PHYSICAL REVIEW RESEARCH 7, L032004 (2025)

Letter

Quantum Lamb model

Dennis P. Clougherty and Nam H. Dinh

Department of Physics, University of Vermont, Burlington, Vermont 05405-0125, USA

(Received 27 March 2025; accepted 17 June 2025; published 7 July 2025)

H. Lamb considered the classical dynamics of a vibrating particle embedded in an elastic medium before the
development of quantum theory. Lamb was interested in how the back action of the elastic waves generated can
damp the vibrations of the particle. We propose a quantum version of Lamb’s model. We show that this model
is exactly solvable by using a multimode Bogoliubov transformation. We find that the exact system ground state
is a multimode squeezed-vacuum state, and we obtain the exact Bogoliubov frequencies by numerically solving
a nonlinear integral equation. A closed-form expression for the damping rate of the particle is obtained, and it
agrees with the result obtained by perturbation theory. The model provides a solvable example of the damped
quantum harmonic oscillator.

DOI: 10.1103/9fxx-2x6n

Introduction. Advances in the fabrication and characteri- tions as a single, effective environmental degree of freedom,
zation of simple mechanical systems in the nanoscopic and but the connection to the microscopic physics is not made.
mesoscopic regime have facilitated experimental and the- Caldeira and Leggett [13] separate the system into a sum
oretical investigations [1,2] into some of the foundational of two subsystems (oscillator and bath) plus an interaction.
principles of quantum mechanics. Prominent examples of Using a path-integral description, the bath degrees of freedom
such systems include vibrating beams and mirrored surfaces can be integrated out to give a general quantum formulation
that interact with laser light through its radiation pressure of dissipative systems. Yurke [14] specifically considered a
(optomechanics) [3], mechanical resonators coupled to elec- Lamb-type model that is a special case of the model consid-
tronic devices (nanoelectromechanics) [4,5], and interacting ered here. (We will recover Yurke’s results by allowing the
mechanical resonators (quantum acoustodynamics) [6–8]. In spring that couples the bead motion to the string to be suitably
addition to providing a path to explore quantum science and stiff.) Yurke considered a string with a point mass at one end.
the limits of precision measurement, such systems might be The point mass is also coupled to a spring with a fixed end.
used to fashion new quantum sensors and devices for manip- The mass-loaded string then has a time-dependent boundary
ulating quantum information [9,10]. condition. As a result, the normal modes are nonorthogonal.
We consider a mechanical system whose first study pre- Yurke overcame this by finding an appropriate weighting fac-
dates the development of quantum mechanics. In 1900, Lamb tor to use in redefining the inner product so that generalized
[11] considered the dynamics of a vibrating particle embedded orthogonality can be applied. He then quantized the model in
in an elastic medium. The back action of the elastic waves the standard way.
generated by the vibrations of the particle work to damp Following Caldeira and Leggett [13], the model considered
those vibrations creating a damped harmonic oscillator. In this in this work expresses the Lamb Hamiltonian as a sum of two
work, we study a quantum version of Lamb’s model and focus subsystems (oscillator and string) plus a coupling term. Since
on the dynamics of the vibrational decay. Figure 1 shows a the coupling is bilinear in operators, the Hamiltonian is ex-
schematic consisting of a vibrating bead coupled by a spring actly diagonalizable with the use of a multimode Bogoliubov
to a long string under tension that serves as the classical basis transformation. We find explicit expressions for the coeffi-
of the model. cients that diagonalize the Hamiltonian. Using the symplectic
There have been other formulations of the damped quan- properties of the transformation, we confirm that our results
tum harmonic oscillator. Feshbach and Tikochinsky [12] satisfy the necessary identities. We then derive a nonlinear
introduced an auxiliary variable into the Lagrangian of a har- equation whose solution yields the Bogoliubov frequencies,
monic oscillator to get the desired effective equation of motion and we use it to numerically calculate the symplectic spectrum
for the damped oscillator. They then proceeded by canonical of the model.
quantization to obtain a quantum description of the damped We show that the ground state of the quantum Lamb
harmonic oscillator. The auxiliary variable presumably func- model is a nonclassical state—a multimode squeezed-vacuum
state—and we relate this ground state to the uncoupled states
(transverse phonons of the string and vibrons of the bead) of
the system. Squeezed states can serve as a quantum resource
Published by the American Physical Society under the terms of the for precision-sensing applications; for example, gravitational-
Creative Commons Attribution 4.0 International license. Further wave detection relies on squeezing to perform displacement
distribution of this work must maintain attribution to the author(s) measurements where uncertainty in momentum is sacrificed
and the published article’s title, journal citation, and DOI. in favor of reduced uncertainty in position. Caldeira and

