Download Schroedinger`s cat states generated by the environment

REVISTA MEXICANA DE FÍSICA S 57 (3) 113–119
JULIO 2011
Schroedinger’s cat states generated by the environment
a
D.D. Bhaktavatsala Raoa , N. Bar-Gillb , and G. Kurizkia
Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel.
b
Physics of Complex systems, Weizmann Institute of Science, Rehovot 76100, Israel.
Recibido el 24 de enero de 2011; aceptado el 15 de abril de 2011
As a rule, the coupling of a quantum system to an uncontrollable thermal reservoir (a “bath”) gives rise to the system’s decoherence, i.e., the
destruction of its unitary coherent evolution [1]. Not less common is the rule that the more complex the quantum system, the more detrimental
are the bath effects [2]. Here we point out that the coupling of a complex quantum system to a bath may actually induce advantageous coherent
dynamics. Namely, an exact solution for a quantum many-body system characterized by large angular momentum (or spin) that is coupled
to a thermal bath reveals a hitherto unexplored general effect: bath-induced effectively nonlinear evolution. This evolution can drive the
large-spin system into a macroscopic quantum-superposition (“Schroedinger-cat”) state [3, 4]. Such counter-intuitive bath-induced effects
should be observable in various setups on long (Markovian) time scales. They change our perspective of non-classicality in open many-body
quantum systems. Namely, the bath may cause rather than impede the formation of distinctly non-classical Schroedinger-cat states, despite
their fragility in the presence of a bath [2, 5].
Keywords: Quantum decoherence; quantum open systems; quantum superpositions.
Como regla general, el acoplamiento de un sistema cuántico con un reservorio térmico no controlable (un “baño”) da lugar a la decoherencia
del sistema, i.e., a la destrucción de su evolución unitaria coherente [1]. No menos común es la regla de que mientras más complejo el sistema
cuántico, más deterioro genera el baño [2]. Aquı́ haremos ver que el acoplamiento de un sistema cuántico complejo con un baño puede, de
hecho, inducir una dinámica coherente ventajosa. Especificamente, una solución exacta para un sistema de muchos cuerpos caracterizada
por un momento angular grande (o espı́n) que se acopla a un baño térmico revela en general un efecto no explorado hasta ahora: la evolución
no lineal efectiva inducida por el baño. Esta evolución puede llevar al sistema de alto espı́n a un estado superpuesto cuántico macroscópico
(“gato de Schroedinger”) [3,4]. Estos efectos contraintuitivos inducidos por el baño deberı́an ser observables en varios arreglos a escalas
largas (markovianas) de tiempo. Ellos cambian nuestra perspectiva de no clasicalidad de sistemas cuánticos abiertos de muchos cuerpos.
Especificamente, el baño puede causar en lugar de impedir la formación de estados no clásicos de gato de Schroedinger, a pesar de su
fragilidad en la presencia del baño [2, 5].
Descriptores: Decoherencia cuántica; sistemas cuánticos abiertos; superposiciones cuánticas.
PACS: 03.65.Yz; 03.65.-w; 03.67.Bg
1.
Introduction
Collective dynamics of ensembles of atoms and spins are
among the few well-studied manifestations of quantum entanglement on macroscopic scales. This comes about because
their quantized collective dynamics can be mapped onto that
~
of an object in an eigenstate of angular momentum (spin) L
with large eigenvalues. Behavior of this kind is exhibited by
spin-polarized ensembles in solids [6] or by atomic ensembles with large pseudospin that collectively emit and absorb
photons [7, 8]. Alternatively, large spin characterizes macroscopic atomic ensembles that are entangled via interaction
with a common light source [9]. The weakness of their interactions with the environment renders large-spin ensembles
robust against decoherence and thus makes them good candidates for quantum memory devices [6, 10].
Here we reveal a hitherto unnoticed basic effect that occurs when a large-spin system is coupled to a bosonic “bath”.
