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Optics Express

  • Editor: J. H. Eberly
  • Vol. 2, Iss. 9 — Apr. 27, 1998
  • pp: 347–354
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Multimode unraveling of master equation and decoherence problem

D. B. Horoshko and S. Ya. Kilin  »View Author Affiliations


Optics Express, Vol. 2, Issue 9, pp. 347-354 (1998)
http://dx.doi.org/10.1364/OE.2.000347


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Abstract

An unraveling of master equation for a set of fields interfering with one another is developed and conditions are found under which decoherence can be avoided for conditional and unconditional evolution of one of this fields.

© Optical Society of America

It is well known that a system coupled to a noncoherent thermal reservoir comes after some characteristic time teq to the state of thermal equilibrium. In quantum physics a system coupled to a reservoir displays yet another fundamental phenomenon: in the timescale much shorter than teq the density matrix of quantum system becomes diagonal in some basis, defined by the system-reservoir interaction Hamiltonian, and then tends to thermal equilibrium preserving this diagonality [1–3

1. W. H. Zurek, “Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?,” Phys. Rev. D 24, 1516 (1981). [CrossRef]

]. This phenomenon, generally known as quantum decoherence, has been widely studied in recent years for such a simple system as single-mode optical or microwave field in an open cavity, for the up to date quantum optical experimental techniques provide a possibility to observe and modify this phenomenon with high precision, and from the other side decoherence causes additional difficulties in such applications of quantum optics as quantum cryptography [4

4. S. M. Barnett and S. J. D. Phoenix, “The principles of quantum cryptography,” Phil. Trans. R. Soc. Lond. A 354, 793 (1996). [CrossRef]

] and development of quantum computers [5

5. D. P. DiVincenzo, “Quantum computation,” Science 270, 255 (1995). [CrossRef]

]. In this work we show that decoherence is not an intrinsic property of a decaying quantum system and study conditions under which it can be completely excluded.

As the first example we consider a single mode of the field in the cavity described by operators of photon annihilation and creation a and a + respectively. One of the cavity mirrors is semi-transparent, so that the field in the cavity interacts with the modes of external, initially vacuum field. The density operator of such a system obeys a well-known master equation of Lindblad form:

ρ˙=i[Ha,ρ]+γa2(2a+a+ρa+a),
(1)

where H = ωa + a is a Hamiltonian of internal evolution, ω is the mode frequency, γa is the decay constant of the cavity and ħ is set to unity. The solution of Eq. (1) can be represented as integral over so-called quantum trajectories [6–8

6. M. D. Srinivas and E. B. Davies, “Photon counting probabilities in quantum optics,” Opt.Acta 28, 981 (1981). [CrossRef]

]:

ρ(t)=n=0+0tdtn0t2dt1ρ(t|t1,tn),
(2)

where

ρ(t|t1,,tn)=S(ttn)JaS(tntn1)JaJaS(t1)ρ,
(3)

and the superoperators Ja and S(t) are defined as follows:

Jaρ=γaa+,
(4)
S(t)ρ=e(iHγaa+a2)tρe(iHγaa+a2)t.
(5)

The representation given by Eq.(2) is usually called an unraveling of the master equation, Eq. (1). In general any equation

ρ˙=,

where L is any Liuvillian superoperator, can be unraveled in the following way:

ρ(t)=n=0+0tdtn0tndt1e(LJ)(ttn)Je(LJ)(tntn1)JJe(LJ)t1ρ,
(6)

for any superoperator J, choice of which determines the physical sense of such an expansion.