2643-1564/2025/7(3)/L032004(5) L032004-1 Published by the American Physical Society

DENNIS P. CLOUGHERTY AND NAM H. DINH PHYSICAL REVIEW RESEARCH 7, L032004 (2025)

The coupling parameters γn can be expressed in terms of

physical parameters of the model [15]
 
ν kn  1
γn = ωs  , (2)
ω0 (kn )2 + (ks )2 1 − ks 
(kn )2 +(ks )2
τ
where ωs = cks = cκτ c and ν = 2mc . The wave number kn is a
solution of the transcendental equation tan kn  = − κτc kn .
Bogoliubov transformation. We diagonalize the Hamilto-
nian in Eq. (1) with the use of a multimode Bogoliubov
transformation. We look for a linear transformation (and its
inverse) with the following form:

bα = (Mαβ aβ + Nαβ aβ† ), (3)
β

aα = (Uαβ bβ + Vαβ b†β ), (4)
β
FIG. 1. Schematic of a generalization of the classical Lamb
model. Bead of mass m at x = 0 is constrained to move in the vertical where b†α (bα ) creates (destroys) a Bogoliubov excitation (bo-
direction. The vibrating bead is coupled by a spring to a long string goliubon) and M, N, U, and V are (N + 1)-dimensional square
under tension τ . The vibrating bead creates transverse acoustic waves matrices whose elements are the coefficients of the transfor-
on the string (  c/ω0 ). The bead subsequently undergoes damped mation. (The string with length  is a system of N discrete
harmonic motion. atoms.)
We require that the transformation preserve the boson
commutation rules [bα , b†β ] = δαβ . As a result [16,17], the
Leggett [13] found, in their studies of the damped quantum coefficients can be grouped to form a 2(N + 1)-dimensional
oscillator, the uncertainty in position for the ground state is symplectic matrix T ∈ Sp(2(N + 1), R):
reduced with increasing damping, strongly so in the over-  
damped regime. This result is consistent with a squeezed M N
T≡ . (5)
ground state. N M
We then study the dynamics of the vibrational decay of
the bead. Dissipation emerges in the thermodynamic limit T then satisfies the symplectic condition TJTT = J where the
where the number of string modes N becomes infinitely large symplectic form J can be represented as
(N → ∞). In this limit, vibrational energy of bead can be  
00 1l
radiated away. We then obtain an explicit expression for the J≡ . (6)
decay rate, and we calculate the spectral distribution of single −1l 00
bogoliubon emission, a product of the bead decay. We show
A number of useful coefficient identities follow [15] from
that for weak coupling strength g, the decay rate calculated
the symplectic structure on this Fock space; for example, the
agrees with both the classical result and the golden rule.
inverse of the transformation matrix T can be obtained simply
Hamiltonian. The Hamiltonian of the system in Fig. 1 is
from the symplectic condition (together with J2 = −1l):


N 
N T−1 = −JTT J (7)
 T 
H= ωα aα† aα − (a0 + a0† ) γn (an + an† ), (1) M −NT
α=0 n=1 = . (8)
−NT MT
where an† (an ) creates (annihilates) a transverse acoustic Hence, we conclude that the coefficients of the inverse trans-
phonon on the string, and a0† (a0 ) creates (annihilates) a vi- formation satisfy U = MT and V = −NT . We summarize the
bron on the bead. [We use index notation where (greek) α = explicit form for the transformation in Table I. The detailed
0, 1, 2, . . . , while (roman) n = 1, 2, . . . , N, and work with calculation of the coefficients is outlined in the Supplemental
natural units where h̄ = 1.] N is the number of vibrational Material [15].
modes for the string. We are ultimately interested in the limit The following Hamiltonian results
N → ∞ in order to obtain a description of the damped bead 
H= α bα bα ,
†
(9)
oscillator. √ √ α
The frequency ω0 = (κ + κc )/m = ωb2 + ωc2 is the
bead vibrational frequency with the string fixed at x = 0, where the Bogoliubov frequencies { α} satisfy the following
while ωn are the vibrational frequencies of the string (tension summation equation
τ , length , mass density σ , transverse speed of sound c)  γq2 ωq
subject to a spring boundary condition at x = 0 and a fixed 2
α = ω02 + 4ω0 . (10)
condition at x = . q
2
α − ωq2

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QUANTUM LAMB MODEL PHYSICAL REVIEW RESEARCH 7, L032004 (2025)

TABLE I. Coefficients for the Bogoliubov transformation Mαβ

and Nαβ (α, β = 0, 1, . . . , N and k, q = 1, 2, . . . , N). Coefficients
for the inverse transformation can be obtained from the transposed
relations Uαβ = Mβα and Vαβ = −Nβα (see Supplemental Material
[15]).

