Interaction without back-action in the context of quantum manipulation
We address the interaction between two quantum systems (A and B) that is mediated by their common linear environment. If the environment is out of equilibrium the resulting interaction violates Onsager relations and cannot be described by a Hamiltonian. In simple terms the action of system A on system B does not necessarily produce a back-action. We derive general quantum equations describing the situation and analyze in details their classical correspondence. Changing the properties of the environment one can easily change and engineer the resulting interaction. It is tempting to use this for quantum manipulation of the systems. However the resulting quantum gate is not always unitary and may induce a loss of quantum coherence. For a relevant example we consider systems A and B to be spins of arbitrary values and arrange the interaction to realize an analogue of the two-qubit CNOT gate. The direction of spin A controls the rotation of spin B while spin A is not rotated experiencing no back-action from spin B. We solve the quantum dynamics equations and analyze the purity of the resulting density matrix. The resulting purity essentially depends on the initial states of the systems. We attempt to find a universal characteristics of the purity optimizing it for the worst choice of initial states. For both spins , the optimized purity is bounded by 1/2 irrespective of the details of the gate. We also study in detail the semiclassical limit of large spins. In this case the optimized purity is bounded by . This is much better than the typical purity of a large spin state . We conclude that although the quantum manipulation without back-action inevitably causes decoherence of the quantum states the actual purity of the resulting state can be optimized and made relatively high.
pacs:85.75.-d, 85.35.-p, 75.78.Fg
The ability to control open quantum systems is important in the context of information processing, that includes information processing with spins Wolf2001 (); Awschalom2002 (); Zutic2004 () and spin quantum computation QuantumInformation (); LDV (); Kane (); HansonRMP (); ZwanenburgRMP (). In general the environment is harmful to quantum systems as it leads to dissipation and the loss of the quantum coherence. On the other hand coupling the system to an environment is essential in the quantum manipulation itselfMoiseyev (); Uzdin2013 (); Muller2012 (); Emary2013 (); Bender2010 (); Evans2015 (); Ketterer2014 ().
Coupling quantum systems together can be achieved in many ways depending on the experimental platform Petta2005 (); Niskanen2007 (); Nowack2011 (); Braakman2013 (); Srinivasa2015 (); Baart2016 (). The environment can mediate the interaction between the subsystems as described by linear response theory Kubo (). If the environment is in thermal equilibrium the environment-mediated interaction between two systems A and B is governed by the Onsager symmetriesOnsager (); VanKampen (): in simple terms the action that system A has on system B is equivalent to the action of system B on system A.
Here we examine the more general case where the environment is out of thermal equilibrium and the Onsager symmetries do not hold. In particular this makes possible the situation where subsystem A influences the dynamics of system B and the reciprocal interaction does not occur. We call this: interaction without back-action. By solving the equations of the dynamics of spins in an environment we show that interaction without back-action inevitably leads to noise and decoherence of the subsystems. We also investigate if this interaction can realize the controlled-NOT (C-NOT gate) that is central to quantum information processingMonroe1995 (); Veldhorst2015 (); Zajac2018 (). We find that this is possible if the number of levels involved is large (large spin quantum number). In contrast for two-level systems (spin 1/2) the coherence is almost entirely lost.
The paper is organized as follows: in the second part we discuss the classical dynamics of two systems in the presence of a non-reciprocal interaction between the systems. In the third part we address the dynamics of classical spins. In the fourth part we derive the equations of dynamics of open quantum systems. In the fifth part we compare the quantum evolution of observables with the classical limit. In the sixth part we compute the time evolution of spin states in a simple model where dephasing is taken into account and we study the feasibility of the C-NOT operation for spin-1/2 and for large spins. The last section is dedicated to the conclusion.
Ii General classical dynamics
Before addressing the environment-mediated interaction at the quantum level, let us consider it at the classical level.
The description of interaction at this level is very simple and compact. The environment can be regarded as a piece of electronics that functionally consists of a set of controls that exert forces on the system variables, a set of meters that measure the variables, and provides feedback setting the control forces corresponding to the measurement results. We will assume an instant and linear feedback. In this case, the environment and the mediated interaction is completely characterized by a matrix of linear susceptibilities , being the proportionality coefficient between the force conjugated to the variable and the value of the variable . In principle, the environment can be always engineered to provide any desired .
The symmetry properties of this matrix are important. If the resulting dynamics is Hamiltonian, and there is a conserving energy characteristic to the dynamics. The symmetry is equivalent to the Newton’s third law: every action produces equal and opposite reaction.
