Hawking radiation from Trojan states in muonic Hydrogen in strong laser field

0.45 MB·English

by  Matt Kalinski

Hawking radiation from Trojan states in muonic
Hydrogen in strong laser field
M. Kalinski
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Department of Chemistry and Biochemistry, Utah State University, Logan, UT 84322
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February 1, 2008
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Abstract – We show that the Unruh-Davis ef- dicted theoretically and confirmed experimen-
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fect is measurable from Trojan wavepackets in tally in single atoms when well confined elec-

1 muonic Hydrogen as the acceleration on thefirst tron on circular orbitin moving almost eternally
v
muonic Bohr orbit reaches 1025 of the earth ac- without dispersion in strong circularly polarized
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7 celeration. It is the biggest acceleration achiev- (CP) electromagnetic field [6, 7]. It has been
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able in the laboratory environment which have also suggested recently that the Unruh-Davis ef-
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been ever predicted for the cyclotronic configu- fect can be observed in radiation of driven elec-
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5 ration. Wecalculatetheratiobetweenthepower trons in ultra strong laser field [8]. The laser
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of Larmor radiation and the power of Hawking field strength necessary for the successful ex-
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h radiation. The Hawking radiation is measurable periment was proposed to be of the order of
p
even for Rydbergquantumnumbersofthemuon 10TeV/cm. The collaborative effect of Coulomb
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t due to suppression of spontaneous emission in fields is known to cause enhanced ionization and
n
a Trojan Hydrogen. the Coulomb ignition of atomic Van der Waals
u
clusters [9]. Nuclear fusion was observed due
q PACS: 42.50.Hz, 36.10.Dr, 04.70.Dy, 03.65.Ge
: to strain Coulomb field acceleration of energetic
v
ions from atomic cluster [10]. The laser disso-
Xi e-mail address: [email protected]
ciation of muonic molecules have been recently
r
a also suggested [11]. In this paper we suggest an
1. INTRODUCTION
alternative method of observing Hawking radia-
Thepossibility to observetheUnruh-Davis ef- tion in strong laser field. In contrast to ultra-
fect using single electron DeWitt detectors has strong laser method it is mainly the Coulomb
been discussed quite long time ago [1, 2, 3, 4]. field which provides the gigantic acceleration of
The measurements of depolarization of electrons the Davis detector and much weaker laser field
in storage ring have been classified confirmingof prepares the quantum particle to be classical. It
spin states heat up by the field vacuum fluctua- isalsothecentrifugal acceleration whichextends
tion [5] The perturbation of Thomas precession the time of observation practically to infinity on
was shown to be weak as the time of the full de- the scale of physical times involved. One may
excitation of the spin state turned out to exceed notice that for classical (Trojan) nonrelativistic
the age of the Universe. Coulomb problem of muonic Hydrogen the cen-
However very recently microscopic cyclotrons trifugal acceleration of classical electron on first
with optical cyclotronic frequency in circularly
polarized electromagnetic field have been pre-
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Bohr orbit is withparametersA,B,C,D. Thestationaryfac-
tor is the eigenfunction of the harmonic Hamil-
1 e2
a= = 1.90 1024g (1) tonian
4πǫ µa2 ×
0 µ H = h¯ω a† a ¯hω a† a +h¯ω a†a (5)
h + + +− − − − z z z
where a is muonic Bohr radium a = a µ/m + const
µ µ 0 e
respectively the estimated Davies temperature
where [13]
on n-th Rydbeg orbit is
ω = ω 2 q 9q2 8q/√2 (6)
±
T = ¯han = 75830K/n4 (2) q − ± ω−= ω√q
2πkc z
wherenisthemuonicRydbergquantumnumber q = e2/4πǫ0µω2x30 (7)
and a is centrifugal acceleration on n-the Ryd-
n end the parameter x is the classical center of
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berg orbit. It is 1000K for n = 3 and therefore
the wavepacket give implicitly
should be easily observable.
