3] between
the states. More recently, we have been studying transient LWI
[4–6] to overcome the decay rate road block. We have also been
investigating a form of Quantum Amplification by Superradiant
Emission of Radiation (QASER) [7], which is free from decay
rate constraints and requires no population in the excited state.
The physics behind QASER operation is counter-intuitive,
and an electronic RLC oscillator analogue has been developed
which supports the basic concept and provides insight.
A key feature of QASER operation is the frequency upconversion of a lower frequency drive. Here, we present a simple
scheme which sheds light on a QASER operation. The model
is depicted in figure 1. In particular, we develop a model which
embodies multiphoton operation, and makes a QASER operation more transparent by taking advantage of the simplicity of
near-resonant operation.
In the next few paragraphs, we discuss the physics behind
collective many-atom emission. As mentioned in [7], one
2. QASER physics from the Maxwell–Schrödinger
equations
Starting with the Maxwell equation for a superradiant field
of amplitude εs(z, t) with frequency νs and wavevector ks as
driven by the polarization P, we write
* It is a pleasure to dedicate this paper to prof I Yevseyev—a true laser
physics pioneer. Our field is richer and our path is wider because of him.
1054-660X/14/094014+6$33.00
1
© 2014 Astro Ltd Printed in the UK
M O Scully
Laser Phys. 24 (2014) 094014
Figure 1. (a) Strong low frequency field drives atoms which cooperatively radiate counter propagating field. (b) Atoms driven from ∣b〉 to
∣a〉 by multiphoton absorption while levels ∣a1〉 and ∣a2〉 are coupled nonresonantly, drive and signal frequencies are νd and νs, respectively.
The drive field may be a single photon or an effective multi-photon field as outlined in the appendix.
Table 1. Table comparing single atom maser and single photon superradiance in which: ℘ is the dipole matrix element, ℏν is the photon
energy, ϵ0 is the vacuum permeability and t is the time.
Probability of atomic excitation
Key variables V and N
Single atom maser
sin2
℘
ℏ
ℏν
ϵ 0V
Nph t
V = E.M. cavity volume, Nph = number of photon in cavity
Single photon superradiance
sin2
℘
ℏ
ℏν
ϵ 0υ
Nat t
υ = atomic cloud volume, Nat = number of atoms in cloud
⎛ ∂2
1 ∂2 ⎞
⎜ 2 − 2 2 ⎟ εs(z , t )e−i(νt − kz ) = μ0 νs2P(z , t )
(1)
⎝ ∂z
c ∂t ⎠
Noting that μ0c2 νs = νs/ϵ0 and using
1 4ν 3℘ 2
γ=
,
(6)
4πε0 3ℏc 3
⎛∂
1 ∂ ⎞⎛ ∂
1 ∂⎞
N
⎜
⎟⎜
⎟ εs(z , t )e−i(νt − kz ) = + μ0 νs2℘ ρab
−
+
⎝ ∂z
V
c ∂t ⎠⎝ ∂z
c ∂t ⎠
(2)
allows us to reduce equation (5) to the final working form
⎛∂
⎡ 3
N⎤ s
∂⎞
⎜ + c ⎟ Ωs(z , t ) = − i ⎢ γλ2c ⎥ ρab
.
(7)
⎣
⎝ ∂t
⎠
8π
V⎦
∂z
where the (complex) macroscopic polarization of the
N
superradiant ensemble is given by P = ℘ ρab in which ℘
V
is the dipole matrix element and the off-diagonal density
matrix for the resonant superradiant collective emission is
given by
And we define the collective many-atom atomic frequency by
3 2 N
Ωa2 =
γλ c .
(8)
8π
V
To make a connection with the many-atom single photon
collective emission of table 1, let us write the single photon
℘ ℏν
Nat in terms of the radiative
superradiant frequency
ℏ ϵ 0v
decay rate γ, which yields
s
ρab (z , t ) = ρab
(z , t )e−i(νst − ksz ).
(3)
We proceed by making the usual slowly varying approximation so that equation (2) becomes
⎛∂
1 ∂⎞
N s i(vt − kz )
⎟ εs(z , t ) = + μ0 νs2℘ ρab
2iks⎜
+
e
.
(4)
⎝ ∂z
V
c ∂z ⎠
℘
ℏν
3 2 Nat
λ γc
Nat =
.