The unraveling given by Eqs. (2–5) obtains its physical meaning if we place outside the cavity a photodetector with unity quantum efficiency collecting all outgoing radiation. In this case ρ(t|t 1,…, tn ) represents the conditional (non-normalized) density operator of the cavity field given exactly n photons were registered by the photodetector in the time interval [0, t) at times t 1,…, tn , while its normp p(t|t 1,…, tn ) = Tr{ρ(t|t 1,…, tn )} gives the probability of the corresponding outcome. In this interpretation Eq. (2) can be considered just as an averaging over all possible detector records. The structure of Eq. (3) shows that the conditional state of the field under continuous measurement consists of jumps at times t 1, …,tn described by superoperator Ja and continuous non-unitary evolution between the counts, described by superoperator S(t). It is easy to see that if the initial state of the field is a pure state ρ(0) = |ψa (0)〉 〈ψa (0)| it remains pure under conditional evolution:

ρ(t|t1,,tn)=|ψacond(t)ψacond(t)|,
(7)

where

|ψacond=(γa)n/2e(iHγaa+a/2)(ttn)aae(iHγaa+a2)t1|ψa(0).
(8)

Because of term e -γata+a/2 in Eq. (8) all conditional states tend to vacuum (equilibrium) state, however without any decoherence, which arises only as a result of averaging over all possible trajectories, given by Eq. (2). This provides the following interpretation of the decoherence process: due to coupling of the cavity mode to external modes on the semi-transparent mirror, photons leave cavity at some times t 1, …,tn and the state of the system decays to vacuum. If we know exactly these times, gathering all photons on the surface of a photodetector, the conditional state of the field preserves its quantum coherence and only the loss of information connected with non-unitary detector efficiency or not perfect photon resolution brings about the decoherence of the system state. Thus, to avoid decoherence caused by interaction with a set of bosonic field modes, it is enough to collect all outgoing radiation on a photodetector and resolve all counts - the conditional state of the system with respect to observed photocount sequence will remain pure.

It seems that such a corpuscular interpretation gives a complete description of field decoherence. However the external field cannot be completely described by photon arrival times t 1,…,tn , it possesses a fundamental property of interference with other fields and itself. Moreover, in many experiments one is interested in phase properties of the cavity field, which are measured with the use of homodyne techniques, that is the external field undergoes interference with the field of local oscillator before being measured. So we should answer the following question: can we avoid in principle de-coherence of the cavity state if the external field is subject to interference with other auxiliary field, so that the times t 1, …, tn at which photons left the cavity cannot be more obtained directly by photodetection? The answer on this question is non-trivial and generally depends on the state of the auxiliary field.

Let us suppose that the auxiliary field is produced by another cavity, the single mode of which described by annihilation operator b is initially exited in the state |ψb (0)〉. The external fields of the two cavities are mixed on a symmetric beam-splitter and measured by two photodetectors. The master equation for the joint density operator of modes a and b is

ρ˙=i[Ha,ρ]i[Ha,ρ]+γa2(2a+a+ρa+a)+γb2(2b+b+ρb+b),
(9)

where γb and Hb , are the decay rate and Hamiltonian of the corresponding cavity. Eq. (9) can be unraveled in such a way that it will provide an expression for the conditional state of two modes for given photocount sequences at two detectors. We make this unraveling using the rule that the jump of the conditional wave function after detecting a photon at a photodetector is given by the operator of the field incident on this detector [7].

Disregarding the phase shifts introduced by beam-splitters we write the field operators on two detectors as

Ea=γa2a+γb2b,
(10)
Eb=γa2a-γb2b.
(11)

Now a photocount at the first detector brings about reduction

ρEaρEa+=Jaρ,
(12)

while a photocount at the second causes reduction

ρEbρEb+=Jbρ.
(13)

Between counts evolution is described by superoperator S(t), satisfying the equation

S˙ρ=(LJaJb)ρ=i[Ha+Hb,ρ]γa2(a++ρa+a)γb2(b++ρb+b),
(14)

that is

S(t)ρ=eKtρeK+t,
(15)

where

K=iHa+iHb+γa2a+a+γb2b+b.
(16)

We see again that all superoperators Ja , Jb and S(t) preserve the purity of the state. So the conditional state of two modes is a pure state at any time provided that it was pure at the beginning.