(α0) (αk)
+ω0 0 γk √ 1
M √α  1
− ( 2ωα −ω  1
4ω0 α  γq2 ωq k ) 4ω0 α  γq2 ωq
1+4ω0 q ( 2 −ω2 )2 1+4ω0 q ( 2 −ω2 )2
α q α q
−ω0 0 γk
N √α  1
− ( 2ωα +ω √ 1  1
4ω0 α  γq2 ωq k) 4ω0 α  γq2 ωq
1+4ω0 q ( 2 −ω2 )2 1+4ω0 q ( 2 −ω2 )2
α q α q

By using the pole-expansion form (Mittag-Leffler) for cotan- τ

FIG. 2. Coupling strength g versus κc d
for N,  → ∞.
gent, it is straightforward to show that in the thermodynamic
ωd
limit (, N → ∞ and N → πc ), Eq. (10) gives Yurke’s tran-
scendental
√ equation [14] for the special coupling case of ωc = We verify this by operating on Eq.(14) with bα and us-
∗
ωc ≡ π νωd :
4
ing the identity S(−ξ )aα S(ξ ) = aα − β ξαβ aβ† . We find that
α
Eq. (14) is satisfied, provided the squeeze matrix has the value
2
α = ωb2 + 2ν . α cot (11)
c ξ = M−1 N. (15)
The frequency ωd is recognized as the Debye frequency of the
string (the high-frequency cutoff for the string). Hence, the ground state of the model is always a multimode
We note that the Hamiltonian is no longer positive definite squeezed-vacuum state [16,19] with squeeze parameter ξ de-
when the lowest Bogoliubov frequency vanishes. Thus, there termined by the Bogoliubov coefficients.
is a constraint on the model; namely, using Eq. (10), we see We note that the position uncertainty of the bead in the
that the following condition must be satisfied to prevent an ground state can be readily evaluated with the Bogoliubov
instability: coefficients. Evaluating the bead variance gives
4  γn2 1 
< 1. (12) {0}|u02 |{0} = (Mα0 − Nα0 )2 (16)
ω0 n ωn 2mω0 α
 γ2 1  1
We define the coupling strength g ≡ ω40 n ωnn and from =  . (17)
2m α γq2 ωq
Eq. (12) conclude that there is a critical coupling strength α 1 + 4ω0 q 2 −ω2 2
gc = 1 above which the model is ill defined. Using the form ( α q )
for γn in Eq. (2), we obtain an expression for g in the thermo- The sum can be evaluated analytically using complex-contour
dynamic limit integration [15] to reveal that the position uncertainty is re-
2 ωc 2
πτ duced with increasing damping ν, a result first obtained by
g= tan−1 , (13) Caldeira and Leggett [13] using the fluctuation-dissipation
π ω0 κc d theorem.
where d = /N, the interatomic distance between atoms in To better understand the ground state, the average number
the string. With increasing string tension τ , g asymptotically of uncoupled excitations (phonons and vibrons) in the mode
approaches g = ( ωω0c ) , a quantity bounded by 1 (see Fig. 2).
2 α contained in the coupled ground state |{0} can also be
Thus, the model is stable over the range of physical param- expressed in terms of Bogoliubov coefficients, with
eters. (As a practical matter, before the tension τ becomes 
comparable to interatomic forces in the string, it is likely that nα = 2
Nβα . (18)
the string would break.) β

Multimode squeezed vacuum. Eigenstates of the system An example is given in Fig. 3 for a coupling strength of g =
can be labeled by the set of Bogoliubov excitation num- 0.7. There is a small fraction of a vibron contributed by the
bers for theN + 1 modes  |{nα } with corresponding energies bead (α = 0), with an equal total amount of phonons on the
E ({nα }) = α nα α + 21 α ( α − ωα ). The ground state of string approximately uniformly distributed across the modes
the coupled system |{0} can be constructed from the un- at this coupling strength.
coupled  ground state | 0 with the squeeze operator S(ξ ) = Vibron decay. We now consider the dynamics of the vibra-
exp(− 21 αβ ξαβ aα† aβ† ): tional decay of the bead. We start in the ground state |{0} ,
|{0} = N S(ξ )| 0 . (14) displace the bead by δ to create the initial state |(0) =
exp(−ip0 δ)|{0} , and compute the expectation of the bead’s
ξ is the (matrix) squeeze parameter and N is a normalization position at time t:
factor. (Using an identity due to Schwinger [18], we obtain the
normalization constant N = √det 1
M
.) u0 (t ) = (t )|u0 |(t ) . (19)