(9)
4π
v
ℏϵ 0 ϵ 0v
This is the same collective frequency as in equation (8) modulo, the factor of 2 encountered when connecting radiation
field amplitude and photon production rates.
Note that the working radiation expression equation (7)
for the (resonant) field Ωs is governed by the external drive
field Ωd as is discussed in detail in the following. A key point
Combining terms and using equation (3) yields
⎛∂
μ cνs2 ℘ 2 N s
∂⎞
⎜ + c ⎟ Ωs(z , t ) = − i 0
ρab ,
(5)
⎝ ∂t
2ks ℏ V
∂z ⎠
where Ωs = ℘εs/ℏ.
2
M O Scully
Laser Phys. 24 (2014) 094014
is that the drive field is detuned from the atomic transition
frequency ωab = νs, and is taken as being much stronger than
the superradiant field in the present analysis.
We proceed to write the density matrix ρab which obeys the
usual equation of motion in the conventional notation
.
ρab = − iωabρab − i [Ωs e−i(νst − ksz ) + Ωd e−i(νdt − kdz )](ρbb − ρaa ).
(10)
s
Hence, we have the equation of motion for ρab
.s
= − iΔs ρab − i [Ωs + Ωd e−i [(νd − νs )t − (kd − ks )z ]](ρbb − ρaa )
ρab
(11)
where Δs = ωab − νs . From (8) and (11), we have
∂⎞
∂⎛∂
s
⎜ + c ⎟ Ωs = − iΩa2Δs ρab
∂t ⎝ ∂t
∂z ⎠
− Ωa2[Ωs + Ωd e−i [(νd − νs )t − (kd − ks )z ]](1 − 2ρaa )
Figure 2. Shape of the superradiant pulse after time t = 100/Ωa
obtained by numerical solution for Δ = 2Ωa and Ω3/Ωa = 0.3, 0.5 and
0.7 (solid lines). Dashed line indicates the initial seed pulse (t = 0).
Vertical axis has logarithmic scale.
(12)
where we have used ρaa − ρbb = 1. Since in a phase matched
medium ks = kd, the second term in the square brackets represents conventional parametric conversion.
Ignoring for the moment the forward parametric term, we
get a QASER equation of motion by noting
We proceed by differentiating both sides of (16) and use
equation (17) to obtain
i ⎛∂
∂⎞
s
ρab
= 2 ⎜ + c ⎟ Ωs
(13)
∂z ⎠
Ωa ⎝ ∂t
(
Likewise, we may obtain a polarization equation of motion by
differentiating (17) and using (16) to write
and use the Rabi result for the drive field
1⎛Ω ⎞
ρaa = ⎜ d ⎟ (1 − cos2μ(t − z / v ))
(14)
2⎝ μ ⎠
(
)
s
d
d
s
= − Ωa2 ρbb
− ρaa
ρ¨ab
ρab
(19)
2
where we have ignored the smaller time derivatives going as
d
d
Ωs ρ˙bb
− ρ˙aa
.
d
d
We use equation (14) to write ρbb
− ρaa
= 1 − η cos 2μt
2
where η = (Ωd/μ) , and thus obtain the same gain equation for
both atoms and photons, namely
(
where μ = Ωd2 + Δd2 /4 with Δd = ω − νd. From (12)–(14)
(ignoring the Ωd term in equation (12)), we have the QASER
equation of motion
⎛∂
⎞⎛ ∂
∂⎞
⎜
+ iΔs ⎟⎜ + c ⎟ Ωs
⎝ ∂t
⎠⎝ ∂t
∂z ⎠
⎡
⎛ Ω ⎞2
= − Ωa2⎢1 − ⎜ d ⎟ +
⎝ μ ⎠
⎢⎣
)
d
d
Ω¨ s = − Ωa2 ρbb
− ρaa
Ωs.
(18)
)
(
)
X¨ = − Ωa2(1 − η cos 2μt )X
(20)
⎛ Ωd ⎞2
⎛
z ⎞⎤
⎜ ⎟ cos 2μ⎜t − ⎟⎥Ωs.
⎝
v ⎠⎥⎦
⎝ μ ⎠
s
where X = Ωs or ρab
.
The solution to equation (20) can be written in the form
(15)
X (t ) = x+eiωt + x−e−iωt
(21)
which upon assuming ω = Ωa and neglecting the small x¨±
terms yields
Equation (15) is essentially the QASER equation of motion
expressed in the form of a Mathieu equation. The backward propagation aspect of QASER operation as depicted in
figure 2 is given in [8].