Namely, upon tracing out the bath, the large-spin system
would evolve from an initial coherent state to a squeezed or
Schroedinger-cat state. This is surprising, since such effects
are generally [11, 12] known to be a result of coherent non~ 2 ) dynamics. Yet, we show that they can arise
linear (here L
from linear coupling to another system acting as the bath, a
situation that is expected to lead to decoherence.
Markovian (Lindblad) master equations [13,14] are commonly used in studies of such macroscopic systems, in analogy with those governing simple two-level systems, where
the only bath parameter entering the dynamics is the decoherence rate. Here, we show that such description does not
account for the dynamics of the large-spin system in the presence of a bath, and it is imperative to study the system-bath
dynamics with exact account for many-body effects.
Exact system-bath solution
We consider an ensemble of N non-interacting spins or
atomic two-level systems (TLS) that are identically coupled
to a bosonic (oscillator) bath. It is described in the collective
basis by the many-body Hamiltonian
H = HS + HB + HI ,
HS = ωx Lx + ωz Lz ,
X
X
~ · ²̂
HB =
ωk b†k bk , HI = L
ηk (bk + b†k ).
k
(1)
k
Here Li (i = x, y, z) are the angular-momentum components
of the effective N -level system (large-spin ensemble), b†k , bk
are the creation and annihilation bosonic operators of the k-th
bath mode, and ηk the corresponding (k − mode) coupling
114
D.D. BHAKTAVATSALA RAO, N. BAR-GILL, AND G. KURIZKI
F IGURE 1. Bath-induced cat formation for a large spin system. (a). Schematic representation of a large spin initialized in a coherent state
which evolves into a Schroedinger cat state via coupling to a bath. (b) The time-dependent functions responsible for the nonlinear (f (t))
and linear (Γ(t)) dynamics of the system. (c) Bath induced squeezing is plotted as a function of time. The regime in which the system is
squeezed is shaded in red. (d1 ) − (d3 ). The state of the system simulated at various times. The system is a spin ensemble composed of
N = 50 particles coupled to an Ohmic bath.
rates. The unit vector ²̂ defines the nature of the system-bath
coupling and the resulting decoherence: ²̂ = x̂ (Lx -coupling)
entails population exchange among the N levels (cooperative
absorption or emission) whereas ²̂ = ẑ (Lz -coupling) causes
pure cooperative dephasing without population exchange.
In general, the dynamics generated by Eq. (1) is insolvable. In order to circumvent this difficulty we may consider
one of the two equivalent scenarios, obtained from each other
by π/2 rotation of the spins in the ensemble (Lx ↔ Lz ). (i)
Lx coupling to the bath (cooperative population exchange):
We prepare the system in a Lz -eigenstate with ωz = 0. (ii)
Lz -coupling to the bath (pure cooperative dephasing): we
prepare the system in a Lx -eigenstate and then switch off ωx .
Let us explicitly consider scenario (ii). The ensuing coupling to a bosonic bath gives upon using the Magnus expansion [15] the following closed-form equation for the timeevolution operator (see Methods)
"
U (t)= exp −itf (t)L2z +Lz
X³
k
αk (t)b†k −αk∗ (t)bk
´
#
(2)
where the coupling to the bath determines the functions
1X 2
ηk (ωk t − sin ωk t)/ωk2 ,
f (t) =
t
k
1 − eiωk t
.
(3)
ωk
We thus obtain a striking exact result: the bath-induced
evolution occurs as if it were driven by both linear and nonlinear terms in Lz . For the well-studied TLS, L2z = σz2 ≡ I,
the nonlinear term yields an overall phase and does not affect
the dynamics. In contrast, for multi-level (or multipartite)
systems where L2z 6= I, the term f (t)L2z in equation (2) can
give rise to nonlinear coherent dynamics that is absent in the
two-level case. Scenario (i) (Lx -coupling to the bath) yields
nonlinear evolution via f (t)L2x , with similar effects.