If the environment is in thermodynamic equilibrium, the symmetry property as well as the energy conservation is guaranteed by Onsager symmetry principle Onsager (). An active environment is not in the equilibrium, for instance, the piece of electronics realizing may be connected to the battery, and it should lead to . In this paper, we concentrate on the cases when Onsager relation is violated.
To illustrate the importance of Onsager symmetry let us consider the classical dynamics of two identical oscillators with coordinates that are weakly coupled via the environment
being the oscillator frequency, . These linear dynamics are easy to analyze.
If , there are two close oscillating frequencies . The absence of Onsager symmetry is only manifested in non-symmetric eigenvectors of the corresponding oscillating modes. If , the oscillators become unstable, there is an exponentially growing oscillation with amplitude . Apparently, the energy is supplied by the environment. Another interesting case is . In this case, the motion of the oscillator is not affected by the coupling. The oscillator experiences the oscillations of as an external resonance force, so its oscillation amplitude increases linearly with time. One can say that the oscillator provides an unobtrusive and very efficient detection and amplification of .
This example illustrates the importance of the Onsager symmetry violation at the classical level: it may drastically change the dynamics, but does not have to do this always.
Let us consider the most general classical equations of motion in the linear response regime. Let us have a set of classical variables (they may be distibuted between two coupled systems). Their evolution is governed by the equation,
The first term includes the Hamiltonian contribution to the system dynamics and stands for Poisson brackets,
being the canonical coordinates of the system. This is the most general linear feedback equation that will be checked upon quantum derivations in Section IV.
The system of equations (2) is closed if the Poisson brackets of can be expressed in terms of the elements of the set of constants. This is obviously the case when are canonical variables. The Poisson brackets in this case are , , (), and the interaction terms are expressed as follows:
While quantum version of these equations are worth exploring, their relation to quantum information processing is not obvious. Another example of a variable set where Poisson brackets are closed is provided by classical spins. Since there is a direct and straightforward analogy between the spins and (collections of) qubits, we concentrate on this case.
Iii Classical spins
Let us define two classical spins through angular momenta expressed in therms of canonical variables (). The Poisson brackets read
with the Levi-Civita antisymmetric tensor and the Kronecker symbol. The susceptibility matrix relates the ”magnetic field” acting on spin to the spin components of spin . The resulting equations of motion read
These equations conserve the moduli of both spins, . If Onsager symmetry holds, the quantity is conserved as well, and the equations have stable stationary solutions. This is not generally the case if Onsager symmetry is violated.
Let us concentrate on a simple extreme case where and . In this case, the spin remains constant in time. It creates a constant magnetic field for the spin that precesses around this magnetic field.
This behavior is reminiscent to that of the C-NOT gate that is of fundamental importance in quantum information processing QuantumInformation (). This is a two-qubit operation that does not modify the spin of the first (control) qubit. The spin of the second qubit does not change if the first spin is ”up” and flips its direction if the first qubit is ”down”
Let us note that we can realize a classical analogue of the C-NOT gate by means of the environment-mediated interaction. We assume that we can switch the interaction on and off. We arrange interaction such that the only non-vanishing element of the susceptibility matrix is . We switch on the interaction for a time interval such that the spin is rotated by about z-axis provided the spin is in direction, . We switch off the interaction and apply ”magnetic field” to the spin to rotate it by .
Thereby we achieve the controlled-NOT gate functionality. It is worth noting that big spins can be seen as collections of aligned qubit spins. The operation is performed on all qubits of the collection and they remain aligned at each stage of the operation.
Iv Open system quantum dynamics
Let us work out the time evolution of a quantum system coupled to a bath. The ensemble system + bath is described by a density matrix whose time evolution is given by the von Neumann equation
The Hamiltonian of the system and the bath reads
It includes the Hamiltonian of the subsystem , the Hamiltonian of the bath , and the Hamiltonian that describes the coupling between the two. The latter is written as
where are the operators of the subsystem, are the operators of the bath, and there is summation over the repeated index. The density matrix of the subsystem is obtained by taking the trace over the bath variables of the full density matrix:
Let us define
Assuming to be time-independent we can check that satisfies the equation
where the transformed Hamiltonian reads
Integrating Eq.(12) leads to
and inserting this into the second term of Eq.(12) gives
At this stage we let . Taking the trace over the bath variables yields
Let . The bath operators vanish on average: , and the correlators decay fast for , being the time scale of the correlations of the bath variables. We also assume , the elements of the density matrix are essentially constant over the time scale , and the correlators are uniform in time. Therefore we let
we arrive at the differential equation that describes the time evolution of the subsystem density matrix:
Let us introduce the (symmetrized) noises and the susceptibilities
Then we can write Eq.(17) as
The matrix is positive-Hermitian. It includes a symmetric part that consists of the noises and an anti-symmetric part that includes the elements of the susceptibility matrix. In order to reveal its physical content we rewrite Eq.(19) as
V Comparison between classical and quantum dynamics
In order to give a quasiclassical interpretation of the equations derived in the preceding section let us compute the time evolution of the expectation value of the operator . Using the cyclic property of the trace we obtain from Eq.(22):
The connection between quantum and classical dynamics is generally obtained by employing the correspondence between the commutators and the Poisson brackets:
In the classical case in addition the noise matrix is symmetric: . With those prescriptions the time-evolution of the classical variables writes:
The first and third lines are consistent with Eq.(2) and there is an additional noise term given by the second line.