ηΩ 2
x˙+ = − i a x−
(22)
4ω
ηΩ 2
x˙− = i a x+
(23)
4ω
3. QASER gain
To gain insight into QASER dynamics, we write equation (7)
for the case of no z-dependence and equation (11) for the case
of Δs = 0 while ignoring the parametric conversion term going
as Ωd. Then, we have the simple coupled field-matter equations
and therefore
x¨± = Gx± ,
(24)
where the gain G is given by
s
Ω˙ s = − iΩa2ρab
(16)
(
η
G = Ωa ,
(25)
4
)
s
d
d
ρ˙ab
= − iΩs ρbb
− ρaa
(17)
and as before we have assumed Ωa = ω. It is important to note
that both the field oscillator and the collective atom oscillator
experience gain.
where we have indicated, by the superscript ‘d’, that the populations ρaa(ρbb) are governed by the driving field.
3
M O Scully
Laser Phys. 24 (2014) 094014
Acknowledgments
(b)
(a)
The National Science Foundation Grant has supported this
work PHY-1241032 (INSPIRE CREATIV) and the R A
Welch Foundation (Awards A-1261). It is a pleasure to thank
O Kocharovskaya, Y Rostovtsev, W Schleich, G Shchedrin,
A Svidzinsky, D Wang and L Yuan for useful discussions.
Figure A1. (a) Odd number of drive photons couple, for example,
the 1S and 2P levels of hydrogen. (b) Even photon number couples
for example 1S and 2S.
Appendix. Multiphoton Hamiltonian analysis
Using the Gibson approach [9], the Hamiltonian in the interaction picture is given by
eiΘ = ∣b⟩⟨b ∣+ cos θ (∣1⟩⟨1 + 2⟩⟨2∣)
Hint (t ) = V12(t ) + Vs(t ) + Vd (t )
(A.1)
V12(t ) = ℏα12[∣2⟩⟨1∣ + adj.] sin (νd t − k dz )
(A.2a)
The essential operator product as it appears in equations (A.2b)
and (A.2c) then reads
Vs(t ) = ℏα v∣1⟩⟨b∣eiωt aŝ e−i(νst − ksz ) + adj.
(A.2b)
+ i sin θ (∣1⟩⟨2 + 2⟩⟨1∣).
(A.9)
̂
̂
(A.10)
eiΘ ∣1⟩⟨b∣e−iΘ = cos θ∣1⟩⟨b ∣+ i sin θ∣2⟩⟨b∣,
i ωt
Vd (t ) = ℏαd∣1⟩⟨b∣e sin (νd t − k dz ) + adj.
(A.2c)
and so from equations (2), (7) and (8), we have
where α12 is the Rabi frequency of the drive field coupling ∣1〉,
∣2〉 and ℏαv is the vacuum Rabi coupling between ∣a1〉, ∣b〉;
note that we usually use the notation ∣a〉, ∣b〉 for the laser levels
[9]. Here, the upper level ∣a〉 is replaced by ∣a1〉 and ∣a2〉, but
we will call the upper levels ∣1〉 and ∣2〉 for convenience. The
other symbols have their usual meaning, e.g. α v = ℘abE 0/ ℏ
where ℘ab is the dipole matrix element and E 0 = ℏωab / ϵ 0V
is the electric field per photon of the weak signal field which
is resonant with the atomic frequency ωab and aŝ is the signal
field annihilation operator. The strong drive field of frequency
νd and strength Ed is treated classically and the corresponding
Rabi frequency is αd.
We proceed by transforming V1,2 away such that the
Schrödinger equation
vs ̂ = ℏα v (cos θ∣1⟩⟨b ∣+ isin θ∣2⟩⟨b∣) as†̂ eiωt e−i(νst − ksz ) + adj.
(A.11a)
vd̂ = ℏαd (cos θ∣1⟩⟨b ∣+ i sin θ∣2⟩⟨b∣) eiωt sin (νdt − k dz ) + adj.