αk (t) = ηk
Bath-induced squeezing
In scenario (ii) (Lz coupling to the bath) two ensembles of
atoms or spins are prepared at t = 0 in a superposition state
via ωx Lx , in equation (1) and then cease to interact upon
setting ωx = 0. This means that the system is prepared in a
Rev. Mex. Fı́s. S 57 (3) (2011) 113–119
115
SCHROEDINGER’S CAT STATES GENERATED BY THE ENVIRONMENT
F IGURE 2. Cat formation in a thermal gas Raman-coupled to a buffer gas. (a) A Vapor cell experimental scheme for inducing a reservoirmediated nonlinear interaction in a large spin system. Cs atoms are the active large-spin system, and a N e buffer gas acts as the thermal
reservoir. (b) Optical Feshbach Raman transition in the presence of a Cs-Ne atom pair. The interaction is resonant with a change of the
internal state of the Ne atom, and no change in the internal state of the Cs atom. (c). The time-dependent functions f (t) and Γ(t). (d). The
noisy cat formed at τcat . The simulated system is a spin ensemble composed of N = 50 particles coupled to a bath with Lorentzian coupling
spectrum.
spin-coherent state [16] labelled by the Bloch-sphere angles
|θ, φi (Lx -eigenstate). The system is uncorrelated at t = 0
with the thermal bath. The state of the system alone at any
later time, found upon tracing over the bath degrees of freedom, is
coherent state (for example | π2 , 0i) to diffuse approximately
as
2
2
hL+ (t)i/N ∼≈ e−tΓ(t) e−t /τd (t) ,
where
τd (t) ≈ √
−itf (t)L2z
ρ(t) = e
(
)
X
2
−tΓ(t)(m−n)2
×
ρmn (0)e
|mihn| eitf (t)Lz . (4)
m,n
The evolution of the system is thus dynamically modified by
the bath in a nonlinear fashion alongside bath-induced decoherence. The decay of the off-diagonal terms increases exponentially with (m − n)2 rendering multilevel or multipartite
coherence vulnerable to decoherence [17]. The time dependence of the decay rate Γ(t) is determined by the bath spectrum as discussed below. This evolution causes the average
single-particle phase-coherence hL+ i/N of an initial spin-
1
.
2N |f (t)|
(5)
Namely, the interaction with the bath induces Gaussian phase-diffusion, which is synonymous with squeezing [16–18], via the L2z term, on a time scale τd . It also causes
exponential decay of the coherence (decoherence) via the linear Lz term.
Bath-induced cat formation
Under the nonlinear term f (t)L2z in Eq. (3), the initial spincoherent state |θ, φi will evolve at
Rev. Mex. Fı́s. S 57 (3) (2011) 113–119
τcat ≡ t =
π
2f (t)
(6)
116
D.D. BHAKTAVATSALA RAO, N. BAR-GILL, AND G. KURIZKI
F IGURE 3. Interaction with a single mode cavity. (a). Schematic representation of an atomic ensemble consisting of N = 100 particles,
interacting with a single mode cavity.The behavior of the time-dependent functions f (t) (b), Γ(t) (c), are plotted for an ideal (zero-linewidth)
mode with resonant frequency ω0 = 4N η. (d1 ), (d2 ) display the state of the initial atomic ensemble and the prefect cat formed at t = τcat
in this idealized model. For a finite linewidth mode the results of Fig. 2 apply.
into the cat state [11]
1
|ψicat = √ [e−iπ/4 |θ, φi + eiπ/4 |π − θ, −φi].
2
In scenario (i) (Lx -coupling to the bath) a cat with
|θ, φi ↔ |
π
− θ, φi
2
(see expression after (7)) will form at the same time.