We would like to consider two special cases: canonical operators and spins. On the one hand the phase space dynamics is obtained from the commutation relations of the canonical operators , , and . The corresponding equations of motion are
The term can be taken out of the summation, leading to
This corresponds to Eq.(9) of Ref.28.
Spin-1/2 operators in particular satisfy the anticommutation relations . In that case the term in the third line is the number . It accounts for the effect of spin pumpingDanon2011 (). The last term of the above equation is the quantum counterpart to Eqs.(6a,6b).
In Eq.(33) the noises lead to relaxation and dephasing of the spins. This is an important feature since in quantum physics the asymmetry of the susceptibility matrix leads to finite noise as required by the positivity of the matrix in Eq.(19). The noise is expected to reduce the purity of the quantum state. In the next section we compute the purity of the spin quantum states and we find its maximal achievable value in the process of interaction without back-action.
Vi Spin states and purity
Having the C-NOT operation in mind we wish spin B to rotate under the action of spin A which is immobile. This is fulfilled when and . Then is conserved for . The equations of motion for spin B are
If spin A is in the state then spin B rotates around spin A with angular velocity . Thus it rotates by an angle over the time
We would like to evaluate how the quantum states are affected by the noise after a certain time of manipulation such as the C-NOT operation. A simple measure of the effect of the noise is given by the purity of the quantum state
The time-dependence of the density matrix is obtain from Eq.(25):
We choose as a basis the products of the eigenstates of and : with
In this basis we obtain
These equations show that the noises lead to the decay of the non-diagonal elements of the density matrix which is a property of dephasing. The solution to Eq.(40) is
The initial state is taken as the product of the density matrices of pure spin states:
and we assume the spin coherent statesLieb73 ():
Here are the binomial coefficients and . These correspond to the states with maximal single-spin projection along the direction , and being the angles that parametrize the sphereLieb73 (). In particular the continuous limit is achieved when . Applying the central limit theorem to the binomial distribution leads to:
Because of the positivity of matrix in Eq. (19) the noises cannot all simultaneously vanish when the interaction between the subsystems is non-reciprocal. Here this translates to the quantum noise inequalityClerk2010 ()
In the large spin regime it is obtained from the Gaussian integrals:
In the conditions specified above the purity is maximized for , , , and its optimal value is
This has a minimum when with value .
In comparison for the purity is
is optimal when either or vanishes, depending on and and independent of finite and finite and non-zero . Thus
This represents a case where one spin does not experience dephasing and the other spin is completely dephased. So the purity equals the initial purity of a single spin . If () then spin B is maximally dephased and the purity equals . Fig.(1) compares Eqs. (48) with (50) for .
We have studied the interaction between two quantum systems A and B that is mediated by their common linear environment. When the environment is out of thermal equilibrium the interaction in general violates the Onsager symmetries and the action of system A on system B may not produce a back-action. We have derived the corresponding quantum equations and obtained their classical limit. In the quantum domain interaction between the systems without back-action necessarily involves minimal noise on the systems. As an application of the formalism we analyzed the quantum manipulation of spin coherent states of arbitrary spin quantum number and the realization of an analogue of the C-NOT gate. As a measure of the decoherence induced by the noise we evaluated the purity of the quantum states after a time of interaction between the spins that corresponds to the time of operation of the C-NOT gate. The final purity depends on the initial states of the spins and we optimized the purity for all initial states. In the worst case of the initial state the optimal purity is for spin 1/2 and for large spins. Thus even though the decoherence is important for two-level systems, it can be made relatively small for systems of large spins.
Acknowledgements. We acknowledge interesting discussions with Lieven Vandersypen. This work was partly supported by the European Research Council (ERC-Synergy).
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