(A.11b)
Recalling that θ = x cos φ (with x = − Ω12/ν) and using the
Bessel generating functions
∞
cos(
x cos φ) = J0(x ) + 2 ∑ (−)ℓ J2ℓ(x ) cos 2ℓφ,
(A.12a)
ℓ= 1
∞
sin (x cos φ ) = 2 ∑ (−)ℓ J2ℓ+ 1(x ) cos[(2ℓ + 1)φ]
(A.12b)
i
Ψ˙ = − Hint Ψ
(A.3)
ℏ
ℓ= 0
yields
is replaced by
cos θ 1⟩⟨b + i sin θ 2⟩⟨b∣
∞
⎧
⎫
= ⎨J0(x ) + 2 ∑ ( − )ℓ J2ℓ(x )cos 2ℓ(νd t − k dz )⎬∣1⟩⟨b∣
⎩
⎭
ℓ= 0
i
Φ˙ = − (vd̂ (t ) + vs(̂ t ))Φ,
(A.4)
ℏ
where
̂
(A.5)
Ψ = e−iΘ (t )Φ,
with
(A.6)
Θ (̂ t ) = θ (t )(∣1⟩⟨2∣ + adj.),
(A.7)
vd̂ (t ) + vs(̂ t ) = eiΘ (t )(Vd (t ) + Vs(t ))e−iΘ (t ).
⎪
⎧ ∞
⎫
+ ⎨2 ∑ ( − )ℓ J2ℓ+ 1(x )cos (2ℓ + 1)(νd t − k dz )⎬∣2⟩⟨b∣.
⎩ ℓ= 0
⎭
(A.13)
⎪
⎪
⎪
⎪
∞
⎧
⎫
vs(̂ t ) = ℏα v ⎨J0(x ) + 2 ∑ (−)ℓ J2ℓ(x )cos 2ℓ(νd t − k dz )⎬ ∣ 1⟩
⎩
⎭
ℓ= 1
We expand the exponential operator in equation (5)
̂
⎪
⎪
From equations (13) and (A.11a, A.11b), we distinguish the
small processes involving an odd and even number of photons
as depicted in figure A1 and listed in equations (A.14a, A.14b)
and (A.15a, A.15b) as
odd
where θ(t) = (α12/νd)cos(νdt − kdz). In such a case, the transformed drive and signal field interaction Hamiltonian are
given by
eiΘ = 1 + iθ (t )(∣1⟩⟨2 + 2⟩⟨1∣)−
⎪
θ (t )
(∣1⟩⟨1 + 2⟩⟨2∣) + ....
2
2
(A.8)
and by using 1 = ∣b〉〈b∣ + ∣1〉〈1∣ + ∣2〉〈2∣ we write this as
⎪
⎪
⎪
⎪
⟨b ∣ eiωt aŝ e−i(νst − ksz ) + adj .
(A.14a)
4
M O Scully
Laser Phys. 24 (2014) 094014
∞
⎧
⎫
⎨
vd̂ (t ) = ℏΩd J0(x ) + 2 ∑ (−)ℓ J2ℓ(x )cos 2ℓ(νd t − k dz )⎬ ∣ 1⟩
⎩
⎭
ℓ= 1
⎪
⎪
⎪
⎪
where p = 2m + 1 as it appears in equation (20) with
Δp = ω − pνd, and the effective coupling frequency for odd p is
2p
Ωp = αd Jp(x ),
odd.
(A.22)
x
⟨b ∣ eiωt sin (νd t − k dz ) + adj .
(A.14b)
even
Likewise, for an even number of drive photons we take from
equation (A.15b) terms having ℓ = m which go as 2mνd ≈ ω.
Such a term is proportional to
⎧ ∞
⎫
vs(̂ t ) = ℏα v ⎨2 i ∑ (−)ℓ J2ℓ+ 1(x )cos (2ℓ + 1)(νd t − k dz )⎬ ∣ 2⟩
⎩ ℓ= 0
⎭
⎪
⎪
⎪
⎪
eiνt − kz iωt
e ,
2i(−)m J2m + 1e−i(2m + 1)(νt − kz )
(A.23)
2i
⟨b ∣ eiωt aŝ e−i(νst − ksz ) + adj .
(A.15a)
and from the term in the series for which ℓ = m − 1, we have
another term going as 2mνd
⎧ ∞
⎫
vd̂ (t ) = ℏΩd ⎨2 i ∑ (−)ℓ J2ℓ+ 1(x )cos (2ℓ + 1)(νd t − k dz )⎬ ∣ 2⟩
⎩ ℓ= 0
⎭
⎪
⎪
⎪
⎪
e−i(νt − kz ) iωt
e ,
2i(−)m − 1J2m − 1e−i(2m − 1)(νt − kz )
(A.24)
−2i
⟨b ∣ eiωt sin (νd t − k dz ) + adj .