The time at which a cat forms is independent of the length
~ = N/2. Hence, its
of the angular momentum vector |L|
formation time is much longer than the squeezing
(phasep
diffusion) time ∼ τd in Eq. (5): τcat ≈
N/2τd . The
cat formation time τcat is typically in the long-time Markov
regime as it is longer than the bath correlation time tc . Hence,
the cat encounters a much lower Γ ≈ ΓM , and much higher
f ≈ fM , than those encountered during τd in Eq. (5), which
typically occurs in the non-Markov regime (see Methods).
The condition for the cat to survive decoherence is, from
Eqs. (4)-(6),
τcat ΓM N 2 < 1.
(7)
This requires |fM | À |ΓM |: the feasibility of this requirement is discussed below for various setups.
Although scenarios (i) and (ii) are mathematically equivalent, both leading to equation (7), they may strongly differ
quantitatively. In scenario (i) ΓM = 1/T1 , whereas in scenario (ii) ΓM = 1/T2 . Likewise, they may have very different fM values and different bath spectra (and thus correlation
times tc ). We may compare the cat states formed by baths
with different coupling spectra, having the same width, i.e.,
inverse correlation time 1/tc (compare Figs. 1 and 2).These
baths lead to different ΓM in the long-time Markov limit and
hence to cat states with different purities. In particular, the
Lorentzian bath spectrum, which rapidly falls off with ω,
gives lower ΓM than the Ohmic bath spectrum (Fig. 1) that
has slower falloff with ω. Yet different bath spectra are seen
to give the same asymptotic value of f (t) = fM .
2. Experimental scenarios
A. Spin ensemble coupled to a phonon bath
Consider N weakly interacting spin-1/2 particles (e.g. electron spin ensembles in fullerenes [6, 19] and quantum
dots [20,21]) that undergo dephasing via coupling to phonons
in a lattice, so that they conform to Eq. (1). We assume that
Rev. Mex. Fı́s. S 57 (3) (2011) 113–119
SCHROEDINGER’S CAT STATES GENERATED BY THE ENVIRONMENT
the spins are initially aligned along their quantization axis (z)
| + zi = | ↑↑ · · · ↑i and rotated by a π/2 pulse to the x plane.
This state of the ensemble then becomes
1
| + xi = √ |(↑ + ↓)(↑ + ↓) · · · (↑ + ↓)i.
2N
The interaction with the bath along the ẑ axis, while destroying the phase, also correlates the initially uncorrelated
spins, thereby creating the required macroscopic superpositions. Ideally, one would like to have
π
1
HS+B
2 (ŷ)
| + zi −−
−→ | + xi −−−−→ √
2
π
(−ŷ)
1
1
× [| + xi − i| − xi] −2−−−→ √ [| + zi − i| − zi] = √
2
2
× [| ↑↑ · · · ↑i − i| ↓↓ · · · ↓i],
thereby creating macroscopic GHZ like states [23] for the
N spin ensemble. Since the bath spectrum is continuous,
the formation of such perfect cat states is not possible. Yet,
the formation of high-purity cat states may still be possible.
For an Ohmic spectrum with the Debye cutoff ωD we find
fM = αωD , and ΓM = α/β, where β is the inverse temperature and α < 1 is a dimensionless coupling constant. The
condition in Eq. (7) may be satisfied at temperature below 1
mK, for fM ∼ ωD ∼ 1013 Hz À ΓM ≥ MHz to obtain a cat
with N ∼ 300 spins.
B. Thermal gas Raman-coupled to a buffer gas
Consider a room temperature gas of non-interacting active
atoms, e.g. 133 Cs (in a standard vapor cell setup - see
Fig. 2(a)). The two hyperfine ground states of each atom, denoted 1 and 2, comprise the pseudospin degrees of freedom.
The ensemble of N such TLS is a large (N/2) spin system,
that does not have an intrinsic nonlinearity (self interaction).