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DENNIS P. CLOUGHERTY AND NAM H. DINH PHYSICAL REVIEW RESEARCH 7, L032004 (2025)

FIG. 5. Spectral probability distribution for single bogoliubon

emission P1 ( α ) from the decay of a vibron |1; {0} . Parameter
FIG. 3. Distribution of the average number of uncoupled exci- values are g = 0.7 and N = 15. The spectrum of single bogoliubons
tations (phonons and vibrons) in the coupled ground state nα = comprise 90.7% of the total emission.
{0}|aα† aα |{0} for parameter values N = 15, g = 0.7.

√
where ωr ≈ ω0 and r ≈ νω0 . For the case of light damping
The expectation can be expressed in terms of Bogoliubov
where r ωr , Eq. (23) gives  ≈ ν, in agreement with the
coefficients [15]
classical result.
  Using Fermi’s Golden Rule, we obtain for weak coupling
u0 (t ) = δ Re 2
U0α − V0α
2
exp(−i α t ) (20)
strength the decay rate GR for a transition |1; {0} → |0; {1n }
α
with the bead losing E = ω0 to the string
 exp(−i αt )
= δ Re  γn2 ωn
. (21) 
α 1 + 4ω0 n 2 −ω2
2 GR = 2π | 1; {0}|Hi |0; {1n } |2 δ(ωn − ω0 )
( α n ) n

We identify the factor 2
(U0α − V0α
2
)
in the summand of Eq. (20)
= 2π dωD(ω)γ 2 (ω)δ(ω − ω0 )
as the spectral density of the decay:
νω0 c 
ρ( α) = U0α
2
− V0α
2
. (22) = 2π = 2ν. (24)
ω0 π c

This spectral density satisfies a sum rule [15] α ρ( α ) = 1,
and the width of this spectral density is the decay rate of the That GR is twice the decay rate for the bead displacement is
bead displacement [20] (see Fig. 4). expected, since GR is the energy decay rate, while  is the
Using contour integration in the complex plane, the sum displacement decay rate. As energy of the bead varies as the
can be evaluated [15] and the decay rate  can be obtained: square of the oscillation amplitude, GR = 2.
We now turn to the radiation spectrum from the vibrating
⎛ ⎞ 21 bead. The decay of the vibrating bead is accompanied by the
ωr ⎝ r4 emission of bogoliubons. The probability of the emission of a
= √ 1 + 4 − 1⎠ , (23)
2 ωr single bogoliubon of frequency α can be expressed in terms
of Bogoliubov coefficients
−2
P1 ( α) = | {0}|bα a0† | 0 |2 = M0α (det M )−1 . (25)

A plot of the spectral probability distribution for single

bogoliubon emission is given in Fig. 5.
Summary. We analyzed the dynamics of a vibrating parti-
cle coupled to an environment by extending a generalization
of the Lamb model to the quantum regime. The model pro-
vides an exactly solvable example of a damped quantum
harmonic oscillator. These results may apply to a variety of
related quantum systems, e.g., a local vibrational mode in a
magnetic insulator (vibron-magnon) or coupled to an electro-
magnetic cavity (vibron-photon).
Our solution explicitly calculates the coefficients of
 ρ( α ) versus α /ω0 for
FIG. 4. Spectral distribution function the multimode Bogoliubov transformation that diagonalize
g = 0.4 and 0.6. It satisfies the sum rule α ρ( α ) = 1 and its width the Hamiltonian, and we use these coefficients to describe the
gives the decay rate  of u0 (t ) . properties of the system. We found that the true ground state

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QUANTUM LAMB MODEL PHYSICAL REVIEW RESEARCH 7, L032004 (2025)

of the system is always a multimode squeezed vacuum state emission has a nearly symmetric lineshape about a slightly
where the displacement uncertainty is reduced with increasing red-shifted peak frequency.
damping rate ν. We then obtained an explicit expression for Acknowledgments. We thank Dennis Krause for bringing
the vibrational decay rate of the bead and found that it recov- Ref. [14] to our attention. This research was supported in
ered the classical damping rate in the light damping regime. part by NSF Grant No. PHY-2309135 to the Kavli Insti-
We examined the acoustic radiation spectrum emitted by tute for Theoretical Physics (KITP) and NASA Grant No.
the vibrating particle. We obtained an expression for the 80NSSC19M0143.
probability of single bogoliubon emission in terms of the Data availability. No data were created or analyzed in this
Bogoliubov coefficients, and we observed that the spectral study.

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