Using Eqs.(12), (13) and (15) we can construct a conditional state of two modes for any given sequences of photocounts at the detectors. For example, if one count was observed at the first detector at time t 1, and one at the second detector at time t 2 > t 1, the conditional (non-normalized) state at time t > t 1, t 2 is

|ψabcond(t|t1;t2)=eK(tt2)(γa2aγb2b)eK(t2t1)(γa2a+γb2b)
×eKt1|ψa(0)|ψb(0).
(17)

This wave function can be considered as a sum of four parts:

ψabcond(tt1;t2)=γa2eK(tt2)aeK(t2t1)aeKt1ψa(0)ψb(0)
+γaγb4eK(tt2)aeK(t2t1)beKt1ψa(0)ψb(0)
γaγb4eK(tt2)beK(t2t1)aeKt1ψa(0)ψb(0)
γb2eK(tt2)beK(t2t1)beKt1ψa(0)ψb(0),
(18)

which describe the four possible ways of obtaining one photon at first and one at second detector: (i) both photons came from the cavity a (first term), (ii) first photon came from the cavity b, second came from the cavity a (second term), (iii) vice versa (third term), and (iv) both photons came from the cavity b (forth term). as we can in no way determine the paths of photons the resulting wavefunction is a superposition of four possible wavefunctions.

So the joint conditional state of two modes under continuous measurement does not decohere on its way to equilibrium (vacuum state). However we cannot say it about the mode a alone! The reason is that the states of two modes become entangled, that is the joint state does not factorize:

|ψabcond(t)|ψa(t)|ψb(t).
(19)

It is most easily seen from Eq. (17), where each term factorizes, while the sum generally not. If we now average over mode b, the state of mode a becomes a mixture.

When such an entanglement is produced? Obviously not between the counts, as the superoperator S(t) does not entangle states:

S(t)ρaρb=e(iHaγaa+a2)tρae(iHaγaa+a2)te(iHbγbb+b2)tρbe(iHbγbb+b2)t.
(20)

It is produced by quantum jumps connected with photon detection, because generally

(γa2a+γb2b)|ψa|ψb|ψa|ψb.
(21)

In the Appendix we show that the equality in Eq. (21) holds if and only if one of the states |ψa 〉 or |ψb 〉 is a coherent state. It is easy to see from the implicit form of the superoperators Ja , Jb , and S(t) that coherent state of mode a or b remains coherent under conditional evolution of the joint state. So we can state that the joint conditional state of two modes under continuous measurement performed after interference on a beam-splitter is disentangled if and only if one of these modes is in a coherent state initially.

This is just the situation met in a homodyne measurement. It means that if in homodyne measurement of cavity field all photons are gathered on photodetectors with unity quantum efficiencies and all photocounts are resolved, then the conditional state of the cavity field does not decohere, and the cause of decoherence, as in case of direct detection, is the loss of information.

The scheme with two interfering beams considered above can be generalized for the case of arbitrary number of modes. Let us consider n single mode fields of the same frequency ω described by operators ai , i = 1, …, n in n cavities with decay rates γi , i = 1, …, n. Such a system is described by master equation

ρ˙=i(i[Hi,ρ]+γi2(2aiρai+ai+aiρρai+ai)),
(22)

where Hi = ωai+ ai is the Hamiltonian of internal evolution of the ith mode. We suppose that the outgoing fields of the cavities are mixed with one another on a set of beam splitters and the resulting n beams are detected by n photodetectors. The field operators on the detectors Eiout are given by relation

Eiout=jUijEjin,
(23)

where Uij = Uji* is a unitary matrix, while

Ejin=γiai,
(24)

is the field operator just outside the cavity. Taking into account that

iγiaiρai+=iEiinρEiin+=ijkUijUki*EjoutρEkout+
=jkδjkEjoutρEkout=kEkoutρEkout+
(25)

we rewrite Eq. (20) in the following form

ρ˙=i[iHi,ρ]+iEioutρEiout+iγi2ai+aiρρiγi2ai+ai.
(26)

The last equation can be interpreted as follows: observing a photon at the ith detector brings about a reduction ρEioutρEiout+ while between counts the state evolution is described by the superoperator S(t):

S(t)=exp{(iiHiiγi2ai+ai)t}ρexp{(iiHiiγi2ai+ai)t}.
(27)

The conditional state of n-mode field is again a pure and generally entangled state.