e2/4πǫ x2+e +µω2x = 0 (8)
− 0 0 Ef 0
2. TROJAN WAVEPACKETS
Note that it is not exactly the Hamiltoniam of
three harmonic oscillators because of negative
We start our detailed analysis from the har-
†
sign of a a but rather Hamiltonian predicting
monic theory of Trojan wavepackets in muonic − −
vacuum collapse if coupled to electromagnetic
Hydrogen [6, 13]. We neglect the spin for sim-
field. The parameters A,B and C are respec-
plification. Note that even through the lifetime
of the muon is 2.1970 10−6s the packet can tively functions of the dimensionless parameter
× q only [13, 15]
make thousands of revolution for the laser opti-
cal frequency, more then the lifetime due to its
A(q) = (1+2q)[4f(q) 9q2]/3q (9)
electromagnetic resonant nature. q −
Since the ratio between the electron velocity B(q) = (1 q)[4f(q) 9q2]/3q
q − −
on the first Bohr orbit to the speed of light is
C(q) = f(q)/3q,
always α = 1/137 for arbitrary mass of quantum
D(q) = √q
particle the system still can be described by the
Schro¨dingerequation. TheSchro¨dingerequation where
for theTrojan wavepacket incircularly polarized
f(q) = 2+q 2 (1 q)(1+2q) (10)
field in the Coulomb field is − q −
p2 1 3. HAWKING RADIATION
( ex +ωL )ψ = i¯hψ˙ (3)
f z
2µ − r − E The probability of the Hawking photon emis-
sion by the non-relativistic Trojan wavepacket
where µ is the muon mass µ = 206.7683m .
e can be calculated similarly to the spontaneous
The standard stability analysis [6, 13] of the tra-
emission rate [15, 14] as
jectory in rotating frame extended to relativis-
tic case gives the following harmonic Gaussian ΓUD = (11)
wavefunction e2
< ψ x ψ >< ψ x ψ >
¯h2 0| i| 1 1| j| 0
ψ = NeIµωx0y/¯he−iE0t/¯h (4) Xi,j
0
(E E )
e−µω[A(x−x0)2+By2+2iC(x−x0)+Dz2]/2¯h Gij[ 1 − 0 ]
× × ¯h
2
where G (ω) is the Fourier transform field-field can be expressed by their fluctuations. Using
i,j
correlation function for the accelerated observer the result of [15] one gets
on the circular orbit
< ψ xψ > = αλ< ψ x2 ψ >
1 0 0 0
∞ | | | |
G (Ω) = e−iωτ˜G (τ˜) (12) αλ¯h
ij Z−∞ i,j = 2Amω
< ψ y ψ > = α< ψ y2 ψ >]
1 0 0 0
| | | |
G (τ) = (13) α¯h
ij = (18)
G (τ τ ) = < 0E (τ )E (τ )0 > 2Bmω
ij 1 2 i 1 j 2
− |
where [15]
The Wightman function (electromagnetic field
tensor - tensor correlation function of the elec- 2µω/¯h 2µω
α = = α˜ (19)
tromagnetic field in the Minkowski vacuum is λp2/A+1/B r ¯h
λ = (p1+C)/(A+ω /ω). (20)
< 0F (x)F (x′)0 > (14) −
µν ρσ
| |
¯hc
= 4 (x x′)−6 (g g g gνρ) The energy difference between the Trojan and
µρ νσ µσ
ǫ0π{ − { − first deexcited Trojan state is
2[(x x′) (x x′) g
µ σ νσ
− − −
(x x′) (x x′) g (E1 E0)= h¯ω− (21)
ν ρ µσ −
− − −
(x x′) (x x′) g
µ σ νρ The probability of emission of Hawking photon
− − −
+ (x x′) (x x′) g ] (thedecayrateofmuonicTrojanwavepacket due
ν ρ µρ
− − }
to the Unruh-Davies effect) is therefore
The DeWitt detector response is due to delay
of the proper time on the noninertial trajectory. 1 π e2ω2 λ2 1
Γ = α˜2( + )θ3 (22)
The space-time trajectory parameterized by the UD 3r2ǫ µc3 A2 B2
0
proper time τ is for Trojan atom
with the dimensionless scaling function
x= (cτγ,Rcos(ωτγ),Rsin(ωτγ),0)
θ(q)= 2 q (9q2 8q)/√2 = ω /ω (23)
where r − −q − −
v In order the effect to be observable it must
β = (15)
be not much less the decay rate of the emission
c
1 whichisqualitativelythesameasduetotheLar-
γ = (16)
(1 β2) morradiation of classical charged particleon the
− circular orbit [15].