A buffer gas, e.g. Ne, acts as a bath on this large-spin system,
if the two gases are coupled by a pair of laser beams in the
Raman configuration (Fig. 2(b)). These beams induce an energy shift of the 1,2 levels, in the presence of the buffer gas
atoms. At Raman resonance between these two species [24],
known as an optical Feshbach resonance, the incoming active
and buffer atoms form a bound molecular state, in which the
internal state of the active (Cs) atoms does not change, nor
does it affect the strength of the interaction, while the internal
state of the buffer (Ne) atoms is flipped. Thus the effective
interaction term conforms to Eq. (1), with L̂z coupled to the
bath modes.
By controlling the frequency difference and the angle between the laser beams and their intensity, the strength of the
system-bath coupling η can be varied proportionally to the
Raman-induced scattering length [24]
¡
¢
Ω21 ∆1 − Ω22 /∆2
1
ρbuf f ,
η∝−
2q (∆1 − Ω22 /∆2 )2 + (γ1 /2)2
117
where q is the wavevector difference of the laser beams, Ω21
(the Rabi frequency squared) is proportional to the intensity
of beam 1, Ω22 is proportional to the intensity of beam 2, ∆i
is the detuning of beam i from resonance, γ1 is the decay rate
of the excited state 1, and ρbuf f is the buffer gas density. The
Markovian decoherence rate is [24] ΓM ' γ1 (Ω2 /∆1 )2 . For
example, η can be estimated to be ∼ 1 kHz, for ∆1 =50 MHz,
∆2 = 10 MHz, Ω1 = 50 kHz, Ω2 = 10 kHz, and
ρbuf f > 1012 [cm−3 ] for Cs in N e buffer gas [24]. We then
obtain: τcat ' 1ms ¿ 1/ΓM ' 2s. According to Eq. (7),
this allows the formation of a cat state with N ∼ 50 atoms.
Low temperature would decrease ΓM , thereby increasing the
number of atoms in the cat state.
C. Atomic ensemble coupled to a single-mode cavity
When a non-interacting atomic gas is strongly Lx -coupled
to a single-mode cavity [21, 22, 25, 26], it can yield a
Schroedinger cat state within the relaxation time of the cavity mode. For a zero-linewidth mode at frequency ω0 , the
parameters
f (t) =
η2
[ω0 t − sin ω0 t]/ω02
t
and
η2
[1 − cos ω0 t]/ω02
t
exhibit strongly oscillatory behavior (see Fig. 3(b), 3(c)). At
times ω0 t = 2mπ, (m = 0, 1, 2, · · · ) Γ = 0, the state of
the system resumes its initial purity and a perfect cat forms.
Remarkably, only a proper choice of η and ω0 yields the
Schroedinger cat, namely:
Γ(t) =
ω02
= integer.
4η 2
Yet, for finite cavity linewidth, this model breaks down and
the cat dynamics is similar to Fig. 2. For atom-cavity
coupling strength η ≥ MHz, cavity linewidth 1 MHz, and
ΓM ∼ 1/T1 ∼ 3 MHz [25] we find from Eq. (7) that cats
with N ' 2.3 may form.
3.
Conclusions
We have used an exactly solvable model to reveal the unexpected nonlinear (squeezing) dynamics of a system with large
angular momentum coupled to a thermal bath. The intriguing
consequence is that a bath may induce rather than impede the
formation of Schroedinger cat states.
Equations (4)-(7) are the main results of this paper. They
imply novel dynamic features in open systems with level
(spin) numbers N > 2 at low temperatures:
(i) In contrast to TLS dynamics (N = 2), nonlinear (L2z or L2x ) terms induced by the bath can dominate the evolution of systems with N > 2.
Rev. Mex. Fı́s. S 57 (3) (2011) 113–119
118
D.D. BHAKTAVATSALA RAO, N. BAR-GILL, AND G. KURIZKI
(ii) While the non-Markovian bath dynamics
affects the squeezing of an initial coherent state, its
Markovian dynamics governs the formation of a high
purity cat.
Since Lz commutes with all its powers implies that only the
first two terms of the expansion are non-zero. We then recover Eq. (2) of the main text.