So we have seen how it is possible to avoid decoherence in conditional state evolution. In some cases we can obtain also a non-conditional evolution without decoherence by means of electro-optical feedback between photodetectors and cavity field. Below we consider how this can be done in a special case of decaying superposition of coherent states in an open cavity.

We consider again a single mode cavity with photon annihilation operator a and decay rate γa . Initially the field inside the cavity is exited in a superposition of coherent states with amplitudes α and -α (a so-called Schrodinger cat state [10

10. V. Buzek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” Prog. Opt. 36 3(1995). [CrossRef]

])

|ψa(0)=N(|α+e|α),
(28)

where φ is arbitrary real phase and N is a normalization constant. The conditional evolution of the state vector of the field under continuous measurement is given by Eq. (8). Substituting Eq. (28) into Eq. (8) we obtain

|ψacond(t)=(γaα)nN(|αeγat/2iωt+(1)ne|αeγat/2iωt),
(29)

where n is number of counts in the time interval [0, t). Eq. (29) shows that the conditional state of field remains a superposition of two coherent states with altering sign between them [11

11. B. M. Garraway and P. L. Knight, “Evolution of quantum superpositions in open environments: Quantum trajectories, jumps, and localization in phase space,” Phys. Rev. A 50, 2548 (1994). [CrossRef] [PubMed]

]. In contrast to this the unconditional evolution of field, i.e. the solution of Eq. (1) for initial state given by Eq. (29) gives fast decoherence of the field into statistical mixture of states |α〉 〈α| and |-α〉 〈-α|. Decoherence again arises as a result of averaging over all possible detector records, however in this special case the conditional state does not depend on times t 1, …, tn of photon arrivals, but only on total number of photons in the interval [0, t), or more precisely there are only two possible wave functions |ψacond (t)〉: one for even and one for odd values of n, which differ only by sign between states |α〉 and |-α〉. When a photon leaves the cavity this sign is switched to the opposite value. This sign switching has a simple physical interpretation only for the case of φπ2, when it corresponds to shifting the phase of the intracavity field by π, and can be realized by the phase shift operator exp{iπa + a}:

ea+a12(|α+i|α)=12(α+iα)=i2(αiα).
(30)

The last relation shows how we can convert a conditional non-decohering evolution into unconditional non-decohering one [12

12. D. B. Horoshko and S. Ya. Kilin, “Direct detection feedback for preserving quantum coherence in an open cavity,” Phys. Rev. Lett. 78, 840 (1997). [CrossRef]

]. It can be done by means of electro-optical modulator placed inside the cavity, controlled by photocurrent in such a way that each photocount is followed by a field phase shift by π, which must be performed during a time small compared to cavity photon lifetime γa1 and average distance between photocounts γa1 |α|-2. In such a scheme the conditional state of the field is given, instead of Eq. (8), by the following expression:

ψacond(t)=(γa)n2e(iHγaa+a2)(ttn)ea+aaea+aae(iHγaa+a2)t1|ψa(0),
(31)

which gives for initial condition Eq. (28)

ψacond(t)=(γaα)n12(|αeγat2iωt+i|αeγat2iωt).
(32)

Now conditional state does not depend on detector records at all. So to obtain an unconditional state we should only normalize it:

ψa(t)=12(αeγat2iωt+iαeγat2iωt)
(33)

i.e. the decaying state inside the cavity remains a superposition of coherent states with opposite amplitudes. Some of the difficulties which a practical realization of this method can meet, such as non-unity efficiency of photodetector, are discussed in Ref. [12

12. D. B. Horoshko and S. Ya. Kilin, “Direct detection feedback for preserving quantum coherence in an open cavity,” Phys. Rev. Lett. 78, 840 (1997). [CrossRef]

].