p
The field-field correlation function for the ob- The decay rate of the spontaneous emission of
server moving on the circular trajectory can be thenonrelativisticTrojanwavepacket isgivenby
estimated [16] (ωτ 1) as [15]
≪
1 4h¯c 1 1 e2 ω2(µ/A 1/B)2
Gij(τ˜−τ˜′) = ǫ0 π c4γ4(τ˜−τ˜′)4δij (17) ΓSP = 4πǫ0µc3 3 (µ2/A−+1/B)(1+θ)3 (24)
The only nonvanishing matrix elements between Fig.1 shows both the Hawking decay rate and
the first deexcited Trojan state and the Trojan thespontaneousemissionrateasfunctionsofthe
state are those of x and y coordinates [6] and scaled electric field = ω−4/3 = (1 q)/q1/3.
f
E E −
3
Figure 1: The relative spontaneous emission Figure 2: The ratio between the Hawking de-
rate (for the reference only, see [15]) and Hawk- cay rate and the spontaneous emission rate. For
ing emission rate due to Unruh-Davies effect as the best confined packet q = 0.9562 the ratio up
functions of scaled electric field for hypothetical to the γ 1 factor is universal for all n and
→
n = 1. Both rates scale like 1/n3 with the reso- equal γ4Γ /Γ = 0.190686 is much smaller
UD SP
nant Rydberg number n. Note that singularities but significant and measurable correction.
[17] at stability border values are nonphysical
but the Hawking decay rate goes to zero when
already for n = 12, almost twice faster then the
there is no electric field present.
decayofthemuonitself,wellsufficienttoobserve
decay of the wavepacket before the decay of the
muon and still during thousands of the packet
The ratio between Γ /Γ is therefore dimen-
UD SP revolutions. It corresponds to the laser inten-
sionless parameter depending only on param- sity I = 5.89 1014W/cm2 and the wavelength
eter q of the harmonic theory of the Trojan ×
λ = 380.782nm Since the spontaneous emission
wavepacket. TheDavis effect is observable when
is the Larmor radiation in the classical limit [15]
powerofspontaneous(Larmor)radiationiscom-
theUnruh-Davies emission shouldbereadily ob-
parable with the power of Hawking radiation.
servableas thecorrection to theonly mechanism
Note that the relativistic parameter γ from the
of radiation. Note that the Unruh-Davies radia-
relativistic Wightman function within the the-
tionvanishesinthelimitofvanishingelectricCP
ory limit presentedhereshouldbeapproximated
field which means that stationary circular states
by 1 for consistency (β = 1/137) since we nei-
do not emit Hawking radiation at all as they are
ther use the Klein-Gordon or Dirac equation for
not localized DeWitt detectors.
the theory of the Trojan electron itself. Fig.2
shows the ratio between the Hawking decay rate
5. CONCLUSIONS
and the spontaneous emission rate as function
of the scaled electric field . For the best con- As quantum detectors accelerate they heat up
E
finedwavepacket wegetΓ = 1.11784 106s−1 due to the vacuum fluctuations of the quantum
UD
×
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electromagnetic field. No detectors of classical [8] P. Chen, and T. Tajima, 1999, Phys. Rev.
masscanreachtheaccelerations necessarytoob- Lett., 83, 256.
serve the effect. Trojan atoms as the smallest
[9] C. Rose-Petruck, K. J. Schafer, K. R. Wil-
cyclotrons ever predicted theoretically seem to
son, and C. P. Barty, 1997, Phys. Rev. A,
be the best experimental systems to observe the
55, 1181.
effect. We show that the Unruh-Davies contri-
bution to the decay of Trojan states is as signifi-
[10] J. Zweiback, R. A. Smith, T. E. Cowan,
cantasspontaneousemissionandthereforeread-
G. Hays, K. B. Wharton, V. P. Yanovski,
ily observable effect as equivalent to the Larmor
andT.Ditmire,2000, Phys. Rev. Lett.,84,
radiation of cyclotronic electrons in the classical
2634.
limit.
[11] S. Chelkowski, A. D. Bandrauk, and P.
6. ACKNOWLEDGEMENTS B. Corkum, 2004, Phys. Rev. Lett., 93,
083602-1.
We thank Iwo Bialynicki-Birula for valuable
comments about electrodynamics in rotating [12] P. Chen, and T. Tajima, 1999, Phys. Rev.
frames and Kenneth Schafer from providingpar- Lett., 83, 256.
tial computer support for initial stages of this
[13] M. Kalinski, and J. H. Eberly, 1996, Phys.
research.
Rev. A, 53, 1715.
[14] H. Terashima, 1999, Phys. Rev. D, 60,
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