(iii) The bath coupling spectrum and its possible “engineering” play a crucial role in allowing the
formation of a cat state. Its purity may be further improved using quantum control techniques to dynamically modulate the system levels [27] and thus modify
the bath effects.
Dependence of cat formation on bath spectrum
(iv) Another intriguing consequence of the
analysis is that only bosonic bath induces L2z or L2x
nonlinearity, whereas a fermionic bath generates also
higher-power nonlinearities via the Magnus expansion [15], thereby modifying the dynamics.
Let us consider the dependence of squeezing and decoherence on the bath spectrum. The temperature-dependent coupling spectrum of the bath is defined as [1, 28]
X Z
GT (ω) =
ηk2 dωδ(ω − ωk )[2nT (ω) + 1],
k
nT (ω) being the temperature-dependent population distribution of the bath modes. The bath correlation time tc is the
inverse width of GT (ω). Rewriting (3) in terms of GT (ω)
yields
Acknowledgments
The support of EC (MIDAS project, FET Open) DIP, GIF
and ISF is acknowledged. G. K acknowledges support by the
Humboldt-Meitner Award.
Derivation of the time-evolution operator induced by a
bosonic bath
In the interaction picture (keeping ωx = 0) in Eq. (1) yields
X
HI (t)=eiH0 t HI e−iH0 t =Lz
ηk (e−iωk t bk +e−ωk t b†k ),
k
where H0 = ωz Lz +HB . To obtain a closed-form expression
for the time-ordered unitary operator


Zt
U (t) = T← exp −i dt0 HI (t0 ) ,
0
we resort to the Magnus expansion [15] of the exponent of
U (t) = exp(Ω(t)). The first few terms of the expansion are
Zt
0
1
+
6
Z∞
G0 (ω)
0
ωt − sin ωt
,
ω2
where G0 (ω) is the zero-temperature (nT = 0) coupling
spectrum to the bath. The corresponding decoherence rate
is:
Methods
Ω(t) =
1
f (t) =
t
1
HI (t1 )dt1 −
2
Zt Zt1
[HI (t1 ), HI (t2 )]dt1 dt2
0
0
Zt Zt1 Zt2
dt1 dt2 dt3 {[HI (t1 ), [HI (t2 ), HI (t3 )]
0
0
0
+ [[HI (t1 ), HI (t2 )], HI (t3 )]} + · · · .
We now take advantage of the remarkable property of bosonic
bath operators, namely, that the commutator of the interaction
Hamiltonian at two different times is a C-number function in
the bath operators:
X
[HI (t), HI (t0 )] = −2iL2z
ηk2 sin ωk (t − t0 ).
1
Γ(t) =
t
Z∞
GT (ω)
0
1 − cos ωt
.
ω2
The decoherence rate Γ(t) is time-dependent in the nonMarkov regime of t < tc [28]. At sufficiently low temperatures, Γ(t) is drastically reduced in the Markovian limit as
opposed to its fast initial non-Markovian increase:
Γ(t ¿ tc ) À Γ(t → ∞) = ΓM
(Fig. 1(b), 2(c)).. Since Γ(t) initially has contributions from
all the bath modes,
Z
GT (ω)dω,
there is a steep initial rise of decoherence, marking the transition from the Zeno to the anti-Zeno regime, which then decreases as the bath oscillators go out of phase in the Markov
regime [29].
By contrast, f (t) increases in the course of transition
from the non-Markov to the Markov regime, where it settles
at its long-time value |f (t → ∞)| = |fM | À |f (t ¿ tc )|.
This can be seen from Fig. 1(b) for an Ohmic bath with
coupling spectrum G0 = αωe−ω/ωc and Fig. 2(c), for
Lorentzian coupling spectrum
k
Rev. Mex. Fı́s. S 57 (3) (2011) 113–119
G0 (ω) = α
ωc2
.
ωc2 + (ω − ω0 )2
SCHROEDINGER’S CAT STATES GENERATED BY THE ENVIRONMENT
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