In general case we may try to find such a feedback, which will preserve the state purity for any given Hamiltonian H and initial state |ψa (0)〉. We may describe the effect of instant feedback by some operator U and write a more general form of Eq. (31):

ψacond(t)=(γa)n2e(iHγaa+a2)(ttn)UaUae(iHγaa+a2)t1ψa(0)
(34)

Now we need the field state (normalized) at time t to be independent of photon arrival times tn and photocount number n, that is

ψacond(tn,t1,tn)=c(tn,t1,tn)|ψa(t),
(35)

where c(t|n, t 1, …tn ) is a c-number defining the probability of the corresponding count sequence and |ψ′a (t)〉 is some vector depending only on time. Letting n = 0 in Eq. (35) we find that |ψ′a (t)〉 can be chosen as

ψa(t)=e(iHγaa+a2)tψa(0).
(36)

It is readily seen from the structure of Eq. (34) that Eq. (35) is satisfied for any photocount sequence if and only if each of the states |ψ′a (t)〉, t ∈ [0, +∞) is an eigenstate of operator Ua:

Ua|ψa(t)=c(t)ψa(t),
(37)

where c(t) is a c-number. So, if for given H and |ψa (0)〉 we find an operator U satisfying Eq. (37) and the corresponding physical process, which can be controlled by feedback loop, then the decoherence of intracavity field induced by decay can be completely removed.

So we have shown using a single-mode field in an open cavity as an example that decoherence is not an intrinsic property of a system coupled to a reservoir and can be avoided if some information about the reservoir state is available. In some cases the necessary information can be much less than the complete information about the reservoir state, and sometimes it can be used to avoid decoherence in unconditional dynamics of the system.

This work was supported by INTAS project ”Single photon generation and quantum state synthesis”.

Appendix

We consider conditions under which equality in Eq. (21) holds for both signs, that is

(γa2a+γb2b)|ψaψb=|ψaψb,
(A1)
(γa2aγb2b)|ψaψb=|ψaψb
(A2)

for some states |ψa 〉, |ψb 〉, |ψ′a 〉 |ψ′b 〉 |ψ″a 〉, and |ψ″b 〉. Multiplying both sides of Eq. (A1) by bra vector 〈φa | such that 〈φa |ψa 〉 = 0 we obtain

γa2φaaψa|ψb=φa|ψa|ψb.

This condition can hold if and only if either (i)

|ψb=c|ψb,

where c is a non-zero c-number, or (ii)

φaaψa=φa|ψa=0.

In the case (i) Eq. (A1) is rewritten as

(γa2a+γb2b)ψaψb=c|ψaψb,

which after multiplying both sides by 〈ψa | gives

γb2b|ψb=(cψa|ψaγa2ψaaψa)|ψb,

i.e. |ψb 〉 is an eigenstate of operator b(coherent state).

In the case (ii) we have: a|ψa 〉 ⊥ |φa 〉 for any |ψa 〉 ⊥ |ψa 〉, therefore a|ψa 〉 = c′ |ψa 〉 where c′ is a c-number. That is |ψa ) is an eigenstate of operator a(coherent state).

So we have proven that Eq. (A1) holds if and only if at least one of states |ψa 〉 or |ψb 〉 is a coherent state.

In the same way we find that Eq. (A2) holds under the same conditions.

References

1.

W. H. Zurek, “Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?,” Phys. Rev. D 24, 1516 (1981). [CrossRef]

2.

W. H. Zurek, “Environment-induced superselection rules,” Phys. Rev. D 26, 1862 (1982). [CrossRef]

3.

W. H. Zurek, “Decoherence and transition from quantum to classical,” Phys. Today 44, 36 (1991). [CrossRef]

4.