(A.15b)
so that equation (A.15b) now reduces to
Consider the case of a hydrogen atom driven by an odd number of photons to an excited Rydberg state. The associated
atomic frequency ω is multiphoton resonant with two terms in
the sum of equation (A.14b). For example, let us take the term
in equation (A.14b) with ℓ = m which goes as
Hence from equations (A.15b), (14) and (A.25), we have
vd̂ (t )
= (−)m [J2m + 1(x ) + J2m − 1(x )]ei(ω − 2mν )t ei2mkz .
(A.25)
ℏαd
4m
vd̂ (t )
J2m(x )eiΔ2m t ei2mkz ∣ 2⟩⟨b ∣ +adj . ,
= (−)m
x
ℏ
α
d
2(−)m J2mcos 2m(νd t − k dz )sin (νd t − k dz )eiωt
≈ 2(−)m J2me−i2mνdt
e−iνdt iωt i(2m + 1) kdz
e e
,
−2i
even
(A.26)
so for 2m = n drive photons, we have the even number effective Rabi frequency
(A.16)
in which (2m + 1)νd ≈ ω. Likewise there is a term in (A.14b)
with ℓ = m + 1 which is nearly resonant, that is
2n
Ωn = αdi n Jn(x ),
even.
(A.27)
x
eiνdt iωt i(2m + 1) kdz
e e
.
−2i
(A.17)
A couple of points should be made: x = Ω12/νd is much larger
than αd = Ω1b/ω since the Rydberg matrix elements 〈1∣r∣2〉 are
larger than the direct matrix elements 〈1∣r∣b〉. Furthermore, νd
(e.g. IR) is substantially smaller that ω (e.g. UV).
The Hamiltonian for the near-resonant (ω ≅ νs) interaction
from equations (A.11a) and (A.12a) is given by
vd̂ (t )
(−)m
≅
[J2m(x ) + J2m + 2(x )]ei(ω − 2(m + 1)ν )t ei(2m + 1) kdz ∣ 1⟩
ℏαd
i
(A.18)
vs ̂ = ℏΩs∣1⟩⟨b∣eiωt e−i(νst − ksz ) + adj. ,
(A.28)
2( − )m + 1J2m + 2 cos 2(m + 1)(νd t − k dz ) sin(νd t − k dz )eiωt
≈ 2( − )m + 1J2m + 2e−i2(m + 1) νdt
Thus, equation (A.14b) reduces to
⟨b ∣ +adj .
where Ωs = J0(x ) α v n (t ) with n (t ) being the average number
of photons in the superradiant pulse at time t.
Thus far, the effort has focused on one three-level atom
driven by a strong drive field having Rabi frequency αd. We
may apply this to the many-atom QASER, e.g. for the case of
odd (e.g. three) photon absorption, we find from equation (22)
with p = 3. We have
Recalling the recursion relation for Bessel functions
2k
Jk − 1(x ) + Jk + 1(x ) =
Jk(x ),
(A.19)
x
and taking 2m = k − 1 in equation (18), we have
vd̂ (t )
(−)m 2(2m + 1)
≅
J2m + 1(x )ei [ω − (2m + 1) νd ]t ei(2m + 1) kdz ∣ 1⟩
x
ℏαd
⟨b ∣ +adj .
(A.20)
6
Ωd = αd J3(x ).
(A.29)
x
In the limit of small x, Jp(x) ∼ (x/2)p/p! and
x2
Ωd ~αd .
(A.30)
8
where we have neglected overall phase factors. Thus, the
effective Hamiltonian for the absorption of p drive photons
may be written as
Recalling that x = Ω12/νd and considering the case 1S → nP
3
1
and using ⟨nP∣r∣1S⟩ ≅ n2a 0 and ⟨2P∣r∣1S⟩ = a 0, then
2
2
vd̂ = ℏΩp eiΔp t eipkdz∣1⟩⟨b∣ + adj. ,
odd
(A.21)
5
M O Scully
Laser Phys. 24 (2014) 094014
Ω12 ∼ 3n αd. Hence, if αd ∼ 10 and νd ∼ 10 , then for n ∼ 10
we have x ≲ 1 and Ω ∼ 1012.
2
13
[4] Kilin S Y, Kapale K T and Scully M O 2008 Phys. Rev. 100
173601
[5] Svidzinsky A A, Yuan L and Scully M O 2013 New J. Phys.
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6

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