S. M. Barnett and S. J. D. Phoenix, “The principles of quantum cryptography,” Phil. Trans. R. Soc. Lond. A 354, 793 (1996). [CrossRef]

5.

D. P. DiVincenzo, “Quantum computation,” Science 270, 255 (1995). [CrossRef]

6.

M. D. Srinivas and E. B. Davies, “Photon counting probabilities in quantum optics,” Opt.Acta 28, 981 (1981). [CrossRef]

7.

H. Carmichael, An Open System Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).

8.

S. Ya. KilinQuantum Optics. Fields and Their Detection (Minsk, 1990).

9.

S. Ya. Kilin, D. B. Horoshko, and V. N. Shatokhin, “Quantum instabilities and decoherence problem,” Acta Phys. Pol. A 93, 97 (1998).

10.

V. Buzek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” Prog. Opt. 36 3(1995). [CrossRef]

11.

B. M. Garraway and P. L. Knight, “Evolution of quantum superpositions in open environments: Quantum trajectories, jumps, and localization in phase space,” Phys. Rev. A 50, 2548 (1994). [CrossRef] [PubMed]

12.

D. B. Horoshko and S. Ya. Kilin, “Direct detection feedback for preserving quantum coherence in an open cavity,” Phys. Rev. Lett. 78, 840 (1997). [CrossRef]

13.

H. Mabuchi and P. Zoller, “Inversion of quantum jumps in quantum optical systems under continuos observation,” Phys. Rev. Lett. 76, 3108 (1996). [CrossRef] [PubMed]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(270.2500) Quantum optics : Fluctuations, relaxations, and noise

ToC Category:
Focus Issue: Control of loss and decoherence in quantum systems

History
Original Manuscript: December 1, 1997
Published: April 27, 1998

Citation
Dmitri Horoshko and Sergey Kilin, "Multimode unravelling of master equation and decoherence problem," Opt. Express 2, 347-354 (1998)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-9-347


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References

  1. W. H. Zurek, "Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?," Phys. Rev. D 24, 1516 (1981). [CrossRef]
  2. W. H. Zurek, "Environment-induced superselection rules," Phys. Rev. D 26, 1862 (1982). [CrossRef]
  3. W. H. Zurek, "Decoherence and transition from quantum to classical," Phys. Today 44, 36 (1991). [CrossRef]
  4. S. M. Barnett and S. J. D. Phoenix, "The principles of quantum cryptography," Phil. Trans. R. Soc. Lond. A 354, 793 (1996). [CrossRef]
  5. D. P. DiVincenzo, "Quantum computation," Science 270, 255 (1995). [CrossRef]
  6. M. D. Srinivas and E. B. Davies, "Photon counting probabilities in quantum optics," Opt. Acta 28, 981 (1981). [CrossRef]
  7. H. Carmichael, An Open System Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).
  8. S. Ya. Kilin Quantum Optics. Fields and Their Detection (Minsk, 1990).
  9. S. Ya. Kilin, D. B. Horoshko, and V. N. Shatokhin, "Quantum instabilities and decoherence problem," Acta Phys. Pol. A93, 97 (1998).
  10. V. Buzek and P. L. Knight, "Quantum interference, superposition states of light, and nonclassical effects," Prog. Opt. 36 3(1995). [CrossRef]
  11. B. M. Garraway and P. L. Knight, "Evolution of quantum superpositions in open environments: Quantum trajectories, jumps, and localization in phase space," Phys. Rev. A 50, 2548 (1994). [CrossRef] [PubMed]
  12. D. B. Horoshko and S. Ya. Kilin, "Direct detection feedback for preserving quantum coherence in an open cavity," Phys. Rev. Lett. 78, 840 (1997). [CrossRef]
  13. H. Mabuchi and P. Zoller, "Inversion of quantum jumps in quantum optical systems under continuos observation," Phys. Rev. Lett. 76, 3108 (1996). [CrossRef] [PubMed]

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